EFFECTS OF DIRECTED ENERGY WEAPONS
EFFECTS OF DIRECTED

ENERGY WEAPONS

Philip E. Nielsen
Library of Congress Cataloging-in-Publication Data

Nielsen, Philip E., 1944–

Effects of Directed Energy Weapons

/Philip E. Nielsen.

p. cm.

Includes bibliographical references and index.

ISBN 0–945274–24–6 (alk. paper)

1. United States. Air Force—Aviation—History. 2. Air Power—

United States.

3. United States—Armed Forces—Management—History.

I. Title

VG93.B36 1994

359.94 0973—dc20 94–1937

CIP
Contents

Page

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi

Preface .......................................... xxvii

Chapter 1. Basic Principles.......................... 1

Overall Theme ................................ 1

A Word About Units ........................... 1

Developing Damage Criteria ..................... 2

The Energy Required for Damage............. 2

Is Energy Alone Sufficient for Damage? ........ 6

Energy Density Effects .................. 6

Energy Delivery Rate Effects ............. 7

Implications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

All-Purpose Damage Criteria . . . . . . . . . . . . . . . . 16

Energy Spread and Loss in Propagation . . . . . . . . . . . . 20

Energy Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Energy Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Where We’re Going . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Chapter 2. Kinetic Energy Weapons. . . . . . . . . . . . . . . . . . . 29

I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

II. Fundamentals of Kinetic Energy Weapons . . . . . . . . . 29

III. The Propagation of Kinetic Energy Weapons in 

a Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

A. Motion under the influence of gravity . . . . . . . . 34

B. The motion of powered weapons . . . . . . . . . . . 42

C. Summary: Propagation in a Vacuum . . . . . . . . . 43

D. Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

IV. Propagation in the Atmosphere . . . . . . . . . . . . . . . . . 44

A. Gravitational forces . . . . . . . . . . . . . . . . . . . . . . 44

B. Drag Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

vii
viii

C. Other forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

D. Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

E. Summary: Propagation in the Atmosphere . . . . 57

F. Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

V. Interaction of Kinetic Energy Weapons with

Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

A. Important Parameters . . . . . . . . . . . . . . . . . . . . 58

B. What is Damage? . . . . . . . . . . . . . . . . . . . . . . . . 60

C. General Principles . . . . . . . . . . . . . . . . . . . . . . . 61

D. Damage in Space—Hypervelocity Impacts . . . . 64

E. Damage in the Atmosphere—Lower Velocity

Impacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

F. Tradeoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

G. Summary: Target Interaction . . . . . . . . . . . . . . . 73

H. Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

VI. Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

VII. Implications and Analogies . . . . . . . . . . . . . . . . . . . . 75

VIII. Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . 77

Chapter 3. Lasers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

II. Fundamental Principles of Laser Light . . . . . . . . . . . . 81

A. Fundamentals of Propagation . . . . . . . . . . . . . . 81

1. Wave Propagation and Electromagnetic

Radiation . . . . . . . . . . . . . . . . . . . . . . . . 82

2. Refraction. . . . . . . . . . . . . . . . . . . . . . . . . . 84

3. Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . 86

4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 88

B. Fundamentals of Laser Interaction with

Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

1. Interaction with Gases . . . . . . . . . . . . . . . . 90

2. Interaction with Solids. . . . . . . . . . . . . . . . 93

III. Laser Propagation in a Vacuum. . . . . . . . . . . . . . . . . . 101

A. Near Field Propagation . . . . . . . . . . . . . . . . . . . 102

B. Far Field Propagation . . . . . . . . . . . . . . . . . . . . . 104

C. Departures from Perfect Propagation. . . . . . . . . 107

D. Summary: Propagation in a Vacuum . . . . . . . . . 108

E. Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
ix

IV. Laser Propagation in the Atmosphere . . . . . . . . . . . . . 110

A. Absorption and scattering . . . . . . . . . . . . . . . . . 110

1. Molecules. . . . . . . . . . . . . . . . . . . . . . . . . . 110

2. Small Particles (Aerosols) . . . . . . . . . . . . . 117

3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 119

B. Index of Refraction Variations . . . . . . . . . . . . . . 120

1. Turbulence and Coherence Length . . . . . . 120

2. Adaptive Optics. . . . . . . . . . . . . . . . . . . . . 126

3. Summary: Index of Refraction

Variations . . . . . . . . . . . . . . . . . . . . . . . . 128

C. Nonlinear Effects . . . . . . . . . . . . . . . . . . . . . . . . 129

1. Thermal Blooming . . . . . . . . . . . . . . . . . . . 130

2. Stimulated Scattering. . . . . . . . . . . . . . . . . 137

3. Air Breakdown . . . . . . . . . . . . . . . . . . . . . 142

4. Aerosol Induced Breakdown. . . . . . . . . . . 149

5. How might Aerosol Affect Air

Breakdown? . . . . . . . . . . . . . . . . . . . . . . . . 149

6. Plasma Maintenance and Propagation . . . 152

7. Summary: Nonlinear Propagation

Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . 167

D. Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

V. Laser-Target Interaction and Effects. . . . . . . . . . . . . . . 169

A. Types of Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 170

B. Effects in the Absence of Plasmas. . . . . . . . . . . . 171

1. Melting. . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

2. Vaporization . . . . . . . . . . . . . . . . . . . . . . . 174

3. Mechanical Effects . . . . . . . . . . . . . . . . . . . 175

4. Energy Requirements for Damage. . . . . . . 179

5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 181

C. Effects of Plasmas on Target Interaction. . . . . . . 182

1. Plasma Effects in a Vacuum. . . . . . . . . . . . 183

2. Plasma Effects on Coupling in the

Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . 185

3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 188

D. Summary of Main Concepts. . . . . . . . . . . . . . . . 189

E. Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

VI. Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . 194
x

Chapter 4. Microwaves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

II. Fundamentals of Microwaves . . . . . . . . . . . . . . . . . . . 208

A. Fundamentals of Propagation . . . . . . . . . . . . . . 209

B. Fundamentals of Interaction with Matter. . . . . . 211

C. Summary: Microwave Fundamentals . . . . . . . . 212

III. Microwave Propagation in a

Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

A. Propagation Tradeoffs . . . . . . . . . . . . . . . . . . . . . 213

B. Diffraction and Interference around Objects . . . 215

C. Summary: Microwave Propagation in a

Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

D. Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

IV. Microwave Propagation in the Armosphere . . . . . . . . 219

A. Losses due to Absorption and Scattering . . . . . . 219

1. Molecular Absorption and Scattering . . . . 219

2. Effect of Liquid Water and Atmospheric

Aerosols . . . . . . . . . . . . . . . . . . . . . . . . . 223

3. Summary: Absorption and Scattering . . . . 226

B. Losses due to Index of Refraction Variations . . .

C. Nonlinear Effects . . . . . . . . . . . . . . . . . . . . . . . . 227

1. Air Breakdown . . . . . . . . . . . . . . . . . . . . . 234

2. Aerosol-Induced Breakdown. . . . . . . . . . . 238

3. Plasma Maintenance and Propagation . . . 238

4. Thermal Blooming . . . . . . . . . . . . . . . . . . . 241

5. Summary: Nonlinear Effects. . . . . . . . . . . 241

D. Summary: Propagation in the Atmosphere . . . . 242

E. Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

IV. Microwave Interaction with Targets . . . . . . . . . . . . . . 243

A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

B. Mechanisms of Soft Kill . . . . . . . . . . . . . . . . . . . 244

1. In-Band Damage . . . . . . . . . . . . . . . . . . . . 244

2. Out-of-Band Damage. . . . . . . . . . . . . . . . . 249

C. Estimates of Damage Thresholds . . . . . . . . . . . . 252

D. Summary: Target Interaction . . . . . . . . . . . . . . . 254

V. Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

VI. Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . 256
xi

Chapter 5. Particle Beams. . . . . . . . . . . . . . . . . . . . . . . . . . . 263

I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

II. Fundamental Principles of Particle Beams . . . . . . . . . 264

A. Electromagnetic Fields and Forces . . . . . . . . . . . 266

B. Relativistic particle dynamics . . . . . . . . . . . . . . 266

C. Major forces affecting charged-particle beams . . 269

D. Particle beam characteristics . . . . . . . . . . . . . . . . 271

III. Propagation in a Vacumm . . . . . . . . . . . . . . . . . . . . . . 275

A. Neutral particle beams in a Vacuum . . . . . . . . . 275

B. Charged particle beams in a Vacuum. . . . . . . . . 277

1. Expansion from electrostatic repulsion . . . 277

2. Effects due to external fields . . . . . . . . . . . 279

C. Summary: Propagation in a Vacuum. . . . . . . . . 281

D. Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

IV. Propagation in the Atmosphere. . . . . . . . . . . . . . . . . . 282

A. Neutral particle beams in the Atmosphere. . . . . 282

B. Charged particle beams in the Atmosphere . . . . 286

1. Charge Neutralization . . . . . . . . . . . . . . . . 286

2. The Evolution of Beam Radius . . . . . . . . . 290

3. Summary: Beam Radius vs Distance . . . . 296

4. Energy Losses . . . . . . . . . . . . . . . . . . . . . . 296

5. Current Losses . . . . . . . . . . . . . . . . . . . . . . 301

6. Hole boring . . . . . . . . . . . . . . . . . . . . . . . . 302

7. Nonuniform atmospheric effects . . . . . . . . 306

8. Summary: Energy and Current Losses . . . 307

9. Nonlinear Effects (Instabilities) . . . . . . . . . 308

C. Summary: Propagation in the Atmosphere . . . . 315

D. Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

V. Interaction with Targets . . . . . . . . . . . . . . . . . . . . . . . . 318

A. Energy Deposition and Flow . . . . . . . . . . . . . . . 318

B. Damage and interaction times . . . . . . . . . . . . . . 323

C. Summary: Interaction with Targets . . . . . . . . . . 325

D. Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

VI. Summary of Main Concepts . . . . . . . . . . . . . . . . . . . . 326

VII. Overall Implications . . . . . . . . . . . . . . . . . . . . . . . . . . 328

VIII. Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . 330

Appendix A—Units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

Appendix B—Some Useful Data . . . . . . . . . . . . . . . . . . . . . . 343
xii

Tables 

Chapter 1

1–1. Thermal Properties of Common Metals ............ 5

1–2. Energy Losses in Propagation . . . . . . . . . . . . . . . . . . . 23

Chapter 2

2–1. Parameters Affecting Target Response and

Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2–2. Effect of Drag on Projectile Range . . . . . . . . . . . . . . . . 54

2–3. Kinetic Energy Required for 7.62 mm Projectiles to

penetrate Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2–4. Stiffness Coefficients of Common Mertals . . . . . . . . . . 63

Chapter 3

3–1. Some Typical Energy Gaps . . . . . . . . . . . . . . . . . . . . . . 96

3–2. Response of Metals and Insulators to Incident

Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3–3 Implications of Propagation Effects in the

Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

Chapter 4

4–1. Rain Rates vs Meteorological Conditions . . . . . . . . . . . 226

4–2. Electron Heating in Microwave and Laser

Frequency Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

4–3. Issues Affecting Microwave and Laser Propagation in

the Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

4–4. Skin Depth and Absorptivity of Copper at Microwave

Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

4–5. Damage Estimates for Microwaves . . . . . . . . . . . . . . . 253

Chapter 5

5–1. Relativistic Parameters for Energentic Electrons and

Protrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

5–2. Quantities Used to Characterize Particle Beams . . . . . 273

5–3. Scaling of Energy Loss Mechanisms . . . . . . . . . . . . . . 301

5–4. Factors Affecting Beam Intensity . . . . . . . . . . . . . . . . . 303

5–5. Effect of a Nonuniform Atmosphere Energy and Current

Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
xiii

List of Symbols

I have tried to keep common symbols consistent throughout the

text. Unfortunately, there aren’t enough letters to go around, and

some symbols are so common in the literature that custom has

been retained even though it results in multiple usage. Context is

usually adequate to sort out any ambiquity , and terms which are

limited to specific sections are identified in the table below.

Many symbols are used with subscripts to identify specific values, such as Ti for initial temeprature. These are identified as they

are used, and will typically have have consistent meaning only in

a specific treatment or argument.

The “units “ mentioned in the tabel below are typical ones used

in the text. They’ll let you know the dimensions associated with a

specific quantity, but are by no means exclusive. Energy, for example, may be expressed in Joules or electron volts (see Appendix A).

Symbol Meaning Units Comments

A Activation energy eV Chapter 3

A Area cm2

a Acceleration cm/sec2 =dv/dt

a Particle radius cm, m

a Speed of Sound cm/sec =3  104 cm/sec

(Chap 3,5)

B Brightness W/sr Chapter 3,4

B Brightness Amp/m2 sr Chapter 5

B Magnetic Field Gauss Chapter 5

b Impact parameter cm Chapter 5

C Heat Capacity J/gm °K

C Stiffness Nt/m2,J/m3 Relates stress to strain

(Chapters 2,3)

Cd Drag Coefficient Dimensionless Chapter 2

CN Refractive Index m–1/3 Chapter 3

structure factor

c Speed of light m/sec = 3  108 m/sec

D,d Diameter cm

D Thermal Diffusivity cm2/sec = k/Cp

D Relative Depth dimensionless See Figure 5–31
xiv

dQ/dx Gradient of Q Q/cm2 The slope of a curve of

any quantity Q

vs distance

d2Q/dx2 2nd Derivative of Q Q/cm2 The slope of a curve of 

dQ/dx vs distance

dQ/dt Rate of changeof Q Q/sec The slope of a curve of

any quantity Q vs time

E Energy Joules, eV

E Electric Field Volts/m Chapter 5

e base of natural

logarithms  2.72

e electron charge Columb 1.6  10–19 Coul

e Strain demensionless Chapters 2, 3

e* Stain at Failure dimensionless Strain corresponding

to P*

F Fluence J/cm2 Energy density on a

surface

F Force Newtons

f focal length km

f fractional ionization dimensionless

G Gravitational Constant Nt m2/kg2  6.67  10–11 Nt m2/kg2

g Acceleration of Gravity m/sec2  9.8 m/sec2

g gain (stimualted cm/W Chapter 3

scattering)

h altitude km  height above earths

surface

ho atmospheric scale ht km  7km

h Target Thickness cm Chapter 2

h Planck’s Constant Joule sec Relates photon energy to

frequncy of light

 6.63  10–34 J sec

I Current Amperes Chapter 5

I Impulse Nt sec  F dt

I* Specific Impulse dyn sec/J Efficiency of

momentum

transfer with lasers

(Chap 3)

Isp Specific Impulse sec Rocket impulse/weight 

of fuel used (Chap 2)

I Ionization Potential eV Chapters 3, 4, 5
xv

j Current Density Amp/m2 Chapter 5

K Kinetic Energy Joules Some authors use

T. Kinetic energy is

used most in Chapters

2 and 5

K Attenuation coeffficent km–1 Used in Chapters 3

and 4

k Thermal Conductivity W/cm °K in u  –k dT/dx

k Boltzmann’s Constant J/°K converts temperature to 

energy = 1.38 x 10-23J/0K

L,l length cm

Lm Heat of Fusion J/gm

Lv Heat of Vaporization

M,m mass grams, kg

N, n number density cm–3

Nt Thermal Distortion dimensionless Chapter 3

factor

n index of refraction dimensionless Chapters 3, 4

P Pressure Nt/cm2, J/cm3

P Stress Nt/cm2, J/cm3 Chapters 2, 3

(stress is the equivalent

of pressure in a Solid)

P* Modulus of Rupture Nt/cm2,J/cm3 Stress at which a solid 

fails Chapters 2, 3

p Momentum gm cm/sec  mv

Q intensity W/cm2 Reradiation from a

plasma Chapter 3

Q Impact parameter ratio dimensionless Chapter 5

q electric charge Coulombs Chapter 5

R,r radius cm used for radial

distances

R reflectivity dimensionless fraction of light

reflected

Chapters 3, 4

Re earth radius km  6370 km

Re* effective earth radius km Chapter 4

Ri ionization rate sec–1

rc cyclotron radius cm Chapter 5

ro coherence length cm Chapter 3

S Intensity of Radiation W/cm2
xvi

t time seconds

tp pulse width seconds

td magnetic diffusion

time seconds Chapter 5

T Wave Period seconds Chapter 3

T Perpendicular Energy Joules = m<v2/2, Chapter 5

T Temperature °K, °C

Tm Melting Point

Tv Vaporization Point

T Perpendicular T Measures random motion

perpendicular to beam

direction

u Energy Flow Rate W/cm2 From Thermal

Conduction:

u  –k dT/dx

u LSD or LSC velocity cm/sec Chapter 3

V, v Velocity cm/sec

V perpendicular V cm/sec velocity perpendicular

to beam motion

(Chapter 5)

W Work Joules

w beam radius cm

Z, z range km

Zr Rayleigh Range km
xvii

Greek Symbols

Symbol Meaning Units Comments

 (alpha) km Constant in equation of 

central motion (Chapter 2)

 Absortivity dimensionless fraction of incident

radiation absorbed

(Chapters 3,4)

 (beta) velocity ratio dimensionless  v/c, Chapter 5

 (Gamma) specific heat ratio dimensionless Chapter 3

 Relativistic factor dimensionless  1/(1 – v2/c2) 1/2, 

Chapter 5

 (Delta) “change in” various T  change in T, etc.

	 (delta) skin depth cm Chapters 3, 4


 (epsilon) orbital eccentricity dimensionless Chapter 2


 electron energy eV, Joules Chapters 3,4


0 permittivity of farad/m  8.85  10–12 fd/m

free space

 (theta) beam divergence radians

 angle radians

 (lambda) Wavelength cm, m Chapters 3, 4


 (nu) Frequency sec–1


o Collision Frequency sec–1 Chapters 3,4

 (rho) mass density gm/cm3 Chapter 5

 charge density Coul/cm3 Chapter 5

 (sigma) Conductivity mho/m  is a more common

notation

 (sigma) Stefan-Boltzmann W/cm2 °K4 Relates radiation from a

Constant Black Body to its

Temperature

 5.67  10–12 W/cm2 °K4

 cross section cm2

 (tau) orbital period hours Chapter 2

 (phi) elevation angle radians

 (omega) Radian Frequency sec–1  2

List of Figures

Chapter 1

1–1 Temperature and State of a 50 gram Ice Cube vs Energy

Deposited 4 

1–2 Energy Deposition from Bombs and Directed Energy

Weapons 7

1–3 Energy Flow Along a Temperature Gradient 9

1–4 Energy Flow and Resulting Change in Temperature 10

1–5 Temperature vs Time and Distance 11

1–6 Effect of Wind on Temperature at a Point 12

1–7 Radiation from a Black Body vs Temperature 14

1–8 Damage Thresholds vs Pulse Width 15

1–9 Depth Vaporized by 104 Joules vs Area Engaged and

Fluence 16

1–10 “Effective Thickness” of Targets 19

1–11 Divergence and Jitter 21

Chapter 2

2–1 Forces and Their Effect on an Object’s Motion 31

2–2 Parameters Locating a Body along its Orbit 36

2–3 Possible Trajectories for Objects near the Earth 36

2–4 An Object in Circular Orbit Around The Earth 37

2–5 Orbital Velocity vs Altitude 39

2–6 Orbital Period vs Altitude 39

2–7 Effect of ∆v on a Weapon’s Motion 40

2–8 Effect of the Earth’s Curvature on Horizontal 

Propagation 45

2–9 Projectile Distance and Velocity Coordinates 46

2–10 Effect of Elevation Angle on Velocity Components 47

2–11 Projectile Range vs Elevation Angle 48

2–12 Drag and Gravitational Forces on a Projectile 49

2–13 Factors Affecting the Drag Coefficient 51

2–14 Drag Coefficient vs Mach Number and Projectile Shape 53

2–15 Trajectories in Vacuum and With Different Drag 

Coefficients 54

2–16 Potential Instability of a Projectile 56

2–17 Angle of Attack 59

xix
xx

2–18 Possible Effects of Kinetic Energy Projectiles 61

2–19 Spring–like Binding Forces Between Atoms in a Solid 62

2–20 Nominal Pressures Induced by Projectiles of Various

Velocities 65

2–21 Hypervelocity Impact of a Sphere on a Plane Target 66

2–22 Ratio of Crater Depth (D) to Projectile Size (L) vs. 

ProjectileVelocity for Hypervelocity Projectiles 68

2–23 Kinetic Energy per Centimeter Penetrated vs Projectile

Velocity 69

2–24 Penetration of a Plate by a Projectile 70

2–25 A Fin–Stabilized Discarding-Sabot Projectile 73

Chapter 3

3–1 Wave Parameters and Propagation 82

3–2 Electromagnetic Spectrum 83

3–3 Material and Wavelength Dependence of Index of

Refraction 84

3–4 Refraction Between Materials with Different Indices of

Refraction 85

3–5 Converging and Diverging Lenses 86

3–6 Diffraction of Light Passing Through an Aperture 87

3–7 Focusing of a Beam of Light and the Rayleigh Range 88

3–8 Rayleigh Range vs Aperture and Wavelength 89

3–9 Electron Orbits and Energy Levels in Hydrogen 91

3–10 Energy Levels of Molecular Nitrogen 92

3–11 Atmospheric Absorption Lines 93

3–12 Energy Levels of Atoms, Molecules, and Solids 95

3–13 Fraction of Photons Transmitted Through Gallium

Arsenide 96

3–14 Plasma Frequency vs Electron Density 99

3–15 Skin Depth vs Conductivity and Wavelength 100

3–16 Reflectivity vs Skin Depth and Wavelength 100

3–17 Aperture vs Range and Wavelength in the Near Field 103

3–18 Beam Power vs Aperture and Intensity 104

3–19 Near-and Far-Field Engagements with a Target 104

3–20 The Concept of Brightness 105

3–21 Range vs Brightness and Intensity 106

3–22 Scattering and Absorption Cross Section 111

3–23 Transmission vs Optical Depth 112

3–24 Atmospheric Attenuation vs Wavelength 113
xxi

3–25 Beam Range and Altitude 105

3–26 Optical Depth vs Range and Elevation Angle 116

3–27 Particle Size Distribution and Variation with Altitude 117

3–28 Attenuation Factor due to Aerosols in Mie’s Theory 118

3–29 2πa/λ vs Wavelength and Particle Size 119

3–30 Index of Refraction of Air vs Temperature and 

Wavelength 121

3–31 Coherence Length and its Effect 122

3–32 Divergence Angle vs Coherence Length 123

3–33 Atmospheric Structure Factor vs Time and Altitude 124

3–34 Effect of Turbulence on Brightness 125

3–35 The Principle of Adaptive Optics 126

3–36 An Adaptive Optics Experiment 127

3–37 Linear and Nonlinear Propagation Effects 129

3–38 The Physics of Thermal Blooming 130

3–39 The Physics of Thermal Bending 131

3–40 Beam Profile with Thermal Blooming and Bending 132

3–41 Time for the Onset of Thermal Blooming vs Beam Radius 133

3–42 Relative Intensity vs Distortion Number 135

3–43 An Instability in Thermal Blooming with Adaptive Optics 136

3–44 Parameter Tradeoffs to Prevent Thermal Blooming 137

3–45 Stimulated vs Normal Scattering of Light 139

3–46 Gain for SRS vs Altitude and Pulse Width at 1.06 m 140

3–47 Critical Range for SRS vs Intensity and Wavelength 

at Sea Level 141

3–48 Electron Heating Rate vs Laser Intensity and 

Wavelengthat Sea Level 145

3–49 Electron Energy Losses in a Model Gas 146

3–50 Breakdown Threshold for 10.6 m Radiation 148

3–51 Breakdown Threshold vs Aerosol Radius 150

3–52 Estimated Aerosol-Induced Breakdown Threshold vs

Altitude (10.6 m) 151

3–53 Plasma Propagation 154

3–54 Pressure, Temperature, and Density in the “Combustion”

Mode of Plasma Propagation 157

3–55 Pressure, Temperature, and Density in the “Detonation”

Mode of Plasma Propagation 158

3–56 Fractional Ionization of Hydrogen vs Temperature and

Density 160
xxii

3–57 Absorption Coefficient vs Temperature and Density 

for Nitrogen, λ = 10.6 m 161

3–58 LSD Velocity vs Intensity and Density 162

3–59 LSD Temperature vs Intensity and Density 163

3–60 Detonation Wave Maintenance Thresholds 164

3–61 Velocities of Combustion Waves Propagating by

Thermal Conduction and Radiation 166

3–62 Target Heating by a Laser Beam 171

3–63 Threshold for Melting vs Pulse Width and

Absorptivity 172

3–64 Hole Erosion with Molten Material Removed 173

3–65 Target Erosion Rate with Molten Material Removed 174

3–66 Threshold for Vaporization vs Pulse Width and 

Absorptivity 175

3–67 Erosion Rate for Target Vaporization 176

3–68 Pressure on a Target vs Intensity and Specific Impulse 177

3–69 Mechanical Damage of a Target by a Laser 178

3–70 Intensity for Mechanical Damage vs Pulse Width and

Target Thickness 179

3–71 Fluence to Damage or Penetrate a Target vs Thickness 180

3–72 Plasmas Helping and Hindering Laser-Target Coupling 182

3–73 Laser-Target Coupling with Ionized Vapor in a vacuum 183

3–74 Thermal Coupling with Plasmas in the Atmosphere 186

3–75 Momentum Transfer from an LSD 187

3–76 Significant Propagation and Target Interaction Effects 191

Chapter 4

4–1 The Microwave Portion of the Electromagnetic Spectrum 207

4–2 Microwave Antennas 210

4–3 Law of Refraction 211

4–4 Rayleigh Range and Divergence 213

4–5 Rayleigh Range vs Aperture and Wavelength 214

4–6 Range vs Brightness and Intensity 215

4–7 Diffraction of Radiation Near a Barrier 216

4–8 Attenuation Coefficient for Microwaves due to 

Oxygen and Water Vapor 220

4–9 Concentration of Water Vapor vs Temperature at 

100% Humidity 221

4–10 Relative Atmospheric Density vs Altitude 221

4–11 Beam Range and Altitude 222
xxiii

4–12 Optical Depth vs Range and Elevation Angle 223

4–13 Attenuation of Microwaves by Rain 225

4–14 Index of Refraction for Dry Air vs Temperature and

Pressure 228

4–15 Correction to Index of Refraction for 100% Humidity

vs Temperature 229

4–16 Propagation of Microwaves in the Atmosphere 230

4–17 Effective Earth Radius vs Refractive Index Gradient 232

4–18 Ducting of a Microwave Beam near the Ocean’s Surface 233

4–19 Breakdown Threshold for Air at Sea Level, 10.6 m 

Radiation 237

4–20 Air Breakdown Threshold at Microwave Frequencies 238

4–21 Engagement between a Radar and a Target 245

4–22 A Wire in an Electrical Circuit 246

4–23 Current to Melt Copper Wire vs Time and Wire 

Cross Section 248

4–24 Resistivity of Silicon Doped with 1013 Carriers 

per Cubic Meter 249

4–25 Microwave Range vs Output Power and 

Damage Criterion 253

Chapter 5

5–1 Electrical and Magnetic Fields and Forces 265

5–2 The Relativistic Factor  vs v/c 267

5–3 An Idealized Charged Particle Beam 269

5–4 Attractive and Repulsive Forces in a CPB 270

5–5 Current Flow in a Wire 272

5–6 Real vs Ideal Beam Velocities 274

5–7 Beam Radius vs Divergence and Propagation Distance 276

5–8 Expansion of a Charged Particle Beam in Vacuum 

withTime 278

5–9 Doubling Distance for Electron and Proton Beams of

Intensity 107 W/cm2 279

5–10 Motion of a Charged Particle in a Magnetic Field 280

5–11 Cyclotron Radius vs Particle Energy for B = 0.5 Gauss 280

5–12 Collisional Ionization of Neutral Particle Beams 

in the Atmosphere 283

5–13 Cross Section for Ionization of Hydrogen in a 

Background Gas of Oxygen or Nitrogen as 

a Function of Kinetic Energy 284
xxiv

5–14 Atmospheric Density vs Altitude 285

5–15 Minimum Operational Altitude for Neutral Hydrogen

Beams as a Function of Range and Particle Energy 286

5–16 Sequence of Events in Charge Neutralization 287

5–17 Electric Field of a CPB Before and After Charge 

Neutralization 288

5–18 Maximum Altitude for CPB Neutralization vs 

Current andRadius 289

5–19 Opposing Forces which lead to an Equilibrium Beam

Radius 291

5–20 Atmospheric Beam Range vs Current and Energy 294

5–21 Charged Particle Beam Range and Altitude 295

5–22 Relative Beam Heating vs Range and Elevation Angle 295

5–23 CPB Radius in Atmospheric Propagation 297

5–24 Ionization through Beam-Electron Interactions 298

5–25 Particle Energy Loss Rate due to Ionization vs Particle

Kinetic Energy 299

5–26 Energy Loss Rate due to Ionization and Bremsstrahlung 300

5–27 Nuclear Collisions vs Ionization and Bremsstrahlung 302

5–28 Sequence of Events in “Hole Boring” 304

5–29 Time Scales Associated with Hole Boring 304

5–30 Energy Deposition in the Air from a Charged Particle

Beam 306

5–31 Relative Depth Factor for use in Adjusting Energy and

Current Losses in a Nonuniform Atmosphere 307

5–32 Hose Instability of a Charged Particle Beam 309

5–33 Instability Avoidance through Beam Chopping 312

5–34 Development of the Sausage Instability 314

5–35 Pulse Structure for Hole Boring with Instability

Avoidance 317

5–36 Particle Range in Solid Targets 320

5–37 Energy Deposition from Lasers and Particle Beams 321

5–38 Steady State Energy Deposition and Loss in a Particle

Beam-Irradiated Target 322

5–39 Target Temperature vs Beam Current 323

5–40 Interaction Time for Target Vaporization vs Beam

Current and Radius 324
Acknowledgments

Philip L. Taylor first taught me the importance of insight as opposed to mathematical elegance in dealing with physical phenomena, and Greg Canavan showed me the power of zero order analysis

in sorting through a problem to see the factors that were driving

the solution and its behavior. While teaching physics at the Air

Force Institute of Technology, I benefited from discussions with

numerous colleagues, most notably Michael Stamm and George

Nickel, who helped me to clarify my ideas and offered numerous

examples. Later, while engaged in operations research analysis at

the Air Force Studies and Analyses Agency, Thomas Hopkins joined

me in many hours of thought-provoking discussion and was also

kind enough to read and comment upon the initial manuscript for

this book.

The concept for the book developed while I was at the National

Defense University (NDU) during 1987–1988. Fred Kiley of NDU

Press and LtGen Bradley Hosmer, who was then NDU president,

encouraged its development and publication. Since then, NDU

Press has been saddled with what is arguably the most complex

book that they have ever published. I am grateful to successive

administrations at the press for putting up with the equations and

graphics and for keeping the project alive during times of tight

budgets and other priorities. Special credit should go to George

Maerz, who has been associated with the project from its inception, and to Jeffrey Smotherman, who worked hard to bring it to

closure, and finally succeeded.

Finally, I should acknowledge my wife, Mary Jane, and children,

Aaron, June, and David. They had the patience to put up with me

when I was researching and writing the text, and the faith to put

up with my optimism that the project would one day come to a

successful conclusion.

xxv
xxvii

Preface

This book is on the effects of directed energy weapons. That is,

how they propagate to and interact with targets. Propagation and

target interaction are the key elements in an analysis of a

weapon’s utility to accomplish a given mission. For example, the

effectiveness of a nuclear missile is determined by the yield of its

warhead and the accuracy of its guidance, and the effectiveness of

a rifle is determined by the type of round fired, the range to the

target, and the skill of the soldier who fires it. Directed energy

weapons are no different. But while there are books and manuals

that deal with the issues affecting the utility of nuclear missiles

and rifles, there is no comparable source of information for directed energy weapons. I have tried to fill that void with this book.

Weapons are devices which deliver sufficient energy to targets to

damage them. Weapon design involves a dialog between weapon

designers, and military planners. Designers create means of projecting energy, and planners have targets that they would like to

destroy. Effective design requires a knowledge of the targets and

the circumstances of their engagement, and effective planning requires a knowledge of the weapons and their characteristics. But in

new and emerging areas of weaponry, designers and planners

often don’t speak the same language. As a result, designers can operate in ignorance of operational realities, and planners can assume

that anything involving new technology will meet all their needs.

This book should also serve as an introduction to the language of

directed energy weapons for military planners and other non-technical persons who need to understand what the engineers and scientists involved in their development are talking about.

Chapter 1 outlines basic philosophies and ideas that are used

throughout the book. The other chapters are each devoted to a specific type of directed energy weapon, and are reasonably self-contained. Therefore, a reader interested primarily in one weapon type

will find it sufficient to read Chapter 1 together with the chapter of

interest. In some cases, duplication is avoided by developing topics

in great detail in one chapter, and presenting them again in a summary form in other chapters. The reader is referred to the detailed

discussion for any elaboration that may be required.
xxviii

I have assumed no technical background other than that associated with an introductory college-level physics course. Some

knowledge of algebra and trigonometry is assumed. A knowledge

of calculus would be helpful but is not required. Equations are

provided so that those with sufficient interest and motivation can

extend the results in the text. Numerous graphs and examples will

enable casual readers to skim over any material which seems too

mathematical.

Weaponry is not a precise science. Propagation paths and target

details are never known precisely. You wouldn’t want to go hunting for bear with a rifle whose bullet was precisely designed to just

penetrate the skin of an average bear, only to come up against a

bear that had just put on weight for the winter! You’d probably

prefer a rifle designed to work against the biggest conceivable

bears. The same is true of directed energy weapons. Too much precision in effects calculations is unwarranted, and a certain amount

of conservatism is required in defining operational parameters.

Therefore, I have kept arguments physical and intuitive at the expense of mathematical rigor. All formulas and expressions should

be considered correct to “zero order”—good enough to produce

answers within an order of magnitude of the “correct” result. No

attempt has been made to incorporate the latest and most accurate

experimental data, as these are under continual revision. Rather,

the material presented here is designed to enable you to place theories and results in the proper context. Extensive notes and references are provided for those who’d care to go into any topic in

greater depth.
EFFECTS OF DIRECTED ENERGY WEAPONS
1: BASIC PRINCIPLES

Overall Theme

This book deals with the effects of directed energy weapons,

treating such diverse types of weaponry as lasers, particle beams,

microwaves, and even bullets. In order to understand these

weapons and their effects, it is necessary first to develop a common

framework for their analysis. It is a thesis of this book that all

weapons may be understood as devices which deposit energy in

targets, and that the energy which must be deposited to achieve a

given level of damage is relatively insensitive to the type of

weapon employed. Nuclear weapons may be characterized in

terms of megatons, bullets in terms of muzzle velocity, and particle

beams in terms of amperes of current, but when this jargon is reduced to common units for energy absorbed by a target, similar

levels of damage are achieved at similar levels of energy deposited.

Of course, energy cannot be deposited in a target unless it’s first

delivered there. Therefore, an important element in understanding

weapons is a knowledge of how they deliver (or “propagate”)

their energy. Some loss of energy is invariably associated with this

propagation, whether it’s the atmospheric drag on a bullet or the

absorption of microwaves by raindrops. A weapon must therefore

produce more energy than needed to damage a target, since some

of its energy will be lost in propagation. As a result, weapon

design depends upon two factors. First, the anticipated target,

which determines the energy required for damage. And second,

the anticipated scenario (range, engagement time, etc.) which

determines how much energy must be produced to insure that an

adequate amount is delivered in the time available. This chapter is

devoted to developing this theme, introducing concepts and tools

which will be used throughout the remainder of the book.

A Word About Units

Since our goal is to reduce the jargon associated with different

types of weaponry to common units, the choice for these common
units is obviously of interest. For the most part, we’ll use metric

units, where length is in meters, mass in kilograms, and time in

seconds. In these units, energy is expressed as Joules.1 A Joule is

approximately the energy required to lift a quart of milk a distance of three feet, or 1/50000 (2 x 10–5) of the energy it takes to

brew a cup of coffee.2 Scientists and engineers frequently prefer to

express quantities in units which depart from the standard, since

units that result in numerical values of order 1–10 are easier to talk

about. For example, it’s easier to say (and remember) that the

ionization potential of hydrogen is 13.6 electron volts than to say

that it’s 2.2 x 10–18 Joules. Since published literature on directed

energy weapons is full of such specialized units and terminology,

we’ll follow convention and depart from standard units when

others are more appropriate to the subject at hand. However, we’ll

always try to bring things back to a common denominator when

summaries and comparisons are made. Appendix A is a summary

of units which are common in the field of directed energy

weapons, along with the relationships among them.

Developing Damage Criteria

If we are to determine how much energy a weapon must produce to damage a target, we need to know two things—how much

energy it takes to damage a target, and what fraction if the energy

generated will be lost in propagating to it. These will be developed

in detail for different weapon types in subsequent chapters. For the

moment, we’ll consider some of the fundamental issues which affect damage and propagation independent of weapon type.

The Energy Required for Damage

In order to be quantitative about the amount of energy necessary for damage, we must first define what we mean by damage.

For a military system, this could be anything from an upset in

a target’s computer, preventing it from operating, to total vaporization. These two extremes are usually referred to as “soft” and

“hard” damage, respectively. Clearly, soft damage is much more

sensitive to specific details of the system under attack than

hard damage. Without knowing the details of a computer, its

circuits, and the hardness of its chips, we won’t know if it’s been

Effects of Directed Energy Weapons

2
upset until we see it in operation, whereas vaporizing it produces immediate feedback on the effectiveness of an attack. On

the other hand, vaporizing a target will require more energy

than degrading its performance. We’ll concentrate in this book

on hard or catastrophic damage for two reasons: it avoids targetspecific details which are often classified, and it provides a

useful first cut at separating weapon parameters which will

almost certainly result in damage from those for which the likelihood of damage is questionable, or for which more detailed

analysis is required.

As a simple example of the kind of energies necessary to achieve

damage, let’s first consider what it takes to vaporize an ice cube.3

If we have taken this ice cube from a freezer, its temperature is

below the temperature at which it will melt. We must first supply

enough energy to raise its temperature to the melting point.

Clearly the amount of energy required to do so will be proportional to both the necessary temperature rise and the amount of

ice in the cube. The mathematical expression for this relationship

is E  mC(Tm–Ti), where E is the energy required (Joules), m the

mass of the ice cube (grams), Ti its initial temperature (degrees

Celsius or Kelvin), Tm the melting temperature, and C a constant

of proportionality known as the heat capacity (J/gm oC). For

water, C is approximately 4.2 Joules per gram per degree, so that if

the ice cube has a mass of 50 grams and is initially at a temperature of –10 oC, 2100 Joules of energy will be required to raise it to

the melting point at 0oC.

Having raised the ice cube to the melting point, we must melt it.

The amount of energy required to convert one gram of solid to

one gram of liquid at the melting point is known as the heat of fusion, symbolized by Lm. For water, this is about 334 Joules per

gram, so that 16,700 additional Joules of energy would be required

to melt our ice cube once we had invested the 2100 Joules necessary to raise it to the melting point.

At this point, our ice cube is a puddle of water at 0oC. Some

might consider this damage enough. If, however, we insist on vaporizing it, we must raise it to the boiling point by supplying an

additional amount of energy E  mC (Tv–Tm)  21,000 Joules,

where Tv is the vaporization temperature (100oC), and we must

convert the molten water at Tv to vapor at the same temperature by supplying the heat of vaporization, Lv. The heat of

3

Basic Principles
vaporization is about 2440 Joules per gram for water, so that an

investment of 122,000 additional Joules is required to finally do

away with our 50 gram puddle of water. The total energy required from beginning to end to vaporize the 50 gram ice cube is

thus 2,100 + 16,700 + 21,000 + 122,000  161,800 Joules. It is interesting to note that the heat of vaporization accounts for about

75% of the energy required. This should not be surprising—a lot

of energy is needed to separate the bonds which hold the molecules of a solid or liquid together and disperse them as vapor.

Figure 1–1 is a plot of how the temperature of the ice cube varies

as energy is deposited, along with its physical state at various

points along the way. If the energy is deposited at a constant rate

or power (Watts  Joules/second), the bottom scale is also proportional to time.4 Clearly, the vast majority of time and energy

are taken up in vaporizing the molten cube.

Having considered the simple example of melting and vaporizing an ice cube, let’s turn our attention to materials of more

interest from a military standpoint. Table 1–1 summarizes the

heat capacities, heats of fusion, and vaporization for some common materials.5 An examination of Table 1–1 reveals some interesting facts. First, it’s clear that there isn’t a great deal of variation from one material to another. Within a column, the entries

are (roughly) within a factor of two or three of one another. This

is fortunate, and means that we’ll be able to make order of magEffects of Directed Energy Weapons

4

WATER + STEAM

STEAM

WATER

ICE

+ WATER

20 40 60 80 100 120 140 160

ENERGY DEPOSITED (pcJ)

Figure 1-1. T emperature and State of a 50 gram Ice Cube vs Energy Deposited

Temperature

(0C)

100

80

60

40

20

-20

 0
nitude damage estimates without our results being sensitive to

the type and construction of target under attack. Second, as in

the ice example, the energy required by the heat of vaporization

represents the greatest portion of the energy budget in vaporizing a target. Third, it appears from this table that about 10,000

Joules will be sufficient energy to vaporize a gram of almost anything. Given that most solid materials have a density on the

order of 1–10 grams per cubic centimeter, this is equivalent to

saying that 10,000 Joules is sufficient energy to vaporize about a

cubic centimeter of anything. 

It’s interesting to note that 10,000 Joules is close to the energy

delivered by a wide range of weapons. A few examples will

serve to illustrate this point. A typical rifle round has a mass

of about 10 grams, and is fired with a muzzle velocity of about

1000 m/sec.6 This corresponds to a kinetic energy (mv2/2) of

5,000 Joules. In its March 1979 issue, Scientific American has an

article on ancient Roman siege catapults, and reports that a typical catapult could throw a stone weighing 20 kilograms over a

range of 200 meters.7 A calculation of the kinetic energy required

for so massive a stone to travel so far results in an estimate of

40,000 Joules. Finally, a more recent issue of Scientific American

(January, 1985) has a report on medieval crossbows, and reports

that a typical bow could launch an 85 gram bolt over a range of

275 meters.8 The energy required to do this is approximately

13,000 Joules.

5

Basic Principles

MATERIAL

ALUMINUM

COPPER

MAGNESIUM

IRON

TITANIUM

11000

4700

5300

6300

8800

400

210

370

250

320

0.9

0.38

1.0

0.46

0.52

2500

2600

1100

3000

3700

MELTING

POINT, Tm

(0C)

660

1100

650

1500

1700

DENSITY

(gm/cm3)

2.7

8.96

1.74

7.9

4.5

VAPORIZATION

POINT, Tv

(0C)

HEAT

CAPACITY

(J/gm0C)

HEAT OF

FUSION

(J/gm)

HEAT OF

VAPORIZATION (J/gm)

Table 1-1. Thermal Properties of Common Metals
Is Energy Alone Sufficient for Damage?

Table 1–1 and the examples above might suggest that something

on the order of 10,000 Joules could be a good “all purpose damage

criterion,” useful as a measure of the amount of energy a weapon

must deliver to damage a target. But there are two observations

which suggest that there is more to weaponry than the mere generation of energy. First, consider the detonation of a nuclear

weapon. A bomb releases lots of energy: one kiloton of yield

corresponds to about 4 1012 (4,000,000,000,000) Joules.9 This far

exceeds a 10,000 Joule damage criterion, yet at a distance of less

than a mile from a one kiloton detonation a concrete structure

would be undamaged.10 Over the same range an artillery shell

with only 10,000 Joules of energy could easily destroy such a

structure. Second, consider the sun. In a 24 hour period, it deposits

about 5,000 Joules of energy over every square centimeter of the

earth’s surface ll, yet we see no evidence of cars melting in parking

lots, people being fried (except voluntarily, on beaches!), or houses

bursting into flame. Clearly, something more than energy is required for damage. The energy must also be delivered over a

small region and in a short time to the target. In other words, energy is not the only factor important in establishing damage criteria. Also important are the density of energy on the target (Joules

per square centimeter, often called “fluence”), and the rate of energy delivery, or power (Joules/second, or Watts).12 Let’s consider

the physical basis for these results.

Energy Density Effects. Figure 1–2 contrasts a kiloton nuclear

detonation and an artillery shell, both of which are used to attack

a structure at a range of one mile. The vast majority of the energy

released by the bomb does not intersect the target and is

“wasted” from the standpoint of damaging it. By contrast, the

artillery shell is a “directed energy” weapon, delivering all its

energy right to the target in question. To be more quantitative, if

we spread the energy in the bomb over the surface of a sphere at

a range of one mile, we find that the energy density is only about

13 Joules per square centimeter, far less than the energy density

of about 10,000 Joules per square centimeter which the artillery

round applies at the point of penetration.

Effects of Directed Energy Weapons

6
Once the effect of area is understood, it is easy to show that nuclear weapon effects are consistent with the energy delivery associated with bullets, siege catapults, and crossbows. Reinforced

concrete buildings suffered severe damage at ranges of about 0.1

mile from the point of detonation of the weapons employed

against Hiroshima and Nagasaki.13 Since these weapons had

yields of about 20 kT, they released about 8 1013 joules of energy.

At a range z of 0.1 mile (= 1.6 104 cm), the energy density would

have been about 8 1013 J/4πz2, or 2.5 104J/cm2 Therefore,

when the spreading of the blast energy is accounted for, a result

consistent with other weapon types emerges. 

7

Basic Principles

NUCLEAR DETONATION

 ENERGY SPREAD OUT

4 X 10 JOULES 12

ARTILLERY FIRE

ENERGY DIRECTED

JOULES 104

1 MI 1 MI

Figure 1-2. Energy Deposition from Bombs and Directed Energy Weapons

Energy Delivery Rate Effects. Next, consider the observation

that if energy is delivered over too long a period, it is not effective in damaging targets. This is because if energy isn’t delivered

in a short time, the target can shed energy as rapidly as it’s

deposited, and so won’t heat up to the point of sustaining damage. Cars in a parking lot heat up in the sun until they become

so hot that thay radiate energy away as rapidly as it’s being

deposited. After that, their temperature remains constant. People

on the beach perspire and cool by evaporation. Only if energy

is delivered more rapidly than the target can handle it will

damage ensue. There are three main mechanisms by which

energy can be carried away from a target: conduction, convection, and radiation.
Conduction or, more properly, thermal conduction, is the process

by which energy flows from hot regions to cold as a result of hot,

agitated molecules bumping into and exciting, or heating, their

neighbors. In this way the hot molecules lose energy and the cold

molecules gain energy until a uniform temperature is reached

throughout. This process is well known through everyday experience. The handle of a spoon in a coffee cup becomes hot as energy

flows from the hot portion of the spoon, in the cup, to the

cold portion, along the handle. Physicists speak of this as a flow

of energy “downhill” along a “temperature gradient,” as illustrated in Figure 1–3.

The term temperature gradient is just a fancy expression for the

slope of the curve of temperature vs distance illustrated in Figure

1–3. The steeper this slope, the faster energy will flow. Physically,

what’s happening is that the energy and temperature are trying

to smooth out and come to equilibrium. Energy flows until the

temperature is the same everywhere, the temperature curve is

flat, and the temperature gradient goes to zero. The mathematical

expression which captures this relationship is:

u  -k (dT/dx)

where u is the rate of flow of energy across a surface (Joules per

square centimeter per second, or (Watts/cm2), dT/dx the slope of

the temperature curve (degrees per centimeter), and k a constant

of proportionality known as the thermal conductivity. The minus

sign in this expression reflects the fact that if the slope is negative

(temperature decreasing in the positive direction) the energy flow

will be in the positive direction, and vice versa. The thermal conductivity, k, can vary greatly from one material to another. Copper, which conducts energy well, has a k of about 4.2 J/ sec cm

deg, while air, a thermal insulator, has a k of about 0.00042 J/sec

cm deg—three orders of magnitude less.

As a result of the energy flow u that results from thermal conductivity, the temperature T in different regions of the target

will change. In some regions, T will increase, and in others, T will

decrease. Figure 1–4 illustrates how a knowledge of u throughout

a target can be used to calculate the rate of change of temperature within it.

Shown in the figure is a thin region within a target having a

cross section A and a thickness dx. There is some flow of energy

Effects of Directed Energy Weapons

8
(Joules per square centimeter per second) into the region, denote

Uin, and some flow out of the region, denoted Uout. If these two

quantities are not equal, then the amount of energy within this

region will increase or decrease, and the temperature within it

will rise or fall. In the example of Figure 1–4, the flow out is less

than the flow in, with the result that the temperature within the

region illustrated will increase. What is the rate at which the

temperature will change? It is a straightforward exercise to use

the heat capacity (C) of the target material to relate the change of

energy within the region shown in Figure 1–4 to a change in temperature, and to use the thermal conduction equation U  -k

dT/dx to relate the difference in energy flow into and out of the

region (Uin–Uout), to a change in the temperature gradient,

dT/dx, across the region. This results in what is known as the

thermal diffusion equation:

dT/dt  (k/C)(d2T/dx2). 

In this expression, k is the thermal conductivity, C the heat

capacity, and  the density of the target material (gm/cm3). Physically, this equation makes a lot of sense. It tells us that the temperature will change in a region if the temperature gradient

changes across that region, so that energy does not merely flow

through it, but increases or decreases within it. The quantity

9

Basic Principles

TEMPERATURE TEMPERATURE

GRADIENT

ENERGY FLOW

DISTANCE

Figure 1-3. Energy Flow Along a Temperature Gradient
d2T/dx2 is the slope of a curve of dT/dx as a function of x, just as

dT/dx is the slope of a curve of T as a function of x. The quantity

(k/C) is known as the thermal diffusivity, and is frequently

denoted D. Interestingly enough, D does not vary much from

one material to another, and is typically on the order of 1–10

square centimeters per second. This is because materials of low

density (), such as air, also tend to have a low thermal conductivity (k), and vice versa, so that the ratio is similar over a variety

of materials.

Mathematically, the thermal diffusion equation is a second

order differential equation, and cannot be solved without the aid

of a computer except in a few special cases. Those special cases

have been studied extensively, however, because of the importance of this equation in engineering problems where understanding heat flow and the resulting changes in temperature is

necessary.l4 One such case which is of interest from the standpoint

of understanding weapon effects is illustrated in Figure 1–5.

Figure 1–5 shows how the temperature on the interior of a

solid varies with time if the surface is maintained at a constant

temperature, T. As you can see on the left hand side of the figure,

Effects of Directed Energy Weapons

10

U

U in

U out

dx x

Figure 1-4. Energy Flow and Resulting Change in Temperature

U in

dx

U out

Joules

cm2 sec

Joules

cm2 sec

e = Energy Density = E/[Adx]

 (Joules per cubic cm)
and as you might expect, the heated region propagates into the

target, which ultimately would all be heated to the temperature T.

On the right hand side, the distance into the target to which the

heat has propagated has been plotted as a function of time. This

distance obeys a rather simple law: x  Dt, where the symbol

 means “approximately equal to.” A similar relationship applies

in almost every problem of heat flow: temperature moves to fill in

up to its equilibrium value at a rate which varies as the square

root of time. This result is frequently of use in developing criteria

for target damage from different weapon concepts, and will be

used extensively in subsequent chapters.

Physically, thermal conduction arises because temperature is

related to the random motion of molecules. It is a microscopic

process. As molecules become hotter, they “wiggle” more, bump

into their neighbors, and agitate them as well. In this way, the

hotter molecules cool down by giving up energy, the cooler molecules warm up by gaining energy, and the whole assembly moves

towards a constant, equilibrium temperature. By contrast, convection is a process in which heat is carried away through the macroscopic motion of molecules. A common example is the wind from

a fan. The macroscopic flow of air induced by a fan can carry hot

air away from an attic, for example, and lower its temperature

much more efficiently than thermal conduction. In the study of

11

Basic Principles

2 sec 3 sec

1 2

Figure 1-5. Temperature vs. Time and Distance

x (cm)

1

2

3

2468 10 t (sec)

t= 1 sec

x (cm)

Temperature

T
weapon effects, convection is an important source of energy loss

in a number of situations. Many targets, such as airplanes, are

moving rapidly through the air. There is a motion-induced wind

across the surface of these targets which can be an important

factor in establishing their damage threshold from weapons such

as lasers which deposit energy primarily on a target’s surface (see

Chapter 3). In other cases, since hot air is lighter than cold air and

tends to rise, the process of heating a region can itself set air into

motion, affecting the threshold and extent of damage.

Mathematically, the change of temperature due to convection

can be handled as illustrated in Figure 1–6. Shown in this figure

is a region of space in which the temperature is varying with

distance with a temperature gradient, dT/dx. Wind of velocity

V comes along and time dt blows this temperature profile

downstream to the point indicated by the dotted line. As a result,

the temperature at some point drops in time dt from T to

T–V (dT/dx) dt. Thus, the rate of change of T in time at point is

dT/dt  –V dT/dx. This expression for the effect of wind on the

temperature at a point clearly makes sense—if the wind velocity V

is stronger, the temperature drops more rapidly, and if the temperature is the same everywhere, so that the temperature gradient

dT/dx is zero, then the wind serves only to replace hot air with

more hot air, and the temperature does not drop. 

For a target to lose energy by thermal conduction or convection,

it must be immersed in the atmosphere, water, or some other

Effects of Directed Energy Weapons

12

T

e

m

p

e

r

a

t

u

r

e

Wind, Speed V

T(t) - T (t+dt)

V dt

= dT/dx

dT/dt = -VdT/dx

T (t + dt)

V dt

T (t)

Distance x

Figure 1-6. Effect of Wind on Temperature at a Point
medium to supply the necessary molecules to carry the energy

away. Yet even targets in the vacuum of outer space can lose energy through radiation. As the molecules and atoms in a target heat

up, some of the energy that their temperature represents resides in

internal degrees of freedom. That is, the molecules are not only

moving randomly in space, but are also vibrating, rotating, and in

other ways incorporating energy into their internal structure. It is

a well established fact that molecules can give up internal energy

of this sort by emitting electromagnetic radiation. Electromagnetic

radiation is discussed in detail in the introduction to Chapter 3,

since it is crucial to an understanding of lasers and their interaction with matter, but we’re all familiar with certain types of radiation, such as light, radio waves, and microwaves. 

As a target becomes hot, the molecules within it begin to give

up some of their energy as radiation. In some cases, this radiation

is visible as light, such as the radiation from the hot filament in a

light bulb, or from a red hot piece of iron in a forge. In other cases,

the radiation may be of a type which we can’t see with our eyes,

such as the infrared radiation emitted by warm objects and

detectable only with special equipment. But all of this radiation is

a source of energy loss, limiting the rise in a target’s temperature

as energy is deposited within it. 

The mathematical details of how much radiation a given target

will emit at a given temperature can be quite complex,15 but one

special case which can be treated in detail is the radiation from a

“black body.” A black body is a mathematically ideal surface

which would absorb all the radiation incident upon it, and therefore would in equilibrium radiate away more energy than any

other object. The total intensity of radiation, S (Watts/cm2) emerging from the surface of a black body at temperature T is S   T4,

where   5.67 10–12 Watts/cm2 K4, and is known as the StefanBoltzmann constant. l6

Figure 1–7 is a plot of the radiation from a black body as a function of temperature. The important point to note is the strong

dependence on temperature. Since radiation is proportional to the

fourth power of T, it does not become important until fairly high

temperatures are reached. Moreover since any real object will radiate to a lesser extent than the ideal black body, Figure 1–7 is an

upper bound to the potential for energy loss through radiation

from a target. 

13

Basic Principles
Implications

In looking at what it takes to damage targets, we’ve seen that

damaging targets depends not only on delivering energy, but also

on concentrating the energy in both space and time. In space, we

need to deliver something like 10,000 Joules per square centimeter of target surface, either at a single point, as with a bullet, or

over the whole surface, as with a nuclear weapon. In time, this

energy must be delivered more rapidly than the target can get rid

of it through such energy loss mechanisms as thermal conduction, convection, and radiation. Our task in subsequent chapters

will be to look at how each weapon type deposits energy in a target, and then to consider energy deposition and loss rates to determine criteria for damaging the target. The fluence

(Joules/cm2) or intensity (Watts/cm2) necessary to damage a target will typically vary with the time or pulse width that the

weapon engages the target, and will have the form shown in

Figure 1-8. For extremely short times, energy is deposited into the

target so rapidly that there is no way for radiation, conduction, or

other energy loss mechanisms to carry it away. For short pulse

widths (less than t1 in the figure), the fluence necessary to damage the target is a constant, and the intensity necessary to damage

Effects of Directed Energy Weapons

14

1010

108

106

104

102

0

10-2

10-4

100 1000 10000 100000

(W/cm2)

S

TEMPERATURE, 0K

FIGURE 1-7. RADIATION FROM A BLACK BODY vs TEMPERATURE
it decreases linearly with pulse width. At longer interaction

times, such as between t1 and t2 in the figure, some of the energy

deposited is carried away before it can contribute to damage, and

so the fluence to achieve damage begins to rise with pulse width.

Finally, beyond some long pulse width such as t2 in the figure,

energy is deposited too slowly to do any damage unless some

minimum intensity is exceeded, and the energy threshold is proportional to pulse width.

Scaling

An important task which will face us as we develop damage

threshold curves like those shown in Figure 1-8 will be to determine how the curves shift or “scale” as important parameters of

the problem are varied. For example, thermal conduction may be

an important factor in establishing the intensity level at which a

target will damage. Knowing how the damage threshold depends on (or scales with) this parameter, we can immediately determine the threshold for targets of different materials if we

know the threshold for one. This is particularly useful when the

mathematics of deriving damage thresholds is so complex that

simplifying assumptions must be made. As a result of these assumptions, we may not have confidence in the magnitude of the

damage threshold derived, yet feel that the scaling is well established through our analysis. In this case, experimental data can

15

Basic Principles

Fluence

(J/cm2)

Intensity

(W/cm2)

t

1 t2

Pulse Width, sec

t

1 t2

Pulse Width, sec

Figure 1-8. Damage Thresholds vs Pulse Width
be taken in the laboratory and used as a starting point for scaling

to situations where the powers, ranges, or target parameters are

quite different from those in the lab. Of course, doing this with

confidence requires a good understanding of how things scale

and where transitions occur from one type of scaling to another. 

“All-Purpose” Damage Criteria

It is beyond the scope of this book to develop damage criteria

for each type of directed energy weapon against all targets of potential interest. There are too many such targets, and the details of

their design and construction are in most cases not well enough

known to permit detailed analysis. Therefore, it would be useful

to have some generic criteria that could be applied to a first approximation in developing the weapon parameters which are

likely to achieve damage. 

We have made a start along these lines by restricting our attention to such “hard” damage mechanisms as target melting or vaporization. As we saw in Table l–l, the energy required to vaporize a cubic centimeter of most materials is about 104 Joules, and

indeed, most weapons damage targets when they are capable of

delivering a fluence of about 104 J/cm2 to them on time scales

Effects of Directed Energy Weapons

16

10.000

1.000

0.100

0.110

0.001

105 104 103 102 10

0.1 1 10 100 1000

Area on Target (cm2)

Fluence (J/cm2)

Depth

(cm)

Figure 1-9. Depth Vaporized by 104 Joules vs Area Engaged and Fluence
too short for the energy to be rejected. The reason for this is

suggested by Figure 1–9, which shows the depth to which 104

Joules can vaporize a target as a function of the area over which

this energy is spread. As you can see from the figure, 104 Joules

is only capable of vaporizing a significant depth of target when

the area over which it is spread is such that the fluence is on

the order of 104 J/cm2. At significantly lower fluences, the depth

vaporized would not be sufficient even to penetrate the skin of

most targets. 

The fact that so many weapons place energies on the order of

104 J/cm2 on target, together with Figure 1–9, suggest that we can

take 104 J/cm2 as an all-purpose damage criterion, and assert that

making a hole in a target to a depth of about 1 centimeter is sufficient to damage almost anything. Is there a rational basis for such

a conclusion, and what are the limitations to keep in mind while

applying it?

First of all, it is important to recognize that in saying that a

weapon must be able to vaporize to a depth on one centimeter

into a target, we are being very conservative. Most targets are less

than a centimeter thick, if we count only the thickness of solid

matter that must be penetrated to prevent the target from functioning. The outer surface of an automobile, for example, is sheet

metal whose thickness is far less than a centimeter. If a weapon

were to penetrate that surface near the gas tank, it would then

propagate through several centimeters of air with little opposition before encountering the surface of the tank, which is itself

less than a centimeter in thickness. Thus, an automobile might be

damaged through rupture of its gasoline tank at fluences much

less than 104 J/cm2. On the other hand, if the weapon were to

penetrate the hood and encounter the engine block, it could encounter considerably more mass, and the fluence necessary for

damage would be correspondingly greater. 

Additionally, the mechanism of target damage need not be vaporization of a hole clear through it. A bullet, for example, penetrates by pushing material aside, rupturing the bonds that hold

the target together along a few lines, rather than throughout the

volume that it passes. Intuitively, it should take less energy to

move material aside than to vaporize it entirely. A laser might vaporize a thin layer of the target’s surface with such rapidity that

the vapor, in blowing off, exerts a reaction force on the target that

17

Basic Principles
deforms or buckles it. In short, we need to consider the specific

mechanisms by which each weapon type interacts with matter,

and take these into account in establishing the fluence or intensity

requirements for damage. Thus, the interaction with matter of the

weapons considered in this book is a significant topic in subsequent chapters. 

We have seen that there are two ways in which an all-purpose

damage criterion might need to be modified to reflect the energy

requirements in a realistic scenario—the effective thickness of the

target, in terms of the mass which must be penetrated, may be

greater or less than a nominal one centimeter, and the mechanism

by which penetration occurs may be different from pure vaporization. The second issue is a matter of physics, and will be

treated in detail for each weapon type. The first issue is a function

of the target to be attacked, and can be dealt with here only in the

most general of terms. 

One way of capturing the relative thickness of material a

weapon would encounter in attacking a target is to look at the

“average thickness” of the target. This number is obtained by

finding the thickness of a plate having the same mass and surface

area as the target. In effect, the target’s innards are plastered

against its walls to give a feel for how much mass would be encountered on a random path through it. Figure 1–10 shows the

average thickness of some typical targets, based on rough available data on mass and surface area.17 

From Figure 1–10, you can see that in satellites and aircraft,

where weight is at a premium, there is relatively little mass for a

weapon to encounter. Tanks, on the other hand, have thick layers

of protective armor, and an ICBM is literally filled with solid propellant. This creates the intuitive picture that the threshold for

damaging a satellite will be less than that for damaging an ICBM

or tank. Published estimates of the energy needed to damage targets suggest that fluences on the order of 104 J/cm2 are required

to damage thick targets, whose thickness is on the order of a centimeter or greater. Therefore, a possible zero-order approach to

establishing damage criteria would be to use that value for thick

targets, and degrade it for thinner targets in proportion to the effective thickness. This would suggest, for example, that the fluence necessary to damage a satellite would be on the order of 100

J/cm2, a value which is also consistent with published estimates.18

Effects of Directed Energy Weapons

18
Will the tank and ICBM be equally difficult to damage? Probably not, since damaging the tank requires penetrating its thick

armor, while damaging the ICBM does not require penetrating

all its solid fuel. You only need to penetrate its relatively thin

skin, and gases will vent through the hole, altering the rocket

thrust so that a successful flight is not possible. Thus, even

measures of average hardness such as the effective thickness

shown in Figure 1–10 need to be tempered with some feeling for

the construction of the target and the mechanisms by which it

may be damaged.

In subsequent chapters, we will generally use 104 J/cm2 as a

nominal damage threshold for hard targets engaged by weapons

on short time scales. This will be done primarily as a means of

establishing the general parameters within which weapon propagation and interaction should be studied. This threshold should

not, of course, be taken as a definitive number, since the thickness of target penetration and the mechanism of interaction

could be different from those which this criterion implies. Wherever possible, results will be provided in such a way that you

can supply whatever damage criterion you feel appropriate, adjusting weapon performance parameters appropriately. And

when the mechanisms of target interaction are considered, their

implications from the standpoint of damage criteria will be

discussed in detail.

19

Basic Principles

20

18

16

14

12

10

 8

 6

 4

 2

 0

Satellites Aircraft ICBMs Tanks

TARGET TYPE

Figure 1-10. "Effective Thickness" of Targets

Thickness

(cm)
Energy Spread and Loss in Propagation

At the beginning of this chapter, we indicated that there are two

essential elements necessary to understand the interaction of

weapons with targets: the energy which must be deposited within

the target if it is to be damaged, and the losses which will be sustained as energy propagates from the weapon to the target. Knowing these two things, it’s possible to design an effective weapon

which will produce sufficient energy in a short enough time

that damage criteria can be met even after propagation losses are

accounted for. In general, there are two types of energy loss in

propagation: the spreading of energy such that some of it does not

interact with the target, and the wasting of energy in interactions

with a physical medium, such as the atmosphere, through which it

passes on the way to the target. The first type of loss will occur

whether the weapon and target are located on earth or in the

vacuum of space, while the second will occur primarily when

either the weapon or the target lies within the atmosphere.l9 Let’s

consider each type of loss in turn.

Energy Spread

In discussing damage criteria, two contrasting weapon types

were discussed: directed energy weapons, in which all the

energy transmitted is brought to bear on the target, and bombs,

in which the energy is spread out indiscriminately over an ever

expanding sphere. All real weapons fall between these extremes,

since even lasers and other weapons which have been characterized as being “directed energy” have some inherent spread associated with their propagation.20 This may be due to physical

reasons which cannot be overcome, such as the diffraction of

light as it emerges from a laser, or due to practical or engineering

limitations, such as the spread of bullets on a target which occurs

even when a skilled marksman aims consistently at a single

point. The concepts of divergence and jitter are used to describe

these effects, and are illustrated in Figure 1–11. In the upper half

of the figure, laser light emerges from a device and, after propagating a distance z to its target, has spread to a beam size R. This

spreading, which is known as “beam divergence,” may be characterized in terms of the angle, , which the beam envelope

Effects of Directed Energy Weapons

20
makes. The beam size R and the range z are related through the

simple geometrical relationship R=z.

In the lower half of the figure, a beam of particles or bullets is

being fired, and due to a lack of shot-to-shot reproducibility the

shot group occupies a size R at range z. The angular spread , resulting from this jitter in the aiming and firing mechanism is related to the range, z, and diameter of the shot pattern, R, through

the same geometrical relationship.

Under realistic circumstances, both divergence and jitter can

contribute to the energy spread from a weapon. For example, a

laser beam will have some divergence related to the wavelength

of the light, and some jitter resulting from the accuracy of its

pointing and tracking mechanism. In either case, this spread in

energy means that the weapon in question may have to put out

much more energy than the nominal 10,000 Joules in order to

insure that 10,000 Joules are actually brought to bear on a small

area of the target. As we have seen, a bomb is an extreme example of this phenomenon. The bottom line is that there is no

perfect directed energy weapon, and an adequate description of

any weapon must include some measure of its departure from

21

Basic Principles

RANGE, z

SPREAD, R

Figure 1-11. Divergence and Jitter

θ
perfection. Considerable time and effort will be devoted in subsequent chapters to the mechanisms which cause energy from

each weapon type to spread on its way to the target, and how

this divergence is affected by parameters which are under the

control of the weapon designer, such as energy, pulse width, and

initial beam radius.

Energy Losses

Not all the energy emitted by a weapon will make it to the target.

Some will inevitably be lost through energy losses to the atmosphere. Table 1–2 provides a summary of some of the loss mechanisms which can affect the weapon types considered in this book.

As you can see from Table 1–2, there are numerous mechanisms

by which energy can be lost from weapons propagating in the atmosphere. Indeed, these mechanisms can feed on themselves to

increase energy losses. For example, the heating of the air through

which a laser beam passes can modify the atmospheric density

within the beam channel in such a way as to increase the divergence of the beam. Such “nonlinear” effects are often a feature of

the propagation of energy that is sufficiently intense to damage

targets. As a result, the subject of propagation in the atmosphere

occupies a larger volume of text than any other subject in this

book. That’s not to say that this is the most important topic in the

book. Indeed, the difficulties associated with atmospheric propagation have caused interest in many weapon types to shift to applications that can be accomplished in the vacuum of space.

Chapter Summary

The study of weapon effects is in essence a study of energy—

how it propagates to, interacts with, and is redistributed within a

target. The goal of this study is to determine under what conditions sufficient energy will accumulate within a target to damage

it. In order to achieve damage, energy must be concentrated both

in space and time. The following fundamental ideas will find application throughout the remainder of this book.

1. The necessary concentration of energy in space (fluence) for

a hard target kill is on the order of 10,000 Joules per square centimeter. This fluence will serve as an upper bound for our

Effects of Directed Energy Weapons

22
analyses, since many targets of interest may be damaged at

lower fluences (see Figure 1–10).

2. For 10,000 Joules per square centimeter to achieve damage, it

must be concentrated in time so that it cannot flow and be

redistributed within the target. Energy deposition is manifested

in a temperature rise which is proportional to the mass and heat

capacity of the region over which the energy is absorbed. The

redistribution and loss of energy occurs through three primary

mechanisms: thermal conduction, convection, and radiation.

When the time scale for weapon-target interaction is such that

these mechanisms can come into play, the fluence necessary for

damage will begin to rise. Eventually, a point will be reached

where the damage threshold is more properly characterized as

an intensity dependent threshold (Watts/cm2) than a fluence

dependent threshold (J/cm2) The main task of each subsequent

chapter will be to determine what these thresholds are and

where in interaction time the transition between them occurs

(see Figure 1–8).

3. Directed energy weapons are those for which the energy is

directed at the target. However, no weapon fully meets this

ideal. All are characterized by some level of beam divergence,

which spreads the energy out as it propagates, and jitter, in

23

Basic Principles

Energy Loss Mechanisms

Atmospheric Drag

Absorption by molecules

Scattering by molecules

Absorption by aerosols (small particles)

Scattering by aerosols

Absorption by molecules

Scattering by molecules

Absorption by water droplets

Scattering by water droplets

Energy losses to electrons

Scattering from nucleii

Scattering from electrons

Radiation

Weapon Type

Kinetic Energy

(bullets, rockets)

Lasers

Microwaves

Particle Beams

Table 1-2. Energy Losses in Propagation
which multiple shots do not follow exactly the same path.

Divergence and jitter are characterized in terms of an angle, 

which relates beam spread, R, to range to target, Z, through the

mathematical relationship R  Z.

4. In the atmosphere, various energy loss mechanisms (absorption, scatter, etc), will cause some fraction of the energy directed

at a target to be lost in propagation to it. This must be accounted for in developing weapon design criteria. A weapon

must be capable of giving up the energy that will be lost in

propagating over the anticipated range to target and still place

sufficient fluence or intensity on the target to damage it. 

Where We’re Going

Each chapter to follow is devoted to a specific type of directed

energy weapon, and is organized into the following main sections.

• An introduction, in which the fundamentals of each weapon

type are developed.

• A discussion of propagation in a vacuum. Here the factors

responsible for the divergence associated with each weapon

type are discussed in detail, and used to develop criteria

for placing a damaging level of fluence on target in the absence

of atmospheric effects.

• A section on propagation in the atmosphere, where energy

loss mechanisms and their implications from the standpoint of a

weapon’s design and its ability to deliver damaging fluence

within the atmosphere are discussed.

• A section on interaction with targets. The specific mechanisms

by which each weapon type deposits energy in targets are

discussed, and the resulting advantages and disadvantages of

that weapon type are highlighted. The implications of weapon

specific target interaction mechanisms from the standpoint of

damage criteria will be addressed.

• Notes and references. These are presented as a guide to more

detailed literature for those interested in examining any topic in

greater depth.

Effects of Directed Energy Weapons

24
Each chapter is self-contained, and may be read without reference

to any other. In some cases, however, topics which are developed in

great detail in one chapter are presented in a more cursory form in

others, with reference to the chapter containing a more detailed

treatment for those who find this less than satisfying.

25

Basic Principles
Notes and References

1. The “Joule” as a unit of energy derives its name from James P.

Joule (1818–1889), whose experiments on heat and work established the equivalence between them, and led to the law of conservation of energy.

2. The estimate of the energy it takes to brew a cup of coffee assumes that it is a 6 oz cup, and that the water must be raised about

80C in temperature.

