Robert M. Manning

Glenn Research Center, Cleveland, Ohio

Beamed-Energy Propulsion (BEP): Considerations 

for Beaming High Energy-Density Electromagnetic 

Waves Through the Atmosphere 

NASA/TM—2015-218726

May 2015
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Robert M. Manning

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Beamed-Energy Propulsion (BEP): Considerations 

for Beaming High Energy-Density Electromagnetic 

Waves Through the Atmosphere 

NASA/TM—2015-218726

May 2015

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
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Beamed-Energy Propulsion (BEP): Considerations for 

Beaming High Energy-Density Electromagnetic 

Waves Through the Atmosphere

Robert M. Manning

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Preface

A study to determine the feasibility of employing beamed electromagnetic energy for vehicle 

propulsion within and outside the Earth’s atmosphere was co-funded by NASA and the Defense 

Advanced Research Projects Agency that began in June 2010 and culminated in a Summary Presentation 

in April 2011. A detailed report entitled “Beamed-Energy Propulsion (BEP) Study” appeared in February 

2012 as NASA/TM—2012-217014. Of the very many nuances of this subject that were addressed in this 

report, the effects of transferring the required high energy-density electromagnetic fields through the 

atmosphere were discussed. However, due to the limitations of the length of the report, only a summary of 

the results of the detailed analyses were able to be included. It is the intent of the present work to make 

available the complete analytical modeling work that was done for the BEP project with regard to 

electromagnetic wave propagation issues. In particular, the present technical memorandum contains two 

documents that were prepared in 2011. The first one, entitled “Effects of Beaming Energy Through the 

Atmosphere” contains an overview of the analysis of the nonlinear problem inherent with the transfer of 

large amounts of energy through the atmosphere that gives rise to thermally-induced changes in the 

refractive index; application is then made to specific beamed propulsion scenarios. A brief portion of this 

report appeared as Appendix G of the 2012 Technical Memorandum. The second report, entitled “An 

Analytical Assessment of the Thermal Blooming Effects on the Propagation of Optical and MillimeterWave Focused Beam Waves For Power Beaming Applications” was written in October 2010 (not 

previously published), provides a more detailed treatment of the propagation problem and its effect on the 

overall characteristics of the beam such as its deflection as well as its radius. Comparisons are then made 

for power beaming using the disparate electromagnetic wavelengths of 1.06 μm and 2.0 mm.

NASA/TM—2015-218726 iii

Contents

Preface .........................................................................................................................................................iii

Chapter 1.—Effects of Beaming Energy Through the Atmosphere ............................................................. 1

1.1 Executive Summary ...................................................................................................................... 1

1.2 Introduction................................................................................................................................... 1

1.3 The Deleterious Atmospheric Propagation Mechanisms at High Energies.................................. 3

1.3.1 Ionization and Electrical Breakdown ................................................................................ 3

1.3.2 Induced Molecular Polarization—The Kerr Effect........................................................... 3

1.3.3 Induced Heating of the Atmosphere—Thermal Nonlinearities and “Thermal 

Blooming” ......................................................................................................................... 4

1.3.4 Atmospheric Aerosols....................................................................................................... 4

1.3.5 Atmospheric Turbulence ................................................................................................... 5

1.4 Modeling the Effects of Atmospheric Thermal Nonlinearities on Beamed Energy 

Propulsion and the Required Level of Their Mitigation............................................................... 6

1.4.1 A Quick Overview of the Model and the Description of Beam Behavior at a 

Target in LEO and GEO due to Atmospheric Thermal Nonlinearities............................. 7

1.4.2 Application of the Foregoing to the Calculation of Beam Spread and Deflection 

due to Atmospheric Thermal Nonlinearities for Power Beaming to LEO and 

GEO................................................................................................................................... 8

1.4.3 Phase Compensation of the Effects of Atmospheric Thermal Nonlinearities................. 13

1.4.4 Using the Propagation Model for the Analysis of Millimeter Wave Beaming 

Within the Earth’s Atmosphere....................................................................................... 18

1.4.5 A Tool for the Quick Assessment of Nonlinear Thermal Effects on Power 

Beaming .......................................................................................................................... 25

1.5 Dynamics of Adaptive Phase Compensation for Power-Beaming Applications........................ 26

1.6 Experiments Assessing the Compensation of Thermal Nonlinearities of High Power 

Beams in the Atmosphere........................................................................................................... 29

1.7 Conclusions and Recommendations ........................................................................................... 29

Appendix A.—The Derivation of the Fundamental Nonlinear Propagation Equations for High Energy 

Transmission Through the Atmosphere and the Associated Critical Powers....................................... 31

A.1 The Critical Power for the Atmospheric the Kerr Effect ............................................................ 33

A.2 The Critical Power for Atmospheric Thermal Effects................................................................ 34

A.2.1 Continuous Wave Source ................................................................................................ 34

A.2.2 Pulsed Source .................................................................................................................. 36

A.3 Conclusion .................................................................................................................................. 37

References............................................................................................................................................ 38

Chapter 2.—An Analytical Assessment of the Thermal Blooming Effects on the Propagation of Optical 

And Millimeter-Wave Focused Beam Waves For Power Beaming Applications................................ 41

2.1 Introduction................................................................................................................................. 41

2.1.1 A Simple Analytical Model for Beam Broadening and Displacement ........................... 42

2.1.2 Application of the Foregoing to Millimeter and Optical Wavelength Power 

Beaming .......................................................................................................................... 46

References............................................................................................................................................ 48

NASA/TM—2015-218726 v

Chapter 1.—Effects of Beaming Energy Through the Atmosphere

1.1 Executive Summary

Power beaming the large amounts of energy through the atmosphere that is required for propulsion 

either within the atmosphere or in transferring payloads from LEO to GEO will be met with the extremely 

deleterious effects of the thermal nonlinearities induced by atmospheric heating from beam power 

absorption. The resulting phenomena, collectively called thermal blooming, will be the source of two 

major effects, i.e., beam steering away from the intended target vehicle as well as beam broadening which 

will make the beam larger at the target than intended. Both of these effects have been modeled and are 

shown here to be remedied by the use of high order (10th order Hermite) phase compensation at the 

transmitter. Although it has been shown that these perturbations can be mitigated in principle, there may 

remain many engineering obstacles that must be overcome, such as elimination of mechanical jitter of the 

transmitter platform, etc., during beam propagation. However, there is nothing in the prevailing physics of 

the situation that would preclude power beaming through the atmosphere as discussed in this report.

It is important to begin to capture the prevailing effects and overall system operation using a scaled 

atmospheric experiment that would simulate the realistic environment in which an adaptive optics system 

must operate, from the variable wind velocity up to the mechanical jitter of the transmitter platform. It is 

recommended that, due to the immediate availability of high power sources, scaled atmospheric 

experiments be implemented (tailored after the Scaled Atmospheric Blooming Experiments (SABLE) 

Project by Lincoln Laboratory in the early 1990s) in which the operation of a closed-loop millimeter wave 

adaptive optics algorithm is assessed in the presence of a moving extended target. Additionally, this 

experimental scenario can also be used to address the issue related to the possibility of air ionization, and 

subsequent breakdown, across the apertures of the combined millimeter wave gyrotron sources as 

discussed elsewhere in this report.

Finally, there is the need to look at various general beam wave profiles such as hypergaussian as well 

as fractional charge (in the topological sense) Laguerre-Gaussian beam waves that show great promise in 

their ability to be robust with respect to atmospheric nonlinearities. However, it is important that the 

modeling effort be kept to the level of yielding analytical results, rather than requiring numerical 

evaluation, so as to capture all the nuances of the physics involved and, at the same time, provide a tool 

for overall system evaluation as well as the design of adaptive optics algorithms. Finally, the model 

should be incorporated into a trajectory analysis program so a cadre of launch geometries can be 

evaluated from the point of view of atmospheric thermal nonlinearities.

1.2 Introduction

When delivering large amounts of power through the Earth’s atmosphere via millimeter or infrared 

‘beams’ (i.e., laser beams or beams formed at the output of a millimeter wave antenna system), many 

propagation mechanisms must be addressed that may be potentially deleterious such power transmission.

The most obvious one is the ever-present random variation of the atmospheric refractive index due to 

local temperature variations known as ‘turbulence’. This naturally occurring phenomena is driven by 

thermal convection of heat from the Earth’s surface; once the resulting air motion exceeds a critical value 

of velocity, laminar flow essentially evolves into turbulent flow and fluctuations in the temperature 

distribution becomes statistically random (Ref. 1). These temperature fluctuations then act directly on the 

prevailing refractive index, thus rendering the refractive index a random quantity. These refractive index 

variations randomly focus and defocus the intervening electromagnetic wave field. Thus, the atmosphere 

NASA/TM—2015-218726 1
can be considered to be composed of ‘lenses’ of random focusing and defocusing characteristics that, due 

to the gross atmospheric motion due to wind, move across the beam. This gives rise to many beam quality 

variations; the major ones being beam broadening and beam steering. The statistical analysis and 

modeling of this type of atmospheric propagation as a long and rich history and has resulted in analytical 

descriptions for the impact of turbulence on the operation of systems relying on such beam propagation.

Many models and descriptions exist for the ‘engineering analysis’ of the operation of transmission 

systems that rely on the propagation of electromagnetic beam propagation in the atmosphere (for a good 

recent treatment, see Ref. 2 and the references therein).

The scenario discussed above may be considered as ‘passive’ electromagnetic wave propagation, i.e., 

the wave field moves through an atmosphere the refractive index of which is determined by other sources, 

not the field itself. However, as the energy density of the beam increases, absorption of the beam energy 

by atmospheric gas components results in local heating of the atmospheric which does indeed act directly 

on the refractive index causing it to decrease in value. The possibility of this situation was first advanced 

in 1966 (Ref. 3). This thermal change of the refractive index field then acts on the electromagnetic wave 

field causing it to also change, and so on. The propagation scenario now becomes an ‘active’ one, 

whereby the propagating field modifies the very medium it which it exists. This heating process is called 

‘thermal blooming’ and substantially differs from that of the passive propagation discussed earlier 

(Refs. 4 and 5 (Ref. 5 contains a very comprehensive review of work and references that existed up to 

1990)). Here, a ‘thermal lens’ is created within the atmosphere by the heating due to the energy density of 

the beam. This ‘self-action’ of the beam will not only bend the beam into regions of higher refractive 

index (beam steering), but convection within the atmospheric fluid will also arise which is the source of 

self-induced turbulent flow of the medium. The situation is further complicated when one includes the 

effects of atmospheric wind and aerosols and the abovementioned passive propagation effects.

Defocusing and other such associated nonlinear thermal blooming distortions of the beam cross-section 

will then result. In extreme cases of very large energy densities, the propagating beam will essentially 

break up into smaller beams, or filaments, which severely constrains the amount of energy density that the 

beam will be able to possess as it travels through the atmosphere. Unlike the situation of passive 

propagation, the thermal blooming mechanism introduces nonlinearities into the analysis of the 

phenomena that substantially complicates a complete mathematical description. Complete analyses of 

these types of propagation scenarios can only be done numerically, which was a major activity within the 

United States and Russia in the late 1980s. Other than the usual ‘order-of-magnitude’ estimates using the 

equations of fluid mechanics and wave propagation, only numerical modeling of the effects of 

atmospheric thermal nonlinearities abound in the literature. Analytical treatments appropriate for an 

engineering analysis of atmospheric propagation systems encountering thermal blooming have been 

lacking, especially those that endeavor to describe the result of adaptive correction of such nonlinear 

effects. This situation makes a comparative assessment of the operation of through-the-atmosphere power 

transmission difficult.

In order to assess the atmospheric effects on the various beamed energy propulsion scenarios 

considered in this report, the propagation environment of high-energy electromagnetic wave transmission 

within the Earth’s atmosphere will be presented. First, a brief review will be given in Section 1.3 of the 

major physical mechanisms that prevail in the atmosphere that will deleteriously affect energy transfer via 

electromagnetic beam waves. The critical power thresholds for the wavelengths of 2.0 μm (infrared) and 

2.0 mm (millimeter wave) will be derived for various transmitter output aperture sizes at which these 

propagation mechanisms arise and must be addressed. Once this has been accomplished, Section 1.4 will 

present a propagation model that will be used to calculate the thermally-induced beam broadening and 

steering that will occur for a 2.0 μm laser beam, for a variety of aperture sizes and output powers, 

NASA/TM—2015-218726 2
transmitted from the Earth’s surface to a target at distances of 800 to 35200 km. The adaptive correction 

to mitigate the thermal nonlinearity effects on the beam will then be considered. Of the various 

performance parameters that an adaptive optics system can be designed to optimize at the target, it will be 

the minimization of the beam radius at target that will be the optimization parameter. Section 1.5 will then 

dwell on the dynamics that an adaptive optics system must satisfy, particularly with regard to the control 

delays inherent with the propagation distances that are involved. Finally, Section 1.6 will highlight some 

scaled ground experiments that can be performed using available high power millimeter wave sources. In 

particular, the effects of using an extended target with a closed-loop adaptive optics approach can be 

assessed. At this same time, the possible electrical breakdown of moist air in the vicinity of the combined 

gyrotron outputs can be studied.

1.3 The Deleterious Atmospheric Propagation Mechanisms at High Energies

At the power densities that are required for beamed energy propulsion, several aspects of propagation 

within the atmosphere must be addressed that can potentially perturb the beam wave and render it 

unreliable for energy transfer to a small target. Once these aspects have been identified, physically 

understood and mathematically modeled, appropriate mitigation procedures can be specified and applied 

to the problem to optimize energy transfer. These propagation features and the critical power levels at 

which they will be seen will now be given.

1.3.1 Ionization and Electrical Breakdown

When considering the transmission of high energy electromagnetic waves through the atmosphere, 

the first phenomenon that is considered is air ionization and the subsequent electrical breakdown (Ref. 6, 

a good introduction and the original work). Within the atmosphere, this occurs via the process of cascade 

ionization whereby a free electron is created by multiquantum absorption by the atmospheric gas and 

through the inverse bremsstrahlung process, accelerates and subsequently collides with an atom. The 

collision produces another electron, and both accelerate, collide with other atoms, and so on. This 

cascading process terminates in the release of light known as electrical breakdown. At a wavelength of 

λ = 10.6 μm (i.e., the wavelength of high power CO2 lasers), electrical breakdown occurs at a power 

density of ≈ 109 W/cm2 at atmospheric pressures and densities typical of those at sea level. This intensity 

is reduced by two orders of magnitude by the formation of the shock front due to the explosive detonation 

of atmospheric aerosols. Thus, for purposes of comparison, one can choose a power density of ≈ 107

W/cm2 as the threshold for electrical breakdown within the atmosphere.

At millimeter wavelengths in the 100 to 200 GHz frequency range, electrical breakdown occurs at a 

smaller power density of ≈ 1010 W/m2

. One can then assume the worst case of a twofold decrease in this 

value due to atmospheric aerosols, thus giving a threshold power density of ≈ 108 W/m2

.

1.3.2 Induced Molecular Polarization—The Kerr Effect

The next process that must be considered is the Kerr effect, i.e., the process whereby the permittivity 

of the atmosphere, which is a function of the polarizability of the constituent gases, varies due the intense 

electric field of the laser beam acting to induce molecular orientation (Refs. 7 and 8). As discussed in the 

Appendix, the Kerr process can be described by a single differential equation, making its analysis 

straightforward as compared to the thermally-induced permittivity variations to be discussed in the next 

section. The associated relaxation times for molecular polarizability within the atmosphere are ~ 1011 sec. 

so in the case of pulsed laser propagation, the medium can be considered in steady state for pulse lengths 

larger than this value. As the brief analysis in the Appendix shows, the threshold value PcrK of beam 

NASA/TM—2015-218726 3
power at which the Kerr effect is only a function of the Kerr constant and the wavelength (see Eq. (A17)); 

it becomes an issue as a perturbing propagation mechanism at PcrK ≈ 1.5×109 W for the wavelength of 

λ = 2.0 μm and PcrK ≈ 1.5×1016 W at λ = 2.0 mm. The Kerr effect serves to increase the permittivity in the 

location of the beam and thus results in what has become to be known as “self-focusing”, i.e., the beam 

tends to move into regions of larger permittivity and thus acts as if it has encountered a lens. The lens is 

essentially induced by the beam through the Kerr effect. This self-focusing phenomena results in a spatial 

instability within the beam causing a break-up of the beam into individual filaments, where each filament 

takes on the size which corresponds to PcrK and the associated electric field strength.

1.3.3 Induced Heating of the Atmosphere—Thermal Nonlinearities and “Thermal Blooming”

The absorption of electromagnetic radiation by atmospheric gases becomes a source for the 

generation of heat. At large field intensities, the resulting temperature increase also changes the 

atmospheric permittivity. However, unlike the Kerr effect, the permittivity decreases and the beam tends 

to defocus (Refs. 4, 5, 9 to 12). Additionally, one must also admit the description of the atmospheric fluid 

dynamic processes that are elicited due to local heating and thus incorporate the Navier-Stokes and heat 

transfer equations for a moving medium; the medium is in motion not only due to the Archimedean forces 

that appear but also due to wind which serves to remove the heat that is generated. The problem in its 

entirety is a severely complicated one and is usually relegated to numerical analysis applied to specific 

cases. However, within certain approximations, analytical results can be obtained. As shown in the 

Appendix, one can obtain an expression for the critical power PcrT the beam must possess above which 

thermally-induced propagation issues can become prominent. Due to the interplay of the several physical 

mechanisms that prevail in this propagation process, one must specify the prevailing wind velocity V as 

well as the effective radius reff of the output aperture of the beam source and the wavelength dependent 

absorption constant α. As given by Equation (A25), one has for V = 4.4 m/s (≈10 mi/hr), reff = 10 m and 

λ = 2.0 μm, for which α ≈ 4×10–6 m–1

, PcrT = 16.7 W, a surprisingly small power. This is mainly due to 

the very large radius subtended by the beam through which the field-induced temperature increase must 

dissipate due to wind. For a more typical radius of reff = 0.1 m, one has PcrT ≈ 1.7 KW. At the much larger 

wavelength λ = 2.0 mm, one has α ≈ 2.2×10–4 m–1 giving for reff = 10 m, PcrT = 3.5×105 W.

In the case where the radiation source is pulsed with time duration of 1 μs, the threshold power 

becomes much higher due to the fact that the atmosphere does not have sufficient time to heat during the 

movement of the pulse through the atmosphere. As obtained from Equation (A31), one has for the critical 

power at λ = 2.0 μm, PcrP = 3.7×107 W and for λ = 2.0 mm, PcrP = 7.7×1011 W.

1.3.4 Atmospheric Aerosols

The appearance of aerosols in the form of fog and clouds serve to be very efficient absorbers of 

electromagnetic radiation (Refs. 13 and 14). At the energy densities that prevail for beamed energy 

propulsion, aerosols will essentially explosively detonate thus diminishing their absorption and scattering 

abilities. The dynamics of this complex process has been well studied and it has been established that, 

once a steady state has been established, propagation ‘channels’ will appear in the medium through which 

the beam can propagate (Refs. 15 and 16).

NASA/TM—2015-218726 4
1.3.5 Atmospheric Turbulence

The effect of random variations of the atmospheric permittivity field due to heating of the 

atmosphere, (the sources of which are solar absorption in the atmosphere and on the Earth’s surface) is 

independent of the beam power level for the lowest critical power thresholds derived above for the 

thermal blooming case. This ever-present deleterious mechanism has the most heritage in terms of study 

and understanding. The effects of this phenomenon on the propagation of a beam wave show themselves 

in terms of loss of spatial coherence, beam steering and beam broadening and are attendant with thermal 

blooming (Ref. 17). Of course, these characteristics are the same as seen with thermal blooming but the 

adaptive compensation in the case of turbulence is straightforward in that the effect being corrected is not 

the result of a nonlinear process.

The results given in the above subsections are condensed in Table 1 and stated in terms of the critical 

power the appropriate propagation mechanism for output aperture sizes (where they matter) of 1, 10, and 

30 m. Also shown, for reference, are the general equations for the critical power and associated critical 

power density. As can be seen, discussions of the use of beam powers up to 100 MW at infrared 

wavelengths easily exclude the effects of electrical breakdown and the atmospheric Kerr effect leaving 

thermally induced nonlinearities and, of course, turbulence, to be dealt with. For millimeter wavelengths, 

the thresholds for electrical breakdown are three orders of magnitude smaller than for the infrared region. 

Hence, aperture sizes larger than 10 m must be used for 100 MW power levels at these wavelengths using 

contiguous apertures.