3. Heating, melting, and vaporization are discussed in any text on

thermodynamics, such as Chapter 11 of Mark W. Zemansky, Heat

and Thermodynamics (New York: McGraw-Hill, 1957). In general, all

thermodynamic quantities such as heat capacity, heat of fusion, and

heat of vaporization are functions of temperature and pressure.

4. The “Watt” as a unit of power derives its name from James

Watt (1736–1819), the inventor of the steam engine.

5. The data in Table 1–1 were taken from “Physical Constants of

Inorganic Compounds” in Robert C. Weast (ed.), Handbook of

Chemistry and Physics, 45th ed. (Cleveland, OH: Chemical Rubber

Co., 1964), and from Tables 22.06 and 22.07 in Herbert L. Anderson (ed.), Physics Vade Mecum (New York: American Institute of

Physics, 1981). Similar data for other materials are available in almost any physics or engineering handbook.

6. See C. J. Marchant-Smith and P. R. Halsam, Small Arms and

Cannons (Oxford: Brassey’s Publishers, 1982).

7. Werner Soedel and Vernard Foley, “Ancient Catapults,” Scientific American 240, 150 (March, 1979).

8. Vernard Foley, George Palmer, and Werner Soedel, “The Crossbow,” Scientific American 252, 104 (January, 1985)

9. More precisely, the nuclear equivalent of one kiloton of TNT is

by definition 1012calories, or 4.184 1012Joules. See Table D, section 1.02, Physics Vade Mecum (note 5). 

10. Chapter V, Samuel Glasstone and Philip J. Dolan, The Effects of

Nuclear Weapons, 3rd ed. (Washington, DC: US Government Printing Office, 1977). This book is an excellent reference on all things

having to do with nuclear weapon effects. 

Effects of Directed Energy Weapons

26
11. The intensity of radiation from the sun, or solar constant, is

0.134 W/cm2. The actual intensity received at any point on the earth

will vary with the time of year and latitude, because in reaching any

given point solar radiation must propagate through different thicknesses of atmosphere and will strike at different angles.

12. The units relating to the strength and density of energy striking a

surface are perhaps less standard than any others, since different conventions have evolved in different branches of physics. We will use

“fluence” for J/cm2, and “intensity” for W/cm2. Be aware that others

may use these terms differently, and it’s good practice to check the

units associated with such terms to be sure of their meaning.

13. See Figures 5.20, 5.22, and 5.23 of Glasstone and Dolan (note 10).

14. The standard reference is H. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford: Clarendon Press, 1959). A

less theoretical approach, with an emphasis on graphical solutions

to problems arising in chemical engineering, can be found in

Aksel L. Lydersen, Fluid Flow and Heat Transfer (New York: WileyInterscience, 1979). 

15. A good treatment of thermal radiation can be found in Section

II of Ya. B. Zel’dovich and Yu. P. Raizer, Physics of Shock Waves and

High-Temperature Hydrodynamic Phenomena, vol I (New York: Academic Press, 1966).

16. See Equation 2.16 in Zel’dovich and Raizer (note 15).

17. The effective thicknesses shown in Figure 1–10 are averages of

calculations for specific weapons made with data available from a

variety of sources. There is, of course, considerable uncertainty involved in estimating a target’s surface area from its published dimensions, especially for something like a satellite or tank. Satellite

data are from Reginald Turnill, ed. Jane’s Spaceflight Directory 3rd

ed. (London: Jane’s Publishing, 1987). Aircraft data are from

William Green and Gordon Swanborough, Observers Directory of

Military Aircraft, (New York: Arco Publishing, 1982), and from

Norman Polmar, The Ships and Aircraft of the U.S. Fleet, 12th ed.

(Annapolis, MD Naval Institute Press, 1981). ICBM data are from

Bernard Blake (ed) Janes Weapon Systems, 1987–88 (New York:

Jane’s Publishing, 1987). Tank data are from United States Army

Weapon Systems, 1987, Department of the Army, 1987.

27

Basic Principles
18. Estimates of damage thresholds for ICBMs and satellites can be

found in the “Report to the APS of the Study Group on Science

and Technology of Directed Energy Weapons,” Reviews of Modern

Physics 59, Part II (July, 1987). See Section 3.1.2 and Chapter 6.

19. Some type of energy loss through interaction with a physical

medium could occur even in space. For example, a weapon may

need to propagate through the exhaust plume of a rocket, or the

target may even eject or be covered by some type of absorbing,

protective material as a countermeasure against attack.

20. Interestingly enough, there have recently been suggestions that

the radiation from a nuclear weapon could be focused, making it

more of a directed energy weapon. See Theodore B. Taylor, “Third

Generation Nuclear Weapons,” Scientific American 256, p. 30

(April, 1987).

Effects of Directed Energy Weapons

28
2: KINETIC ENERGY WEAPONS

The word kinetic comes from the Greek verb to move, and kinetic

energy weapons are those for which it is the energy of a moving

projectile, such as a bullet or rocket, which damages the target.

Kinetic energy weapons are the oldest form of directed energy

weapon, spears and catapult stones being early examples of

weapons in this category. In some classification schemes, the term

directed energy weapon is reserved for modern, high technology devices such as lasers or particle beams, and kinetic energy weapons

are kept in a class by themselves. Nevertheless, they properly fit

the definition which we have adopted for directed energy

weapon—their energy is aimed or directed at a target, and intercepts a small fraction of the target’s surface area. Including them

in this book is appropriate from the standpoint of completeness,

and serves as a useful point of departure for the more esoteric discussions in later chapters. The general approach taken in this

chapter is also the same as that we will use throughout. We’ll first

discuss some of the fundamental concepts needed to understand

kinetic energy weapons, then their propagation or travel towards

a target, and finally their interaction with a target and the mechanisms by which the target is damaged.

Fundamentals of Kinetic Energy Weapons

Kinetic energy is the energy which an object has by virtue of its

motion.1 Mathematically, the kinetic energy of an object having a

mass M and velocity v is K  Mv2/2. This definition makes sense

on physical grounds. We’d expect that at the same velocity (say 55

mph) a more massive object (such as a semi truck) would have

more energy, and be more likely to damage an object it encountered, than a less massive object (such as a motorcycle). Similarly,

if two objects have the same mass, we’d expect the one moving at

the greater velocity to have more energy.

An important principle of physics is that energy is a conserved

quantity. That is, the energy of an object can’t increase unless it

gains this energy from some outside source. An object is given
kinetic energy when outside forces act on it, doing work and

accelerating it. An object loses kinetic energy when it, in turn,

exerts forces and does work on a second object.The energy lost can

appear as kinetic energy in the second object, such as when one

billiard ball strikes another. It can also appear as random energy

(heat), or disrupt the second object’s structure, such as when a

bullet pierces a target.

Mathematically, the kinetic energy gained or lost when an object

is accelerated can be found from Newton’s law: F  Ma, where F

is the force acting on the object, M is its mass, and a is the acceleration (rate of change of velocity, dv/dt) which the object experiences. From Newton’s law, you can see that a greater force will

accelerate an object more rapidly, and that more massive objects

are more resistant to acceleration than lighter ones. In metric

(MKS) units, mass is expressed in kilograms, acceleration in

m/sec 2, and force in Newtons.2

It is important to recognize that force, velocity, and acceleration

are all what is known as vector quantities—those which require

both a magnitude and a direction to be completely specified.3 For

example, a bullet which is moving at 1,000 m/sec toward us is

quite different from one which is moving at 1,000 m/sec away

from us. Therefore, to specify the velocity of an object we need to

provide both its speed and the direction in which it is moving.

Similarly, the force applied to a object might be in the same direction as its velocity, in which case the object will be accelerated, or

it might be in a direction opposite to its velocity, in which case it

will be decelerated. The force on an object might even be in a

direction which is unrelated to its velocity. For example, gravity

exerts a force on all objects which accelerates them towards the

center of the earth, regardless of their direction of motion.

The forces which act on an object are independent of one another, and the object’s motion is simply the sum of the motions

which each force acting alone would have produced. For example,

a bullet fired from a gun feels both the force of gravity, which

makes it move downward, and a drag force from the resistance of

the air through which it propagates, which slows its velocity in the

forward direction. The study of how the projectile from a kinetic

energy weapon propagates is simply an analysis of the forces

which act on it and their resultant effect on the projectile’s motion.

Figure 2–1 illustrates how forces affect the motion of an object.

Effects of Directed Energy Weapons

30
31

Kinetic Energy Weapons

a. An object moving at velocity v will continue to move at that velocity unless acted upon by some outside force.

b. The force of gravity acts downward on the object, producing an additional component of velocity in the downward direction,

with the result that the object follows a curved path.

c. Atmospheric drag induces a force which is opposite to the object's velocity. This force decelerates the object,

reducing its velocity. Together with the force of gravity, it produces a curved path of shorter range.

Figure 2-1. Forces and Their Effect on an Object's Motion.

v

v

v Fd

Fg

Fg

When a projectile encounters a target, interest shifts from the

forces felt by the projectile to those felt by the target. The two are

related, of course. The forces which the projectile exerts on the target, possibly damaging it, are mirror images of the forces which

the target exerts on the projectile, slowing it down and probably

damaging it as well. Two principles are of value in evaluating the

interaction between a kinetic energy weapon and its target. One is

conservation of energy, which says that any energy lost from the

projectile must be given to the target. Therefore, if a projectile enters a target with one velocity and emerges from the other side

with a lower velocity, the energy transferred to the target is the

difference between the kinetic energies on entry and exit. A second principle is conservation of momentum. The momentum of

an object is the product of its mass and velocity, Mv. Conservation

of momentum requires that the total momentum of the projectile

and target be the same before and after they interact. Therefore, if

a target is initially at rest and is struck by a projectile whose momentum is Mvo, the target’s momentum and that of the projectile

after the interaction will sum to Mvo. In applying the principle of

conservation of momentum, it must be remembered that the velocities are all vector quantities, so that their direction, as well as

their magnitude, must be taken into account.

When two bodies such as a projectile and target interact,

conservation of energy and momentum are useful in evaluating

the energy and velocity of each following the interaction. They
are not, however, generally sufficient to predict the damage done

to the target. This is because damage is usually some internal

disruption of the target, rather than a change in such gross features as velocity or energy.4 Physical damage of a target is usually

determined by the pressure the target feels, the area over which

this pressure is applied, and the time for which the pressure

applied. Pressure is the force a target feels, divided by the area

over which that force is applied.

Pressure is a useful concept because the response of a target to

an applied force depends upon the area over which that force is

applied. For example, if a hammer strikes a wooden table, it will

probably only dent the wood. But if the hammer strikes a nail, and

the nail in turn strikes the table with the same force, the smaller

area represented by the nail will result in a greater pressure, and

the nail will probably penetrate the table’s surface.

The total force felt by a target is the pressure applied multiplied

by the area over which it is applied. A target’s response will depend on total force as well. A man may lean against a structure,

applying a certain pressure to it with his palm, and do no damage.

Yet the wind may apply the same pressure over the whole structure, and the total force could be sufficient to topple it.5

Finally, the time over which a force is applied is important in

determining a target’s response. Squeezing an object with a sufficiently high pressure will gradually alter its shape, with the

amount of deviation being proportional to the length of time the

squeeze is applied. The product of force times the time over

which the force is applied is known as impulse.

6 The momentum

and energy of a kinetic energy weapon, together with the resulting force, pressure, and impulse transmitted to a target, are the

key parameters which enter into any discussion of target damage.

Table 2–1 is a summary of these parameters and the relationships

among them.

Of course, the specific damage to be anticipated in a given situation depends not only on the parameters of the projectile, but also

on the nature of the target. How structures respond to forces

applied to them, and the criteria for their damage, is an engineering subject known as strength of materials. It’s well beyond the scope

of this book to go into this subject in any detail. For purposes of

understanding the effects of kinetic energy weapons, it will be

sufficient to deal with general principles and generic results.

Effects of Directed Energy Weapons

32
33

Table 2-1. Parameters Affecting Target Response 

ParameterKinetic EnergyMomentumForcePressureImpulse

Symbol Definition Comm

Units

K

p

F

P

I

Joules (J)

kg m/sec

Newtons (Nt)

Nt/m2

Nt sec

Mv2/2

Mv

M dv/dt

Force / Area

Force X Time

M,v = ProK and p aalso F = dForce / Ar Nt/m2 
Propagation in a Vacuum

Any weapon must reach a target before it can damage it, and

propagation can alter the physical parameters with which the

weapon engages the target. For example, atmospheric drag will

slow a bullet, so that it will be less effective at a given range. Additionally, the environment through which weapon propagates can

affect its motion, and must be compensated for in aiming the

weapon. For example, a strong wind will deflect a bullet. Therefore, understanding the constraints which propagation places on a

weapon is the first step in examining its utility for a given application. Two cases are of interest—propagation in a vacuum, which

characterizes kinetic energy weapons employed in space, and

propagation in the atmosphere, which characterizes their employment on earth. Of course, there are cases in which the propagation

of a weapon takes it through both environments, such as when a

ballistic missile launches and reenters through the atmosphere, yet

spends a substantial portion of its propagation path in the vacuum

of space. Situations like this are easily treated by analyzing each

phase of the weapon’s flight in turn. We’ll first consider propagation in a vacuum.

Two types of force affect the propagation of a kinetic energy

weapon in the vacuum of space—internal forces resulting from

any rocket motor which the weapon may have, and external forces

resulting from the environment through which it propagates. In a

vacuum, the only significant external force is that of gravity. We’ll

first consider the propagation of a projectile which has no propulsion of its own, and whose motion is determined only by gravity

and the initial conditions given it by the weapon launcher. We’ll

then consider necessary modifications for a projectile with onboard propulsion.

Motion Under the Influence of Gravity

The effects of gravity on the motion of an object are described

by the law of universal gravitation.

7 This law says that there is an

attractive force between any two bodies, whose magnitude is

proportional to the product of their masses, and inversely proportional to the square of the distance between them. Mathematically, this is expressed as F  GMm/r2, where M and m are the

Effects of Directed Energy Weapons

34
masses of the two bodies, r is the distance separating them, and G

is a constant (= 6.67 10–11 nt m2/kg2) known logically enough as

the gravitational constant. The gravitational force is a relatively

weak one, even though it’s responsible for such useful features as

our being held to the surface of the earth, and not drifting off into

space. For example, two bowling balls, each having a mass of

about 4.5 kg and separated by a distance of 1 meter, are attracted

to each other with a force of only 1.4 10–9 nt (3 10–10 lb). It is

only because of the very large mass of the earth (6 1024 kg) that

the bowling balls are held to the earth with the much more substantial force of 44 nt (10 lb).

As the example of the bowling balls makes clear, gravitational

forces are important when a kinetic energy weapon interacts with

massive astrophysical objects such as the sun, earth, or moon.

They are not important and need not be considered when objects

of ordinary size interact with one another. Therefore, a kinetic

energy weapon will not home in on its target through attractive

gravitational forces, but will be affected in its flight to the target by

a gravitational pull towards the earth. This effect which must be

accounted for in aiming the weapon.

When dealing with the propagation of directed energy

weapons in a vacuum, we’re of course considering applications

in outer space. The atmosphere gradually decreases in density

with altitude, and becomes negligible at altitudes on the order of

100–200km.8 Therefore, propagation in a vacuum is of interest

for “strategic” applications, where the target and the weapon

are in outer space, and the distance between them can be very

large. A geosynchronous satellite, for example, orbits at an altitude of 40,000 km. The study of how objects move in space near

the earth is known as orbital mechanics, since the first practical application was found in predicting the motion of artificial

earth satellites.

It is straightforward to determine the path of a body of mass m

moving under the influence of the gravitational force from a much

larger body of mass M, such as the earth.9 All paths are of the form

/r  1 +  cos where  and  are constants, r is the distance of

the body from the center of the earth, and  is the angle which

locates the body along its path, as illustrated in Figure 2–2.

The parameter  is known as the eccentricity of the body’s path.

Some of these paths are very familiar. For example, if  = 0, then r

35

Kinetic Energy Weapons
Effects of Directed Energy Weapons

36

Figure 2-2. Parameters Locating a Body along its Orbit

r

θ

is a constant  independent of  and the path or trajectory is

simply a circular orbit with the earth at its center. If  lies between

0 and 1, the trajectory is an ellipse, with the center of the earth at

one focus. If   l, the trajectory is a parabola, and if  is greater

than one, it is a hyperbola. Some of these trajectories are shown in

Figure 2–3.

The elliptical and circular trajectories shown on the left hand

side of the Figure 2–3 are those of satellites orbiting the earth. Parabolic or hyperbolic trajectories are those of bodies which have

sufficient energy to escape the gravitational pull of the earth. It is

also possible for the trajectory of an object in space to intersect the

surface of the earth, such as that shown on the right hand side of

the figure, which is characteristic of an intercontinental ballistic

missile (ICBM).

With kinetic energy weapons, the object is to make the weapon

intersect the target at some point on its trajectory. In this case, interest lies in determining how the angle  shown in Figure 2–2

varies with time, since  determines the position of an object on its

circle

Hyperbola Elliptical trajectory

which intersects

the earth's surface

ellipse

Figure 2-3. Possible Trajectories for Objects near the Earth
path. To locate the trajectory in space, we also need to know how

that trajectory is oriented with respect to the earth. It requires six

different parameters to completely specify an object’s trajectory

and the location of the object on that trajectory.10 The specific data

which serve to locate a body on its trajectory around the earth are

known as orbital elements or ephemeris data. These are cataloged for

satellites which are in constant orbit around the earth, and must

be rapidly calculated for objects like ICBMs or kinetic energy

weapons, whose trajectory may be a partial orbit of short duration. Attacking an object in space with a kinetic energy weapon involves predicting the motion of the object, and then firing the

weapon with orbital elements that will result in its intersecting the

target object at some time in the future.

Let’s be more quantitative for the simple case of an object in a

circular orbit around the earth, illustrated in Figure 2–4. From

the law of gravitation, there is an attractive force between the object in orbit and the earth, having a magnitude F  GmM/r2

where M is the mass of the earth, m the mass of the satellite, and

r the distance from the satellite to the center of the earth. The altitude h of the satellite is less than r by an amount equal to the

radius of the earth, since the distance r which enters into the law

of gravitation is that between the centers of the objects concerned.11 The velocity v of the satellite is parallel to the surface of

the earth, and therefore perpendicular to F, as illustrated in the

37

Kinetic Energy Weapons

Figure 2-4. An Object in Circular Orbit Around the Earth

F

h

r

v
figure. This force accelerates the satellite in a downward direction, with the result that it follows a circular path, continually

“falling” towards the center of the earth. The situation is entirely

analogous to swinging a rock on a string—the rock would like to

fly off in a direction away from the swinger, but the string is continually pulling it back, as gravity does in the case of the satellite. The acceleration felt by an object constrained to move in a

circular path of radius r at a constant velocity v is v2/r, so that

Newton’s law, F  ma, becomes GmM/r2  mv2/r in this case.12

This expression is easily solved for the velocity which an object

must have to orbit the earth at a radius r: v  (GM/r) 1/2. Interestingly, this velocity is independent of the satellite’s mass—all

objects orbiting at a given radius will have the same velocity,

regardless of how big they are. This is because the gravitational

force on an object is proportional to its mass. As it becomes more

massive and resistant to acceleration, the gravitational force increases proportionally to compensate. The time  that it takes the

satellite to go around the earth once is called the period of the

orbit, and is just the circumference of the circular path divided

by the satellite’s velocity: 

  2 r/v  2r3/2/(GM)1/2. 

Figures 2–5 and 2–6 are plots of the orbital velocity, v, and period, , as a function of altitude (h). From these figures, you can

see that as an object’s orbit gets further from the earth, its velocity slows down and the time it takes to go around goes up. A

case of particular interest is a geosynchronous satellite. At an

altitude of about 42,000 km, such a satellite has a period of

24 hours, the same as the rotational period of the earth. In an

equatorial orbit, it would appear from the surface of the earth to

remain fixed over a single point. Communication satellites have

orbital altitudes in this range. The space shuttle and many other

satellites operate at much lower altitudes, on the order of 200km.

This region is known as “low earth orbit.”

You can see from Figure 2–5 that the velocities of objects in orbit

around the earth are on the order of several thousand meters per

second. Speeds like this are greater than the muzzle velocities

from typical infantry weapons, which are on the order of 1,000

m/sec. Therefore, a kinetic energy weapon in space probably has

more than enough kinetic energy to damage a target. The trick is

Effects of Directed Energy Weapons

38
39

Kinetic Energy Weapons

LOW EARTH ORBIT GEOSYNCHRONOUS

100 300 1000 3000 10000 30000 100000

Altitude, h (km) r = h + 6370 km

10000

 1000

Velocity

(m/sec)

Figure 2-5. Orbital Velocity vs Altitude

100 300 1000 3000 10000 30000 100000

100

10

1

Figure 2-6. Orbital Period vs Altitude

Altitude, h (km) r = h + 6370 km

Period

(hr)
to arrange the trajectory of the weapon so that it intersects the trajectory of the target object.

If the launcher for a kinetic energy weapon is in a given orbit,

and a round is to be launched to intersect a target in another orbit,

the kinetic energy round must be given an impulse to launch it on

a new orbit that will intersect the target’s orbit at a place and time

when the target is there. You will recall (see Table 2–1) that impulse is simply the force applied to an object multiplied by the

time that force is applied. When a kinetic energy round is

launched, the force is generally quite high, and the time of application quite short, so that the round may be considered to have

been given an instantaneous kick in some direction. Mathematically, the impulse given to an object is equal to the change in its

momentum p, and since momentum is p  mv, this is equivalent

to some change in velocity, v. Therefore, it is traditional among

those who deal in orbital mechanics and orbital transfer to speak

of how much “v” a launcher or rocket has as a means of specifying its potential to intersect other objects in a timely manner. A

large v implies a potential for more maneuvering, and the ability

to encounter a targeted object more rapidly. Figure 2–7 is suggestive of how v affects the ability to engage targets.

Shown in Figure 2–7 is a kinetic energy weapon following some

trajectory in space, as indicated by the center line. At some time,

an impulse can be delivered to this weapon, changing its velocity

v by an amount v. Now v can be applied in the same direction

as v, in which case the velocity of the weapon increases and it

moves up to a higher orbit, or it can be in the opposite direction,

dropping it down into a lower orbit, or it can be in any direction

in between, as indicated by the sphere of possible directions in

which v may be applied. For each choice of direction for v, the

round will arrive somewhere different at some later time t1. The

Effects of Directed Energy Weapons

40

t3 t2 t1 V v

Figure 2-7. Effect of v on a weapon's motion
surface labeled t1 locates those points which the round might

reach at t1 given an appropriate choice in the application of v. At

some later time t2, the round has the potential to reach points

within the surface labeled t2 and so on.

Clearly, the reach of the weapon within a given time span

depends on the amount of v which its launcher can provide to it.

Therefore, there is considerable interest in developing rocket fuels

which have a high specific impulse, or ratio of impulse to weight of

propellant employed. If a small weight of propellant can burn to

yield a large impulse, there is a high specific impulse, and a very

efficient weapon. For space-borne systems, keeping the specific

impulse high is especially important, since the cost of launching

things into orbit places a premium on weight reduction. Mathematically, specific impulse is defined as Isp  Ft/mg, where Ft is

the impulse (Force time) given to the round, m the mass of

propellant employed to deliver the impulse, and g is a constant

(= 9.8 m/sec2) which converts mass to weight.13 Most rocket

fuels have an Isp on the order of 200–400 sec. The advantages of

advanced technology fuels are usually expressed in terms of their

potential for achieving higher Isp.

Figure 2–6 makes it clear that even though objects in orbit

around the earth have considerable speed, the large distances

involved mean that considerable time, on the order of several

hours, can be required for them to traverse the distance from one

point to another. Therefore, increasing the reach of a weapon by

increasing the impulse it can provide to its rounds is not sufficient to insure that targets can be engaged in a timely manner.

The weapon may well be on the other side of the earth at the time

a target emerges. Thus, multiple weapons must be deployed in

space to insure that one is in place near the point where a target

is expected to appear at the time that it does. The arrangement of

weapons and the orbits they’re deployed in is known as a constellation. Establishing the constellation of kinetic energy weapons

for a given scenario involves tradeoffs among the altitude of the

orbit, the impulse which can be given to the weapon rounds, and

timelines available for target engagement. It is beyond the scope

of this book to discuss these issues, but the literature is full of

lively discussion on constellation sizing.14

41

Kinetic Energy Weapons
The Motion of Powered Weapons

In the previous section, we assumed that a kinetic energy

weapon moved towards its target only under the influence of

gravity and an initial impulse from a launcher. It is also possible

for the weapon to have on board an engine and control devices to

steer it towards its target. Such an engine could be used throughout the weapon’s flight, or could be reserved for fine tuning the

trajectory as the weapon neared its target, making the initial calculation of orbital parameters for the weapon less difficult. Kinetic

energy weapons which have some measure of on-board terminal

guidance to their target have been referred to as “smart rocks.”15

An object in powered flight feels two forces: that of gravity and

that from its engine. In this case, Newton’s law, F=ma, becomes

GmM/r2 + Fr  dp/dt, where the first term is that of gravity and

Fr is the additional force from the engine. The acceleration term,

ma, must be expressed in terms of the rate of change of momentum, dp/dt  d(mv)/dt, since the mass of the object is not fixed,

but decreases with time as fuel is expended.

The solution to Newton’s law when a rocket motor adds significantly to the acceleration of gravity is scenario-dependent, and a

general solution cannot be presented. However, it is easy to show

that a substantial amount of fuel is required for an on-board engine to add to the forces experienced by a weapon throughout its

flight. The specific impulse of an engine is the ratio of the total impulse delivered to an object divided by the weight of fuel expended: Isp  Frt/mg. Therefore, the force contributed to the

weapon by its motor is Fr  mg Isp/t. This force will be comparable to the gravitational force if FrGmtM/r2, where mt is the total

mass of the weapon, to distinguish it from the mass of the fuel, m.

Therefore, if Fr is to rival gravity as an influence on the weapon’s

motion, mg Isp/t  Gmt M/r2. This expression can be solved for

the ratio of fuel mass to total vehicle mass: m/mt =(Re/r)2 t/Isp.

l6

What does the expression m/mt = (Re/r)2 t/Isp tell us? The radius of the earth is about 6400 km, and satellites orbit at altitudes

from about 100–20,000 km above this. Therefore, the factor

(Re/r)2 is somewhere between 0.1 and 1. This means that the

ratio of fuel to total mass will approach unity if the duration of

powered flight is between one and ten times Isp. Since Isp is on the

order of 200–400 seconds (3–7 minutes), you can see that unless a

Effects of Directed Energy Weapons

42
substantial fraction of the weapon’s mass is devoted to fuel, the

amount of time a kinetic energy weapon can engage in powered

flight is limited compared to orbital time scales (see Figure 2–6).

This means that weapons with on-board propulsion would

generally use it in two phases—an initial phase, to aquire the

orbital parameters necessary to approach the target; and a final

phase, to “fine-tune” the trajectory to strike the target. Therefore,

even the flight of powered weapons is described over most

of their path as the flight of weapons given an initial impulse

toward their target.

Summary: Propagation in a Vacuum

1. Two forces affect the motion a body in space—the gravitational attraction of the earth, and any propulsive force. The

propulsive force can be either an impulse from a launcher or a

reaction force from an on-board rocket.

2. Gravitational forces are very strong because of the large

mass of the earth. Therefore, bodies in motion under the influence of the earth’s gravitational field have substantial velocities (5000 m/sec), and sufficient kinetic energy to meet damage criteria. This means that the weapon launcher or on-board

propulsion are designed more to bring about the intersection

of the weapon and the target than to impart sufficient kinetic

energy for damage.

3. Because of the large distances involved in space, flight

times from one point to another can be very long. Therefore,

large numbers of orbiting weapon launchers are required to

insure the timely interception of targets. Requirements can be

reduced as propulsion systems are developed with a capacity

to impart a greater velocity change (v) to weapons on launch.

Implications

Kinetic energy weapons could be among the first weapons

employed in space because of their proven technology. Their disadvantages lie in long flight times from weapon to target, the

need for accurate target trajectory data to insure interception,

and the number of deployed launchers needed to insure timely

target engagement. These disadvantages have spurred interest in

43

Kinetic Energy Weapons
such high technology directed energy weapons as lasers and particle beams, which project energy at or near the speed of light,

and have the potential to defend a greater volume of space.

Propagation in the Atmosphere

Gravitational Forces

Just as in space, kinetic energy weapons within the atmosphere

are subject to the force of gravity. However, the fact that we are

now within the atmosphere and near the surface of the earth enables certain simplifying assumptions to be made. For all practical

purposes, the earth’s atmosphere extends to an altitude no greater

than about 100 km. At this altitude, atmospheric density is only

about 10-7 of its value at sea level, and can have very little effect on

the flight of a projectile, at least in the near term. Since the radius

of the earth is about 6,400 km, you can see that the extent of the atmosphere is but a small perturbation, about 1.5%, to the radius of

the earth. This means that the force of gravity, F  GmM/r2, does

not vary significantly for projectiles whose flight is limited to altitudes within the earth’s atmosphere. Therefore, it is adequate for

most calculations to assume that the product GM/r2 is a constant,

roughly equal to GM/Re

2, for projectiles in the atmosphere. This

constant is known as the acceleration of gravity, and is commonly

denoted by the symbol g. Within the atmosphere, then, the force of

gravity is towards the earth’s surface, and its magnitude is given

by the simpler relationship F  mg. The value of g is about 9.8

m/sec2 in MKS units.l7

Additionally, the ranges of kinetic energy weapons within the

atmosphere are much less than the radius or circumference of the

earth. Artillery rounds, for example, might have ranges of 30 km

or so, and the employment of rifle rounds is limited to the user’s

visual range or less. Over distances this short, the curvature of the

earth is not an important factor. Figure 2–8 illustrates that in propagating a distance z along a horizontal path, the earth’s surface

drops by a distance d  Re–(Re

2–z2)1/2. Over a 100 km range, this

distance is less than a kilometer. Therefore, the drop-off d can usually be neglected for an atmospheric projectile.

For the propagation of kinetic energy weapons in the atmosphere, then, it can be assumed to a first approximation that the

Effects of Directed Energy Weapons

44
force of gravity is a constant, and the curvature of the earth may

be neglected. This second approximation is equivalent to assuming that the earth is flat, although of course terrain features, such

as mountains or trees, could well interfere with the propagation of

a kinetic energy projectile. This set of assumptions has been used

for many years in ballistics, the study of projectile motion in the at45

Kinetic Energy Weapons

z

z

r e

r e

d

[re

2 - z2]

1/2

Figure 2-8. Effect of the Earth's Curvature on Horizontal Propagation

mosphere.l8 Let’s first consider the motion of a projectile under

these assumptions alone, neglecting any further atmospheric effects such as wind or drag.

For a projectile in motion under the constant force of gravity,

Newton’s law (F  ma) becomes simply mg  ma, or g  a. In

other words, the projectile is accelerated towards the surface of the

earth at a constant rate, g, which is independent of its mass. This

says that light objects should fall as fast as heavy objects. Experience tells us that this is not so: the reason for this discrepancy is

drag, or the friction between objects in motion and the air through

which they pass. The refinement which drag makes to the equations of motion will be discussed next. For the moment, we’ll

focus on the effect of gravity alone.

The solution to Newton’s law for the motion of a projectile

under the influence of gravity is best understood by breaking its

motion into two parts—in the downrange direction, denoted by z,

and up or down in altitude, denoted by h. There is a corresponding velocity downrange, vz, and of rising or falling, vh. These are

illustrated in Figure 2–9.
As you can see from Figure 2–9, at some point in time the projectile has propagated a distance z downstream, and is at an altitude h

above the surface of the earth. The projectile’s velocity, v, may be

broken up into two components: vz, the rate at which z is increasing,

and vh, the rate at which h is increasing or decreasing. When vh is

positive, the projectile is rising, and when it’s negative, the projectile

is falling. 

The solutions to Newton’s law for vh, vz, h, and z when a projectile is acted upon by the force of gravity are as follows:

vh  voh – gt

vz = voz 

h  voh t – gt2/2

z  vozt

In these equations voh and voz are the projectile’s initial velocities

in the h and z directions, respectively. Since gravity acts only to

pull the projectile towards the surface of the earth, it does not

affect vz, and the projectile moves downrange at a constant velocity, with z growing linearly in time. On the other hand, vh is

steadily decreasing in time, due to the force of gravity, and eventually becomes negative, so that the projectile falls. The range of the

projectile can be found by solving the expression for h for the time

to at which h  0 and the projectile strikes the earth. This time can

then be inserted into the equation for z to see how far downrange,

zr, the projectile will have gone when it strikes the earth. The result

of this procedure is that to  2 voh/g, and zr  2 voh voz/g.

The initial velocity components (voh, voz) given to the projectile

are crucial in determining its range—the greater they are, the further the particle will go. From a practical standpoint, voh and voz

Effects of Directed Energy Weapons

46

z

h

h

V

V V

z

Figure 2-9. Projectile Distance and Velocity Coordinates

Vh=dh/dt Vz= dz/dt
are not independent. The device which launches the projectile will

give it some total velocity, such as the muzzle velocity of a bullet

emerging from a rifle. This velocity is then apportioned between

voh and voz by setting the elevation angle of the launcher. As

shown in Figure 2–10, a given elevation angle 	 produces velocity

components voh = v sin	 and voz  v cos 	. In the extreme case

where 	  90°, the projectile goes straight up and comes back

down at the point of launch: when 	  0°, the projectile strikes the

ground almost immediately, and similarly has no range. The maximum range is achieved when 	  45°, and voh  voz  0.707 v.

Figure 2–11 is a plot of range as a function of elevation angle.

In summary, if gravity alone were to act on a projectile fired

near the surface of the earth, its range would be determined solely

by the velocity imparted to it and by the elevation angle with

which it was launched. The shape and mass of the projectile do

not enter into the analysis, and do not affect its propagtion.

Drag Forces 

Many of the results from the previous section make sense—our

everyday experience tells us that if we throw a stone faster it will

47

Kinetic Energy Weapons

V oh

V

v oz

Zr = 2voz voh /g

Voz = Vcos φ

Figure 2-10. Effect of Elevation Angle on Velocity Components

Voh = Vsin φ

φ
go farther, and that if it is thrown with some elevation between

0° and 90°, it will go farther than it will at either of these extremes. Other results are at odds with our experience. A massive

rock will propagate farther than a light styrofoam ball of the

same shape, and an aerodynamically shaped projectile will go

farther than poorly shaped one of the same mass. Clearly, propagation in the atmosphere means more than propagation under

gravity with assumptions appropriate to being near the surface

of the earth. As a projectile moves through the atmosphere, it

must push aside the air through which it passes. This results in a

force on the projectile which is known as drag. Drag acts in a

direction opposite to that of the projectile’s motion, as shown in

Figure 2–12. 

Unlike gravity, which always pulls towards the surface of the

earth, drag opposes the projectile’s motion through the air, and is

directed opposite to its velocity. Therefore, drag forces have a

component both downrange and in altitude. When a progectile is

rising, drag slows its ascent. When it is falling, drag slows its rate

of fall. And in the downrange direction, drag lowers a projectile’s

velocity and reduces its range.

Effects of Directed Energy Weapons

48

Relative Range = Range / Max Range

Elevation Angle (deg)

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

 0

 0 5 10 15 20 25 30 35 40 45 50 55 60

Figure 2-11. Projectile Range vs Elevation Angle

Relative

Range
What factors affect the magnitude of the drag force? The crosssectional area A that the projectile presents to the air through

which it passes will be important. The greater this area, the more

air must be pushed aside as the projectile propagates, and the

greater the drag. This principle is employed in parachutes, which

open to a large cross-sectional area, increasing drag and slowing

the descent of a falling body. The density of the atmosphere, 
,

will also be important. The lower 
 becomes, the less air must be

pushed aside, and the lower drag should be. This effect is taken

advantage of by long distance aircraft, which prefer to operate at

high altitudes, where the density of the air and drag are less, improving fuel efficiency.

The speed of the projectile will be important, since one which is

moving rapidly will encounter more air in a given amount of time.

Finally, the shape of the projectile will be important. One which is

streamlined can slip through the air with less drag than one which

presents a flat surface on its forward edge. All of these commonsense features are embodied in the expression for the drag force:

Fd  Cd A 
v2/2. In this expression, A is the area of the projectile

as viewed end-on, 
 is the density of the air through which it trav49

Kinetic Energy Weapons

Drag Force, F = ρCd V2 A/2

Fz

F

h

cross-sectional

area A

Gravitational Force

F = mg

Figure 2-12. Drag and Gravitational Forces on a Projectile
els, and v is its velocity. The combination of parameters 
v2/2 has

a simple physical interpretation as the average kinetic energy density represented by the air molecules which are, from the projectile’s point of view, rushing forward to meet it. Physically, this kinetic energy density is the pressure which a flat plate would feel if

it were rushing through the air with velocity v. When multiplied

by the cross-sectional area A, the total drag force felt by such a flat

plate results. The drag coefficient, Cd, is a measure of how much

better a given projectile is than a flat plate from the standpoint of

its shape reducing drag.

In the presence of atmospheric drag, Newton’s Law (F  ma)

leads to the following equations which need to be solved for the velocity and position of the projectile, both downrange and in altitude:

m dvh/dt  –mg –Cd A 
 v vh/2

m dvz/dt= –Cd A 
 v vz/2

In these coupled equations, v is the projectile’s total velocity, (vz 2

+ vh 

2)l/2. There is no general solution to these equations, since they

depend on parameters which vary with the environment (
), the

shape of the projectile (Cd, A), its mass, and flight profile. Nevertheless, a few general conclusions are immediately apparent. Most

importantly, the mass of the projectile is now a big player in its

trajectory. Dividing the equations of motion through by m, you

can see that the magnitude of the drag term relative to the gravitational term depends upon Cd A/m, and can be reduced if m can

be increased while the other factors are held constant. For projectiles of a similar shape and material, A and m will rise in proportion to one another—an increase in mass will be accompanied by a

corresponding increase in size and cross-sectional area. The only

way to reduce the ratio of A to m is to go to materials of high density, where a large amount of mass occupies a relatively small volume. This is one reason why high density materials, such as lead

or depleted uranium, are favored for kinetic energy projectiles.

Other than reducing the ratio of A to m through the use of high

density materials, the only way to reduce the effect of drag is to

lower the drag coefficient, Cd. The drag coefficient is a function both

of the shape of the projectile and its velocity. Some of the factors

which affect Cd are illustrated in Figure 2–13.19 The lines in the

Effects of Directed Energy Weapons

50
figure indicate what happens to the air as the projectile passes

through it. First of all, the air which is ahead of the projectile must

be pushed aside. In a sense, the projectile acts like a piston as it

rushes through the air. This compresses the air directly ahead of the

projectile, and the work expended in doing so is reflected in drag. A

longer, sharper tip will reduce the amount of drag which results

from compression of the air ahead of the projectile.

After a projectile has pushed the air aside, it must pass

through it. There will be a contribution to drag from the friction

of the air which passes along the surface of the projectile. The

contribution of this “skin friction” to drag will be greater as the

surface area of the projectile increases, and is therefore of less

importance for things like rifle or artillery rounds, than for larger

objects such as rockets.

Finally, after a projectile has passed through a given region of

space, the air must return to fill in the hole left by its passage. This

results in a region of turbulence behind the projectile. In this region, the air does not flow smoothly, but collapses back in a series

of eddies, just as a ship leaves a turbulent wake behind in the

water. Just as with ships, this base drag can be reduced by shaping

the trailing edge of the projectile, so that it doesn’t end as abruptly

as the one illustrated in Figure 2–13.

The expression for the drag force, Fd = Cd A 
v2/2, appears to

suggest that Cd is independent of the projectile’s velocity, v. This

51

Kinetic Energy Weapons

Friction

Compression

Turbulence

Figure 2-13. Factors Affecting the Drag Coefficient
is only a rough approximation over a narrow range of velocities.

The drag coefficient is a function of velocity, and becomes much

greater as a projectile’s velocity exceeds the speed of sound.

This is because the speed of sound is the velocity with which pressure disturbances can propagate through the atmosphere. The

compression of the air ahead of the projectile creates such pressure

disturbances. If the projectile is supersonic, these disturbances

cannot relieve themselves by moving out ahead of it, but must

pass to the side. This results in the generation of strong shock

waves, along with an increase in the drag coefficient, when a projectile exceeds the speed of sound. Most modern projectiles do, in

fact, exceed the speed of sound ( about 0.3 km/sec). The sharp

crack of a rifle bullet is the “sonic boom” which results from the

shock waves generated by its passage. The energy carried by these

shock waves is reflected in a loss, through drag, from the kinetic

energy of the bullet. Figure 2–14 is a plot of the drag coefficient as

a function of velocity for two different shapes of projectile. You

can see the increase in Cd which accompanies passage through the

speed of sound, as well as the effect of different shapes in changing the amount of drag, particularly at supersonic speeds.20

Because drag and its effect on propagation depends on a projectile’s mass, shape, and velocity, the form a kinetic energy projectile

takes will depend upon its projected mission. A shape which is

optimal for propagation over a given range may not be optimal for

acceleration in the projectile launcher, and a shape which is optimal

for acceleration or propagation may not be optimal from the standpoint of delivering kinetic energy to a target. For example, the ball

shaped projectile shown in Figure 2–14 has a lower drag coefficient

at supersonic velocities than the bullet shaped projectile, yet cannon

balls have long been abandoned as kinetic energy weapons, since

the bullet shape seals better within the barrel of a gun, and is more

effectively accelerated by a given amount of propellant. We’ll have

more to say about these tradeoffs later in the chapter. For the

moment, it’s sufficient to note that drag can have a significant effect

on the propagation of a projectile in the atmosphere.

Newton’s laws for the motion of a projectile under the influence

of both gravity and atmospheric drag may be solved quantitatively

through numerical integration on a computer. From a qualitative

standpoint, drag has the effect of reducing both the maximum altitude and the range achieved by a projectile. Figure 2–15 is a plot of

Effects of Directed Energy Weapons

52
a projectile’s trajectory both in vacuum and in air under the assumption of two different drag coefficients, and Table 2–2 contrasts

the maximum range for two types of kinetic energy projectile in air

with what their range would be if there were no atmospheric drag.

Other Forces

To this point we have looked at the roles that gravity and drag

play in the propagation of a kinetic energy projectile in the atmosphere. While these are the forces that determine the gross features

of a projectile’s trajectory, there are other forces as well. One of the

most important of these is the force resulting from any cross-wind,

or air motion perpendicular to the projectile’s path.

Drag results from the pressure exerted by the wind which results

from the relative motion between the projectile and the air. Any

naturally occurring wind will also exert a pressure on a projectile.

Since the natural wind is not necessarily parallel to a projectile’s

path, there will be a force tending to deflect it from its aimpoint.

This force will be less than that of drag, since typical wind velocities are much less than the velocities with which kinetic energy

53

Kinetic Energy Weapons

A

B

A B

Drag

Coefficient

(Cd)

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

0 .5 1 1.5 2 2.5 3 3.5 4 4.5 5

Mach Number (Speed/Sound Speed)

Figure 2-14. Drag Coefficient vs Mach Number and Projectile Shape
weapons propagate. Nevertheless, it does not take too great a

deviation from the desired path for a kinetic energy round to miss

its target. Consider, for example, the mortar round listed in Table

2–1. It has a muzzle velocity on the order of 400 m/sec, and drag

in the forward direction shortens its range by about 5 km. Suppose

there is a cross wind of about 10 km/hr (6.2 mi/hr or 3 m/sec).

Since this wind speed is about 0.008 of the wind speed resulting

from forward motion, an initial guess at the sideways deflection

would be about 0.008 5 km, or 40 meters (about 120 ft). deflection of this magnitude could easily cause the mortar to miss its

target, and would have to be accounted for in directing the

weapon’s fire. Of course, a precise prediction of the deflection

would have to account for the fact that the cross section of the

round as viewed from the side is greater than that viewed from

the front, the wind velocity may vary from point to point along

the projectile’s trajectory, and the effective “drag coefficient” at veEffects of Directed Energy Weapons

54

Altitude

h (km)

Vacuum

Cd = 0

Cd = 0.01

Range Z, km

Cd = 0.1

30

20

10

10 20 30 40 50 60 70 80 90 100

Figure 2-15. Trajectories in Vacuum and with Different Drag Coefficients

Type of Projectile

300 mm Morter

7.62mm Rifle

Max Range

in Vacuum

16 km

70 km

Max Range

in Air

11 km

4 km

Table 2-2. Effect of Drag on Projectile Range (21)
locities characteristic of natural winds will be different from that

at typical muzzle velocities. Therefore, from a practical standpoint,

wind forces require that aiming be done on a “shoot-look-shoot”

basis, with rounds initially fired according to theory, and adjustments then made based on impact point observations.

Additional forces result from the rotation of the earth. Newton’s

law (F  ma) assumes that the forces and accelerations are measured against a set of fixed coordinates, not moving in space. Yet

the earth is spinning from west to east, and a projectile launched

to the east has a greater velocity than one launched to the west.

The motion of the earth adds an effective force to the equations of

motion for a projectile which varies with the latitude of the

launcher, as well as the bearing along which projectile is fired.

Weapons whose sights have been adjusted for operation in the

northern hemisphere will miss their targets in the southern hemisphere, simply because some of these rotational forces differ in

sign above and below the equator.22

Instabilities

To this point, we have treated the propagation of a projectile as

though it were a single point in space of mass m, accelerated according to Newton’s law by various forces. However, a real projectile has a finite shape. The interaction of this shape with the air

can cause it to become unstable, tumbling so that drag is much

increased and propagation prevented. How this can happen is illustrated in Figure 2–16. The single point on a projectile which

follows the trajectory predicted by Newton’s law is known as the

center of mass. Suppose the projectile is not pointed directly

along the trajectory, but has some angle relative to it, as illustrated in the figure.

Since the projectile is moving in the air, there is an effective wind

on it. As shown in the top part of the figure, this wind causes a

pressure on the projectile. In the average, this pressure results in

drag. Looking more closely, you can see that if the projectile has

some angle relative to its direction of propagation, pressure above

the center of mass will tend to flip the projectile over, while pressure below the center of mass will tend to straighten it out. An instability will occur if the pressure above the center of mass exceeds

the pressure below the center of mass, and the projectile flips over.

55

Kinetic Energy Weapons
In tumbling, aerodynamic stream-lining is lost, drag increases by

orders of magnitude, and useful propagation ceases.

In more technical terms, the average point at which wind pressure appears to be applied is known as the center of pressure, and

if the center of pressure is ahead of the center of mass the projectile will be unstable, while if it is behind the center of mass it will

be stable.23 The bottom portion of the figure illustrates two projectiles. The first is unstable, since the broad head on the projectile

represents a large area over which pressure can be applied ahead

of the center of mass. The second is stable, since a larger, counterbalancing area has been added to the rear of the projectile. This is

why arrows have feathers on their trailing edge.

In some cases “feathers,” in the form of fins, are still employed

for stability on projectiles such as rockets or bombs. For gun

launched projectiles, this approach is not practical since fins

would prevent the seal between the projectile and gun barrel

which is necessary for the detonating charge to accelerate the

round. Early guns used balls as projectiles, since they are absolutely stable, with their center of pressure and center of mass at

the same point regardless of orientation. In the search for greater

range, however, a better seal within the barrel is required, and this

demands that the rear of the projectile be flat. Reduction of drag

demands that the front be pointed. In consequence, most of the

Effects of Directed Energy Weapons

56

CENTER OF MASS

CENTER OF MASS

CENTER OF PRESSURE

WIND PRESSURE

WIND PRESSURE

TRAJECTORY

UNSTABLE STABLE

Figure 2-16. Potential Instability of a Projectile
mass in a modern projectile is toward its rear, and it is very easy

for the center of mass to lie behind the center of pressure, causing

an instability. This problem has been overcome by spinning the

projectile, which imparts gyroscopic forces that counter the tendency to tumble, just as a spinning top resists the efforts of gravity

to tumble over on its side.

Gyroscopic stabilization is not without penalty, however. Some

of the energy which would otherwise go into the kinetic energy

of the projectile must be invested in rotational energy, and the

projectile shape which is best stabilized in this way may not be

desirable from the standpoint of other propagation parameters.

For example, short, fat tops are easier to stabilize than long,

skinny ones. Yet a short, fat projectile would have a large cross

sectional area and a large drag force. Accordingly, the design of

kinetic energy rounds results from tradeoffs among competing

factors which affect range, stability, and even, as we’ll see later,

target interaction.

Summary: Propagation in the Atmosphere

1. Gravity and drag are the main forces which affect the propagation of a projectile in the atmosphere. Because propagation in

the atmosphere means that a projectile is near the earth’s surface, gravitational forces may normally be considered constant

and directed downward at a flat earth. Drag opposes the forward motion of a projectile, and is proportional to its area as

viewed from the front, the density of the air, and the square of

its velocity.

2. In the absence of drag, a projectile will have its maximum

range when launched at an elevation of 45°. Drag reduces both

the range and altitude achieved by a projectile launched at a

given elevation angle.

3. The constant of proportionality between the drag on a flat

plate and that on a given projectile is known as the drag coefficient. It measures how well streamlining has reduced the pressure of the air on the front of the projectile. The drag coefficient

is generally greater for projectiles moving at supersonic speeds.

4. Other factors which affect propagation in the atmosphere include winds, which blow a projectile off course, and the rotation

57

Kinetic Energy Weapons
of the earth, which imparts latitude and azimuth dependent

forces to it.

5. Projectiles will be unstable, and tumble in flight, if their center of pressure lies ahead of their center of gravity. The design of

a projectile for atmospheric use must include stability-enhancing features, such as tail fins or spinning. These features may

detract from other desirable attributes for the projectile.

Implications

The propagation of kinetic energy weapons in the atmosphere is

much more complex than in a vacuum, where the rules of the

game are well known and can be accounted for ahead of time in

sending a projectile to its target. Propagation in the atmosphere is

upset by random factors, such as wind, which can’t be accounted

for ahead of time. The shape of a projectile intended for atmospheric use may not be ideal from the standpoint of target interaction, and energy may be “wasted” to insure propagation through

spinning the projectile or overcoming drag.

These general features will be seen throughout this book. Propagation in a vacuum for kinetic energy weapons, lasers, microwaves, and particle beams proceeds by well defined and understood physical laws. In the atmosphere, additional effects come

into play which make putting energy on a target much more difficult. It is unfortunate that we and most of the targets we’d care to

engage exist within the atmosphere. Only in recent years has the

technology existed to place objects in space, making the question

of propagation in a vacuum of more than academic interest.

Interaction with Targets

Important Parameters

After its propagation either through space or the atmosphere, a

kinetic energy projectile will (hopefully) strike its target. At this

point, we hope that a significant amount of the projectile’s kinetic

energy will be transferred to the target, damaging it. What are

some of the factors which will affect the probability of exceeding

the threshold for damage?

Effects of Directed Energy Weapons

58
Pressure and Impulse. Pressure is the force applied per unit

area as the projectile strikes the target, and impulse is the integral

of force over time (roughly speaking, the force applied multiplied

by the time for which it is applied). High pressure and high impulse will be more effective in damaging targets than low pressures or impulses. How much pressure and impulse a round delivers to a target will depend upon its kinetic energy, shape, and

the material of which it is made. For example, a very skinny or

sharply pointed projectile will apply high pressures to the the

point of impact, since its forces will be concentrated over a very

small area. A projectile made of a soft, deformable material such

as lead may deliver a greater impulse than a very rigid one, which

could strike and bounce off the target in a short period of time.

Angle of Attack. A glancing blow will be less effective than one

which is head-on. Therefore, the angle with which the projectile

strikes the target is important in determining the response of both

projectile and target. As Figure 2–17 illustrates, a projectile may

ricochet and have little effect at a high angle of attack, yet penetrate the target and cause severe damage at a low angle of attack.

Target Material and Shape. The material and shape of a projectile will affect the pressure and impulse it delivers; the material

and shape of a target will affect its response. Some materials are

softer than others, and targets can be protected by covering them

with a material (armor) that is very hard and resistant to damage.

Table 2–3, for example, shows the amount of kinetic energy re59

Kinetic Energy Weapons

≈

Angle of Attack

 O

Figure 2-17. Angle of Attack

θ

θ
Effects of Directed Energy Weapons

60

quired for a given projectile (a standard 7.62 mm NATO rifle

round) to penetrate different materials of varying thickness.

Proper shaping of a target can affect the angle of attack with

which a round is likely to strike. Thus, armored objects are made

very smooth and rounded, increasing the probability of a large

angle of attack. There is little chance of a round striking head-on

to a flat surface if the target is properly shaped.

What is Damage? 

Recognizing that various parameters will affect damage, how

can we determine the optimal combination of these parameters

to achieve a given level of damage? First of all, we need to recognize that a wide variety of effects are possible when a projectile

strikes a target, all of which might be adequate for damage.

Some of these possible effects are illustrated in Figure 2–18.

A projectile may simply tear the target material, pushing it

aside and penetrating the surface, as illustrated on the left hand

side of the figure, or, it may shatter the target material so that

both projectile and target fragments emerge from the rear surface,

as illustrated in the center. It’s even possible that while the projectile may not penetrate the target, shock waves will propagate

through the target and throw flakes of material off of the back

surface, as illustrated on the right hand side of the figure. This is

known as spallation, and can damage a target even in cases

where its surface is not penetrated.

From a military standpoint, all three of the outcomes illustrated in Figure 2–18 may be equally effective in damaging or

Target Type Energy for Penetration (Joules)

Unprotected Man

23 cm Timber

Very light armor

0.5 cm concrete

1.2 cm brick

80

200

770

1500

3000

Table 2.3. Kinetic Energy Required for 7.62 mm Projectiles to Penetrate Targets (24)
negating a target. Yet the criteria for achieving them might be

quite different, and scale differently with the shape, energy, and

other parameters of the projectile. Thus, damage criteria can be

more of an art than a science. There is more than one way to

achieve a given end. As a result, there is a danger that in optimizing a projectile for a specific type of damage, a solution may

be found which is not optimal when other damage mechanisms

are considered. In addition, the entire range of issues related to

projectile acceleration, propagation, and target destruction must

be considered in weapon design.

Figure 2–18 shows just a small sample of the wide range of phenomena that can occur when a projectile strikes a target. Typically,

what will happen depends upon the kinetic energy of the projectile, the material and shape it has, and the material and thickness

of the target. Predicting what will happen is a complex problem in

predicting the response of a solid to a time varying pressure and

impulse. Since exact predictions of damage are highly dependent

on the type of target and what it’s made of, the most we can attempt to do in this book is look at some of the general principles

and orders of magnitude involved.

General Principles

The solid material which a projectile strikes differs from the atmosphere in having an internal structure and resistance to any

61

Kinetic Energy Weapons

Penetration Fragmentation Spallation

Figure 2-18. Possible Effects of Kinetic Energy Projectiles
change in the configuration of its atoms or molecules. In a gas,

molecules are free to wander about at will, and have no significant interaction with one another. In a solid, on the other hand,

molecules are so close that they are bound to one another. The

forces which bind them can be thought of simplistically as represented by the springs shown in Figure 2–19.These springs have

some equilibrium length, as well as some inherent stiffness, or

resistance to stretching or squeezing. As a result, a solid resists either compression, an attempt to squeeze it into a smaller volume,

or tension, an attempt to stretch it out. In general, the length of

the springs and their stiffness may be different in different directions, so that the response of a material may depend on how

forces are oriented relative to the arrangement of atoms within it.

Forces applied to a solid are referred to as stress. Under stress, the

springs in a solid will be compressed or stretched, and its volume

will change slightly. The change in volume of a solid under stress

is known as strain.

If stresses are sufficiently low, the strain is proportional to the

stress, just as the length of a spring is proportional to the force applied to it. Under these conditions, the response of the material is

said to be elastic. However, if the stresses become too great, the

material can become permanently deformed, just as a spring

which is stretched beyond its elastic limit will be permanently

deformed. When this happens, strain is no longer proportional to

stress, the response of the material is said to be plastic. Finally,

Effects of Directed Energy Weapons

62

Figure 2-19. Spring-like Binding Forces between Atoms in a Solid
if extreme stresses are applied to a solid, the springs or bonds

between the atoms can actually be broken, and the atoms will be

free to move around at will under the influence of these forces,

just as a gas would. At this point, the response of a solid is no

different from that of a gas.

Dimensionally, the stress in a solid is the force applied per unit

area at a point, and has the same dimensions as pressure. Strain

is the fractional change in volume, V/V, resulting from a given

stress, and is dimensionless. Mathematically, the relationship

between stress and strain is of the form P  Ce, where P is the

applied stress, e is the strain, and C is a constant of proportionality known as the stiffness coefficient, which has the dimensions of

a pressure. It is important to note that while the pressure in a gas

can only be positive, the stress in a solid can be either negative

(compression) or positive (tension).25

Since P  Ce, the fractional change in dimension induced by a

given stress on a solid will be small as long as P << C. The fractional change will be large, with the likelihood that the elastic

limit will be exceeded, when P becomes comparable to C. If P is

much greater than C, the strength of the bonds holding the

atoms together in a solid are almost irrelevant, and the solid will

respond as a gas. Therefore, the first step in analyzing the response of a target to a kinetic energy projectile is to compare the

stresses or pressures anticipated when the projectile strikes the

target with the stiffness coefficients that characterize the target

material. Table 2–4 provides stiffness coefficients for some common metals at room temperature. The data in the table reflect

63

Kinetic Energy Weapons

Material Stiffness Coefficients, C11 (at 300°K)

(x1011 Nt/m2

 or J/n3)

Tungsten

Copper

Aluminum

Lead

5.233

1.684

1.068

0.495

Table 2.4. Stiffness Coefficients of Common Metals (26)
what we already know from experience—that lead is a relatively

soft material, easy to deform, for example, with the stresses that

might be applied by a hammer. And tungsten, a stiff, brittle material, finds much of its application in the high temperature, aerospace environment.

In contrast with the stiffness coefficients in Table 2–4, atmospheric pressure is only about 105 Nt/m2. Therefore, pressures in

excess of about 106 atmospheres are required to make a solid

respond like a gas to the impact of a projectile. How much

pressure is expected from the impact of a projectile? You may

recall that the drag pressure felt by a projectile traveling through

an atmosphere of density 
 is on the order af 
v2/2—the kinetic

energy density of the air rushing to meet the projectile. In a similar manner, if a target sees a projectile of density 
o and velocity

V rushing at it, we’d expect it to feel a pressure on impact of

order 
 v2/2. This estimate of pressure on impact is shown in

Figure 2–20 as a function of projectile velocity for projectiles of

density 11.6 and 2.7 gm/cm3 (lead and aluminum, respectively).

Figure 2–20 shows that pressures in excess of typical stiffness

coefficients will be generated when projectile velocities exceed

something on the order of 5–10,000 m/sec. For purposes of

comparison, the speed of sound is about 300 m/sec, the muzzle

velocities of small arms are about 1000 m/sec, and the speed of

objects in low earth orbit is about 8000 m/sec. Therefore, you

can see that kinetic energy weapons in space are likely to induce

high pressures on targets. At these pressures, targets respond

like a high-density gas, their internal binding energies being

negligible, at least initially. Kinetic energy weapons in the atmosphere, on the other hand, will create stresses within targets

that lead to a plastic or elastic response. 

Damage in Space—Hypervelocity Impacts

Impacts of projectiles with targets at velocities well in excess of

the speed of sound are known as “hypervelocity” impacts. The

target’s binding energies can initially be neglected, and it responds

as a compressible fluid (gas). The binding energies of the projectile

are comparable to those of the target, so it, too, behaves like a fluid

on impact, and target and projectile material intermingle. The target’s initial response is similar to the impact of a drop of water on

Effects of Directed Energy Weapons

64
a puddle, as shown in Figure 2–21. This figure shows the sequence

of events when an aluminum sphere strikes an aluminum target at

a velocity of about 7000 m/sec. The projectile is severely deformed

in striking the target, and at late times the distinction between

projectile and target material has disappeared.

If the projectile and target were indeed fluid and not solid,

the disruption caused by the impact would go away, just as the

crater formed when a stone strikes the surface of a pond fills in.

However, in a solid, target material resolidifies in whatever configuration it finds itself at the time when externally applied

stresses become small compared to the internal stiffness constants. In this way, the impact shown in Figure 2–21 would leave

a crater in the target, whose appearance would be roughly like

the last of the four shown in sequence.

Beyond the crater shown in Figure 2–21, there is a region of

plastic deformation, where the density of the material might be

permanently altered, and beyond that a region into which elastic

waves will propagate. These waves, which are analogous to

sound waves in air, can be quite intense, and if the rear surface of

the target is not too far away they can cause spallation, as they reflect off the rear and head back towards the front surface. This is

because at the surface of a target its strength is less than on the interior, since molecules on the surface have no molecules beyond

65

Kinetic Energy Weapons

Velocity (m/sec)

Pressure

(Nt/m2)

300 1000 3000 10000 30000

10

10

10

10

10

10

13

12

11

10

9

8

ρo = 11.6 g/cm 3

ρo = 2.7 g/cm3

Figure 2-20. Nominal Pressures Induced by Projectiles of Various Velocities
them to inhibit their motion away from the boundary. This situation is somewhat analogous to a whip. A wave started at the handle of the whip can cause its tip to move supersonically, creating

the characteristic crack of a “sonic boom,” as it reflects at the tip

and heads back towards the handle.

The depth of the crater in Figure 2–21 is a measure of the penetrating power of the spherical projectile. If the target’s surface were

thinner than this crater, the projectile would penetrate, and a spray

of projectile and target material would fly into the target’s interior.

It is these secondary projectiles which would likely cause militarily

significant damage, since the surface of most targets merely covers

and protects important components which lie on the interior. It

would be of value to have a way of predicting the requirements for

a projectile to penetrate a target of a given material and thickness.

For hypervelocity impacts, it’s easy to develop empirical formulas

to relate projectile parameters to crater depth. This is because the

energies are so great, and the projectile so severely deformed, that

its initial shape is not an important factor in the process. Rather,

the projectile can be modeled simply as a source of momentum

and energy deposited on the target’s surface.28

How can we develop an expression to correlate projectile parameters with target damage? The procedure is analogous to that

Effects of Directed Energy Weapons

66

t = 0 t = 0.69μsec

t = 1.89 μsec

t = 13.3 μsec

Figure 2-21. Hypervelocity Impact of a Sphere on a Plane Target (27)

0.2 cm
employed for nuclear weapons and other explosives. In essence,

an explosion represents the release of a large amount of energy in

a small amount of space. Since the energy is released in a volume

which is small compared to the range of the anticipated effects, the

explosion may be treated as a point source of energy. The effect of

an explosion is overpressure—strong shock waves which travel

out from the source of an explosion, damaging objects which they

encounter. Pressure is force per unit area, or energy per unit volume. Therefore, if the shock wave from an explosion has traveled

outward a distance r, the overpressure at that distance should be

approximately E/(4r3/3), where E is the energy released by the

explosion, and 4r3/3 is the volume of a sphere of radius r. If a

given target is destroyed by an over- pressure P, this relationship

can be solved for the range to which such a target could be destroyed: r  (3E/4P)1/3. This result, that the range of the effects

from an explosive scales as the cube root of the energy released, is

well established as a method of estimating the destructive power

of nuclear weapons.29

In an analogous way, we might imagine that the range of some

effect (such as crater size) from a projectile will scale with the kinetic energy it carries. A projectile whose characteristic size is L

has a volume on the order of L3, and a mass on the order of 
L3,

where 
 is the density of the projectile material. If the projectile

has velocity V, its kinetic energy, MV2/2, will be on the order of


L3V2/2. Thinking that the kinetic energy (K) is the primary factor responsible for the creation of a crater of depth D (and volume on the order of D3), it would be reasonable to suppose that

D3=bK, where b is an appropriate constant of proportionality.

This reasoning leads to the conclusion that D3 is proportional to


L3V2, or that D/L is proportional to 
l/3 V2/3. Alternatively, if we

were to believe that the primary factor responsible for a crater

of depth D was not the energy of the projectile, but rather its

momentum ( 
L3V), we would conclude that D/L should be

proportional to 
1/3 Vl/3. In either case, if we believe that crater

depth is determined by projectile mass times some power of its

velocity, we conclude that the ratio of crater depth to projectile

size, D/L, should vary as the cube root of the projectile’s density

multiplied by some power of its velocity.

Such a relationship has been demonstrated through “numerical

experiments,” in which detailed computer calculations were

67

Kinetic Energy Weapons
made of a target’s response to projectiles of different density,

shape, size, and velocity.30 These showed that if the projectile kinetic energy density 
V2/2 exceeds about four times the target’s

stiffness constants, the ratio of crater depth to projectile size, D/L,

scales as 
l/3 V0.58. The scaling with velocity is somewhat less than

the 2/3 power that results from scaling with energy, and greater

than the 1/3 power that results from scaling with momentum.

This suggests that both energy and momentum are factors in the

response of a target to the impact from a hypervelocity projectile.

Figure 2–22 is an approximate plot of D/L as a function of V. The

figure assumes that scaling with V0.58 is valid, and that the projectile density is the same as that of the aluminum target. The curve

in the figure may be scaled as (
/
t)l/3 to cases where the projectile density, 
, differs from that of the target, 
t.

Figure 2–22 may be used to estimate the projectile size necessary to penetrate a target of a given thickness at a given velocity.

Suppose that we need to penetrate a l cm target with a projectile

whose velocity is 10 km/sec. From Figure 2–22, D/L is about 6

at this speed, so that the projectile size must be L  D/G  0.17

cm. You can see from this example that even small debris in

space can damage satellites at the velocities characteristic of

earth orbit.

Effects of Directed Energy Weapons

68

10

9

8

7

6

5

4

3

2

1

0

8 9 10 11 12 13 14 15

Velocity (km/sec)

D/L

Figure 2.22. Ratio of Crater Depth (D) to Projectile Size (L) vs

Projectile Velocity for Hypervelocity Projectiles
It is interesting to compare the kinetic energies of the projectiles responsible for the craters in Figure 2–22 with an allpurpose damage criterion of 10,000 Joules (see Chapter 1). Figure

2–23 is a plot of the kinetic energy required per centimeter of

crater depth as a function of projectile velocity, assuming that the

projectile has a characteristic size, L, of 1 centimeter. As the

figure demonstrates, the “all-purpose criterion” is a pretty good

guess of the energy required for a hypervelocity projectile to

place a cm–radius hole in a cm–thick target.3l

Damage in the Atmosphere—Lower Velocity Impacts

You will recall from Table 2–4 and Figure 2–20 that when the velocity of a projectile falls below about 5,000 m/sec, the pressure it

exerts as it impacts a target will fall below the target’s stiffness coefficients. In this velocity range, which is characteristic of projectiles fired within the atmosphere, the energy and momentum of

the projectile are no longer sufficient to determine the target’s response. Rather, the material, shape, and angle of attack of the projectile, along with specific details of the target’s construction, become much more important in establishing the target’s response. It

is therefore more difficult to establish general rules and scaling relationships for target response in this case than for hypervelocity

69

Kinetic Energy Weapons

50000

46000

42000

8 9 10 11 12 13 14 15

Velocity (km/sec)

J/cm

38000

34000

30000

26000

22000

18000

14000

10000

Figure 2-23. Kinetic Energy per Centimeter Penetrated vs Projectile Velocity
impacts. Figure 2–24 presents a simplified picture of some of the

factors which need to be accounted for when a lower-velocity projectile strikes a target and attempts to penetrate it.

Figure 2–24 shows a plate of thickness h, supported by a structural member of diameter D. This plate is struck at its center by

a projectile of diameter d and length l, which bends the plate

inward a distance x. Obviously, this produces a strain P in the

plate. The strain is a tensile one, which tries to pull the plate apart.

The direction of this strain is along the thickness of the plate, at

an angle  to the plate’s original, flat orientation. Therefore, the

component of strain in the upward direction, resisting the motion

of the projectile, is P sin   P 2x/D.32 The total force which the

projectile feels is this strain (force/area) multiplied by the area

over which the tension is applied. This area is that of the cylindrical plug under the projectile, dh. Therefore, the force on the

projectile is P(2x/D)dh.

As the projectile interacts with the plate, x and P increase.

At some point, P exceeds a critical value P* at which the plate

will rupture. P* is known as the modulus of rupture, and can be

found tabulated for different materials in engineering handbooks.33 The work (Joules) which the projectile expends to

Effects of Directed Energy Weapons

70

Figure 2-24. Penetration of a Plate by a Projectile

d

l

0 x h

D
71

Kinetic Energy Weapons

rupture the plate is the integral of force over distance34, or

roughly W  P*(2xc

2/D) dh, where xc is the value of x at which

the plate ruptures. This work must be done at the expense of the

projectile’s kinetic energy K, so that a criterion for the projectile

to penetrate the plate is K=P*(2xc

2/D)dh.

The only unknown in this expression is the distance xc. It can be

shown that e*  2xc

2/D2, where e* is the strain which corresponds

to the stress P*.35 Using this result, the criterion for projectile penetration becomes K=P*e*dhD, where d is the diameter (caliber) of

the projectile, h the thickness of the target, and D the diameter of

the structural member supporting the target plate. This implies

that target penetration is easier for

• Thinner projectiles, which apply a greater pressure at the

point of impact. This is another reason why pointed projectiles

are preferred even if not required for purposes of drag reduction in propagation.

• Thinner targets, which have less mass to penetrate.

• Rigid targets, whose structural supports are closer together.

This is because a very rigid target has less room to “give” before

it is penetrated.

For aluminum, P* is about 3 108 Nt/m2, and e* is about 0.15.36

Therefore, the kinetic energy K  P*e*dhD necessary for a 1 cm

projectile to penetrate a l cm thick sheet of aluminum supported at

a distance of 0.1 m is about 1400 Joules. This value is consistent

with the penetration energies shown in Table 2–2, suggesting that

this model for target penetration is not unrealistic, and contains the

essential features of damaging targets whose response is plastic.37

It’s also interesting to note that this energy and those in Table 2–2

are about an order of magnitude less than the energies necessary

for penetration by hypervelocity impact (see Figure 2–23). This is

because fewer bonds between the atoms in a solid have to be broken when a projectile pushes its way through than when it must effectively vaporize a whole plug of material underneath it.

Even in the simple case just considered, plate penetration depends on details of the target’s construction, such as the diameter

D of the plate’s support. Damage criteria are obviously more difficult to develop when penetration is through plastic deformation of
the material, rather than through hypervelocity impact. The fundamental reason for this is that at slower projectile velocities, stress

waves can travel from the point of impact to structural members,

whose rigidity affect the target’s response. In a hypervelocity impact, all of the action is over before any information about that

impact can be transmitted to structural members. This transmission of information occurs at the speed of sound within the

material, which is proportional to the square root of the stiffness

constant. Further details of plate penetration by projectiles may be

found in Reference 37.

In most cases, it’s not possible to know in advance such details as

the exact materials a target may be made of or how it is constructed. Even if these were known, the physical parameters these

materials possess and the resulting damage criteria can only be determined through detailed experiment and analysis. Therefore, it is

impractical to determine in advance the exact response of a given

target to a particular type of projectile of a given kinetic energy.

The most we can hope to do is use theory as a guide in scaling experimental results to projectiles of different energies and targets

of different thickness or structure. The validity of this process

depends upon the extrapolation not being too great. In general, different formulas are required to correlate weapon effects in different

parameter regimes. The subject of projectile-target interaction is

known as terminal ballistics. There is considerable literature on this

subject to enable you to correlate and extrapolate weapon effects

under different circumstances.38

Tradeoffs

The specific design of a kinetic energy round for use in the atmosphere will inevitably involve tradeoffs. If it is accelerated in a

gun barrel, it will be accelerated to a greater velocity if it has a large

base for the pressure of gases in the weapon to act on, and is short,

so that its total mass and inertia are not too great. As it propagates,

however, a short, squat projectile will feel a lot of drag, and lose

considerable energy. Drag can be reduced with a long, pointed projectile, but such a round may be unstable and require spin or fins.

Finally, when striking a target, the shape which will penetrate most

effectively may not be the one which would propagate best. The

design of kinetic energy projectiles must therefore take into account the total mission profile.