TABLE 1.—THRESHOLD POWER LEVELS BEYOND WHICH PROPAGATION EFFECTS OCCUR AT

INFRARED AND MILLIMETER WAVELENGTHS FOR VARIOUS OUTPUT APERTURE RADII reff

Propagation Mechanism λ = 2.0 μm λ = 2.0 mm General relation for critical power Pcr

and power density pcr

Electrical Breakdown, 

1 m

10m

30m

eff r



 = 





11

13

14

3.1 10 W

3.1 10 W

2.8 10 W

×

×

×

8

10

11

3.1 10 W

3.1 10 W

2.8 10 W

×

×

× No General Relation

Kerr Effect 9 1.5 10 W × 16 1.5 10 W × 2 8 crK

K

c P

k = ε , 2 2 8 crK

eff K

c p k r = π ε

Thermal Blooming (CW),

1 m

10m

30m

eff r



 = 





Thermal Blooming (Pulsed, 1 μs)

166.7 W

16.7 W

5.6W

7 3.7 10 W ×

6

5

5

3.5 10 W

3.5 10 W

1.2 10 W

×

×

×

11 7.7 10 W ×

2

p

crT

eff T

C V

P

k r

πρ = ε α , 2 3

p

crT

eff T

C V

p k r

ρ = ε α

2

p

crP

T p

C

P

k t

πρ = αε , 2 2

p

crP

eff T p

C

p kr t

ρ = αε

Turbulence > 0 > 0

NASA/TM—2015-218726 5
1.4 Modeling the Effects of Atmospheric Thermal Nonlinearities on Beamed Energy 

Propulsion and the Required Level of Their Mitigation

Although it is not the prevue of this work to physically model the thermal nonlinearities attendant 

with the propagation of high energy radiation through the Earth’s atmosphere, it has been found necessary 

to introduce a simple analysis employing the peculiarities of the propagation scenarios of beamed energy 

propulsion in order to analytically assess and compare the overall effects of the atmosphere in these 

various scenarios as well as evaluate the level of the corrective adaptive optics that will be required to 

make feasible the goals of beamed energy propulsion. Two such power beaming cases will be considered 

here, i.e., 1) orbit transfer from a low Earth orbit (LEO) to a geosynchronous Earth orbit (GEO) from a 

ground based laser transmitter operating at a wavelength of 2.0 μm in the infrared spectrum and 

2) beaming to a vehicle being launched within the Earth’s atmosphere using a ground based array of 

gyrotrons operating at the wavelength of 2.0 mm within the far infrared spectrum. As orbital transfer from 

LEO to GEO presents the worst-case situation in these two beaming cases due to the distances to be 

traversed, it will be used as the example to form an atmospheric propagation model upon which the 

overall operation can be assessed. It will then be shown how the model can be used to analyze the 

millimeter power beaming case for which the results will also be presented.

The major property of the use of such propulsion to transfer payloads from LEO to GEO is, of course, 

the proportion of the diffraction length Ld associated with the beam wavelength and radius, to that of the 

Focal length F and the thickness H of the atmosphere. Here, one has that the diffraction length 2 d eff L kr ≡

(where k is the wave number of the radiation, reff is the effective radius of the transmitting aperture; see 

Appendix A) and F is such that Ld/F  H/Ld. This circumstance allows for the natural diffraction of the 

beam to be neglected within the region in which the beam is perturbed be induced nonlinear effects. This, 

along with the associated occurrence of the very large values of parameters describing the nonlinear 

interaction of the electromagnetic radiation with the fluid dynamics of the atmosphere (see the Appendix 

A), makes for the creation of an easily implementable model to describe the major beam parameters that 

are of interest, i.e., the radius of the beam at the target and the displacement of the beam from the target 

when no adaptive correction is applied. Once these have been secured, the corrections to the various 

orders of aberrations of the initial phase of the beam can be applied and their effect at the target be 

assessed. It is important to note here that only the effects of thermal nonlinearities of the atmosphere are 

considered. The additional effects of the ever-present turbulent fluctuations of the atmosphere will not be 

included here as this problem has much heritage and well understood and techniques for its mitigation are 

established. However, due to the very large power that the beams will possess that are considered here, it 

is expected that the induced convective velocity within the atmosphere in the beam path will swamp any 

turbulent velocity fluctuations. Thus, the turbulent mechanism affecting beam wave propagation at these 

power levels will be rendered negligible. The goal of the present analysis is to isolate the effects of the 

much more deleterious thermal nonlinearities on very long-range power beaming and as well as estimate 

the required level of adaptive compensation to make the orbital transfer mechanisms discussed elsewhere 

in this report realistic and feasible.

NASA/TM—2015-218726 6
1.4.1 A Quick Overview of the Model and the Description of Beam Behavior at a Target in LEO 

and GEO due to Atmospheric Thermal Nonlinearities

At the outset, the basis of the propagation model will be reviewed, leaving the details of the 

derivation to a forthcoming publication. The initial equations that will be employed are given by 

Equations (A22) and (A26) of the Appendix A, viz.,

( ) 2 2 2 V d 0, exp 2 , E T i E R TE E E L z E

z x ρ′

∂ ∂ ′′ ′ + ∇ + = = −α ′′ ′ ′′ ′′ ′ ′ = ′ ∂ ∂ ′ ′  (1)

where the dimensionless nonlinearity parameter RV is given by

2

eff T 0

V

p

kr P

R

C V

ε α ≡ πρ

(2)

connecting the total incident power P0 in the beam with the effective radius reff of the transmitted beam 

and the prevailing wave number k ≡ 2π/λ of the radiation of wavelength λ. The atmospheric absorption 

coefficient at the particular wavelength is α, εT ≡ ∂ε/∂T is the variation of the permittivity with respect 

to the associated temperature variation T, ρ is the density of the atmosphere, Cp is its specific heat at 

constant pressure, and V is the total relative velocity of the beam slewing across the atmosphere and that 

of atmospheric wind. (All other variables and their normalization are introduced and discussed in the 

Appendix A.) Using the parameter values discussed in the Appendix A, one has, for a 1 MW beam, i.e., 

P0 = 1.0×106 W, the value RV = 6.0×104 W for λ = 2.0 μm. In this instance, the second term in the first 

relation of Equation (1) can be neglected relative to the third term, i.e., the evolution of the beam within 

the atmosphere due to natural diffraction is negligible as compared to the phase perturbation due to the 

thermal nonlinearity acting on the permittivity function. Hence, the model equations become

( ) 2 2 0, exp 2 , V d

E T i R TE E E L z E

z x

∂ ∂ ′′ ′ + = = −α ′ ′′ ′′ ′ ′ = ′ ∂ ∂ ′ ′ (3)

The solution of these relations must now be augmented with the prevailing boundary condition for the 

initial radiation profile of the beam at the output aperture of the transmitter at z′ = 0. Using a Gaussian 

beam profile, one has from Equation (A11)

( )

2

222 0 ,0 exp , 1 , 2

Ld E A i xy

F

    ρ′ ′′ ′ ′ ′ ′ ′ ′ ρ = − γ γ≡ + ρ = +        

 (4)

However, in most power beaming applications, the focal length F of the beam will be set to the distance L

to the target, i.e., F = L. Additionally, the full normalized representation for a Gaussian beam wave must 

be employed in the last relation of Equation (3), viz,

( ) 0 2 1 , exp 1 21

A E z

i z i z

′ ′      γ ′ ′′ ρ= − ρ′      + γ′ ′   + γ′ ′    

 (5)

Finally, one must consider the effective thickness H of the atmosphere. Although there are several 

models that can be used to represent the absorption constant α with respect to the height within the 

NASA/TM—2015-218726 7
atmosphere (the most popular is an exponential variation with height), it proves expedient here to use a 

single value for H and a corresponding effective value for α.

The solution of Equations (3) and (4) can now be considered in the case for Ld/F  H/Ld; to be sure, 

one has H ≈ 10 km, F < 35,200 km, and for the aperture size for reff ≈ 30 m, Ld ≈ 2.8×109 m at infrared 

wavelengths.

1.4.2 Application of the Foregoing to the Calculation of Beam Spread and Deflection due to 

Atmospheric Thermal Nonlinearities for Power Beaming to LEO and GEO

Again, leaving the details concerning the derivations, etc., to a forthcoming NASA Technical 

Memorandum, one can employ Equations (3) to (5) to obtain expressions for the two very important 

‘performance parameters’ for power beaming to a target, i.e., the amount that the beam radius is widened 

due to the induced thermal de-focusing (beam spread) and the amount that the entire beam is steered away 

from the intended target due to the fact that the propagation path of the beam tends to favor regions of 

larger permittivity, i.e., steer away from regions of higher atmospheric temperature caused by absorption 

(beam displacement). It is important to note that, as discussed in the Appendix A, the wind direction has 

been taken to occur along the x-axis of a coordinate system whose origin is situated at the transmitter 

aperture with beam propagation occurring along the z-axis. Also, the radius of the transmitter aperture at 

z = 0 is given by an effective radius reff(0) which is related to the corresponding beam waist radius W0 by 

eff (0 2 ) 0 r W= .

Because of the asymmetry introduced into the problem by the wind velocity V moving along the xaxis of the originally circular beam, there are two effective radii reff,x(L) and reff,y(L), that characterize the 

beam in the x and y axes, respectively, at the distance L to the target vehicle. Again, this circumstance is 

brought about by the removal of heat from the beam channel along the x-axis. However, heat is not 

convectively removed from the beam along the y-axis; it diffuses in the y direction, which is a much 

slower process than convection. The result is that the de-focusing in the y direction is much greater than 

in the x direction and the beam becomes elliptical as it leaves the atmosphere and continues to enlarge 

until it interacts with the target in space. 

The results of the calculation for the effective radii and the target are given by

( ) ( ) ( ) 2 , 0

1 1

3 eff x eff NLV r LRL P = +Θ (6)

( ) ( ) ( ) ( ) 2 , 1 0.866 0 0 1.612 d

eff y eff NLV NLV

L r LRL P P

L

= + Θ +Θ (7)

where 

( ) ( )

0

0 4 0

T

NLV

p eff

k HP P

C Vr

ε α Θ ≡

πρ

(8)

is yet another nonlinear parameter that enters the problem and characterizes, within the thin phase screen 

approximation, the level of the thermal nonlinearity effect on the beam wave, of initial power P0, through 

an atmosphere of thickness H. (See Appendix A for the definitions of the remaining variables.) The 

NASA/TM—2015-218726 8
quantity Reff(L) ≡ L/(kreff(0)) is what the radius of a beam focused on the target at a distance L would be if 

the atmosphere were not present, i.e., if transmission occurred entirely within a vacuum. (The reader is 

asked to excuse the rather excessive notations related to the various beam radii; the initial radius of the 

beam at the transmitter, denoted here as reff(0), is usually stated in terms of the waist radius W0 that is a 

factor of 2 smaller than the former. However, it is desired here to show how the effective radii ‘evolve’ 

throughout the propagation process.) Thus, the quantities given by the radicals in Equations (7) and (8) 

are the factors by which the beam radii increase at a space borne target due to the induced thermal 

nonlinearities within the atmosphere.

Additionally, as discussed above, there occurs a deflection ∆(L) of the beam axis into the direction 

opposite to that of the wind. This is given by

∆= Θ (L RL P ) 2 eff NLV ( ) ( 0 ) (9)

The implications of these results to the behavior of a beam wave to targets at LEO (L = 800 km) as well 

as GEO (L = 35,200 km) will now be given. The operating wavelength of the continuous wave case 

considered here is taken to be λ = 2.0 μm and the atmospheric wind velocity along the x-axis is taken to 

be Vx = 4.4 m/s ≈ 10 mi/hr. Three transmitter aperture radii of reff(0) = 5 m, reff(0) = 10 m, and reff(0) = 30 

m are considered. Table 2 displays the beam radii that would occur at a target in LEO as well as GEO if 

the atmosphere were not present.

Only a subset of these possibilities will be considered in what is to follow; to keep things realistic, 

reff(0) = 5 m at GEO and reff(0) = 30 m at LEO will not be subject to further analysis.

Figure 1 to Figure 4 display the plots of reff,x (L), reff,y (L) and Λ(L) versus P0 for transmission to LEO 

and GEO for the various initial beam radii at the output aperture using Equations (6) to (9). In addition to 

the values quoted in Appendix A, the nominal value of H = 10 km has been used in conjunction with the 

average absorption coefficient α ≈ 4×10–6 m–1

. As can easily be seen in all the examples, the thermal 

action of the atmosphere severely impacts the integrity of the beam wave at the target locations. 

TABLE 2.—BEAM RADII AT LEO AND GEO TARGETS FOR VARIOUS

OUTPUT APERTURE SIZES THE WAVELENGTH IS λ = 2.0 μm

reff (0) = 5 m reff (0) = 10 m reff (0) = 30 m 

LEO (L = 800 km) 0.1 m 0.051 m 0.017 m

GEO (L = 35,200 km) 4.48 m 2.24 m 0.75 m

NASA/TM—2015-218726 9
(a)

(b)

Figure 1.—(a) Beam radii (in m) along semi major and minor axes versus power (in MW) for a 5 m beam into LEO. 

(b) Beam deflection from target (in m) versus power (in MW) for a 5 m beam into LEO

NASA/TM—2015-218726 10
(a)

(b)

Figure 2.—(a) Beam radii (in m) along semi major and minor axes versus power (in MW) for a 10 m beam into LEO. 

(b) Beam deflection from target (in m) versus power (in MW) for a 10 m beam into LEO.

NASA/TM—2015-218726 11
(a)

(b)

Figure 3.—(a) Beam radii (in m) along semi major and minor axes versus power (in MW) for a 10 m beam into GEO. 

(b) Beam deflection from target (in m) versus power (in MW) for a 10 m beam into GEO.

NASA/TM—2015-218726 12
(a)

(b)

Figure 4.—(a) Beam radii (in m) along semi major and minor axes versus power (in MW) for a 30 m beam into GEO. 

(b) Beam deflection from target (in m) versus power (in MW) for a 30 m beam into GEO.

Comparing the radii that result at the targets in LEO and GEO, due to a beam that has traversed 

thermal nonlinearities within the atmosphere, to those quoted in Table 2 shows the rather unsatisfactory 

but not unexpected results that deem beaming power to such locations as unfeasible without the proper 

compensation to the beam applied at the transmitter. The mitigation of the induced deflection of the beam 

can easily be removed by a tracker but the remaining aberrations that remain that result in the beam 

broadening can only be removed by appropriate phase compensation applied at the transmitter.

1.4.3 Phase Compensation of the Effects of Atmospheric Thermal Nonlinearities

The first research into the compensation of thermal blooming effects considered the application of a 

Zernike polynomial expansion of the transmitted phase front (Refs. 18 and 19). This technique provides 

for an optimal representation of the phase front of the beam wave so long as the cross-section of the beam 

NASA/TM—2015-218726 13
remains circular; after all, Zernike polynomials are, by design, orthogonal on a unit circle. The problem 

encountered in the above development is different; a beam asymmetry arises due to the differences in 

convective versus diffusion cooling of the atmospheric channel in which the beam propagates. The model 

briefly reviewed in Section 1.4.1 allows for the analytical assessment of phase compensation of the 

deleterious effects shown in Section 1.4.2.

The different behavior of the beam along the transverse axes suggest the use of a more general set of 

orthogonal polynomials than those of Zernike. Here, since the initial form of the beam wave is given by a 

Gaussian function, the transmitted phase front S(x,y) at the output aperture of the transmitter can be 

expanded in a set of orthogonal polynomials associated with a Gaussian weight function, i.e., the Hermite 

polynomials Hn(x) and Hm(y) (Ref. 20). Thus, 

( , ) ( ) ( )

M N

nm n m

n m

S xy a H xH y =∑∑ (10)

The expansion coefficients anm are determined by applying one of several ‘performance metrics’. For 

example, for power beaming applications considered in this study, it is desired to shape the phase front so 

as to minimize the radius of the beam at the target so as not to have the beam interfere with the adjacent 

structures of the vehicle. Hence, the performance metric to be minimized through the selection of the anm

is related to the electric field E′′(ρ′,L) of the beam at the target vehicle by

( )

( )

( )

( )

2 3

0 2 2

2

0

,

,

,

EL d

L x y

ELd

∞

∞

′′ ′ ′ ′ ρ ρρ

Π ≡ ρ= + ′ ′′

′′ ′ ′ ′ ρ ρρ

∫

∫

(11)

in which E′′(ρ′,L) results from the application of the boundary condition

( ) ( )

2

0 ,0 exp , , 1 2

Ld E A iS x y i

L

    ρ′ ′′ ′ ′ ρ = − γ−     ′ ′ ′ ′ γ≡ +

   

 (12)

to the solution of Equation (3). Ideally, the number of terms of the expansion given in Equation (10) is 

infinite, i.e., N,M → ∞. The linear terms for which n + m = 1 determine the inclination correction of the 

beam wave; the quadratic terms for which n + m = 2 determine the focusing correction and the higher 

order terms for which n + m ≥ 3 give the higher order aberration corrections. However, in realistic 

applications, the number of aberrations N and M is finite and, in fact, it is desired to find the smallest 

number of aberration corrections that can be used to represent the compensated phase. The specific 

expressions for the coefficients anm are found by equating to zero the derivative of Π(L) with respect to 

these coefficients.

The propagation problem defined here, as well as the associated minimization problem, can be solved 

analytically and will also appear in a forthcoming NASA Technical Memorandum. The results applied to 

the calculation of the phase-corrected beam radii are as follows. Due to the elliptical shape of the beam 

which is characterized by two radii at the target vehicle, it facilitates the calculations to consider a single 

effective radius r L eff ′ ( ) defined by

NASA/TM—2015-218726 14
( ) ( ) ( ) 2 2 , ,

2

eff x eff y

eff

r Lr L

r L

+ ′ ≡ (13)

Applying the modeling procedure as outlined above, one finds for the radius r L eff Corr ′ , ( ) of a focused 

beam wave corrected for the first M phase aberrations represented by the model of Equation (10)

( ) ( ) ( ) ( ) ( ) ( ) ( )

( )

2

1 2 2 , 0 0 3( )/2

1 0

2

3 /2

0

0 0

1 2.95 4 2 ! !

2 !!

0

2 !

42!

M M n m n m eff Corr eff NLV NLV n m n m

M n n

n n

H H

r L R L P P n m nm

n m

H

n n

n

− +

+ = =

=

     ′ = + Θ −Θ  +      

 



π  

+       

∑ ∑

∑

(14)

Figure 5 to Figure 8 show the results of this relation applied to the various LEO and GEO beaming 

cases considered in the last section.

Figure 5.—Beam radius (in m) at target versus power (in MW) for a 5 m beam into LEO for various orders of phase 

correction.

NASA/TM—2015-218726 15
Figure 6.—Beam radius (in m) at target versus power (in MW) for a 10 m beam into LEO for various orders of phase 

correction.

Figure 7.—Beam radius (in m) at target versus power (in MW) for a 10 m beam into GEO for various orders of phase 

correction.

NASA/TM—2015-218726 16
Figure 8.—Beam radius (in m) at target versus power (in MW) for a 30 m beam into GEO for various orders of phase 

correction.

These plots display the fact that for initial beam powers of 10 MW or less, phase correction using 

M = 10 order aberrations returns the radius of the beam to its desired value at the target vehicle. Beyond 

this power level, only the case shown in Figure 8 using a beam radius of 30 m at the transmitter can a 10th

order phase correction be effective up to about 70 MW.

One assumption that has been prevailing in all these calculations is that the wind velocity V is a 

constant along the entire propagation path. In reality, this is certainly not the case and, in fact, there may 

be regions in which the V ≈ 0, i.e., a stagnation zone, and the convective heating scenario assumed here 

will transform into a diffusive heating one characterized by much lower critical power thresholds. This 

case will not be studied here, as it is the subject of a recent study (Ref. 21), but it must be kept in mind 

that a statistical description of the wind profile along the propagation path will be required in a more 

careful examination.

Finally, there is the whole issue of the effect that the beam profile will have on the level of thermal 

nonlinearity perturbations. In fact, very high-energy laser beams are characterized by initial beam profiles 

that are ‘tubular’, i.e., have a local minimum of intensity in the center which increases toward the outer 

portions of the beam. Such beams, which can be modeled by hyper Gaussian profiles, tend to be 

characterized by higher critical power thresholds. This is due to the fact that the center of the beam, with a 

lower intensity, is thermally cooler than the periphery and thus tends to bend toward the center and 

remain stable. This effect should be studied for the particular high-energy sources that will be used as the 

design of the power-beaming scenarios evolve.