Effects of Directed Energy Weapons

72
Modern projectile design occasionally makes use of projectiles

which change their configuration during the different phases of

their operation, and can be more nearly optimal throughout their

flight. These designs are frequently found in armor piercing

rounds, which have an inherently difficult job to accomplish. An

interesting example is shown in Figure 2–25.

This projectile has features designed for each portion of its flight

from tube to target. Within the barrel of the gun, the sabot (a

French word, which in origin means “boot”) provides a seal

against the wall, and a large surface area for gases to act upon. This

sabot is discarded as the projectile leaves the barrel. During flight,

the fins at the rear provide stabilization and the ballistic cap on the

front provides a low drag coefficient. Finally, as the round encounters the target, the ballistic cap breaks away, and the penetrating

cap on the interior engages the target. The shape of the penetrating

cap is optimized for target effect and damage. In ways such as this

a kinetic energy weapon can be designed which is near optimal

throughout its flight profile. However, energy is wasted accelerating mass which does not encounter the target, and a complex design invites problems in reliability.

Summary: Target interaction

1. Unlike propagation, which follows well understood physical

laws, interaction is highly scenario dependent. The effect of a

projectile striking a target will depend on such weapon parame73

Kinetic Energy Weapons

Discarding Sabot

Ballistic Cap

Penetrating Cap

Stabilizing Fins

Figure 2-25. A Fin-stabilized, Discarding - Sabot Projectile (39)
ters as momentum, energy, and shape; such target parameters as

material, thickness, and construction; and such scenario parameters as the angle of attack between the projectile and the target.

2. The response of a target is determined from the stress applied to it, and the resulting strain or deformation it suffers.

Stress and strain are related through stiffness coefficients. When

the stress is numerically much less than the stiffness coefficient,

target response is elastic, with strain proportional to stress.

When stress is comparable to stiffness, target response is plastic,

and the target may be permanently deformed or penetrated. Finally, if stress far exceeds stiffness, a target responds as though

it were a high density gas.

3. At velocities characteristic of engagements in space, projectiles produce stress far in excess of stiffness. For these hypervelocity impacts, simple scaling laws for target damage and

penetration can be developed (see Figures 2–22 and 2–23). For

engagements in the atmosphere, stresses are more likely to

result in a plastic response. Predicting penetration under these

conditions requires more detailed knowledge of projectile,

target, and engagement parameters.

4. Projectile design involves tradeoffs among factors which

influence acceleration, propagation, and interaction. A realistic

projectile will therefore not be optimal for any one of these

areas.

Implications

For engagements in space, the lack of atmospheric drag and

the large velocities involved result in energies more than sufficient to cause damage. Projectile design is therefore not a critical

issue in space as it is in the atmosphere. Even in the atmosphere,

it is generally not advisable to optimize a weapon to defeat a

specific target. In military engagements, a weapon may be employed against a variety of targets, the enemy may harden his

targets, and so forth. Therefore, it is more appropriate to design a

weapon to exceed estimated damage criteria throughout its useful range. What this means in practice is that our all-purpose

damage criterion remains a useful estimate of the energy which a

weapon must reliably place on target to achieve damage. It is inEffects of Directed Energy Weapons

74
teresting to note that this estimate is not too bad even for infantry weapons, which for the most part are employed against

such soft targets as enemy personnel.

Chapter Summary

1. Kinetic energy weapons damage targets with their energy of

motion.This energy is proportional to a projectile’s mass and

the square of its velocity.

2. In space, projectile motion is determined by the gravitational

force of the earth, along with forces from the projectile’s

launcher or on board engine. Gravitational forces dominate a

projectile’s trajectory, and kinetic energies far exceed damage

criteria. As a result, stresses in a target exceed its internal

strength, and it responds like a dense gas. Details of projectile

and target construction are therefore of minor importance.

3. In the atmosphere, ranges are shorter and energies less due

to atmospheric drag. At these lower energies, forces internal to

a target are important, and its response depends on details of

construction and the engagement scenario. Projectile design for

efficient propagation and interaction therefore becomes a primary concern. Optimization for propagation may conflict with

optimization for target interaction, and projectile design is a

compromise among competing factors.

Implications and Analogies

The finite speed of kinetic energy weapons (10 km/sec or less)

means that the time to engage goes up with increasing distance,

and moving targets can be engaged only if they are “led,” with

calculations made in advance on how to bring the weapon and

target together. Therefore, considerable interest has arisen in more

exotic forms of directed energy weapon, such as lasers or particle

beams, in which engagement can occur at or near the speed of

light. Indeed, the main focus of this book will be on these newer

weapon concepts, since kinetic energy weapons have been studied

for centuries and documented in many texts, both theoretical, experimental, and empirical. Nevertheless, it is worthwhile to include information on kinetic energy weapons here so that it will

75

Kinetic Energy Weapons
be apparent in later chapters that the concepts dealt with there are

in essence no different than those treated here. For example, the

following truths hold throughout the book:

1. Propagation in a vacuum follows well defined physical laws,

which can be accounted for to insure that a weapon will place

adequate energy on target. However, the long ranges associated

with engagements in space place severe constraints on the energy which the weapon launcher requires to insure that lethal

energies are brought to bear on the target.40

2. In the atmosphere, ranges are much less than in space. At the

same time however, interaction with the atmosphere results in

much greater energy losses. Therefore, weapon parameters

(bullet shape, laser pulse width, etc) must be tailored to minimize these energy losses.

3. When the weapon encounters its target, energy must be efficiently absorbed for damage to occur. This places constraints on

weapon parameters which may be at odds with those necessary

for efficient propagation.

Effects of Directed Energy Weapons

76
Notes and References 

1. Any basic physics text will deal with the concept of kinetic energy. See, for example, Chapter 7 in David Halliday and Robert

Resnick, Physics for Students of Science and Engineering (New York:

John Wiley and Sons, 1962).

2. The Newton (Nt) as a unit of force is named for Sir Isaac Newton (1643–1727), whose law of acceleration, F  ma, underlies all

mechanical analysis. One of the greatest physicists of all time,

Newton is also remembered for the law of gravitation, for inventing calculus, and for fundamental contributions to optics. Every

chapter in this book makes use of his insights and theories.

3. Vector quantities are defined by the three numbers which represent their components along three independent directions. For

example, a car moving to the southeast is moving south with some

speed, east with some speed, and vertically with no speed! A good

discussion of vectors and the mathematics by which they are combined can be found in Chapter 2 of Halliday and Resnick (note 1).

4. Under some circumstances, deflection can be militarily adequate for “damage.” For example, a missile which is diverted

from its course and therefore misses its target will have been

adequately damaged.

5. A man can easily exert a pressure of two pounds per square

inch by pressing with his thumb on a wall. This will do no damage, but a similar pressure over whole wall will blow it in or shatter it. See Table 5.145 in Samuel Glasstone and Philip J. Dolan

(eds.) The Effects of Nuclear Weapons (Washington, DC: U.S. Government Printing Office, 1977).

6. More precisely, impulse is the integral of force over time. See

Chapter 10 in Halliday and Resnick (note 1).

7. The law of universal gravitation was developed by Newton

(note 2), who in folklore, at least, was inspired by an apple falling

on his head.

8. When the atmosphere becomes negligible depends to some

extent on the mission under consideration. Satellites in orbit at an

altitude of 200 km still feel some atmospheric drag, and, over a period of years will slow and fall to earth. For kinetic energy

weapons, on the other hand, flight times are not for years, but for

77

Kinetic Energy Weapons
hours or less, so that this effect can be neglected. A plot of atmospheric density as a function of altitude can be found in Chapter 5

(Figure 5–14).

9. Satellite motion under the influence of gravity is a special case of

what is known as central force motion. This subject is discussed in

detail in any book on mechanics, such as Jerry B. Marion, Classical

Dynamics of Particles and Systems (New York: Academic Press, 1965).

The specific results shown here may be found in Chapter 10.

10. A good discussion of how orbits are specified and the resulting paths traced over the surface of the earth may be found in

Charles H. MacGregor and Lee H. Livingston (eds.), Space Handbook, AU–l8 (Maxwell AFB, AL: Air University, 1977).

11. Strictly speaking, the distance r which appears in the law of

gravitation is the distance between the centers of mass of the two

objects. The center of mass is found by averaging all points within

an object with the mass density at those points and their location

in a coordinate system. See Marion (note 9), Chapter 3.

12. The force mv2/r necessary to keep an object of mass m in a circular orbit at a radius r is known as a centripetal force. These forces

are discussed in Section 6–3 of Halliday and Resnick (note 1).

13. Weight is the force which gravity exerts on an object of mass m

at the surface of the earth. Since F  GmM/r2, the “constant” g is

GM/Re

2, where M is the mass of the earth, and Re its radius.

14. A good introduction to constellation sizing may be found in

Appendix I of William D. Hartung, et al., The Strategic Defense Initiative: Costs, Contractors, and Consequences (New York: Council on

Economic Priorities, 1985).

15. More recently, as increasing circuit miniaturization has permitted greater computing capability in smaller volumes, there

have been proposals for “brilliant pebbles”—very small weapons

which are scattered out almost like a shotgun to seek and destroy

targets on their own. If these trends continue, we may some day

attack objects with “genius dust!”

16. Deriving the final expression, m/mt  (Re/r)2 t/Isp makes use

of the definition g  GM/Re

2 to eliminate g from the previous expression.

17. The constant g introduced here is the same one used in the definition of specific impulse. See note 13.

Effects of Directed Energy Weapons

78
18. C. L. Farrar and D. W. Leeming, Military Ballistics: A Basic

Manual (Oxford: Brassey’s Publishers, 1983) is a good introductory

text on ballistics.

19. The factors affecting drag are discussed in detail in Chapter 4

of Farrar and Leeming (note 18).

20. Figure 2–14 has been adapted from Figure 4.16 in Farrar and

Leeming (note 18).

21. The data in Table 2–2 are taken from Figure 4.11 in Farrar and

Leeming (note 18).

22. See Marion (note 9), Section 12.4.

23. Just as the center of mass in an object is the point arrived at by

averaging positions in the object by their mass, the center of pressure is a point arrived at by averaging points on the surface with

the pressure felt at those points. In effect, it is the average point at

which pressure is applied to the object. Since pressure is an area

phenomenon, and mass is distributed throughout an object, it

should not be surprising that in general the centers of mass and

pressure will be different.

24. The data in Table 2–3 were taken from Table 3.1 in C. J.

Marchant-Smith and P.R. Hulsam, Small Arms and Cannons (Oxford: Brassey’s Publishers, 1982).

25. Stressr strain, and stiffness are discussed in any text on solid

state physics. See Chapter 4 of Charles Kittel, Introduction to Solid

State Physics, 3rd ed. (New York: John Wiley and Sons, 1966).

26. The data in Table 2–4 were taken from Table 1, Chapter 4, of

Kittel (note 25).

27. Figure 2–21 has been adapted from Figure 1 in J. K. Dienes

and J. M. Walsh, “Theory of Impact: General Principles and the

Method of Eulerian Codes,” Chapter III in Ray Kinslow (ed), HighVelocity Impact Phenomena (New York: Academic Press, 1970).

28. The insensitivity to projectile shape in hypervelocity impacts

is apparent from the rapid deformation and loss of shape which

the projectile suffers, as shown in Figure 2–21.

29. See Chapter III in Glasstone and Dolan (note 5).

30. See Dienes and Walsh (note 27).

31. It would be more accurate to look at the energy expended per

unit volume of crater. However, as you can see from Figure 2–21,

79

Kinetic Energy Weapons
most of the crater’s volume is reflected in its depth. The width of

the crater is not substantially different from that of the projectile.

32. This approximation makes use of the fact that sin   tan 

for small .

33. See Section 5 in Ovid W. Eshbach (ed), Handbook of Engineering

Fundamentals 2nd ed (New York: John Wiley and Sons, 1961), and

Section 1.08 in Herbert L. Anderson (ed), Physics Vade Mecum

(New York: American Institute of Physics, 1981).

34. See Chapter 7 in Halliday and Resnick (note 1).

35. Deriving this result makes use of the definition of e as V/V.

The fractional change in volume can be related to the plate’s

change in diameter, which goes from D to 2[(D/2)2  x2]1/2. For x

small, this is approximately equal to D(l  2x2/D2).

36. See section 6.3.5 in the “Report to the APS of the Study Group

on Science and Technology of Directed Energy Weapons,” Reviews

of Modern Physics 59, Part 59 (July, 1987).

37. The approach illustrated here (and others) are discussed in G.

E. Duvall, “Applications,” Chapter 9 in Pei Chi Chou and Alan

K. Hopkins (eds.), Dynamic Response of Materials to Intense Impulsive Loading (Wright-Patterson AFB, OH: A. F. Materials Laboratory, 1973).

38. A variety of expressions for the prediction and correlation of

damage from lower-velocity projectiles can be found in Farrar and

Leeming (note 18), as well as K.J.W. Goad and D.H.J. Halsey, Ammunition (including Grenades and Mines) (Oxford: Brassey’s Publishers, 1982).

39. This type of projectile and other novel approaches are discussed in Farrar and Leeming (note 18), and in Goad and Halsey

(note 38).

40. It may seem as though kinetic energy weapons in space violate

this principle, since much of their energy is that of orbital motion.

However, even this energy isn’t “free,” and ultimately comes from

the energy in the rocket engines which placed the weapons in orbit.

Effects of Directed Energy Weapons

80
3: LASERS

This chapter examines the effects of lasers—one of the first

exotic directed energy weapons to capture public attention. The

acronym laser stands for “light amplification through stimulated

emission of radiation,” and thus a laser is fundamentally nothing

more than a device which can produce an intense, or highly energetic, beam of light. The theory of how lasers are constructed,

and descriptions of the different types currently available, can be

found in a wide range of texts and journal articles.l For purposes

of this book, which deals with the effects of laser light, these

details are for the most part not important.

We’ll begin by looking at the fundamental principles of laser

light—those features which laser light has in common with all

light, and those which make it unique. With these fundamental

principles in mind, we’ll then look at how the energy emitted by a

laser propagates to a target, both in a vacuum and within the atmosphere. Finally, we’ll look at how laser energy interacts with

and damages a target. Our goal will be to develop criteria which

can be used to estimate how much laser energy or power would

be required to damage a target at a given range, either in vacuum

or in the atmosphere. Particular emphasis will be placed on how

these criteria scale with such beam parameters as wavelength,

pulse width, pulse energy, and so forth.

Fundamental Principles of Laser Light

Propagation

We are all familiar with light, which is a special case of what is

known as electromagnetic radiation. Other examples of electromagnetic radiation include radio waves, x-rays, and microwaves.

Electromagnetic radiation propagates through space from its

source as a wave, much as a wave of water propagates through a

pond from its source. The source of a water wave is obvious,

such as a pebble which has been dropped into the pond, and its

propagation can be easily visualized as ripples spreading out
from the source. Unlike water waves, electromagnetic waves do

not require a physical medium for their propagation—they can

propagate even through the vacuum of space.

They can be detected when they are absorbed in an appropriate

detector, such as a radio for radio waves, or the retina of the eye

for light. The theory of electromagnetic radiation was first developed by James C. Maxwell in the 1860s. He recognized that a large

body of seemingly diverse phenomena in electricity and magnetism could be explained and summarized in four equations, appropriately known as Maxwell’s Equations. These equations and their

implications are the subject of graduate-level physics courses, and

obviously cannot be dealt with in any detail here.2 Nevertheless,

key concepts are relatively straightforward.

Waves. First, let’s consider what we mean by a wave.3 A wave

is a periodic disturbance which propagates through space. It may

be characterized by the amplitude of the disturbance, the periodic

distance over which it repeats, and the velocity with which it

propagates, as illustrated in Figure 3–1. This figure shows a wave

propagating from left to right with a velocity v.

The amplitude, or height of the disturbance illustrated in Figure

3–1 is a measure of its strength, just as a high water wave is more

likely to knock you over than a small one. The distance over

Effects of Directed Energy Weapons

82

t

Wavelength,

Amplitude

Direction of Propagation,

velocity v

vt Distance

0

Figure 3-1. Wave Parameters and Propagation

y

Time
which the disturbance repeats itself is known as the wavelength.

Finally, the wave moves with some velocity, v, so that any feature,

such as the trough shown in Figure 3–1, moves a distance vt in

time t. If you are standing at a given spot and the wave is passing

by, what you see will repeat in a time equal to the wavelength divided by the velocity, since over a wavelength’s distance the wave

repeats itself. This time in which the wave repeats is known as the

period of the wave. The inverse of a wave’s period is known as its

frequency. By common convention, wavelength, period, and frequency are denoted by the symbols , T, and , respectively. Mathematically, the relationship among these quantities is  v/, or

T  /v. The wavelength of an electromagnetic wave determines

the nature of the radiation. Figure 3–2 shows the type of radiation

associated with different regions of wavelength.

You can see from Figure 3–2 that electromagnetic waves span a

broad range of wavelengths. The span is so great that only the

logarithmic scale in the figure suffices to capture even a part of it,

with wavelengths varying from the submicroscopic (X-rays and

gamma rays) to the macroscopic (radio waves). Despite this diversity in wavelength, all electromagnetic radiation obeys the same

physical laws. We’ll briefly summarize those which are important

for understanding lasers, their propagation, and their interaction

with targets.

In a vacuum, all electromagnetic radiation travels at the same

speed—the speed of light, commonly denoted by the symbol c.4

Because of the general relationship between frequency and wavelength for waves, the frequency of electromagnetic radiation propagating in a vacuum is given by  c/.

83

Lasers

Microwaves

Radiowaves

X - Rays

Gamma Rays Ultraviolet Infrared

Visible

10-12 10-10 10-8 10-6 10-4 10-2 100 101 102

Figure 3-2. Electromagnetic Spectrum

Wavelength, Meters
The value of c is approximately 3  108 meters per second, so

that the frequencies corresponding to the extremes of wavelength

in Figure 3–2 are about 1020 cycles per second for gamma rays to

about 106 cycles per second for radio waves. A cycle per second is

known as a Hertz, abbreviated Hz.5 Thus, 106 cycles per second is a

Megahertz, a unit you may recognize from your FM radio dial.

Refraction. In a medium other than vacuum, such as air, water,

or glass, electromagnetic radiation will travel at a speed less than

c. The ratio of c to the speed in a particular medium is known as

the index of refraction, and is denoted by n. In general, n will vary

with wavelength and material, as illustrated in Figure 3–3.6

The reason the ratio between light speed in vacuum and light

speed in some material is called the index of refraction is that this

ratio is related to a bending which a beam of light undergoes

when it travels from a medium with one index of refraction to another. This bending obeys what is known as the law of refraction,

and is illustrated in Figure 3–4. The law of refraction states that

when a ray of light passes from one material of index n1 to another

of index n2, then the angles, 1 and 2, which the ray makes with a

line perpendicular to the surface between the two materials, are

related by n1 sin 1  n2 sin 2.

Effects of Directed Energy Weapons

84

Light Flint Glass

Crystal Quartz

Index of

Refraction

(n)

1.710

1.689

1.668

1.647

1.626

1.605

1.584

1.563

1.542

1.521

1.500

0.4 0.5 0.6 0.7 0.8 0.9 1

Wavelength,μm

Figure 3-3. Material and Wavelength Dependence of Index of Refraction
85

Lasers

n2

Figure 3-4. Refraction Between Materials with Different Indices of Refraction

θ1 θ1

θ2

n1 n1

n2

θ2

n1 < n2 n1 > n2

Mathematically, the trigonometric sine function increases with

the angle , varying from 0 when   0 to 1 when   90°. Thus,

if n2 > n1, as illustrated on the left hand side of Figure 3–3, the

light will be bent towards a line perpendicular to the surface,

while if n2 < n1, as illustrated on the right hand side of Figure

3–3, the opposite will occur. This phenomenon is responsible for

what happens when a beam of white light, containing light

of all wavelengths, passes through a prism: since the index of

refraction for different wavelengths is different, different

wavelengths (colors) of light are bent to different degrees and

separated from one another. Similarly, it is the refraction of

light passing through water droplets in the air which is responsible for rainbows.

Refraction is put to good use in lenses.7 Figure 3–5 illustrates

two types of lenses: converging and diverging. It’s easy to

convince yourself by use of the law of refraction and Figure 3–4

that a lens which is thicker in the middle than at the edges will

tend to make light rays focus to a point, while one which is thicker

at the edges than the middle will make them diverge. Nearsighted

persons, whose eyes focus in front of the retina, have diverging

lenses in their glasses to move the focal point back, while farsighted persons, whose eyes focus behind the retina, have their

vision corrected with converging lenses. Refraction and lens

theory are important to the understanding of laser propagation

because, as we shall see, small density fluctuations in the atmosphere which arise due to turbulence can act as lenses and cause a

light beam to be bent in random ways.
Effects of Directed Energy Weapons

86

Converging Lens Diverging Lens

Figure 3-5. Converging and Diverging Lenses

θ1

θ1

θ2

θ2

Diffraction. Another phenomenon of real importance from the

standpoint of the propagation of laser light is that of diffraction.

Diffraction refers to the spreading, or divergence, of light which

emerges from an aperture of a given diameter, as shown in Figure

3–6. In this figure, a beam of light of essentially infinite beam

width is passed through an aperture of diameter D. It is easily verified experimentally that the light which passes through this aperture is no longer collimated, propagating in a straight line, but

rather now has some divergence angle, . It may be shown both

theoretically and experimentally that this angle is related to D and

to the wavelength, , of the light by the relationship   /D.

The exact relationship between  and /D depends on whether

the aperture is square, a circle, or has some other shape (which

is why we use “” instead of “”), but the expression given is

approximately true in any case and will be sufficiently accurate for

our purposes.8

You will recall from Chapter 1 that in propagating a distance z, a

beam of divergence  will expand to a width w  z. Therefore, the

implication of diffraction is that if you want a beam of light to

travel a long range without much spreading, you must make 

small by choosing either a small wavelength or a large aperture.

This is why lasers are popular as beam weapons: with wavelengths

on the order of 10–5 cm, an aperture of modest size, such as 10 cm,

will produce a divergence of only 10–6 radians (about 0.0000060). By

contrast, a typical radar system, with a wavelength of 10 cm,

would diverge with an angle of 60 if beamed from a 10 centimeter

aperture. This is why radar antennas must be quite large if they are
to beam over long distances with relatively little divergence. Chapter 4 discusses the implications of these results from the standpoint

of microwaves, which have wavelengths of 1 – 10 cm.

One way of reducing the effects of diffraction to achieve a

longer effective beam range without appreciable divergence is by

focusing the beam as illustrated in Figure 3–7.

In Figure 3–7, a converging lens has been placed in the path of

the beam. This lens, as we have seen, serves to bend the light rays

inward, focusing them to a spot of radius W.9 The width of the

focal spot depends upon the focal length, f, of the lens. If f is very

short, the light will be focused to a small spot, and will diverge

rapidly beyond that spot. If f becomes too long, the light will diverge from the lens as though it were just an aperture.

In between these extremes, there is a focal length at which

the beam will “hang together” and remain essentially collimated

for the greatest distance. This focal length is known as the

Rayleigh Range, Zr. The beam radius at the Rayleigh Range is

W  D/32, and the Rayleigh Range is given by Zr  W2/.

10

Therefore, in practical applications, laser light can be used as a

collimated beam over a distance of about twice the Rayleigh

Range, or about D2/, where D is the aperture from which the

light emerges from the weapon, and  the wavelength of the

light. Beyond this distance, diffraction and divergence at an

angle of about /D must be taken into account in evaluating the

energy density on target.11 Figure 3–8 shows how the Rayleigh

Range varies with wavelength and aperture, and reinforces the

87

Lasers

D

Figure 3-6. Diffraction of Light Passing through an Aperture

θ
Effects of Directed Energy Weapons

88

f < Zr

f = Zr

f > Zr

D

D

D

W <

D

3 2

y/D

y θ ≈ /D

y/D

Figure 3-7. Focusing of a Beam of Light and the Rayleigh Range

θ ≈

θ >

2

W >

D

3 2

W =

D

3

idea that shorter wavelengths and larger apertures result in

longer propagation distances without spreading.

Summary. To this point, we have seen that laser light is a special case of electromagnetic radiation, and is characterized by

some wavelength, frequency, and a speed of c in vacuum or c/n in

some other medium. When propagating from one medium to another, the light is bent according to the law of refraction, and when

passing through an aperture of diameter D will spread by diffraction with a divergence angle of about /D. Focusing can limit

diffraction spreading, but only over a range of about D2/.

Laser Interaction with Matter

Having considered the fundamental features of laser propagation, we’ll turn our attention to the fundamentals of laser interaction with matter—either the atmospheric gases through which the

beam may propagate, or the solid targets it may engage. The interaction of electromagnetic radiation with matter was poorly under-
stood until the present century, when a host of experimental data

and theory came together in what is now known as quantum mechanics.12 One of the fundamental features of this theory is that

light, while it propagates as a continuous wave, interacts with matter

in discrete units. That is, light may be treated as a wave, subject to

refraction, diffraction, and all those effects while it propagates, but

when it’s finally absorbed or scattered from a target, it needs to be

treated as a stream of tiny little bullets. 

How small are the bullets? Quantum theory tells us that light

of frequency is absorbed in units of h, where h is a constant

(Planck’s constant), equal to 6.63  10–34 Joule seconds. This

means that for red light, with a wavelength of about 0.7 m and

a frequency of about 4  1014 Hz, the energy of a single bullet is

h 3  10–19 Joules. This is a very small number—to absorb

the 10,000 Joules we have taken as a zero order criterion for

damage, about 3  1022 units (commonly called photons) of red

89

Lasers

Aperture (cm)

Figure 3-8. Rayleigh Range vs Aperture and Laser Wavelength

105

104

103

102

101

1

10-1

10-2

10-3

Rayleigh

Range

(km)

0.33

0.69

1.06

10.6

1 10 100 1000

λ,μm
light would need to be absorbed. This is why in our everyday

experience the absorption of light seems continuous, just as the

flow of water appears continuous—because the individual photons of light, or water molecules, though discrete, are too small

to be picked up individually.

Interaction with Gases. If the discrete nature of light absorption is so small, what difference does it make? It makes a big difference, because light can only be absorbed if the energy of a photon is exactly equal to the difference between two allowed energy

states in the absorbing material. This sounds complicated, but it

really isn’t. Just as light exists in discrete photons of energy, matter

exists in discrete energy states, corresponding to different configurations of the atoms and molecules of which it is made. This is illustrated, in a rather simplistic form, in Figure 3–9. This figure

shows an atom of hydrogen, which consists of a single proton

being “orbited” by a single electron. Quantum mechanics tells us

that only certain discrete orbits are allowed for the electron, each

corresponding to a different energy.13 The energy levels corresponding to the allowed orbits in hydrogen are shown on the right

hand side of Figure 3–9.14

If we take the lowest energy orbit of hydrogen, or ground state to

be the zero of energy, the next highest orbit is at 10.2 eV. (An electron volt [eV] is the energy which one electron gains in falling

through a 1 volt potential—about 1.6  10–19 Joules.) The next

highest is at 12.l eV, and so forth. As we go up in energy, the levels

get closer together, until at 13.6 eV, the ionization potential of hydrogen, the electron is no longer bound to the nucleus, and is free to

go its own way and do its own thing.

If a photon of light encounters an atom of hydrogen, it can only

be absorbed if its energy connects an occupied electronic level

with an unoccupied one. Thus, if the hydrogen is in its ground

state, photons of energy greater than 13.6 eV can be absorbed,

since they can ionize the atom, sending its electron into one of an

infinite number of free and unbound states. Photons of energy less

than 13.6 eV will have much greater difficulty being absorbed.

Only those of energy 10.2 eV, 12.1 eV, and so on can be absorbed as

they promote an electron from the ground state to one of the

higher levels. Since a photon of red light has an energy of 3  10–19

Effects of Directed Energy Weapons

90
Joules  1.9 eV, you can see that red light of this wavelength cannot be absorbed by hydrogen in its ground state. Hydrogen gas

should be transparent to red light, and indeed it is.15

Laser light consists of photons in the infrared and visible light

portions of the electromagnetic spectrum, with energies from

about 0.1 to 3 eV. Since the ionization potentials of almost all materials lie between 10 and 20 eV, it’s obvious that absorption of light

in gases is generally unlikely—only photons unlucky enough to

connect a few discrete energy states will be absorbed. This is in accord with everyday experience—our gaseous atmosphere is transparent to light in a broad range of wavelengths from the infrared

to the ultraviolet.

As materials become more complex, their energy states become

more diverse, allowing more possibilities for absorption. This is illustrated in Figure 3–10, which shows some of the energy states of

molecular nitrogen (N2).16 You can see that there is much more

structure than there is for atomic hydrogen, because the fact that

two atoms have come together to form a molecule has introduced

new degrees of freedom. An atom can be given energy only by exciting its electrons. A molecule can also be set to vibrating, as the

atoms which make it up move back and forth against the forces

which hold them together, or rotating, as they tumble around one

another. As Figure 3–10 illustrates, this makes what would be single energy states for an isolated atom split into multiple energy

states: for a given level of electronic excitation, there are multiple

levels of vibrational excitation, and for a given degree of electronic

91

Lasers

Proton

Electron

Energy (eV)

13.6

12.1

10.2

0

E∞

E3

E2

E1

E1 E2 E3

Figure 3-9. Electron Orbits and Energy Levels in Hydrogen
and vibrational excitation, there are multiple levels of rotational

excitation. As the figure also illustrates, there is a difference in

energy scale among the different type of excitations. Typically,

electronic excitations are on the order of 1 – 10 eV, vibrational excitations on the order of 0.1 – 1 eV, and rotational excitations on the

order of 0.01 eV.

The mixing of different molecules, as in the atmosphere, introduces more and more wavelengths at which light might be absorbed. Nevertheless, the absorption properties of gases may be

adequately characterized as discrete absorption lines, separated by

relatively broad regions of transmission. Figure 3–11 shows a portion of the absorption spectrum of the atmosphere, and indicates

the molecules responsible for some of the absorption lines shown.

In interacting with a gas, light may not only be absorbed, but

also scattered. Depending on the mechanism of scattering, it can

be treated either as a macroscopic phenomenon and analyzed

using wave theory, or as a microscopic phenomenon, and analyzed as an interaction between photons and atoms. For example,

scattering from suspended particulate matter in the atmosphere is

Effects of Directed Energy Weapons

92

ELECTRONIC VIBRATIONAL ROTATIONAL

E,eV

7.5 .75

5

2.5

0 0

.25

.50

.18

.17

.16

.15

.14

Figure 3-10. Energy Levels of Molecular Nitrogen
a macroscopic phenomenon, because the particles in question are

of a size (about 10–4 cm) equal to or greater than the wavelength of

light. On the other hand, photons may be “absorbed” and “reemitted” by atomic and molecular species in a gas, resulting in

what is effectively a scattering process by particles whose size is

about 10–8 cm—much smaller than the wavelength of visible light.

Scattering phenomena will be discussed in greater detail later in

the chapter.

In summary, gases are comprised of atoms and molecules which

absorb light at only certain discrete wavelengths. It would therefore be wise to choose a laser for atmospheric applications whose

wavelength is not coincident with one of these absorption lines.

Interaction with Solids. How do the absorption characteristics of solids and liquids differ from those of gases? In principle,

not at all, but in practice there’s a big difference due to the high

density of atoms which make up a solid. In essence, a solid is a

macromolecule, in which about 1023 atoms per cubic centimeter

are bound together into one huge assembly. What this means is

that many, many more degrees of freedom are introduced, there

are many more ways of exciting the material, and therefore there

are many more wavelengths which might be absorbed. It would

seem that this would make the absorption properties of solids

very, very complex: but as often happens, they become so complex

93

Lasers

1.0

0.8

0.6

0.4

0.2

 0

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Fraction

Transmitted

(0.3km path)

H2O H2O

02

H2O H2O H2O H2O

H2O

H2O

CO2

Wavelength (μm)

Figure 3-11. Atmospheric Absorption Lines (17)
as to be simple from a macroscopic point of view. This is analogous to the way in which the motions of the individual molecules

in a gas, though chaotic and impossible to predict on an individual basis, are well described in a macroscopic sense by the equations of fluid flow. We do not need to know how each molecule is

moving to know which way the wind is blowing! Figure 3–12

shows schematically what happens to the allowed energy levels as

atoms of some substance are brought together to form molecules,

and as molecules are brought together to form a solid.

As we have seen, the energy levels of an atom become more diverse when two or three of them are brought together into a molecule, and the individual electronic energy levels develop a lot of

fine structure due to rotational and vibrational degrees of freedom. As many, many molecules are brought together to form a

solid, this process continues. Eventually, the discrete energy levels

become so dense as to form what are, in effect, energy bands: regions within which an electron may have any energy. These energy bands are separated by energy gaps: regions within which

none of the energies are permitted to an electron. This type of

structure is shown at the extreme right hand of Figure 3–12. The

band of highest energies in a solid is known as the conduction

band, because these energy states, as we shall see, are responsible

for the conductivity of metals. The lower bands of energy states

are known as the valence bands: these arise from the tightly

bound, innermost electrons in the isolated atoms.l8

What are the implications of the energy level structure for

solids which is illustrated in Figure 3–12? How these energy

bands are filled with electrons determines whether the solid is a

metal or an insulator, and this, in turn, determines how it responds to light which is incident upon it. An insulator is a solid in

which all the energy bands are either completely occupied with,

or free of, electrons. It’s easy to see why this is so: if an electric

field is applied to such a solid, it tries to accelerate the electrons

within it, increasing their energy. But the electrons at the top of

the band can’t increase in energy, since energies above them

aren’t allowed. Electrons just below these can’t increase in energy,

because the energy levels above them are filled, and so on. It’s

rather like a line of people, with the first person up against a

brick wall: no matter how hard someone pushes at the rear of the

Effects of Directed Energy Weapons

94
line, it won’t go anywhere. Consequently, insulators, as their

name implies, don’t conduct electricity. The electrons within them

simply can’t flow in response to an applied electric field.

What happens to the electrons in an insulator when they

encounter a photon of light? There are two possibilities. If the

photon energy is less than the energy gap, the photon can’t be

absorbed, since it can’t raise any electrons from a filled to an

empty energy level. On the other hand, if the photon energy exceeds the energy gap, it can raise electrons from the valence to

the conduction band, and be absorbed in the process. Thus, we

anticipate that the absorption of light by insulators will be highly

energy dependent, taking a great leap forward when the photon

energy exceeds the energy gap. To continue the analogy introduced earlier, a line of people up against a brick wall can move

forward if the first person in line is kicked over the wall! This

phenomenon is well documented experimentally, as illustrated

by Figure 3–13, which shows the fraction of photons transmitted

through a l0–4 cm sample of gallium arsenide (GaAs) as a

function of photon energy.

You can see from Figure 3–12 that the transmission of light by

GaAs does indeed fall rapidly for photon energies above the energy gap. For all but very thin samples, GaAs is opaque to photons of energy greater than 1.521 eV, the energy gap. Table 3–1

provides the energy gaps for some common insulators, and indicates where the photons that can bridge these gaps lie in the electromagnetic spectrum.

It is easy to relate the energy gaps shown for the materials in

Table 3–1 to their physical appearance. Since diamond, for example, has an energy gap which corresponds to the energy of

95

Lasers

ATOM ATOM

MOLECULE

Energy Band

Energy Gap

SOLID

Figure 3-12. Energy Levels of Atoms, Molecules, and Solids
photons in the ultraviolet region of the electromagnetic spectrum,

it should not be able to absorb photons of lesser energy, such as

those in the visible region of the spectrum. The fact that diamonds

appear transparent to us confirms this result. Similarly, silicon and

gallium arsenide appear opaque to us—their energy gaps are less

than the energy of visible photons—yet these materials are used as

windows for infrared detectors, since they are transparent to photons of infrared energy.

In summary, insulators will be transparent to light whose photons have less energy than their energy gap, and opaque to light

whose photons exceed the energy gap. Let’s consider next the type

of behavior we expect from conductors—materials in which the uppermost energy band is only partially filled with electrons.

Having a partially filled energy band greatly affects the electrical and optical properties of a material. We saw that when a

band was entirely filled with electrons, an electric field couldn’t

Effects of Directed Energy Weapons

96

1.0

0.8

0.6

0.4

0.2

0

1.51 1.52 1.53 1.54 1.55 1.60

Transmission

(10-4 cm sample)

Energy Gap of GaAs

Photon Energy (eV)

Figure 3-13. Fraction of Photons Transmitted through Gallium Arsenide (19)

Table 3-1. Some Typical Energy Gaps

Material Energy Gap(eV) Special Location

Diamond 5.33 Ultraviolet

Zinc Selenide 3.6 Visible

Silicon 1.14 Infrared

Gallium Arsenide 1.521 Infrared

Lead Telluride 0.30 Infrared

(20)
accelerate any electrons, since they had nowhere to go in energy

space. If, however, the band isn’t full, there are states of greater

energy available within the band, and an electric field can

accelerate the electrons within it. This results in the flow of electricity. These materials are therefore called conductors: they

conduct electricity.

We might be tempted to conclude that conductors could absorb

light of any frequency, since there are energy states available for

electrons to be promoted to in absorbing photons of any energy.

However, this is not the case, due to the collective effect of all

these electrons flowing about. You may recall from sophomore

physics that “there can be no electric fields on the interior of a conductor.“ 21 What does this general principle mean? It means that

when an electric field is applied to a conductor, the electrons

within it flow so as to prevent that electric field from penetrating

to the interior. This was first recognized by Michael Faraday, who

proved it by sitting inside a metal box while his assistants charged

up the exterior of the box to a high potential, with electric fields of

such magnitude that sparks were flying all over the place. He

emerged alive, thus demonstrating that the electric fields had not

penetrated into the interior. Even today, the name “Faraday cage”

is given to metal boxes which are used to shield electronic components from external electromagnetic fields—a typical military application is to protect electronics from external fields, including

the “electromagnetic pulse” (emp) which accompanies the detonation of nuclear weapons.22

What does all this have to do with the absorption of light? Remember that light is a wave of electric and magnetic fields, propagating through space. If these fields can’t penetrate to the interior

of a conductor, how can they be absorbed? And if they can’t be absorbed or propagate through, what happens to them? A clue lies

in the fact that metals, which are good electrical conductors, appear shiny: they reflect waves of light which are incident upon

them. Indeed, metallic mirrors are a common optical element in

many lasers.

In the last two paragraphs we have flipped between two extremes. First, we thought that since there were energy states available for absorption in conductors, they would absorb laser light.

Then, recognizing the shielding effect which flowing electrons

could have, we changed our mind and thought that they would

97

Lasers
reflect laser light. The second extreme is closer to the truth, but it’s

still not the whole truth. There are frequencies of light which will

penetrate to the interior of a conductor, and even for those frequencies which are reflected, some small portion will be absorbed.

What is the physical basis for these results?

We have related the reflection of light by conductors to the

flow of electrons within them. Suppose the frequency of the light

is so great that its electromagnetic field changes sign more

rapidly than the electrons can flow in response. In this case, the

electrons do not have time to move and produce their shielding

effect, and we’d expect the light to penetrate into the conductor.

How great a frequency is required for this to happen? If there

are n free (conducting) electrons per cubic meter in a material,

then the maximum frequency at which they can respond is called

the plasma frequency, p. The plasma frequency is proportional to

the square root of n, and is equal to 1015 – 1016 Hz at the density

of electrons found in typical conductors.23 These frequencies are

in the ultraviolet and X-ray portions of the electromagnetic

spectrum, which is why visible light is reflected from metals, but

X-rays can be used to probe them to examine the quality of

welds and in other applications. A similar phenomenon occurs

in the atmosphere. The density of electrons in the ionosphere

is such that low frequency radio waves are reflected and can

thus bounce over the horizon, enabling us to communicate

overseas. Higher frequency microwaves, above the plasma frequency, will propagate through the ionosphere and can be used

to communicate with spacecraft. Figure 3–14 is a plot of plasma

frequency as a function of electron density, with key frequencies

and densities identified.

With the concepts developed so far, we can conclude that when

light encounters a conducting material, it will be mostly transmitted if its frequency exceeds the plasma frequency, and mostly reflected if its frequency is below the plasma frequency. The word

“mostly” appears in the previous sentence because nothing in the

world is perfect, and some small fraction of the light which is incident on a conductor will be absorbed. What is the physical basis

for this absorption? Fundamentally, it arises because no conductor

is perfect. There is some resistance to the flow of electrons in any

material due to imperfections in its structure, the thermal vibrations resulting from its finite temperature, and so forth. What this

Effects of Directed Energy Weapons

98
99

Lasers

1016

1014

1012

1010

108

106

104 108 1012 1016 1020 1024

X-Rays

Visible Light

Infared

Microwaves

Radio Ionosphere Air @ Sea Level Solids

Singly-Ionized Conducting

Plasma

Frequenct

(Hz)

Figure 3-14. Plasma Frequency vs Electron Density

resistance means is that, while the electrons would like to flow to

prevent the penetration of electromagnetic radiation, they will be

retarded from doing so, with the result that light will penetrate to

some degree. The depth to which light will penetrate is known as

the skin depth, . It can be shown that the skin depth is inversely

proportional to the square root of the product of the light frequency times a material’s conductivity. Figure 3–15 is a plot of

skin depth as a function of a metal’s conductivity and the frequency of light incident upon it.

As you can see from the figure, the skin depth is quite small—

micrometers or less—for infrared and visible wavelengths. As

Metals Insulators

Transmissive Light Frequency>Plasma Frequency Photon Energy<Energy Gap

Absorptive Light Frequency<Plasma Frequency* Photon Energy>Energy Gap

Reflective Light Frequency<Plasma Frequency*

 *Over 90% reflected, remainder absorbed

Table 3-2. Response of Metals and Insulators to Incident Light
Effects of Directed Energy Weapons

100

n

q

s

l

s

s

s

s

s

s

s

s s s s s s

q

q

q q q q q q q q q q q

l

l l l l l l l l l l l l

n n n n n n n n n n n n n

4.2

3.8

3.4

3.0

2.6

2.2

1.8

1.4

1.0

0.6

10 7 2 x 10 7 3 x 10 7 4 x 10 7 5 x 107 6 x 107

Wavelength, mm

Conductivity, Mho/m

Skin

Depth

(cm)

x106

10.6

1.06

0.69

0.33

Cu Ag Fe Al

Figure 3-15. Skin Depth vs Conductivity and Wavelength

wavelength, mm

2.0

Reflectivity

10.6

1.06

o.69

0.5 1.0 1.5 2.5 3.0 3.5 4.0

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Skin Depth, cmx106

Figure 3-16. Reflectivity vs Skin Depth and Wavelength
conductivity decreases, the skin depth increases and light penetrates further into a conductor, so that a greater fraction will be

absorbed.24 The fraction of light reflected from the surface of a

conducting material is given by R  1 – 4  /c.

Figure 3–16 shows reflectivity as a function of skin depth and

wavelength. For example, if radiation with a wavelength of

10.6m is incident upon aluminum, which has a skin depth of

about 1.6  10–6 cm, the reflectivity is R  0.98. Thus, very little of

the light incident on a metallic surface will be absorbed. This is

good, if you want to use it as a mirror; and bad, if you hope to

damage it with the light incident upon it.

If there’s anything like a simple bottom line to the physics of

light absorption in gaseous materials or solids it’s this: in gases, a

photon is lucky to be absorbed. Only a few, discrete frequencies

will be in resonance with the gaseous absorption lines. In solids,

on the other hand, a photon will be lucky to be transmitted. Its

energy must lie below the energy gap of an insulator, or its frequency above the plasma frequency of a conductor. This contrast

reflects our everyday experience that gases seem to be transparent,

while solids are for the most part opaque.

Laser Propagation in a Vacuum

Let’s now use the basic ideas developed in the previous section

to investigate the propagation of laser light, first in a vacuum,

and then in the atmosphere. In a vacuum, there are no gases or

particles to absorb or scatter the beam, and so beam propagation

involves only its spreading through diffraction. A beam of electromagnetic radiation of wavelength  emerging from an aperture of diameter D and focused to a radius w0  D/3 2  D/4

will travel a distance roughly equal to the Rayleigh Range,

Zr   w0 

2/, before diverging at an angle   /D (Figure 3–7).

Therefore, there are two propagation limits, one in which the

beam is essentially collimated, with a (roughly) constant radius

wo, and another in which it diverges, with its radius growing

linearly with distance. These two regimes are sometimes called

the near field and far field respectively, for the obvious reason

that far field propagation occurs at a further distance from the

laser source than does near field propagation.25 In the near field,

101

Lasers
beam intensity is roughly independent of range, while in the far

field, intensity will decrease as the square of the range.

Near Field Propagation

It’s obvious from the expression for the Rayleigh Range that

shorter wavelengths and larger apertures are favored for longer

propagation as a collimated beam (see Figure 3–8). We can be

more quantitative in this assessment by developing criteria for the

aperture, wavelength, and power necessary to deliver a given intensity (W/cm2) to a target at a given range without appreciable

beam spreading. You will recall from Chapter 1 that damaging targets requires an energy density (fluence) on the order of 104 J/cm2,

delivered in a time shorter than that over which the target can redistribute or reject the energy. For longer times, the fluence necessary to damage the target increases, and ultimately it is a constant

power density (intensity) that is required for damage (see Figure

1–8). In designing a laser for use as a weapon in a given scenario,

the parameters which are available to play with are its power

(Watts), pulse duration (pulse width, seconds), aperture, and

wavelength. If it is desired to operate in the near field, some combination of these must be chosen that will allow the necessary fluence or intensity to be placed on target within the Rayleigh Range.

The Rayleigh Range, Zr   wo

2/, is directly proportional to the

beam area, A   wo

2, and the beam radius is in turn proportional

to the output aperture D of the laser system. Figure 3–17 is a plot

of the aperture necessary to achieve a given range in the near field

for several laser wavelengths.

In looking at the Figure 3–17, it is useful to bear in mind that there

are two general scenarios of interest. These are tactical applications, in

which ranges are characteristic of those likely to be found on a battlefield (1 – 100 km), and strategic applications, in which ranges are characteristic of intercontinental distances or the orbital altitude of geosynchronous satellites (104

–105 km). From the figure, you can see that over

tactical ranges it’s possible to propagate in the near field with systems

whose apertures are of a reasonable size—a meter or less. Over strategic ranges, aperture sizes can approach hundreds of meters, especially

for longer wavelength infrared radiation.

A laser whose output power is P (Watts) delivers an intensity S

(Watts/cm2) equal to P/ w0

2. Since meeting damage criteria deEffects of Directed Energy Weapons

102
pends upon the intensity S and the pulse duration tp, a less powerful laser with a small spot area might achieve damage equivalent

to a more powerful laser with a larger spot size. On the other

hand, the beam with the smaller area will have a smaller Rayleigh

range. Figure 3–18 is a plot of the power necessary to achieve various intensities as a function of aperture size.

Figure 3–18 reinforces the idea that engaging targets in the near

field is practical only at tactical ranges. Only when output apertures are less than a meter are power levels of 106 W (MW) or less

sufficient to place damaging intensities on target. Figure 3–19

shows the reason for this, and also suggests why at strategic

ranges it would be wiser to propagate in the far field, accepting

the resulting beam divergence.

In the upper portion of Figure 3–19, a large aperture laser is seen

engaging a target at a strategic range. With an aperture size of

10–100 meters, such a beam will actually be larger than many typical targets, and power and energy will be wasted. In the lower

portion of the figure, a smaller diameter aperture is employed.

This laser will diverge, but its spot size at the target may actually

be smaller than that of the large aperture laser! As a result, an engagement in the far field can result in a smaller, less cumbersome

laser design as well as more efficient energy delivery to the target.

Therefore, it’s appropriate to turn our attention next to propagation in the far field.

103

Lasers

Wavelength, m

10.6

1.06

0.69

0.33

μ

100 1000 100000 1 10 10000

100.00

10.00

1.00

0.10

0.01

Aperture

(m)

Range (km)

Figure 3-17. Aperture vs Range and Wavelength in the Near Field

◆

◆

❑

❑

❅

❅

■

■

◆

❑

❅

■
Far Field Propagation

In the far field, the beam has a divergence angle   /D

where D is the diameter of the final output element (lens, mirror,

etc.) for the laser. Recall (Chapter l) that divergence at an angle 

results in a beam radius w  z after propagation over a range Z,

Effects of Directed Energy Weapons

104

Figure 3-19. Near-and-Far-Field Engagements with a Target

Engagement in the Near Field

Engagement in the Far Field

z

POWER

(WATTS)

Aperture (m)

S = 107

W/cm2

S = 104

S = 10 104

102

106

108

1010

1012

1014

1016

1

0.01 0.1 1 10 100

Figure 3-18. Beam Power vs Aperture and Intensity
so that the intensity S of a laser having power P is S  P/ w2 

PD2/ Z22. Unlike propagation in the near field, where intensity

is roughly constant with the range z, in the far field it decreases

as l/Z2. 

The decrease in intensity as l/Z2 reflects the fact that in the far

field the beam and its energy are spreading with distance, and the

energy is no longer as “directed” as in the near field. The concept

of brightness, illustrated in Figure 3–20, is used to quantify the

extent to which a given laser beam falls between the extremes of

being perfectly directed, as in the near field, and perfectly divergent, as is the detonation of a bomb. 

Shown in the figure is a laser of wavelength  with an output

power P, propagating and spreading with a divergence angle

  /D. At some range z, the beam has a cross-sectional area

A  w2   z2 2/D2. If the laser’s power were totally undirected, like that from a bomb, the beam would occupy a sphere

of area 4 z2 at a range z. In either case the area increases as z2,

it’s just the constant of proportionality that’s different. The

smaller that constant, the tighter the beam is, and the less it’s

spreading. The constant of proportionality between z2 and the

area of the beam is known as the solid angle that the beam

occupies.26 The unit of solid angle is called a steradian. The

brightness of the beam illustrated in Figure 3–19 is defined to be

B  PD2/ 2, or the ratio of its power to its degree of spreading

in steradians. The units of brightness, therefore, are Watts per

steradian, abbreviated W/sr.

105

Lasers

Area of Sphere

= 4πz2

Area A

Power P

Z



Figure 3-20. The Concept of Brightness
It is easy to see that this definition of brightness agrees with

what we might think on an intuitive basis. Clearly, the brightness

of a light source should be proportional to its output power—we

know that a 100 Watt light bulb is brighter than a 60 Watt light

bulb. Thus, the definition of B involves the power, P. Brightness

should also increase as the angular cone through which the light

is funneled decreases—a 60 Watt flashlight is much brighter

(when it intersects our eyes) than a 60 Watt bulb, because it

sends those 60 Watts in one direction, rather than in all directions. Brightness is a useful “figure of merit” for a directed energy weapon such as a laser, because it depends only on the

characteristics of the laser device itself (P, D, and ), and is independent of scenario-dependent factors such as the range, z.27 The

intensity S (W/cm2) at a range z from a laser of brightness B is

simply B/z2. In this way, it’s possible to find the range at which a

laser with a given brightness can engage a target with a given intensity, as shown in Figure 3–21.

Effects of Directed Energy Weapons

106

Figure 3-21. Range vs Brightness and Intensity

Range

(km)

108

106

104

102

1

Brightness (Watts/sr)

1016 1018 1020 1022 1024 1026

107

104

S = 10W/cm2
Figure 3–21 may be used to evaluate the tradeoffs associated

with laser propagation in the far field just as Figures 3–17 and

3–18 may be used in the near field. Suppose, for example, that we

wish to engage a satellite at a range of 104 km, and that damaging

this satellite requires that an intensity of 104 W/cm2 be applied

to the target for a period of 0.1 second. From Figure 3–21, this

requires that we have a laser with a brightness of 1022 W/sr. Using

the definition of brightness, B  PD2/p2, this requirement can be

further developed into operational parameters for the laser to engage the satellite. Suppose, for example, that we feel constrained

for engineering reasons to using a 1 m (100 cm) aperture, and a

wavelength of 0.7 m (7  10–5 cm). Then D2/2  6.5  1011, and

the power of the laser must be 1.5  1010 W (15,000 MW). This is a

pretty hefty power, and the technical challenges associated with

building so powerful a device may cause us to rethink the constraints on the problem. For example, if we were to increase the

aperture to 10 m, then D2/2 would increase by two orders of

magnitude to 6.5  1013, and the necessary power would decrease

by two orders of magnitude to 150 MW.

Departures from Perfect Propagation

One final point needs to be made before we can wrap up our

discussion of propagation in a vacuum. To this point, we have

treated our laser beam as if it were an ideal wave of electromagnetic radiation. Real lasers do not produce ideal waves for a variety of reasons, such as inhomogenieties in the active elements of

the laser itself or in the lenses and mirrors which serve to couple

the beam into the outside world. The result of this lack of perfection is to make a real laser beam spread more rapidly than an

ideal beam. The difference between the real and the ideal is expressed as the ratio of the actual spot size to the ideal or diffraction-limited spot size. Thus, if you see a beam described as 1.5

times diffraction limited, you know that you can predict the spot

radius by diffraction theory, multiply by 1.5, and get the actual

beam radius for this particular device. For such real beams,

Figures 3–17, 3–18, and 3–21 remain adequate to determine the

relationships among damage criteria, wavelength, aperture, and

power as long as the spot size is degraded by a factor to account

107

Lasers
for the effects of beam spreading through factors other than diffraction. Thus, for a beam which is said to be “n” times diffraction limited, the intensity at the Rayleigh Range is not P/w0

2,

but rather P/ (nw0)2, and so on. Therefore, to maintain the same

intensity for a beam which is “n” times diffraction limited, the

power must be increased by a factor of n2.

Another definition which is sometimes used to express the extent to which a given beam deviates from perfection is to use the

ratio of the intensity of the real beam at its center to that of an

ideal beam. This is called the Strehl ratio. If a laser beam is n times

diffraction limited, the Strehl ratio is roughly l/n2. An increase in

the Strehl ratio or a reduction in n is often used as a measure of

merit for optical systems whose purpose is to clean up a beam

whose quality would otherwise be unacceptable. 

Summary: Propagation in Vacuum

The following key points serve to summarize our discussion of

laser propagation in a vacuum: 

1. A laser will propagate as a collimated beam, with little

spreading, over a distance on the order of D2/, where D is

the diameter of the last optical element in the laser and its associated beam steering devices. Figure 3–17 shows the apertures

necessary to achieve propagation without spreading as a function of range. Limiting the aperture to a modest size limits

propagation to tactical ranges, on the order of 100 km or less.

The power necessary to place a given intensity on target also

becomes very large at long ranges, as illustrated in Figure 3–18.

Figures 3–17 and 3–18 may be used together to evaluate tradeoffs in aperture, wavelength, and power in this propagation

limit, which is known as the near field. 

2. Over ranges greater than that for which it remains collimated, the beam radius grows with a divergence angle  

/D, so that the intensity decreases with range Z as 1/Z2. The

constant of proportionality between the beam intensity S

(W/cm2) and 1/Z2 is known as the “brightness” of the beam, B

 PD2/2. figure 3–21 relates brightness to the range over

which a given intensity may be placed on target. This propagaEffects of Directed Energy Weapons

108
tion limit, which is appropriate for strategic applications with

ranges greater than 1,000 km, is known as the far field. 

3. Imperfections invariably present in real lasers cause them to

spread at a rate greater than that predicted by diffraction theory.

The ratio between real and ideal spot radius is known as the

number of times “diffraction limited” the beam is, and the ratio

of the real intensity at a given point to the ideal intensity is

known as the “Strehl ratio.”

Implications

As a directed energy weapon, a laser offers a lot of advantages. In tactical applications, propagation is with relatively little

beam spread, so that the intensity on target is roughly independent of range. This is a useful feature, since in tactical engagements there can be considerable variation in the ranges at which

targets may appear. Over strategic ranges, the beam will spread

and its intensity will decline with distance, but this is somewhat

compensated for by the fact that strategic scenarios are generally

more stylized, with ranges and time scales for engagements predictable in advance.

In either case, there is a clear advantage from the standpoint

of propagation in using as large an aperture and as small a wavelength as possible. Increasing the aperture is limited by engineering considerations—it may not be possible to construct a huge

lens or mirror of sufficient quality to keep the beam close to

diffraction limited, and large optical elements will have a lot of

inertia, making the pointing and tracking of the beam on its

target difficult. Reducing the wavelength is limited more by

physical considerations—the materials and technology to develop lasers of arbitrarily short wavelength either do not exist

or have yet to be discovered. And as wavelength gets shorter,

photon energy increases and the atmosphere becomes increasingly opaque, as photons become able to connect energy states

of the atmospheric gases and be absorbed. Therefore, it is appropriate to consider beam propagation in the atmosphere next,

and consider the additional constraints this will have on beam

parameters.

109

Lasers
Laser Propagation in the Atmosphere.

In the atmosphere, beam propagation and divergence are to a

first approximation the same as in a vacuum, with the added feature that interaction of the beam with atmospheric constituents

causes it to lose photons. The intensity of the beam then decreases

with range for two reasons: divergence increases the beam size,

and atmospheric interactions reduce the energy that it carries.

Having already quantified the first of these effects, we will turn

our attention to the second. Photons may be lost from the beam in

several ways. They may be scattered or absorbed by atmospheric

gases or particulate contaminants. They may be bent from the

beam by the lensing effect of density fluctuations in the atmosphere. And at high intensities, they may cause the air through

which the beam passes to break down into an absorbing plasma.

These and related effects will be considered in this section.

Absorption and Scattering

Earlier in this chapter we discussed the absorption and scattering of light by both gases and solids. While the atmosphere is

composed primarily of gases, solids are present, too, in the form of

suspended particulate matter (water droplets and dust, or

aerosols). Both contribute their part to energy losses from a propagating laser beam. In our discussion of fundamentals, we concentrated our attention on the interaction of a single photon of light

with a single molecule of gas. Our task now is to extend that

analysis to the case where many photons of light encounter many

molecules of gas, as well as small, suspended particles.

Molecules. When a photon encounters a molecule of gas, it

may be absorbed or scattered. The probability of this happening

is expressed in terms of the cross section, 	, for such an event to

occur.28 This concept is illustrated in Figure 3–22. Imagine that a

laser beam of area A is propagating through a thickness dz of atmosphere in which there are N molecules per cubic centimeter.

The total number of molecules that photons within the beam

will encounter is NAdz. If each of these molecules has an effective “size” or cross section 	, the area blocked off by the molecules will be a  N 	Adz. Therefore, the probability that a phoEffects of Directed Energy Weapons

110
ton will collide with a molecule and be lost from the beam

through absorption or scattering is the ratio of the area blocked

off to the total area, a/A  N 	 dz. This means that if n photons

enter the region shown in the figure, nN 	 dz will be lost from

the beam. Since the beam intensity, S, is proportional to the

number of photons n, it follows that S decreases by an amount

dS  –SN 	 dz in propagating a distance dz.

The equation dS  –SN 	 dz is well known in mathematics. Its

solution for the intensity S(z) which a beam whose original intensity was S(0) will have after propagating a distance z is S(z) 

S(0) e–N	z. This result is known as Bouguer’s Law or Lambert’s

Law.

29 It simply states that as light propagates through the atmosphere (or any substance, for that matter) its intensity decreases

exponentially over the distance traveled. The quantity N	 is

traditionally denoted K and is called the attenuation coefficient.

The distance over which a beam’s intensity will decrease by a

factor of l/e (about 1/3) is l/K, called the absorption length. The

product Kz  N	Z is known as the optical depth, and is a measure

of the effective thickness, from the standpoint of absorption, of

the medium through which the light has traveled.

111

Lasers

n photons

in, intensity

S

n(1-Nσ dz) photons out,

intensity S(1-Nσ dz)

N molecules / cm3

cross section σ

Fraction of area blocked = Nσ A dz/A = Nσ dz

Figure 3-22. Scattering and Absorption Cross Section

dz
Figure 3–23 is a plot of S(z)/S(0), the fraction of a beam’s intensity

transmitted over a range z, as a function of optical depth Kz. You can

see from this figure that for ranges z much greater than l/K, large

amounts of energy will be lost from the beam. Clearly, we must

choose the parameters of a laser so that K is as small as possible, and

the effective propagation range, l/K, as large as possible.

Our derivation of the absorption law looked at the probability of

a photon interacting with a single type of molecule, with a cross section 	. Within the atmosphere, there are many types (species) of

molecules present (N2, O2, CO2, etc). The probability of interaction

with one is independent of the probability of interaction with another. That is, photons lost through interaction with one type of

molecule may be added to those lost through interaction with another. As a result, the attenuation coefficients attributable to each

type of molecule may simply be added: K  K(N2) + K(O2) +

K(CO2) + .......(etc). Furthermore, the attenuation coefficient due to

each molecular type is in turn comprised of two parts, one for the

absorption of photons by that type, and one for their scattering:

K(CO2)  K (Absorption by CO2) + K(Scattering by CO2), and so on.

Clearly, attenuation in the atmosphere can be quite complex,

with a variety of terms contributed from different molecular

species, whose relative abundance and importance might change

with latitude, longitude, relative humidity, and other climatic

factors. Each of these terms may have a different dependence

Effects of Directed Energy Weapons

112

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0.5

0.6

0.7

0.8

0.9

1.0

0.4

0.3

0.2

0.1

0.0

Transmission

S(Z) / S(o)

Optical Depth, Kz

Figure 3-23. Transmission vs Optical Depth
upon laser wavelength, since a given wavelength may not be

absorbed by one type of molecule, yet be strongly absorbed by

another. Therefore, it should not be surprising that a considerable body of literature has developed in this area, ranging from

detailed studies of the absorption by a single molecule to gross

measurements of how much light penetrates the atmosphere as a

function of frequency under given climactic conditions.30 We can

only scratch the surface and provide a general feeling for atmospheric absorption and scattering.

Figure 3–24 is an overview of atmospheric attenuation over a

broad range of wavelengths.31 This figure shows some of the broad

windows for the propagation of laser light. However, there is considerable fine structure which the scale of this figure does not reveal. This is shown in the bottom portion of Figure 3–24, which is

an expanded view of one narrow region in the upper half.

From Figure 3–24, you can see that even within what appears to

be a propagation window, there may be narrow absorption bands

at specific frequencies. Therefore, the choice of laser wavelength

can be critical for propagation. For example, recent measurements

of the output frequencies from a DF chemical laser have resulted

113

Lasers

Fraction

Transmitted

(1.8 km path)

Fraction

Transmitted

(10 km path)

1.0

0.8

0.6

0.4

0.2

1.0

0.8

0.6

0.4

0.2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

3.76 3.77 3.78 3.79 3.80 3.81 3.82 3.83 3.84

Figure 3-24. Atmospheric Attenuation vs Wavelength

Wavelength,μm

Wavelength,μm
in a changing one wavelength from 3.7886 to 3.7902 m.32 This

change has been sufficient to alter the assessment of how much of

this light would penetrate a 10 km path at sea level from about

90% to about 50%. This is one reason why free-electron lasers have

recently received considerable interest. Unlike most lasers, whose

output frequencies are fixed by the active, light- producing material in them, free-electron lasers are tunable in wavelength, offering

greater flexibility in adjusting the output for efficient atmospheric

propagation.

To this point, we have seen that the intensity of a laser beam

will decrease with distance as S(z) S (0) e–Kz, where K, the attenuation coefficient, is a sum of terms representing absorption and

scattering by the different species present in the atmosphere.

Given K and knowing the range z required for a given application, Figure 3–23 can be used to evaluate the resulting decrease in

intensity or brightness. This decrease can then be used to modify

our results for propagation in a vacuum, allowing us to develop

new criteria for target damage in atmospheric applications. For

example, Figure 3–18 says that a laser power of 10 kW is required

to deliver an intensity of 104 W/cm2 in a collimated beam with a

3 cm aperture. If the beam is propagating in the atmosphere, and

K is such that only 50% of the intensity is transmitted over the

range to the target, we would need to use a 20 kW laser, so that

after 50% attenuation we would have 10 kW left over to meet the

intensity requirement on target. Alternatively, we might choose a

laser with a different wavelength, for which the attenuation

would be less.

A further complication arises in longer range strategic applications. The atmospheric parameters which determine K may

change over a long range, so that K is not constant, but varies

with distance. This would occur, for example, in using a groundbased laser to attack the moon. As the beam goes up through the

atmosphere, K, which is proportional to the density of molecules,

is steadily decreasing. Eventually, the beam leaves the atmosphere and over the greater part of its range is propagating in a

vacuum. We would greatly overestimate the amount of beam attenuation by using exponential absorption with a K appropriate

to the atmosphere at sea level, and a z equal to the range to the

moon! In cases like this, we must modify our treatment of attenuation and allow K to be a function of z. If K is a variable, depenEffects of Directed Energy Weapons

114
dent upon z, the solution to our original equation, dS/dz  –KS,

becomes S(z)  S(0) exp[–0 z K(z) dz].

This “improved” version of the exponential attenuation law

looks complicated, but its interpretation is straightforward. It says

we must integrate K over the path length. In effect, we split the

beam’s path into many small segments. Over each, K is effectively

constant, and exponential attenuation can be used. The total effect

is then given by the sum of the optical depths over each small

path segment. Doing this in any realistic case requires the use of a

computer model which can keep track of how the distribution of

molecules and their density varies with altitude, and can use these

data to calculate an altitude dependent attenuation coefficient.

There is, however, a simple model which is reasonably accurate,

can be solved analytically, and gives a good feel for the effect of altitude dependence upon beam attenuation.

Within the lower atmosphere (0–120 km), density varies exponentially with altitude.33 That is, the density of molecules N(h) at

altitude h is related to the density N(0) at sea level by the relationship N(h)  N(0) exp(-h/ho), where the constant ho is about 7 km.

Since K ( 	N) is also proportional to N, we can to a first approximation say that K(h)  K(0)exp(-h/ho).34

Suppose that we are fire a laser into the air at some angle 
, as

illustrated in Figure 3–25. The beam’s altitude h is related to its

range z and the elevation angle 
 through the simple geometrical

relationship h  z sin 
. When 
  0, h  0 for any z, and when


  90°, h and z are identical. Using this relationship between h

and z, and assuming K(h)  K(0) exp(–h/ho),we can evaluate the

optical depth to any range z. The result is shown in Figure 3–26.35

115

Lasers

Figure 3-25. Beam Range and Altitude

h

z

φ
Figure 3–26 is a plot of the optical depth to a range z, normalized to K(0)ho, as a function of z, normalized to ho. At 
 0o, the

beam is propagating horizontally, the atmospheric density is constant, and the optical depth increases linearly with distance, as we

would expect. At 
  90°, the beam is propagating straight up,

rapidly emerges from the atmosphere, and beyond that point the

optical depth no longer increases. At intermediate angles, the

beam has greater and greater lengths of atmosphere to propagate

through before it emerges from the atmosphere, and so the optical

depth approaches a limiting value later and of a higher value. For

ranges less than ho, the altitude over which atmospheric density

changes significantly, the optical depth is roughly independent of

elevation angle.

Figure 3–26 can be used together with our results for vacuum

propagation to estimate attenuation for long range applications

and its impact on laser requirements. For example, let’s return to

the case where we wish to attack the moon. If at sea level the attenuation coefficient at the frequency of our laser is 0.1 km–l, and if

we are able to shoot when the moon is directly overhead (
 

90°), then by Figure 3–26 the optical depth will be K(o)ho, or 0.7.

Effects of Directed Energy Weapons

116

Ú 0zK(z)/[K(o)h 0]

Relative

Depth

Relative Range (z/h0)

150

300

900

Figure 3-26. Optical Depth vs Range and Elevation Angle 

0.0

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5

Elevation Angle,F =00

(z/h0)

 k(z) dz / [K(0)h0] z

0
We can then use Figure 3–23 to see that for this optical depth, laser

intensity and brightness will be about 0.6 of what they would

have been in a vacuum. Therefore, the brightness requirements

necessary to place a given intensity on target, obtained from

Figure 3–20, must be increased by a factor of 1/0.6, or about 1.7.

Small Particles (Aerosols). To this point, we’ve looked at the

attenuation of a laser beam due to the gases (molecules) which

comprise the majority of the atmosphere. We must next consider

the effect of small solid or liquid aerosols which are invariably

suspended in the atmosphere, especially near the surface. Figure

3–27, for example, shows the number density of suspended particles as a function of particle radius at sea level, along with the way

in which the density of particles in different size ranges varies

with altitude.36 This figure must, of course, be considered somewhat notional, since the actual particle size distribution can vary

greatly, depending on the local climate and wind conditions.37

A number of important things are apparent from Figure 3–27.

First, particles in excess of 1m are quite rare, and largely confined to regions near the surface of the earth. Second, the range

of particle sizes is comparable to the wavelengths of lasers operating from the visible to far infrared (0.4 – 10m). The absorption

and scattering of light by solid particles becomes quite complex

to analyze when the particle size is comparable to the

117

Lasers

a n (a) (cm-3)

10-4 10-2 1 101 102

Altitude

(km)

10-4

30

Particle Density, ( cm-3)

Particle radius, a

0.1 m<a <1 m μ μ

a<0.1 mμ

Sea Level Particle Size Distribution Variation of Particle Density with Altitude

Figure 3.27. Particle Size Distribution and Variation with Altitude

25

20

15

10

 5

10-2 1 102 104

Particle Radius, a ( m) μ

n(a) da = density of

 particles with radius within da of a

106

104

102

1

10-2

10-4

10-6

a>1 mμ
wavelength of the light. The relevant theory is known as Mie

Scattering Theory for its developer, a German meteorologist.38 The

development of this theory is beyond the scope of this book, but

its essence is summarized in Figure 3–28.

Figure 3–28 shows how the actual attenuation cross section for a

dielectric aerosol (in this case water) compares to its physical size,

a2, as a function of 2a/, where a is the aerosol radius and  the

wavelength of the light. There are various dips and bumps in the

cross section, reflecting resonances between particle size and light

wavelength, but for the most part 	 is on the order of 2a2, especially when a is much larger than . Somewhat crudely, you might

think that each particle contributes twice its physical cross section

to light attenuation because it can contribute to attenuation in two

ways—through absorption and scattering.39

Curves similar to Figure 3–28 are available in the literature for

a variety of different particle types, both dielectric and metallic.40

As a general rule, they exhibit behavior similar to that shown in

Figure 3–28—	 falls to zero as 2a/ goes to zero, but for 2a/

 1, 	 is in the neighborhood of 2a2. Given Figure 3–27, which

suggests that the majority of aerosols are of a size less than 1m,

we can conclude that the effect of aerosols on light attenuation

will be greater for visible lasers ( 0.4 – 0.7 m) than for those

operating in the infrared (  1–10m). The relative particle size,

2a/, is shown in Figure 3–29 as a function of wavelength and

particle size. This figure may be used with Figure 3–28 to estimate the contribution of aerosols to light attenuation. However,

Effects of Directed Energy Weapons

118

4

Attenuation Factor

(	/πa2)

4

3

2

1

0 5 10 15 20 25

Relative Particle Size (2πa/)

Figure 3-28. Attentuation Factor due to Aerosols in Mie's Theory
it must be emphasized that under realistic circumstances the

density and type of aerosols can vary greatly from day to day.

Thus, the operational use of lasers will require that site-specific

surveys be made and that beam brightness be increased to enable meeting damage criteria under “worst case” conditions.

Just as the attenuation coefficient for a mixture of molecules

may be obtained as the sum of the coefficients contributed by each

molecule present, the total attenuation coefficient when aerosols

are suspended in the atmosphere is the sum of contributions from

the molecules and the aerosols taken separately. Molecules and

aerosols can contribute to attenuation in roughly equal amounts,

but for quite different reasons. The attenuation coefficient K is 	N,

where 	 is the attenuation cross section and N the density of the

attenuator. Molecules have a very small attenuation cross section

10–25 – 10–26 cm2), but their density is quite large (3  1019/cm3 at

sea level). By contrast, particles can have a very large cross section

( 3  10–8 cm2), but their density is quite low (1/cm3 or less).

Summary: Absorption and Scattering.

1. Beam intensity S (Watts/cm2) decreases exponentially as a

function of the distance of propagation, z: S(z)  S(0)e–Kz. Figure

119

Lasers

1000.00

100.00

10.00

1.00

0.10

0.01

0.3 1 3 10

Relative

Particle Size

(2Πa/λ) Particle Size

10μm

Wavelength (μm)

Figure 3-29. 2Πa/λ vs Wavelength and Particle Size

1.0μm

0.1μm
3–23 is a plot of this relationship. K is known as the attenuation

coefficient, and Kz as the optical depth.