NASA/TM—2015-218726 17
1.4.4 Using the Propagation Model for the Analysis of Millimeter Wave Beaming Within the 

Earth’s Atmosphere

The methodology developed above for the infrared power beaming case for LEO/GEO orbital transfer 

can be applied to the millimeter wave case for power beaming to vehicles launched from within the 

Earth’s atmosphere. The wavelength of concern here is λ = 2.0×10–3 m = 2.0 mm with an output aperture 

radius reff = 50 m. This gives Ld = 7.9×106 m. The propagation distances involved range from L = 20 km

to L = 120 km. Although the beam will not be directly focused on the target vehicle (the dimensions being 

envisioned will require the beam to have a focal length just beyond the vehicle), for the purposes of this 

discussion, F ≈ L. The effective distance within the atmosphere responsible for most of the absorption at 

these wavelengths is H ≈ 1.8 km. Hence, the condition Ld /F  H/Ld is easily met. Additionally, there is a 

characteristic length LT that is a measure of the distance at which thermally induced diffraction effects 

will occur. This characteristic length can be defined using the length Ld as well as the nonlinear distortion 

parameter RV given by Equation (A24). In fact, a detailed analysis similar to that given in Appendix A

gives for the thermal diffraction distance

3

0

eff p

T

T

r CV

L

P

π ρ = ε α

(15)

At the wavelength considered in this section, the associated absorption coefficient is α ≈ 2.2×10–4 m–1

which gives for the total transmitter power of P0 = 800 MW (which will be considered below), 

LT ≈ 73 km. Hence, thermal diffraction effects will certainly not prevail within the region H ≈ 1.8 km or, 

for that matter, within the entire troposphere and the model constructed above can be adopted here. (This 

circumstance is solely due to the very large output aperture that is being used.)

Equations (6) to (9) can now be used to consider four scenarios for a typical power-beaming launch:

1) Horizontal beaming (θ = 0°) at L = 20 km 

2) Beaming at θ = 30°, L = 50 km

3) Beaming at θ = 50°, L = 100 km

4) Beaming at θ = 90°, L = 120 km

In each case, the target is to be illuminated in an area with a radius of Reff(L) ≈ 1.8 m. Thus, a dynamic 

focus is to be employed; the focal length is determined by

( ) 2

2 , 1 1

2

d eff

d

L L kL R L

F

L L

= Φ= ± − Φ (16)

where the ‘-‘ sign is used in front of the radical for a beam focused behind the target. The results of 

applying the propagation model to these cases are displayed in Figure 9 to Figure 12.

NASA/TM—2015-218726 18
(a)

(b)

Figure 9.—(a) Beam radii (in m) along semi major and minor axes versus power (in MW) for a 50 m beam. (b) Beam 

deflection from target (in m) versus power (in MW) for a 50 m beam.

NASA/TM—2015-218726 19
(a)

(b)

Figure 10.—(a) Beam radii (in m) along semi major and minor axes versus power (in MW) for a 50 m beam. 

(b) Beam deflection from target (in m) versus power (in MW) for a 50 m beam.

NASA/TM—2015-218726 20
(a)

(b)

Figure 11.—(a) Beam radii (in m) along semi major and minor axes versus power (in MW) for a 50 m beam. 

(b) Beam deflection from target (in m) versus power (in MW) for a 50 m beam.

NASA/TM—2015-218726 21
(a)

(b)

Figure 12.—(a) Beam radii (in m) along semi major and minor axes versus power (in MW) for a 50 m beam. 

(b) Beam deflection from target (in m) versus power (in MW) for a 50 m beam.

As can be seen from these results, the worst case is, of course, the horizontal beaming at an elevation 

angle θ = 0° as displayed in Figure 9(a) and (b). Here, the beam traverses 20 km of atmosphere. The beam 

severely broadens along the y-axis, perpendicular to the wind velocity, since the heat that is generated is 

not convectively removed in this direction as it is along the x-axis. Related to this phenomenon is the fact 

that the beam is deflected in the x-direction up to 12 m from the target at a power level of 800 MW. Of 

course, as the elevation angle of the beam path increases, the distance within the atmosphere decreases up 

to the best case in which θ = 90° as depicted in Figure 12(a) and (b). It must be cautioned, however, that 

the model used here, as mentioned earlier, does not include viscous and Archimedean forces that occur 

during the thermally induced motion of the atmosphere within the vertical column in which the beam 

NASA/TM—2015-218726 22
propagates. For situations in which large elevation angles are realized, these effects must be included in a 

future extension of the analysis. Finally, it must be noted that in order to follow the relatively close target 

vehicle, the beam will have a discernable slewing velocity as compared to that of the LEO/GEO transfer 

case discussed earlier. This slewing velocity may (depending on the relative wind velocity) tend to lessen 

the thermal effects. This will certainly be the case for the upward movement of the beam. As for the 

horizontal component of the slewing velocity, it can either detract or worsen the thermal effects. Such 

aspects of the problem can only be assessed via a simulation of the particular launch environment.

Just as with the infrared beaming case, the adaptive correction of these deleterious effects is 

accomplished through the use of phase compensation at the transmitter output aperture. Again, because of 

the asymmetry induced within the structure of the beam, it is advantages to represent the corrected phase 

front as an expansion of Hermite polynomials. Figure 13 to Figure 16 show the results of applying 

various orders of Hermite correction to the millimeter wave beam. The goal is, of course, to return the 

beam radius to the desired value of 1.8 m. As expected, the θ = 0° case requires the most compensation; 

correction up to 10th order aberrations is needed, after the removal of the beam deflection via a tracker.

The other cases only need correction up to 5th order.

Figure 13.—Beam radius (in m) at target versus power (in MW) for a 50 m beam into GEO for various 

orders of phase correction.

Figure 14.—Beam radius (in m) at target versus power (in MW) for a 5 m for various orders of phase 

correction.

NASA/TM—2015-218726 23
Figure 15.—Beam radius (in m) at target versus power (in MW) for a 5 m for various orders of phase 

correction.

Figure 16.—Beam radius (in m) at target versus power (in MW) for a 5 m for various orders of phase 

correction.

The additional nuance that this millimeter wave power-beaming scenario has over the LEO/GEO 

orbital transfer case is that the target vehicle will have a much larger apparent velocity in both the traverse 

and longitudinal directions than an object executing a LEO/GEO transfer. This circumstance brings to the 

fore the conditions of stability of the control algorithms which compose a closed-loop adaptive optics 

system. As mentioned above, the beam will also have a significant slewing velocity that, in some respects, 

eases the effects of thermal nonlinearities, but in others, places a burden on the operation of the adaptive 

optics control system. This will be discussed below in Section 1.5.

It finally should be mentioned that for the power levels considered in this millimeter wave case, one 

should consider the mechanism of explosive detonation (Ref. 16) of small atmospheric aerosols that may 

traverse the beam, especially in the horizontal beaming case. This will give rise to a time dependent 

absorption coefficient that my indeed lessen the long-term heating effects within the atmosphere.

NASA/TM—2015-218726 24
1.4.5 A Tool for the Quick Assessment of Nonlinear Thermal Effects on Power Beaming

Appendix A introduced several expressions for threshold power levels at which thermal effects will 

become prevalent. Although these derivations were essentially based on normalization and dimensional 

analysis, they can be given credence by using the results obtained above. However, these critical power 

levels need to be related to a specific property or characteristic of a beam wave that will impact the 

performance of the power beaming system. As there are many such characteristics, there too will be as 

many corresponding critical power levels; there is no single universal quantity that can be stated to 

convey the specific impact thermal blooming will have on the entire performance of a beamed energy 

system. However, the most important characteristic in power-beaming situations is the variation of the 

beam radius due to the defocusing that occurs during the heating of the atmosphere. As shown by 

Equation (7) and, e.g., in Figure 1 to Figure 4, the radius of a beam in an atmosphere with convection is 

severely perturbed in the direction perpendicular to the wind direction. In order to attempt to secure an 

analytical estimate at which the radius of such a beam will begin to increase due to thermal blooming, one 

can expand Equation (7) and get

, 0 ( ) ( ) ( ) 0.866 1

2

d

eff y eff NLV

L r LRL P

L

  = + Θ+      (17)

and require the value for P0 at which 

( ) ( ) 2 0

0

0.866 0

0.1 1

2

d eff T

NLV

p

L k r PH

P

L C VL

ε α

Θ ≈ = πρ

(18)

Setting L = H, one finally obtains for the critical power PcrR at which the beam radius begins to expand

( ) 2 10

0

p

crR

eff T

C V

P

k r

πρ = ε α

(19)

which is just 10PcrT as obtained in the Appendix A. One can now use this relation for millimeter 

wavelengths and compare to the infrared case. Figure 17 and Figure 18 show plots of Equation (19), as 

well as the associated power density ( ) 2 crR 0 eff P r π , versus aperture radius for both cases. Here, the 

nominal wind velocity is taken to be V = 0.5 m/s. Of course, Equation (19) is simple enough to apply a 

statistical model describing the value for V.

Evaluating Equation (19) using the numerical values of the parameters that do not depend on the 

wavelength, one has for this critical power

2

crR 200 MW

eff

P

r

λ = α

(20)

where PcrR is in MW and λ and reff are in meters. (The units of α are m–1

.) These results clearly show that 

propagation at millimeter wavelengths within the atmosphere can sustain much larger power levels than at 

infrared wavelengths. 

NASA/TM—2015-218726 25
Figure 17.—Critical power and power density at which beam radius is affected due to thermal defocusing versus 

initial aperture radius for millimeter wavelengths.

Figure 18.—Critical power and power density at which beam radius is affected due to thermal defocusing versus 

initial aperture radius for infrared wavelengths.

1.5 Dynamics of Adaptive Phase Compensation for Power-Beaming Applications

Of the two major approaches that can be used to implement phase compensation in order to mitigate 

atmospheric effects, viz., wave front ‘reversal’ or conjugation, and wave front control employing the 

optimization of a specific system metric for performance, the first presents a fundamental problem for the 

mitigation of thermal nonlinearities in that it relays upon a principle of reciprocity, a basic tenant of which 

is violated in thermal blooming applications. The use of such a principle essentially requires, through a 

wave front conjugation condition, that the amplitude and phase be corrected. However, it is much easier 

to correct the phase perturbations and not the associated amplitude perturbations. This is fine if amplitude 

variations are small, e.g., astronomical imaging. In thermal blooming scenarios, amplitude variations 

cannot be neglected. Using this approach in power beaming is not warranted. What is recommended here 

is the use of a method using the actual radiant intensity at the target. What used to be called aperture 

NASA/TM—2015-218726 26
optimization or ‘tagging’ (Ref. 22), has now come to be known as target-in-the-loop and its 

implementation is known as ‘gradient descent optimization (GDO) wave front control’ (Refs. 23 to 27)

and is suggested for the power beaming applications addressed in this report. Here, the control rule is 

based on the direct optimization of an easily measured system performance metric, such as the radiant 

intensity at the target.

The model governing the operation of such an adaptive optics system is given by first identifying a 

performance metric Π≡Π  S a( )    as defined by Equation (11) that is dependent upon the phase given by 

Equation (10) and the array a a = [ nm ]  of associated expansion coefficients (Ref. 24). The coefficients in 

Equation (10) were selected for modeling purposes based on the minimization of the beam radius on the 

target vehicle. Here, the optimal selection of the values of anm during the actual operation of the system is 

given by the control rule

( ) ( ) nm

nm a

nm

at t

t a

∂ ∂Π

τ = −γ ∂ ∂ (21)

in which τnm are the time constants and γ is the control (or ‘update’) coefficient. Here, J will be taken to 

be the size of the focal spot at the target vehicle. The intensity can be inferred by recording the reflected 

radiation form the target at a point separated from the transmitter so that a slightly different and 

unperturbed propagation path is used. It must now be established that such a control structure will 

dynamically operate in the two extreme cases of power beaming considered here, i.e., infrared beaming to

a vehicle in GEO and millimeter wave beaming to a near Earth vehicle.

Consider first power-beaming to GEO. For purposes of a brief analysis in which the required 

temporal characteristics of the adaptive optics system (and thus the temporal stability) are to be derived, 

one can consider the isolated case for which n + m = 2, i.e., according to the discussion in Section 1.4.3, 

the dynamic focusing of the beam wave. Thus, one has from Equation (10) for the associated (now time 

dependent) aberration coefficient 

20 ( ) ( )

Ld a t

F t = (22)

where, e.g., H2(x) ~ x2

. For the case in which the focus is to be placed at the target at a distance L, i.e., 

F(t) = L, and the one has from Equation (21),

20 ( ) 20 (0 exp ) d d

a

L L t

a t a

L L t

   

=+ + −         (23)

where ( ) 2 2 a 20 d t LL =τ γ is the adaptation time of the adaptive optics system. This very important time 

constant is, although proportional to τ20, inversely proportional to the distance to the target. This wellknown property has it that the adaptive system will possess a faster response the farther the target from 

the transmitter. If this were all that the description of the adaptation system required, then the state of 

focusing on the target will monotonically approach the desired result Ld/L. However, there is a limiting 

factor placed on this circumstance by the delay inherent in overall system response do to the propagation 

time to and from GEO. When the propagation delay is allowed to enter into the control rule, Equation 

(21) now becomes

NASA/TM—2015-218726 27
( ) ( ) nm

nm d a

nm

a t tt

t a

∂ ∂Π −

τ = −γ ∂ ∂ (24)

where td is the delay time (in this case, light time to and from the target). Expanding the right side of this 

equation in a Taylor series in td and keeping the first few terms yields

( ) ( ) ( ) ( ) 2 2 2

2 2 nm

nm d

a d

nm nm d nm d

at t t t t

t

t a at a t

∂ ∂Π ∂ Π ∂ Π  

τ = −γ − +   − ∂ ∂ ∂∂ ∂∂   (25)

The corresponding characteristic equation ς for the resulting differential relationship for a20 (t) = exp(–ςt) 

in the example considered above is given by

2 2

2

a d a ad 2 d

d

t t t tt t

t

±

−± − −

ς = (26)

shows that the system converges to a stable state so long as ta > td, i.e., the adaptation period of the system 

is longer than the delay time. For a target in LEO, td ≥ 0.005 sec and for GEO, td ≈ 0.24 sec (for the round 

trip time). If, on the other hand td > ta, the system does not converge to the stable state and 

lim 20 ( ) ta t →∞ → ∞ . Thus, the adaptive system must be continuously ‘tuned’ (by adjusting γ in response to 

changes in td as well as the evolving values for L in the case of a moving target) so that the condition 

ta > td is satisfied.

To these considerations must also be added the temporal delay in the overall system response. This 

will not be discussed here as these considerations can be found in the literature. Suffice it to say that the 

delay that is met with in beaming to GEO can be easily factored into the operation of the adaptive system.

It must be remembered that the distance to the target is always increasing in time for a beamed power 

scenario and this, above and beyond that of the propagation delay, must be properly treated in the design 

and specification of the adaptive optics control system.

Work on the actual implementation of the close-loop control of the gradient descent algorithm of 

Equation (21) has rapidly progressed. The technology is now mature and forms the basis for reliable 

adaptive control. The only possible nuance to its application is for power beaming to locations within the 

Earth’s atmosphere for surface or near-surface launching. Here, the target vehicle, unlike the LEO/GEO 

transfer case, is rapidly moving and is considered an extended target, i.e., its spatial extent cannot be 

relegated to a point object and its possible rotation and orientation becomes an issue for use as a reflector 

of radiation bask to the adaptive optics receiver. Here, the randomly rough surface of the target reflector 

creates a speckle field characterized by bright and dark regions of intensity has been found to complicate 

the implementation of a closed loop adaptive optics scenario. This can lead to the slowing of the 

convergence speed of the system and, ultimately, to the efficiency of its operation in terms of the 

optimization of the focal spot at the target. Methods using the pre-compensation of the beam show 

promise (Ref. 28).

Although it has been shown above that the basic physics of the situation does not preclude beaming 

energy to a prescribed area on a target vehicle in GEO, the engineering implementation will offer some 

major obstacles to be negotiated. In particular, the beaming into a GEO location will require 30 

nanoradian pointing accuracy on the part of the transmitter optics. The presence of mechanical jitter of the 

transmitter platform must be continuously mitigated. Additionally, on the other side of the spectrum, 

NASA/TM—2015-218726 28
launching from within the atmosphere will challenge the application of the adaptive optics at millimeter 

wavelengths. The fact that the beam is rapidly slewing due to the relatively rapidly moving object makes 

it difficult for the closed-loop adaptive optics system to converge to a stationary value since the medium 

within the column of atmosphere in which the millimeter wave beam exists is constantly being 

exchanged. Here, instead of employing the target as the beacon source, the use of an artificial beacon 

placed ahead of the moving vehicle may help in characterizing the atmosphere ahead of the beam. This 

will assess the phase perturbations of the nonheated atmosphere but the problem still remains concerning 

just how the atmosphere will respond to the heating from the beam as it arrives at that particular column.

This suggests that, due to the availability of high power sources at millimeter wavelengths, one configure 

an in situ experiment to assess the operation of both closed-loop versus artificial beacon adaptive optics 

approaches. This will be discussed further below.

1.6 Experiments Assessing the Compensation of Thermal Nonlinearities of High Power 

Beams in the Atmosphere

Experiments dealing with the induced effects due to thermal nonlinearities elicited by high power 

propagation are usually performed in a laboratory using liquids placed in cells in which the thermal 

nonlinearity thresholds are much smaller than in air. In these scaled laboratory experiments, where 

atmospheric turbulence effects are simulated by transmission phase screens, various adaptive optics 

algorithms have been tested and evaluated (Refs. 29 and 30). Some experiments have been performed in 

the open atmosphere along horizontal paths in a program called the Scaled Atmospheric Blooming 

Experiments (SABLE) directed by Lincoln Laboratory (Ref. 31). Work is also continuing along these 

lines in other countries (Refs. 32). All these experiments endeavor to evaluate the in situ operation of 

adaptive optics systems on a scaled basis. The same must be recommended for the operation of a beamed 

power scenarios discussed above. Due to the maturity and availability of high power sources in the 

millimeter range, it is recommenced that a scaled atmospheric experiment be performed on a moving 

target to assess the operation of various adaptive optics algorithms. Such an experiment should be 

modeled after the SABLE project. It is important to note that there currently does not exist a database that 

addresses the power beaming cases considered here. It is important to begin to capture the prevailing 

effects and system operation using a scaled atmospheric experiment which would simulate the realistic 

environment in which an adaptive optics system must operate, from the variable wind velocity up to the 

mechanical jitter of the transmitter platform. Additionally, this experimental scenario can also be used to 

address the issue related to the possibility of air ionization, and subsequent breakdown, across the 

apertures of the combined millimeter wave gyrotron sources as discussed elsewhere in this report.

1.7 Conclusions and Recommendations

The various deleterious propagation mechanisms associated with high-power electromagnetic wave 

propagation through the atmosphere have been discussed. In addition to turbulence, thermal nonlinearities 

associated with the absorption of radiation by the atmospheric gas will contribute to the major effects of 

beam wave propagation for beamed energy propulsion at the power levels considered here. The simplified 

propagation model advanced here shows that the beam radius as well as its deflection are severely 

affected by the phenomena of thermal blooming. However, the model also showed that appropriate phase 

compensation at the transmitter output aperture can mitigate these effects and return the propagation 

situation to one that is acceptable for the power transfer requirements that are needed to be satisfied for 

beamed propulsion. For infrared transmission out of the atmosphere for LEO/GEO beaming, up to 10th

order (once the tilt has been removed) aberration correction will be needed in order to maintain a minimal 

NASA/TM—2015-218726 29
focal spot at a GEO location using a 30 m diameter transmitter aperture. For millimeter wave beaming 

within the atmosphere, 5th order aberration correction will suffice with the exception of the horizontal 

case in which, once again, a 10th order correction would be requires. In principle, these corrections allow 

the beaming system to operate within the prevailing specifications. In practice, however, some challenges 

prevail in the implementation.

The adaptive optics approach recommended here for the LEO/GEO launch case is a closed-loop 

system that uses the specular reflection from the target vehicle as a ‘beacon’ source (i.e., a target-in-theloop system). The target vehicle is seen by the adaptive optics system as a simple point reflector. The 

round-trip propagation time delay inherent in this scenario can be tolerated so long as the adaptation time 

of the adaptive optics system is properly set to a slightly larger time than that of the delay to assure proper 

convergence. 

In the case of millimeter wave beaming to a moving vehicle within the Earth’s atmosphere, the object 

is now considered as an extended target the rotation and orientation of which can complicate application 

of the closed loop system. The reflection from the vehicle will posses a speckle structure that can severely 

impact the wave front sensor used by the adaptive optics system. Here, it may be that the simpler artificial 

beacon method can be used but there is still an additional complication, viz., the rapid movement of the 

beam across the atmosphere. The changes induced in the beam column through the atmosphere will not 

all be those due to the adjustment by the adaptive optics and the system will not be able to properly adapt.