2. The attenuation coefficient, K, is a sum of contributions resulting from absorption and scattering by both molecular and

aerosol species in the atmosphere. Figure 3–24 is representative

of the wavelength dependence of K due to the molecules in the

atmosphere, and Figure 3–28 of the wavelength dependence of

K due to aerosols.

3. The attenuation coefficient, K, is proportional to the density

N of molecules or aerosols present. When the propagation distance, z, is such that K varies over z, the optical depth is not Kz,

but the integral of K over z: o

z K(z)dz. Figure 3–26 provides an

example of how the variation of atmospheric density with altitude affects optical depth as a function of range and elevation

angle for a ground based laser.

4. At sea level, the contributions of molecules and aerosols to

attenuation are roughly equal. However, large aerosols fall off

rapidly with altitude, as indicated in Figure 3–22.

Index of Refraction Variations

Turbulence and the Coherence Length. In the previous section,

we saw that the variation of atmospheric density with altitude had

an effect on energy absorption and scattering. This was a very largescale effect, occuring over distances comparable to that over which

the atmospheric density varies to a significant extent (7 km). In this

section, we’ll turn our attention to the effect on propagation of

small-scale fluctuations in atmospheric density, which result from

the turbulence induced as the sun heats the atmosphere each day.

Small-scale fluctuations do not affect absorption or scattering

because they tend to average out when the optical depth is computed over a long path. But they can still have a profound effect

on beam propagation through the variations they cause in the

index of refraction of the air. When light passes through regions of

differing refractive index, as in a lens, it is bent (see Figures 3–4

and 3–5). Density fluctuations can have the effect of introducing

many tiny “lenses” into the beam.

Effects of Directed Energy Weapons

120
Though the index of refraction of air is close to the value of 1

which characterizes a vacuum, there is a small deviation which

varies with density and wavelength as shown in Figure 3–30.41

As you can see from Figure 3–30, the index of refraction of

air is pretty close to the value of 1 which is appropriate for a

vacuum. For example, at 0.4 m wavelength, n is 1.00028 at

15 oC, and 1.00026 at 30 oC. Nevertheless, small fluctuations of

this order can have a significant effect over long propagation

distances. As a simple example, suppose that light of 0.4 m

wavelength is propagating through air at a temperature of 30 oC,

and enters a region where the temperature is 15 oC at an angle of

45o. The law of refraction (n1 sin 1  n2 sin 2) can be used to

find that the angle 2 with which the light emerges from the interface between these regions is 44.99885442o. This is very little

deflection, the difference from 45o, being only 0.0011°, or about

2 x10–5 radians. Yet over a propagation path of 100 km this would

mean a beam deflection of about 2 m—more than enough to

cause the beam to miss a target which it might otherwise hit. Of

course, in the real world the beam will encounter regions of

higher and lower temperature and will wander back and forth,

and the sizes of the regions of temperature fluctuation may be

smaller than the beam itself, sending different portions of the

121

Lasers

3080

3022

2964

2906

2848

2790

2732

2674

2616

2558

2500

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

P = 760 mm Hg

T = 0oC,h=1.29 x 10-3g/cm3 

T = 15oC, h=1.22 x 10-3g/cm3

T = 30oC

(n - 1) 107

Wavelength, m

Figure 3-30. Index of Refraction of Air vs Temperature and Wavelength

h=1.16 x 10-3g/cm3
beam front in diverse directions. The net result will be a lessening of the beam intensity on target which is difficult to predict in

advance. We must see if we can quantify this effect and, if possible, compensate for it in the design of our laser beam.

In principle, it’s possible to know and account for fluctuations

in the index of refraction if we know how temperature and

density vary along the beam path. In practice, of course, it’s impossible to know these quantities everywhere within the beam

path at all times—they’re constantly shifting. What we need is a

macroscopic parameter which can capture the effect of microscopic density fluctuations in a straightforward way. There is

such a parameter, known as the coherence length and denoted ro.

42

The physical meaning of the coherence length is illustrated in

Figure 3–31.

Figure 3–31 shows a laser beam of diameter D propagating

through the air. In a vacuum, the beam would diverge through

diffraction at an angle   /D, as indicated by the heavy outline

in the figure. However, density fluctuations in the atmosphere

have the effect of introducing many tiny “lenses” into the beam

volume. These are suggested by the small bubbles which appear

within the beam volume, and they have the effect of breaking

up the unity, or coherence, of the beam front. In effect, the beam

is split up into many tiny beams, each of which encounters a

slightly different environment. The coherence length, ro, may be

Effects of Directed Energy Weapons

122

D

r o

Figure 3-31. Coherence Length and its Effect
thought of as the average size of these little lenses, or as the average diameter of the small sub-beams into which the main beam

is split.

If a laser beam is split into sub-beams of size ro, we would expect its divergence to be governed by the size ro of the sub-beams,

rather that the size D of the original beam. The divergence angle

of a beam of diameter D propagating through an atmosphere with

coherence length ro is shown in Figure 3–32. You can see that our

supposition is correct, and when ro << D, the divergence angle of

the beam becomes not /D, but /ro.

Figure 3–32 gives us the capability to account for the effect of atmospheric density fluctuations on beam propagation, provided

we know the coherence length. A look at Figure 3–31 suggests that

the coherence length will depend on a number of factors:

• The degree of turbulence along the beam path, which establishes the physical size of the little density inhomogenieties.

• The wavelength of the light, which establishes the optical

depth of the inhomogenieties, since index of refraction depends

on wavelength.

• The total path length from beam to target, since the further

the beam travels, the more inhomogenieties of various sizes will

be encountered.

123

Lasers

D/ r0

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

.05

0.0

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3

Figure 3-32. Divergence Angle vs Coherence Length 

y q/ ( /D)

y q= /r0

y q= /D
All of these factors do, in fact, affect ro. A theoretical expression

which takes all of them into account is 

ro  [0.423 (2/)2 o

z CN2 (z)dz]–3/5. 

In this expression, the quantity CN(z) is known as the refractive

index structure coefficient, and characterizes the turbulence at a point

z along the beam path.43 By integrating the square of this quantity

over the whole beam path, the total integrated effect of turbulence

is taken into account. If CN is known, ro can in principle be calculated over any beam path, and the resulting effect on beam propagation accounted for. Unfortunately, this is easier said than done.

Measurements of CN have been obtained experimentally by probing the atmosphere with a laser, examining the effect of turbulence

on its propagation. Typical data are shown in Figure 3–33.44

As you can see from the figure, turbulence is least during the

hours of darkness, when there is no solar heating to introduce temperature inhomogenieties. There is usually a large rise right after

sunrise, followed by a pattern of rough proportionality to solar intensity. Superimposed on this, however, are fluctuations of as much

as 30–50% on time scales of less than an hour.45 As a function of altitude, CN decreases, since solar heating is greatest near the ground,

where the greatest proportion of the solar energy is absorbed.

On the basis of data such as that shown in Figure 3–33, it has

been calculated that ro will be on the order or 5–10 cm for visible

wavelength light propagating from ground to space.46 Since the

beam aperture D would probably be greater than this for propagaEffects of Directed Energy Weapons

124

0.2

0.4

0.6

0.8

1

00 04 08 12 16 20 24

1

12 3

10-1

10-2

10-3

10-4

10-5

Time of Day (hr) Altitude (km)

C2n (m-2/3) x 1012

C2n (m-2/3) x 1012

Figure 3-33. Atmospheric Structure Factor vs Time and Altitude
tion over such a long range, you can see that beam divergence is

much increased by turbulence. This could make achieving damage

criteria in the presence of turbulence virtually impossible. For

example, we saw in Figure 3–21 that to place an intensity S of

104 W/cm2 on a target at a range of 104 km we would need a beam

of brightness 1022 W/sr. This requires a power of 150 MW at an

aperture of 10 m. If turbulence reduces the effective aperture to 10

cm, power requirements increase by a factor of 104, to 1500 GW.

Practically speaking, this type of fix is not possible, because it

leads us into intensities where laser design becomes increasingly

difficult, and more fundamentally because when intensities become too great, there are nonlinear propagation effects, such as air

breakdown, which effectively prevent beam propagation. Nonlinear effects are discussed later in this section.

Figure 3–34 provides a convenient summary of our discussion,

showing how brightness varies with the ratio of ro to D. Using

Figure 3–34, you can determine the effect of a given degree of turbulence, as measured by ro, on the brightness if a given laser. To

regain the original brightness, laser power must be increased appropriately. Since this will typically take us into intensities which

are unacceptable for one reason or another, there has arisen considerable interest in being able to compensate in some way for the

125

Lasers

0.0

0.1

0.2

0.3

0.4

0.5

0.8

0.9

1.0

0.6

0.7

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

r

0/D

Relitive Brightness

(with/without turbulence)

Figure 3-34. Effect of Turbulence on Brightness
effects of atmospheric turbulence on propagation. The techniques

for doing so are known as adaptive optics.

Adaptive Optics. Adaptive optics makes use of the fact that if we

know what the atmosphere is like along the beam path, it’s possible

to send out the beam distorted in such a way that turbulence will in

fact straighten it out! This may seem far-fetched, but the general

principle is straightforward, and is illustrated in Figure 3–35. In the

upper portion of the figure, a beam of light encounters a lens which

might represent a cell of turbulence. This lens focuses the light, so

that it diverges. In the bottom portion of the figure, a second lens

has been introduced, identical to the first, and positioned at twice

the focal distance from it. This lens has the effect of presenting a diverging beam to the original lens, which then proceeds to focus it

back into a parallel beam. The second lens compensates for the first,

so that the net result is as though the first lens were not present. The

idea behind adaptive optics is to compensate for the many little

“lenses” of turbulence in a similar way. All we need to do is know

what’s out there and compensate the beam, mixing up its initial parameters in such a way that the optical path which it traverses acts to

convert the mixed-up beam into the beam we’d have in the absence

of that optical path.

The practical problem in carrying out this approach is knowing

what’s out there so we can compensate for it, since turbulence

changes from moment to moment. We need a way to obtain realtime feedback on the environment, and to use that information in

adjusting the beam appropriately. This is a formidable task, but not

impossible, and it has been accomplished experimentally.47

Effects of Directed Energy Weapons

126

Figure 3-35. The Principle of Adaptive Optics
Figure 3–36 snows schematically how this has been done. The

key to the technique is a deformable mirror—one in which small

actuators move the surface up and down to distort the outgoing

beam in such a way that turbulence will compensate for this distortion, resulting in a nearly diffraction-limited beam on target.

The degree of distortion required is found by examining some of

the light which is reflected back from the target. This light, indicated by the dashed line in Figure 3–36, is fed into a phase sensor,

which compares the quality of the returning light with that in the

outgoing beam. The difference between these is, of course, related

directly to the turbulence along the beam path. A computer uses

this information to provide instructions to the actuators which

then deform the mirror appropriately.

There are, of course, practical considerations which may limit

the ability to do this in realistic scenarios. First, the number of

actuators required may be quite large. Since the beam is being

broken up into segments of a size ro, it follows that the mirror

surface must be broken up into segments of area less than ro

2. As

we have seen, propagation over large distances requires large

mirror sizes—1–10 m. Since ro is typically 5–10 cm, the mirror

must be divided into something on the order of 10,000 segments,

with a corresponding number of actuators. These actuators must

be able to produce the necessary degree of distortion sufficiently

fast to compensate for a continually shifting environment as well.

Typically, this requires that any section on the mirror surface be

able to move 1–10 m in times on the order of 10–3 sec.48

127

Lasers

Target

Beam Splitter

Laser

Deformable

Mirror

Appropriate Instructions

Phase Sensor/Computer

Figure 3-36. An Adaptive Optics Experiment
Additionally, the incoming and outgoing beams must pass

through the same region of air and see the same turbulence. While

the speed of light is fast, it is finite, and in tracking fast moving

targets at large distances (as may be the case in ground-to-space

propagation) it may be necessary to lead the target so that the

light arrives at a given point when the target does. In this case,

you can’t rely on reflected light from the target for turbulence information. This is why it is generally envisioned that a spacebased relay mirror for a ground based laser would have a beacon

laser, ahead of the mirror, which would propagate down and

provide turbulence information for the region of space the beam

would traverse in going to the mirror. A useful parameter for use

in evaluating the need for such a scheme is known as the isoplanatic angle. This is the maximum angle by which a beam can steer

and still see essentially the same turbulent environment. If the

angle by which a target must be led exceeds this angle, then the

beam can no longer be used to provide its own reference, and a

separate beacon is required. Typically, the isoplanatic angle is on

the order of 1–10 rad.49

Summary: Index of Refraction Variations.

1. Atmospheric turbulence has the effect of introducing small

“lenses” into the volume of the beam as it propagates. The characteristic size of these lenses is known as the coherence length, ro.

This quantity depands upon the propagation length, path, time

of day, and laser wavelength, and is typically 5–10 cm.

2. The effect of turbulence is to change the beam divergence

angle from /D to /ro. This reduces beam brightness by a factor of (ro/D)2. Figure 3–34 summarizes the effect of turbulence

on brightness, and may be used together with Figure 3–20 to estimate the impact of turbulence on meeting damage criteria.

3. In many scenarios of interest, particularly those which may

be characterized as strategic rather than tactical, turbulenceinduced beam divergence is unacceptable, and adaptive optics

is required to compensate for it.

4. Adaptive optics is a technique in which the beam is intentionally distorted so that the turbulence which is present restores it

to a diffraction-limited configuration. This is done by evaluating

Effects of Directed Energy Weapons

128
the turbulence present in real time, probing the beam path with

a reference beam and using the resulting information to distort a

deformable laser output mirror. If adaptive optics techniques are

successfully employed, turbulence may essentially be ignored in

assessing damage criteria.

Nonlinear Effects

To this point, everything we’ve considered has been independent of the laser’s intensity. Absorption, for example, reduces the

intensity at a given range z by a fraction, S(z)/S(0)  e–Kz, which is

independent of S(0). This means that the transmitted intensity,

S(z), is directly proportional to the intensity out of the laser, S(0). If

we were to plot S(z) as a function of the S(0), the plot would be a

straight line, as shown in Figure 3–37(a). For this reason, the propagation effects we’ve looked at so far are known as linear effects.

As intensity increases, it is usually found that at some point the

relationship between S(z) and S(0) is no longer linear—there is a

sudden shift in behavior, like that shown in Figure 3–37 (b). This

typically occurs when S(0) exceeds some threshold value, and beyond that threshold the relationship between S(z) and S(0) can be

quite compex and not at all linear, so that nonlinear propagation effects are said to have occurred. The physical reason for nonlinear

effects is that when intensities are strong enough, the beam actually modifies the environment through which it propagates in

such a way that its physical characteristics are altered. For example, a very intense beam might vaporize the aerosols in its path,

129

Lasers

Threshold for

Nonlinear Effects

(a) Linear Propagation

S(z) S(z)

S(0) S(0)

Figure 3-37. Linear and Nonlinear Propagation Effects

(b) NonLinear Propagation
and suffer less degradation due to aerosol absorption and scattering than a lower intensity beam. Or it might ionize the atmosphere

in its path, increasing absorption to the point where propagation

ceases. Unfortunately, most nonlinear effects degrade, rather than

enhance the intensity on target. In this section, we’ll consider nonlinear effects which affect both propagation (thermal blooming

and bending) and attenuation (stimulated scattering, breakdown,

and absorption waves). Our emphasis will be on those effects

which are of concern from the standpoint of beam propagation in

weapon applications.49

Thermal Blooming. One of the first nonlinear phenomena recognized as likely to affect the propagation of a high power laser,

thermal blooming results from the energy which a laser deposits

in the air through which it propagates. The beam loses energy

as a result of absorption. This energy is deposited within the

beam path, where it causes a temperature rise in the air. This

temperature rise modifies the air’s density, alters its index of

refraction, and can severely affect the beam’s propagation. The

sequence of events which results in thermal blooming is illustrated in Figure 3–38.

Figure 3–38 (A) shows the intensity profile of a typical laser

beam viewed end-on: higher in the center than at the edges, where

the intensity falls to zero. The temperature of the air through

which the beam propagates will exhibit a similar profile, as suggested by Figure 3–38 (B). This is because absorption of energy by

Effects of Directed Energy Weapons

130

Intensity Temperature Density Index of Refraction

"Blooming"

(A) (B) (C) (D)

Figure 3-38. The Physics of Thermal Blooming
the air is a linear phenomenon—as intensity goes up, the amount

of energy absorbed, being a constant fraction of the the incident

intensity, goes up as well. The absorbed energy manifests itself as

in increase in temperature. But hot air is less dense than cold air—

at constant (atmospheric) pressure, an increase in temperature implies a decrease in density. Thus, the density profile in the air

through which the beam propagates assumes a form inverse to the

intensity profile, as shown in Figure 3–38 (C). The implication of

this is that the index of refraction of the air through which the

beam propagates, shown in Figure 3–38 (D), mirrors the density

profile, since (n–1) is proportional to density (see Figure 3–30).

This sequence of events has the effect of introducing within the

volume of the beam what amounts to a diverging lens, with a

greater optical density at the edges than at the center. As shown

at the bottom of Figure 3–38, this causes the beam to bloom, or

diverge, at a rate greater than would otherwise be expected.

In practical scenarios, wind sweeps across the beam—either a

naturally occurring wind or one which results from the relative

motion of beam and atmosphere as the beam slews to keep itself

on target. Such a cross-wind causes the beam to bend as well as

bloom. The physical reason for this is illustrated in Figure 3–39. In

this figure, we begin with the temperature profile which produced

thermal blooming. However, wind is blowing across the beam and

introduces cold air. As a result, the upwind portion of the temperature profile becomes cooler and the downwind portion warmer. In

effect, the wind tries to push the hot air downstream. As a result,

the index of refraction profile assumes a shape like that shown, and

131

Lasers

Wind

T T n

Figure 3-39. The Physics of Thermal Bending
the beam sees what looks like a wedge inserted into it. This wedge

causes the beam to bend into the wind, as indicated. Another way

of looking at it might be to think that the wind displaces the diverging lens of thermal blooming, so the the beam sees only one

half of that lens, and bends in a single direction.

Looking at Figures 3–38 and 3–39, you can easily imagine that

the analysis of thermal blooming and bending will be quite difficult, involving gas flow, laser heating, and the temperature and

density dependence of the index of refraction of air. In any realistic scenario, wind velocity will vary along the beam path, and

beam blooming and bending will occur simultaneously. The resulting distortions of the beam’s intensity profile can be quite

complex, as shown in Figure 3–40, which compares a beam’s intensity on target in the presence of these effects with what it

would have been in their absence. Our goal will be to determine

the thresholds for thermal blooming, the magnitude of the effect,

and the potential for dealing with it through techniques such as

adaptive optics in those cases where target range and damage criteria prevent operation below threshold.

We must first recognize that there is a pulse width threshold for

thermal blooming. Even though the atmosphere absorbs energy

and its temperature begins to rise almost immediately, some finite

time is required for the heated air to expand and move out of the

beam, creating the density “hole” shown in Figure 3–38 (C). The

characteristic velocity at which disturbances propagate in air is the

Effects of Directed Energy Weapons

132

200 cm/sec

Wind

Beam Profile

without

Thermal

Blooming

Figure 3-40. Beam Profile with Thermal Blooming and Bending(50)
speed of sound, a ( 3  104 cm/sec). Therefore, the time for air to

move out of a beam of radius w and for blooming to begin is approximately w/a. Figure 3–41 is a plot of the time for thermal

blooming to develop as a function of beam radius.

You can see from Figure 3–41 that thermal blooming is not likely

for pulsed lasers, where the goal is to place all the energy on target

in short time scales—10–5 seconds or less, for example. On the

other hand, if seconds of interaction time with the target are required, there is a potential for thermal blooming to be a problem,

even for strategic applications, where the beam radius may be relatively large. Therefore, we must consider next the magnitude of

the effect, to see if thermal blooming will pose a serious threat to

mission accomplishment should it occur.

The quantitative analysis of all nonlinear phenomena is quite

complex and difficult, since it depends on a beam intensity which

is itself changing in response as the interaction proceeds. Predicting how the spot size and intensity vary with time on target for a

beam of arbitrary intensity distribution therefore requires a computer program in which all these effects are modeled.5l Considerable insight can be gained, however, by looking at simplified examples for which exact solutions exist. One case which has been

133

Lasers

Blooming

Time

(sec)

Beam Radius, m

1.00000

 0.1000

0.01000

0.00100

0.00010

0.00001

0.01 0.1 1 1.0 100

Figure 3-41. Time for the Onset of Thermal Blooming vs Beam Radius
extensively studied is that of a beam in a uniform crosswind of velocity v, having an intensity profile52 which varies with radius as

S(r)  So exp(–2r2/w2). Such a beam can be characterized through

a thermal distortion factor, Nt, which is given by53 

Nt  –(dn/dT)  KSZ2

ncp vw

The first factor in the expression for Nt contains parameters related

to the gas through which the laser is propagating, and the second

contains parameters related to the laser and scenario in which it is

employed. The individual terms have the following interpretation:

• (dn/dT) is the slope of a curve of index of refraction, n, as a

function of temperature, T. The greater the dependence of n on

T, the more pronounced will be the lens or wedge introduced as

the beam heats the air.

• Cp is the heat capacity of the air (J/gm 0K), and  its density

(gm/cm3). Their product, Cp, is the number of Joules of energy

which must be absorbed to heat a cubic centimeter of air by

one degree.

• K is the absorption coefficient of the air (cm–1), and S the laser

intensity (W/cm2). Their product, KS, is the number of Joules

being deposited in a cubic centimeter of air each second.

• z is the range to target, w the beam radius, and v the wind

velocity. Nt increases as z goes up, because the thermal lens has

a longer distance over which to act. It decreases as v and w

increase. A stronger wind will cool the beam volume, perhaps

even blowing the heated air out of it. Since the intensity

changes the most near the edge of the beam, a larger w reduces

the relative importance of these edge effects and the blooming

or bending which results from them.

As Nt increases, the beam becomes more and more distorted,

and its intensity falls off as shown in Figure 3–42. Since Nt is proportional to the beam intensity S, it’s not possible at large distortion numbers to overcome the effect of thermal blooming by increasing S. For example, if Nt is 10, the intensity on target will be

about 0.1 of what it would have been in the absence of blooming.

If we try to compensate for this by increasing S by a factor of 10,

we’ll increase Nt to 100, since it’s proportional to S. But at an Nt of

Effects of Directed Energy Weapons

134
100, the relative intensity is 0.001, and the net effect of increasing S

by a factor of 10 will have been to reduce the intensity on target!

This illustrates one of the more unpleasant features of nonlinear

effects. Actions taken to correct them can have an effect opposite

to that intended, because of the many feedback loops which affect

how the system responds to its inputs.

It is interesting to note as another example that an instability

may occur when adaptive optics is employed in the presence of

thermal blooming. The reason for this is sketched in Figure 3–43.

In the upper portion of the figure, we see what an adaptive

optics system perceives as thermal blooming begins—that a diverging lens has been inserted into the beam path. As a response,

the system, tries to do what is shown in the bottom portion of the

figure—send a converging beam into the lens so that the diverging

nature of the lens will only serve to straighten the beam out. Unfortunately, this serves to increase the intensity in the center of the

beam, increase the temperature in this region, and aggravate the

diverging lens effect, leading to further focusing, further divergence etc, etc. Thus the algorithm responsible for adaptive optics

must be capable of adapting to and compensating for nonlinear

phenomena such as thermal blooming, as well as linear phenom135

Lasers

Relative Intensity

Intensity with blooming

Intensity without blooming ( )

10-1

10-2

10-1 10

1

Thermal Distortion Number, Nt

Figure 3-42. Relative Intensity vs Distortion Number

1
ena such as turbulence. The spatial scale for thermal blooming is

on the order of the beam width, and may be much greater than the

coherence length which characterizes turbulence. This can create

both hardware and software challenges in dealing with the two

phenomena together. Developing adaptive optics schemes capable

of handling both thermal blooming and turbulence is an ongoing

area for research.54

If we want to avoid the complexities of dealing with thermal

blooming, Figure 3–42 tells us that the distortion number must be

of order unity or less. What are the implications from the standpoint of beam design? Figure 3–44 shows the relationships among

intensity, range, velocity, and beam radius subject to the constraint that Nt  1. The curve may be used in the following way.

Suppose we wish to engage a target at a range of 10 km with a

beam whose radius is 0.5 m (50 cm), and anticipate that the crosswind will be 5 mi/hr (about 200 cm/sec). Then the product vw is

104 cm2/sec. As the lines drawn on the figure show, the product

KS must therefore be less than or equal to about 3  10–6W/cm3 if

the distortion number is to be kept less than one. If we know that

the beam intensity needs to be 104 W/cm2 in order to meet our

damage criteria, this implies that the absorption coefficient K

must be less than 3  10–10 cm–l (3  10–5 km–l) for no thermal

blooming or bending to occur. Since absorption coefficients

within propagation “windows” are more like 10–3 – 10–2 km–l, you

can see that it’s very unlikely that we can accomplish this mission

without thermal blooming.55 On the other hand, if the target is an

aircraft moving at 500 mi/hr, vw will be increased to 106 cm2/sec

and KS to 3  10–4 W/cm3. Under these circumstances, a 104

Effects of Directed Energy Weapons

136

Figure 3-43. An Instability in Thermal Blooming with Adaptive Optics

Increased Intensity

in Center of Beam
137

Lasers

10-13

10-11

10-9

10-7

10-5

10-3

10-1

100

Range, km

KS

(W/cm3)

KS = vw/Az2,A = 2.5x10-3cm3/j

vw = 106 cm2/sec

1 10 1000 10000

105

104

103

Figure 3-44. Parameter Tradeoffs to Prevent Thermal Blooming

W/cm2 laser can do the job if K is less than 3 x 10–3 km–l, a more

reasonable value.

You can play with Figure 3–44 and look at the likelihood of

avoiding thermal blooming under a variety of scenarios.56 Such an

examination will convince you that there are far more circumstances where thermal blooming needs to be considered, than

where it does not. The only sure-fire way around the problem is to

shorten the interaction time to the point where blooming can’t develop (Figure 3–41). This solution can bring problems of its own

with it, however, since at short pulse widths the intensity necessary for damage can become quite high, and may exceed the

threshold for other nonlinear effects, such as stimulated scattering

and air breakdown.

Stimulated Scattering. Thermal blooming is a nonlinear propagation effect, which causes the beam to diverge more rapidly and

follow a different path than it would in a vacuum. Stimulated

scattering is a nonlinear form of scattering. Normal scattering

events are independent of one another. When we derived the absorption equation, the probability of a photon scattering when it

encountered a molecule depended only on the molecule’s scattering cross section, 	, and had nothing to do with what other molecules were present, or how many other photons had scattered. As

a result, the attenuation coefficient K was simply a sum of contributions from the different molecules present, and was indepen-
dent of the intensity, S. By contrast, stimulated scattering occurs

when one photon is induced (or stimulated) to scatter because

other photons have done so. In a sense, it’s as though the photons

exert peer pressure on one another, and when the number which

have scattered reaches some critical number, suddenly everybody

wants to get into the game, and the amount of scattering increases

dramatically.

Mathematically, the increase in the intensity of the stimulated scattered light, SS, with distance obeys the relationship

dSs/dz  g Ss S, where S is the intensity of the laser light, and g

is a constant of proportionality, known as the gain, having units

of cm/W. The fact that dSs/dz is proportional to Ss reflects the

stimulation provided by photons which have already scattered.

The solution to this equation is an exponential growth in the

scattered light: Ss(z)  Ss(0) exp (gSz). By contrast, the normal

scattering of light obeys the relationship dSs/dz  	sNS, where

	s is the cross section for scattering. The solution to this equation

is a linear growth in Ss: Ss(z)	sNS z. Under normal circumstances, the intensity of scattered light, Ss, is small, and stimulated scattering is not important. However, over a sufficiently

long range and at a sufficiently high intensity of laser light S, Ss

can build up to the point where stimulated scattering exceeds

normal scattering, and due to its exponential growth, the laser

light is rapidly depleted. The contrast between normal and stimulated scattering is illustrated in Figure 3–45.

In Figure 3–45, you can see that as long as the scattering is

normal and exceeds the stimulated scattering, there is relatively

little decline in the laser intensity. However, the stimulated scattering grows exponentially with distance, and at some critical distance, zc, exceeds normal scattering, rapidly depleting the laser

beam. Therefore, the analysis of stimulated scattering involves determining zc as a function of laser parameters, so that these may be

chosen to avoid the problem over the range to target.

There can be different types of stimulated scattering, according

to the nature of the scattering mechanism. The general theory of

stimulated scattering relies on quantum mechanics and is quite

complex. The theory is discussed in detail in many sources.57 For

our purpose, it will be sufficient to focus in on the form of scattering which is considered to be most limiting from the standpoint of

high power propagation in the atmosphere. This is known as stimEffects of Directed Energy Weapons

138
ulated Raman Scattering, and is often referred to as SRS in the literature.58

It is generally accepted that a good rule of thumb for SRS to

grow to the point where it exceeds normal scattering is that the

intensity of the SRS-scattered light must grow by 11 orders of

magnitude. That is, the SRS scattered light, which is growing as

Ss (z)  Ss (0) exp(gSz), must increase to the point where exp (gSz)

 1011, or gSz  ln (1011)  25. We can use gSzc  25 as a criterion

for SRS to prevent the propagation of a high power laser, and find

the critical range, zc, as a function of the laser intensity, S, and the

gain, g. This requires that we have an understanding of g and how

it varies with such parameters as altitude and wavelength. Figure

3–46 is a plot of the SRS gain for a 1.06 m laser propagating in

the atmosphere.59

As you can see from Figure 3–46, the gain for SRS declines

rapidly above about 40 km, so that from a practical standpoint

stimulated scattering is of concern only at lower altitudes. As a

function of pulse width, there is a modest decline at higher

altitudes. For short pulses; near sea level, there is essentially no

dependence on pulse width. The gain for SRS may be shown on

theoretical grounds to be proportional to the laser frequency, .

Therefore, Figure 3–46 can be scaled to different frequencies or

139

Lasers

Intensity

Normally Scattered Ligth

Laser Ligth, s

Zc

Range

Figure 3-45. Stimulated vs Normal Scattering of Ligth

Ligth from

stimulated

scattering, Ss
wavelengths by multiplying by the ratio /o, or o/, where o

and o are the frequency and wavelength for 1.06 m radiation.

For example, if the light were in the far infrared at 10.6m, the

gain would be reduced by a factor of 0.1 from that shown in

Figure 3–46.

Using the data in Figure 3–46, we can plot the range zc to which

a given intensity S can propagate before SRS becomes a concern,

using as a criterion gzcS  25. This is done in Figure 3–47.

Figure 3–47 tells us that for tactical applications at ranges of 100

km or less, SRS limits us to intensities of about 107 W/cm2 or less

in the far infrared, and about 3  105 W/cm2 or less in the visible.

For strategic applications, the beam might be shooting up into the

atmosphere with some elevation angle 
 (see Figure 3–25). In this

case, since the gain for SRS is significant only below 40 km, the effective range in the atmosphere is z  40 km/sin
. For 
 close to

90°, the limitations on intensity are roughly the same as at tactical

ranges, and at very long slant paths, as 
 approaches zero, intensity becomes ever more limited.

If damage criteria are such that we believe we’ll need to operate

above the threshold for stimulated scattering, there are various

ways by which this limitation might be overcome. In strategic appliEffects of Directed Energy Weapons

140

Gain (10-12 cm/watt)

t

ρ = 50 nsec

CW laser (tρ = ∞)

Altitude (km)

0 20 40 60 80 100

10

1

10-1

10-2

10-3

10-4

Figure 3-46. Gain for SRS vs Altitude and Pulse Width at 1.06 μm
cations, we could increase the laser’s aperture and focus the beam

towards the target. In this way, the intensity could be below the

threshold for SRS within the atmosphere, yet above the threshold

for damage on target. This approach would not work, of course, for

a beam in space fired at a target on the earth, where the highest intensities would be in the lower regions of the atmosphere.

It might also be possible to shorten the pulse width. As you

can see from Figure 3–46, the altitude at which the gain for SRS

begins to decline is lower at shorter pulse widths. Of course, the

penalty for doing this is that the intensity will be higher if damage criteria require a constant energy in target, and it may be that

the higher intensity will again exceed the threshold for excessive

stimulated scattering.

Beyond the obvious remedies of playing with the diameter and

pulse width of the laser, various more esoteric schemes at reducing SRS have been proposed.60 These generally rely on quantummechanical techniques to suppress the stimulated scattering by injecting into the main beam other beams at the slightly shifted

frequencies of the SRS scattered radiation, to “encourage” the scattered photons to propagate along with the main beam, not diverg141

Lasers

Intensity S (W/cm2)

Figure 3-47. Critical Range for SRS vs Intensity and Wavelength at Sea Level

105

104

103

102

101

10

10-1

10-2

10-3

Range

(km)

0.33

0.69

1.06

10.6

104

λ,μm

105 106 107 108 109 1010
ing from it. Whether any of these ideas will work in practice has

yet to be demonstrated experimentally.

Air Breakdown. Stimulated scattering is a nonlinear scattering

effect, and air breakdown, in which the neutral gases of the atmosphere are transformed into a highly absorbing plasma, is a nonlinear absorption effect. It is somewhat ironic that this dramatic and

terminal effect in beam propagation begins with the absorption of

laser light in something so small that it makes no discernable contribution to the absorption coefficient—free electrons in the air.

Free electrons (those not bound to atoms or molecules) interact

strongly with laser light. They’re electrically charged, and laser

light, after all, is electromagnetic radiation. They have a small

mass, only about 10–4 that of a molecule, and so they are easily accelerated by electromagnetic radiation. If there were a substantial

number of free electrons in the air it would strongly absorb any

electromagnetic radiation.61 Fortunately for those who contemplate using lasers as directed energy weapons, there are very few

free electrons in the air—the vast majority are bound into atoms or

molecules. There are about 100 free electrons per cubic centimeter,62 as compared with the 3  1019 molecules per cubic centimeter

present in air at sea level. These few arise from relatively rare

events, such as cosmic rays penetrating the atmosphere and

knocking electrons free from their parent molecules.63

Air breakdown is in essence the runaway growth of electron density to the point where almost all of the molecules in the air have

lost at least one of their electrons. At this point, the air is highly absorbing and no laser can propagate. Molecules or atoms which have

lost an electron are said to have been ionized, and a gas of ions and

electrons is known as a plasma (not to be confused with the blood

plasma that medical people like to talk about, of course!).

How does this runaway growth of electrons occur? The fundamental process is known as an electron cascade, and proceeds by

the following steps:

• The few electrons which are naturally present in the atmosphere

heat up by absorbing laser light, and become very energetic.

• Eventually, they gain sufficient energy that when they collide

with an atom or molecule they can ionize it. This requires that

Effects of Directed Energy Weapons

142
they have at least an amount of energy called the ionization potential, typically 10–20 eV.

• When an electron succeeds in ionizing an atom, the electron

population has grown. The newly born electron joins its predecessors in gaining energy, and is eventually able to ionize an

atom itself.

• This process repeats until almost all of the atoms have been

ionized, and the laser light is totally absorbed.

From the description of air breakdown just outlined, you can see

that it will depend on two things—the rate at which electrons can

gain energy in the presence of electromagnetic radiation of intensity S, and the rate at which they may lose that energy to ionization and to other competing processes that can serve as energy

sinks, limiting the amount available for ionization. We’ll consider

how electrons gain and lose energy in turn.

We’ll call the energy of an individual electron 
, and denote the

rate at which it gains energy from laser light as d
/dt. The electron heating rate d
/dt may be related to the intensity and frequency of the laser light as well as appropriate parameters characterizing the gas through which it propagates. The appropriate

expression is64

d
  e2 Sc

dt 2m 
o (2 + c

2)

This formidable expression has a reasonably simple physical interpretation. It is not surprising that d
/dt should be proportional

to the laser intensity S—we’d expect that the more intense the

laser beam, the more rapidly it could heat the electrons within its

volume. The quantity e is the charge on the electron, and m its

mass. If electrons had a greater charge, we’d expect that they’d

heat at a greater rate, and if they were heavier, they’d be harder to

accelerate and would heat more slowly. Thus, the proportionality

of d
/dt to 
2/m seems appropriate. The quantity c is the rate at

which electrons collide with atoms in the gas. This factor is important because as the electrons move up and down under the influence of the laser’s electromagnetic fields, they are progressively

accelerated and decelerated, gaining and losing energy. Periodically, however, they collide with an atom or molecule. When they

143

Lasers
do so, the energy of motion which they have at that time is converted into random energy, or heat. Thus, it is the friction between

the electrons and the background gas which results in their

heating, just as the electrons in a metal heat as a result of their

collisions with the impurities, vibrations, and other things responsible for the metal’s electrical resistance.

In the denominator of the expression for d
/dt there appears a

factor, (2 + c

2), which merits some discussion. This term reflects

the time scale over which the electrons gain kinetic energy from

the applied electric field before they collide with an atom and convert that energy into heat. The quantity  is known as the radian

frequency of the laser light—2, where is the frequency in Hertz

of the light.65 Two limits are of interest in evaluating d
/dt— > c,

and  < c. When we’re dealing with typical laser frequencies and

gas densities characteristic of the atmosphere,  > c. In this case,

1/ is clearly the amount of time an electron has to gain energy

before the laser field turns around, decelerates it, and gives it the

same amount of energy in the opposite direction. Thus, the electron is accelerated for a time on the order of l/. If we recall that

kinetic energy is mv2/2, and that v is proportional to the acceleration time, it stands to reason that the kinetic energy an electron

gains would be proportional to l/2. If, on the other hand,  < c,

then the acceleration process will be terminated by a collision with

an atom before the field has a chance to turn the electron around,

and the energy gain is proportional to 1/c

2. This limit is the one

which applies at microwave frequencies, and will receive considerable attention in Chapter 4. The factor (2+c

2) in the expression for

the electron heating rate captures both limits.

The other factors which appear in the expression for the electron

heating rate are simply constants which arise as a result of our

choice of metric units: c is the speed of light, 3  108 m/sec, and 
o

a constant known as the permittivity of free space, 8.85 x10–12

farad/m. Figure 3–48 is a plot of the electron heating rate at sea

level as a function of intensity and wavelength.

As you can see from Figure 3–48, electrons will gain energy very

rapidly in the absence of any energy loss mechanisms, reaching energies on the order of 10 eV that are necessary to begin an electron

cascade in times on the order of 10–9 seconds at intensities on the

order of 109 W/cm2 or greater. Since the electron collision frequency

c is proportional to the atmospheric density, Figure 3–48 may be

Effects of Directed Energy Weapons

144
scaled to altitudes above sea level by multiplying the heating rate

shown in the figure by the ratio of the density at the desired altitude

to that at sea level. As we have already seen, this correction factor is

to a first approximation exp(-h/ho), where ho  7 km.

It is important to note that rapid electron heating does not imply

strong absorption of energy from the laser beam. Figure 3–48

shows the heating rate per electron, and at the start of an electron

cascade there are very few electrons around. The absorption coefficient due to electron absorption can be found from KS 

n(d
/dt), where n is the density of electrons. Initially, this is small

compared to molecular or aerosol attenuation, but as breakdown

proceeds, n increases and eventually dominates beam absorption.

With Figure 3–48, we know how free electrons in air will gain

energy. Our next task is to examine how they may lose energy,

since the balance between energy gain and loss will determine

their average energy and the likelihood that they’ll ionize molecules and grow in number. Energy loss is possible whenever an

electron strikes a molecule with greater energy than is necessary

to ionize it or to excite one of its degrees of freedom. You will re145

Lasers

Intensity (W/cm2)

Figure 3-48. Electron Heating vs Laser Intensity and Wavelength at Sea Level

1017

1015

1013

1011

109

107

105

Heating

Rate

(eV/sec)

0.33

0.69

1.06

107

λ = 10.6 μm

108 109 1010 1011 1012 1013 1014
call from earlier in the chapter (see Figure 3–9) that molecules

have ionization potentials on the order of 10 eV, electronic levels

separated by energies on the order of 1eV, vibrational levels separated by about 0.1 eV, and rotational levels separated by about

0.01 eV. If an electron with 5 eV of energy strikes a molecule, it

doesn’t have the energy to ionize the molecule, but it may excite

one of the electronic, vibrational, or rotational levels. If it does, it

will lose the appropriate energy of excitation. For example, if the

electron loses energy in exciting a vibrational level, it will lose energy on the order of 0.1 eV. Figure 3–49 is a plot of the average energy loss rate to various types of excitation as a function of average electron energy in a gas of molecules having thresholds of 0.1,

l, and 10 eV for vibrational, electronic, and ionization energy

losses.66 The actual atmosphere, with many species of molecules

and many different levels to excite, is far more complex. Detailed

analyses are available elsewhere; the simple model shown in Figure 3–49 is sufficient to illustrate the physics involved in breakdown, and is surprisingly accurate from a quantitative standpoint

as well.67

As Figure 3–49 illustrates, electron energy losses become

greater as their average energy increases, and they are able to excite more degrees of freedom in the gas molecules. At the lowest

Effects of Directed Energy Weapons

146

Total

Vibrational Excitation

Electronic Excitation Ionization

Average Energy

Loss

Rate

(eV/sec)

1013

1012

1011

1010

109

107

105

0.1 0.3 1 10 eV

Figure 3-49. Electron Energy Losses in a Model Gas

108

107
energies, most of the electrons are exciting vibrational levels, and

at the highest energies, they’re primarily ionizing the molecules

in the gas. Clearly, the threshold for breakdown lies between

these two extremes.

Figure 3–49 may be used together with Figure 3–48 to estimate

the minimum intensity at which electrons can cascade and the gas

will break down. There will be no growth of electrons unless there

can be some ionization. From Figure 3–49, almost all of the energy

goes into vibrational excitation at average energies below about 0.1

eV, for which the energy loss rate is about 5  1010 eV/second. If

the electrons are to gain energy to the point where breakdown can

begin, the rate of energy gain from the laser must exceed this energy loss rate. From Figure 3–48, you can see that an energy gain of

5  1010 eV/sec requires an intensity of about 3  109 W/cm2 at a

wavelength of 10.6 m. Therefore, about 3 x 109 W/cm2 should be

the threshold for gas breakdown with 10.6 m radiation.

Figure 3–50 shows more detailed calculations of the breakdown

threshold for air as a function of pulse width.68 As you can see, our

simple estimate closely approximates the result of much more

elaborate analysis at longer pulse widths. More importantly, experimental data are also consistent with our analysis and that

shown in Figure 3–50.69 This gives us confidence that we understand the physics of the gas breakdown phenomenon.

The other features shown in Figure 3–50 may also be understood with the use of Figures 3–48 and 3–49. As the pulse width

decreases, the breakdown threshold rises, and at the shortest

pulses the intensity necessary for breakdown is inversely

proportional to the pulse width. At very short pulse widths,

we need very high ionization rates if the gas is to break down

before the laser turns off. This means that almost all of the electron energy losses are to ionization. If all of the energy input

from the laser is going into ionization, it must be true that

d
/dt  I Ri where I is the ionization potential, and Ri the rate at

which electrons ionize neutral molecules. In other words, every

collision is an ionizing one, and causes the colliding electron to

lose an energy I.

If breakdown is to occur in a time tp, the initial electrons must

be able to ionize and multiply themselves g times, where g is the

number of generations required for the electrons to cascade to

breakdown, and ionize all the neutral molecules. About 56

147

Lasers
generations are required in air at sea level.70 Therefore, at short

pulses the ionization rate must be such that Ritp  56, or Ri 

56/tp. Thus, the short pulse criterion for breakdown becomes

d
/dt  I Ri 56 I/tp, which implies that the heating rate and

intensity for breakdown must be inversely proportional to tp, as

seen in Figure 3–50. We can be more quantitative by observing

from Figure 3–48 that at 10.6 m the electron heating rate in

eV/sec is about 10 times the intensity in W/cm2. The criterion

for breakdown at this wavelength can therefore be written as

10 S  56 I/tp. For an ionization potential I of about 10 Joules,

this implies Stp  56 J/cm2. This estimate is shown on figure

3–50, and considering the simplicity of our analysis is amazingly

close to the limit of 10 J/cm2 obtained in more detailed work.

The altitude dependence exhibited by Figure 3–50 derives from

the fact that the electron heating rate is proportional to the collision rate c, and is therefore proportional to gas density. If the

criterion for breakdown is d
/dt  I Ri  56 I/tp, and if c is

reduced by an order of magnitude, S must increase by an order

of magnitude to keep d
/dt constant. This behavior is observed

in the short-pulse, high intensity limit of Figure 3–50. On the

other hand, at long puses the threshold for breakdown is deterEffects of Directed Energy Weapons

148

1012

1011

1010

10-9 10-8 10-7 10-6 10-5 10-4 10-3

3x109W/cm2

10 J/cm2 56 J/cm2

Intensity

S(watts/cm2)

Pulse Width, tp (sec)

N = 3x1019/cm3, sea level

N = 3x1018/cm3, ~ 20km

N = 3x1017/cm3, ~ 40km

 = 10.6 μm y

Figure 3-50. Breakdown Threshold for 10.6 μm Radiation
mined by a balance between d
/dt and the energy losses shown

in Figure 3–49. Since both of these are proportional to c, there is

effectively no density (altitude) dependence in this limit.

While Figure 3–50 has been sketched for laser radiation of

wavelength 10.6 m, it can be used to predict the air breakdown

threshold at any other frequency simply by scaling as 1/2.

Thus, at   1.06 m, the intensity at which breakdown will

occur for a given pulse width will be 100 times greater than that

shown in the figure.

Aerosol Induced Breakdown. To this point, we have treated

breakdown as occuring in clean air—a mixture of oxygen, nitrogen, CO2, H2O, and smaller amounts of other gases. The real air

which is encountered in weapon applications differs in having

small solid particles suspended in it. Since these aerosols

contribute to the absorption and scattering of laser light, it’s

probably not surprising that they can affect the breakdown

threshold as well.

How Might Aerosols Affect Air Breakdown?

• Many aerosols are comprised of materials which are easier to

ionize than the atmospheric gases. Salt, for example, is a common aerosol in a maritime environment, and sodium, one of the

constituents of salt, has an ionization potential of only 5.1 eV, as

compared with 14.6 eV for Nitrogen and 13.6 eV for Oxygen.

Our expression for the breakdown threshold in the short pulse

limit is directly proportional to I. Thus, this factor alone might

lower the breakdown threshold by a factor of 2–3.

• Effects occurring at the surface of a solid aerosol, such as the

boiling (thermionic emission) of electrons from the interior, can

increase the local density of free electrons.71 This increase in the

initial electron density reduces the number of generations that

the electrons have to go through to achieve breakdown, and

may create a small mini-plasma that can grow to block and absorb the beam.

• As the aerosols absorb the laser light, they become hot. At

higher intensities, they may even vaporize explosively, sending

shock waves through the surrounding air. This will induce high

temperatures in the region around an aerosol, and excite many

149

Lasers
of the vibrational and rotational degrees of freedom which limit

electron energy growth at low intensities. In effect, in the hot region near an aerosol many of the energy loss mechanisms

which would otherwise inhibit the growth of electron energy

are bleached out, and no longer effective.

All of these factors would argue that in dirty, aerosol laden air,

the breakdown threshold may be reduced from that which characterizes clean air. This is, in fact, observed experimentally. Indeed,

early breakdown experiments were frequently difficult to interpret

because the combined effects of gas and aerosol-induced breakdown contributed to the measured breakdown thresholds.72 Figure 3–51 is illustrative of calculations performed to look at the effect of aerosols on the air breakdown threshold.73 The fact that the

impact of aerosols is size-dependent should not be surprising. The

smaller an aerosol is, the more rapidly it vaporizes and dissipates

when heated by a laser beam, and the less it can affect the breakdown threshold. An interesting corrolary to Figure 3–51 is that

solid targets, representing aerosols of essentially infinite size,

might be expected to lower the breakdown threshold to the greatest degree. We’ll return to this point later in the chapter, when we

discuss laser-target interaction.

Comparing the range of radii shown in Figure 3–51 with typical

particle sizes seen in the atmosphere (Figure 3–27), it’s apparent

Effects of Directed Energy Weapons

150

10.6 mm Radiation

Aerosol Properties:

K = 104 cm-1

Tv = 2700k

Lv = 2.8x104 J/gm

Breakdown threshold

(w/cm2)

109

108

10-3 10-2 10-1

Aerosol Radius (cm)

Figure 3.51. Breakdown Threshold vs Aerosol Radius

10.6 m Radiation
that only very large particles will lower the breakdown threshold

to a significant extent. Nevertheless, when propagating beams of

large diameter over long distances, there is a very large probability that at least one giant particle will be found within the volume

of the beam to serve as an initiation site for breakdown. For example, Figure 3–27 shows that the density of particles having a size of

about 10–3 cm is about 10–6/cm3. Thus, to be sure of finding a particle this big we’d need a beam volume in excess of 106 cm3. For a

beam of 1 m radius, this volume is exceeded in less than a meter

of propagation distance. Since Figure 3–51 suggests that particles

of this size might lower the breakdown threshold by two orders of

magnitude, it seems clear that aerosols may dominate the initiation of breakdown in realistic scenarios.

Knowing the density of aerosols of a given size as a function of

altitude (Figure 3–27), as well as the extent to which a given size

aerosol will reduce the breakdown threshold (Figure 3–51), it’s

possible to estimate the extent to which aerosols will reduce the

breakdown threshold as a function of altitude. Such an estimate

has been made by the author, and is shown in Figure 3–52.

As common sense suggests and Figure 3–52 indicates, aerosols

will have the greatest effect on the air breakdown threshold near

the surface of the earth, where winds and other local phenomena

can loft very large particles, over 10 m in size, into the air. Particles this large rapidly settle through the force of gravity, how151

Lasers

Breakdown Threshold

(w/cm2)

"Clean Air" Breakdown Limit

0 5 10 15 20 25

1010

109

108

107

h (km)

Figure 3-52. Estimated Aerosol-Induced Breakdown Threshold vs Altitude (10.6 μm )
ever, and aren’t found above a few km altitude. The error bars

shown in the figure are indicative of the range of breakdown

thresholds which have been ascribed to particles of a size prevalent at that altitude, and indicate clearly the lack of precision in

so dirty a business as determining the effect of lofted dirt on the

air breakdown threshold.

Figures 3–51 and 3–52 show the effect of aerosols in lowering

the breakdown threshold of 10.6 m radiation. The next logical

question to ask is how these results might scale with frequency or

wavelength. To a first approximation, we might think that everything should scale as 2, since the electron heating rate is proportional to this factor, and it is the heating of electrons by the laser

light which ultimately results in breakdown. The clean air limit to

breakdown scales in this way, as we have already seen. However,

when considering solid aerosols, we need to recognize that the interaction of light with solid matter has no simple dependence on

wavelength. The absorption characteristics of a solid depend upon

its energy gap (if an insulator) or its conductivity (if a conductor),

as well as the wavelength of the light. Thus, a change in wavelength may take a given aerosol from being a major contributor to

breakdown to being a non-player.

Experimental data on the frequency dependence of aerosol induced breakdown are sparse, but suggest the following: The number and variety of aerosols present in any naturally occurring environment is sufficiently great that there are likely to be aerosols

which are strongly absorptive of any wavelength.These, then, will

heat, vaporize, and serve as the source of small seed plasmas,

which will either grow to encompass the beam or will die out.

The first process—the initiation of small plasmas from the

aerosols—will be essentially independent of wavelength, while the

second—the growth or decay of the plasmas—will be wavelength

dependent, as it results from a wavelength-dependent interaction

between the light and the expanding plasma. This motivates us to

examine the extent to which laser light will maintain a pre-existing

plasma or allow it to cool, recombine, and cease to exist.

Plasma Maintenance and Propagation. Once a plasma has

been created as a result of breakdown in the air, from aerosols, or

at a target surface, it will continue to interact with and absorb

the laser light. From the standpoint of propagation, this interacEffects of Directed Energy Weapons

152
tion is important because if a small plasma initiated from an

aerosol grows to encompass the beam, it will prevent further

propagation. The maintenance, growth, and propagation of a

plasma is also important in target interaction. Plasmas created at

a target’s surface can greatly affect how (and if) any energy is deposited within it. As a nonlinear effect, plasma propagation is

certainly the most dramatic, replacing the normally invisible

propagation of light through the atmosphere with lightning-like

balls and streaks of glowing plasma.

It requires less laser intensity to maintain a pre-existing plasma

than to create one from a neutral gas. The reason is relatively simple. In a neutral gas, electrons can absorb energy from the laser

when they collide with neutral gas molecules. In a plasma, the

neutral molecules have become ions—electrically charged particles. Because of their electrical charge, ions have a much greater

collision cross-section and collision frequency with electrons

than neutral molecules. Typical electron-ion collision cross sections are on the order of 10–12 cm2, while typical electron-neutral

cross sections are on the order of 10–15 cm2—about three orders of

magnitude smaller.74 The effect is to increase d
/dt, the electron

heating rate, by about three orders of magnitude at a given laser

intensity S. Once a gas has been ionized, a laser intensity three

orders of magnitude below that required for breakdown of the

neutral gas may be sufficient to maintain the plasma which has

been created.

Since the transition from neutral gas to plasma is accompanied by

a rapid increase in the power absorbed, it should not be surprising

that once created, a plasma won’t stand still, but will tend to grow

and propagate. How this happens is illustrated in Figure 3–53.

Shown in Figure 3–53 is a small plasma, formed, for example, at

the site of an aerosol in the volume of a laser beam. The plasma is

by nature very hot, since to remain ionized a gas must have a temperature on the order of an electron volt (about 12,000o K!) or

greater. The resulting temperature gradient between the hot

plasma and the surrounding neutral gas causes heating in the region surrounding the plasma through thermal conduction, radiation, or other energy transfer mechanisms. As this region heats it,

too, becomes ionized, transforms into plasma, and heats regions

still further out. In this way, the plasma expands radially. This

process can continue only as long as sufficient energy can be ab153

Lasers
sorbed from the laser to replace the energy used up in heating and

ionizing additional layers of gas. If the plasma becomes too large

its leading edge will absorb most of the laser light, shielding the

rear end from any energy deposition. At this point, the plasma can

no longer grow symmetrically. Instead, the front end can propagate towards the laser, while the rear end must cool and die. The

plasma then propagates preferentially up the beam towards the

source of light, as illustrated in Figure 3–53. The characteristic

thickness of the propagating plasma will be l/K, where K is the

absorption coefficient (cm–l) of the plasma for laser light.

In poetic terms, the plasma shown in Figure 3–53 has become a

wave of absorption, moving towards the laser. Therefore, a plasma

propagating in this way has come to be known as a laser absorption

wave, or LAW. Understanding absorption waves requires a knowledge of the mechanisms by which energy is transferred from the hot

plasma to the surrounding gas, how a neutral gas becomes a

Effects of Directed Energy Weapons

154

Laser Light

Laser Light

Laser Light

Envelope of Laser Beam

Plasma absorbs energy, heats. Heat is transferred into surrounding gas by

thermal conduction or other energy transport mechanisms.

Plasma grows as surrounding regions heat sufficiently to become an

ionized plasma themselves

Leading edge of plasma absorbs laser light and continues to grow.

Energy is not replaced in the trailing edge, which cools.

Leading edge propagates towards the light.

Figure 3-53. Plasma Propagation
plasma as it heats, and how the absorption of laser light varies with

the temperature, density, and degree of ionization in the gas

through which it propagates. We’ll look at each of these issues and

develop criteria for the propagation of absorption waves.

The propagation of absorption waves is best understood by

analogy with a common, everyday phenomenon—the burning, or

combustion, of a fuel in an exothermic chemical reaction.75 Paper,

for example, would like to burn, or combine with the oxygen in

the air to form carbon dioxide and other by-products which we

perceive as smoke. It can’t do this at room temperature, however,

because there is a certain amount of energy, known as the activation energy, which the molecules of carbon in the paper must have

in order to react with oxygen. The reaction rate for a combustible

fuel is therefore a strong function of its temperature T, and is proportional to e- where A is the activation energy. The reaction

rate becomes significant, and combustion can begin, at a temperature known as the flash point, kT  A/l0.

Once combustion begins it will continue, because the energy to

sustain it is released in the combining of carbon with oxygen (the

reaction is exothermic). We can burn a sheet of paper by touching a

match only to one corner. The match serves to ignite the fuel, raising that corner to the flash point. As the corner reacts, it releases

energy and heats adjacent regions of the paper to the flash point.

These react in turn, and provide the energy to ignite regions of the

paper still further from the corner. Thus, after igniting the corner

of the paper with a match, we observe a wave of combustion,

which propagates over the paper, consuming it.

In a similar manner, once a plasma is formed within the volume

of a laser beam, it absorbs considerable energy. This energy heats

adjacent regions of the gas, which themselves become ionized and

absorbing when their temperature exceeds some critical level. The

absorption of laser light in regions which have become ionized

gives them the energy to heat still further portions of the gas

which, in turn, ionize and become absorbing. In this way, a laser

absorption wave (LAW) propagates through the gas. There is one

significant difference between a LAW and a wave of combustion,

however. The energy which serves to propagate a LAW does not

come from within the gas itself, but is absorbed from the laser.

Therefore, a LAW propagates towards the source of laser light. If

we ignite a piece of paper in the middle, it will burn in all direc155

Lasers
tions. By contrast, if we create a plasma in the middle of a laser

beam its rear edge cannot receive enough energy to sustain its

temperature. The rear regions cool and recombine into a neutral

gas. The forward edge, which can absorb energy, heats regions

ahead of it. In this way the LAW propagates up the beam, as illustrated in Figure 3–53.

Because of the strong analogy between the propagation of

exothermic chemical reactions and LAWs, much of combustion

theory has been of value in understanding LAW propagation. One

of the most significant insights to emerge from this theory is that

there are two mechanisms by which LAWs might propagate, just

as there are two mechanisms by which exothermic chemical reactions can propagate—detonation and combustion.76

There are two ways in which a combustible fuel might react. In

one of these, the reaction proceeds relatively slowly, such as burning a piece of paper. On the other hand, if we were to grind the

paper up into fine dust, confine it in some type of enclosure, and

then ignite it, something much more explosive would result. Similarly, various types of explosives can either burn more or less gently or, if properly ignited, burn much more rapidly and blow up.

These two ways in which reactions proceed correspond to two

physically different mechanisms of propagation, known as combustion (or more precisely, deflagration) and detonation, respectively.

The essential feature of absorption wave propagation in the

combustion mode is that it occurs at atmospheric pressure, and

the mechanism of propagation is thermal conduction or radiation.

Figure 3–54 shows how the temperature (T, oK), pressure (P,

J/cm3), and molecular density (N, cm–3) of a gas vary as we move

along the axis of a laser beam into a region of plasma propagating

in the combustion mode. You may recall that the temperature

pressure, and density of a gas are related by P N kT.77 Therefore,

if P is held fixed at the pressure of the surrounding atmosphere,

and T rises going into the plasma, the density, N, must be less on

the interior of the plasma than in the surrounding gas. The length

of the plasma will be on the order of the absorption length 1/K.

Over greater distances the light is essentially all absorbed, and

none is available for plasma maintenance.

As you can see in Figure 3–54, the temperature in the ambient

gas actually begins to rise in the region ahead of the plasma—this

is because energy is flowing into this region from the hot plasma

Effects of Directed Energy Weapons

156
core by thermal conduction or radiation. When the temperature

becomes high enough for ionization to begin, a plasma forms, the

laser light begins to be absorbed, and the temperature rises

rapidly. The heated gas expands, and the density drops, as shown

in the figure. An essential feature of this combustion mode of

propagation is that the plasma moves subsonically, at a velocity

less than the speed of sound in the surrounding gas. The reason

for this is straightforward. As we discussed in the context of thermal blooming, the speed of sound is the characteristic velocity

with which pressure disturbances can propagate in a gas. If the

plasma were to move more rapidly than this, there would be no

time for the pressure balance shown in Figure 3–54 to be established, the pressure would rise rapidly going into the plasma, and

a transition would be made to the detonation mode of propagation, illustrated in Figure 3–55.

In Figure 3–55, a plasma is propagating supersonically as a detonation. A number of consequences are immediately apparent. Since

the plasma advances more rapidly than gas can flow from its interior, density remains high and the pressure rises as we go into the

absorbing plasma region. Because of the greater density within the

plasma, its absorption coefficient K is greater, and the plasma’s

157

Lasers

1/K

Plasma Laser Light

T begins to rise ahead of the Plasma

Ambient Values

Pressure

P

Temperature T

Density

N

Figure 3-54. Pressure, Temperature, and Density in the

 "Combustion" Mode of Plasma Propagation
physical length is much reduced. Since there is no time for subsonic energy transfer processes like thermal conduction to propagate energy ahead of the wave, the temperature rises almost instantaneously at the point where the plasma meets the neutral gas.

If thermal conduction can’t heat the region ahead of the plasma,

how is it possible for the plasma to advance as an absorption

wave? The mechanism of propagation is known as shock heating.

It is a well known result in fluid mechanics that a rapid increase in

pressure will propagate through a gas at supersonic speed as what

is known as a shock wave.78 Behind this wave, the temperature rises

rapidly, since the pressure increases, the density doesn’t have time

to decline, and P  N kT. It is this rapid rise in pressure and

temperature which enables the region behind the shock front to

ionize, absorb energy from the laser, and continue to propagate. In

a similar manner, if you ignite the detonator on an explosive, it

causes a rapid rise in pressure which propagates supersonically

through the explosive as a shock wave, raising the temperature at

each point above the flash point so that the explosive is rapidly

consumed.

Because of the analogy with detonation and combustion in

chemical reactions, laser absorption waves have been classified as

laser supported combustion (LSC) waves or as laser supported

Effects of Directed Energy Weapons

158

1/K

Ambient Values

Laser Light

Pressure

Temperature

Density

P

T

N

Figure 3-55. Pressure, Temperature, and Density in the "Detonation" Mode

of Plasma Propagation
detonation (LSD) waves, depending on whether their propagation

is sub- or supersonic. A quantitative analysis of the conditions

under which LSC or LSD waves can exist and propagate depends

upon the hydrodynamic equations for gas flow, coupled with

an understanding of how the degree of ionization and absorption

coefficient for laser light depends upon the density and temperature of a gas.79

Calculating the degree of ionization in a gas with a given temperature and density is a familiar problem in high temperature

physics. As molecules heat, they move more rapidly, their average

energy of motion (kinetic energy) being a measure of temperature.

Periodically, the molecules collide with one another, and if they’re

hot enough, some of these collisions will result in ionization. The

equilibrium level of ionization is that for which the probability of

ionization equals the probability of recombination, the recapturing

of an electron by an ion to create a neutral molecule.80 Figure 3–56

is a plot of the fraction of atoms ionized in a gas of hydrogen of

density N at temperature T.81

You can see from Figure 3–56 that as temperature rises, the transition from a gas being essentially neutral to almost fully ionized

occurs rapidly, and is more abrupt at lower densities. In fact, the

initial rise in degree of ionization with temperature is proportional

to e–I/kT, and the transition from a neutral gas to an ionized plasma

occurs at a temperature which is on the order of one-tenth the

ionization potential.82 In a sense, the ionization potential I plays a

role in plasma formation analogous to the role of the activation

energy in chemical reactions.

Once we know the degree of ionization at a given temperature,

it is straightforward to find the absorption coefficient for laser

light in a gas with that level of ionization. You will recall that the

absorption coefficient for light which is heating electrons at a rate

d
/dt is simply K  n (d
/dt)/S  e2nc/2mc
o2, where n is the

density of electrons, c is the frequency with which electrons collide with heavy particles (neutral atoms or ions), and  is 2

times the frequency of the light in Hertz. In a neutral gas, c is

simply 	cN, where 	c is the collision cross section for electrons

with gas molecules of density N. In a partially ionized gas, c

becomes 	cN (l–f) + 	iNf, where 	i is the collision cross section

with ions, and f the fraction of molecules which have been ionized. Since 	i is much greater than 	c, the second term will domi159

Lasers
nate the absorption coefficient whenever the fractional ionization,

f, is greater than about 10–3.

Figure 3–57 shows the absorption coefficient for nitrogen gas as a

function of temperature and density for 10.6 m wavelength radiation.83 You can see that in an ionized gas whose density is that of air

at sea level, almost all the laser light will be absorbed within a very

short distance. At lower densities, of course, the absorption length

becomes progressively longer. Knowing the absorption characteristics of a propagating plasma, it becomes possible to see how it propagates as a function of the energy absorbed, and make quantitative

assessments of propagation velocities and thresholds as a function

of laser intensity and wavelength. While detailed analysis is beyond

the scope of this book, we’ll provide the highlights of analyses

which provide good insight into plasma propagation as detonation

or combustion waves.

The detonation mode of LAW propagation is easiest to treat

mathematically, because the transition from “non-plasma” to

“plasma” is made so rapidly that its details are not important. Instead, the shock front can be treated as a discontinuity, with ambient values for T,P, and N ahead of the front, and plasma values

for these quantities behind the front. The flow of the gas may also

Effects of Directed Energy Weapons

160

Figure 3-56. Fractional Ionization of Hydrogen vs Temperature and Density

T. eV (1eV=11,605 K)

Fractional

Ionization, f

N=1013 /cm3

1016

1019

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
be treated as one-dimensional, since the absorption length for the

laser light, l/K, is typically much less than the radius of the laser

beam. This means that any radial flow of hot gases occurs behind

the region in the shock front where the deposition of laser light

occurs, and cannot affect wave propagation. The results of such a

one dimensional LSD analysis84 show that the velocity of wave

propagation, u, and the temperature behind the wave front are

given by u  2[(2–1)S/]1/3 and kT  Mu2/ (+l)2. In these expressions,  is a constant with a value of about 1.2,85 M is the mass

of a gas molecule, and  is the mass density of the ambient gas

(  MN, gm/cm3). It is interesting to note that the velocity and

temperature of the LSD do not depend upon the absorption coefficient of the plasma, K. That’s because of an assumption that K is

so large that all the laser light is absorbed within the shock front,

and serves to propagate the plasma. Figure 3–58 is a plot of the

predicted LSD propagation velocity as a function of laser intensity and gas density, and Figure 3–59 is a plot of the predicted

temperature within the plasma.

In Figure 3–58, you can see that LSD velocities are predicted to

be quite high. At an intensity of 3  107 W/cm2, for example, the

velocity is about 6  105 cm/sec. Since the speed of sound is only

161

Lasers

Temperature, eV (1 eV = 11, 605 0K)

K

cm-1

107

105

103

101

10-1

10-3

10-5

10-7

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

N = 3x1018

N = 3x1019/cm3 (sea level)

N = 3x1017

Figure 3-57. Absorption Coefficient vs Temperature and Density for Nitrogen,

λ = 10.6 μm
3x104 cm/sec, the velocities in Figure 3–58 are highly supersonic—

about Mach 20 or greater. Note, too, that as the gas density  goes

down, the propagation velocity, which varies as the one-third

power of S/, goes up. LSD waves are predicted to propagate

more rapidly at higher altitudes.

Superimposed on Figure 3–58 are experimental data on LSD velocity as a function of intensity at sea level.86 The data are in pretty

good agreement with the theoretical curve, except that below

about 107 W/cm2 the velocity appears to be dropping off more

rapidly than theory would predict. The reason for this is apparent

from Figure 3–59.

You will recall from Figure 3–57 that the absorption coefficient

of a gas begins to decline below temperatures of about 1 eV. From

Figure 3–59, you can see that the data on Figure 3–58 begin to deviate from the theoretical curve in precisely the region where at

sea level the temperature of an LSD is predicted to fall below 1 eV.

In other words, we are seeing the beginnings of a threshold for

LSD maintenance—an intensity below which high levels of ionization can’t be maintained, and the theoretical assumption that all of

the laser’s intensity is absorbed in and helps to propagate the

wave is no longer valid.

An LSD maintenance threshold may be estimated by assuming

that a supersonic wave will no longer propagate when the absorption length within the plasma, l/K, becomes comparable to the

beam radius.87 The reasoning is that when laser energy is absorbed

Effects of Directed Energy Weapons

162

Velocity

(cm/sec)

108

107

106

105

3x106 3x107 107 108 3x108 3x109 109

Intensity, S (W/cm2)

Density, gm/cm3 = 1.29x10-5

Experimental Data

1.29x10-4

1.29x10-3

(sea level)

Figure 3-58. LSD Velocity vs Intensity and Density
over distances greater than a beam radius, some of the absorbed

energy will be carried away as heated gases expand in a radial direction, and will not help to move the beam forward. Figure 3–60

is a plot of the LSD maintenance threshold, determined in this

way, as a function of ambient gas density (altitude), laser frequency, and beam radius.88 The dependence on 2/R as a scaling

factor reflects the fact that the absorption coefficient, K, is proportional to 1/2.

Figure 3–60 may be used in the following way to estimate LSD

maintenance thresholds. The bottom curve is appropriate for a

beam radius of 10 cm and a wavelength of 10.6 m. The scaling

parameter is 2/R, so that if the wavelength were 1.06 m and the

radius remained at 10 cm, 2/R would increase by a factor 100. At

sea level, the threshold would rise from about 6  106 W/cm2 to

about 3  107. If the radius were 1 m and the frequency 1.06 m,

2/R would be back to its original value, and the threshold under

these conditions would be the same as for a 10 cm, 10.6 m beam.

As density decreases, a point is reached where the absorption coefficient is so low that an LSD can’t be maintained even at full single ionization. This point is indicated by the thick cut-off bars, and

occurs at a density of about 3 x10–5 g/cm3 for 2/R  l. The thresholds indicated on Figure 3–60 are in good agreement with those

suggested by the data shown in Figure 3–58.

163

Lasers

Temperature

(eV)

1000.0

10.0

1.0

0.1

3x106 3x107 107 108

3x108

3x109 109

Intensity, S (W/cm2)

N=3x1019/cm3

(sea level)

Figure 3-59. LSD Temperature vs Intensity and Density

100.0

3x108

3x107
If a laser’s intensity falls below the LSD maintenance threshold,

the shock front will empty out, and the plasma might continue to

propagate in the combustion mode, as an LSC. The analysis of LSC

propagation is much more complex than that of the LSD, since gas

flow in the radial direction, thermal conduction, and the transport

of radiation emitted from the plasma, all of which can be ignored

for an LSD, must now be considered. Additionally, these factors

can all contribute in roughly equal measure to the energy balance

in a combustion wave. Therefore, an LSC’s characteristics are the

net result of competing terms of comparable magnitude and differing signs. This means that a simple analysis, accurate to within factors of two or so, cannot hope to be quantitatively correct. Nevertheless, simple analyses have been used to provide valuable

insights into the LSC mode of absorption wave propagation.89

There are two mechanisms by which subsonic plasma propagation can occur—thermal conduction and reradiation, or the absorption of photons emitted from the plasma core. Both of these

energy transport mechanisms were discussed in Chapter 1, but

reradiation is particularly complex for an LSC, and deserves a

little more discussion here. Plasmas are very hot and energetic—

Effects of Directed Energy Weapons

164

S (watt/cm2)

R = 10cm

y = 10.6 μm

2

R

 = laser frequency

R = beam radius

= 1 (relative units)

(gm/cm3)

sea

level

Ambient Density

(Altitude)

40 km 20 km

108

107

10-5 10-4 10-3p

Figure 3-60. Detonation Wave Maintenance Thresholds
within them ionization and recombination are continually occurring, and with each recombination a photon is emitted with an

energy corresponding to the energy difference between the free

electron which existed prior to recombination and the bound

electron which resulted from the process. Photons are also emitted as a result of transitions from higher to lower electronic,

vibrational, or rotational states among the atoms and molecules

in the gas. As a result of all this turmoil, photons are emitted

from a plasma over a broad range of wavelengths.90 Some of the

photons emitted by a laser supported plasma will be at wavelengths which are strongly absorbed in the cold gas ahead of the

wave, and will serve to heat this region. In a sense, the plasma

acts as a “photon converter,” absorbing light at the laser frequency and reemitting it at wavelengths which can be absorbed

in the atmosphere. This process can also be important from the

standpoint of target interaction.