This situation presents itself for a scaled atmospheric high-power millimeter wave transmission 

experiment in which both closed loop and artificial beacon-based adaptive optics systems are tested and 

evaluated.

The propagation modeling for the various beamed propulsion scenarios presented in this report should 

be extended beyond that employed here in Section 1.4 by incorporating additional fluid mechanical 

descriptions of the atmosphere as well as more general beam wave profiles such as hypergaussian as well 

as fractional charge (in the topological sense) Laguerre-Gaussian beam waves which show great promise 

in their ability to be robust with respect to atmospheric nonlinearities (Ref. 33). However, it is important 

that the modeling effort be kept to the level of yielding analytical results, rather than requiring numerical 

evaluation, so as to capture all the nuances of the physics involved and, at the same time, provide a tool 

for overall system evaluation as well as the design of adaptive optics algorithms. Finally, the model 

should be incorporated into a trajectory analysis program so a cadre of launch geometries can be 

evaluated from the point of view of atmospheric thermal nonlinearities.

NASA/TM—2015-218726 30
Appendix A.—The Derivation of the Fundamental Nonlinear Propagation

Equations for High Energy Transmission Through the 

Atmosphere and the Associated Critical Powers

For the propagation of scalar electromagnetic waves through a medium characterized by a variable 

permittivity ∆ε, one has for the equation describing the evolution of the electric field strength 

2 2

0

1 2 0 E E ik E k E

z t ρ

  ∂ ∂   ∆ε   + +∇ + =     ∂ υ∂ ε  

 (A1)

where ρ

 is a position vector in the plane transverse to the direction of propagation along the z axis, 

k ≡ 2π/λ is the wave number of the radiation field of wavelength λ, υ is the velocity of the wave field in 

the medium ( υ= ε0 c where c is the velocity of light), and ε0 is the nominal permittivity (for the 

atmosphere, ε0 ≈ 1). As will be discussed below, the function describing ∆ε is, in general, dependent on 

the value of E, i.e., ∆ε = ∆ε(E), thus making the propagation problem a nonlinear one; only in the case of 

weak fields (the typical situation for atmospheric optics) is ∆ε independent of E. To simplify the analysis, 

one can, to a very good approximation, assume that υ  l/τ where l is the longitudinal size of the spatial 

variation of ∆ε and τ is its temporal variation; this will preclude the description of the propagation pulses 

of very short temporal duration where dispersion will become important but will suffice for the purposes 

of this presentation. Thus, Equation (A1) reduces to

2 2

0

2 0 E ik E k E

z ρ

∂   ∆ε +∇ + =   ∂ ε  

 (A2)

which is known in the literature as the quasi-optical approximation. The first two terms of Equation (A2) 

describe the propagation (within the quasi-optical approximation) of a beam wave (see Eq. (A4)) in a 

vacuum; the last term is the perturbation added to the evolution of the beam wave through a medium 

described by the factor ∆ε. It is through this factor that the various effects of the atmosphere on 

propagation come into play.

For the power ranges of interest in high energy laser propagation, the total permittivity function can 

be written as

( ) ( ) 2 2

0 ET i E T , K T k

α ∆ε = ε + ∆ε + ε (A3)

in which α is the wavelength dependent absorption coefficient; absorption within the propagation medium 

gives rise to the increase in the associated temperature T from the nominal value, which, in turn, modifies 

the value of the permittivity by the amount εTT where εT ≡ ∂ε/∂T is the variation of the permittivity with 

respect to T. (Strictly speaking, the temperature increase T is also a function of |E|

2 but, as will be 

discussed below, it is not a direct function as is the Kerr effect.) This gives rise to thermal nonlinearities 

and the associated thermal ‘blooming’. Also, due to the large value of E in cases of very high energy 

propagation, induced orientation of anisotropically polarized molecules of the atmosphere occurs due to 

the interaction of the induced dipole moment of the molecule due to the intense electric field. This 

contributes another perturbation ∆εK to the permittivity through the quantity |E|

2

. This phenomenon is 

NASA/TM—2015-218726 31
known as the Kerr effect. The functional relations of both ( ) 2 ∆εK E and T to the quantity |E|

2 are also 

given through the prescription of a differential equation. Each of these contributions will be separately 

discussed below and the critical field powers at which they will occur within the atmosphere will be 

derived. Before this is done, however, it will prove to be convenient to make the following digression 

concerning the description of a beam wave.

The consideration of the general solution for Equation (A2) makes it necessary to have an expression 

for the associated boundary condition. This brings to the fore the normalization of spatial coordinates.

The boundary condition is given by the expression for the general form of a Gaussian beam wave, viz.,

( )

2 0 , exp 1 21

A k E z

i z i z

    γ ρ ρ= −   + γ   + γ  

 (A4)

where A0 is the initial amplitude of the field and

121 2 2

0

2 1 i , , kW F γ≡γ + γ γ ≡ γ ≡ (A5)

with W0 being the waist radius of the beam at the exit of the output aperture and F is the focal length. The 

initial condition for the field at the output aperture is thus given by

( )

2

2 2 0 0 2

0

1 2 ,0 exp exp 2 2k k EA A i

W F

  γ ρ         ρ = − ρ = − ρ+                     

 (A6)

At this point, it is convenient to normalize the spatial coordinates with respect to the parameters of the 

problem. For example, if one defines the ‘effective’ radius of the beam wave at the exit of the output 

aperture as r W eff ≡ 0 2 , (this definition is consistent with the definition for the power distribution in 

the transverse plane) one can define the dimensionless transverse coordinate ρ′ ≡ ρ/reff and write 

Equation (A6) as

( )

2

2 2 0

0

1 1 ,0 exp 2 2

kW

EA i

F

        ρ = − ρ+ ρ ′ ′′                 

 (A7)

One can now apply the same coordinate normalization to Equation (A2). That is, since

( ) ( ) 2 2

2

0

2 Ez E z , , W ρ ρ′ ∇ ρ= ∇ ρ   ′ ′   (A8)

Equation (A2) can be written (letting ε0 = 1 for atmospheric scenarios)

2 2 2

0 0 2 2 0

2 2

kW E k W i EE

z ρ′

∂ ∆ε +∇ + =

∂

 (A9)

NASA/TM—2015-218726 32
Now, normalizing the longitudinal coordinate z with respect to the quantity defined by 2

0 L kW d ≡ 2 , i.e., 

z′ = z/Ld, and noting that

( ) ( )

2

0

Ez E z , , 2

z kW z

∂ρ ∂ρ′ ′ = ∂ ∂ ′

 

(A10)

Equations (A7) and (A9) become

( )

2

0 ,0 exp 1 2

Ld EA i

F

    ρ′   ρ= − + ′          

 (A11)

and 

( ) 2 2 2 2 2 , 0 eff

E i E kr E TE

z ρ′

∂ + ∇ + ∆ε =

∂ ′  (A12)

This form of the quasi-optical equation will facilitate the developments that follow. The quantity Ld is 

called the diffraction length and, as seen from Equation (A11), is the length that the focus must achieve 

for the associated diffraction to become appreciable.

A.1 The Critical Power for the Atmospheric the Kerr Effect

To isolate the effect of the Kerr nonlinearity from the thermal effects due to absorption, one simply 

lets ∆ε = ∆εK (|E|

2

) in Equation (A12) and obtains

( ) 2 2 2 2 2 0 eff K

E i E kr E E

z ρ′

∂ + ∇ + ∆ε =

∂ ′  (A13)

The differential equation that governs the evolution of ∆εK is given by

K 2

K K E

t

∂∆ε τ + ∆ε = ε

∂ (A14)

where τ is the relaxation constant for the Kerr process and εK is the Kerr constant for the atmosphere; 

typically, τ ≈ 10–11 sec. Relative to the other processes to be considered (and the fact that very short pulse 

lengths are not being considered here), one can ignore the first term on the left of Equation (A14). In this 

case, ∆εK = εK|E|

2 and Equation (A13) becomes

2 2 2 2 2 0 eff K

E i E kr E E

z ρ′

∂ +∇ + ε =

∂ ′  (A15)

In the literature, Equation (A15) is sometimes stated to describe a ‘cubic’ medium due to the occurrence 

of E in the last term. Connecting |E|

2 to the associated intensity I of the beam using the relation 

|E|

2 = (8π/c)I and then to the corresponding beam power P0 using 2 0 eff P rI = π , Equation (A15) becomes

NASA/TM—2015-218726 33
2 2 0 K

E i E RE

z ρ′

∂ +∇ + =

∂ ′  (A16)

where RK ≡ P0/PcrK is the nonlinear parameter (also called the distortion parameter) for the Kerr 

Nonlinearity defined in terms of the critical power PcrK associated with the Kerr nonlinearity,

2 8 crK

K

c P

k = ε

(A17)

For Earth’s atmosphere at sea level, εK ≈ 2.5×10–16 CGSE units (Ref. 8). Thus, for a laser wavelength of 

λ = 2.0 μm, one has that the Kerr effect has a threshold at PcrK ≈ 1.5×109 W. This value is less than one 

order of magnitude from the powers that are being considered (~70×106 W, giving RK ≈ 0.05) for power 

beaming applications. Hence, the nonlinearities associated with the Kerr effect may become a minor 

factor in power beaming applications (and can be easily removed via adaptive optics) but, as will be 

shown below, RK is several orders of magnitude below the corresponding parameter for thermal 

nonlinearities which dominate the beaming process in the atmosphere. 

In the case of millimeter wave propagation at λ = 2.0 mm, one obtains PcrK = 1.5×1016 W and at 

P0 = 70×106 W, RK ≈ 4.6×10–9 thus alleviating any concerns about the Kerr effect at these wavelengths.