Thermal conduction is inherently a surface phenomenon—energy

is transported across the surface which separates a hot region from a

cold one. By contrast, photon emission from a plasma is a volumetric phenomenon—the more volume a plasma has, the more space

there is from which photons can be emitted. Therefore, thermal conduction will dominate the propagation of beams of small radius,

which have a large surface to volume ratio. Radiation transport

(photon emission and absorption) will dominate the propagation of

beams of large radius. As a result, thermal conduction has been important in understanding the LSCs seen in laboratory scale experiments, while radiation transport will likely be of greater importance

in any LSCs propagating within the large radius beams necessary

for the long ranges required in weapon applications.

Simple theory predicts that the velocity of an LSC propagating through thermal conduction should be on the order of

u  (KSD/Cp  T)l/2, where K is the plasma’s absorption coefficient, D is the thermal diffusivity of the ambient air, about

1 cm2/sec, Cp is its heat capacity at constant pressure,  its

density, and T the temperature, about 1 eV, at which the air

becomes a plasma. If LSC propagation is dominated by radiation transport, theory suggests that its velocity should be on

the order of u  S/ CpT.9l These velocities are plotted as a

function of intensity at sea level in Figure 3–61.

165

Lasers
The most striking feature to be seen in Figure 3–61 is that the

velocities are quite low. Not only are they low compared to LSD

velocities ( 106 cm/sec), they are also low compared to naturallyoccurring wind velocities (10 mi/hr  447 cm/sec). This suggests

a strong possibility that for LSCs propagating in a natural environment, either local winds or those which result from slewing the

beam will introduce cool air into the LSC faster than it can be

heated, resulting in the LSC being blown out. This is completely

analogous to blowing out a chemical flame, as on a match, by introducing cool air so fast that the flame temperature cannot be

maintained, and cools below the flash point.

Detailed analyses of LSC propagation are in general agreement

with these ideas and with experimental data. However, the comparison between theory and experiment is not as precise as in the

LSD case, since theoretical predictions are strongly dependent

upon the assumptions made, and experimental data are often difficult to interpret. An observed LSC velocity is that of the relative

motion between laboratory air and the LSC, and may be substantially different from a theoretical calculation, which assumes propagation into still air.92

Effects of Directed Energy Weapons

166

10000

1000

100

10

1

103 3x103 104 3x104 105 3x105 106

Radiation

Thermal Conduction

Intensity, S(W/cm2)

Velocity

(cm/sec)

Figure 3-61. Velocities of Combustion Waves Propagating by Thermal

Conduction and Radiation
Summary: Nonlinear Propagation Effects.

1. Nonlinear effects are those in which the beam modifies the environment through which it propagates, and this in turn affects

its propagation characteristics.

2. Thermal blooming results when the absorption of laser energy

causes the air density to change in a way which mimics a diverging lens. This causes the beam to spread as it propagates, and to

bend into a wind flowing perpendicular to the beam’s direction.

Thermal blooming may be characterized in terms of a thermal

distortion number, which relates to a decrease in the beam’s onaxis intensity as shown in Figure 3–42. It will only occur if air

has time to flow from the beam volume, a time roughly equal to

the beam radius divided by the speed of sound (Figure 3–41).

3. Stimulated Scattering occurs when the intensity of scattered

light is so great as to influence more photons to scatter in the

same manner. If stimulated scattering becomes too great over

the propagation path of the laser, significant energy will be lost

from the beam. The critical range for this to occur is given as a

function of wavelength and intensity in Figure 3–47.

4. Air Breakdown occurs when free electrons gain sufficient energy from the beam to ionize neutral molecules and multiply

themselves. When this happens, electrons will multiply until the

air is ionized, absorbs the beam, and propagation ceases. Figure

3–50 shows the air breakdown threshold as a function of density

and pulse width for a 10.6 m wavelength laser. The thresholds

shown in that figure may be scaled to other wavelengths as 1/2.

5. Aerosols can reduce the breakdown threshold by up to two

orders of magnitude. This effect is dependent upon the radius

of the aerosols, and is particularly severe near the surface of the

earth, where the largest aerosols are to be found.

6. Absorption Waves occur because an ionized plasma is highly

absorbing, and can propagate towards the source of laser light

at intensities below those necessary for plasma production

through breakdown. There are two modes of plasma propagation: supersonic (detonation), driven by shock heating, and subsonic (combustion), driven by thermal conduction and radiation

transport. Figure 3–60 shows the threshold for detonation

waves as a function of ambient density, frequency, and beam

167

Lasers
168

Nonlinea

Wavelength

Radius/Aperture

Pulse Width

Intensity

Linear Effects

Ga

Breakd

------

shorte

lowe

dε/d

Stimulated

Scattering

---

Shorter

lowers gain

---

longer λ

reduces gain

Thermal

Blooming

Same as

absorption

largerlowers ND

Shorterreduce w/a

Lower

Turbulence/

Adaptive

Optics

------

---

longer λ

preferablereduces ro

Absorption/

Scattering

-aerosols

---------

longer λ

preferable2πa/λ less

Absorption/

Scattering

-molecules

Pick λ in a

"window,"

and not on

a molecular

line

---------

Lower Lower Low

Power --- --- ---

Table 3-3. Implications of Propagation Effects in t
radius. Figure 3–61 shows the velocities at which combustion

waves are predicted to propagate as a function of laser intensity.

If naturally occurring winds exceed these velocities, they will

blow out the combustion wave, and it will cease to exist.

7. The dynamic nature of nonlinear effects makes them difficult

to predict and account for in advance. Therefore, one must either compensate for them in real time, as in using adaptive optics to handle thermal blooming, or operate at intensities below

their threshold for occurrence, as with air breakdown. In either

event, it is important to recognize the limitations which these

effects have on the laser parameters which may be available for

use in achieving damage criteria and target effects.

Implications

If there is a single conclusion to be drawn from this section on

propagation in the atmosphere, it’s that there are many more variables that enter into placing energy on target than the D/ which

was the key ingredient in vacuum propagation. Additionally, the

different effects that are possible can drive beam design in different directions. For example, the gain for stimulated scattering is

less at longer wavelengths, but the air breakdown threshold is less

at higher wavelengths. It is therefore not obvious without more

detailed analysis what general wavelength region may be desired

in a given application, let alone the specific wavelength to be chosen within a region. The same is true of other parameters, such as

the beam’s radius, pulse width, power, and intensity. They are all

interrelated, and one effect can’t be dealt with in isolation from the

others. Table 3–3 is suggestive of the way in which the different effects considered in this section drive beam design. Any such summary is of necessity less than complete, but it is hoped that it can

at least help you to keep in mind potential interactions among effects, rather than dealing with single effects in isolation. Finally, of

course, all of these things must be judged by the constraints they

place on damaging targets, which is the subject of the next section.

Laser-Target Interaction and Effects

To this point, our only consideration of target effects has been

to assume that something on the order of 10,000 J/cm2, delivered

169

Lasers
in a sufficiently short time, would be adequate to damage most

targets. We must now be more quantitative, and consider in

greater detail the types of effects which can occur and the criteria

for their occurrence. We’ll begin with a brief survey of the effects

considered in this section. Broadly speaking, target effects may

be classified as thermal—resulting from the heating of the target,

or mechanical—resulting from pressure and impulse delivered to

the target through the reaction forces of laser-produced vapor.

Many of these effects occur above the intensity at which plasmas

may form and propagate as laser-supported combustion or detonation waves. Therefore, we’ll need to consider the effect of

plasma formation and propagation on laser effects, both in vacuum and in the atmosphere.

Types of Effects

The simplest effect is heating. Some fraction of the laser light incident upon a target will be absorbed in a thin layer (1/K  10–4 –

10–8 cm) near its surface. The absorbed energy will appear as heat.

Since heating begins as soon as a laser engages a target, there is no

threshold for this effect. However, heating will by itself be insufficient to damage a target of military interest unless it is very soft.

There are, however, industrial applications where heating is the

desired effect, such as annealing semiconductor materials into

which ions have been implanted by bombardment.93

As a laser’s intensity on target rises, it can begin to melt a hole.

There is a threshold for this effect, since energy must be delivered

to the target more rapidly than thermal conduction or radiation

can carry it away. Once melting begins, it is possible to melt

through the target’s surface, drilling a hole. There is a threshold

for this as well. The laser must engage the target for a pulse width

or dwell time greater than the target thickness divided by the rate

of hole growth.

At still greater intensities, vaporization becomes possible.

Above the threshold for vaporization, the reaction force which

the jet of emerging vapor exerts on the surface delivers a mechanical impulse. This impulse may deform or damage the target, and may work together with target heating to produce levels

of damage that either heating or impulse could not have

achieved alone.

Effects of Directed Energy Weapons

170
Effects in the Absence of Plasma

Melting. When the surface of a target is heated, the energy

deposited will propagate into the surface a distance of about Dt

in time t, where D is the thermal diffusivity of the target material

(see Figure 1–5). We can use this result to estimate the threshold

for melting. A laser of intensity S and radius w will deposit an

amount of energy E  w2 St into the target in time t, where  is

the thermal coupling coefficient—the fraction of the incident laser

energy that is absorbed.94 During this same time, the energy absorbed will have propagated a distance Dt into the target, as

shown in Figure 3–62. Thus, the mass of target material which has

been heated is M w2 Dt, where  is the density (g/cm3) of

the target material. We can use the target’s heat capacity C to find

the degree of temperature rise T above the target’s initial temperature To which results from the energy deposited in a time t:

E  MC T, or  w2  St   w2 DT C T.

The target surface can begin to melt when T  Tm – To, and

the heated region reaches Tm, the melting point of the target

material. This must happen in a time less than the duration

of the laser pulse, tp. Therefore, the threshold for melting is

Sm  C(Tm – To)(D/tp)l/2/. Figure 3–63 shows this threshold

as a function of pulse width and thermal coupling coefficient.

171

Lasers

W Laser

Light,

Intensity

S

Dt

ρπw Dt 2

Mass Heated =

Target Material

Air or Vacuum

αS absorbed, (1- α) S reflected

Energy Deposited in

time t =

πw2αSt

Figure 3-62. Target Heating by a Laser Beam
A number of features are apparent in Figure 3–63. First, you can

see that the thermal coupling coefficient  plays a big role in

melting (and, indeed, in all target effects). Typically,  ranges

from a few percent in the infrared to a few tens of percent at visible wavelengths. Second, as the pulse width gets longer, the

melting threshold decreases, falling by one order of magnitude if

tp increases by two orders of magnitude. This would imply that

for a CW (continuous wave) laser with an essentially infinite

pulse width, the melting threshold could be made arbitrarily

small. There are both practical and physical reasons why this

can’t be done. In military applications, few targets are likely to be

in view or to sit idly by as they are irradiated for times in excess

of, say, ten seconds. More fundamentally, as the intensity on target gets smaller and the heating becomes more gentle there

are other energy loss mechanisms, such as convection and the

reradiation of energy, that come into play. These are discussed in

Chapter l, and will typically limit interaction times to seconds or

less. Therefore, Figure 3–63 implies that if the damage criterion

is to initiate melting on a target’s surface, intensities in excess of

1 kW/cm2 will be required.95

The threshold for melting shown in Figure 3–63 is the intensity

at which the target just begins to melt. In military applications,

this is usually not sufficient for damage—we’d like to drill a hole

through the target. Therefore, we need to determine how rapidly a

Effects of Directed Energy Weapons

172

α = 0.01

0.03

0.01

0.3

10-7 10-5 10-3 10-1 10

109

108

107

106

105

104

103

Pulse Width (sec)

Intensity

(w/cm2)

Figure 3-63. Threshold for Melting vs Pulse Width and Absorptivity
hole will grow, once we’ve exceeded the threshold for melting.

The rate at which a hole increases in depth is known as the target’s

erosion rate (cm/sec).

There are two assumptions we could make in calculating a target’s erosion rate. If we assume that all molten material is somehow flushed from the hole, then fresh material will be exposed to

the laser light, melt, and in turn be flushed out, so that the depth

of the hole increases in time. If we assume that molten material remains in the hole, it must continue to heat to the point where it is

vaporized if it is to be removed. This requires much more energy.

Therefore, these two assumptions will result in upper and lower

bounds to the erosion rate. In practical engagements, it’s likely

that some of the molten material will be removed, and some vaporized.

Molten material might be removed from a hole through the

force of gravity or, if the hole is shallow enough, through the

flushing effect of a wind across the target’s surface. A relatively

simple argument, based on conservation of energy, is sufficient to

estimate the erosion rate.96 Figure 3–64 illustrates a hole of depth

 (cm), growing at a velocity Vm (cm/sec).

In a small time dt, an amount of mass dM  w2 Vm dt is

removed from the hole, where  is the density of the target material, and w the beam radius. The amount of energy which must be

supplied to melt this material is dE  [Lm + C(Tm–To)]dM, where

Lm is the heat of fusion, C the target’s heat capacity, and Tm its

melting temperature (see Chapter l). The amount of energy absorbed from the laser during this same time period is dE   w2

S dt. At a constant rate of erosion, energy is supplied from the

laser at the same rate that it is carried off by the molten material,

173

Lasers

X

V

S W

Molten Material Being Removed

Air or Vacuum

Target Material

Figure 3-64. Hole Erosion with Molten Material Removed
and these two energies are equal. Equating them and solving for

Vm, we can easily see that the rate at which a laser of intensity S

will erode a surface whose thermal coupling coefficient is  will be

Vm  S/[Lm + C(Tm – To)]. Figure 3–65 is a plot of this erosion

rate as a function of intensity and absorptivity.97

As you can see, both from the expression for the erosion rate

and from Figure 3–65, the hole grows at a rate proportional to the

absorbed laser intensity. Figure 3–65 can be used to estimate the

intensities necessary to penetrate a target of a given thickness in a

given time. Suppose, for example, that our goal is to penetrate a

target of thickness 0.1 cm in less than 0.1 seconds. As shown by

the lines on the figure, this requires an erosion rate greater than 1

cm/sec, and an intensity greater than 3  104 W/cm2 if the target’s

absorptivity is 0.1.

Vaporization. If molten material is not removed from the hole,

we have no choice but to vaporize it. This requires that the intensity exceed the threshold Sv for vaporization. This threshold is

Sv  [C(Tv – To) + Lm] (D/tp)l/2/, and the rate Vv with which a

vaporizing surface erodes is Vv  S/(Lm + Lv + C(Tv – To)] 

S/Lv. This threshold and erosion rate are shown in Figures

3–66 and 3–67.98 You can see from these figures that required

intensities are about an order of magnitude greater, and the

Effects of Directed Energy Weapons

174

α = 0.3

0.1

0.03

0.01

Erosion

Rate

(cm/sec)

Intensity (W/cm2)

104

102

 1

10-2

103 104 105 106 107 108

Figure 3-65. Target Erosion Rate with Molten Material Removed
erosion rates about an order of magnitude slower, if target material must be vaporized. This reflects the fact that heats of vaporization are about an order of magnitude greater than heats of

fusion (see Table 1–1).

In using Figures 3–64 through 3–67 to estimate requirements for

target damage in realistic scenarios, you should keep in mind that

realistic erosion rates will probably lie between the extremes of

Figures 3–65 and 3–67, and that the intensities given in these figures

are the intensities at the target surface. In general, propagation

losses will require that the laser fire with much greater intensities, in

order to hit the target with the intensity necessary for damage.

Mechanical Effects. Mechanical effects result when momentum

is transferred to a target by vapor shooting from it. In effect, the

vapor serves as a small jet, and exerts a reaction force back on the

target. If sufficiently intense, this force can deform the target, or

punch a hole through it without the necessity of physically vaporizing and removing the bulk of the material. Thus, mechanical effects may require less energy to damage a target than thermal

effects. That’s the good news. The bad news is that higher intensities are required, and may take the beam above the threshold for

nonlinear propagation effects. The threshold for mechanical effects

is the threshold for vaporization. If we are to go further and quantify the magnitude of effect to be anticipated at a given level of

175

Lasers

Pulse Width, Sec

Intensity

(W/cm2)

α = 0.3

 0.1

 0.03

 0.01

10-7 10-5 10-3 10-1 10

1010

108

106

104

Figure 3-66. Threshold for Vaporization vs Pulse Width and Absorptivity
intensity, we must first ask what pressure the surface feels at a

given vaporization rate. Then we can consider the effect of that

pressure in terms of the target’s response.

It is relatively straightforward to show99 that the pressure

exerted by the vapor streaming from a target as a result of its irradiation by a laser of intensity S is approximately 

P  (kTv/M)l/2(S/Lv), 

where Tv and Lv are the target’s vaporization temperature and

heat of vaporization, M is the mass of a vapor molecule, and  is

the target’s thermal coupling coefficient. The ratio P/S 

(kTv/M)l/2(/Lv) depends only on parameters for a given target,

and is known as the specific impulse, I*, for the target material. The

parameter I*derives its name from the fact that it’s also the ratio of

the impulse delivered to the target (I   w2 Ptp) to the total energy in the laser pulse (E  w2 Stp). You may recall from Chapter

2 that impulse and pressure are two parameters related to target

damage by kinetic energy weapons, which also rely on mechanical

effects for damage. The specific impulse is on the order of 1–10

dyne sec/J for most materials.100 Figure 3–68 shows the pressure

on a target surface as a function of intensity and specific impulse.101 Intuitively, pressures in excess of 10 atm will be needed to

damage a target with any reasonable degree of strength. Therefore, Figure 3–68 indicates that intensities in excess of 107 – 108

Effects of Directed Energy Weapons

176

Erosion

Rate

(cm/sec)

Intensity (W/cm2)

105 106 107 108

102

 1

104

α = 0.3

 0.1

 0.03

 0.01

104 109

10-2

Figure 3-67. Erosion Rate for Target Vaporization
W/cm2 will be needed to achieve damage through mechanical effects. Our next task is to be more quantitative in estimating the intensities and pulse widths necessary for mechanical damage.

How much pressure and impulse are required to damage a target? The answer to this question revolves around the definition of

damage. In some applications, it may be adequate to deform the

target. In others, it may be necessary to punch a hole through it.

What will happen when a given pressure is applied to a target

over a given area for a given amount of time is highly dependent

upon the nature of the target and its construction. Results may be

quite different, for example, if there is a structural support below

the irradiated area than if there is not. Quantitative analysis for

specific targets is beyond the scope of this book. We can only treat

the target in a generic sense as a slab of material with some given

thickness. However, just as in Chapter 2, we can outline some of

the main features of target damage by mechanical means, and

examine how damage might scale with intensity and pulse width.

Figure 3–69 shows the physical situation when a laser beam of

intensity S engages a target of thickness d.

After a brief time delay vaporization begins, and the emerging

vapor exerts a pressure P  I* S and an impulse I  w2 I* Stp to

the irradiated area, where I* is the specific impulse for the target

177

Lasers

Pressure

(Atm)

Intensity (W/cm2)

103

107 10

1

0.1

10-1

Figure 3-68. Pressure on a Target vs Intensity and Specific Impulse

 10

105

106 107 108 109 1010 1011 1012

Specific Impulse (dyn sec /J)
material. The impulse delivered transfers momentum Mv to a

portion of the target whose mass is M  w2d, where  is the

density (g/cm3) of the target material. This momentum drives it

down with a velocity v  I/M. The target therefore receives a

blow of energy Mv2/2  I2/2M. However, the irradiated portion

of the target can’t just fly away. It’s connected to the rest of the target, and there are internal forces which seek to hold it together.

Damage will be achieved when the energy in the blow from the

laser exceeds the strength of the bonds which maintain the target’s

shape, and it deforms or ruptures.102

How much energy will be required for a damaging blow? In a

solid, the forces that seek to deform it are known as stresses, and

the stretching or compression that results from these stresses

are known as strains. You will recall from Chapter 2 that stress

has the same units (J/cm3 or Nt/cm2) as pressure, and is analogous to pressure in a gas. Strain is the fractional change in volume, e  V/V, which results from an applied stress. Just as

the work done when a pressure P causes a gas to expand from a

volume V to V + V is P V, the work done when a stress P

causes a strain e is VPe, where V is the volume of material which

has been stressed.103 The energy necessary to strain the solid to

the point where it damages is VP*e*, where e* is the amount of

strain necessary for damage, and P* is the stress needed to exceed that strain.104

Effects of Directed Energy Weapons

178

Beam

Radius, w Laser Light

Intensity S

Emerging Vapor,

Pressure P = I*S

d

Target Thickness

Target Motion, velocity v

Figure 3-69. Mechanical Damage of a Target by a Laser
The criterion for damage is that the energy in the laser’s impulsive blow, I2/2M, equal or exceed the energy needed to strain the

target to damage, VP*e*. Using I   w2 I* Stp, M  [w2d], and

V   w2d, it’s easy to show that a laser of intensity S and duration tp will damage a target if Stp > (2P*e*) l/2 d/I*. Figure 3–70 is a

plot of the intensity, S, necessary to damage a target as a function

of pulse width and target thickness for I*  3 dyn sec/J.105 You can

see from the figure that intensities of at least 107 W/cm2 and

fluences on the order of 103 – 104 J/cm2 are required to damage

targets of reasonable thickness.

Energy Requirements for Damage. It’s interesting to compare

the fluences shown in Figure 3–70 with those necessary to melt or

vaporize a hole through targets of similar thickness. The rate of

erosion for a target is between Vm  S/[Lm + C(Tm – To)], and Vv

 S/Lv. These erosion rates can be divided into a target’s thickness d to determine the time required for penetration. This time

can be multiplied by the intensity to find the fluence necessary

for penetration. These fluences are Fm  d[Lm + C(Tm – To)]/

179

Lasers

●

●

●

●

●

●

●

▼

▼

▼

▼

▼

▼

▼

●●

▼ Fluence (J/cm2)

1 cm

0.1

0.01

Target Thickness

1.7 x 104

1.7 x 103

1.7 x 102

1014

1012

1010

108

106

10-9 10-8 10-7 10-6 10-5 10-4 10-3

Intensity

(w/cm2)

Pulse Width (sec)

Figure 3-70. Intensity for Mechanical Damage vs Pulse Width and

Target Thickness
and Fv  dLv/ for target melting and vaporization, respectively. Figure 3–71 is a comparison of these fluences for a

coupling coefficient  of 0.1 with the fluence necessary for

mechanical damage, Fd  (2P*e*)l/2 d/I*. You can see that it may

be more energy–efficient to damage a target of a given thickness

by mechanical means, rather than by chewing all the way

through it. The reason is simple—in mechanical damage. Target

material is not physically removed, it’s just the target’s structure

that’s deformed or ruptured. Of course, the relative positions of

the lines shown in Figure 3–71 may change in any specific case,

depending on the coupling coefficients, target construction, and

degree of thermal or mechanical damage required.

If it’s energetically more efficient to damage targets mechanically, why is there any interest in thermal damage? One reason is

that thermal damage is a relatively sure thing, depending only

on the thermal properties of the target material. By contrast,

mechanical damage thresholds can be greatly affected by details

of the target’s construction. These may not be known with any

precision. Another reason is because of the time scales and intensities involved. If we are to damage a target as shown in Figure

3–69, the impulse must be delivered in a time shorter than the

stresses can relax in a radial direction and be spread out over the

target. This time is on the order of the sound speed in the target

material divided into the beam radius. Sound speeds in solids

Effects of Directed Energy Weapons

180

Figure 3-71. Fluence to Damage or Penetrate a Target vs Thickness

Target Thickness

Fluence

(J/cm2)

Vaporization

Melting

Mechanical Damage

1000000

100000

10000

1000

100

0.01 0.03 0.1 0.3 1 3 (cm)
are fairly high, on the order of 106 cm/sec. For a one meter

beam radius, this implies a pulse width of less than 10–4 seconds,

and intensities of about 108 W/cm2 (see Figure 3–70). By contrast,

you can melt through targets with pulse widths on the order of

seconds, and intensities on the order of 104 W/cm2 (see Figure

3–63). At the higher intensities associated with mechanical

damage, plasma effects are much more likely to occur. These can

drastically affect the interaction, and are discussed next.

Summary.

1. The effects that can occur when a laser engages a solid target include heating, melting, vaporization, and momentum

transfer. Heating begins as soon as laser light is absorbed at a

target’s surface. The other effects begin at progressively

higher intensity thresholds.

2. The threshold for melting and the resulting erosion rates

are shown in Figures 3–63 and 3–65. As the pulse width tp is

increased, the threshold drops. Radiation, convection, and

other energy loss processes limit the pulse width to times on

the order of seconds or less.

3. If molten material can’t be flushed from the irradiated region on a target, it must be vaporized. The threshold for vaporization and the resulting erosion rate are shown in Figures

3–66 and 3–67.

4. The reaction force imparted to a surface by the vapor as it

blows off produces a pressure, shown in Figure 3–68. This

pressure can serve to damage the target, provided the impulse

is great enough to rupture the target’s surface. The threshold

for this type of “mechanical” damage is shown in Figure 3–70.

5. The advantage of target erosion (melting or vaporization)

as a damage mechanism is that the effects are well understood

and require lower intensities (W/cm2). Mechanical damage

through impulse delivery requires less fluence (J/cm2), but is

more difficult to characterize, and requires higher intensities

and shorter pulse widths. Plasmas are more likely to occur

and influence the interaction at these higher intensities.

181

Lasers
Effects of Plasmas on Target Interaction

At short enough pulse widths, the intensities required for any of

the effects discussed in the previous section may lie above the

threshold for plasma production, particularly from aerosols. Indeed, the target may be considered an aerosol with a very large

(effectively infinite) radius, having a correspondingly low threshold for plasma initiation (Figure 3–51). It’s possible to imagine

ways in which plasmas might either help or hinder laser-target effects, as suggested in Figure 3–72. 

In Figure 3–72(a), a plasma exists in close contact with a target

surface. Since plasmas are very hot and absorb the laser light well,

the absorption of energy in this plasma and its transfer to the target by thermal conduction or radiation could deposit a greater

fraction of the laser’s energy in the target than if there were no

plasma. The heavy black lines in the figure suggest such a transfer

of energy.

On the other hand, if a plasma propagates away from the target,

it will continue to absorb almost all of the laser light, but very little

energy will find its way into the target. This is shown in Figure

3–72(b). Clearly, plasmas may be good or bad from the standpoint

of laser effects. Whether they are good or bad will depend upon

the laser intensity, which determines the type of plasma and its

mode of propagation, and the pulse width, which determines how

far a plasma can travel as the beam engages the target. In this secEffects of Directed Energy Weapons

182

Figure 3-72. Plasmas Helping and Hindering Laser-Target Coupling

(a) Plasmas near the target

 can aid in coupling energy

(b) Plasmas separated from the

 target decoupled the laser

 from it
tion, we’ll consider the effect of laser-supported detonation and

combustion waves on both thermal and mechanical effects in air

and vacuum.

Plasma Effects in a Vacuum. At first glance, you might question the need for this topic, since a vacuum contains no molecules

to ionize into a plasma. Nevertheless, even in a vacuum, plasmas

can occur if vapor emerging from the target becomes ionized. Of

course, a plasma formed in this way can’t completely shield the

target from the laser. If it did, vaporization would cease and the

plasma would disperse. The analysis of target coupling in the presence of a vapor plasma in vacuum requires a self-consistent theory

that accounts for energy reaching the target either as laser light

which has penetrated the plasma, or as reradiation or thermal conduction from within the plasma. Figure 3–73 summarizes the results of such a calculation.106

In the calculation represented in Figure 3–73, it was assumed

that a fraction p of the radiation absorbed in ionized vapor was

coupled back into the surface, aiding its erosion and augmenting

the fraction c of the laser radiation absorbed at the target surface.

The effective coupling coefficient is p f + c (l–f), where f is

the fraction of the laser light absorbed within the vapor plasma. If

p is small compared to c, the effective coupling coefficient behaves like the lowest curve in Figure 3–73. It is always less than or

equal to c, and as the intensity rises and the rate of erosion in183

Lasers

Effective

Coupling

Coefficient

Intensity, S (w/cm2)

3 x 104 3 x 105 105 106

10-2

10-1



αp = 0.25

αp = 0.5

αp = 0.1

αp = 0

Figure 3-73. Laser - Target Coupling with Ionized Vapor in a Vacuum
creases, there is a gradual decline in , roughly as S–2/3. By contrast, if p is large compared to c, the effective coupling coefficient rises with intensity to a value approximately equal to p.

This behavior can be related to two self-consistent ways in which

targets can vaporize in the presence of a plasma. If reradiation from

the plasma contributes little to target erosion, then as S increases

the additional energy accelerates the vapor, leaving its density unchanged. This is because if the vapor density were to increase, the

plasma would become more absorbing, target erosion would decrease, and the increase in density could not be maintained. On the

other hand, if reradiation from the plasma is most important in target vaporization, an increase in S will increase the vapor density.

That’s because in this case an increase in density is beneficial, serving to increase the rate of erosion. Therefore, the effective coupling

of a laser to a target at intensities where ionization can occur in the

evolving vapor will depend critically on the degree of coupling to

the bare metal and how this compares with the absorption of radiation emitted from the ionized vapor. As a general rule, it’s clear

from Figure 3–73 that if a plasma is created in the vapor emerging

from a target in a vacuum there is a strong potential for enhanced

coupling, and in the worst case there will be at most a small decline

in the effective coupling coefficient. From Figure 3–16, you can see

that the coupling of laser light to metals is least for long wavelength infrared radiation. Therefore, the potential for enhanced

coupling through plasma ignition is greatest at long wavelengths.

Experimentally, increases in  by factors of 3–10 have been seen

upon plasma ignition with infrared lasers.107

Since plasma ignition can affect the thermal coupling coefficient ,

it will affect the rate of erosion and the specific impulse I* as well.

The momentum transferred to the target by the vapor is on the order

of v2, where  is the vapor density and v its velocity: the energy it

carries away is on the order of v2. Thus, the ratio of momentum to

energy is proportional to l/v. Therefore, in the case where p is small

compared to c, and increased intensity increases the vapor’s velocity, plasma ignition results in a gradual decline in I* as S–1/3.

108 In the

case where p is large and there is enhanced coupling, the fact that

an increased intensity primarily increases the vapor’s density means

that it should be enhanced along with the effective coupling coefficient, and be roughly independent of intensity.

Effects of Directed Energy Weapons

184
In summary, when plasmas are produced over a target surface

in vacuum, the result is on average beneficial for laser effects. Both

the thermal coupling coefficient and specific impulse may increase, and in any case they won’t decline precipitously. The logical next step is to ask what can happen when plasmas are produced within the atmosphere.

Plasma Effects on Coupling in the Atmosphere. In the atmosphere, any plasmas which are created at the target surface may

propagate away from the target as an absorption wave, decoupling

the radiation from the target. Thus, plasmas could have a much

more profound effect on  and I* in the air than in a vacuum.

Quantitative analysis is quite involved, and must include the effect

of light transmitted through the plasma, light reradiated from the

plasma to the target, and the propagation characteristics of the

plasma as a function of laser intensity. Figure 3–74 is a good example of the type of calculations that have been performed, and illustrates many of the features exhibited by analyses of this type.109

The graphs shown in Figure 3–74 show the evolution in time of

three quantities—the intensity of radiation being deposited in the

target (Q) the integral of Q over time, and the instantaneous thermal coupling coefficient, .

110 The intensity Q includes both laser

light and plasma radiation which reach the surface and are absorbed there. The two graphs correspond to two different laser

intensities, S. The first of these, 106 W/cm2, supports plasma

propagation as a laser supported combustion (LSC), while the second, 107 W/cm2, supports propagation as a laser-supported detonation (LSD). In the upper graph on Figure 3–74, we see that when

an LSC is ignited  is initially very high. This reflects the close

proximity of the plasma to the target and its effectiveness in

absorbing laser energy and reradiating it to the target. As the LSC

moves away from the target,  decreases and remains relatively

constant at about 10% for a fairly long time. This is because the

LSC plasma is moving slowly, and does not begin to decouple the

radiation from the target until about 15 sec. The point at which

decoupling begins is evident as Q begins to fall off and the integral of Q over time becomes constant.

In the lower half of the figure, you can see how differently

things behave when an LSD is ignited. The plasma propagates

rapidly away from the target surface, Q falls to zero, and  drops

185

Lasers
to a very low value—less than 1%. (Note that the scale for  in the

lower graph is an order of magnitude less than that in the upper

graph.) From these and similar calculations, it can be concluded

that plasma ignition in the atmosphere will only serve to enhance

thermal coupling if the laser intensity lies below the threshold for

LSD propagation, and then only if the laser pulse width is sufficiently short that LSC propagation can’t ultimately decouple the

beam from the target as well. 

Next let’s turn our attention to the influence of plasmas on mechanical coupling in the atmosphere. You will recall that the pressure in an LSD can be quite high, on the order of 10–100 atmosEffects of Directed Energy Weapons

186

S = 106 W/cm2

ALUMINUM TARGET

”Qdt

”Qdt (J/cm2)

α

Q (105 w/cm2)

Coupling Coefficirnt (%)

10

 8

 6

 4

 2

 0

 0 10 20

 2

 0

100

80

60

40

20

 0

Q

TIME (μ sec)

S = 107 W/cm2

ALUMINUM TARGET

”Qdt

”Qdt (J/cm2)

α

Q (105 w/cm2)

Coupling Coefficirnt (%)

10

 8

 6

 4

 2

 0

 0 10 20

 2

 0

10

8

6

4

2

0

Q

TIME (μ sec)

Figure 3-74. Thermal Coupling with Plasmas in the Atmosphere
pheres. This high pressure, as it relaxes towards equilibrium with

its surroundings, will transfer some momentum and impulse to a

target over and above that which the target receives as a result of

vaporization. Theory and experiment are consistent in demonstrating such an effect.

A relatively simple model, illustrated in Figure 3–75, can be

used to capture the physics of momentum transfer in the presence

of detonation waves.

Figure 3–75 shows an LSD propagating away from a target. As

the thin, absorbing shock front moves forward, it leaves behind hot,

high-pressure gases. These gases are initially at the high pressure Po

associated with an LSD wave. Since the LSD propagates at a supersonic velocity, u, the hot gases form what is effectively a long cylinder, and expand radially to pressure balance as a cylindrical wave187

Lasers

HOT Pressure

= Po u

W

R

u

W

P = P0 (R/w)2

Figure 3-75. Momentum Transfer from an LSD

RT

RT
front. As the gases expand from their initial radius, w (the beam radius), to some greater radius, R, their pressure is reduced by a factor

of (w/R)2. This is entirely analogous to the expansion of the high

pressure gases resulting from the detonation of a bomb (Chapter l),

where the pressure falls off as 1/R3 due to its three-dimensional

(spherical) expansion. In this case, the pressure falls off as l/R2 because the expansion is two dimensional (cylindrical).111

The expanding gases exert a pressure on the target surface. The

force exerted on the target is a constant, even though the pressure

is decreasing as the gases expand. This is because force  pressure

 area, and while the pressure is decreasing as 1/R2, the area over

which it is applied is increasing as R2, so that the product of these

two quantities is constant. Therefore, the impulse delivered to the

target, which is the integral of force over time, increases linearly

with time, until the expanding gases either reach the edge of the

target at a radius RT and relax around it, or the pressure in them

decreases to the point where it equals the surrounding atmospheric pressure, and expansion ceases. An interesting consequence of this is that for small targets, the impulse delivered

should scale with the target area, rather than the beam area.112 On

the other hand, for large targets we’d expect the impulse delivered

to be independent of target size.

What are the implications of these results from the standpoint of

target damage? Both pressure and impulse are important in establishing damage thresholds. In the presence of detonation waves, a

large measured impulse may reflect a low pressure applied over a

large area, as opposed to a large pressure being applied over the

irradiated area. Because the energy transferred to a target area of

mass M by an impulse I is I2/2M you can see that as the area and

mass over which the pressure is applied increase, the energy and

stress delivered to the target decrease. Accordingly, measured data

on the specific impulse I* in the presence of detonation waves

must be carefully interpreted before drawing conclusions from

them relative to the likelihood of exceeding damage thresholds.

Summary.

1. There are mechanisms through which plasmas can both help

and hinder either thermal or mechanical effects. The help arises

from the fact that plasmas are generally more strongly absorbEffects of Directed Energy Weapons

188
ing than the bare target surface, and can serve as an efficient

means of energy transfer to the surface. The hindrance arises

from the fact that plasmas may propagate away from the target

surface, decoupling the absorbed energy from it.

2. In a vacuum, plasma ignition within the evolving target

vapor generally serves to enhance thermal coupling, particularly when the intrinsic absorptivity at the target surface is low. 

3. In a vacuum, plasma ignition will enhance momentum transfer to the extent that increased thermal coupling enhances the

target ablation rate. 

4. In the atmosphere, plasma ignition will enhance thermal

coupling at intensities below the threshold for detonation wave

propagation and pulse widths too short to permit combustion

wives to propagate substantial distances.

5. In the atmosphere, plasma ignition below the threshold for

LSD propagation will enhance momentum transfer to the extent

that thermal coupling and target ablation are enhanced. Above

the threshold for LSD propagation, target erosion ceases but

momentum is transferred as high pressure gases left behind the

advancing LSD wave front relax to pressure balance. Substantial impulse may be delivered in this way, but it is qualitatively

different from that delivered in a vacuum and results from

lower pressures applied over larger areas.

Summary of Main Concepts. We have covered considerable

territory in this chapter. Let’s summarize the main ideas we have

presented.

1. Lasers are intense sources of electromagnetic radiation, with

wavelengths from about 10 to 0.4 m and frequencies from

about 3  1013 to 8 x1014 Hz.

2. The materials with which lasers might interact are characterized by an index of refraction, n, and an attenuation coefficient,

K. When light passes regions of different n, it is bent according

to the law of refraction. This can occur either deliberately, in

lenses, or inadvertantly, since density fluctuations in the atmosphere are accompanied by fluctuations in n. When light propagates a distance z through a region whose attenuation coefficient is K, its intensity is decreased by a factor exp(–Kz).

189

Lasers
3. A laser of wavelength  emerging from an aperture of

diameter D can propagate a distance on the order of D2/  as a

collimated beam. Beyond this distance, it will diverge at an

angle   /D through diffraction.

4. Decreases in intensity resulting both from diffraction and attenuation will reduce the fraction of a beam’s energy which can

be brought to bear on a target. Beam parameters which may be

adjusted to compensate for these effects and enable the delivery

of damaging intensities to a target include the energy, pulse

width, wavelength, and diameter of the beam.

5. In the atmosphere, K is highly wavelength dependent,

containing contributions from absorption and scattering from

both molecules and particles (aerosols). If a beam becomes too

intense, free electrons in the atmosphere will multiply and

the air will break down, forming an ionized plasma which

will absorb the beam. Following breakdown, plasmas can

propagate towards the source of laser light as combustion or

detonation waves.

6. In the atmosphere, n can vary through turbulence or through

expansion induced by the absorption of laser light. The second

effect results in beam expansion (thermal blooming) or bending.

These effects must generally be compensated for in real time

through adaptive optics.

7. When laser light encounters a target, a fraction of the light

is absorbed in the target surface, and appears as heat. Thresholds for melting and vaporization are established by the criterion that energy be deposited so rapidly that it cannot be

carried away within the pulse width of the laser. Targets can

be damaged either through the erosion which results from

melting or vaporization (thermal damage), or through the

momentum transferred to the target surface by an evolving

vapor jet (mechanical damage).

8. The ignition of plasmas at a target surface, and their subsequent propagation as detonation or combustion waves, can

greatly affect the thermal and mechanical coupling of a laser to

a target, either in air or vacuum. In a vacuum, coupling will

most likely be enhanced, while in the air it will most likely be

degraded.

Effects of Directed Energy Weapons

190
Implications

Having seen how lasers interact with targets, what can we conclude regarding the optimal intensity and pulse width for target

damage? Any such optimum needs to consider propagation as

well as interaction effects, since these work together to constrain

the operating parameters which are available. Figure 3–76 is an attempt to draw together and plot on a single graph some of the

main effects considered in the last two sections.

Shown on Figure 3–76 are the thresholds for melting, vaporization, and mechanical damage which we developed in this section,

together with approximate thresholds for thermal blooming, stimulated Raman scattering (SRS), and plasma production. All of the

lines on the figure should be considered only approximate, of

course, because they will shift up or down with different assumptions regarding range, frequency, coupling coefficients, and so on.

However, you can see that those shown are reasonably representative by comparing the lines in the figure with the more detailed

figures presented earlier for each effect.

191

Lasers

Figure 3-76. Significant Propagation and Target Interaction Effects

Intensity, W/cm2

Mechanical Damage

Plasmas

SRS

Thermal Blooming

Melting

Vaporization

1010

108

106

104

10-8 10-6 10-4 10-2 1 Pulse

(sec)
You can see from Figure 3–76 that there is very little opportunity to damage targets in the atmosphere without operating at intensities where potentially deleterious propagation effects must

be dealt with. Even melting through targets in times less than seconds will likely require dealing with thermal blooming, and if

mechanical damage is contemplated the full range of propagation

effects could constrain the interaction between a laser and its target. Because of these potential constraints, it is not yet clear which

approach is best for damaging targets—longer pulses of lower intensity or shorter pulses of higher intensity. Lower intensity

phenomena are easier to understand and model, and are often

favored for this reason alone. However, the interaction times can

become so long that it may not be possible to deal with multiple

targets in stressing scenarios. Higher intensities and shorter

pulses permit the rapid engagement of targets, and may even be

more energy efficient, but are more difficult to model in a reliable

way, and are possibly more challenging from the standpoint of

device construction.

Throughout this chapter, the analysis has been kept simple by

assuming that the laser is a pure device, putting out a single intensity S for a given time period tp. While this is useful from the

standpoint of understanding interaction and propagation phenomena, there is no reason why this needs to be so. It might be

possible, for example, to obtain enhanced coupling with a pulse

that starts off with a high intensity to insure plasma ignition, and

then drops down to a lower intensity where the plasma can be

maintained in close proximity to the target. Thermal blooming

might be used to advantage, with a long duration, low intensity

pulse used to develop an underdense channel with an elevated air

breakdown threshold, followed by a short, high intensity pulse

which damages the target. Combination of low and high intensity

pulses could alternately melt target material and flush it away

with mechanical pressure. In short, there is almost no limit to combinations of effects that might be considered as an aid in achieving

target damage, and the material presented here needs to be

viewed with an eye toward the possibilities, as well as the limitations that it represents.

In realistic engagements, it must also be recognized that true

damage means more than drilling holes or buckling plates. There

Effects of Directed Energy Weapons

192
are many cars on the road that attest to the fact that holes and

dents do not necessarily prevent a target from functioning. And a

sailboat will continue to function if someone shoots a small hole in

the sail, but will be dead in the water if a wind of much lower

pressure rips the sail or snaps the mast. Therefore, a serious tradeoff among different damage mechanisms needs to account for the

specific vulnerabilities of the intended target. The material we

have presented can only serve as a general guide in contemplating

different damage mechanisms.

It is also possible to trade the constraints of physics for the constraints of engineering. Because of the many constraints that the atmosphere places on laser propagation and target interaction, interest has arisen in space-based lasers as weapons. In the vacuum of

space, the only potential effect would be the ignition of plasmas at a

target’s surface, and this could actually improve the coupling of energy to it. Another possibility for attacking targets in space would

be to use ground based lasers with very large apertures, actually focusing the beam so that its greatest intensity is not achieved until it

has exited the densest portion of the atmosphere. Solutions like

these trade physics problems for engineering problems, which may

be more amenable to ingenuity and capital investment.

193

Lasers
Notes and References

1. A good general discussion of the principles of laser operation

and of some of the more common types of lasers can be found in

Bela A. Lengyel, Lasers,2nd ed. (New York: Wiley Interscience,

1971. Jeff Hecht, The Laser Guidebook (New York: MCGraw-Hill,

1986) is a simpler book, aimed at more general audiences. More detailed discussion of specific laser types can be found in more specialized texts, such as J. D. Anderson, Jr., Gas Dynamic Lasers (New

York: Academic Press, 1976), or T.C. Marshall, Free Electron Lasers

(New York: Macmillan, 1985). Unfortunately, the state of the art in

laser technology is advancing so rapidly that published textbooks

are generally inadequate for the purpose of obtaining current performance capabilities and limitations. Information of this sort must

usually be found in conference proceedings and journal articles,

though occasionally a good summary appears in the literature. The

most up-to-date summary available in 1988 was Chapter 3 of the

“Report to the American Physical Society of the Study Group on

Science and Technology of Directed Energy Weapons,” published

in Reviews of Modern Physics vol. 59, pt II (July 1987).

2. John D. Jackson, Classical Electrodynamics (New York: John

Wiley and Sons, 1962) is a good text on electromagnetic theory at

the graduate level.

3. Wave phenomena are discussed in great detail and with minimal mathematical complexities in John R. Pierce, Almost All About

Waves (Cambridge, MA: MIT Press, 1974).

4. The speed of light, c, is a very important parameter in

physics. Einstein’s theory of relativity asserts that the measured

value of c in a vacuum will always be the same, regardless of any

relative motion between the source of radiation and the observer.

In addition, no physical object can exceed the speed of light, so

that electromagnetic radiation is the fastest bullet possible for

weapon use.

5. The Hertz is named for Heinrich Hertz (1857-1894). Hertz

was the first to broadcast and receive radio waves, and to demonstrate that these waves could be reflected and refracted like light.

6. Figure 3–3 has been adapted from Figure 3.13 in Eugene Hecht

and Alfred Zajac, Optics (Reading, MA: Addison-Wesley, 1976).

Effects of Directed Energy Weapons

194
7. The theory of lenses is discussed in any text on optics, such

as Chapter 5 of Hecht and Zajac (note 6).

8. The divergence angle, , of light from a circular aperture is

1.22 /D. For other types of apertures,  is less well defined, since

the dimension of a square aperture, for example, is different along

a diagonal than along a side. In every case, however,  is approximately /D. See Hecht and Zajac (note 6), Chapter 10.

9. The fact that a converging lens can’t focus light to an infinitely small point is another manifestation of the wave nature of

light and its diffraction.

10. See section 3.4 in Joseph T. Verdeyen, Laser Electronics (Englewood Cliffs, NJ: Prentice-Hall, 1981) for a discussion of laser beam

expansion and the concept of a beam’s “spot size.”

11. Propagation within the Rayleigh Range is known as “near field”

propagation, and at greater distances as “far field” propagation.

12. The word quantum comes from the Latin for “how much,”

and refers to the fact that when seen on a fine scale no physical

processes are continuous, but occur in small steps, or quanta.

Quantum mechanics is the mathematical theory that describes

such phenomena. The dual wave and particle nature of light is

discussed in any text on modern physics, such as Paul L.Copeland

and William E. Bennett, Elements of Modern Physics (New York, Oxford University Press, 1961).

13. Interestingly enough, the allowed orbits are those for which

an integral number of particle “wavelengths” will fit around the

path. Just as waves exhibit some of the properties of particles in

quantum mechanics, so also particles exhibit some wavelike properties. See section 4.7 in Arthur Beiser, Concepts of Modern Physics,

3rd ed (New York: MCGraw-Hill, 1981).

14. Figure 3–9 is adapted from Figure 4–20 in Beiser (note 13).

15. If the hydrogen atom is not in its ground state, with its electron in the lowest level, then the photon energies which can be

absorbed are those which will promote the electron from the level

where it resides to some other. For example, an electron in the

10.2 eV level can be excited to the 12.1 eV level by absorbing a 1.9

eV photon. Thus, the absorption properties of a hot, excited gas

will be different from those of a cold gas, and the absorption and

195

Lasers
emission of light from a gas can be used as a measure of its temperature, or degree of excitation.

16. Figure 3–10 is based on Figure 9.28 of Lengyel (note l).

17. Figure 3–11 is based on Figure 3–23 in Robert J. Pressley (ed),

CRC Handbook of Lasers (Cleveland, OH: Chemical Rubber Co, 1971).

18. The band structure of solids and how it relates to their electrical and optical properties is discussed in any text on solid state

physics, such as Chapter 9 of Charles Kittel, Introduction to Solid

State Physics, 3rd ed. (New York: John Wiley and Sons, 1966).

19. Figure 3–13 has been adapted from Figure 3b, p 540, of Kittel

(note 18).

20. The data in Table 3–1 were taken from Table 1, p 302, of Kittel

(note 18).

21. The fact that the electric field must vanish on the interior of a

conductor is a consequence of Maxwell’s equations in the limit

where nothing varies with time, and is discussed in almost any

undergraduate physics text, such as Chapter 28 of David Halliday

and Robert Resnick, Physics, Part II (New York: John Wiley and

Sons, 1967).

22. Shielding electronic components with metal boxes is especially important as a countermeasure against microwave weapons.

See Chapter 4.

23. The plasma frequency is discussed in detail in any text on

plasma physics, and its implication from the standpoint of light

penetrating a conductor is developed in books on electromagnetic

theory. See Jackson (note 2), Chapter 7.

24. This expression for the reflectivity assumes that the skin

depth is small compared to the wavelength of the radiation, which

is the case at all frequencies of interest here and in Chapter 4. See

Chapter 6 of Jerry B. Marion, Classical Electromagnetic Radiation

(New York: Academic Press, 1968).

25. More precisely, the definitions “near” and “far” field relate to

approximations which can be made in diffraction calculations.

Diffraction in the far field is known as Fraunhofer diffraction, and

in the near field as Fresnel diffraction. See Hecht and Zajac (note

6) Chapter 10.

Effects of Directed Energy Weapons

196
26. This definition of solid angle is analogous to the mathematical definition of a plane angle. Let two lines diverge from a point,

and draw a circle about that point. The constant of proportionality

between the arc of the circle the lines cut off and the radius of the

circle is the angle between the lines in radians.

27. The utility of the brightness concept is attested to by the fact

that similar concepts have emerged for almost every form of

beamed energy. See Chapters 4 and 5.

28. The treatment of binary (one-on-one) interactions in terms of

cross sections is common in physics, and will be used throughout

this book. A good discussion of the cross section concept can be

found in Chapter II of M. Mitchner and Charles H. Kruger, Jr., Partially Ionized Gases (New York: Wiley Interscience, 1973).

29. Pierre Bouguer (1698–1758) made some of the earliest measurements of the absorption of light in the atmosphere. Many scientists have corrupted Bouguer’s name to the point where the ansorption law is sometimes referred to as “Beer’s Law.” Johann

Lambert (1728–1777) also studied heat and light, but (1728–1777)

was at heart more of a mathematician. He was the first to prove

that the number  is not a rational number.

30. A good summary of absorption as a function of frequency,

adequate for zero-order analysis, can be found in Section 14, “Optical Properties of the Atmosphere,” in Waiter G. Driscoll, ed.,

Handbook of Optics, (New York: MCGraw-Hill, 1978).

31. The upper portion of Figure 3–24 is based on a figure on

p.115 of R. D. Hudson, Jr., Infrared Systems Engineering (New York:

John Wiley and Sons, 1969). The lower, expanded portion of the

figure is based on Figure 2 in Frederic G. Gebhardt, “High Power

Laser Propagation, “ Applied Optics 15, 1484 (1976). Gebhardt’s

paper is a good summary of many of the phenomena discussed in

this chapter, at a somewhat higher level of technical detail.

32. This example is from Gebhardt (note 31).

33. This result is seen experimentally and may be derived using

statistical mechanics. It assumes that temperature is roughly independent of altitude, and that the acceleration due to gravity is a

constant. Therefore, it is most accurate near the surface of the

earth. Since this is where absorption is the greatest, the “exponential atmosphere” is often adequate for “zero-order” analysis. See

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Lasers
Section 6.3 in F. Reif, Fundamentals of Statistical and Thermal Physics

(New York: MCGraw-Hill, 1965).

34. Since different atmospheric constituents have different molecular weights, they each fall off differently with altitude. The “7

km” value for ho is an average over all constituents. Species whose

weight is lighter than the average will fall off less rapidly, and

those whose weight is heavier than the average will fall off more

rapidly. Since absorption may depend on a single species at a

given wavelength, the exact scale length for absorption may differ

from the nominal value in a specific application.

35. Figure 3–26 is a plot of the expression oz K(z)dz 

[K(0)ho/sin
][1-exp(-zsin
/ho)].

36. Figure 3–27 is adapted from figures found in C. E. Junge, Air

Chemistry and Radioactivity (New York: Academic Press, 1963) and

in J. E. Manson’s article in S.L. Valley (ed), Handbook of Geophysics

and Space Environments (Hanscom AFB: Air Force Cambridge Research Laboratories, 1965).

37. A commonly used mathematical expression for the density of

particles of size r is n(r)  ar exp(-br). The constants a, b, , and

 will vary depending on climate and other conditions. Representative values may be found in Section 3.14 of V. E. Zuev, Laser

Beams in the Atmosphere (New York: Consultant’s Bureau, 1982).

38. All of the gory details of Mie’s theory can be found in Max

Born and Emil Wolf, Principles of Optics, 5th ed. (0xford: Pergamon

Press, 1975).

39. Figure 3–28 is based on Figure 13.14 in Born and Wolf (note 38).

40. A number of representative curves, along with references to

the original literature, can be found in Born and Wolf (note 38).

41. Figure 3-30 is based on data from p. E-373 in Robert C. Weast

(ed) CRC Handbook of Chemistry and Physics, 67th ed. (Boca Raton,

FL: CRC Press, 1987).

42. A good description of the two quantities which affect propagation and its correction through adaptive optics, the coherence

length and isoplanatic angle, can be found in Section 5.4.4 of the

APS Report on Directed Energy Weapons (note 1).

43. A good discussion of the refractive index structure coefficient

and its variation with altitude and time of day can be found in S.F.

Effects of Directed Energy Weapons

198
Clifford, “The Classical Theory of Wave Propagation in a Turbulent

Medium,” Chapter 2 in J. W. Strohbehn (ed), Laser Beam Propagation

in the Atmosphere (Berlin: Springer-Verlag, 1978).

44. Figure 3–33 Is based on Figures 2.1 and 2.4 in Clifford’s article (note 43).

45. See Table 4.1 in Zuev (note 37) for the fluctuations about its

average value which CN undergoes during various times of day.

46. APS Report on Directed Energy Weapons (note 1), section 5.4.4.

47. J. E. Pearson, “Atmospheric Turbulence Compensation Using

Coherent Optical Adaptive Techniques,” Applied Optics 15, 622

(1976.)

48. The time scale for mirror motion is related to the frequency

with which the turbulent environment shifts from one configuration to another, and the degree of motion is related to the wavelength of the light. See Pearson (note 47).

49. There are many nonlinear effects not considered here which

may be important at the intensities and powers appropriate to

other applications, such as laser fusion, or which are of interest

from a scientific standpoint for insight into the structure of matter.

50. Figure 3–40 has been adapted from Gebhardt’s paper (note

31). This figure has become a classic, and can be seen in almost

any discussion of thermal blooming.

51. See P. B. Ulrich, “Numerical Methods in High Power Laser

Propagation,” AGARD Conference Proceedings No. 183, Optical

Propagation in the Atmosphere, Paper No. 31 (27–31 October, 1975).

52. A beam having this intensity profile is known as a gaussian

beam. Many lasers, especially lower-power “research” devices,

have an intensity profile of this shape. See H. Kogelnick and T. Li,

“Laser Beams and Resonators,” Applied Optics 5, 1550 (1966).

53. See Gebhardt (note 31). Figure 3–42 is based on Gebhardt’s

Figure 7.

54. See Section 5.4.8 of the APS Report on Directed Energy Weapons

(note 1).

55. See, for example, Figure 3 in Gebhardt (note 31).

56. In realistic scenarios, of course, the wind is unlikely to be

constant over the whole path, and the beam profile may not be

gaussian. An effective distortion number can nevertheless be

199

Lasers
calculated by integrating over factors which change along the

beam path and over the beam front, much as we did in integrating

K(z) dz to obtain the optical depth when K was not a constant.

The procedure is described in Gebhardt (note 31). The relative intensity varies with distortion number as in Figure 3–42 for a surprisingly broad range of experimental conditions.

57. Zuev (note 37) discusses some of the types of stimulated scattering that can occur in Section 5.8.

58. APS Report on Directed Energy Weapons (note l), section 5.8.

59. Figure 3–46 is based on Figure 5.23 in the APS Report on Directed Energy Weapons (note l).

60. Some of the potential cures for SRS are discussed in Section

5.4.9.4 of the APS Report on Directed Energy Weapons (note 1).

61. Of course, if the electron density were so great that it exceeded

the plasma frequency at the wavelength of the light, the laser would

primarily be reflected, and not absorbed. However, as you can see

from Figure 3–14, laser light at infrared and shorter wavelengths

will exceed the plasma frequency of singly-ionized air at sea level.

This is not, however, the case for microwaves (see Chapter 4).

62. For large beams propagating in the atmosphere, there will be

no problem in finding some free electrons to start a breakdown

cascade. Early breakdown experiments, however, were often difficult to interpret because the low powers of the available lasers required that the beam be focused to a very small spot. The probability of finding an initial electron in these small focal volumes

was small, and breakdown was often very statistical in nature.

63. A more extensive discussion of the ionization of atmospheric

molecules by energetic particles can be found in Chapter 5.

64. See Section 5.1 in Zuev (note 37).

65. The “radian frequency” derives its name from the fact that

going around a circle once (one cycle) encompasses an angle of

360° (2 radians).

66. Figure 3–49 assumes that the rate for electrons of average energy 
 to ionize a molecular degree of freedom whose threshold is

 is c exp(–X/
). The implicit assumption is that the distribution

of energies among the electrons is roughly that of particles in thermal equilibrium. In actuality, the distribution of energies in a gas

Effects of Directed Energy Weapons

200
of electrons having an average energy 
 can be quite different from

that assumed, though this simple model is more than adequate to

illustrate the general features of the interactions between the electrons and the neutral gas.

67. The general theory of gas breakdown is described in P. E.

Nielsen and G.H. Canavan, “Electron Cascade Theory in Laser Induced Breakdown of Preionized Gases,” Journal of Applied Physics 44,

4224 (September 1973). Cross sections for the excitation of various

degrees of freedom in many of the species present in the atmosphere

are presented as a function of energy in L. J. Kieffer, “A Compilation

of Electron Collision Cross Section Data for Modeling Gas Discharge

Lasers,” JILA Information Center Rept 13, Joint Institute for Laboratory

Astrophysics, Boulder CO (September, 1973). Data such as these are

a prerequisite for detailed breakdown calculations.

68. Figure 3–50 is based on calculations for air which are analogous

to those described for helium in Nielsen and Canavan (note 67).

69. For example, see D. E. Lencioni, “Laser-Induced Air Breakdown for 1.06 m Radiation,” Applied Physics Letters 25, 15 (1 July,

1974). The data in this paper must be scaled by a factor of 100 to

compare with the 10.6 m calculations shown in Figure 3–50.

70. If no is the initial number of electrons and N the density of

the neutral gas, the number of generations required for breakdown is found from 2gno  N, or g  log2(N/no).

71. Thermionic emission, field emission, and other mechanisms

by which electrons may be drawn out from a solid are discussed

in Chapter V of James D. Cobine, Gaseous Conductors, (New York:

Dover, 1958).

72. David C. Smith and Robert T. Brown, “Aerosol-Induced Air

Breakdown with CO2 Laser Radiation,” Journal of Applied Physics

46, 1146 (3 March 1975).

73. Figure 3–51 is based on Figure 2 in 6. H. Canavan and P. E.

Nielsen, “Focal Spot Size Dependence of Gas Breakdown Induced by

Particulate Ionization,” Applied Physics Letters 22, 409 (15 April 1973).

74. See Mitchner and Kruger (note 28).

75. A thorough technical discussion of combustion can be found

in Irvin Glassman, Combustion (New York: Academic Press, 1977).

201

Lasers
76. See Glassman (note 75). In addition, a very readable account

of the detonation mode of exothermic chemical reaction is in

William C. Davis, “The Detonation of Explosives,” Scientific American 256, 106 (May, 1987).

77. The relationship P  N kT is known as the equation of state for

the gas. This is only an approximate form, reasonably accurate at

intermediate temperatures. The actual equation of state differs at

low temperatures, where intermolecular forces become important,

and at high temperatures, where molecules may dissociate and

more internal degrees of freedom may be excited.

78. See Chapter I, Section 2 in Ya. B. Zel’dovich and Yu. P. Raizer,

Physics of Shock Waves and-High Temperature Hydrodynamic Phenomena, Vol I (New York: Academic Press, 1966).

79. A good review of the theory for LAW propagation, along

with comparisons with early experimental data, can be found in

Yu. P. Raizer, “Propagation of Discharges and Maintenance of a

Dense Plasma by Electromagnetic Fields, Soviet Physics—Uspekhi

16, 688 (May-June, 1973). A review aimed more at atmospheric

propagation is in P. E. Nielsen and G. H. Canavan, “Laser Absorption Waves in the Atmosphere,” Laser Interactions and Related

Plasma Phenomena, Vol 3, p 177 (New York: Plenum, 1974).

80. A detailed discussion of ionization as a function of temperature can be found in Chapter III of Zel’dovich and Raizer (note 78).

81. Figure 3–56 is a plot of Equation 3–46 in Zel’dovich and

Raizer (note 78).

82. See Chapter V in Zel’dovich and Raizer (note 78).

83. Figure 3–57 was developed from Equations 3.46 and 5.21 in

Zel’dovich and Raizer (note 78).

84. Yu. P. Raizer, “Heating of a Gas by a Powerful Light Pulse,”

Soviet Physics—JETP 21, 1009 (November, 1965).

85. The constant  is the ratio of the heat capacity at constant

pressure to that at constant volume. It is not a true constant, but

varies with temperature, so that a value must be chosen which is

appropriate in the temperature range of interest. See Table 3.2 in

Zel’dovich and Raizer (note 78).

86. The data on Figure 3–58 were taken from R. B. Hall, W. E.

Maher, and P. S. P. Wei, “An Investigation of Laser-Supported

Effects of Directed Energy Weapons

202
Detonation Waves,” Air Force Weapons Laboratory Technical Rept.

No AFWL-TR–73–28 (Kirtland AFB, NM: Air Force Weapons Laboratory, June, 1973).

87. This approach was originally suggested by Raizer (note 84).

88. Figure 3–60 has been adapted from Figure 2 in Nielsen and

Canavan (note 79).

89. Yu. P. Raizer, “Subsonic Propagation of a Light Spark and

Threshold Conditions for the Maintenance of Plasma by Radiation, “Soviet Physics—JETP 31, 1148 (December, 1970).

90. An everyday example is the light emitted from the sun,

which is a high temperature plasma. This light is emitted over a

broad range of frequencies from the infrared to the ultraviolet.

91. J. P. Jackson and P. E. Nielsen, “Role of Radiative Transport in

the Propagation of Laser Supported Combustion Waves,” AIAA

Journal 12, 1498 (November, 1974).

92. The effect of the relative motion between laboratory air and a

propagating LSC can make orders of magnitude differences in the

observed velocity. This is especially true when the initial breakdown event that forms the plasma sets the surrounding air into

motion. See Raizer (note 79) and Jackson and Nielsen (note 91).

93. The advantage of lasers in annealing semiconductors is that

energy is deposited in a thin layer near the surface, where it removes imperfections caused by the implantation of dopants, but

doesn’t heat the bulk of the material to the point where the distribution of implanted ions will change through thermal diffusion.

Heating alone can also be of value in some military applications,

such as the blinding of heat-sensitive infrared sensors.

94. The thermal coupling coefficient, , is approximately l–R,

where R is the reflectivity shown in Figure 3–16. In general, however,  is a function of temperature, and may undergo abrupt

changes as a surface melts or vaporizes. The value that should be

used in analysis is an average value, approximately correct for the

duration of the laser pulse. Representative values are provided for

different materials and wavelengths in Appendix B.

95. Figure 3–63 was calculated under the assumption that

 = 3g/cm3, Tm – To  700 oK, C  l J/g oK, and D  1 cm2/sec.

The curves in the figure may be scaled as C(Tm – To) D for different values of these parameters.

203

Lasers
96. This argument assumes that the time required to heat the surface to the vaporization point is small compared to the total pulse

width. See J. F. Ready, “Effects Due to Absorption of Laser Radiation,” Journal of Applied Physics 36, 462 (February, 1965). Ready’s

book, Effects of High Power Laser Radiation (New York: Academic

Press, 1971) is a good source of general information on laser effects.

97. The parameters assumed in Figure 3–65 are the same as those

in Figure 3–63, along with Lm 350 J/g. The erosion rates shown

in the figure may be scaled as l/(Lm + C(Tm – To)).

98. The threshold for vaporization and the erosion rate when target material is vaporized may be derived in a manner analogous

to the approach used in deriving the threshold and erosion rate for

melting. The threshold for vaporization may be scaled as

(Lm + C)( (Tm – To) )D, and the erosion rate as 

l/(Lm + Lv + C (Tv – To)). 

The parameters assumed in Figures 3–66 and 3–67 are the same as

those assumed in Figure 3–65, along with Lv  8  103 J/g.

99. S.I. Anisimov, “Vaporization of a Metal Absorbing Laser Radiation,” Soviet Physics—JETP 27, 182 (July, 1968).

100. Representative values for I* are provided in Appendix B. See

also the APS Report on Directed Energy Weapons (Note 1), Figures

6.11–6.14.

101. The specific impulse I* should scale as (kTv/M)l/2(/Lv) in

going from a material where I* is known to another where it must

be estimated.

102. A good qualitative discussion of impulsive damage can be

found in Michael S. Feld, Ronald E. McNair, and Stephen R. Wilk,

“The Physics of Karate,” Scientific American 240, 150 (April, 1979).

103. See Kittel (note 18), Chapter 4. In reality, the expression is

somewhat more complex, since a solid may strain differently in

different directions.

104. The stress P* necessary to damage the plate is known as the

modulus of rupture. Representative values are provided in Appendix B.

105. The derivation of the threshold for mechanical damage presented here is attributed to G.H. Canavan in the APS Report on Directed Energy Weapons (note 1), section 6.3.5.

Effects of Directed Energy Weapons

204
106. P.E. Nielsen, “High-Intensity Laser-Matter Coupling in a Vacuum,” Journal of Applied Physics 50, 3938 (June, 1979).

107. J.A. McKay et al, “Pulsed CO2 Laser Interaction with Aluminum in Air: Thermal Response and Plasma Characteristics,”

Journal of Applied Physics 50, 3231 (May, 1979). Note especially

Figures 3 and 4.

108. The classic example of I* decreasing as S–1/3 is D.W. Gregg

and S.J. Thomas, “Momentum Transfer Produced by Focused

Laser Giant Pulses,” Journal of Applied Physics 37, 2787 (1966).

109. Michael R. Stamm, “The Formation, Propagation, and Structure of Laser Supported Detonation Waves and their Effect on

Laser-Target Interactions,” PhD Dissertation, University of

Nebraska, 1977 (unpublished).

110. The instantaneous coupling coefficient is defined as 

  Q dt/S dt.

111. P.E. Nielsen, “Hydrodynamic Calculations of Surface Response in the Presence of Laser-Supported Detonation Waves,”

Journal of Applied Physics 46, 4501 (October, 1975).

112. The dependence of impulse on target area is reported in S.A.

Metz, et al “Effect of Beam Intensity on Target Response to High

Intensity Pulsed CO2 Laser Radiation,” Journal of Applied Physics

46, 1634 (April, 1975).

205

Lasers
4: MICROWAVES

You may recall from Chapter 3 that the light from lasers is a

special case of electromagnetic radiation—waves traveling through

space, carrying energy, and characterized by a specific frequency

and wavelength. Microwaves are another type of electromagnetic

radiation, having a much longer wavelength and much lower frequency than light. For example, red light has a wavelength of

about 0.7 m and a frequency, c/, of about 4  1014 Hertz. By

contrast, microwaves have wavelengths of about 1 cm, and

frequencies on the order of 1010 Hertz, or 10 Gigahertz (GHz).

Since microwaves and lasers are both electromagnetic radiation,

most of the results developed in Chapter 3 will apply here as well.

Therefore, many topics in this chapter are treated by reference to

those in Chapter 3. A more detailed discussion of the physics

behind many phenomena in this chapter can be found in the

equivalent section of Chapter 3.

Microwaves have been around much longer than lasers, and are

used in many devices. Figure 4–1 shows the microwave portion of

the electromagnetic energy spectrum, along with some of the

Frequency Band Designations, Current Search Radar UHF TV Mobile Radio

ATC TransponderSpace TelemetryMicrowave RelaySatilite DownlinkSatellite UplinkFire Control RadarMicrowave RelaySatelllite DownlinkSatellite UplinkMissile Seeker

Frequency Band Designations, WW II - Derived

0.3 1 3 10 30 100

100 10

C D E F GI J K L M

UHF L S C X Ku K Ka mm waves

1 λ (cm)

Figure 4-1. The Microwave Portion of the Electromagnetic Spectrum

υ (GHz)
applications that are common at different frequencies.1 From Figure 4–1, you can see that many devices of military significance operate with microwaves. Typical applications include radars, communication links, and missile seekers.

Also shown in Figure 4–1 are some of the letter designations which are used to define different regions (“bands”) of the

microwave spectrum.2 Letter designations are a shorthand way

of specifying frequency ranges, but there is ambiguity in them

since some letters have referred to different frequencies over the

years. To avoid any ambiguity, we’ll state frequency or wavelength explicitly.

Most microwave devices are designed to detect and amplify

a weak microwave signal. A radar, for example, sends out

microwaves which bounce off a target and return with much less

intensity, since they diverge and are attenuated as they propagate from the radar, to the target, and back. The radar receiver

must detect this signal, amplify, and analyze it. A very powerful

microwave source can overwhelm the radar’s signal, jamming it.

At high enough intensities, it may even permanently damage the

radar. This is analogous to the human eye, which detects and

amplifies weak light signals, and can be temporarily or permanently blinded by an intense light source. It should not be

surprising, then, that intense microwaves have been suggested

as weapons against military systems which have built in vulnerabilities to them. Unfortunately, target vulnerability to

microwaves is more difficult to quantify than vulnerability to

other weapons.

Whether the potential of microwaves as weapons can be exploited depends, of course, on bringing the energy produced by a

weapon to bear on a target. Accordingly, we’ll follow the pattern

established in previous chapters. After discussing the fundamental features of microwaves, we’ll look at how they propagate in

vacuum and air, and use this information to see how much intensity a microwave weapon must develop to engage targets of different vulnerability at different ranges.

Fundamentals of Microwaves

Microwaves differ from lasers only in their wavelength and frequency. This difference can, however, have profound implications.

Effects of Directed Energy Weapons

208
For example, the window on a microwave oven is lined with a

metallic screen. Visible light can penetrate this screen because its

wavelength is much smaller than the size of the screen’s mesh. Microwaves, on the other hand, cannot penetrate. Their wavelength

is larger than the mesh, and it appears to them to be a solid sheet

of metal. In this way, you can see what is cooking in the oven,

without danger of being roasted by microwaves escaping through

the window. We’ll look in this section at some of the fundamental

features of microwave propagation and interaction with matter.

Fundamentals of Propagation 

Like all electromagnetic radiation, microwaves travel at the

speed of light, c, ( 3  108 m/sec) in vacuum. They have a

frequency, , and wavelength, related through the expression

  c/. As you can see from Figure 4–1, microwave frequencies

lie in the range 0.1–100 GHz, and the associated wavelengths lie

in the range 100–0.1 cm. Microwaves are unique in that their

wavelengths are similar to the size of the physical objects with

which they interact. This means that their “wavy” nature can be

important in analyzing how they propagate around or interact

with objects. Microwaves are therefore more difficult to analyze

than either light waves, whose wavelength is much smaller than

most objects, or radio waves, whose wavelength is much larger.

You will recall from Chapter 3 that when collimated electromagnetic radiation of wavelength emerges from an aperture

of diameter D, the resulting beam has an angular divergence

  /D. For lasers, with wavelengths on the order of 10-5 cm, a

10 cm aperture produces a beam of divergence 10-6 radian, or

1 rad. For microwaves, with wavelengths on the order of 1 cm,

a 10 km aperture would be required to achieve the same divergence! Such large apertures are clearly impractical, especially for

military systems, which often must be mobile and capable of deployment. Therefore, microwaves diverge much more than laser

light, so that their energy spreads and their intensity drops much

more rapidly with distance.

For light, the aperture through which the radiation emerges is

obvious—it’s the last lens or optical element in the system. For

microwaves, the transmitting antenna is the “window” or aperture through which they are sent into the world. The divergence

209

Microwaves
  /D is applicable to directional antennas, such as the parabolic dish illustrated in Figure 4–2. Other types of antennas, such

as the “whip” on a police car, are designed to send radiation in all

directions, so that communication is possible between transmitter

and receiver regardless of their relative orientation. Antennas of

this sort have a much greater divergence than /D. Since our

emphasis is on directed energy weapons, we’ll assume that the

intent of the weapon designer is to be energy efficient, making

divergence as low as possible. The divergence   /D is the best

that can be done, corresponding to the diffraction-limited optical

system discussed in Chapter 3.

When propagating in a medium such as air, microwaves travel

at a velocity less than that in a vacuum, and the ratio of the velocity in vacuum to that in the medium is known as the index of refraction, n. As with lasers, microwaves traveling from a region characterized by one index of refraction to another will be bent in

accordance with the law of refraction, n1 sin1  n2 sin2. As illustrated in Figure 4–3, the direction of travel is bent towards the

“normal” (a line perpendicular to the surface) when going into a

region where n is greater, and away from the normal when going

into a region where n is less.

With lasers, refraction finds application in lenses, where the

change in n going from air to glass can be used to expand or focus

the beam (see Figure 3–5). This is not very practical for microwaves, since the radius of the beam is very large, and expands

rapidly with distance. Nevertheless, lensing effects can occur as a

beam of microwaves travels through the atmosphere, since the

index of refraction of air depends on both density and water

Effects of Directed Energy Weapons

210

Non-Directional

Figure 4-2. Microwave Antennas

θ λ/D

θ λ/D

θ λ/D

Directional
vapor content, and these can change over distances comparable to

the beam size. We will see that this can result in some unusual behavior from the standpoint of beam propagation.

Fundamentals of Interaction with Matter 

Electromagnetic radiation can be absorbed when the energy

associated with a unit or photon of electromagnetic energy is

equal to the difference in energy between two allowed energy

levels in the absorbing material. A photon has energy h, where

h is Planck’s constant, 6.63  l0-34 Joule sec, and  is the frequency

of the radiation in Hz. Microwave photon energies lie in the

range 10-25 to 10-22 Joules, or 10-6 to l0-3 eV (1 eV  1.6  10-19 J).

Since the rotational, vibrational, and electronic energy levels in

atmospheric gases are separated by energies on the order of 10-2,

10-1, and 1 eV respectively, microwaves are unlikely to be absorbed by most of the gases in the atmosphere. Indeed, this lack

of attenuation makes microwaves useful as radars and communication devices. However, there are some important exceptions to

this general rule. Both oxygen and water vapor can absorb radiation in the microwave region of the electromagnetic spectrum,

and even liquid water interacts well with microwaves.3 Therefore, humidity and rain can be important factors in microwave

propagation. The interaction between microwaves and water is

employed in microwave ovens. Since water is a major constituent

of organic matter, microwaves are absorbed in, and heat, foods

placed in the oven. Yet they are not absorbed in ceramic dishes,

211

Microwaves

n2

Figure 4-3. Law of Refraction

θ1 θ1

θ2

n1 n1

n2

θ2

n1 < n2 n1 > n2
which don’t contain water. In this way, the food in a dish is

heated while the dish itself remains cool.

When they encounter solid targets, microwaves will to a first

approximation pass through insulators and be reflected from metals. This is because, as you may recall from Chapter 3, electromagnetic radiation is not absorbed in dielectric (insulating) solids unless the photon energy is greater than the band gap in the

material, and is primarily reflected from metallic solids unless its

frequency exceeds the plasma frequency. The frequency of microwaves is far below the plasma frequency of metals, and their

photon energy is far below the band gap of most insulators. In this

way the window or radome through which a radar beam passes is

an insulating solid that is opaque to visible light, and sensitive

electronic components can be shielded from microwaves by enclosing them in metal containers.

Summary: Microwave Fundamentals

1. Microwaves are electromagnetic radiation with a wavelength

of 0.1–100 cm. This wavelength is comparable to the size of the

physical objects with which they might interact, so that the

wavy nature of microwaves is important in their analysis.

2. Like all electromagnetic radiation, microwaves emitted from a

directional antenna of diameter D spread with a divergence angle

of about /D. Due to the large wavelength () of microwaves,

they spread much more than light for realistic aperture sizes.

3. When crossing regions with different indices of refraction, microwave beams are bent just as laser beams are. Because of the

large size of microwave beams, only large scale fluctuations in the

index of refraction have an effect on propagation. Typically, these

effects occur between different layers within the atmosphere.

4. Microwave photon energies are quite small, less than l0-3 eV.

As a result, they pass through most insulating materials. And

their frequency is quite low, less than 100 GHz. As a result,

most metals reflect them.

Effects of Directed Energy Weapons

212
Microwave Propagation in a Vacuum

Propagation Tradeoffs

A beam of electromagnetic radiation of wavelength emitted

from an aperture of diameter D will remain collimated for a

distance zr  D2/, before diverging at an angle   /D The

distance zr is known as the Rayleigh Range, and is illustrated in Figure 4–4. Figure 4–5 is a plot of the Rayleigh Range as a function of

aperture (that is, antenna) size for wavelengths in the microwave

region.4 You can see from this figure that for microwaves to be

beamed over substantial distances, large apertures or short wavelengths must be employed.

For microwaves with a nominal 1 cm wavelength, apertures in

excess of 10 m are required for a beam to travel distances in excess of 10 km without spreading. Apertures this large are practical

only for fixed installations, rather than mobile systems. Even for

fixed installations, a large antenna is a very vulnerable element for

a weapon system. Consequently, microwave propagation will

probably be in the far field, with a divergence of /D, even over

tactical ranges. As a result, if we need to place a given Intensity

(W/cm2) on target, it will be necessary that a greater intensity be

emitted by a microwave weapon. In Chapter 3, we used the concept of brightness to examine tradeoffs among energy, wavelength,

and aperture in an electromagnetic radiation weapon.5

213

Microwaves

Figure 4-4. Rayleigh Range and Divergence

"Near Field" "Far Field"

"Rayleigh Range"

D

=D

Zr = D2/The brightness of an electromagnetic beam having power P and

wavelength , emitted from an aperture of diameter D, is B 

PD2/ 2. Physically, brightness (Watts per steradian, W/sr) is the

power in the beam, divided by the solid angle into which the beam

is spreading.6 If B is divided by the square of the range to target, z,

the result is the intensity of the beam (W/cm2) at the target’s surface. If that intensity is in turn multiplied by the time the beam illuminates the target, t, the result is the total energy density, or fluence (Joules/cm2), delivered to the target. Damage criteria are

typically expressed in terms of the fluence or intensity required on

target, along with time constraints to insure that the target doesn’t

dissipate energy faster than it is absorbed (see Chapter 1).7

Figure 4-6 shows the range to which microwave beams can deliver various intensities (W/cm2) as a function of brightness. This

figure is the same as Figure 3–21, since the range to a given intensity for a given brightness is independent of the type of radiation.

Of course, since microwaves have a much greater wavelength

than lasers, much greater apertures are required to provide the

same brightness and intensity on target.

Figure 4–6 can be used to examine the tradeoffs associated

with placing a given intensity on a target at a given range.

Suppose, for example, that we need to place 10 W/cm2 on a target at a range of 100 km. As the arrows on Figure 4–6 indicate,

brightness must be at least 1015 W/sr to meet these criteria. Since

B  PD2/  2, we can go further and investigate tradeoffs in the

Effects of Directed Energy Weapons

214

105

103

10

10-1

10-3

10-5

0.1 1 10 100

Strategic Range

Tactical Range

Wavelength (cm)

0.1

1.0

10

100

Figure 4-5. Rayleigh Range vs Aperture and Microwave Wavelength

Aperture (m)

Rayleigh

Range

(km)
design of a microwave weapon. For example, suppose  1cm

(  30 GHz). Then achieving a brightness of 1015 W/sr requires

that PD2 2 B  3  1015 cm2 W. If we are constrained to a 1

meter aperture, then D2  104 cm2, and our weapon must transmit 1011 Watts. If we didn’t feel that this power level was an

achievable goal, we could reduce power requirements by increasing the aperture or reducing the wavelength.

Figure 4–6 summarizes all we need to know about propagation

of microwaves along a clear path in a vacuum. From it, we can

find the brightness necessary to damage a target at a given range,

and then investigate the implications of that brightness from the

standpoint of the power, wavelength, and aperture required in a

microwave weapon.

Diffraction and Interference Around Objects

Imagine that a beam of microwaves or other electromagnetic radiation encounters a physical object on its way to a target. A simple example might be a screen such as that illustrated in Figure

4–7, which reflects or absorbs the radiation incident upon it. We

would expect a screen to cast a shadow, so that regions behind it

would be shielded from the radiation. This is approximately true,

but a detailed investigation would show that some radiation does

get into the shadowed or shielded region, and that the intensity

near the border of the geometrical shadow undergoes a series of

fluctuations, also illustrated in Figure 4–7. This spreading of radia215

Microwaves

107

105

103

10

10-1

10.14 1018 1022 1024

Figure 4-6. Range vs Brightness and Intensity

Brightness (W/sr)

Range

(km)

10.16 1020

10.7

10.4

S = 10W/cm2
tion into regions which would appear shielded is referred to as

diffraction, and is another manifestation of the diffraction responsible for the spreading of radiation which passes through or is

emitted from an aperture.

The scale of the fluctuations in intensity, and the distance radiation penetrates into the region of geometrical shadow, is on the

order of (z)l/2, where is the wavelength of the radiation and z the

distance from the shield to the point of observation.8 For infrared or

visible light from a laser, this is of little importance. In this case, the

wavelength is 10-4 cm or less, and at a point 1 m from a shield, fluctuations in intensity would occur over a distance of 1 mm or less.

For microwaves, on the other hand, a wavelength of 1 cm produces

fluctuations on the order of 10 cm at a 1 m range. This means that

microwave energy can reach objects which to a first approximation

are shielded from a microwave transmitter by natural barriers.

The situation becomes more complex when microwaves pass

through a slit or series of barriers. This results in a variety of fluctuations in intensity, whose pattern is difficult to predict except in a few

Effects of Directed Energy Weapons

216

Incident Beam

Shield

Intensity

''Shadow"

Z distance from shield

to observation plane

=

Figure 4-7. Diffraction of Radiation Near a Barrier

λΖ
simple cases.9 From the standpoint of target interaction, this means

that predicting the actual intensity on a target and the vulnerability

of that target can be quite difficult. Microwave energy will enter a

target through gaps and slits in the plates or other shields which surround it.10 Diffraction of the beam as it passes these points will lead

to fluctuations in intensity on the interior of the target which occur

on space scales similar to the separation of elements within the target. Thus, a vulnerable element might be damaged in one interaction

and undamaged in another whose gross features (range, etc.) are essentially identical. Similarly, target vulnerability will be highly dependent upon the details of the target’s construction, which will determine where target elements lie relative to various shields and

entry points for radiation into the interior.

Summary: Microwave Propagation in a Vacuum

1. Like all electromagnetic radiation, microwaves of wavelength emerging from an aperture or antenna of diameter D

diverge with an angle on the order of /D in the far field—

distances greater than about  D2/. Because of the size (lcm)

of wavelengths in the microwave region, almost all propagation

is in the far field, and divergence is much greater than for laser

light from an aperture of similar size.

2. The brightness of a beam of radiation in Watts per steradian,

along with its dwell time on target, will determine the intensity

and fluence on a target at a given range. Figure 4–6 can be used

to determine the range at which beams of different brightness

can place different intensities on target. This information can

then be used to perform tradeoffs among power, wavelength,

and aperture for specific applications.

3. When microwaves pass around objects, diffraction spreads

the radiation by a substantial amount into regions which on the

basis of geometry would be shielded. Therefore the actual intensity within a target is in practice difficult to quantify.

Implications

The theory of microwaves is in essence no different than that

for lasers—only the wavelengths are different. Since beam diver217

Microwaves
gence is on the order of /D, why would anyone be interested

in microwaves as weapons? Lasers, with their much shorter

wavelength, have a much smaller divergence, and are therefore

more directed in their energy. There are several reasons why

microwaves are seen to have potential as weapons.

Lasers are a relatively new invention, the first having been

constructed in 1960. By contrast, the microwaves have been used

for radar and communication since the 1940s. The long ranges

needed for these microwave applications, together with the large

divergence angles, required the development of high power

sources. As a result, the technology for high power microwaves

is more advanced than that for high power lasers.11 Indeed, some

high power lasers, such as the free electron laser, are themselves

powered by microwave energy. Thus, it is possible that in a

given application the higher available power in a microwave

device could compensate for a larger divergence to place more

energy on target.

Many interesting military targets are themselves microwave

receivers. They are sensitive to the weak signals they have to

receive and interpret, and are vulnerable to microwaves of

much greater intensity than they were designed to encounter.

With a few exceptions (such as the human eye or an optical

detector), most military targets are not as vulnerable to light.

Therefore, a microwave weapon may achieve damage at a lower

intensity than a laser.

Diffraction is much greater for microwaves than for laser light.

This means that if microwave energy penetrates a target at any

small point, such as an opening where a wire emerges from a

black box, the energy will spread within the box, possibly finding vulnerabilities which neither the designer nor the attacker

contemplated. Laser light, on the other hand, is largely confined

to straight line propagation beyond the point of entry, and is

much less likely to find targets of opportunity on the interior of

a target.

In short, lasers damage targets through a frontal assault—melting

their way through the surface and on into the interior of a target.

Microwaves damage targets by more subtle means, going on

through an already prepared entry point and wandering around on

the interior. The entryway may be obvious such as an antenna, or it

may be some gap or slit in the target’s surface. Damage when entry

Effects of Directed Energy Weapons

218
is through an antenna is known as front door damage, if entry is

through some less obvious point it’s known as back door damage.

Front door damage is more strongly wavelength dependent, since

antennas and receivers are tuned to accept radiation in specific

wavelength ranges. Because of the way in which they damage targets, microwaves may be more effective as weapons than they

would appear on the basis of propagation alone.

Microwave Propagation in the Atmosphere

In Chapter 3, we saw that an atmosphere had many effects on

laser propagation. Molecules or aerosols could absorb or scatter the

light. Self-induced index of refraction fluctuations (thermal blooming) could cause the beam to bend and wander. And at sufficiently

high intensities, air breakdown could prevent propagation entirely.

Analogous phenomena occur for microwaves, and we’ll consider

these here, contrasting our results with those in Chapter 3.

Losses due to Absorption and Scattering

You may recall from Chapter 3 that the intensity, (W/cm2), of

electromagnetic radiation propagating a distance z in an absorbing medium decays as S(z)  S(0) exp(-Kz), where K is known as

the attenuation coefficient. The attenuation coefficient can in turn

be written as a sum of coefficients arising from absorption or scattering from the various molecules or small particles (aerosols) present in the atmosphere. For microwaves, the most important contributions to K come from oxygen, water vapor, and liquid water

(rain or other precipitation).12 We’ll consider the effect of molecules first, and then look at liquid water and other aerosols.

Molecular Absorption and Scattering. Figure 4–8 shows the

attenuation coefficients due to oxygen and water vapor for radiation at microwave frequencies.

Figure 4–8 assumes that the density of oxygen is that at sea

level, and that there are 7.5 gm/m3 of water vapor. At higher altitudes and different levels of humidity, the attenuation coefficient

Scales with the density of molecules. Thus, at an altitude where

the density of oxygen is half that at sea level, the attenuation coefficient due to oxygen will be half that shown in Figure 4–8. Figure

219

Microwaves
4–9 gives the concentration of water vapor at 100% humidity as a

function of temperature.14 Figure 4–9 may be used to scale the

contribution of water vapor from that assumed in Figure 4–8. For

example, at 0 oC and 50% humidity, the concentration of water

vapor is half that shown in Figure 4–9, or about 2.5 gm/m3. Since

this is one third of the vapor density assumed in Figure 4–8, the

attenuation coefficient for water vapor under these conditions will

be one third of that shown in Figure 4–8.

In scaling Figure 4–8 to altitudes above sea level, you can

use the fact that at lower altitudes, atmospheric density scales as

	  	o e–h/ho, where 	 is the density at altitude h, 	o the density

at sea level, and ho is approximately 7 km.15 This relationship

is plotted in Figure 4–10. You can use figures 4–8, 4–9, and

4–10 to estimate the atmospheric attenuation coefficient for

microwaves due to oxygen and water vapor at any temperature,

altitude, and humidity.

In many cases, a microwave beam propagates up into the atmosphere, and the attenuation coefficient is not constant along

the beam path. As we saw in Chapter 3, the intensity at a range

z in this case is no longer S(z)  S(0) exp(–Kz), but rather

S(0) exp[–oz K(z) dz].16 In order to solve this equation, we need

to know the attenuation coefficient, K(z), at all points z along the

beam’s path. In general, this isn’t possible without making simEffects of Directed Energy Weapons

220

total

100

10

 1

 0.1

 0.01

 0.001

10 20 30 40 50 60 70 80 90 100

K(H2O)

K(O2)

K

(km-1)

Frequency, GHz

Sea Level Air, 7.5g/m3 Water Vapor

Figure 4-8. Attenuation Coefficent for Microwaves due to Oxygen and

Water Vapor (13)
plifying assumptions. For example, we might assume that a

beam of microwaves is fired into the air at some angle 
, and

that both oxygen and water vapor fall off with altitude as shown

in Figure 4–10.

This case is illustrated in Figure 4–11, and the solution for the

relative attenuation factor or optical depth is shown in Figure

4–12. You will recognize that Figures 4–11 and 4–12 are essentially

the same as Figures 3–25 and 3–26. The physics is the same as that

discussed in Chapter 3; only the sea level attenuation coefficient,

K(0), is different. However, the assumptions involved in this

analysis are not as valid for microwaves as they are for lasers. This

is because water vapor is a major contributor to microwave attenuation, and humidity can vary with altitude. In addition, we shall

221

Microwaves

Figure 4-9. Concentration of Water Vapor vs Temperature at 100% Humidity

Vapor

Concentration

(gm/m3)

40

36

32

24

20

16

12

8

4

0

-20 -10 0 10 20 30

Temperature, 0C

28

Figure 4-10. Relative Atmospheric Density vs Altitude

Altitude (km)

Density

Ratio

(	(h) / 	(0))

1,000

0.100

0.010

0.001

0 5 10 15 20 25 30 35 40
see later in this chapter that refractive effects in the atmosphere

can cause a microwave beam to travel in paths which are not

nearly as straight as that illustrated in Figure 4–11.

Because the assumptions used in deriving it are not fully realistic, Figure 4–12 is probably not suitable for quantitative analysis.

Nevertheless, there are useful qualitative conclusions to be drawn

from the figure. First, optical depth and attenuation are virtually

independent of elevation angle for ranges less than the distance

over which atmospheric properties vary significantly. This distance is 7 km in the example, but variations in humidity and water

vapor concentration may shorten it. Over short ranges, then, attenuation is simply S(z)  S(0) exp (-Kz). Second, at larger elevation angles, attenuation virtually ceases as the beam emerges from

the lower, high density regions of the atmosphere. Thus, only

these lower levels of atmosphere are significant in beam attenuation. This suggests that given information regarding the extent in

altitude of significant water vapor, it would be possible to add the

appropriate attenuation coefficient for water vapor over that

lower altitude range only. Some authors suggest that absorption

due to water vapor will occur primarily at altitudes below 2 km.17

A final point should be made regarding microwave absorption

from oxygen and water vapor. Looking at Figure 4–8, and recalling

that 1/K is the distance over which a beam’s intensity will fall by a

factor of l/e (approximately 1/3), you can see that depending on

the wavelength, microwaves in the atmosphere will travel between

Effects of Directed Energy Weapons

222

Figure 4-11. Beam Range and Altitude

h

Z

h = z sin 




100 and 0.1 km without significant attenuation. There is an interesting tradeoff between absorption and beam divergence. Lower frequency beams are absorbed less, but have a longer wavelength and

therefore diverge more rapidly than higher frequency beams.

Therefore, it is not immediately obvious which frequency will enable the greatest energy to be placed on target at a given range. Of

course, target vulnerabilities may also be frequency dependent, further complicating the choice of frequency for a given application.

Effect of Liquid Water and Atmospheric Aerosols. Water

molecules interact strongly with microwaves, and water vapor

contributes to their attenuation in the atmosphere. Therefore, it’s

logical to expect that when water is concentrated into a liquid as

rain it will have a severe effect on propagation. The effect of rain

on microwave propagation depends upon the scattering and

absorption from a drop of water of a given radius, together with

the number of drops of various radius that are likely to be encountered. This last information is related to the type of rain

storm—a mist or fog has many drops of small radius, while an

223

Microwaves

∫

0

z

K(z)dz / [K(o)h0]

Relative

Depth

Relative Range (z/h0)

150

300

900

Figure 4-12. Optical Depth vs Range and Elevation Angle

0.0

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

1.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5

Elevation Angle,Φ=00
afternoon thundershower has fewer drops of much larger radius.

Let’s first consider the attenuation from a given rain drop, and

then the distribution of rain drops.

In Chapter 3, we discussed the fact that when electromagnetic

radiation of wavelength encounters a small particle (aerosol) of

radius a, there is an effective cross section or size which the

particle presents to the radiation from the standpoint of absorption and scattering. The attenuation coefficient, K, is given by

N, where N is the concentration of particles (cm–3), and  the

cross section (cm2). When the particle size, a, is much greater

than the wavelength of the radiation, the cross section is roughly

2a2. This is the case for laser light encountering most atmospheric aerosols. On the other hand, when the particle is smaller

than the wavelength of the radiation, the cross section deviates

from this general rule, and is highly wavelength dependent (see

Figure 3–28).

When the particle size is much less than the wavelength of the

radiation, the cross section goes to zero. For this reason, solid

particles in the atmosphere, whose size is 10 m or less, have

negligible effect upon microwaves, whose wavelengths are millimeters or greater. Water droplets come in sizes that approach

microwave wavelengths, however, and can be an important

source of attenuation. Since microwaves have wavelengths on

the order of 1 cm, and raindrops rarely exceed 2 mm in radius,

quantitative analysis from first principles is quite difficult. You

can’t assume that the wavelength is much larger than the particle

size, so that  goes to zero, or that the wavelength is much

smaller than the particle size, so that   2a2. Therefore, research has focused on developing empirical attenuation formulas

or curves by measuring attenuation as a function of wavelength

and rain conditions. Figure 4–13 is a typical plot of the attenuation coefficient of rain as a function of microwave frequency and

rate of rainfall.18 Comparing this figure with Figure 4–8, you

can see that at higher rates of precipitation, rain can be more

important than atmospheric molecules in attenuating microwaves. Table 4–1 provides the rain rates associated with different

types of precipitation, so that you can relate to Figure 4–13 on

more familiar terms.19

You need to be careful in using Figure 4–13 and Table 4–1 to estimate microwave attenuation under a given set of conditions.

Effects of Directed Energy Weapons

224
The figure is based on average conditions, and as Table 4–1

points out, there is a range of drop sizes and precipitation rates

associated with a given type of rain. Since attenuation is a strong

function of the ratio of drop size to wavelength, there could be

large variations from the average given in the figure in any

particular case. Additionally, it is unlikely that the rate of precipitation will be constant over the entire path from microwave

source to target. Therefore, even if the estimates shown in Figure

4–13 were known to be accurate, it would be necessary to

measure the rain rate all along the propagation path, integrating

the attenuation coefficient to find the total attenuation along the

path. This procedure might be possible for experiments over a

well-instrumented range, but is impractical for use in the

real world. Therefore, the most you can hope to have in realistic

situations are general guidelines based on an accumulation of

experimental data. In weapon design, then, the available energy

needs to be sufficient to insure that damage criteria are met

even if climactic conditions are near the extreme end of their

range of probability.

225

Microwaves

Attenuation

Coefficient

K

(km-1)

Figure 4-13. Attenuation of Microwaves by Rain

Frequency, GHz

10

1

10-1

10-2

10 100 1000

Precipitation Rate (mm/hr)

50 mm/hr

5 mm/hr

1.25 mm/hr

0.25 mm/hr
Summary: Absorption and Scattering. 

1. Like all electromagnetic radiation, microwaves are removed

from a beam through absorption and scattering from molecules

and aerosols in the atmosphere. For microwaves, the most important molecules are O2 and H2O, and the most important

aerosols are those of precipitation—solid ice or liquid water.

2. Energy loss from the beam follows the attenuation law

S(z) = S(0) exp(–Kz), where K, the attenuation coefficient, is the

sum of individual coefficients for each contributor to attenuation. Figure 4–8 provides attenuation coefficients for oxygen

and water molecules, and Figure 4–13 provides them for liquid

water aerosols as a function of precipitation rate.

3. Attenuation from atmospheric molecules scales with their

density. Figure 4–12 is an estimate of the effective Kz, or optical

depth, to be used in the attenuation law when a beam is aimed

into the atmosphere at some elevation angle. Most attenuation

occurs below 7 km in the atmosphere.

4. Attenuation from water vapor scales with relative humidity

and temperature. Figure 4–9 provides data on water vapor concentration as a function of temperature which may be used to

scale Figure 4–8 to different conditions.

5. Attenuation due to rain scales with the rain rate and drop

size. Table 4–1 may be used to estimate these from observed

precipitation conditions, and the appropriate attenuation coefficient can be estimated from Figure 4–13.

6. Estimates made using the tables and figures in this section

should be considered order of magnitude only. Physical conditions can vary greatly along a beam’s propagation path, and

Effects of Directed Energy Weapons

226

Table 4-1. Rain Rates vs Meteorological Conditions

Condition Precipitation Rate (mm/hr) Drop Size (mm)

Drizzle

Light Rain

Moderate Rain

Heavy Rain

<0.5

1 - 2

1 - 2

1 - 2

0.2 - 0.5

< 2

2 - 6

>7
data are typically available at only a limited number of points

along the path. Microwave weapons must therefore be designed

to be capable of placing damaging energy on target even at the

extreme end of conditions considered operationally realistic.

Losses Due to Index of Refraction Variations

In Chapter 3, we saw that the index of refraction of air depends

upon its density and temperature, as well as the wavelength of the

radiation (see Figure 3–30). At optical frequencies, small fluctuations in density and temperature resulting from atmospheric turbulence act as mini-lenses, breaking up the coherence of the beam

front, and causing it to diverge at a rate much greater than would

be expected from its aperture. This requires that active measures

(adaptive optics) be taken to insure that a laser beam will propagate over long ranges in the atmosphere without substantial loss

of intensity. At microwave frequencies, the situation is considerably different. The wavelength is long compared to the size of turbulent density fluctuations in the atmosphere, so that they do not

affect a beam’s propagation to the same extent. Thus, adaptive

optics and other techniques which are needed to enable long

range laser propagation in the atmosphere are not required for

microwaves. This is fortunate, since otherwise microwaves would

not have found so many applications as a means of communication, where accurate phase and amplitude information are required if a coherent signal is to be received.

We cannot conclude from this, however, that density fluctuations within the atmosphere do not affect the propagation of microwaves from source to target. While this is true for the smallscale fluctuations resulting from turbulence, there is another

density variation which can affect microwave propagation—the

large scale variations in density that occur as a beam passes between different atmospheric layers.

At microwave frequencies, the index of refraction of air is insensitive to wavelength, and is a function of pressure, temperature,

and the amount of water vapor present, as shown in Figures 4–14

and 4–15. Figure 4–14 shows the index of refraction as a function

of temperature for dry air (no water vapor present), and Figure

4–15 shows correction factors which can be added to the curves in

Figure 4–14 to account for the amount of water vapor present.20

227

Microwaves
The correction factor in Figure 4–15 is appropriate for 100% humidity at any pressure. This factor scales with humidity, so that at

50% humidity, for example, the appropriate correction factor is

simply half that shown in Figure 4–15.

From Figure 4–14, you can see that normally the index of refraction of air for microwaves decreases with altitude, and its deviation from l changes substantially over altitude ranges on the order

of a kilometer. What is the implication from the standpoint of

propagation in the atmosphere? Basically, this variation results in

a beam of microwaves following a curved, rather than a straight

path, as illustrated in Figure 4–16.

Shown in Figure 4–16 is a simplified example in which a microwave beam travels between two regions of the atmosphere—

the lower one having index of refraction n1, and the upper one

having index of refraction n2, with n1 greater than n2. A ray of electromagnetic radiation passing from one region into another where

the index of refraction is less is bent away from the normal to the

surface at the point where the ray crosses between the regions (see

Figure 4–2). You can see in the blow up of the point where the

Effects of Directed Energy Weapons

228

340

306

272

238

204

170

136

102

68

34

0

Index

(n-1)x106

1000

Pressure (mbar)

Figure 4-14. Index of Refraction for Dry Air vs Temperature and Pressure

900 800 700 600 500

0 1 2 34 5 h, km

1 Atm = 1000 mbar

233

273

300

313

-40

0

27

40

-40

32

80

104

0K 0C 0 Temperature F
beam crosses regions in Figure 4–16 that this results in the beam

being bent back towards the surf ace of the earth. The tall antenna,

which would otherwise not be in line of sight from the microwave

transmitter, can now receive its signals. In other words, the range

of the microwave beam over the surface of the earth has been extended by refraction.

In a real situation, the index of refraction decreases continuously,

rather than in discrete regions as shown in Figure 4–16. As a result,

a beam will follow a curved path rather than one composed of discrete line segments, but the net result will be the same, the beam’s

range will be extended. As you can see from this example, the exact

path a microwave beam will travel in the atmosphere will depend

on how the index of refraction varies with altitude in the region between a microwave source and receiver. A variety of attempts have

been made to capture this effect in a simple formula to approximate

beam propagation in practical circumstances. One such approach is

known as the effective earth radius transformation.

Because microwaves for the most part travel from source to receiver by line of sight, it is the horizon, or curvature of the earth,

which limits propagation between stations located on the earth’s

229

Microwaves

400

360

320

280

240

200

160

120

80

40

0

Correction

to

(n-1)x106

230 240 250 260 270 280 290 300 310 320

Temperature. 0K

Figure 4-15. Correction to Index of Refraction for 100% Humidity vs Temperature
surface. This is evident from Figure 4–16. The range within which

line of sight communication can be maintained is limited by the radius of the earth Re; if Re were greater, the range would be longer.

As we have seen, index of refraction changes in the atmosphere

have the effect of bending a microwave beam, giving it a longer

range along the surface than it otherwise would have. The effective

earth radius transformation is one in which this effect is modeled as

an increase in the radius of the earth—Re is changed to a new, effective value Re*, such that if Re* is substituted into an expression for

the range of a beam in the absence of refraction, the true range in its

presence results.21 If dn/dh is the slope of the curve of index of refraction as a function of altitude, the effective earth radius is given

by Re*  Re/[l  Re(dn/dh)], where Re is the true radius of the earth

(6370 km), and dn/dh is the rate of change of n—the amount by

which n changes for each kilometer increase in altitude.22

Looking at Figure 4–14, you can see that in going from sea level

to 1 km altitude, the quantity (n–1)  106 changes from about 240

to 200. Thus, dn/dh is about –40  10–6/km, and the ratio Re*/Re

can be found from the expression above to be about 4/3. In other

words, under normal circumstances microwaves travel line of sight

Effects of Directed Energy Weapons

230

n1 n2

n2

n1

n2 n1

Figure 4-16. Propagation of Microwaves in the Atmosphere
between points on the earth as though the earth had a radius 4/3

times its true radius, or about 8500 km. Figure 4–17 is a plot of

Re*/Re as a function of –dn/dh. As the magnitude of the index of

refraction gradient increases, Re* becomes greater and greater, and

is infinite for dn/dh  –160  10–6/km. When the magnitude of

dn/dh exceeds 160  10–6, Re* actually becomes negative! The

physical meaning of these results are shown on the figure. At a rate

of change of n of –160  10–6/km, the curvature of the beam equals

that of the earth, so that the earth is essentially “flat” from the

standpoint of microwave propagation. When the index gradient

exceeds this critical value the beam actually bends back towards

the earth, and its range is less than it would be in a vacuum.

You can see from Figure 4–17 that a variety of interesting effects

can occur as a result of the change or gradient in refractive index

with altitude. These can be particularly dramatic if the index of refraction gradient is very large, and this can actually occur under

certain conditions. The nominal value for the index of refraction

gradient is about –40  10–6/km, and the resulting effective earth

radius is about 4Re/3, but there can be considerable variation in

dn/dh, both over time at a given point and at different points at the

same time. One of the biggest factors affecting n and dn/dh is humidity. At high temperature and humidity, the “correction” to (n–l)

from Figure 4–15 can be as large as the value for dry air in Figure

4–14. Worldwide values of dn/dh vary from –30  10–6 in dry climates to over –100  10–6 in hot, humid climates. Greater extremes,

even exceeding the critical value of about –160  10–6, can be found,

for example, when dry inland air blows over damp, humid air in a

coastal region. Because of the importance of dn/dh to microwave

propagation, there are actually published tables of this parameter

for different regions of the earth and different times of the year.23 Of

course, such values are averages and only an indication of what

might occur. There can be considerable variation from the value in a

table at any given time and place.

The mere bending of microwaves is not all that can occur. Under

appropriate conditions, variation in n can lead to ducting, or the

channeling of microwaves along a given direction in the atmosphere. This can happen, for example, when dn/dh exceeds –160 

10–6 per km over the ocean. Salt water is a fairly good conductor of

electricity, and reflects microwaves as a metallic surface would do.

Beams which have been bent back to earth may therefore reflect off

231

Microwaves
the ocean’s surface, skipping their way over much greater

distances that would be predicted on the basis of simple theory, as

illustrated in Figure 4–18.

Ducting can also occur above the surface of the earth, if a region

for which dn/dh exceeds –160  10–6/km lies above a region in

which dn/dh has a more normal value. Whether or not a particular

beam or portion of a beam will be trapped in a duct depends upon

the angle with which it enters the ducting region. Obviously, beams

directed more vertically than horizontally are less likely to be

trapped in this way, since at normal incidence the degree of bending due to refraction goes to zero (see Figure 4–3). These effects are

somewhat analogous to the internal reflection which is evident to

swimmers underwater. As viewed from below, the surface of the

water appears silvery. Many of the light rays striking the surface

from below are reflected back, since n decreases in going from

water to air. Nevertheless, some rays do escape, and someone in a

boat can see swimmers below the surface (assuming, of course, that

the water is clean enough!).

Effects of Directed Energy Weapons

232

Figure 4-17. Effective Earth Radius vs Refractive Index Gradient

-20 -40 -60 -80 -100 -120 -140 -160 x 106 dn__ /km-1

dn

20

15

10

5

Re*

Re

__ Max Range as Bending

"circles the Earth"

Extreme Bending

Shortens Range

Bending Increases

Range
Summary: Index of Refraction Variations. 

1. At microwave frequencies, the index of refraction of air (n)

depends strongly on temperature, pressure, and water vapor

content. All of these can vary with altitude, resulting in largescale fluctuations in n that act as huge lenses to bend microwave

beams from the straight line path which they would otherwise

follow in a vacuum. As a result, the intensity at a given point

distant from a transmitter can vary greatly from what would

have been predicted from vacuum propagation theory.

2. A key factor in microwave propagation is dn/dh, the rate at

which n changes with altitude. Figure 4–17 shows how the effective earth radius varies with dn/dh. The effective radius obtained

from this figure can be used to estimate the range over which microwaves can reach targets via line of sight propagation.21

Nonlinear Effects

In nonlinear propagation, a beam’s intensity is so great that it

modifies the environment through which it propagates, altering

its propagation characteristics. In chapter 3, we saw that there

were many nonlinear effects in the propagation of laser light.

Among these were thermal blooming, stimulated scattering, and

air breakdown. There are similar nonlinear effects in the propagation of microwaves. However, they are different in character or

233

Microwaves

Figure 4-18. Ducting of a Microwave Beam near the Ocean's Surface

Beam bent back toward surface

since dn/dh < -160x10-6/km

Beam reflects from

ocean surface

Ocean's Surface
magnitude from those affecting lasers, due to the different frequencies and spatial scales involved.

Air Breakdown. The ultimate limit to atmospheric propagation

is air breakdown, in which a beam’s intensity is sufficient to heat

free electrons within its volume to the point where they can ionize

neutral atoms, multiplying their number in an electron cascade.

The electron density grows until the air has been transformed into

an absorbing plasma and propagation is no longer possible.24

The electron cascade which leads to breakdown occurs in the

following way: 

• Free electrons which are naturally present within the beam

volume heat up by absorbing electromagnetic radiation from

the beam. These initial electrons (about 100/cm3) arise through

cosmic rays and other natural sources.

• As the electrons heat, they may gain sufficient energy to ionize a neutral atom or molecule when they collide with it. To do

so, an electron’s energy must exceed the ionization potential of

a molecule in the air. Ionization potentials are on the order of

10–20 eV. (1 eV, or electron volt,  1.6  10–19 Joules.)

• The electron density grows as ionization occurs and the

newly born electrons also heat and ionize molecules. Eventually, almost all the neutral molecules are ionized, and the gas

becomes a conducting plasma which absorbs the incident radiation, making propagation impossible.

The key to a quantitative analysis of breakdown lies in knowing

the rate at which electrons will heat by absorbing electromagnetic

radiation, the rate at which they may lose energy through collisions

with atmospheric particles, and their probability of ionizing molecules as their average energy increases. Of these factors, only the

heating rate depends upon the type of electromagnetic radiation.

The rate at which electrons heat in the presence of electromagnetic

radiation of intensity S (W/cm2) is25

d
  e2Sc

dt 2mc
o (2  c

2)

Effects of Directed Energy Weapons

234
where 
 is the electron energy (Joules), e and m are the electron

charge (Coulombs) and mass (kg), c is the frequency with which

electrons collide with neutral molecules (sec–l, or Hz), c is the speed

of light (m/sec),  is 2 times the frequency in Hertz of the radiation, and 
o is a constant known as the permittivity of free space

(8.85  10–12 fd/m). The factors which appear in this expression for

the electron heating rate are discussed in detail in Chapter 3, and

make sense on physical grounds. We certainly expect d
/dt to be

proportional to the intensity of radiation, S. The factor c in the

numerator is the frequency with which an electron collides with a

heavy molecule, and reflects the fact that it is the friction between

the electrons and the background gas which results in their heating.

The factor (2 + c

2) in the denominator of the electron heating

rate expression is particularly interesting and significant from the

standpoint of the difference between electron heating at optical and

microwave frequencies. This factor is related to the average gain in

energy which an electron achieves every time it collides with a neutral molecule. If an electron is simply bobbing up and down at the

radiation frequency  between collisions, the longest time it will

gain energy is about 1/—the time an electron moves in one direction before the field reverses direction and begins decelerating the

electron, reducing its energy. Thus, if  is very large, the electron

simply quivers in place, gaining very little energy between collisions. On the other hand, if the collision frequency c is greater than

, the electron cannot gain energy for a full cycle of the radiation

frequency—its energy gain will be interrupted by the collision.

Thus, the factor (2 + c

2) in the denominator of the heating rate expression reduces the energy gain appropriately in the limits of high

radiation frequency or high collision frequency.

The difference between lasers and microwaves from the standpoint of air breakdown lies in the factor (2 + c

2). A typical value

for c at sea level is about 4  1012 Hz.26 By way of comparison, the

frequency of infrared laser light is about 1014 Hz, and that of microwaves is about 107 Hz. Thus, laser-induced breakdown occurs

in the limit where the radiation frequency exceeds the collision

frequency, and microwave-induced breakdown occurs in the limit

where the radiation frequency is less than the collision frequency.

What are the implications of these two limits from the standpoint

of breakdown thresholds and how they scale with frequency and

atmospheric density (altitude)? Looking at the expression for the

235

Microwaves
electron heating rate, you can see that If  > c, the heating rate is

approximately d
/dt  e2Sc/2mc
o2; if  < c, it is approximately

d
/dt  e2S /2mc
oc. These two limits behave quite differently as

frequency or altitude are varied. In the laser limit, there is a strong

frequency dependence, and d
/dt decreases as the square of the

light frequency. Since the collision frequency is proportional to the

gas density, you can also see that in this limit the heating rate will

decline as altitude increases and density decreases. By contrast, in

the microwave limit d
/dt is independent of the radiation frequency, and is inversely proportional to the gas density. The ratio

of the laser to the microwave heating rates is (c/)2. Table 4–2

compares these two limits to electron heating by electromagnetic

radiation in a background gas.

Once we’re comfortable with the electron heating rate for microwaves, we’re prepared to deal with air breakdown. The

other factors which determine the breakdown threshold, such

as ionization rates and electron energy loss mechanisms, do not

depend on the frequency of the applied radiation. Indeed, all

of the results that were developed in Chapter 3 can be used

here, as long as we use the proper electron heating rate for

microwave frequencies.

In Chapter 3, we identified two limits to the breakdown

problem: a high energy limit, in which almost all of the energy

absorbed by the electrons was effective in ionizing molecules;

and a low energy limit, in which the input of energy to the

electrons was just equal to the rate at which energy was lost in

collisions that excited vibrational or electronic energy levels of

the gas molecules, leaving none of it available to further the

cause of breakdown. For 10.6 m laser radiation at sea level, the

first of these limits was about 10 J/cm2, and the second was

Effects of Directed Energy Weapons

236

Microwaves Lasers

Electron Heating

rate, dε/dt

Frequency Scaling

Density Scaling

Intensity Scaling

e2

S/2mcεoνc

Independent of ω

Proportional to 1/N

Proportional to S

e2

Sνc/2mcεoω2

Proportional to 1/ω2

Proportional to N

Proportional to S

Table 4-2. Electron Heating in Microwave and Laser Frequency Limits
about 3  109 W/cm2, with the transition occurring at a pulse

width of about 10–8 sec. Figure 4–19 is a summary of these results

(see also Figure 3–50).

We can modify Figure 4–19 to make it appropriate for microwaves simply by scaling the electron heating rate appropriately.

Since the ratio of laser to microwave heating rates is just (c/)2, it

follows that a lower microwave intensity, Sm  (c/)2 S1, will

heat electrons to the same degree as a laser of intensity Sl. Figure

4–19 is for sea level air (c  4 x1012 Hz), with a 10.6 m laser

(  1.8  1014 sec-1). In this case, Sm  5  10–4 Sl. This means that a

microwave beam with an intensity (W/cm2) almost 4 orders of

magnitude less than a 10.6m laser beam will heat electrons to

the same extent. Using this scaling, it follows that at short

pulses the microwave breakdown threshold should be about

(5  10–4)10 J/cm2  5  10–3 J/cm2, and that for long pulses, it

should be about (5  10–4) 3  109  1.5  106 W/cm2.

You can see from these results that microwaves will induce air

breakdown at very low intensities compared to lasers. Moreover,

the microwave breakdown threshold scales differently with altitude. As you can see from Figure 3–50, the high intensity limit to

optical breakdown is inversely proportional to the gas density N,

while the low intensity limit is virtually independent of N. The

ratio of laser to microwave heating rates, (c

2/2), is proportional

to N2, since c is proportional to N. Therefore, the high intensity

limit to microwave breakdown is proportional to N, and the low

intensity limit is proportional to N2. These results are summarized

in Figure 4–20, a curve for microwaves which is analogous to

Figure 3–50 for lasers.

237

Microwaves

Figure 4-19. Breakdown Threshold for Air at Sea Level, 10.6 m Radiation

Intensity,

S(W/cm2)

10 12

10 11

10 10

10 9

10 J/cm2

3 x 109 W/cm 2

10.6 μm Radiation

Sea level air density

10 -11 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 Pulse Width,

t

p. (sec)
Aerosol-Induced Breakdown. In Chapter 3, much attention

was paid to the effect of dirty air on breakdown. There, we saw

that small, micrometer sized particles in the atmosphere could

absorb energy from a laser beam, vaporize, and lower the breakdown threshold from that expected for clean air. Is this effect important in microwave induced breakdown? You may recall that

the cross section for attenuation of radiation of wavelength by

particles of radius a falls rapidly to zero if 2a/<< l (see Figure 3–28). Atmospheric particles are 10m or less in size (see Figure 3–27), while microwaves have wavelengths of about cm.

Therefore, 2a/is about 10–2 or less, and very little microwave

energy is absorbed in these aerosols. Accordingly, they do not

heat as much as they would in the presence of laser light, and are

less likely to serve as a source for the initiation of breakdown.

Additionally, the microwave breakdown threshold is quite low

even in pure air—on the order of 106 W/cm2 or less, as compared

to 109W/cm2 or more at laser frequencies. Thus, initiation of

breakdown from aerosols, which is roughly wavelength independent, is unlikely to be an important effect at microwave

wavelengths, where the pure gas breakdown threshold lies

below the threshold associated with the impurities in a dirty gas.

Plasma Maintenance and Propagation. Once breakdown has

taken place, how does the resulting plasma interact with the radiation? In Chapter 3, we saw that plasmas could be maintained

Effects of Directed Energy Weapons

238

Figure 4-20. Air Breakdown Threshold at Microwave Frequencies

Pulse Width, sec

Intensity,

S(W/cm2)

10 8

10 6

10 4

10 2

3 x 1019/cm 3, sea level

3 x 1018/cm 3, 20 km

3 x 1017/cm 3, 40 km

10 -10 10 -8 10 -6 10 -4
at intensities below the breakdown threshold, and could propagate towards the source of radiation as supersonic detonation or

subsonic combustion waves. Do similar phenomena occur when

microwaves interact with a breakdown plasma?

A plasma will propagate if sufficient microwave energy can be

absorbed to replace energy it loses through radiation, thermal

conduction, and heating up the cold, ambient air which it entrains as it propagates. Thus, as with breakdown, the key factor

in plasma maintenance and propagation is the absorption coefficient of a plasma with a given electron density for microwave radiation of a given frequency. Let us consider, then, how a plasma

interacts with microwaves.

The interaction of microwaves and plasmas has been studied

for some time, and indeed microwaves have long been used as

diagnostic tools to probe and measure plasma properties.27 If

air at sea level is fully ionized, the electron density will be

about 1019/cm3. At this density, the plasma frequency, p, is about

1013 Hz (see Figure 3–14). You will recall that if the frequency of

radiation incident on a plasma lies below the plasma frequency,

electrons within the plasma can shield its interior from the incident radiation. The radiation will primarily be reflected, and absorption will occur only within a short distance of the plasma’s

surface (the skin depth). Lasers, having frequencies of 1014 Hz and

greater, are able to penetrate to the interior of a breakdown

plasma, where they are efficiently absorbed. Microwaves, with

frequencies of 1011 Hz and less, are largely reflected from a

plasma whose electron density equals that of air at sea level. This

means that microwaves cannot support plasma propagation as a

detonation wave. This type of wave propagates supersonically,

with a density roughly equal to that of the surrounding atmosphere (see Figure 3–55). The threshold intensity for such a wave,

even assuming that all the incident radiation is absorbed in the

shock front and serves to propagate it, is about 107 W/cm2. This

is well below the threshold for laser-induced breakdown, but

well above the threshold for microwave induced breakdown.

This means that microwave-supported detonation waves are not

a real problem from the standpoint of microwave propagation.

Provided the intensity of the beam has been kept below the air

breakdown threshold, these waves will not occur.

239

Microwaves
Microwaves can, however, support plasma propagation as a

subsonic combustion wave. This type of wave is in pressure balance with its surroundings, so that at the high temperature associated with a plasma, its density is much less than that in a detonation wave (see figure 3–54). Combustion waves are a common

phenomenon in waveguides for high power microwaves, where

an impurity within the waveguide, such as a flake of metal, can

ignite a plasma which absorbs or reflects most of the radiation,

preventing further propagation through the guide.28 These

waves can be maintained at intensities below that for breakdown, and have been seen at intensities of a few hundreds

W/cm2. By contrast, the microwave breakdown threshold in air

is on the order of 106 W/cm2 (see Figure 4–20). Typical wave velocities are on the order of 100 cm/sec, similar to laser-supported

combustion waves (see Figure 3–61).

The quantitative analysis of microwave-supported combustion

waves can be performed in a manner analogous to that which we

outlined in Chapter 3 for laser-supported combustion waves.

The major difference lies in the fact that since the plasma frequency may exceed the microwave frequency, a substantial portion of the incident radiation will be reflected, rather than absorbed. You may recall from Chapter 3 that simple solutions for

wave thresholds and velocities are not possible except under

simplifying and largely unrealistic assumptions. The same situation applies in the microwave case.

Fortunately, combustion waves are unlikely to have an effect

on propagation for two reasons. First, there are no sources for the

ignition of these waves other than at the surface of a target.

While laser-supported combustion waves could be ignited in

dirty air breakdown from aerosols, we have seen that these

aerosols do not interact with microwaves to any appreciable extent. Second, even if they should be created near a target surface,

they are unlikely to propagate away from the surface, decoupling the radiation from it. This is because the velocity of a combustion wave is so slow that either naturally occurring winds or

those generated by the motion of the target are likely to blow it

out as it tries to propagate towards the microwave transmitter.

Therefore, microwave supported plasmas are more of an issue

from the standpoint of target interaction than beam propagation.

Effects of Directed Energy Weapons

240
Thermal Blooming. In thermal blooming, heating of the air

through which the beam travels leads to density and index of refraction gradients that cause the beam to expand at a greater rate

than it would in a vacuum (see Figure 3–40). Is this likely to be a

significant effect for microwaves? The answer is no, due to the

much larger spatial scale of a microwave beam. You may recall

from Chapter 3 that there are two ways to prevent thermal

blooming from affecting beam propagation. The first is to use a

beam whose pulse width is shorter than the time necessary for

blooming to develop (Figure 3–41), and the second is to choose

beam parameters such that the thermal distortion number is less

than unity (Figures 3–42 and 3–44). Both of these approaches are

easier as the beam radius increases, and a microwave beam will

have a radius at least 104 greater than a laser beam with the same

divergence. Moreover, the intensity of a microwave beam is

likely to be less than that of a laser beam, since the air breakdown threshold is less for microwaves, and microwaves are

more likely to be employed for soft damage that can be accomplished at lower intensities. All of these things combine to remove thermal blooming as an issue from the standpoint of microwave propagation.

Summary: Nonlinear Effects.

1. Air breakdown is the most significant nonlinear effect for

microwaves. Its threshold is many orders of magnitude less for

microwaves than for lasers, and it scales differently with altitude and frequency as well, (see Figure 4–20)

2. Once breakdown has occurred, plasma evolution is more

complex for microwaves than for lasers, since the plasma

frequency in breakdown plasmas may exceed the frequency of

the radiation. In this case, much of the radiation is reflected

rather than absorbed in the plasma. A strongly-absorbing

detonation wave plasma cannot be maintained by microwaves, though a low density combustion wave can.

3. Other non-linear effects which are important in the propagation of lasers are not significant for the propagation of microwaves, either because the spatial scales are too large (e.g. thermal blooming), or because their thresholds lie above the

241

Microwaves
threshold for microwave-induced air breakdown (e.g. dirty air

breakdown).

Summary: Propagation in the Atmosphere

For microwaves as for all weapon types, propagation in the atmosphere is considerably more complex than propagation in a vacuum. A microwave beam will lose energy through absorption, with

water, both liquid and vapor, contributing significantly to the attenuation coefficient. The natural variation of the atmospheric index of

refraction with altitude will make a microwave beam bend, extending its range along the surface of the earth beyond that achievable

in a vacuum. Proper atmospheric conditions can even cause the radiation to proceed over great distances along “channels” in the atmosphere. And if a beam’s intensity becomes too great, the atmosphere will break down, propagation ceasing. None of these effects

are truly new; they affect lasers as well, and were introduced in

Chapter 3. However, the relative importance and scaling of these effects differs from lasers to microwaves. Table 4–3 is a summary of

those things which affect both laser and microwave propagation,

highlighting their similarities and differences.

Implications

Microwaves are qualitatively different from lasers as directed energy weapons. In the popular mind, lasers are seen as bullets—

flashes of light which proceed in a straight line toward targets

which are destroyed by the holes drilled in them. While not strictly

correct, this view is close enough to reality that it’s not terribly misleading. If a laser beam is like a bullet shot from a rifle, a microwave beam is more like the shot from a shotgun. Through divergence, the energy spreads with distance, and the effective range

is less. Moreover, the unusual paths which a beam may take in the

atmosphere, together with the intensity limitations imposed by

breakdown, suggest that microwaves are unsuited for the precision

destruction of threatening targets. Rather, they’re more likely to be

fired at an array of targets, hoping that most will receive a dose of

energy sufficient to exploit internal vulnerabilities and result in target negation even at intensities below that required for physical destruction. Therefore, a crucial and difficult to quantify problem in

microwave weaponry is to understand the mechanisms by which

they might damage targets.

Effects of Directed Energy Weapons

242
Microwave Interaction with Targets

Microwaves have an inherent divergence, rather unpredictable

propagation in the atmosphere, and a low threshold for atmospheric breakdown. These things make them unsuitable for hard

kill of a target through the deposition of sufficient energy to melt

or vaporize it. Therefore, the thought of microwaves as directed

energy weapons usually begins with the assumption that the target is soft, having a built-in vulnerabilities which microwaves can

exploit.

Unfortunately, any vulnerabilities a target may have to microwaves will depend on its construction—the frequencies it is

designed for, the arrangement of its components, their shielding

by the black boxes which encase them, and so forth. Therefore, the

243

Microwaves

Microwaves

λ= 0.1-100cm

Lasers

λ= 0.4-10μm

Attenuation

(absorption/scattering)

Main causes

Main cures

Index of Refraction

Changes/gradients

Main effect

Main cures

Atmospheric molecules

liquid water drops (<1mm)

Increase intensity

Bending and ducting

None

Atmospheric molecules

dust particles (<10μm)

Increase intensity

choose λ in a propagation

"window"

Increased divergence

Adaptive Optics

Self-induced

Refractive effects

Main effect

Main cures

Thermal Blooming

Adaptive Optics

None

None

Atmospheric Breakdown

Main causes

Main cures

Breakdown Plasma

maintenance

electron cascade

dust (aerosol) initiation

Operate below threshold

Combustion Waves

Detonation Waves

electron cascade

Operate below threshold

Combustion Waves

Table 4-3. Issues Affecting Microwave and Laser Propagation in the Atmosphere
data needed to assess a target’s vulnerability are not generally

available. If we want to drill a hole in a target with a laser, the

thermal properties that we need to know to calculate a damage

threshold are available. But if we want to damage the logic circuit

in the guidance section of an anti-aircraft missile, we will have

little information to use in predicting its damage threshold. Therefore, this discussion of microwave interaction with targets must

be general and qualitative. Detailed analysis of microwave-target

interaction cannot be done on a zero-order basis as it can for the

other directed energy weapons discussed in this book.

Mechanisms of Soft Kill

Since we anticipate that we can’t use microwaves to blow

targets out of the sky, it’s logical to ask what we can accomplish.

How can targets be damaged or negated below the intensity

necessary for physical destruction? There are two ways in

which microwave radiation can get into and damage a target.

The easiest way is if the target is itself a microwave receiver,

such as a radar or communication link. In this case, the target is

designed to detect, amplify, and process microwaves at some

specific frequency. If the attacking microwaves have the same

frequency and a much greater intensity than those which were

anticipated, you can easily imagine that the target’s circuitry

might be damaged. This is known as in band damage, since the

microwave weapon operates in the same frequency band as the

target. The opposite case, in which the attacking microwaves are

at a frequency unrelated to the nature of the target, is known as

out-of-band damage. Both modes of damage have advantages and

disadvantages.

In-Band Damage. In an in-band attack, the idea is to exploit the

circuitry within the target. Consider, for example, the situation illustrated in Figure 4–21, where a radar illuminates a target at

some range z, and the target returns fire with microwaves of the

same frequency. Of necessity, the radar receiver must detect and

process low intensity radiation, since only a small fraction of the

power that it transmits is received in the return signal. If the radar

beam has a brightness B, the intensity of its radiation at a range z

is B/z2. Only a small fraction of the radiation received at the target

Effects of Directed Energy Weapons

244
is reflected back towards the radar. This fraction is quantified by

defining the target’s effective brightness as seen by the radar as

B’  B4z2, where  is the radar cross section of the target.29 The

intensity of the radiation returned to the radar is then S  B’/z2 

B/4z4. The important point to note is that the intensity of the

received signal decreases as the fourth power of the range. This

makes physical sense, since the intensity falls off by a factor of

1/z2 in propagating to the target, and the reflected radiation again

falls off by a factor of 1/z2 in returning to the radar. If it is to be

245

Microwaves

Figure 4-21. Engagement between a Radar and a Target

c. The Target illuminates the radar with microwaves

b. Energy reflected from the target is received and processed

a. Radar illuminates target

Z

Z

Z

Transmitted Brightness B Intensity at Target S = B/z2

Small fraction of power

sent is received

Received intensity S = Bt/z2 Brightness Bt

Radar Target

Radar Target

Radar Target
useful, the radar must detect targets at long range, so that the received intensity could be quite low.

Now suppose that the target, recognizing that it has been

“painted” by the radar, returns fire with a beam of microwaves of

brightness Bt. When it arrives at the radar, this beam will have an

intensity St  Bt/z2. The ratio of the signal sent by the target to the

signal returned from the target is simply St/S  Bt 4 z2/B. This

ratio will be greater than one, and the radar will be “jammed” and

incapable of discerning its signal from the noise it receives, as long

as Bt/B >/4 z2. Typical radar cross sections are quite low

(square meters or less) compared to typical ranges, which are kilometers or greater.30 Thus, the jammer can be effective even if its

brightness is much less than that of the radar, an advantage which

becomes greater as the target and radar move farther apart. The

situation is analogous to looking for someone in the dark with a

flashlight. If the person sought has a flashlight of his own and

shines it in your eyes, you will be unable to identify or locate him

with the reflected light from yours.

In-band attack is not limited to jamming. If the power of the

attacking microwaves is great enough, it can damage the radar’s

circuitry. This is because the receiver must detect and amplify

very weak signals. If the received signals are too great, currents

can be induced which are so great that they will burn out eleEffects of Directed Energy Weapons

246

Figure 4-22. A Wire in an Electrical Circuit

W

L

d
ments in the radar’s circuit boards. Modern equipment is particularly sensitive to this type of damage, since the wires and other

circuit elements are thin strips of metal or semiconductor etched

on a board. It does not take too much current before they can

heat to the point of self destruction. Consider, for example, the

wire illustrated in Figure 4–22. The electrical resistance of such a

wire is R  L/wd where  is the conductivity (mho/m) of the

wire’s material and L, w, d are its length, width, and thickness.

The resistance of the wire goes up as its cross sectional area, wd,

goes down, and in modern circuits there is motivation to make

these dimensions as small as possible.

If a current I flows through the wire, the power dissipated

within it is simply P  I2R.31 This means that in time t an energy E

 Pt  I2Rt will have been deposited in the wire by the flowing

current. In any circuit design, allowances must be made for cooling or heat sinks to get rid of this energy before wires and components are damaged by the associated rise in temperature. The design is made based on assumptions regarding the maximum

power to be dissipated, and if that maximum is exceeded, temperature can rise to the point where damage will occur. If the wire

shown in Figure 4–22 were thermally insulated and could not get

rid of any energy, it would begin to melt when the energy deposited was sufficient to raise its temperature to the melting

point, or when I2Rt  I2Lt/wd  	wdLC(Tm – To), where C is

the heat capacity of the wire’s material, 	 its density, Tm its melting point, and To its initial temperature (see Chapter 1). This expression may be solved to find the current I necessary to melt the

wire in time t: I  wd [C	(Tm – To)/t)l/2. This expression is plotted in Figure 4–23.

On a printed circuit board, a wire might be less than a micrometer in thickness, and about a micrometer in width, for a cross sectional area of less than 10–8 cm2. Less than a milliamp of current

could be sufficient to melt conducting paths on a circuit board

with wires of this size.

The solid state devices that these wires connect can be even

more sensitive to transient currents and voltages. Unlike metals,

whose resistance increases with temperature, semiconductors

have a resistance which decreases with temperature if the temperature exceeds some critical value, as illustrated in Figure

4–24. You may recall from Chapter 3 that a semiconductor is a

247

Microwaves
insulator with a relatively narrow band gap, so that at higher

temperatures electrons can be thermally excited from the valence band into the conduction band. Once this happens, the

current flow increases, since there are more current carriers, and

the rate of energy deposition P  I2R increases, since it is proportional to the square of the current. Thus, an instability can

develop in a semiconductor at elevated temperatures, in which

a temperature rise leads to increased current flow and an even

greater heating rate and more rapid temperature rise. This

process is known as thermal runaway, and has been the subject of

considerable research as a mechanism for semiconductor failure.

Clearly, excessive transient currents have the potential to produce a destructive rise in temperature. Circuit designers try to anticipate such events, and include in their design circuitry to shunt

off into harmless channels current and voltage signals which exceed some threshold. But neither the attacker nor the defender

knows how effective these measures might be in the face of pulse

forms and intensities which are unknown in advance. And the

present trend towards ultra miniaturization and large scale integrated circuits means that space on a circuit board is at a premium, leaving less room for such luxuries as protective circuitry

and wider, more robust current paths.

Effects of Directed Energy Weapons

248

Figure 4-23. Current to Melt Copper Wire vs Time and Wire Cross Section

102

1

10-2

10-4

10-6 10-4 10-2 1

wd =

10-6cm2

10-8cm2

time (sec)

Current

(Amps)
The advantage to an in-band attack on a microwave device is

that weapon power is most efficiently used. It enters the target via

the front door, an antenna which is designed to receive microwaves, and the target’s internal circuits may amplify the received signal to even greater and more destructive power levels.

The disadvantage to such an attack is that you need to know in

advance the frequency at which your untended target is operating, or you need to obtain that information in real time and adjust

the weapon’s output appropriately. The target may not sit idly by

while this done. A common countermeasure to in-band attack is

frequency hopping, in which a radar, for example, changes frequency with each pulse, and thus is not jammed by a return signal

at a frequency it is no longer looking for. The counter to this countermeasure is to attack over a broad band of frequencies, but this

carries with it the disadvantage of spreading the weapon’s available energy over a broad band, so that only a fraction of the total

output power will be received by the target. In the end, if the desire is to be able to attack generic targets, it may be more appropriate to consider an out-of-band attack.

249

Microwaves

Figure 4-24. Resitivity of Silicon Doped with 1013

Carriers per Cubic Meter(32)

Temperature, 0C

0 400 800

Resitivity

(ohm-cm)

103

102

1

10-1

10-2

Room Temperature
Out-of-Band Damage. In an attack on a target with radiation

not in its frequency band, the damage mechanism is simply raw

power absorbed by the target’s circuits. It might seem as though

this would be difficult to do, but there are many examples of

random electromagnetic radiation damaging electronic circuits.

Lightning storms, for example, can induce fluctuations in the line

voltage supplied to houses, so that computers, microwave ovens,

and similar devices are frequently damaged if not protected by

surge suppressors. Even the small electrical discharge associated

with someone walking across a carpet in dry weather has been

associated with loss of data or damage in computer systems, so

that in the workplace computer stations are frequently located on

grounded surfaces, and operators are asked to touch such a

grounded surface before touching the equipment. And the military has invested considerable time and effort in protecting its

systems against EMP—the electromagnetic pulse of radiation

which accompanies the detonation of a nuclear weapon, and

which can damage electronic equipment at great ranges from the

location of the detonation.33

How does microwave energy get into a target, and what does it

do when it gets there? Energy cannot enter through metallic surfaces, which are highly reflective of microwaves, and typically

enters through windows or gaps in the target’s surface. Due to

their long wavelength, microwaves are strongly diffracted as they

pass through small apertures, and can irradiate areas which

would seem to be well shielded by the target’s external surface. A

target designer seeks to keep potential entry points to a minimum,

but will never know without extensive testing how successful he

has been. An elaborate program of shielding and testing against

microwave attack may be practical for a small number of extremely important targets, such as strategic bombers or missiles,

but is impractical for a large number of military systems such as

trucks, small radios, and so forth. All such items contain solid

state circuitry and may be vulnerable to attack, though the level of

vulnerability is virtually impossible for either the attacker or defender to quantify.

Once microwaves enter a target, they can be absorbed in electronic circuit elements and damage them. Since the microwave

radiation is a time-varying electromagnetic field, it can induce

currents in the target’s circuits, resulting in heating and thermal

Effects of Directed Energy Weapons

250
damage. Even metallic strips will be subject to absorption and

damage. Their thickness is typically comparable to the skin

depth to which microwaves can penetrate metal, and they are

laid down on a dielectric surface which may be a thermal insulator. This inhibits the efficient transport of energy away from

them. Table 4–4 shows the skin depth and absorptivity of copper

as a function of microwave frequency.34

From the table, you can see why metals are considered effective shields against microwaves—they don’t penetrate very far,

and only a small fraction of the incident intensity, on the order of

10–4, will be absorbed. Nevertheless, this small amount of penetration and absorption can damage the thin conducting paths on

a circuit board, which are only 1–10 m in thickness, and are frequently thermally isolated, so that their temperature can build

up rapidly.

As an example, consider the conducting wire of Figure 4–22. If

a fluence F is incident on the wire, and it has absorptivity , the

energy absorbed will be FwL. The mass of the wire is wLd	,

and so the energy required to bring it to the melting point is

wLd	C(Tm – To), where Tm is the melting temperature and To is

the initial temperature of the wire. Equating the energy absorbed

to the energy necessary to bring the wire to the threshold of

melting and solving for F, we find F  d	C(Tm – To)/. If we assume as typical values that the thickness d is 0.5 m and that 

 10–4, we find that for copper35 the necessary fluence is 1800

J/cm2. This value is an order of magnitude less than the all purpose damage criterion of Chapter 1, even though only a small

fraction of the incident energy is absorbed. The wire is so thin

and tenuous that there is little mass to melt, and it doesn’t take

much absorbed energy to melt it.

251

Microwaves

Table 4-4. Skin Depth and Absorptivity of Copper at Microwave Frequencies

0.1

1.0

10

100

6.6

2.1

0.66

0.21

2.8x10-5

8.8x10-5

2.8x10-4

8.8x10-4

Frequency (GHz) Skin Depth, δ (μm) Absorptivity, α
The fluence just derived would be much reduced if the absorptivity were greater. In Chapter 3, we saw that the coupling of radiation to a target could be enhanced if plasmas were ignited at

the target surface (see Figure 3–74). The criteria for this enhanced

coupling were that the plasma not propagate away from the surface, and that the bare target surface couple poorly to the incident radiation (see Figure 3–73). Since microwaves have a very

low absorptivity at metal surfaces, and since they will only support slow moving combustion waves, these criteria are likely to

be satisfied in microwave-target interaction. It would be necessary, of course, for the microwave intensity to exceed the threshold for the ignition of such plasmas at the target surface. This

threshold lies in the range 102–103 W/cm2.

36

Estimates of Damage Thresholds

Damaging targets with microwaves is quite different from damaging them with the other directed energy weapons discussed in

this book. If we were after hard kill, it would be a straightforward exercise to calculate damage thresholds and then incorporate propagation constraints to design a weapon for a particular

application. But when dealing with soft kill, specific details of the

target’s construction come into play. For microwaves, the problem

is exacerbated since their long wavelength means that diffraction

and interference effects within the target make the specific environment that it sees difficult to predict, and difficult to duplicate

under operational conditions even if it has been measured in laboratory experiments.

In view of these uncertainties, we cannot develop definitive

damage criteria for microwave weapons. About the most we can

do is review some published estimates for the levels of energy or

power necessary to achieve damage, either in or out-of-band. A

single number for the damage threshold for a target is not of much

value, unless we know the conditions under which that number

was obtained. Published damage thresholds are most useful if

presented along with the mechanism of damage and the limits on

pulse width for which that mechanism is likely to determine the

threshold. Typically, however, this is not the case. At least in the

more popular literature, it is more likely that a single damage

estimate will be published. In this circumstance, it should be

Effects of Directed Energy Weapons

252
assumed in the absence of other data that a quoted fluence level is

appropriate for short pulses, and a quoted intensity level is appropriate for long pulses, with the definition of short and long being

undefined. About the most that can be said is that times less than

10–8 sec are almost certainly short, and times greater than 10–3 sec

are almost certainly long (see Chapter 1).37

253

Microwaves

In-Band

Out-of-Band

Thermal (target

melting, vaporization)

10-5 10-2

10-8 - 10-6

10 - 100

103 - 104

Type of Damage Fluence Threshold

(J/cm2)

Intensity Threshold

(W/cm2)

Table 4-5. Damage Estimates for Microwaves(38)

-

Figure 4-25. Microwave Range vs Output Power and Damage Criterion

Output Power (W)

Assumes λ/D=3x10-2

Range

(km)

106

104

102

1

10-2

105 106 107 108 109

Strategic Ranges

Tactical Ranges

In-Band

Out - of - Band
Keeping the uncertainties associated with published damage

levels in mind, Table 4–5 provides a summary of some damage

threshold estimates for microwaves which have appeared in the

literature.

As you can see from the table, there is considerable uncertainty

in the energy and power levels at which microwaves might damage targets. Bearing this in mind, Figure 4–25 provides a graphical

summary of some of the damage criteria in Table 4–5, showing the

range to which targets might be damaged to different levels with

microwave devices of different wavelength and output power.

Figure 4–25 has been drawn for /D  3  10–2—the value appropriate for 3 cm microwaves emitted from a 1 m aperture. The

ranges in the figure scale inversely as /D, so that if the aperture

were 10 m (or the wavelength 3 mm), the curves would all move

up by an order of magnitude. Clearly, the greatest potential for microwave devices of modest output power is in achieving in-band

damage of electrical equipment. Only for such soft damage can

damage levels be exceeded over ranges of interest. There is a large

region of uncertainty, where powers exceed those necessary for inband damage, yet fall below those necessary for out-of-band damage. Within this region, it is likely that in some engagements targets would be damaged, and in others not, depending on the

nature of the target and its vulnerabilities.

Summary: Target Interaction

1. Microwaves are likely to damage targets through soft kill

mechanisms—those which exploit inherent target vulnerabilities. There are two types of soft kill: in-band and out-of-band.

2. With in-band damage, microwaves enter a target through its

own antenna. This requires that the attacking microwaves be of

the same frequency as those the target is tuned to receive. Damage occurs when the target’s circuits are loaded beyond their design capacity.

3. In out-of-band damage, microwaves enter the target

through the back door—apertures which were not designed for

their entry. Damage occurs as the microwaves are absorbed in

thin, sensitive electronic components, heating them to the point

of damage.

Effects of Directed Energy Weapons

254
4. Because of uncertainty in knowing by what paths microwaves

will enter the target, and in predicting the intensity that components will see on the interior, it is difficult to develop definitive

damage criteria for microwaves. Table 4–5 and Figure 4–31 provide a summary of some limited data, and suggest that microwaves can certainly be used for in-band damage, can occasionally be used for out-of-band damage, and are unlikely to be used

for hard or thermal damage of target structural components.

Implications

Anecdotal evidence has always suggested that microwaves have

great potential as weapons. Radiation at microwave frequencies

from radars, lightning, and microwave ovens has been associated

with the damage of various types of equipment. Unfortunately,

the difficulty in quantifying these effects and in knowing the vulnerabilities and shielding in potential targets makes microwaves

difficult to rely on from a military standpoint. The major exception

is in the area if in-band attack, where jammers are an established

element in the arsenal of electronic warfare. If microwaves find

broader uses as weapons, it will be because devices developed as

jammers are found through experience to have broader application in achieving out-of-band damage. As technology makes

greater powers available, it will be possible to exploit targets of

255

Microwaves
opportunity and see what happens. The uncertainty in performing

a priori calculations of damage makes it unlikely that a microwave

weapon will be developed from first principles for the attack and

damage of a specific target.

Notes and References

1. Figure 4–1 was adapted from Stephen Cheung and Frederic

H. Levien, Microwaves Made Simple: Principles and Applications.

(Dedham, MA: Artech House, 1985). This book is a good introductory reference on the principles of microwave generation

and engineering.

2. The older frequency band designations originated during WW

II as a means of keeping the frequency of radars classified. From

this beginning, they developed acceptance as a convenient form of

shorthand. However, it’s easy to become confused when comparing articles in which band designations are used, since some of the

letters have changed meaning over the years, and some engineers

persist in using the older designations. A “C band” device operates at about 6 GHz using the WW II-derived band designations,

and at about 0.5 GHz using the newer designations. Authors are

not always clear on the convention they’re using. The older designations are common in radar engineering, and the newer ones in

other applications.

3. One reason why water interacts strongly with microwaves is

that it has a permanent electric dipole moment. This means that the

centers of positive and negative charge are separated in a water

molecule, so that it interacts with an applied electric field just as a

permanent magnet interacts with an applied magnetic field.

4. Because of their larger wavelength, there are diffraction effects

with microwaves that make the pattern of radiation from an

antenna less than the ideal illustrated in Figure 4–4. The main

portion of the beam (or main lobe) behaves as illustrated in the

figure, but some fraction of the energy goes out to the side just as

diffraction carries energy into regions of geometric shadow. This

diffracted energy is called side lobe radiation.

Effects of Directed Energy Weapons

256
5. Brightness, as we use the term here, refers to main lobe radiation. In microwave engineering, the concept of “antenna gain” is

usually used instead of brightness. The gain, G, of a directional

antenna is the ratio of the intensity along its preferred direction to

what the intensity would be if the antenna radiated uniformally in

all directions (isotropically). In terms of G, the intensity at a range

z is S  PG/4  z2. Thus, brightness may be expressed in terms of

antenna gain as B  PG/4, and the gain of a directional antenna

having aperture D is G  4B/P  4D2/2. A good discussion of

antennas and their radiation patterns can be found in Chapter 16

of Cheung and Levien (note l).

6. If you imagine a spreading beam passing through a sphere that

surrounds the beam’s source, the solid angle that the beam encompasses is the constant of proportionality between the area through

which it passes and the square of the sphere’s radius. Therefore,

an isotropic source, spreading equally in all directions, is spreading into a solid angle of 4 sr. This definition is analogous to that

of plane angles, for which the angle in radians between two diverging lines is the ratio of the arc of a circle they cut off to the radius of the circle.

7. There are a variety of different measures of beam strength to

be found in the literature. Our definitions of fluence

(energy/area), intensity (power/area) and brightness (power/

solid angle) are common in directed energy work. Other terms

may be used for these quantities, so it’s useful to check on the

units of any quantity used in the literature as a measure of beam

strength. See Table C in Chapter 17 (Optics) of Herbert L. Anderson (ed.), Physics Vade Mecum. (New York: American Institute of

Physics, 1981).

8. A good quantitative discussion of diffraction from a straight

edge can be found in section 18.11 of Francis A. Jenkins and Harvey E. White, Fundamentals of Optics, 3rd ed. (New York: MCGrawHill, 1957).

9. Some nice pictures of different diffraction patterns can be

found in Chapter 10 of Eugene Hecht and Alfred Zajac, Optics

(Reading, MA: Addison-Wesley, 1976).

10. The gaps or slits through which microwaves enter targets

need not be physical holes. Most non-metallic materials are trans257

Microwaves
parent to microwaves, so they can penetrate through windows,

rubber grommets, weather stripping, and so on.

11. Microwave tubes are commercially available with average

powers at 3 GHz of about 1MW and peak powers of about 10 MW

(see Figure 10.2 of Cheung and Levien, note l). By contrast, the

“Report to the American Physical Society of the Study Group on

Science and Technology of Directed Energy Weapons” Rev. Mod

Phys 59, pt II (July 1987), reports that only recently has MW-class

pulsed power become available from lasers that stress the state of

the art. Of course, since laser technology is more in its infancy, it

may be that there is more potential for growth and breakthroughs

in laser power output.

12. A good summary of issues affecting microwave propagation

can be found in S. Parl and A. Malaga “Theoretical Analysis of Microwave Propagation,” Rome Air Development Center Technical Report no. RADC–TR–84–74 (April, 1984). [AD–A143–762].

13. Figure 4–8 has been adapted from Figure 2–9 of Parl and

Malaga (note 12). In reviewing this and other sources of data on

microwave attenuation, you’ll note that data are often presented in

terms of dB/km rather than km–l. A “dB” (decibel) is one tenth of

an order of magnitude. Thus, a one dB drop in intensity means that

log(Io/I)  0.1, or Io/I  1.26. This notation is common in radio and

radar engineering. The absorption length, l/K, is the distance over

which intensity falls by a factor of l/e, or Io/I  2.72. This notation

is more common in optical engineering. The appropriate conversion factor is 1 km–l  4.3 dB/km. The use of decibels as a way of

treating quantities which can vary by many orders of magnitude is

discussed in Chapter 2 of Cheung and Levien (note l).

14. Figure 4–9 is based on data from a table on p E–23 of Robert C.

Weast (ed), Handbook of Chemistry and Physics, 45th ed (Cleveland,

OH: Chemical Rubber Co, 1964).

15. The exponential decay of density with altitude is a common

approximation for the lower atmosphere. See Chapter 4 in the

APS Report on Directed Energy Weapons (note 11). This type of behavior may also be derived theoretically. See Section 6.3 in F. Reif,

Fundamentals of Statistical and Thermal Physics (New York: MCGraw Hill, 1965).

Effects of Directed Energy Weapons

258
16. The physical meaning of this integral is that the propagation

path is broken up into many small segments. Along each of these,

the attenuation coefficient is roughly constant. The total attenuation is the product of the attenuations along each segment, and the

effective “Kz” is the integral oz K (z) dz.

17. See Section 2 of Parl and Malaga (note 12).

18. Figure 4–13 has been adapted from Figure 2–10 of Parl and

Malaga (note 12).

19. Table 4–1 was developed from data in the Encyclopedia Britannica, 15th ed. (Chicago: Encyclopedia Brittanica, Inc, 1978). See

“Rain” (Micropedia, vol VIII, p. 390), and “Precipitation” (Macropedia, vol 14, p. 960). There can, of course, be considerable variation in the characteristics of any given rainfall.

20. Figure 4–14 is based on Eq. 2.2, Parl and Malaga (note 12). Figure 4–15 is based on a table of vapor pressure of water found on p.

D–92 of the Handbook of Chemistry and Physics (note 14).

21. The maximum line or sight range between a transmitter of

height ht and a receiver of height hr is Z  (2Re)l/2[ht

l/2 +hrl/2],

where Re is the radius of the earth.

22. See Parl and Malaga (note 12), Eq. 2–6.

23. For example, see C.A. Sampson, “Refractivity Gradients in the

Northern Hemisphere,” ITS Report no. OTR–75–59 (1975)

[AD–A009 503]

24. Almost everything you ever wanted to know about breakdown with microwaves can be found in A.D. MacDonald, Microwave Breakdown in Gases (New York: John Wiley and Sons, 1966).

25. See Equation 5.3 in V. E. Zuev, Laser Beams in the Atmosphere

(New York: Consultant’s Bureau, 1982).

26. See Figure 43 in M. Mitchner and Charles H. Kruger, Jr., Partially Ionized Gases (New York: John Wiley and Sons, 1973). The collision rate is   Nv, where  is the collision cross section and N

the density of molecules.

27. A brief description of plasma diagnostics with microwaves can

be found in section 4.6 of Nicholas A. Krall and Alvin W. Trivelpiece, Principles of Plasma Physics (New York: MCGraw Hill, 1973).

259

Microwaves
28. See Section 7 in Yu. P. Raizer, “Propagation of Discharges and

Maintenance of a Dense Plasma by Electromagnetic Fields,” Soviet

Physics-Upsekhi 15, 688 (May-June, 1973).

29. The idea of a radar cross section comes from the idea that if the

intensity on target is B/z2

(W/cm2), then the power reflected back

at the radar will be some area, , multiplied by that intensity. The

radar cross section is a measured quantity, and need not be the

same as the physical cross section of the target. Indeed, the idea behind stealth technology is to make  as small as possible by absorbing radar waves or reflecting them in directions away from the

radar. In general  will depend on the frequency of the radar, the

nature of the target, and the orientation between radar and target.

30. For example, fighter aircraft have radar cross sections on the

order of 2 m2, and missiles on the order of 0.5 m2. See Chapter 14

in Cheung and Levien (note 1).

31. This expression for the power dissipated on a wire can be

found in any discussion of elementary circuit theory, such as

Chapter 32 of David Halliday and Robert Resnick, Physics, Part II

(New York: John Wiley and Sons, 1967).

32. Figure 4–24 has been adapted from Figure 3 in D.C. Wunsch,

“The Application of Electrical Overstress Models to Gate Protective Networks,” in the Proceedings of the 16th Annual International

Reliability Symposium, San Diego, CA, 18–20 April, 1978. Published

by the Electron Devices Society and the IEE Reliability Group,

these annual proceedings are a good source of information on the

failure modes of circuit elements and techniques which can be

used to prevent failure from current or voltage transients.

33. EMP arises because a nuclear burst ionizes the air in its vicinity. The electrons which result from this ionization are lighter and

move more rapidly than the ions from which they were separated.

This results in a separation of charge which is asymmetrical, since

the density of the ionized air decreases with altitude. The resulting

currents generate strong electromagnetic fields. Malfunction and

damage of electronic equipment as a result of EMP were first observed during atmospheric nuclear tests in the 1950’s. See Chapter

XI of Samuel Glasstone and Philip J. Dolan (eds.), The Effects of Nuclear Weapons, 3rd ed. (Washington, DC: US Government Printing

Office, 1977).

Effects of Directed Energy Weapons

260
34. See Figures 3–15 and 3–16 in Chapter 3. Expressions for skin

depth and absorptivity may be found in any text on electromagnetic theory, such as Section 16–4 of John R. Reitz and Frederic J.

Milford, Foundations of Electromagnetic Theory (Reading, MA: Addison-Wesley, 1960).

35. The thermal properties of copper can be found in Table 1–1,

Chapter 1.

36. See Raizer (note 28).

37. These estimates of long and short time scales are of necessity

soft, since in any given interaction between a weapon and a target

the specific mechanisms of interaction and energy loss, and the

time scales associated with them, must be taken into account.

38. Generic damage thresholds such as those in Table 4–5 are always suspect, but are provided here simply to give a feeling for the

wide range of levels at which microwaves may damage targets.

The intensity thresholds are from Chapter 8 of Jeff Hecht, Beam

Weapons: The Next Arms Race (New York: Plenum Press, 1984). The

Fluence threshold is from Theodore B. Taylor, “Third Generation

Nuclear Weapons,” Scientific American 256, 38 (April, 1987).

261

Microwaves
5: PARTICLE BEAMS

Particle beams are composed of large numbers of small particles

moving at speeds approaching that of light. By “large numbers,”

we mean densities on the order of 1011 particles per cubic centimeter, and by “small” particles we mean the fundamental particles

which comprise matter: electrons, protons, or perhaps neutral

atoms, such as hydrogen. Particle beams are not easy to analyze

and understand in detail. The atomic and sub-atomic scale of the

particles in the beam means that their interactions with each other

and with the target can be understood only on the basis of atomic

and nuclear physics. The fact that the particles are traveling at velocities approaching that of light means that their propagation is

governed by Einstein’s theory of relativity. And the large number

of particles in the beam means that their interactions with one another need to be taken into account—the behavior of an individual

particle is affected by behavior of the other particles in the beam.

Relativity, nuclear physics, and many of the other concepts needed

to understand particle beams are outside the scope of everyday

experience. Perhaps for this reason, particle beams have not attracted public attention to the same extent as lasers, whose description as intense beams of light involves concepts which are

much more intuitive.

Therefore, we will devote a significant portion of this chapter to

some of the fundamental physics which affects particle beams as

they propagate and interact with matter. We cannot treat these

fundamentals in detail, but we’ll try to develop a physical feeling

for the factors which determine the effectiveness and utility of

particle beams as directed energy weapons. As always, we will

emphasize the relative magnitude of important factors and how

they scale with the parameters which characterize the beam, the

atmosphere through which it propagates, and the target with

which it interacts.
Fundamental Principles of Particle Beams

Electromagnetic Fields and Forces

There are two types of particle beams: charged and neutral. A

charged-particle beam (CPB) is made up of particles such as

electrons or protons which possess an electrical charge, and a

neutral-particle beam (NPB) is made up of particles such as

atomic hydrogen which are electrically neutral. Even an NPB

begins its existence as a CPB, since only charged particles can be

accelerated to high velocities and energies through electromagnetic forces in a particle accelerator.l Therefore, any discussion of

particle beams must begin with a discussion of electromagnetic

fields and the forces they produce on charged particles.

There exist in nature electrical fields, commonly denoted

E, and magnetic fields, commonly denoted B. These fields are

vector quantities, which means that they have associated with

them at every point in space both a magnitude, or strength, and

a direction in which they point.2 Like the gravity field, electric

and magnetic fields cannot be seen, but their presence can be

inferred from their effect on matter. When a particle whose electrical charge is q encounters an electric field of strength E, the

particle feels a force F  qE, pushing it in the same direction as

E. When a particle whose charge is q, moving with velocity v, encounters a magnetic field of strength B, the particle feels a force F

 qvB sin , where is the angle between the direction of motion

of the particle and the direction of B. The direction of this magnetic force is perpendicular to the directions of both B and v, as

shown in Figure 5–1.3

Where do these electrical and magnetic fields come from?

Interestingly enough, they originate from charged particles. A

particle of charge q (coulombs) produces an electric field which

points radially outward, and whose magnitude a distance r from

the particle is E  q/4or2 (Volts/m). A current (flow of charged

particles) of magnitude I (Amperes) produces a magnetic field

which encircles the direction of current flow, and whose magnitude at a distance r is B  I/2c2or (Webers/m2). The constant o

is known as the permittivity of free space (8.85  10–12 fd/m in

MKS units), and c is the speed of light (3  108 m/sec). These relationships are also illustrated in Figure 5–1.

Effects of Directed Energy Weapons

264
The study of charged particles and their interaction with electric and magnetic fields can be quite complex: any fields which

are present affect the particles, and particle motion, in turn,

affects the fields which are present. The complex relationships

between these fields and their forces were first fully described

mathematically by James C. Maxwell in 1873. He developed four

interrelated equations, appropriately known as Maxwell’s Equations, which form the basis of electromagnetic theory even today.

These equations quantify the experimental observation that

charged particles produce electric fields, and that moving

charged particles (currents) produce magnetic fields. In addition,

Maxwell’s equations predict that a time—varying electric field

265

Particle Beams

Figure 5-1. Electrical and Magnetic Fields and Forces

Electrical Field

Strength E.

Charged

particle,

charge q

Force, strength qE

(a) Force due to an Electrical Field

(b) Force due to Magnetic Field

Magnetic Field,

strength B

Charged particle,

charge q, velocity v Magnetic force,

strength qvB sinΘ

direction v and B

F

E

q

+ =

B

+ =

F

q,v

θ

(c) Electrical Field due to charge q

(d) Magnetic Field due to a current I

E =q/4πε0r

2

r

B=I/2πc

2

ε

0

r

Current I

q
will produce a magnetic field, and that a time-varying magnetic

field will produce an electric field. This last phenomenon results

in the propagation of electric and magnetic fields coupled together as electromagnetic radiation (Chapter 3).

Relativistic Particle Dynamics

Since electromagnetic fields produce forces on charged particles,

we’ll need to understand how these particles respond to the forces

they feel. In classical mechanics, the response of a particle of mass

m to an applied force of strength F is given by Newton’s Law: F 

ma, where a  dv/dt is the acceleration, or rate of change of the

velocity, v, of the particle. We see this law in action every day. A

force accelerates a heavier, more massive, object to a lesser extent

than a lighter one, and as long as a force is applied, the acceleration of an object continues, with its velocity becoming greater and

greater. However, this common experience fails us as an object’s

velocity approaches the speed of light, c ( 3  108 m/sec). No

force, however long applied, can induce a particle to exceed that

speed. Einstein’s theory of relativity explained this strange behavior and other contradictions to common experience that appear

when distance, length, and time scales are such that the speed of

light is an important parameter.4

Einstein’s theory extended Newton’s law to account for motion

at speeds approaching that of light. This generalization is to write

F  ma as F  dP/dt, expressing it in terms of the rate of change

of a particle’s momentum P = mv. The factor  is defined as  

l/(l – v2/c2)1/2. If v << c,  is essentially l, and dP/dt becomes

d(mv)/dt  m dv/dt  ma, since the mass m of the particle is a

constant, and the acceleration, a, is the rate of change of velocity,

dv/dt. Thus, at velocities characteristic of everyday life, Newton’s

law applies as it always has. On the other hand, as v approaches c,

you can see that  gets bigger and bigger, and the effective mass of

a particle, m, becomes greater and greater. As m increases, a

particle’s resistance to further acceleration increases, so that an applied force can never push it over the edge into velocities exceeding c. The relativistic factor, , appears as a modifier in many formulas for relativistic particles, and is plotted in Figure 5–2.

From Figure 5–2, you can see that that we don’t really need to

concern ourselves with  and its effect on particle dynamics until

Effects of Directed Energy Weapons

266
velocities get very close to the speed of light. However, once we

do approach that speed,  becomes the dominant factor in shaping

a particle’s response to applied forces.

At this point, we need to introduce some common terminology

that has evolved for use in discussing relativistic particles—those

whose velocities approach the speed of light. The ratio v/c is frequently of interest, and by convention has come to be called .

Therefore,  may be written as 1/(1 – 2)1/2. The mass of a particle

is often expressed in terms of an energy, using Einstein’s most popular formula, E  mc2. Using this relationship, the mass of an electron, 9.11  10–31 kg, corresponds to an energy of 8.2  10–14 Joules,

and the mass of a proton, 1.67  10–27 kg, corresponds to 1.5  10–10

Joules.5 The energies corresponding to the masses of electrons, protons, and other sub-atomic particles are very small, and it’s common when dealing with small quantities to introduce new units

which make them appear larger. The electron volt (1 eV  1.6 

10–19 Joules) is the traditional unit of energy in particle physics.

Physically, it represents the amount of energy which a particle having the electrical charge of an electron (1.6  10–19 Joules) gains

267

Particle Beams

Figure 5-2. The Relativistic Factor  vs v/c

10

9

8

7

6

5

4

3

2

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 v/c

γ
when accelerated by an electrical potential of one volt. In terms of

this unit, the mass of an electron is 0.511 MeV, and that of a proton

is 938 MeV. (1 MeV  106eV.) Therefore, a traditional rule of thumb

is that the mass of an electron is about half an MeV, and that of a

proton is about one GeV (1 GeV  1000 MeV.)

In addition to its mass, a particle has some kinetic energy associated with its motion. You may recall from Chapter 2 that the kinetic

energy of a bullet with mass M and velocity v is K  Mv2/2. The

kinetic energy of a relativistic particle is K  (  – 1) mc2. This definition may seem strange at first, but it can easily be shown6 that for

v << c it reduces to the traditional expression, K  mv2/2. The kinetic energy of a particle, ( – l)mc2, is equal to its mass expressed

in energy units, mc2, when   2. A good rule of thumb is that particles must be treated relativistically when their kinetic energy is

comparable to their rest energy, or mass expressed in units of energy. This corresponds to a velocity of about 0.7c. The sum of the

kinetic energy and rest energy of a particle, mc2, is often referred

to as its total energy, though of course only the kinetic energy of a

particle may be deposited in a target, damaging it.

When particles are relativistic, some of the distinctions between

them tend to disappear. This is illustrated in Table 5–1, which

shows parameters for an electron and a proton, each having a kinetic energy of 2 GeV.

The mass of a proton is about 2,000 times greater than that of an

electron. If it were non-relativistic, the momentum of a proton

would be about 45 times greater than that of an electron at the

same kinetic energy. Yet at energies where both are relativistic,

there is hardly any distinction between the momentum of an electron and that of a proton. This is because the lighter electron has a

Effects of Directed Energy Weapons

268

Kinetic Energy

K (GeV)

 γ = 1 + K/mc2 v/c = (1-1/ γ2)1/2 Momentum

P = γmv

(kg m/sec)

Particle

Electron

Proton

2

2 3 0.943 1.4 x 10-18

4000 0.999999938 1.1 x 10-18

Table 5-1. Relativistic Parameters for Energetic Electrons and Protons
much greater , making its effective mass, m, much closer to that

of the heavier proton. In this and other ways, we’ll often encounter surprising results when dealing with the relativistic particles which comprise particle beams. These results may run

counter to our experience, since velocities near the speed of light

are not a part of that experience. This can make the interpretation

of particle beam phenomena difficult.

Major Forces Affecting Charged-Particle Beams

We’re now in a position to look at the major forces affecting a

charged-particle beam (CPB) as it propagates. Let’s consider the

idealized CPB shown in Figure 5–3.

The CPB shown in the figure is an assembly of n particles per

cubic centimeter, each having a charge q and a velocity v. This assembly is confined to what is effectively an infinitely long cylinder

of radius w. What are the electric and magnetic fields resulting

from these particles, and how do these fields affect their motion?

Using Maxwell’s equations, it can be shown that the electric field

associated with these particles grows linearly with the distance, r,

from the center of the beam, reaching a maximum value of E 

qnw/2o at the beam’s surface. For r > w, E falls off from this maximum as l/r. The direction of this field is radially outward, and the

resulting force pushes the particles apart, making the beam expand. This behavior is an example of the well known fact that like

charges repel one another. The particles in the beam, having the

same charge, exert repulsive forces on one another.

269

Particle Beams

Particles of charge q, density n, velocity v

Figure 5-3. An Idealized Charged Particle Beam

W
Similarly, Maxwell’s equations may be used to show that the

magnetic field resulting from the current flow that these moving

particles represent is proportional to r for r < w, and decays as l/r

for r > w, reaching a maximum value of B  qnvw/2c2 at r=w.

The direction of the magnetic field is circular, surrounding the

beam, and results in a force which is radially inward, tending to

make the beam contract. This phenomenon represents the attraction of parallel currents that is commonly discussed in elementary

physics texts. Figure 5–4 is a plot of these electric and magnetic

fields, along with a sketch to indicate why they are repulsive and

attractive, respectively.7

Because of the electric and magnetic fields shown in Figure 5–4,

a particle at the surface of the beam feels a net force F = nq2w(1 –

v2/c2)/2o  nq2w/2o2. As you can see, this force involves a factor of l – v2/c2. The first term in this factor results from the repulsive electric force, and the second from the attractive, magnetic

force. Since v is always less than c, the repulsive term will always

dominate, and the beam will tend to spread. This is why neutral

particle beams, whose particles are not charged and therefore do

not repel one another, are favored for applications in the vacuum

of space. We’ll see later that other phenomena which arise when a

charged particle beam (CPB) propagates in the atmosphere can

serve to negate the repulsive force just discussed, making a CPB

more suited to atmospheric applications.

Effects of Directed Energy Weapons

270

2ε

0

nqw

E

W

nqvw

B

Attractive Magnetic Force

( both v and B)

2c2

ε

0

W

Figure 5-4. Attractive and Repulsive Forces in a CPB

V V

B

V
What are the magnitudes of the electric and magnetic fields

we’ve been discussing? Suppose that the CPB shown in Figure 5–3

has particles with a charge, q, equal to the electron charge: a density, n, of 1017/m3 (8 orders of magnitude below the atmospheric

density of 1025/m3), a radius, w, of 1 cm, and a velocity, v, of 0.9c.

These values result in an electric field at the surface of the beam of

E  qnw/2o  9  106 V/m, and a magnetic field of B 

nqvw/2c2o  270 Gauss. By contrast, naturally occurring electric

and magnetic fields at the earth’s surface are on the order of 100

V/m and 0.5 Gauss, respectively.8 Electric fields on the order of 3

 106 V/m are sufficient to induce the electrical breakdown of the

earth’s atmosphere and produce lightning. Clearly, the electric and

magnetic fields of a particle beam will be important in determining its behavior.

Particle Beam Characteristics

So far, we’ve described particle beams in terms of particle mass,

density, charge, velocity, and so on. These are the fundamental

quantities which enter into Maxwell’s equations for electric and

magnetic fields, and into the force equation which tells us how the

particles respond to these fields. But these aren’t the quantities

typically quoted by beam engineers, who prefer to speak in terms

of the current their beams carry, the energy of their particles, and

the brightness their beam possesses. We need to relate this traditional terminology to the more fundamental quantities we’ve been

using, as well as to parameters which can be related to damage

criteria, such as the number of Joules or Watts deposited by the

beam on its target.

Current, as electricians use the term, is a measure of the flow of

electricity. The unit of current is the Ampere. One Ampere is a

Coulomb of charge passing through a wire in one second. Imagine that current is being carried through a wire of radius w by

some charged particles, as shown in Figure 5–5. If there are n particles per cubic centimeter, and each has a charge of q Coulombs,

then there are nq Coulombs per cubic centimeter. The quantity nq

is known as the charge density, and is commonly denoted by 	. If

the particles in Figure 5–5 are moving at a velocity v, nqv

Coulombs will pass through each square centimeter of the wire in

one second. The quantity nqv is commonly called the current

271

Particle Beams
density, and denoted j. The current in the wire, or the total number of Coulombs passing through its cross-section each second, is

I  w2j   w2nqv.

Of course, the particles in a particle beam aren’t confined to a

wire, but nevertheless they are generally flowing in a straight line

with a reasonably well defined radius. Therefore, the current in a

particle beam can be defined as I  w2nqv. The particle beam of

radius 1 cm whose surface fields we calculated earlier had particles of density 1017/m3, velocity 0.9c, and charge 1.6  10–19 Coul.

This corresponds to a current of 1,360 Amp (1.36 kAmp).

It requires more than current to characterize a particle beam.

Since I  w2nqv, the same current can be achieved in a beam

with particles of high density and low velocity, or of low density

and high velocity. This ambiguity is resolved by providing the kinetic energy of the particles in electron Volts.9 As we have seen, K

can be related directly to the velocity of the particles, through the

relationship K  ( – 1)mc2, where   1/(1 – v2/c2)l/2. In the beam

used previously as an example, a velocity of 0.9c corresponds to a

 of 2.29 and a kinetic energy of about 0.7 MeV if the particles are

electrons, for which mc2  0.5 Mev.

How can the energy of individual particles and the beam’s current be related to such macroscopic parameters as the total energy

or power carried by the beam? If each particle has an energy K,

then the energy density (J/cm3) in the beam is nK, just as the

charge density is nq when each particle carries a charge q. Just as

the charge density, nq, becomes the current density, nqv, when

Effects of Directed Energy Weapons

272

Figure 5-5. Current Flow in a Wire

In time t, particles within a distance vt of the surface A will pass it. The total charge

passing through is thus nq vt A. Since A = πw2, the charge passing through A per

unit time is I=πw2 nqv

vt

W

Particles of density n, charge q, velocity v
multiplied by the particle velocity v, so also the energy density, nK,

becomes the power density or beam intensity, Knv. If K is in Joules,

n in cm–3, and v in cm/sec, then Knv will be in Joules/cm2/sec, or

Watts/cm2. So, the beam parameters we’ve been using for illustrative purposes (n  1017/m3, v  0.9c, K  0.7 MeV) correspond to

an energy density of 1.1  104 J/m3., and an intensity of S  3 

1012 W/m2 (3  108 W/cm2). If the beam has a total duration (pulse

width) of tp, the fluence (J/cm2) it delivers is just F  Stp. Table 5–2

provides a translation guide which can be used for conversions

among the quantities commonly employed in fundamental

physics, beam engineering, and weaponeering.

The quantities in Table 5–2 characterize an ideal beam, in which

there is no variation in K or v among the particles. In reality, no

beam is perfect, and a beam will consist of particles having a distribution of energies and velocities about their nominal, or average, values. Therefore, the complete characterization of a particle

beam requires some measure of its deviation from perfection to

supplement the nominal beam parameters in Table 5–2. The most

significant way in which a beam can deviate from the ideal is if

the velocities of its particles have a significant component perpendicular to the direction of beam motion. In this case, the beam will

diverge, or spread with distance, as illustrated in Figure 5–6.

In Figure 5–6 we see the contrast between a perfect beam, with

each particle’s velocity identical and oriented in the same direction,

and a real beam, where each particle’s velocity is slightly different,

both in magnitude and direction. It’s convenient to express any in273

Particle Beams

Fundamental Physics Beam Engineering Weaponeering

Particle charge, q

Beam radius, w

Particle density, n

Particle velocity,v

Current,1

I = nqv πw2

Kinetic Energy, K

K = (γ - 1)mc2

Pulse Width, tp

γ = 1/(1-v2/c2)1/2

Beam intensity, S

S = nKv

Beam Fluence, F

Pulse Width, tp

F = S tp

Table 5-2. Quantities Used to Characterize Particle Beams
dividual particle velocity, v, as the sum of the velocity vo which an

ideal particle would have, and an additional velocity, v which represents the deviation of that particle’s velocity from the ideal, as illustrated in the figure.10 When the square of v is taken and averaged over all the particles in the beam, we have a quantity, < v 2> ,

which represents in an average way the degree to which the particles deviate from ideal motion. From this quantity, an average perpendicular temperature, T   m< v 2>/2k can be defined, where k

(1.38  10–23 Joules/oK) is Boltzmann’s constant. This quantity

has a simple physical interpretation as the temperature associated

with the random motion of the beam particles in directions perpendicular to the direction of beam transport, just as the temperature

of a gas is a measure of the average motion of the particles in the

gas, a random energy divided by Boltzmann’s constant. This concept of a random beam temperature is of considerable utility, and

will allow us to discuss some aspects of beam motion and expansion by analogy with similar behavior in an ideal gas.

Other concepts of value in characterizing the deviation of a particle beam from perfection are the divergence and brightness of the

beam. A beam’s divergence is just the average11 of the angle, 

v/vo, by which the direction of motion of a particle deviates from

the beam axis (Figure 5–6). As we saw in Chapter l, a beam of

divergence will expand to have a radius w  zafter it has

propagated a distance z. This is the case for a neutral particle

beam. However, the repulsive and attractive forces that a charged

particle beam feels as a result of the charges on its particles tend to

dominate in determining how a charged beam’s radius changes

Effects of Directed Energy Weapons

274

Figure 5-6. Real vs. Ideal Beam Velocities

Perfect Beam:

Velocities uniform and

in one direction

Real Beam:

Velocities slightly different in both

magnitude and direction

θ

v

v0

v = vo + v

v
with distance. Therefore, divergence is not as useful a concept for

a charged particle beam as for a neutral particle beam.

In Chapter 3, the brightness of a laser was defined as the beam’s

power (Watts) divided by a measure of the beam’s divergence, the

solid angle into which the beam was spreading (see Figure 3–20).

Thus, a bright beam will send a lot of power into a narrow cone as

it propagates. There is a related concept for particle beams. The

brightness of a particle beam is defined as the beam current per

unit cross-sectional area per unit solid angle, or B  I/[ w2 (2)],

where I is the beam’s current, w its radius, and its divergence.

This parameter is different from the brightness defined for lasers,

although it still relates to the fundamental idea of how much

power is being sent into how small an angle. Current is a more

natural parameter for beam engineers, and is equivalent to power,

since power is energy per time and current is charge per time.

Power is related to current by a factor of K/q, where K is the kinetic energy and q the charge carried by a beam particle. fn this

way, you can convert the brightness quoted in the beam literature

to that quoted in the laser literature, making appropriate connections between these quantities.12

Propagation in a Vacuum

Having discussed some of the fundamental concepts and terminology associated with particle beams, we can now consider how

they propagate. The propagation characteristics of charged and

neutral particle beams are different, both in space (a vacuum) and

in the atmosphere. Therefore, we’ll treat each of these cases separately, beginning with the simplest case of neutral particle beams

propagating in a vacuum.

Neutral Particle Beams in a Vacuum

The concept of divergence is sufficient to understand neutral

beam propagation in a vacuum. In propagating a distance z, a

neutral beam’s radius will grow to a size w  wo 
 z, where is

the beam’s divergence and wo is the beam’s radius as it emerges

from the particle accelerator. Consequently, a beam’s intensity

will decrease by a factor of [wo/(wo 
 z)]2 in propagating a distance z. This implies that a beam of a given output intensity

275

Particle Beams
(W/cm2) can propagate only a limited distance before its intensity

falls below that necessary to damage a target in the time available. Suppose, for example, that wo is 1 cm, and that the beam can

expand by no more than a factor of 10 before its intensity becomes unacceptably low. Then the criterion z 10 cm would define the maximum range, z, achievable with a beam of divergence

. Figure 5–7 shows combinations of z and which will satisfy

different criteria for z. Since propagation in a vacuum implies

that the beam and its target are in the vacuum of space, strategic

ranges are clearly of interest in this section. These ranges might

lie somewhere between low earth orbit, 200 km, and the altitude

of a geosynchronous satellite, about 40,000 km. If we don’t want

the beam to waste energy by expanding to a size that exceeds a

typical target, its radius needs to be kept below something on the

order of 10 m. Figure 5–7 tells us that divergences on the order of

microradians or better are required if the beam is not to expand

beyond 10 m over strategic ranges.

We have seen that the divergence of a particle beam is given

by q  <v>/vo, where <v> is the average particle’s speed

perpendicular to the direction of beam propagation, and vo is the

average velocity along the direction of beam propagation.13 From

Effects of Directed Energy Weapons

276

Range (km)

Figure 5-7. Beam Radius vs Divergence and Propagation Distance

Divergence

(radians)

10-3

10-5

10-7

10-9

10-11

10-13

Zθ = 10m

10 cm

1mm

10 102 103 104 105 106 107
this definition, you can see that there are two ways in which you

can reduce a beam’s divergence . You can decrease <v>, or you

can increase vo. The second approach is limited—the particle’s

speed cannot exceed the speed of light no matter how much energy is given to it. Therefore, a beam divergence of 1 rad or less

requires that <v> be less than 10–6c, or 300 m/sec. As a point of

comparison, this ratio is approximately that of the speed of

sound to the speed of light. Clearly, propagation of a neutral particle beam without significant divergence requires that the beam

quality be very good, with little deviation of any individual particle’s velocity from that of the main beam. Current technology

results in neutral particle beams with a divergence of just about a

rad, and there is clear research interest in reducing this value so

as to increase useful beam ranges.

Charged Particle Beams in a Vacuum

Expansion from Electrostatic Repulsion. Divergence due to differences in the motion of individual particles is responsible for

the spread of neutral particle beams. For charged particle beams,

there is also the more important effect of the mutual repulsion that

the particles in the beam feel. You will recall that the net force

felt by particles at the surface of a charged particle beam is given

by F = nq2w/2o(l – v2/c2)  nq2w/2o 2, where n is the density of

particles in the beam, v their velocity, w the beam radius, and o a

constant (8.85  10–12 fd/m). This force is directed radially outward, and pushes the particles apart, so that w increases and n

decreases. As with the neutral beam, w can’t be allowed to increase too much, or the energy carried by the beam will disperse,

and little of it will intersect the intended target.

The growth of a charged particle beam’s radius, w, is not as easy

to calculate as that of a neutral particle beam, since the forces

which make it expand are themselves dependent on w. As w increases, n decreases as 1/w2, and the net force F decreases as well.

The equation of motion for particles near the surface of the beam

is F  dP/dt  d(mv)/dt, where v is the velocity which these

particles have in the direction perpendicular to the beam’s motion.

The beam’s radius w grows at a rate dw/dt  v. The solution to

these equations is shown in Figure 5–8. This figure shows the rela277

Particle Beams
tive beam radius, w/wo, as a function of time, measured in units

of td, the “doubling time” that it takes for w to grow to 2wo.

Initially, the beam expands quadratically, with w/wo 

(1 + t/td)2. Later in time, as repulsive electric forces decrease,

the expansion becomes more linear, with the beam radius growing

at a constant velocity. The doubling time, td, is given by td

2 

4mo3/noq2  4(mc2)2 co3( – l)3/Sq2, where m and q are the

particle mass and charge, and no is the initial density of particles

in the beam. In the second form, the beam’s density no has been

written in terms of its intensity, S  ( – 1) mc2nov (W/cm2). This

is a more useful form, since it’s more straightforward to relate intensity to damage criteria than particle density. The expression for

the doubling time agrees with what we would expect on an intuitive basis. The doubling time decreases, and a charged particle

beam expands more rapidly, if the beam is comprised of light particles, which are easily accelerated, or is very dense which makes

repulsive electrostatic forces greater. Conversely, the doubling

time increases if the beam’s particles are highly relativistic for two

reasons. The greater relativistic mass, m, increases a particle’s resistance to acceleration, and attractive magnetic forces more

nearly compensate for repulsive electric forces.

If we multiply the doubling time td by the speed of the particles

in the beam, we get the range to which the beam can propagate in

that time. It is clear from Figure 5–8 that within a few doubling

times a beam will expand to the point where it is no longer useful

as a directed energy weapon. Therefore, zd  vtd can be taken as a

Effects of Directed Energy Weapons

278

0 0.5 4 1 1.5 2 2.5 3 3.5

20

18

0

2

15

4

14

12

6

10

8

QUADRATIC APPROXIMATION, W/W0 = 1+ (t/td)

2

Exact Solution

Relative Time (t/t ) d

Figure 5-8. Expansion of a Charged Particle Beam in Vacuum with time

Relative

Radius

(w/w ) 0
measure of the useful range for a charged particle beam in vacuum. Figure 5–9 is a plot of zd for electrons and protons as a function of kinetic energy for an intensity of S  107 W/cm2.

You can see from Figure 5–9 that electrostatic repulsion severely

limits the range and utility of charged particle beams in a vacuum.14 Only for energies in excess of a GeV do ranges even begin

to approach the strategic ranges for which a space based particle

beam would have some utility. And since zd varies as 1/S1/2, it’s a

rather weak function of beam intensity. This means that we can’t

substantially improve the picture shown in Figure 5–9 by going to

a less intense beam, and still hope to damage targets within reasonable time scales. Therefore, neutral particle beams are a more viable

alternative for strategic applications in space.

Effects due to External Fields. Charged particles are affected

not only by the electric and magnetic fields which they themselves

generate, but also by any externally applied electric or magnetic

fields. The most significant of these is the magnetic field of the

earth. You will recall that a magnetic field produces a force which

is perpendicular to both the magnetic field and the particle’s direction of motion (see Figure 5–1). This causes charged particles to

move in curved paths. Figure 5–10 illustrates the simple case

279

Particle Beams

Electron

Proton

S=10 W/cm

Kinetic Energy (Mev)

1 3 10 30 100 300 1000 3000

Range

 (km)

10

10

10

10

10

10

2

4

-2

-4

-6

-8 10

7 2

Figure 5-9. Doubling Distance for Electron and Proton

Beams of Intensity 10 W/cm 7 2
where the particle’s motion and the direction of the magnetic field

are perpendicular to one another.

Shown in Figure 5–10 is a particle of charge q, mass m, and

velocity v. Throughout the region of space in which the particle

travels there is a magnetic field of strength B, pointing into the

plane of the paper. The force which a magnetic field produces on

a charged particle is perpendicular to both v and B and has a

magnitude qvB sin , where is the angle between v and B. In the

situation sketched in Figure 5–10, the force is qvB, since B is perpendicular to v, and sin 90°  1. This force will cause the particle

to turn, and as it turns the force will turn as well, orienting itself

so that it’s always perpendicular to the particle’s instantaneous

velocity. As a result, the particle will travel in a circle. A good

Effects of Directed Energy Weapons

280

Magnetic Field

(Uniform and into Paper) F = qvB

Figure 5-10. Motion of a Charged Particle in a Magnetic Field

v

Figure 5-11. Cyclotron Radius vs Particle Energy for B = 0.5 Gauss

Particle Energy (MeV)

Cyclotron

Radius

(km)

10 3

10 2

10

1

10 -1

10 -2

10 -3

1 3 10 30 100 300 1000 3000

r c

Accelerator

beam

Target

Protons

Electrons
analogy is that of a stone swung in a circle on a string. The string

exerts a force which is always inward and perpendicular to the direction the stone is moving, and the stone follows a circular path.

It’s a straigntforward task to solve the equation of motion for a

charged particle moving in a uniform magnetic field as illustrated

in Figure 5–10. The radius of the circle traveled is rc  mv/qB,

where rc is known as the cyclotron radius.15 Figure 5–11 is a plot of

rc as a function of particle energy for electrons and protons in a

magnetic field of strength 0.5 Gauss, a value roughly equal to the

magnetic field of the earth. This figure gives a feeling for the type

of deflections which a particle beam might experience while propagating. You can see clearly that except at very high energies rc

will actually be less than the ranges over which we might wish to

employ a charged particle beam. This may prevent the beam from

reaching its target, as indicated in the sketch shown in Figure 5–11.

Clearly, the curvature of a charged particle beam propagating in

the magnetic field of the earth must be taken into account in aiming the beam.

Of course, the magnetic field of the earth is far more compex

than a uniform value of 0.5 Gauss, and in general a particle beam

won’t be propagating in a direction perpendicular to this field,

whose orientation changes with latitude, longitude, and altitude.

Indeed, the magnitude of the earth’s magnetic field changes with

time of day and season as well, since its affected by the solar

wind—energetic particles emitted from the sun.16 Since the precise

nature of the earth’s magnetic field can’t be known or predicted in

advance, charged particle beams in a vacuum would need to engage targets through a “shoot-look-shoot” technique, with a rough

guess made of the beam’s propagation, the beam path followed,

and adjustments made to bring it toward the target. This process

of iteration and adjustment would be analogous to the adaptive

optics employed with lasers to deal with the bending and distortion of the beam resulting from thermal fluctuations in the atmosphere (see Chapter 3).

Summary: Propagation in a Vacuum

1. A neutral particle beam’s radius, w, will grow with distance

as w  z, where is the beam’s divergence. This divergence

arises from the small, sideways components of velocity which

281

Particle Beams
the particles in any real beam will have. If w becomes too great,

the energy in a beam will no longer be efficiently directed at a

target. Figure 5–7 shows the resulting limitations on beam range

if w is to remain less than a specified value.

2. Charged particle beams will expand more rapidly than neutral

particle beams because charged particles repel one another. The

distance over which a charged particle beam will double in size is

shown in Figure 5–9 for a beam of intensity 107 W/cm2. The distances shown in this figure scale as l/S to other intensities.

3. Charged particle beams are also affected by external electric

and magnetic fields. The most significant of these is the magnetic field of the earth, which makes the beam particles travel in

a curved path. Figure 5–11 provides a rough estimate of the radius of curvature of this path as a function of particle energy.

The specific path followed by a charged particle beam in the

magnetic field of the earth will vary with latitude, longitude, altitude, time of day, time of year, and even sunspot activity, due

to the sun’s effect on the earth’s magnetic field.

Implications

A charged particle beam is impractical for use in the vacuum of

space. Even at energies well in excess of those produced by

today’s particle accelerators, a charged beam will expand by an

unacceptable amount in a very short distance—kilometers or less.

Therefore, only neutral particle beams are a viable option for

space applications. Even these, as Figure 5–7 shows, will require a

very good (< 1 rad) beam divergence if they are to be useful over

the long ranges associated with engagements in space.

Propagation in the Atmosphere

Neutral Particle Beams in the Atmosphere

Just as charged particle beams are impractical for applications

in space, so neutral particle beams are impractical for applications

in the atmosphere. This is because the neutral particles in the

beam will collide with molecules of oxygen, nitrogen, and other

substances in the atmosphere. As a result of these collisions the

beam particles will become ionized, losing the electrons which

Effects of Directed Energy Weapons

282
make them electrically neutral. Thus, as it propagates through the

air, a neutral particle beam becomes a charged particle beam. It is

not a very good one, since the collisions which ionize the beam

particles also impart momentum and velocity in the sideways

direction, increasing the divergence of the beam. This process is

shown in Figure 5–12. The probability of the event illustrated in

Figure 5–12, and its threat to NPB propagation, depends upon the

density of the atmosphere. Collisions are more likely if there are

more molecules to collide with. Therefore, a key problem in

neutral beam propagation is to find the greatest density, or lowest

altitude, at which the beam will propagate.

The probability for the collision illustrated in Figure 5–12 is

determined by the collision cross section, .

17 Imagine that the molecule shown in the figure has a cross section  (m2), and that there

are N molecules per cubic meter. If the neutral particle beam has a

cross-sectional area A ( w2), and propagates through a thickness dz of atmosphere, it will encounter AN dz molecules. These

will close out a fraction  AN dz/A   N dz of the beam’s crosssectional area. Therefore, the fraction of beam particles that will

collide in propagating through a distance dz of the atmosphere is

N dz (see Chapter 3, Figures 3–22 and 3–23). Since the current

and intensity of the beam are proportional to the density of particles in it, these quantities will be reduced by the same fraction. For

example, the decrease in current I in propagating a distance dz

must be dI  –I N dz. This equation has the solution I  Io e–Nz.

This means that the beam’s current decreases exponentially with

propagation distance, and is reduced by a fraction l/e (about 1/3)

in propagating a distance l/N. Therefore, a good criterion for

neutral beam propagation in the atmosphere is that the intended

range of the beam be on the order of l/N or less.

283

Particle Beams

e

p

Neutral Particle

(Hydrogen)

Encounters Atmospheric Molecule

(Oxygen or Nitrogen)

Electron is stripped from the Hydrogen,

leaving a charged proton behind. The

electron and proton both receive a

"kick", giving them a greater velocity

perpendicular to the beam's motion.

Figure 5-12. Collisional Ionization of Neutral Particle Beams in the Atmosphere

e

p
What is the cross section for a neutral particle such as hydrogen

to be ionized as it collides with particles in the atmosphere? Since

the atmosphere is primarily oxygen and nitrogen, the relevant cross

sections will be those for ionizing collisions with these molecules.

Figure 5–13 shows the ionization cross section for neutral hydrogen

encountering O2 and N2 as a function of kinetic energy.18 The cross

section vanishes if the available kinetic energy is less than the ionization potential, or energy required to strip an electron from a hydrogen atom (13.6 eV). Above that energy,  rises rapidly to a few

times 10–16 cm2 in the neighborhood of 100 keV, after which it gradually declines, and is about an order of magnitude less between 1–10

MeV. These cross sections lead to the general rule of thumb that the

“size” of an atom or molecule is about 10–8 cm ( 1 Ångstrom).19

Let’s use Figure 5–13 to estimate the minimum altitude at which

a 1 MeV neutral particle beam might be used to engage a target

100 km away. This requires that l/N be 100 km or greater. From

the figure,  is about 10–16 cm2 for both N2 and O2 at 1 MeV kinetic

energy. Therefore, in making an estimate, we can consider both

oxygen and nitrogen the same, and treat the atmosphere as a homogeneous gas with a cross section of 10–16 cm2 and density N.

The requirement 1/N > 100 km corresponds to the requirement

that N be less than about 109 molecules per cubic centimeter. This

corresponds to a density of about 10–10 that at sea level (3 

1019/cm3), which is a pretty rarefied atmosphere, and a pretty high

Effects of Directed Energy Weapons

284

Figure 5-13. Cross Section for Ionization of Hydrogen in a Background

Gas of Oxygen or Nitrogen as a Function of Kinetic Energy

Particle Energy, KeV

10

-15

10

-16

10-17

3 30 10 100 300 1000 3000 10000

Cross

Section

(cm2

)

Nitrogen

Oxygen
altitude. Figure 5–14 is a plot of atmospheric density as a function

of altitude.20 From this plot, you can see that 109/cm3 is the density

at an altitude of about 200 km. This is the altitude range (known

as low earth orbit) at which the space shuttle operates. Clearly,

neutral particle beams have little utility for applications in which

significant penetration of the atmosphere is required.

The criterion l/N > z, where z is the intended range of a neutral particle beam, may be used together with data on the cross

section for neutral beam ionization as a function of particle energy

(Figure 5–13) and data on density as a function of altitude (Figure

5–14) to plot the minimum altitude for neutral beam employment

as a function of range and particle energy. Such a plot is shown in

Figure 5–15. This figure makes it clear that neutral particle beam

employment is limited at reasonable particle energies and realistic

ranges to altitudes on the order of 100 km or greater. Clearly, the

neutral particle beam is a device suited to vacuum, as opposed to

atmospheric, applications.

285

Particle Beams

Figure 5-14. Atmoshperic Density vs Altitude

Altitude, km

50 100 150 200 250 300

10

10

10

10

10

10

1

-2

-4

-6

-8

-10

-12

Density Ratio

ρ(h) / ρ(o)
Charged Particle Beams in the Atmosphere

Charge Neutralization. From our earlier description of

charged particle beam expansion (Figure 5–91) it might seem as

though there would be no way in which a charged particle beam

(CPB) could find application as a directed energy weapon. This

is not the case, however, since a CBP in the atmosphere experiences a phenomenon known as charge neutralization. Charge

neutralization has the effect of eliminating beam expansion

through electrostatic repulsion. The sequence of events involved

is illustrated in Figure 5–16.

Effects of Directed Energy Weapons

286

Figure 5-15. Minimum Operational Altitude for Neutral Hydrogen Beams as

a Function of Range and Particle Energy

360

324

288

252

216

180

144

108

72

36

0

10 30 100 300 1000 3000 10000

Minimum

Altitude

(km)

Engagement Range,(km)

Particle Energy = 0.1MeV

1MeV

10 MeV

As a charged particle beam enters the atmosphere, it encounters

atmospheric molecules (Figure 5–16a). Just as these molecules

could ionize the particles in a neutral beam through collisions, so

the particles in a CPB can ionize them through collision. (The opposite can’t occur in this case since the particles comprising a CPB,

electrons or protons, are already ionized, and have no further

charges to be stripped off through collision.) As the beam passes

through the air, the air becomes ionized, going from a neutral gas

to an ionized plasma. The charged electrons and nuclei in the atmospheric plasma are free to carry electrical current (Figure 5–16b).
We saw earlier in the chapter that there is a strong electric field

associated with the charge carried by the particles in the beam (see

Figure 5–4). This field is responsible for the repulsive force that the

particles in the beam feel, and it also exerts a force on the newly

ionized atmospheric particles. Particles whose charge is the same

as that of the particles in the beam are repelled from the beam volume, while those whose charge is opposite are attracted into it

(Figure 5–16c). The net result of all this action is to move the

charge imbalance to the surface of the beam, shorting out the

electric field in the beam’s interior. This is an illustration of the

general principle that there can be no electrical fields on the interior of a conducting medium.21

287

Particle Beams

c. If the beam particles have negative charge, electrons are repelled from within the beam volume,

while nuclei are attracted into it. Net result is that there is charge neutrality within the beam, and all

excess charge resides on its surface.

electron

nucleus

Air Molecules

b. Some molecules are struck by particles in the beam. This can ionize them, separating electrons

from the atomic nuclei. The air becomes an electrically conducting plasma behind the head of the

beam.

Beam Particles

electron

nucleus

Air Molecules

a. Beam particles encounter air molecules as they exit the accelerator and enter the atmosphere

Beam Particles

Air Molecules

Figure 5-16. Sequence of Events in Charge Neutralization
The configuration of electrical charges and the resulting radial

electrical field in a charge neutralized CPB are illustrated in Figure

5–17.

If charge neutralization is to permit stable CPB propagation in

the atmosphere, it must occur more rapidly than the beam can expand through the mutual repulsion of its particles. Therefore, we

need to compare the time scale for charge neutralization with the

beam doubling time illustrated in Figure 5–9, which is the time

scale for beam expansion.

It is relatively straightforward to derive from Maxwell’s equations that in a medium whose conductivity is 
, any excess electrical charge density, 	o will decay in time exponentially as 	(t) 

	oe–
t/o. The electrical conductivity, 
, is just the constant of proportionality between the electrical field at any point in space and

the resulting current density, j: j  
E.22 From this relationship,

you can see that the charge density will decay by a factor of l/e in

a time tn  o/
, where tn may be considered a charge neutralization time.23

The conductivity of an ionized gas is given by 
  ne2/m,

where n is the density of current carriers (primarily electrons,

since they’re lighter and can respond more quickly to an applied

electrical field), e and m the charge and mass of these carriers, and

 the frequency of collisions between the current carriers and particles which retard their flow, such as ions or neutral molecules.

The collision frequency  depends upon the density of the background gas and its degree of ionization. For singly-ionized air at

sea level, it’s approximately 1014/sec. The resulting conductivity is

Effects of Directed Energy Weapons

288

Before Charge Neutralization

Figure 5-17. Electric Field of a CBP Before and After Charge Neutralization

E

2ε0

nqw

W

Uniform

charge

density

After Charge Neutralization

W

Excess charge confined

to a thin surface layer -

No electrical field on the

interior of the beam

E

nqw

2ε0
about 8000 mho/m, and the corresponding neutralization time,

o/
, is about 10–15 sec.24 By contrast, Figure 5–9 shows beam doubling times which are about 5 orders of magnitude longer. At sea

level, therefore, charge neutralization will occur far more rapidly

than beam expansion. This means that for CPB propagation in the

atmosphere, there is a mechanism with the potential to negate the

undesirable effects of beam expansion.25

What happens at altitudes above sea level? To a first approximation, the conductivity of the ionized atmosphere is unaffected,

since 
  ne2/m, and both n and  are proportional to the density of the atmosphere. As a practical matter, however, charge neutralization is not possible if the atmospheric density falls below

that of the particles in the beam. In this case, ionizing the atmosphere cannot provide sufficient charge for neutralization. You’ll

recall that a beam’s current, I, is related to its radius, w, and the

density of its particles, n, through I  nqv w2. Therefore, the density of particles in a beam of current I and radius w is n  I/qv w2.

This density may be compared with a plot of atmospheric density

as a function of altitude (Figure 5–14) to find the maximum altitude at which a beam of current I and radius w can propagate, as

shown in Figure 5–18.

You can see from Figure 5–18 that charged particle beams with

sufficient current to be of interest from the standpoint of weapon

289

Particle Beams

Figure 5-18. Maximum Altitude for CPB Neutralization

vs Current and Radius

1000

100

10

0.1 100 0.3 1 3 10 30

Beam Current, kAmp

Beam Radius = 10 cm

1 cm

01. cm
Effects of Directed Energy Weapons

290

applications are limited to operation at altitudes below about 200

km. At altitudes above this, the air is too thin to permit charge

neutralization, and beam expansion will limit propagation as in a

vacuum. It is interesting to note (Figure 5–15) that neutral particle

beams are for the most part constrained to operate at altitudes

above 200 km, since at lower altitudes they are ionized, and lose

their integrity. The utility of particle beams as weapons therefore

falls into two distinct categories: neutral beams in space and

charged beams within the atmosphere.

The Evolution of Beam Radius. A charged particle beam feels

two forces: an electrical force which pushes it apart, and a magnetic force which squeezes it together. In space, the electrical force

always dominates, and a CPB will expand. We have just seen that

in the atmosphere, charge neutralization can eliminate the electrical force. However, charge neutralization won’t eliminate the

magnetic force, which depends upon the beam current. Even

though the net charge on the interior of the beam is zero, the net

current is not, since only the beam particles have a net motion

downstream. The atmospheric electrons and ions which neutralize the charge are standing still, relatively speaking, as the particles in the beam stream through them. Therefore, a charge-neutralized beam will initially pinch down in radius under the

influence of the magnetic force which remains.

Of course, the pinching and reduction of a beam’s radius under

the influence of its self-generated magnetic field can’t go on forever. Something must arise to counteract that force and establish

an equilibrium radius. That something is the small, random sideways components to the velocities of the particles in the beam.

This sideways motion arises from fluctuations and inhomogeneities in the performance of the accelerator which creates the

beam, as well as from collisions in which beam particles encounter air molecules. An equilibrium radius is established when

the outward pressure resulting from motion perpendicular to the

beam is sufficient to counteract the inward magnetic force, as illustrated in Figure 5–19.

The problem of determining the equilibrium beam radius, w,

when an inward “magnetic pressure” opposes an outward kinetic

“temperature” is a standard one in plasma physics. It arises not

only in particle beam propagation, but also when charged parti-
cles are to be contained by externally applied magnetic fields, as

in plasma fusion reactors.26 In the case of a charged particle beam

propagating in the atmosphere, there is an additional complication due to the fact that the beam’s temperature is not a constant,

but grows with distance as the beam propagates. As a beam propagates through the atmosphere, its particles continually encounter atmospheric molecules. They interact with one another,

as in the collisions which produce the ionization that permits

charge neutralization. In each of these collisions, a beam particle

receives some small, sideways deflection. The effect of these interactions is to increase the beam’s temperature as it propagates

downstream. In effect, there is friction between the propagating

beam and the atmosphere through which it moves. As the beam’s

temperature rises, its internal pressure increases, and its radius

grows as it expands against the compressive magnetic force. In

order to analyze the evolution of beam radius following charge

neutralization, we need to consider the rate at which random energy or “temperature” is added to the beam, and how this energy

is expended in pushing out the beam’s boundary.

A simple estimate can be made by equating the energy added

to the beam to the work done in expanding against the pinching

magnetic force. Consider the situation shown in Figure 5–19. The

total pinching force F per unit beam length is the magnetic force

per particle, nq2w2/2o, multiplied by the number of particles

per unit length,  w2n. Since the beam current is I   w2nqv,

this force may be written as F  I2/2o wc2. If the temperature of

the beam rises, and its radius increases from w to w + dw, the

small increment of work, dW, done through this expansion is just

dW  F dw  [I2/2o c2](dw/w). This last expression is a useful

291

Particle Beams

Inward magnetic forces

Outward "pressure"

from slight sideways

motion

(greatly exaggerated)

Figure 5-19. Opposing Forces which lead to an Equilibrium Beam Radius
form, since the total current, I, is unaffected by the beam radius.

Thus, the only variable in the expression is the fractional change

in the beam radius, dw/w.

The work done in beam expansion must equal the energy

given to the beam through collisions between its particles and

those in the atmosphere. Suppose each particle’s “perpendicular

energy,” T  m<v2>/2, increases at a rate dT/dz. Since there

are w2n particles per unit length, the beam’s energy per unit

length will be enhanced by an amount dE   w2n (dT/dz) dz in

propagating a distance dz. Using the fact that the beam current is

I  w2nqv, we can write dE  (dT/dz)(I/qv) dz. Equating the

energy deposited, dE, to the work done in expansion, dW, yields

an equation for the beam radius as a function of propagation distance, (dT/dz) (I/qv) dz  [I2/2o  c2) (dw/w). This equation is

of a form we’ve seen several times before, and is easily solved to

find w(z)  w(o) exp[(2o  c2/Iqv)(dT/dz)]z. The beam radius

will grow exponentially with distance, and will grow by a factor

of e (roughly 3) in a distance zg  Iqv/[2o c2(dT/dz)]. The radial

growth distance zg increases linearly with I, since high-current

beams have stronger magnetic fields to hold them together, and

is inversely proportional to (dT/dz), since a greater heating rate

results in a more rapid expansion.

All we need to do to complete our analysis is to estimate

dT/dz, the rate of increase in the perpendicular energy of the

beam particles. It is a straightforward but tedious exercise to arrive at the estimate dT/ dz  4 NZ q2e2 lnQ/[M (4o)2 2].

The various terms which appear in this expression and their

physical interpretation are as follows:

• N is the density of molecules in the atmosphere, and Z

is the number of electrons associated with each of these

molecules. Beam particles collide primarily with the orbital

electrons of atmospheric molecules, so that the rate of these

collisions is proportional to NZ.

• The quantity q is the charge on a beam particle, and e the

charge on an electron. The factor q2e2 reflects the fact that

beam particles of greater charge will interact more strongly

with electrons orbiting the molecules in the atmosphere.

Effects of Directed Energy Weapons

292
• M is just the relativistic mass of the beam particles. More

massive and relativistic particles will be less deflected in their

encounters with atmospheric particles.

• v is the velocity of the beam particles. The rate of energy

addition is proportional to l/v2 because beam particles which

are moving more rapidly spend less time in the vicinity of

a given atmospheric particle, and thus have less time to be

affected by it.

The other quantities appearing in the expression for dT/dz are

simply constants or factors which vary so little that we need not

concern ourselves with them. A thorough discussion of the

physics which underlies dT/dz may be found in most texts on

electromagnetic theory.27

Our expression for dT/dz may be substituted into our expression for the distance over which the beam radius will grow by a

factor of e, zg  Iqv/[2 c2(dT/dz)], to obtain an estimate of a

charged particle beam’s useful range as a function of particle

type, energy, and beam current: zg  [23o/Zcqe2 lnQ] 

[Mc2I/N]. This range is proportional to the total energy of the

particles in the beam and the beam current, and is inversely proportional to the atmospheric density. As a numerical example, zg

 0.42 meters for a 1 kAmp beam of MeV electrons at sea level,

for which   3, Mc2  0.5 MeV, and N  3  1025/m3. It is a

straightforward exercise to scale this result to other cases; Figure

5–20 is a plot of zg as a function of beam current for different particles and energies.

You can see from Figure 5–20 that for a charged particle beam

to achieve substantial range in the atmosphere requires high currents and either heavy or very relativistic particles. For example,

mc2  1 GeV corresponds either to protons of 1 MeV kinetic energy ( 1) or to electrons of 1 GeV kinetic energy ( 2,000).

Currents of a few kiloamps are required as a minimum for engagements even over tactical ranges. The curves in Figure 5–20

scale as l/N, where N is the atmospheric density, so that at an altitude where N is 1/10 of its value at sea level, the beam expansion distance will be greater than that shown in the figure by a

factor of ten.

293

Particle Beams
Effects of Directed Energy Weapons

294

Beam Current, kAmp

 mc2 = 2GeV

1 3 10 30 100

100

10

1.0

0.1

 mc2 = 1GeV

Figure 5-20. Atmospheric Beam Range vs Current and Energy

Range

(km)

γ

γ

More generally, it may be that the atmospheric density varies

significantly over the propagation path of a charged particle beam.

In this case, dT/dz can no longer be considered a constant in integrating the expression for beam radius as a function of distance.

The appropriate generalization is 

w(z)  w(o) exp[(2o c2/Iqv) o

z (dT/dz)dz], 

where the integral is taken over the propagation path. You may recall that a very similar situation was encountered in Chapter 3,

when variations in atmospheric density along a laser beam’s path

had to be accounted for in evaluating the attenuation in beam intensity in propagating over that path. The simple model for the

variation of atmospheric density with altitude which was used

there may also be used in this case.

Within the lower atmosphere (0 – 120 km), atmospheric density varies exponentially with altitude.28 That is, the density of

molecules N(h) at altitude h is related to the density N(o) at sea

level by the relationship N(h)  N(o) exp(–h/ho), where the constant ho is about 7 km. Since dT/dz is also proportional to N, we

can say that dT/dz at some altitude h is equal to dT/dz at sea

level multiplied by exp(–h/ho).29 Suppose a charged particle

beam is fired into the air at some elevation angle , as illustrated in Figure 5–21.
The altitude h is related to the beam range z and the elevation

angle  through the simple geometrical relationship h  z sin .

Using this relationship between h and z, we can evaluate o

z

(dT/dz)dz to any range z. The result is shown in Figure 5–22.30

Figure 5–22 shows that the degree of heating and expansion experienced by a charged particle beam is not strongly affected by

elevation angle as long as the range is less than ho, the distance

over which atmospheric density varies significantly. Beyond that,

elevation angle makes a considerable difference. At   0o, the

beam propagates horizontally, density is constant, and the amount

of heating increases linearly with distance. At   90o, the beam

propagates straight up and rapidly emerges from the atmosphere,

295

Particle Beams

h

0

Figure 5-21. Charged Particle Beam Range and Altitude

Figure 5-22. Relative Beam Heating vs Range and Elevation Angle

o ∫

z(dT/dz)dz

(dT/dz)oho

Relative

Heating

Relative Range (z/h0)

15o

30o

90o

0.0

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

1.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5

0 = 0o
after which no further expansion due to heating should be expected.31 At intermediate angles, the beam has greater and greater

lengths of atmosphere to traverse, the total heating is greater, and

it approaches a limiting value later.

You can use Figure 5–22 together with the.equation for beam expansion, w(z)  w(0)exp [(2oc2/Iqv)(dT/dz)]z to estimate a

beam’s growth at any range and elevation angle. All you need to do

is evaluate (dT/dz)z for propagation over a range ho in a constant

density atmosphere at sea level, and then modify this result with

the appropriate factor from Figure 5–22, substituting the result into

the expansion equation. Suppose, for example, that the range to a

target is z  21 km, at an elevation angle of 30°. The relative range is

z/ho  3. From Figure 5–22, the relative heating at this range and elevation angle is about 1.5. This means that the beam radius at this

range is w(o) exp[(2o c2/Iqv)(dT/dz)]z, where dT/dz is evaluated

at sea level and z  1.5 ho  10.5 km. The sea level value of dT/dz

is about 8.2  104 eV/m for MeV electrons in sea level air. This

value scales as l/mc22 to other particles and energies.

Summary: Beam Radius vs Distance. The initial radius for a

charged particle beam undergoes considerable modification as it

propagates. Following a brief initial period of expansion through

electrostatic repulsion, charge neutralization occurs and the

beam begins to pinch down in radius. As the beam continues to

propagate, its random energy rises as a result of collisions with

atmospheric molecules, and its radius grows exponentially with

distance. Figure 5–23 provides a summary of this behavior. The

figure is not to scale, since neutralization and any initial contraction occur very rapidly relative to the prolonged period of continued beam expansion.

Energy Losses. We have seen that as a charged particle beam

propagates through the atmosphere, encounters with atmospheric

particles give it some random motion and cause it to expand. In

each of these encounters, a beam particle gives up some of its directed energy as well. Therefore, particle energy decreases with

distance. Both of these effects lower the intensity of the beam, and

limit its useful range for weapon applications.

There are two primary mechanisms through which particles

propagating in the atmosphere lose energy. One is ionization, in

Effects of Directed Energy Weapons

296
which beam particles give up energy to strip electrons off of atmospheric molecules. The other is radiation, in which the deflection of

beam particles causes them to lose energy as electromagnetic radiation. Therefore, there is both good news and bad news associated

with the interaction between charged particle beams and the atmosphere. Ionization causes the atmosphere to become a plasma.

Without it, charge neutralization could not occur and there would

be no propagation. At the same time, however, it causes a loss of

energy from the beam. The scattering of beam particles gives them

a “perpendicular temperature,” and prevents the beam from pinching down to so small a radius as to be unstable. At the same time,

however, it leads to energy losses in the form of radiation. We’ll

consider each of these energy loss mechanisms in turn.

Ionization—Figure 5–24 shows how ionization occurs. A beam

particle passes within some distance b of an electron orbiting the

nucleus of an atom in the atmosphere. In ionization, the beam

particle gives the electron a sufficient kick to rip it free of its parent atom. With every ionization, a beam particle loses an amount

of energy at least equal to the ionization potential, the degree of

energy by which an electron is bound to an atom. Ionization potentials are typically on the order of 10–20 eV.

The analysis of energy losses through ionization may be found

in many advanced texts on electromagnetic theory.32 The details

are mathematically complex, but the physics is straightforward.

The goal of the analysis is to calculate dK/dz, the rate at which the

kinetic energy of the beam particles decreases as they propagate

297

Particle Beams

Figure 5-23. CPB Radius in Atmospheric Propagation (not to scale)

Expansion from

mutual repulsion

Charge neutralization,

initial contraction

Expansion through atmospheric

collisional "heating"

Ioniation

Time

Neutralization

Time

Expansion

time

Propagation Distance

= particle velocity x time
and lose energy in ionizing atoms. This is done in two steps. First,

the energy loss K(b) associated with a single encounter between

a beam particle and an electron at a distance b is calculated. It

should not be surprising that K(b) is proportional to (qe/vb)2—

particles with a large charge q will interact more strongly with an

electron of charge e, particles moving more rapidly (greater v) will

spend less time in the vicinity of the electron and be less affected,

and particles approaching more closely to an electron (smaller b)

will interact more strongly.

After K(b) has been calculated, it must be averaged over all

values of b, weighted by how many electrons are likely to be encountered at each value of b. This result depends, of course,

upon the density N of particles in the atmosphere, and leads to

an energy loss rate, dK/dz, which is proportional to N(qe/v)2. It

is apparent from this expression that dK/dz will be minimized at

relativistic energies, where v approaches its maximum value—

the speed of light, c. Figure 5–25 is a plot of dK/dz as a function

of (–l), which is proportional to particle kinetic energy.33

For air at sea level,   2, and q =e, dK/dz is about 7  105

eV/m. At this rate of energy loss, a 1 MeV particle can propagate

about 1.5 m, and a 1 GeV particle about 1.5 km. From the figure,

you can see that it is adequate to use this estimate at all relativistic

energies (>2), scaling it as l/v2 at non relativistic energies. Since

dK/dz is proportional to N(qe/v)2, it can be scaled for different atEffects of Directed Energy Weapons

298

Figure 5-24. Ionization through Beam-Electron Interactions

Beam particle,

 mass M, charge q, v ≈ c

b

Electron,

 mass m, charge e v ≈ 0

Nucleus

(a) Before Interaction

Nucleus

(b) After Interaction

Deflection Beam particle,

 mass M, charge q,v ≈ c

b

Electron,

 mass m, charge e

 v ≈ 0

Deflection

θ
mospheric densities and charged particle species as well. It is interesting to note that dK/dz due to ionization does not depend

upon the mass M of the beam particles—electrons and protons of

the same  will lose energy at the same rate. Of course electrons,

being less massive, will have a much larger  at a given K than

will protons. The fact that dK/dz is minimized at relativistic energies is another reason why relativistic particle beams are envisioned for weapon applications.

Radiation—When charged particles are accelerated or decelerated, they lose energy as electromagnetic radiation.34 This fact is

employed in radio transmitters, which drive electrons up and

down in an antenna, so that they radiate the radio signal. When

charged particles in a CPB encounter particles in the atmosphere

and are deflected or accelerated to the side, they will radiate some

of their energy away. Because the particles lose energy and are

slowed down, this radiation is referred to as bremsstrahlung, the

German word for “braking radiation.”

As with ionization, detailed treatments may be found in any

standard text.35 The qualitative treatment which follows will give

you a feeling for what happens. If a particle of mass M and charge

q is accelerated in a direction perpendicular to its direction of motion, energy is radiated at a rate proportional to the square of its

acceleration. The acceleration can be obtained as a function of the

particle’s distance of approach, b, to an electron or nucleus in an

atom. It is proportional to qe/vbM since particles which approach

299

Particle Beams

Energy Loss

Rate

(MeV/m)

Figure 5-25. Particle Energy Loss Rate due to Ionization vs Particle Kinetic Energy

0.01 0.03 0.1 0.3 1 10 100 1000 10000 3 30 300 3000

10

1.0

10

100

(γ - 1) = k/mc2
one another closely feel a greater force, those which are moving

more quickly feel the force for a shorter time, and heavier particles

are accelerated less. Using these results, the energy radiated can be

calculated as a function of b, and averaged over all b. The

bremsstrahlung energy loss rate derived in this way is shown and

contrasted with the ionization energy loss rate in Figure 5–26.

From Figure 5–26, you can see that at non-relativistic energies,

bremsstrahlung is less important than ionization as an energy loss

mechanism. However, at relativistic energies, dK/dz due to

bremsstrahlung is proportional to K, so that for higher energies (

>300) it exceeds ionization in importance for electrons. At these

higher energies, dK/dz due to bremsstrahlung is of the form dK/dz

 -K/zo where zo, the so-called “radiation length,” is proportional

to M2/Nq4. This equation has the solution K(z)  K(0)exp(–z/zo).

Particle energy decays exponentially with distance, and falls by a

factor of l/e in a distance equal to the radiation length, zo. The radiation length scales inversely with atmospheric density, N, and is

proportional to the square of the particle mass. Electrons, being

very light, will suffer much more energy loss due to bremsstrahlung

than other particles. For electrons propagating in air at sea level, the

radiation length is approximately 300 m, while for protons it is approximately 106 km. Thus, bremsstrahlung is an important energy

loss mechanism only for electrons.

Table 5–3 contrasts ionization and bremsstrahlung as energy

loss mechanisms for charged particle beams. Ionization must be

considered for both electrons and heavy particles, while

bremsstrahlung need only be considered for electrons.

Effects of Directed Energy Weapons

300

Energy Loss

Rate

(MeV/m)

Figure 5-26. Energy Loss Rate due to Ionization and Bremsstrahlung

100

10

1

0.100

0.010

0.001 0.01 0.03 0.1 0.3 1 10 100 1000 10000 3 30 300 3000

Bremsstrahlung

(electrons)

Ionization

(γ - 1) = k/mc2
Current Losses. Ionization and bremsstrahlung cause the particles in a beam to lose energy. But as long as the particles are relativistic, with a velocity close to the speed of light, the current in

the beam, nqv, is unaffected. By contrast, collisions between beam

particles and the nuclei of molecules in the atmosphere decrease a

beam’s current while leaving the energy per particle relatively unaffected.

In a nuclear interaction, a particle in the beam approaches close

enough to an atomic nucleus to interact with it through the short

range strong nuclear force.36 These collisions therefore occur only

for heavy particles, since electrons do not possess this force. Nuclear interactions are relatively rare, because the nucleus occupies

a small fraction (about 10–12) of an atom’s volume. On the other

hand, when they occur, nuclear interactions are quite catastrophic.

Since the nucleus contains almost all the mass in an atom, it is a

much more efficient scatterer than the very light orbital electrons.

Interactions with electrons merely reduce the energy in a beam

particle slightly, leaving it still within the beam, while collisions

with nuclei actually send particles out of the beam, as illustrated

in Figure 5–27.

With each ionizing collision, a particle loses energy on the order

of the ionization potential—l0–20 eV. Since particle energies are in

the MeV to GeV range, many collisions are required before a particle loses sufficient energy to drop out of the beam. This type of interaction is relatively frequent, so that all particles in the beam lose

energy at the same rate, dK/dz. By contrast, nuclear collisions are

less frequent, but when they occur, the affected particle is lost

from the beam. Nuclear collisions are the sort of atom smashing

interactions used by physicists to probe the inner structure of

nuclear matter. Frequently, the target nucleus is shattered by the

301

Particle Beams

Energy Dependence Particle Mass Dependence

Ionization

Bremsstrahlung

Roughly independent

of γ for γ > 2

Mass Independent

Proportional to γ

γ >2

Proportional to 1/M2

Table 5-3. Scaling of Energy Loss Mechanisms
Effects of Directed Energy Weapons

302

collision, and various collision products, characteristic of both the

beam and target particles, are produced.37

The effect of nuclear collisions on a beam’s current may be

determined from the nuclear collision cross-section. A zero-order

estimate for the cross section for a collision between a relativistic

proton and a nucleus of atomic number A is   4  10–30 m2 

A2/3.

38 For nitrogen, whose atomic number is 14, this is about 2.3 

10–29 m2. By contrast, we saw earlier that the cross section for neutral beam particles to be ionized in the atmosphere is about 10–20

m2. Clearly, nuclear encounters are much less likely than atomic

encounters. You’ll recall that in an atmosphere of molecules whose

collision cross section is  and whose density is N, beam intensity

or current falls off exponentially with distance as I  Io e–Nz.

Therefore, nuclear collisions will cause the current to fall to l/e of

its initial value in a distance zN  l/N. For air at sea level, N  3

 1025/m3, and zN is about 1.45 km. You Can see from this that the

effect of nuclear interactions on particle beam propagation is not

trivial, and must be considered in calculating the intensity of the

beam as it propagates. At higher altitudes, of course, the effect of

these collisions becomes progressively less, since zN scales as l/N.

Hole-Boring. A particle beam’s intensity (W/cm2) is given by S

 IK/qw2, where I is the beam current, K the particle kinetic enA. In Ionization and Bremsstrahlung, beam particles lose a small amount of energy with each interaction,

but are generally not lost from the beam. Beam current, therefore, is relatively unaffected as energy

per particle decreases.

B. Nuclear interactions deflect particles from the beam, reducing its current. The energy per particle

is unchanged in those particles which remain in the beam.

Figure 5-27. Nuclear Collisions vs Ionization and Bremsstrahlung

Ε Ε −∆Ε Ε − 2 ∆ Ε Ε − 3∆ Ε

Ι Ι − ∆Ι Ι − 2 ∆ Ι Ι − 3∆ Ι
ergy, q the particle charge, and w the beam radius. We have seen

that there are phenomena which will affect each of the variables in

this expression for intensity. These are summarized in Table 5–4,

along with the characteristic length over which each variable is

likely to be severely affected at sea level. When all the phenomena

we have considered act together, charged particle beams are unlikely to have useful ranges in the atmosphere at sea level, even for

tactical applications.

All of the phenomena listed in Table 5–4 are reduced at higher

altitudes, since zg, zo, and zN are proportional to 1/N, and dK/dz

is proportional to N. Therefore, propagation over useful ranges

might be possible if the density of the air along the beam path

could somehow be reduced. This has led to the concept of holeboring. In hole boring, a particle beam initially serves to heat a

channel of air. The hot air in the channel is then allowed to expand

to a lower density, permitting propagation over a greater

distance.39 This process is analogous to the rarification of the atmosphere that produces thermal blooming in a high power laser

(Chapter 3). However, the degree of heating and density decrease

is much greater for a particle beam, which interacts strongly with

the atmosphere, than for a laser, whose frequency is chosen to

have as little interaction as possible with the air. The sequence of

events that occurs in hole-boring is illustrated in Figure 5–28.

As particles propagate through the atmosphere, they lose energy to ionization and bremsstrahlung. Much of this energy

appears along the beam path as a rise in the temperature of the

atmosphere. Since the pressure of a gas is proportional to its temperature, there is a corresponding rise in pressure. This pressure

303

Particle Beams

Relationship Characteristic Length

(Sea Level Air)

Radius, w zg ≈ 3km( Fig 5-20)

Table 5-4. Factors Affecting Beam Intensity

Affected by

Kinetic Energy, K

Current, I

Expansion w(z) = w(o) exp(z/zg)

K(z) = K(o)-(dK/dΖ) Z K/(dK/dz) ≈1.5km (K ≈ GeV)

K(z) ≈K(o) exp(-z/zo)

I(z) = I(o) exp(-z/zN)

Ionization

Bremsstrahlung

(electrons only)

Nuclear Collisions

(heavy particles only)

Beam Attribute

Ζο≈0.3km

ΖN ≈1.5km
pushes gas out of the beam center, reducing the density there.

Eventually, a quasi-equilibrium is established, in which the underdense beam channel is at pressure balance with its surroundings. Since pressure disturbances travel at a velocity on the order

of the speed of sound, a (= 3  104 cm/sec), the time required for

hole boring is about w/a, where w is the beam radius. For a

beam whose radius is 1 cm, this time is about 30 sec.

If hole boring is to be used for propagation over long ranges,

particle beams may require complex pulse structures. There are

two time scales involved: the time necessary to heat the beam

channel to a point where it will expand to the desired lower density, and the time necessary for the heated air to expand out of the

beam volume. These two time scales are illustrated in Figure 5–29.

Effects of Directed Energy Weapons

304

B. Pressure relaxes towards

balance with its surroundings,

and pushes mass from the

center of the beam channel

C. A density "hole" is left

in the beam channel after

a time on the order of w/a,

where w is the beam radius

and a the speed of sound

Figure 5-28. Sequence of Events in "Hole-Boring"

A. Temperature and

Pressure rise in the

channel through which

the CPB passes

T, P P, Density Density

Time to heat

beam channel

over its initial

propagation

range Z h

Time for the heated

channel to clear out

Time to heat the

channel out to a

range 2Z h

Pulse structure:

Figure 5-29. Time Scales Associated with Hole Boring

Accelerator Zh = K/(dK/dZ) Target
When the particle beam initially penetrates into the atmosphere,

it is limited by energy losses in atmospheric density air to a range

zh on the order of K/(dK/dz), where K is the kinetic energy of the

beam particles, and dK/dz the total energy loss rate per unit path

length due to ionization and bremsstrahlung. Therefore, we can

only heat a segment of the beam path of length zh. If the goal is to

reduce the density in the beam channel by a factor of 10, the beam

must be on for a time period sufficient to increase the temperature

in this channel by a factor of 10. After that, the beam should be off

for a time w/a, to allow the heated zone to expand, lowering its

density.40 The beam can then be turned on again, and as a result of

the reduced density over its initial range it will now propagate

further, to a distance of about 2zh. The newly encountered air must

in turn be heated and allowed to expand, and the process repeated

until the beam has been able to chew its way, segment by segment,

to its target. If the total range to the target is Z, then there are Z/zh

channel segments which must in turn be heated and allowed to

expand before the target can be engaged. The total time necessary

to bore a hole to the target at range Z is therefore (te + th)Z/zh,

where te is the time for each heated segment to expand, and th is

the time to heat it to the desired temperature.

We know that the channel expansion time te is on the order of

w/a, where w is the beam radius and a the speed of sound. What is

the time span th for each segment to be heated to a given temperature? The approach to use in calculating th is shown in Figure 5–30.

Figure 5–30 shows a portion of the beam channel of thickness z.

Particles enter this region with kinetic energy K, and leave with energy K –(dK/dz), z. The flow of particles into the region is just nv

(particles per unit area per unit time), so that the total energy deposited in the region per second is nv(dK/dz) zw2, and the energy deposited per unit volume per second is just nv(dK/dz). But

energy per unit volume is just pressure, so if we want to raise the

pressure in the region by a factor of 10, we must leave the beam on

for a time th such that nv(dK/dz)th  10po where Po is the initial

(atmospheric) pressure. Therefore, the necessary heating time is th

 10po/[nv(dK/dz)]. For a relativistic beam (v c) of density n 

1017/m3 in air at sea level, the energy loss rate is dK/dz 7  105

eV/m, and th is about 0.3 sec. Scaling this value to different temperature rises and energy loss rates is straightforward. Since

305

Particle Beams
dK/dz and the pressure in the atmosphere are both proportional to

the atmospheric density N, th is independent of altitude.

In reality, of course, the dynamics of the atmosphere as it interacts with a charged particle beam is more complex than that

sketched above. Some expansion of the air occurs as it is being

heated, and over the range zh  K/(dK/dz) the beam energy and

atmospheric heating rate are decreasing. Nevertheless, since the

heating time is generally small compared to the expansion time

(0.3 sec vs 30 sec for a 1 cm radius beam), it is usually adequate

to think of the interaction and subsequent expansion as occurring

in discrete phases as discussed here.

Non-uniform Atmospheric Effects. When we discussed beam

expansion, we used the simple model of an atmosphere whose density decreases exponentially with altitude to examine the effect of a

nonuniform atmosphere (Figure 5–22). This same approach may be

used to model the effect of a nonuniform atmosphere on ionization,

bremsstrahlung, and nuclear collisions. The mathematics is straightforward and completely analogous to our treatment of beam expansion; the results are summarized in Table 5–5 and Figure 5–31.

Energy and current losses in a nonuniform atmosphere may be

treated just as in a uniform atmosphere, except that the total range,

z, is replaced by a relative depth, D, which depends upon the elevation angle and ratio of Z to the atmospheric decay height ho as

shown in Figure 5–31. The ionization loss rate, dK/dz, the radiation

length, zr, and the atmospheric density, N, are all to be evaluated at

the beam’s initial propagation altitude (normally sea level), as implied by the “o” subscripts in the equations appearing in the table.

Effects of Directed Energy Weapons

306

Figure 5-30. Energy Deposition in the Air from a Charged Particle Beam

K - (dK/dZ) ∆Z K

∆Z

nv particles per

square cm per

second
Summary: Energy and Current Losses. As a charged particle

beam propagates from an accelerator through the air, many factors

will affect the intensity that ultimately strikes a target. The beam

radius changes as a result of increases in the beam’s perpendicular

temperature, and particles in the beam lose energy through ionization of atmospheric particles and possibly bremsstrahlung. Ionization is important both for electrons and heavy particles. It causes

307

Particle Beams

Ionization

Bremsstrahlung

Nuclear Collisions

K(z) = K(o) - (dK/dz)oz

K(z)= K(o) exp(-z/zr)

I(z) = I(o) exp(-Noσz)

K(z) = K(o)-(dK/dz)oD

(D = relative depth

 from Fig 5-31)

K(z)= K(o) exp(-D/zr)

I(z) = I(o) exp(-NoσD)

Table 5-5. Effect of a Nonuniform Athmosphere on Energy and Current Losses.

 The relative depth, D, may be obtained from Figure 5-31.

At Sea Level In an Nonuniform,

Exponential Atmosphere

D = ∫ zN (z)dz / Noho

Relative

Depth

(D)

Relative Depth (z/h0)

15o

30o

90o

Figure 5-31. Relative Depth Factor for use in Adjusting Energy

and Current Losses in a Nonuniform Atmosphere

0.0

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Elevation Angle,Φ=0o

o
particles to lose energy at a rate which at relativistic energies is

roughly independent of their energy. Bremsstrahlung is important

only for electrons, causing them to lose energy at a rate proportional to their energy. Heavy particles may collide with the nuclei

of atmospheric atoms, an event so catastrophic that they are removed from the beam entirely.

All of these effects are proportional in their severity to the atmospheric density. Therefore, the heating of a beam’s path, and its

subsequent expansion to a lower density (hole boring), can result

in longer propagation ranges. The price to be paid is having to

wait to engage a target until a channel can be cleared over the

beam’s entire range. In evaluating all these propagation phenomena, the effect of the decline in atmospheric density with altitude

is easily accounted for, at least at the lower altitudes where density decays exponentially, by using the relative depth provided in

Figure 5–31 in place of the actual range.

Nonlinear Effects (Instabilities). There is a final type of phenomenon which can affect the propagation of a charged particle

beam in the atmosphere—beam instabilities. Ionization,

bremsstrahlung, and nuclear collisions are all one-on-one effects,

arising from interactions between single particles in the beam and

single particles in the atmosphere. By contrast, instabilities arise

through collective effects, in which the total assembly of particles in

the beam, acting in concert, leads to its breakup and destruction.

An instability occurs when small perturbations in the beam’s parameters grow without bound as a result of nonlinear interactions

between the beam and its environment. Ultimately, instabilities

destroy a beam’s integrity, and propagation is no longer possible.

A good analogy might be found in a school bus. A bus rocks at a

natural frequency, determined by its mass and suspension system.

As it propagates, it jostles to and fro, both from bumps in the road

and the motions of its occupants. However, these motions occur

randomly and are not in phase with the bus’ natural frequency, so

that the perturbations decay and do not affect its motion. However, if school children in the bus conspire to rock together in

phase with the bus’s natural frequency, their perturbations will

add to the displacement of the bus with each cycle. Ultimately, the

bus will tip over, its propagation ceasing. Under appropriate circumstances, perturbations to a particle beam can in a similar way

provide feedback which is in phase with a natural frequency of

Effects of Directed Energy Weapons

308
the beam, causing those perturbations to grow to the point where

propagation ceases.

Any survey of the literature on particle beams and their propagation in the atmosphere will reveal that instabilities seem far

more likely than unlikely. It is probably more meaningful to ask

what beam configurations can propagate in a stable manner, than

to ask under what circumstances instabilities can occur.41 Stable

propagation of a charged particle beam over a substantial range in

the atmosphere has yet to be demonstrated, and the area of instabilities, their growth rates, and how beams can be configured to

avoid them is one of active research. Accordingly, we cannot provide a detailed compendium of instabilities that might occur and

of known techniques for avoiding them. Rather, we’ll provide a

qualitative description of two instabilities currently thought to

limit propagation in the atmosphere, along with some indication

of how these instabilities might be avoided.

The resistive hose instability results from an interaction between

the beam particles and the magnetic field that they generate. It can

occur when the beam receives a small, sideways perturbation

from its original direction as a result, for example, of jitter in the

device which aims the beam at a target. This instability can be understood by analogy with a fire hose directing a stream of water,

as illustrated in Figure 5–32. Suppose a small kink occurs somewhere in the hose. The flowing water will exert forces which tend

to straighten the hose out again. ff these forces over-react, an even

309

Particle Beams

Figure 5-32. Hose Instability of a Charged Particle Beam

Fire Hose

Particle Beam

Restoring forces overshoot

An even greater kink in the opposite direction

results
greater kink in the opposite direction will result. After several cycles of this destructive action and reaction, the hose will be so

twisted that its effectiveness in directing the stream of water will

be limited. This is one reason why fire hoses require a team of firemen to keep them pointed in the proper direction.

For a charged particle beam, the encircling magnetic field plays

a role analogous to that of the fire hose for a stream of water. It

confines the beam, pinches its particles into a narrow channel, and

opposes their tendency to expand due to random, sideways motions (perpendicular temperature). If the beam wanders from its

original path, restoring forces will be set up that tend to drive the

beam back to where it was.

However, these can overshoot and induce an even greater kink

in the opposite direction. Under the proper conditions, small kinks

will grow in time until the beam loses its integrity.

Having seen qualitatively how a hose instability could occur in

a propagating particle beam, let’s look more closely at the mechanism for its growth. In this way, we can gain insight into how a

beam can be structured to avoid the instability. The hose for a particle beam is the magnetic field which surrounds it and confines

its particles. Maxwell’s equations predict that a magnetic field

will be “frozen” into a background plasma. That is, if a magnetic

field exists inside of a plasma, it will continue to exist and remain

fixed in the plasma, even if the original source of the magnetic

field should move or decay. Physically, this occurs because if a

plasma moves relative to a magnetic field within it, the resulting

forces on the charged particles in the plasma induce currents

that in turn generate a magnetic field which mimics the initial

field. From the perspective of an outside observer, the magnetic

field has remained unaltered, frozen into the plasma. Of course,

this situation cannot be maintained indefinitely. Eventually, the

induced currents that are responsible for freezing the magnetic

field into the plasma will decay due to the electrical resistance

which is present in any plasma. As these currents decay, the magnetic field will thaw, returning to the value and location dictated

by currents external to the plasma.42 In Chapter 3, this effect is responsible in another form for the fact that light, as electromagnetic radiation, cannot penetrate into the interior of a metallic conductor. Since there is no magnetic field in the conductor before the

Effects of Directed Energy Weapons

310
light impinges, there can be none afterward—the light is frozen

out by the flow of electrons on the interior of the conductor.

What does all of this have to do with the resistive hose instability? A charged particle beam in the atmosphere is propagating in a

background plasma of ionized air. When a kink develops in the

particle beam, or if the beam tries to move sideways, the encircling

magnetic field will not immediately move along with it, but will

remain fixed in this background plasma. This disparity between

what is and what should be regarding the relationship between the

beam and its magnetic field leads to restoring forces that try to return the beam to its original direction.

There would be no problem if a particle beam developed a kink

and its magnetic field were permanently frozen into the surrounding plasma. Restoring forces would drive the beam back to its original position. The beam might oscillate about its initial position like a

pendulum, but no instability would occur. But the magnetic field is

not permanently frozen. In time, it begins to thaw, and rejoin the

particle beam which created it. But by this time, the kink has moved

downstream, and the field lines are moving down on a previously

“unkinked” portion of the beam, perturbing it and initiating another kink. The delay between the perturbation of the beam and the

response of the encircling magnetic field causes the feedback between the beam and its field to be out of phase, so that an instability

develops. That’s why this instability is known as the resistive hose instability. The resistance of the background plasma is responsible for

the delayed response that can lead to an instability.43

This insight into the origin of the resistive hose instability can

suggest ways in which it might be prevented. Since the instability

arises because of feedback between one portion of the beam and another, it might be avoided by chopping the beam into segments,

where the length of these segments is shorter than the distance over

which the instability grows, and the time between them is great

enough for any frozen magnetic field lines to thaw, so that each

beam segment enters an environment with no memory of what has

happened to previous beam segments. This approach is illustrated

in Figure 5–33. In order to carry out this scheme, we need to quantify the necessary length of the segments and separation between

them in terms of appropriate beam and atmospheric parameters.

How can we quantify the pulse width, tp, and pulse separation,

ts, shown in Figure 5–33? The necessary separation time is straight311

Particle Beams
forward to estimate, using results which can be found in any book

on plasma physics. If a magnetic field whose characteristic size is

w is frozen into a background plasma of conductivity 
, and if the

current responsible for the magnetic field goes away, then the field

will decay in time as B  Bo exp(-t/td), where td  
w2/c2o is

known as the magnetic diffusion time.44 The expression for the magnetic diffusion time td makes sense on physical grounds. In plasmas of little resistance (high conductivity), B will take longer to

thaw, and a physically larger magnetic field (w large) will persist

for a longer time. For a beam of radius 1 cm and a conductivity

characteristic of singly-ionized air at sea level 

(
  8.4  103 mho/m), td  1 sec. 

Thus, for these conditions, the pulse separation time shown in Figure 5–33 should be several microseconds. This will enable the magnetic field associated with the beam to decay to zero between

pulses. This criterion scales to other altitudes with the atmospheric

density, N, since 
 is proportional to N.

Effects of Directed Energy Weapons

312

A. Continuous Beam. Communication between one portion of the beam

and another via the encircling magnetic field lines results in an instability

B. Chopped Beam. The beam is cut into setments shorter than the distance

(wavelength) over which the instability grows. The time between segments

is sufficiently long that the magnetic field decays, providing no memory of

what has gone before to subsequent segments.

Wavelength

Pulse separation,

Pulse

width, tp

ts

Figure 5-33. instability Avoidance through Beam Chopping
What should be the pulse width, tp, shown in Figure 5–33? There

is a characteristic wavelength at which the instability grows. This

wavelength is the distance between the perturbation of the magnetic field and the associated response of the propagating beam.

Clearly, the pulse width should be chosen so that the length of a

beam segment, vtp, is less than this wavelength. In this way, each

individual beam segment is shorter than the distance over which

the instability can grow. The determination of the wavelength or

range of wavelengths at which an instability will grow is a tedious

exercise, in which the beam is assumed to be perturbed at a given

wavelength, and the equations for the response of the beam and

its magnetic field under this assumption are studied to see if the

perturbation grows or decays. It can be shown from such an

analysis that the wavelength for growth of the hose instability is

proportional to (vE/I)l/2w, where v is a beam particle’s velocity, E

its total energy (Mc2), I is the beam current, and w the beam radius.45 This length is about 4 meters at v  0.9c, E  2 GeV, w  1

cm, and I  1 kAmp. This means that for a beam with these parameters, the pulse width must be less than 4 m/0.9c, or about 1.5 

l0–9 sec. Therefore, the hose instability can be avoided in a kAmp

beam of GeV particles by chopping the beam into segments about

a nanosecond in length, and separating these segments by several

microseconds. Scaling these criteria to other beam sizes, currents,

and energies is straightforward, using the scaling relationships

provided above.

Another instability of some concern from the standpoint of

charged particle beam propagation in the atmosphere is the

sausage instability.46 While the resistive hose instability results from

perturbations in the beam’s position perpendicular to its direction

of motion, the sausage instability results from longitudinal perturbations in the beam’s density. You will recall that the equilibrium

beam radius results from a balance between the pinching magnetic field which the beam generates and the outward pressure of

its perpendicular temperature. Both the outward pressure and the

inward magnetic field depend on the density, n, of particles in the

beam. Should this density increase somewhere within the beam,

the outward pressure will increase, while the magnetic pinching,

being frozen into the background plasma, cannot respond immediately. Therefore, the beam radius in this region will begin to

grow. At a later time (on the order of the magnetic diffusion time,

313

Particle Beams
td), the magnetic field will increase to reflect the increased density.

But by this time the density increase has moved downstream, and

at the point on the beam where the magnetic field finally responds, the density has not increased. At this point, the magnetic

compression exceeds the outward pressure, and the radius tends

to decrease. Under these circumstances, the beam will alternately

grow and pinch, attaining the appearance of a string of sausages,

as illustrated in Figure 5–34.

As with the hose instability, avoiding the sausage instability involves chopping the beam into pulses separated by several magnetic diffusion times, and restricting each pulse to a length shorter

than that over which the instability can develop. In general, the

sausage instability places limitations on pulse width and separaEffects of Directed Energy Weapons

314

Figure 5-34. Development of the Sausage Instability

C. Out of phase action-reaction continue, and the beam develops a sausage-like pertubation.

B. Beam radius grows in region of increased density due to unbalanced forces as beam propagates.

Behind the perturbation, the magnetic field strength increases as it "thaws" and attempts to adjust to

the increased density. In this region, however, there is now a net inward force, and the beam radius

decreases.

A. Particle density increases somewhere in the beam. Outward pressure increases, but inward

magnetic force cannot immediately respond, since its value is "frozen" into the background plasma.
tion that are not as severe as those resulting from the hose instability. Therefore, choosing beam parameters to avoid the hose instability should prevent the sausage instability as well. However, the

theoretical analysis of instabilities, the criteria for their growth,

and techniques to avoid them can be quite complex. Existing treatments invariably involve numerous simplifying assumptions. Accordingly, theoretical predictions of beam structures that can propagate without instabilities must be validated and extended

through a careful experimental program. To date, technology limitations to available accelerators have not permitted a comprehensive investigation of beam propagation or a complete validation of

theoretical analysis. Indeed, the greatest problem in the design of

high-current, high-energy accelerators is the development of instabilities within the accelerators themselves.47

Summary: Propagation in the Atmosphere

1. Neutral particle beams can’t propagate in the atmosphere because collisions with atmospheric particles ionize the particles

in the beam, converting it into a charged particle beam of poor

quality. Figure 5–15 shows how the minimum altitude at which

a neutral beam can be employed varies with particle energy and

beam range. Practically speaking, neutral beams are limited to

altitudes above 100 km.

2. As charged particle beams (CPBs) propagate through the

atmosphere, they will ionize atmospheric gases. This permits

charge neutralization, a flow of charge which shorts out the internal electric fields which would otherwise prevent CPB propagation. Figure 5–18 shows the maximum altitude at which

charge neutralization can occur as a function of beam radius and

current. Practically speaking, charged particle beams are limited

to altitudes below about 200 km.

3. Following charge neutralization, a CPB contracts to a radius

at which the self-generated magnetic field, which tends to pinch

it, is balanced by internal pressure from the small, sideways

motions (perpendicular temperature) of its particles. This radius then grows as w(z)  w(o) exp(z/zg) because collisions

with atmospheric particles increase the beam’s internal pressure

as it propagates. Figure 5–20 shows the beam expansion range

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Particle Beams
zg as a function of beam current and particle energy. In air at sea

level, this range is on the order of 10 km.

4. As a charged particle beam propagates, its particles lose energy through ionization and bremsstrahlung, and its current declines through nuclear collisions. Energy losses to ionization are

independent of energy for relativistic particles, so that a particle’s energy decreases in proportion to its range. Ionization

losses limit the range of a 1 MeV particle at sea level to about 1.5

m. The energy loss to bremsstrahlung is proportional to a particle’s energy, so that its energy decreases exponentially with distance by this mechanism. The range for a l/e decrease in an electron’s energy through bremsstrahlung at sea level is about 300

m. This range scales inversely with atmospheric density and as

the square of a particle’s mass. Accordingly, bremsstrahlung is

not an important energy loss mechanism for particles other than

electrons. Nuclear collisions occur only for heavy particles, not

electrons, and cause a beam’s current to decline exponentially

with distance. For protons in air at sea level, the l/e distance is

about 1.45 km, scaling inversely with atmospheric density.

5. All the factors which affect the intensity of a charged particle

beam in the atmosphere become less severe as the atmospheric

density decreases. Therefore, hole boring may increase a

charged particle beam’s range in the air. This involves heating a

channel of air as far as a beam will go, then turning off the

beam as the heated air expands to a lower density. Following

this expansion, the beam can then propagate through the heated

channel, heating another region of air further downstream.

Thus, the beam must be chopped into segments whose width is

long enough to heat the air to the desired degree and whose

separation is great enough for the heated air to expand out of

the beam channel. Heating times are typically on the order of 1

sec, and channel expansion times in excess of 10 sec.

6. Over long ranges, the decrease in atmospheric density with

altitude must be accounted for in evaluating the effect of expansion, energy losses, and current losses on a beam’s intensity.

Figures 5–22 and 5–31 provide correction factors to apply to sea

level analysis for a beam aimed into the air at a given range and

elevation angle. The bottom line is that most beam losses occur

in the lowest 7 km of the atmosphere.

Effects of Directed Energy Weapons

316
7. Particle beams have no rigid structure, and may wiggle, both

along and perpendicular to their axis. Because of the interaction

between a beam’s magnetic field and the ionized air plasma

through which it propagates, these wiggles may be magnified,

with the beam becoming unstable. Prevention of instabilities in

a propagating beam requires that the beam be chopped into

segments whose length is less than the distance over which an

instability can grow, and whose separation is sufficiently great

that each segment is independent of its predecessor. This requires pulse widths on the order of nsec, and pulse separations

on the order of μsec.

Implications

Propagation in the atmosphere places severe constraints upon

the energy, current, and pulse structure of a charged particle

beam. If energy losses are to be minimized, the particles must be

relativistic. If expansion is to be minimized, high currents are required. If hole boring is to be attempted and instabilities avoided,

extremely complex pulse structures may result, as illustrated in

Figure 5–35.

The division of the beam into “macropulses” and “micropulses”

as shown in Figure 5–35 can be a challenge in designing a particle

accelerator to create the beam. Moreover, much of the theory of instability growth and suppression has yet to be tested experimentally. While there can be no doubt that the broad outlines of our discussion are valid, specific details and operating constraints cannot

be validated until weapons grade particle accelerators are built.

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Particle Beams

On time less than the time for instability growth

Off time longer than the magnetic diffusion time

Time for the heated channel to expand, boring a hole

Time to heat a channel sufficiently to permit hole boring

Figure 5-35. Pulse Structure for Hole Boring with Instability Avoidance
The complexities of propagation in the atmosphere have in recent years caused interest in particle beams to shift to applications

in space, where the more straightforward problems of propagation

in a vacuum must be dealt with. Here, of course, it is neutral particle beams that find application, and the primary technical challenge is to reduce beam divergence to the point where targets can

be damaged over the ranges anticipated. It is unfortunate that in

space the distances involved are of necessity great, so that this

challenge is not trivial. Nevertheless, the primary challenges in

space are technical and engineering—reducing divergence, orbiting and maintaining particle accelerators and associated equipment, and so forth. By contrast, the primary challenges in the atmosphere are physical. You can’t repeal Maxwell’s equations or

Einstein’s theories, and so the constraints these place on beam design must be accounted for. It is not yet clear which applications in

the atmosphere have a solution consistent with these constraints.

Interaction with Targets

Energy Deposition and Flow

Knowing how neutral and charged particle beams propagate, we

can turn our attention to their interaction with targets. All the

physics we need is already in place, since the deposition of beam

energy in targets occurs by the same mechanisms that cause a loss

of beam intensity in air: ionization, bremsstrahlung, and nuclear interactions. The only difference is that target material is denser than

atmospheric gases. Since all the energy loss mechanisms are proportional to the density of molecules the beam encounters, we’ll need

to scale our atmospheric results appropriately. In essence, then,

beam-target interaction is simply beam-atmospheric interaction,

scaled with density. The interaction of beams with targets is not

qualitatively different from their interaction with the atmosphere.48

From the standpoint of target interaction, it makes no difference

whether the particle beam is charged or neutral. The two types of

beams propagate in different environments: air for a charged particle beam and vacuum for a neutral particle beam. In interacting

with targets, however, they’re both alike, since a neutral beam is

collisionally ionized immediately upon encountering a target

Effects of Directed Energy Weapons

318
(l/N 3 m), and the small amount of energy required (10 eV)

is insignificant relative to the total particle energy.

Despite the fact that solid targets are denser than atmospheric

air, it is nevertheless true that from the standpoint of energy loss

from the beam, the air is thicker than the target. This is because

the range to the target is much greater than the target thickness, so

that a beam propagating in the atmosphere sees more molecules

on its way to the target than it sees within the target itself. Consider, for example, a particle beam firing over a range of only 1 km

to a target which is 10 cm thick.49 Since the density of air is about

10–3 gm/cm3, and solid densities are on the order of 1 gm/cm3,

there are about 1,000 times as many molecules encountered per

unit path length in a solid than in the atmosphere. But the range to

the target is 10,000 times greater than the target thickness, even at

this short range. This means that the beam encounters ten times as

many molecules on the way to the target than within the target itself, and ten times as much energy will be lost in propagation than

in target interaction. The implication of this result is that target interaction is of much less concern for particle beams than for other

types of directed energy weapons. For particle beams in the atmosphere, if you have sufficient energy to propagate to the target,

you’ll have enough energy to damage it. For particle beams in

space, there are no propagation losses other than through beam

divergence, and damage requirements can play a greater role in

establishing beam parameters.

We have seen that relativistic particles lose about 7  105 eV/m

to ionization in traveling through the atmosphere. There are additional energy losses to bremsstrahlung if the particles are electrons, and losses from the beam’s current through nuclear collisions if they are heavy particles. While all of these effects would

need to be considered in a detailed treatment of energy deposition

from a specific type of beam in a specific target, it is adequate to

look at the generic features of beam-target interaction using ionization losses alone. Solid matter is about 1,000 times more dense

that air at sea level, and the energy loss rate is proportional to density, so that the energy loss rate of relativistic particles in solids

should be about 7  108 eV/m, or 7 MeV/cm. Figure 5–36 is a plot

of particle range as a function of energy at this energy loss rate.

You can see from Figure 5–36 that the ranges shown exceed the

thickness of most targets. This means that a particle beam will

319

Particle Beams
penetrate deeply into a target; some particles will even pass

through it. This is in contrast to a laser beam (Chapter 3), which

deposits its energy in a very thin layer on the target surface. From

the standpoint of damaging targets, this means that a particle

beam will do serious damage sooner, reaching and engaging vital

parts rapidly. A laser must heat, vaporize, and penetrate a target’s

outer surface before it can reach these vital parts.

Now let’s consider the resulting differences in energy flow

within a target. With a laser, energy flow is mostly from the surface

of the target towards the interior. With a particle beam, it’s mostly

radially outward from the deposition region, which penetrates well

into the target. This contrast is illustrated in Figure 5–37.

Heat flows downhill along a temperature gradient. With a laser,

the greatest gradient is in the direction into the target, since all the

laser’s energy is deposited within a thin layer on the surface. A particle beam, on the other hand, penetrates into the depth of the target, and the greatest gradient is in the radial direction. What limitation will the flow of energy out of the heated region place on

damage to targets with particle beams? Figure 5–38 illustrates the

mechanisms of energy deposition and loss when a particle beam engages a target. You will recall that the intensity (W/cm2) in a particle beam is just S  nKv, where n is the density of particles in the

beam (cm-3), v their velocity, and K their kinetic energy. In passing

through a target of thickness d, each particle loses energy (dK/dz) d,

so that the beam’s intensity declines to nv[K-(dK/dz) d]. This

means that the rate of energy deposition within the solid (W/cm3) is

Effects of Directed Energy Weapons

320

Figure 5-36. Particle Range in Solid Targets

Kinetic Energy (MeV)

10000.0

0.1

1000.0

100.0

10.0

1.0

Range

(cm)

1 3 10 30 100 300 1000 3000 10000

dK/dz = 7 MeV/cm
just dS/dz  nv(dK/dz). And since the volume of target through

which the beam passes is w2d, the total rate of energy deposition

in that region is w2d nv(dK/dz) (Watts).

Now let’s consider how energy is carried out of the irradiated

region either by thermal conduction or radiation. The flow of energy (Watts/cm2) across a surface due to thermal conduction is

given by u  –k(dT/dx), where k is the thermal conductivity and

dT/dx is the slope of a curve of temperature vs distance. Suppose

we were to look end-on at the target shown on the left hand side

of Figure 5–38. A radial temperature profile will look like that

shown on the right hand side of figure 5–38. The temperature

slope in the radial direction, dT/dr, is on the order of T/w, where

T is the temperature within the beam volume and w is the radius

of the beam. The resulting flow of energy through the surface of

the energy deposition region and into the surrounding target material is about kT/w (Watts/cm2). Since the total surface area of

that region is 2wd, energy flows out of the irradiated volume at a

rate of about 2wd(kT/w) (Watts).50

As we saw in Chapter 1, energy can also be lost from the front

and back surfaces of the target by radiation. If we assume that the

target radiates as a black body, the intensity of radiation (W/cm2)

emitted will be T4, and the total energy loss rate by radiation will

be 2w2 T4 (Watts), where  is the Stefan-Boltzmann constant

(5.67  10–12 W/cm2 K4).

We have argued that the particle beam deposits energy in a target

at a rate w2dnv(dK/dz), that thermal conduction carries energy

away at a rate 2wd(kT/w), and that radiation carries it away at a

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Particle Beams

Figure 5-37. Energy Deposition from Lasers and Particle Beams

Heated Zone

(on surface)

Laser Beam

= Energy Flow

Particle Beam

Heated

Zone
rate 2w2 T4, where T is the temperature in the irradiated region.

We can set the rates of energy deposition and loss equal to one another to find the temperature at which energy is carried away as

fast as it is deposited. For thermal conduction, that temperature is T

 w2nv (dK/dz)/2k = I (dK/dz)/2kq, where I  w2 nqv is the

beam current (Amperes). For radiation, it’s [d I(dK/dz)/2w2

 q] 1/4.

Figure 5–39 is a plot of these temperatures as a function of beam current for q  1.6  10–19 coul, (dK/dz)  7 MeV/cm, w  l cm, and k

 2.4 W/cm K (the thermal conductivity of aluminum).

As a particle beam engages a target, the temperature will rise

until the target is vaporized or energy is carried away as fast as it

is deposited, whichever occurs first. Most materials vaporize at

temperatures on the order of 3000 oK (see Table 1–1). And from

Figure 5–39, you can see that for almost any reasonable beam current, the vaporization point will be reached well before thermal

conduction or radiation can limit the temperature rise in a beamirradiated target. This means that we need not be concerned about

the radial flow of energy from the heated zone or radiation from

the target surface limiting the target’s temperature. Particle beams

with high currents and energies deposit energy too rapidly for

energy loss mechanisms to affect the interaction. For all practical

purposes, the irradiated region of the target may be assumed

vaporized as soon as sufficient energy has been deposited. Our

next task is to determine how long a time that is.

Effects of Directed Energy Weapons

322

Figure 5-38. Steady-State Energy Deposition and Loss in a

Particle Beam-Irradiated Target

surface area of irradiated volume = 2πwd

volume of irradiated volume = πw2d

End View

dT/dr T/w

d

nv

w

r
Damage and Interaction Times

There are two times of interest from the standpoint of damaging

targets with particle beams—the damage time, or the time the

beam must engage the target to damage it, and the interaction

time, the actual time the beam must deposit energy within the

target. In general, these two times will not be the same for charged

particle beams. Atmospheric propagation constraints, such as the

times the beam must be off to permit hole boring or to prevent instability growth must be considered in the total time to damage

the target. Even for neutral particle beams, constraints in accelerator design may prevent pulse widths from being arbitrarily long,

so that multiple pulses on target may be required. If the off time is

too long relative to the on time, thermal conduction or radiation

could begin to affect the energy density within the irradiated area,

and the interaction time may increase.

How long will typical interaction times be? We know the rate of energy deposition (W/cm3)—dS/dz  nv(dK/dz) =I(dK/dz)/w2q,

where (dK/dz) is the particle energy loss rate, I is the beam current, q

the particle charge, and w the beam radius. From Chapter l, we also

know the energy density (J/cm3) needed to vaporize the target material—	(CTv+Lm+Lv) 104 J/cm3. The time t to heat the target to the

point where it vaporizes is simply the ratio of the energy required to

the energy deposition rate, or t  w2q 	(CTv+Lm+Lv)/I(dK/dz). Figure 5–40 is a plot of this time as a function of beam current and radius.

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Particle Beams

0.01 0.1 1 10 100 1000 10000

Thermal Conduction

Radiation

Beam Current, Amperes

Temperature

(K)

Figure 5-39. Target Temperature vs Beam Current

1010

108

106

104
As you can see from Figure 5–40, interaction times are quite

short. A 10 kAmp, 1 cm beam can vaporize a target in about 1

sec. But even over short ranges, the total engagement time

will be greater. You will recall, for example, that to avoid the hose

instability the beam must be split up into segments about a nanosecond (10–9 sec) in length, separated by periods of several microseconds. It takes 1,000 pulses of 10–9 seconds each to add up to a

microsecond of interaction time. If each of these pulses is separated by a microsecond, the total time the beam must engage the

target will be at least 1000  10–6  10–3 seconds. In this simple example, the total time to damage the target is about three orders of

magnitude greater than the actual time the beam interacts with the

target. More generally, if a particle beam of current I is composed

of pulses of width tp separated by a separation time ts, the effective

average current on target will be I tp/(tp+ts), and the actual engagement time can be found by using Figure 5–40 with this effective current. Should the effective current fail below the level where

Figure 5–39 shows thermal conduction or radiation to be important, these loss mechanisms will increase the energy necessary for

damage, and must be accounted for as well. In general, then, the

current per pulse should be kept large enough that the effective

current doesn’t fall below about 10–2 – 10–1 Amperes.

Effects of Directed Energy Weapons

324

Beam Current, Amperes

Interaction

Time

(Sec)

10-1

10-2

10-3

10-4

10-5

10-6

w = 10 cm

100 1000 10000

Figure 5-40. Interaction Time for Target Vaporization vs Beam Current and Radius

1 cm
Summary: Interaction with Targets

1. Particle beams interact with solid matter through one-on-one

interactions, where individual particles in the beam lose energy

through encounters with individual particles in the target. Since

this is the same way in which particle beams interact with the atmosphere, energy deposition in targets may be estimated by

scaling atmospheric energy losses with density.

2. Since the distance to a target is much greater than the thickness of a target, more energy is generally lost in propagation

through the atmosphere than is deposited within the target itself. Therefore, beams with sufficient energy to propagate

through the atmosphere will have sufficient energy to damage

their target. As a result, target interaction is an important factor

in beam design only for neutral particle beams propagating in

a vacuum.

3. Relativistic particles have ranges in solids comparable to a

typical target thickness. Therefore, a beam deposits energy

throughout a target’s volume, and not on its surface. This results

in lower temperature gradients, and neither thermal conduction

nor radiation are effective as mechanisms to limit target damage.

4. Relativistic beams of high current deposit energy in a target

rapidly, and the time to achieve damage is short, on the order of

microseconds. However, various factors affecting propagation,

such as the need to avoid instabilities or bore through the atmosphere, can require that energy be delivered to a target in

short bursts over a longer period. This decreases the average

rate of energy delivery and increases the time necessary for the

beam to engage its target.

Implications

Because of the way in which particle beams interact with targets, shielding by applying extra layers of material to a target’s

surface is not a practical way of limiting target damage. Relativistic particles penetrate so deeply that an inordinate weight and volume of material would be required for shielding. Consider, for example, a target whose nominal size is 1 meter. Such a target has a

total surface area on the order of 12 m2. To shield this target from

particles of energy 10 MeV would require about a 1 cm layer of

325

Particle Beams
material over the entire surface of the target, for a total volume of

0.12 cubic meters of shielding material. At a density of 3 grams per

cubic centimeter, this shielding would add about 360 kg (780 lb) to

the weight of the target. This would be an unacceptable weight

penalty for most targets, and it could easily be countered by increasing the energy in the beam to 20 MeV, which would necessitate doubling the amount of shielding. Thus, countermeasures

against particle beams are much less practical than against lasers,

where energy deposition is a surface effect, and a small amount of

shielding can have a much greater protective effect.

Summary of Main Concepts

In looking at the propagation and interaction of charged and

neutral particle beams, we’ve introduced numerous physical concepts. We’ll summarize the main concepts here to give you a final

glimpse of the forest after having wandered among the trees for so

many pages.

1. Particle beams are large numbers of atomic or sub-atomic

particles moving at relativistic velocities (velocities approaching that of light). Because of the large number or density of

particles in these beams, their interactions among themselves

can be as important as their interactions with the atmosphere

and with targets.

2. There are two types of particle beams: charged and neutral.

Charged particle beams consist of particles such as electrons

and protons which have an electrical charge. These beams tend

to spread because of the mutual repulsion of their particles.

Neutral particle beams consist of electrically neutral particles,

such as hydrogen atoms.

3. A particle beam is characterized by the current it carries, the

energy of its particles, and its radius. These quantities may be

related to more weapon related parameters such as intensity,

through the relationships shown in Table 5–2.

4. Real particle beams deviate from perfection, in which all the

particles propagate in the same direction with the same velocity.

This lack of perfection may be expressed in many ways, such

as the beam’s brightness (current per area per solid angle),

divergence (the angle which the beam’s envelope makes as it

Effects of Directed Energy Weapons

326
expands), or temperature (small, random fluctuations in energy

about the average value).

5. Neutral particle beams can propagate only in a vacuum (altitudes greater than about 100 km). In the atmosphere, their particles will ionize through collision with atmospheric particles.

Propagation over reasonable distances requires that a neutral

beam’s divergence be on the order of microradians or less.

6. Charged particle beams can propagate only in the atmosphere (altitudes less than about 200 km), since in a vacuum they

rapidly diverge through electrical repulsion. In the atmosphere,

charged beams ionize the gas through which they propagate,

which enables charge to flow in such a way that electrical repulsion is neutralized.

7. Charged particle beams which have been charge neutralized

pinch down to a smaller radius through magnetic forces, then

expand again because of the random motion their particles receive in colliding with atmospheric particles. The resulting expansion is unacceptable except at currents in excess of a kiloamp. Figure 5–20 shows the distance over which the beam

radius grows as a function of beam current and particle energy.

8. In propagating through the atmosphere, particles in a beam

lose energy by ionizing the background gas, as well as through

radiation (bremsstrahlung) induced by the acceleration they

suffer in collisions. The rate of energy loss to ionization is independent of particle type, and at relativistic energies is roughly

energy-independent. The rate of energy loss to bremsstrahlung

is roughly proportional to energy, and inversely proportional to

the square of the particle mass. Therefore, bremsstrahlung is of

concern only for light particles (electrons).

9. In propagating through the atmosphere, a beam of heavy

particles (protons or atomic nuclei) loses current from collisions

with the nuclei of particles in the atmosphere.

10. All the adverse effects associated with atmospheric propagation (expansion, ionizaton, bremsstrahlung, and nuclear collisions) are reduced in magnitude as the atmospheric density is

reduced. Losses are therefore greatest within the lowest 7 km of

the atmosphere, and may be reduced by hole boring, in which

the beam is initially used to heat the channel of air through

327

Particle Beams
which it passes, and this channel then expands, becoming less

dense and allowing propagation to greater distances.

11. Particle beams in the atmosphere can become unstable and

cease to propagate when feedback from one portion of the beam

to another enables small perturbations to grow catastrophically.

These instabilities can be eliminated by chopping the beam into

segments whose duration is less than the time it takes the instability to grow, and whose separation is sufficient for each beam

segment to be an independent event.

12. Particle beams interact with targets just as they do with the

atmosphere—through ionization, bremsstrahlung, and nuclear

interactions. The energy deposited within the target may therefore be calculated by scaling with density. Typically, energy

losses from a particle beam propagating through the atmosphere to a target are less than those within the target itself, so

that target interaction is not of as much concern as propagation

in this case.

13. The range of relativistic particles in solid matter is typically

centimeters or greater, so that these particles deposit their energy in depth through the target, rather than on its surface. This

reduces temperature gradients, and the resulting transfer of energy by thermal conductivity, to the point where it generally

need not be considered in developing damage criteria.

14. For particle beams in the atmosphere, the total time it takes

to destroy a target may be greater than the time required for a

constant beam to deposit sufficient energy on it, since time must

be allowed for hole boring, the suppression of instabilities, and

so forth.

Overall Implications

In principle, particle beams should be ideal as directed energy

weapons. Unlike lasers or microwaves, their propagation in unaffected by clouds, rain, or other meteorological effects, which add

very little to the mass a particle beam might encounter on the way

to its target. Therefore, they are in effect all-weather devices, capable of engaging targets under almost any circumstances. Once

they encounter a target, the long penetration range of relativistic

particles ensures that critical components on the interior of the

Effects of Directed Energy Weapons

328
target will be rapidly engaged. Time need not be wasted in eroding away protective layers of matter on the target’s surface in

order to reach them. Shielding targets as a defensive countermeasure is not a practical alternative.

In practice, the apparent difficulties in achieving stable propagation through the atmosphere have caused interest to focus on

space-based neutral particle beams, where the physical problems

of atmospheric propagation are replaced with the engineering

problems associated with deploying and maintaining large constellations of particle accelerators in space. Both charged and neutral particle beam research has been hampered by the lack of

weapon grade accelerators which might test our largely theoretical understanding of propagation and interaction issues. The theoretical potential of particle beams as directed energy weapons is

clear, but the experimental demonstration of that potential awaits

advances in the state of the art for particle accelerators.

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Particle Beams
Notes and References

1. A good summary of the different techniques used to accelerate

particles can be found in Waldemar Scharf, Particle Accelerators and

their Uses (New York: Harwood Academic Publishers, 1986).

2. Electric and magnetic fields, and the forces which they exert

on charged particles, are discussed in any introductory physics

text. See, for example, David Halliday and Robert Resnick, Physics

(New York: John Wiley and Sons, 1967).

3. The way in which the vector quantities v and B are combined

to yield the resultant force on a charged particle is known mathematically as the vector product, or cross product. Detailed treatments

of vectors and the ways in which they are combined can be found

in junior- or senior-level texts on electromagnetism or mechanics,

such as Jerry B. Marion, Classical Electromagnetic Radiation (New

York: Academic Press, 1968).

4. A very readable introduction to Einstein’s theory of relativity

and to the seemingly contradictory experimental data which that

theory served to clarify can be found in Chapter VI of G. Gamow,

Biography of Physics (New York: Harper and Row, 1964). A more

technical treatment is in Robert Resnick, Introduction to Special Relativity (New York: John Wiley and Sons, 1968).

5. The physical meaning of the energy E = mc2 is that this is the

energy that would be liberated if a particle of mass m were totally

annihilated. This is observed experimentally when particles and

their antiparticles are brought together. See Chapter VII in Gamow

(note 4).

6. The equivalence of the relativistic and traditional definitions

of kinetic energy in the limit where v << c can be demonstrated by

making use of the definition of , together with the approximation

1/(1–x)1/2 1 
 x/2 for x << 1.

7. The electric and magnetic fields shown in Figure 5–4 are special cases of the more general situation where the particle beam

propagates parallel to an external, guiding magnetic field, or has

some fraction of its charge neutralized due to the presence of stationary background particles with a charge of the opposite sign. A

detailed treatment of these fields can be found in R.C. Davidson,

Theory of Nonneutral Plasmas (Reading, MA: W. A. Benjamin, 1974).

Effects of Directed Energy Weapons

330
8. The earth’s naturally occurring electric field results from a

continual charging of the atmosphere through thunderstorms. See

Chapter 9 in Volume II of Richard P. Feynman, Robert B. Leighton,

and Matthew Sands, The Feynman Lectures on Physics (Reading,

MA: Addison-Wesley, 1964). The earth’s magnetic field arises from

currents circulating in its molten core. The field strength of 0.5

Gauss is appropriate for the latitude of Washington, DC. See Halliday and Resnick (Note 2), Appendix B.

9. Electron volts are a convenient unit in particle physics because

an electron of charge e  1.6  10–19 coul gains 1 eV of energy if it is

accelerated by a potential of 1 Volt. Thus, the energy of particles

whose charge is ze gain an energy in electron volts numerically

equal to zV when passed through an accelerator of potential V.

10. The subscript is meant to convey the fact that most deviation

from ideal motion is in a direction perpendicular to the beam’s

main direction of motion. When particles are highly relativistic,

their speed is so close to the speed of light that minor fluctuations in

energy will make little difference in their speed of forward motion.

11. Strictly speaking, the average of is zero, since as many particles have deviations down as up. The divergence is really the average of the magnitude of .

12. Unfortunately,while the concept of brightness is relatively

straightforward, there has been no movement towards a universally accepted definition. Various definitions, differing by factors

of  or  from one another, may be found in the literature. This

can make it difficult to compare beam characteristics as reported

by different laboratories. A detailed discussion of the brightness

concept can be found. J. D. Lawson, The Physics of Charged Particle

Beams (Oxford: Oxford University Press, 1978).

13. Just as the average of should be zero (note 11), the average

perpendicular velocity, <v>, should be zero, since there are generally as many particles with vgoing up (positive) as there are

with vgoing down (negative). What is really meant here by the

symbol <v> is the root mean square average of v, the square root

of the average of v

2, or [<v2>]l/2.

14. It’s interesting to note that electrons, the lighter particles, expand less rapidly than do protons at a given energy. This is because electrons have a much higher  than protons at the same en331

Particle Beams
ergy. For example, a I GeV electron has a  of about 2000, while a

GeV proton has a  of about 2.

15. The cyclotron radius is discussed in almost any physics text,

such as Halliday and Resnick (note 1). It receives its name from

the fact that this type of circular motion in a magnetic field is employed in a particle accelerator called a “cyclotron.”

16. An excellent review article on the earth’s magnetic field and

that of other planets in the solar system can be found in Louis J.

Lanzerotti and Stamatios M. Krimigis, “Comparative Magnetospheres,” Physics Today volume 38, 24 (November, 1985). When

charged particles propagate in a direction which is not perpendicular to a magnetic field, their motion is helical, a combination of

circular motion around the magnetic field lines and forward motion along the field lines.

17. The concept of a cross section is used throughout atomic, molecular, and nuclear physics to describe interactions between two

particles (binary interactions). An introductory text in any of these

areas will discuss the subject in detail. The probability for reactions to occur are typically presented in the literature in terms of

cross sections. A good physical discussion of the cross section concept can be found in Feynman (note 8). See Volume I, section 32–5.

18. A good review of ionization cross section data can be found in

H. Tawara and A. Russek, Reviews of Modern Physics volume 45,

178 (1973).

19. This rule of thumb can also be related to the fact that the distance at which electrons orbit nuclei is on the order of 10–8 cm. For

example, the lowest orbit of electrons around the proton that is the

nucleus of a hydrogen atom (the first Bohr orbit) is at a distance of

0.53  10–8 cm. Anders J. Ångstrom (1814–1874) was a Swedish

spectroscopist who first discovered that the sun’s atmosphere contained hydrogen.

20. Figure 5–14 is based on standard atmosphere data found in R.

C. Weast (ed), Handbook of Chemistry and Physics, 45th ed. (Cleveland, OH: Chemical Rubber Co, 1964).

21. Halliday and Resnick (note 1), section 28–4.

22. Common notation is to use  for conductivity. We’ve used the

capital 
 to avoid confusion with the collision cross section, which

is also commonly denoted by .

Effects of Directed Energy Weapons

332
23. The expression for the decay of excess charge density is derived in Marion (note 2), section 4.2.

24. A discussion of the conductivity of an ionized gas can be found

in Chapter II, Section 13 of M. Mitchner and C. H. Kruger, Jr., Partially Ionized Gases. (New York: John Wiley and Sons, 1973). Table 1

in Chapter IV provides typical conductivities for various plasmas.

25. The initial portion of a particle beam doesn’t have the benefit

of charge neutralization, since it is in the process of ionizing the

air through which it passes. This leads to a phenomenon known as

beam head erosion, in which the initial portions of the beam disperse

as they make way for subsequent portions. It has been suggested

that the beam channel can be pre-ionized with the use of a laser, so

that the beam can have a path already prepared through which it

can propagate. See Section 4.2.1 of the “Report to the APS of the

Study Group on Science and Technology of Directed Energy

Weapons,” Reviews of Modern Physics 59. Part II (July, 1987).

26. See Davidson (note 7), section 2.5. A discussion of the dynamics by which the equilibrium radius is approached can be found in

S. R. Seshadri, Fundamentals of Plasma Physics. (New York: American Elsevier, 1973), section 2.10.

27. A pretty comprehensive treatment may be found in chapter 13

of J. D. Jackson, Classical Electrodynamics. (New York: John Wiley

and Sons, 1963). The quantity Q which appears in the expression

for dT/dz is the ratio of the maximum to minimum values of the

distance of approach of the beam particle to the atmospheric particle which scatters it. See Figure 5–24. The exact value of this parameter is not very important, since it only appears as a logarithm.

28. The exponential variation of density with altitude may be inferred from Figure 5–14, since an exponential function plots as a

straight line on log-linear paper. The scale length can be found from

the slope of this plot. The 7 km length used here is a common approximation for the lower atmosphere. See Chapter 4 of the APS Report on Directed Energy Weapons (note 25). This type of behavior may

also be derived theoretically. See Section 6.3 in F. Reif, Fundamentals

of Statistical and Thermal Physics (New York: MCGraw-Hill, 1965).

29. This relationship for dT/dz is not exact, since the different atmospheric constituents fall off differently with altitude. However,

it is sufficiently accurate for our purposes here.

333

Particle Beams
30. Figure 5–22 is a plot of o

z (dT/dz)dz/[dT/dz(o)ho] 

[l– exp(-z sin/ho)]/sin.

31. Of course, as the beam gets too high in the atmosphere, expansion will begin again as a result of electrostatic repulsion. See Figure 5–18.

32. See Jackson (note 27), Section 13.3.

33. Figure 5–25 is a plot of Equation 13.44 in Jackson (note 27).

34. A good discussion of radiation from accelerated charges can

be found in Marion (note 3), Chapter 7.

35. See Jackson (note 27), Chapter 15. Figure 5–26 is based on

Jackson’s equations 15.26 and 15.45.

36. Particles interact through four forces: gravitational, electromagnetic, strong, and weak. Only the first two have a range sufficiently great to be a part of our everyday experience. The second

two, however, are responsible for such fundamental things as how

nuclei are formed and decay. In recent years, unified theories of

these forces have emerged, in which the last three are seen as different manifestations of a single theory. These theories have implications both for the small-scale structure of matter and the largescale structure of the universe. See A. Linde, “Particle Physics and

Inflationary Cosmology,” Physics Today 40, 61 (September, 1987).

37. Any text on nuclear physics will discuss nuclear collisions. for

example, see Chapter 16 of I. Kaplan, Nuclear Physics (Reading,

MA: Addison-Wesley, 1963).

38. Ibid Chapter 3.

39. While hole boring where the beam produces its own channel

has not yet been observed experimentally, electron beams have

been guided through an ionized channel created by a laser in low

pressure gas. See S.L. Shope, et alt “Laser Generation and Transport of a Relativistic Electron Beam, Transactions on Nuclear Science

NS–32, 3092 (October, 1985), and Theresa M. Foley, “Sandia Researchers Report Progress on Delphi Beam Weapon Technology,”

Aviation Week and Space Technology, 129, 79 (August 22, 1988).

40. It is, of course, possible to leave the beam on for the whole

time that it’s boring through the air to the target. However, this

wastes energy compared to the pulsed approach discussed in the

Effects of Directed Energy Weapons

334
text. The actual approach used would depend on energy, accelerator, and other constraints.

41. A good survey of instabilities can be found in Chapter 4 of R.

B. Miller, An Introduction to the Physics of intense Charged Particle

Beams. (New York: Plenum Press, 1982).

42. A derivation of the fact that magnetic field lines are frozen into

a perfectly conducting plasma can be found in Seshadri (note 26),

Section 2.9.

43. The hose instability is discussed in Miller (note 41), section

4.4.1. Figure 4.4 in that section is a nice picture of a hose instability

in a relativistic electron beam.

44. A derivation of the time scale for magnetic field lines to thaw,

or diffuse to their final position in a resistive plasma, can be found

in Section 3.9 of N. A. Krall and A. W. Trivelpiece, Principles of

Plasma Physics. (New York: MCGraw Hill, 1973).

45. Miller (note 41), Section 4.4.1.

46. The sausage instability is discussed both in Seshadri (note 26),

Section 2.11, and in Miller (note 41), Section 4.4.2.

47. See Chapter 7 in Scharf (note 1).

48. This is in contrast to lasers, whose interaction with targets is

quite different from that with the air (Chapter 3). That’s because

collective effects among the molecules in a target dominate the target’s response to laser light. The energies which bind molecules

together into solids are so small compared to relativistic particle

energies that they play no role in beam-target interaction.

49. Ten centimeters is a very large target thickness. You can see in

Chapter 1 (Figure 1–10) that the amount of matter which must be

penetrated to damage most targets is on the order of centimeters

or less. Therefore, the conclusion that energy losses in propagating

to a target are greater than those in propagating through a target is

a fairly robust one.

50. If the target is in the atmosphere, there could be some loss of

energy by thermal conduction from the surface of the target into

the air. However, the thermal conductivity of air is so low compared to that of metals that this contribution to the energy loss

may safely be neglected.

335

Particle Beams
Appendix A

Units

Hopefully, the concepts in the text have been developed with

sufficient clarity to give you a good physical feeling for the issues

associated with the propagation of directed energy weapons and

their interaction with targets. The many graphs and figures should

also enable you to make quantitative estimates of important parameters, such as the distance a particle beam can be expected to

propagate in the atmosphere, or the time it will take a laser to melt

a hole through a target of a given thickness.You may, however,

want to evaluate some of the formulas in the text for yourself, either to extend our results beyond what is covered in the charts or

to compare our results with those in other references or reports. If

you do, you’ll come up against the problem of units. Scientists and

engineers in any specialized area like to express things in units

which are easy to remember or which make their formulas easy to

write. This is great for the initiated, but can cause confusion when

one reference is compared with another. For example, in Chapter 5

the electric field a distance r from a particle of charge q is given as

E  q/4or2. But if you consult many of the references at the end

of Chapter 5, you’ll see that the same field is given as E  q/r2.

Both are correct, but the units of q and E in the two forms are different. And if quantities are not substituted into equations with

the correct units, the answers obtained will be numerically incorrect. This could cause problems, if you calculated a weapon to

have an effective range of 10 km, and the true answer was 10 cm!

There are two ways around this problem. The first, and easiest,

is scaling. Both E q/4or2 and E  q/r2 scale as q/r 2. This

means that if q doubles, E doubles, and if r doubles, E decreases by

a factor of 4. Therefore, if you have an answer you trust, say from

one of the charts or examples in the text, you can scale it by using

ratios of parameters, and ratios are independent of the choice of

units. For example, in Chapter 3 we estimate the rate at which a

laser can melt through a target as Vm  S/[Lm  C(Tm–To)].

Figure 3–65 is a plot of this relationship for four specific values of

. If you need to evaluate the erosion rate for a different value of ,

you need only multiply one of the curves in Figure 3–65 by the

ratio of the absorptivities for the new case and the one on the

chosen curve.
If it’s not possible to scale from a known answer, self consistency is the key to numerical accuracy. In this book, our preference

is for “MKS” or “SI” units, where lengths are in meters, masses in

kilograms, time in seconds, and electrical charge in Coulombs. In

these units, energy will be in Joules, power in Watts, and current

in Amperes—the familiar units of everyday experience. Formulas

in the text will work out correctly as long as input quantities are in

these units. For example, the rate at which electrons heat when interacting with microwaves is d/dt  e2S/2mcoc. This expression will come out correctly in Joules/sec, provided e is in

Coulombs, S in Watts/m2, m in kg, c in m/sec, and c in sec–l. Typically, of course, we express S in W/cm2, since beam sizes for lasers

and particle beams are more likely to be on the order of a centimeter, and particle beam people will express m in electron volts. But

to evaluate the expression, you need to convert these to the self

consistent set of meters, kilograms, seconds, and Coulombs. Having obtained the answer in J/sec, you can then convert it to some

other more convenient unit, such as eV/sec, if you choose.

Almost any science or engineering text will have as an appendix

tables of conversion factors for different units.l The following tables should be adequate for converting to the units we have used

in the text. In keeping with our zero order approach, no attempt

has been made to provide highly accurate conversion factors,

which are available elsewhere.2 If the general concept of constructing self-consistent sets of units is of interest, books on the subject

are also available.3 The official SI units for evaluating formulas are

in the right hand column.

Effects of Directed Energy Weapons

338
Dimensionless Units

1 degree (°)  1.745  10–2 radians (rad)

1 circle  2rad 

1 sphere  4steradians (sr)

Units of Length 

1 Ångstrom  10–10 m 

1 micron  10–6 m (1 	m) 

1 inch (in)  0.0254 m 

1 foot (ft)  0.3048 m 

1 mile (mi)  1609.3 m 

1 nautical mile  1852 m

Units of Velocity 

1 ft/sec  0.305 m/sec

1 km/hr  0.278 m/sec

1 mi/hr  0.447 m/sec

1 knot  0.514 m/sec

339

Appendix A

Power of 10 Prefixes 

(e.g. 109 Watt = l Gigawatt (GW), 10–3 m = 1mm)

Power Prefix Abbreviation

12 Tera T

9 Giga G

6 Mega M

3 Kilo k


2 Centi c 


3 Milli m 


6 Micro 	


9 Nano n 


12 Pico p
Units of Area 

1 cm2  10–4 m2

1 in2  6.45  10–4 m2

1 ft2  0.093 m2

1 mi2  2.59  106 m2

Units of Volume 

1 cm3  10–6 m3

1 in3  1.64  10–5 m3

Units of Mass 

1 gm  10–3 kg 

1 MeV  1.78  10–30 kg*

1 GeV  1.78  10–27 kg*

1 lb  0.454 kg**

*The eV is not really a unit of mass, but of energy. This is the mass equivalent using

m = E/c2.

**The pound is a unit of force. This is the mass that exerts a 1 lb force on the earth’s surface.

Units of Density 

1 gm/cm3  103 kg/m3

1 lb/ft3  16 kg/m3

1 lb/in3  2.77  104 kg/m3

Units of Energy 

1 BTU  1055 J 

1 kw hr  3.6  106J

1 calorie  4.19 J

1 eV  1.6  10–19 J 

1 erg  10–7 J 

Effects of Directed Energy Weapons

340
Units of Power 

1 erg/sec  10–7 W 

1 horsepower  746 W

1 BTU/hr  0.293 W 

Units of Fluence and Intensity

1 J/cm2  104 J/m2

1 W/cm2  104 W/m2

Units of Pressure (Stress)

1 Atmosphere  105 Pascal (1 Pa  1 Nt/m2)

1 bar  105 Pascal 

1 torr  133 Pa 

1 dyn/cm2  0.1 Pa 

1 psi  6895 Pa 

Units of Electromagnetic Theory 

1 Gauss  10–4 Tesla (1 T  1 Weber/m2)

1 Statvolt  300 Volt

1 Statcoul  3.33  10–10 Coul

1 Abcoul  10 Coul

1 Statamp  3.33  10–10 Amp

1 Abamp  10 Amp 

1 Amp hr  3600 Coul

1 ohm–cm  10–2 ohm–m

1 mho  1 Siemens (1 S  1 ohm–1)

341

Appendix A
References

1. One of the most convenient and comprehensive sets of tables is

in Appendix G of David Halliday and Robert Resnick, Physics for

Students of Science and Engineering, 2nd ed. (New York: John Wiley

and Sons, 1967).

2. Comprehensive, up-to-date, and accurate conversion factors

may be found in Section 1.02 of Herbert L. Anderson (ed), Physics

Vade Mecum (New York: American Institute of Physics, 1987).

3. The general theory of units and their dimensions is discussed

in the “Appendix on Units and Dimensions” to John D. Jackson,

Classical Electrodynamics (New York: John Wiley and Sons, 1963).

For a more popular account, see William D. Johnstone, For Good

Measure (New York: Holt, Rinehart, and Winston, 1975). Every obscure unit you could ever be interested in is contained in Stephen

Dresner, Units of Measurement (Aylesburg; Harvey Miller and

Medcalf, 1971).

Effects of Directed Energy Weapons

342
Appendix B

Some Useful Data

Much of the data needed to calculate weapon effects, such as the

thermal conductivities and heat capacities of different materials,

are readily available in handbooks of science and engineering data.

But other information, such as the energy loss rate for charged particles penetrating different materials, can be found only be searching through a variety of sources. Some of these less available data

have been gathered here for your convenience in evaluating or extending results presented in the text. As with all data in this book,

these are presented to give you a feeling for orders of magnitude

and to enable you to make simple estimates. They should not be

considered definitive numbers, since many of the details and assumptions involved in their derivation have been glossed over.

Characteristics of Common Kinetic Energy Rounds 

Weapon Type Bullet Mass Velocity Kinetic Reference

Type (Caliber) (g) (m/sec) Energy (J)

Handgun .38 Special 6.16 361 400 1

Handgun .45 Automatic 11.99 272 444 1

Handgun 9 mm 7.45 363 491 1

Rifle 7.62  51 mm 9.33 838 3276 2

Rifle 7.62  39 mm 7.97 715 2037 2

Machine gun 9  19 mm 7.45 390 567 2

Machine gun 11.4  23 mm 15.16 280 594 2

Specific Impulse of Rocket Fuels 

Fuel/Oxidizer Isp (sec) Reference

Hydrogen/Oxygen 391 3

UDMH*/Oxygen 310 3

Ammonia/N2O4 269 3

*Unsymmetrical dimethylhydrazine
Failure Data for Common Materials 

Material Modulus of Rupture Strain at Reference

(J/m3) Rupture

Aluminum 3.0  108 0.15 4

Wood 3.6  106 0.03 5 

Concrete 4.5  106 0.002 5

Thermal Coupling Coefficients for Lasers*

Material Wavelength(	m)  Comments Reference

Titanium (6 A1–4V) 3.8 0.23 6

10.6 0.14 6

2.8 0.24 8

Stainless Steel (604) 3.8 0.17 6

10.6 0.10 6

2.8 0.25 8

Aluminum (abraded) 10.6 0.09 without plasma 7

ignition

10.6 0.18 with plasma 7

ignition

Aluminum 2.8 0.06 8

10.6 0.03

Nickel 2.8 0.13 8

*Thermal coupling coefficients can vary with sample, surface preparation, and laser intensity, especially if plasmas are ignited. Theoretically, the coupling coefficient should scale

the square root of frequency. See Chapter 3.

Effects of Directed Energy Weapons

344
Specific Impulse Coupling Coefficients for Lasers*

Material Wavelength I* Comments Reference

(	m) (dyn sec/J)

Aluminum

10.6 0.3 without plasma ignition 9

10.6 7.5 with plasma ignition 9

1.06 1.0 10

0.248 3.0 10 

0.53 1.0 10 

Titanium

10.6 0.2 9

0.248 3.0 10

Stainless Steel

10.6 0.2 9

*Specific impulses can show considerable variability shot-to shot, and are strongly

affected by the ignition of plasmas.

Relativistic Particle Ionization Energy Loss Rates*

Material dK/dz (MeV/cm)

Aluminum 4.05

Copper 13.44

Iron 11.85

Magnesium 2.61

Titanium 6.74

*Calculated values, based on Eq 6.40 in Reference 10. Assumes  > 2.

Electron Radiation Lengths*

Material Radiation Length (cm)

Aluminum 8.25

Copper 1.82

Lead 0.39

Magnesium 18.50

Iron 1.49

Titanium 3.08

*Calculated values, scaled from the value for lead by Equation 15.48 in Reference 11.

Assumes  > 2.

345

Appendix B
References

1. Mason Williams, Practical Handgun Ballistics (Springfield, IL:

Charles C. Thomas, 1980).

2. C. J. Marchant-Smith and P. R. Hulsom, Small Arms and Cannons (Oxford: Brassey’s Publishers, 1982).

3. Charles H. MacGregor and Lee H. Livingston (eds) Space

Handbook, AU–18 (Maxwell AFB, AL: Air University, 1977).

4. “Report to the APS of the Study Group on Science and Technology of Directed Energy Weapons,” Reviews of Modern Physics

59, Part II (July, 1987).

5. Michael S. Feld, Ronald E. MCNair, and Stephen R. Wilk, “The

Physics of Karate” Scientific American 240, 150 (April, 1979).

6. T.J. Wieting and J. T. Schriempf, “Infrared Absorptances of

Partially Ordered Alloys at Elevated Temperatures, “Journal of Applied Physics 47, 4009 (September, 1976).

7. J.A. MCKay, et al., “Pulsed CO2 Laser Interaction with Aluminum in Air: Thermal Response and Plasma Characteristics,”

Journal of Applied Physics 50, 3231 (May, 1979).

8. R.B. Hall, W. E. Maher, D. J. Nelson, and D. B. Nichols, “High

Power Laser Coupling,” Air Force Weapons Laboratory, Kirtland

AFB, NM, Report no AFWL–TR–77–34 (June, 1977).

9. S.A. Metz, L. R. Hettche, R. L. Stegman, and J.T. Schriempf, 

“Effect of Beam Intensity on Target Response to High-Intensity

Pulsed CO2 Laser Radiation,” Journal of Applied Physics 46, 1634

(April, 1975).

10. “Report to the APS of the Study Group on Science and Technology of Directed Energy Weapons,” Reviews of Modern Physics

59, Part II (July, 1987).

11. John D. Jackson, Classical Electrodynamics (New York, John

Wiley and Sons, 1963).

Effects of Directed Energy Weapons

346
About the Author

347

Philip E. Nielsen is a director and senior technical advisor for

MacAulay-Brown, Incorporated, a defense engineering services

firm headquartered in Dayton, Ohio. Prior to joining MacAulayBrown, Dr. Nielsen served on active duty with the U.S. Air Force

(USAF) for 26 years, retiring as colonel. During this period, he

served in a variety of positions related to the research, development, and acquisition of advanced weapon systems. He received

the USAF Research and Development Award for contribution to

high energy laser physics in 1975.