A.2 The Critical Power for Atmospheric Thermal Effects

A.2.1 Continuous Wave Source

Using Equation (A3) in Equation (A12) and neglecting the Kerr contribution gives

2 22 2 0 eff T

E i E kr i T E

z k ρ′

∂ α   +∇ + +ε =   ∂ ′  

 (A18)

The temperature variation from the nominal atmospheric temperature is given by the application of the 

conservation of energy to the atmospheric heat budget, given in its entirety by

2

p

T C VT T I

t

  ∂ ρ + ⋅∇ = κ∇ + α     ∂

  (A19)

where ρ is the density of the atmosphere, Cp is its specific heat at constant pressure, κ is its thermal 

conductivity, V

 is the total relative velocity of the motion (slewing) of the beam across is the atmosphere 

and that of the atmospheric wind, and α is the absorption coefficient for the radiation of intensity 

I ≡ (c/8π) |E|

2

. (Strictly speaking, one should also include the set of Navier-Stokes equations in the 

Boussinesq approximation to account for the viscous and Archimedean forces that occur during the 

thermally induced motion of the atmosphere. However, for the purposes of this discussion and subsequent 

analysis, a description at this level is not required.) Equation (A19) can be simplified straight away by 

first specializing the slewing and wind velocity to be along the x axis of the coordinate system, V Vx = ˆ  , 

where xˆ is the unit vector. Further, the diffusivity ∇2

T will be taken to have contributions only in the 

direction transverse to the laser beam propagation, i.e., 2 2 T T

ρ ∇ ≈∇ . Finally, only the stationary heating 

case will be considered for this particular discussion whereby the time derivative can be dropped. (That is, 

the CW radiation is taken to be acting long enough for the steady state case to be achieved. This is not the 

NASA/TM—2015-218726 34
case for pulsed radiation which will be discussed below.) Applying these considerations to Equation 

(A19) allows it to be written

2 2

8 p

T c VTE

x C ρ

∂ α = χ∇ + ∂ πρ  (A20)

where χ ≡ κ/ρCp is the associated thermal diffusivity. As was done earlier with Equation (A2), one can 

normalize the transverse coordinates with respect to reff. Additionally, the field value E can be normalized 

with respect to E′ = E/E0 where 2 0 0 8 8 eff E Ic P r c ≡π = (this is what was essentially done to 

Equation (A15) to obtain Equation (A16)). Equation (A20) then becomes

2

2 2

2 0

eff

p

eff eff

r V T C T E

P r xr ρ′

  π   ∂ χ ρ    −∇ = ′   α ∂ ′   

 (A21)

Now, the two heat transfer mechanisms that are described here can be isolated. Consider the ratio of the 

coefficients of the two terms within the brackets of Equation (A21), i.e., η ≡ Vreff/χ. This dimensionless 

ratio is known in fluid mechanics as the Peclet number. Taking the nominal wind velocity V = 4.4 m/s 

(10 mi/hr), reff = 10 m and using the documented value for the thermal diffusivity of the atmosphere 

χ = 2.12×10–5 m2

/s, one has that η = 2.1×106  1 allowing one to neglect the second term in the brackets.

In this instance, one can write

2

0

p eff Cr V T T E

P xx

  πρ ∂ ∂ ′   = = ′   α ∂∂ ′ ′  

(A22)

where the temperature has been normalized with respect to the characteristic temperature 

TV ≡ αP0/πρCpreffV. Finally, applying this prescription to Equation (A18) gives

2 2 2 2 0 d eff T V

E i E iL E k r T T E

z ρ′

∂ ′

+∇ + α + ε = ′ ′ ′ ′ ∂ ′  (A23)

As done with Equation (A15), one can define the distortion parameter for the thermal nonlinearity

2 0 2 2 0 eff T

V TV eff p crT

kr P P R kr T

CV P

ε α

≡ ε= =

πρ

(A24)

where

2

p

crT

eff T

C V

P

k r

πρ ≡ ε α

(A25)

is the critical power associated with the thermal nonlinearity in the case of large Peclet numbers. Using 

Equation (A24) and noting that the third term in Equation (A23) can be transformed away, one has

NASA/TM—2015-218726 35
( ) 2 2 0, exp 2 V d

E i E R TE E E L z

z ρ′

∂ ′′

+ ∇ + = = −α ′′ ′ ′′ ′′ ′ ′ ∂ ′  (A26)

The problem of propagation through a thermal nonlinearity thus reduces, to within the 

approximations adopted, to the solution of Equations (A22) and (A26). For the Earth’s atmosphere, 

ρ = 1.2 kg/m3

, CP = 1.0×103 J/K-kg, and εT ≈ –2.3×10–6 K–1. (That is, the atmospheric permittivity 

decreases with an increase in temperature. This indicates that the laser beam will steer toward regions 

with larger permittivity, i.e., cooler regions. In the calculations, the absolute value of this quantity is 

used.) At λ = 2.0 μm, α ≈ 4×10–6 m–1 and using the earlier stated values for the other quantities, one finds 

that PcrT = 16.7 W. Thus, for P0 = 70×106 W, RV ≈ 4.2×106

. The very small value for the critical power 

occurs due to the rather large beam radius. The temperature must diffuse over the large cross section of a 

cylindrical column. Of course, larger values of V will increase PcrT. For a smaller, more typical value for a 

beam radius, reff = 0.1 m, one finds that PcrT = 1667 W and RV ≈ 4.2×104 for P0 = 70 MW, still a rather 

large value for the nonlinearity parameter. As a comparison, a more typical laser power of P0 = 100 KW 

corresponds to RV ≈ 60. It is thus seen that nonlinearities due to thermal effects at infrared wavelengths 

within the atmosphere will dominate those of the Kerr effect.

The small values for PcrT at the large output aperture radii and the corresponding very large values for 

RV do indeed have severe ramifications for the long distance propagation of beam waves at P0 ≈ 70 MW 

as shown earlier. However, these values facilitate the use of approximation procedures to be used in 

solving Equations (A22) and (A26) from which specifications for the adaptive correction of the various 

aberrations are derived and it is shown that complete mitigation is possible.

The situation for millimeter wave propagation is better. For λ = 2.0 mm, α = 2.2×10–4 m-1 and one has 

PcrT = 3.5×105 W and for P0 = 70 MW, RV = 200.8. Although the absorption coefficient α is two orders of 

magnitude larger than that for infrared wavelengths, the overall optical effects due to the thermally 

induced refractive index variations are much smaller due to the larger wavelength; the ‘thermal lens’ that 

is induced has weaker focusing capabilities at millimeter wavelengths.

A.2.2 Pulsed Source

In the case of a source that produces radiation pulses of temporal duration tp, the prevailing fluid 

dynamics of the situation differs from that of a CW source. Here, for tp < reff/V, the convection/diffusion 

equation, Equation (A19) is approximated as

p

T C I

t

∂

ρ =α ∂ (A27)

Coupling this equation with the propagation equation, Equation (A18), and following the same 

normalization procedure as outlined above for the CW case and normalizing the time to tp, one obtains in 

place of Equation (A26), 

( ) 2 2 P d 0, exp 2 E i E R TE E E L z

z ρ′

∂ ′′

+ ∇ + = = −α ′′ ′ ′′ ′′ ′ ′ ∂ ′  (A28)

NASA/TM—2015-218726 36
where 

2

2

0

p eff

p

C r T T E

Pt t t

  πρ ∂ ∂ ′   = = ′   α ∂∂ ′ ′  

(A29)

and the prevailing nonlinear parameter is now

2 T p 0 0

P

p crP

k tP P R

C P

ε α ≡ = πρ

(A30)

in which the corresponding critical power is

2

p

crP

T p

C

P

k t

πρ ≡ αε

(A31)

which is independent of reff.

Taking a nominal pulse width of tp ≈ 1 μs, one has for λ = 2.0 μm, PcrP = 3.7×107 W and for 

λ = 2.0 mm, PcrP = 7.7×1011 W. Thus, temporally modulating the source does not elucidate the convective 

thermal effects that are inherent in the CW case and allows for much larger critical values of peak power.

Pulse lengths smaller than 1 μs require the use of Equation (A1) which takes into account pulse dispersion 

through the medium. This effect is especially important when atmospheric turbulence is taken into 

account. However, for the purposes of the present analysis, which is to obtain the critical powers at which 

various nonlinear propagation effects show themselves, these effects are not considered.

The issue of the repetition rate of a pulse train within the atmosphere in conjunction with scanning 

(slewing) the beam through a region of the atmosphere to aid in heat dispersion (for high pulse repetition 

rates) is characterized by yet another critical power threshold which will be larger than that given by 

Equation (A31).

A.3 Conclusion

The parameters RK, RV and RP provide for a comparative assessment of the nonlinear propagation 

mechanisms that are induced due to the large field power densities. Thus, at the power levels considered 

here, the effect of Kerr nonlinearities are almost negligible relative to that of thermal effects and it is the 

later that must be completely understood so as to be effectively mitigated via properly designed adaptive 

optics algorithms. They do not, however, give an assessment of the behavior of specific laser beam 

properties, such as beam steering and beam broadening that accompany propagation. This can only be 

done through modeling using the specific propagation geometries that are being considered. As 

mentioned above, the very large values for RV in the case of the power levels considered for beamed 

propulsion are a benefit in that they allow for the use of approximation procedures to be applied to 

analytically evaluate the propagation and adaptive optics models. 

NASA/TM—2015-218726 37
References

1. R. M. Manning, Stochastic Electromagnetic Image Propagation and Adaptive Compensation, 

McGraw-Hill Book Co., New York, NY. 1993, Chapter 2.

2. A. D. Wheelon, Electromagnetic Scintillation: Volume 2, Weak Scattering, Cambridge University 

Press, 2003.

3. A. G. Litvak, “Self Focusing of Powerful Light Beams by Thermal Effects,” Sov. Phys. JETP Letters, 

4 (9), pp. 230-232, (1966).

4. D. C. Smith, “High Power Laser Propagation – Thermal Blooming,” Proc. IEEE 65 (12), pp. 1679-

1714 (1977).

5. V. V. Vorob’ev, “Thermal Blooming of Laser Beams in the Atmosphere,” Prog. Quantum Electron. 

15, pp. 1-152 (1991). 

6. L. V. Keldysh, “Ionization in the Field of a Strong Electromagnetic Wave,” Sov. Phys. JETP 20, pp. 

1307-1314 (1965). 

7. R. W. Boyd, Nonlinear Optics, Academic Press, San Diego, CA, 1992, Chapter 4.

8. R. L. Sutherland, Handbook of Nonlinear Optics, Marcel Dekker, Monticello, NY, 1996.

9. J. D. Reichert, W. G. Wagner, and W. Y. Chen, “Instabilities of Intense Laser beams in Air,” J. Appl. 

Phys. 44 (8), pp. 3641-3646 (1973).

10. J. D. Reichert, W. G. Wagner, and W. Y. Chen, “Propagation of an Intense Gaussian Laser Pulse in 

Air,” J. Appl. Phys. 44 (8), pp. 3647-3658 (1973).

11. J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-Dependent Propagation of High Energy Laser Beams 

Through the Atmosphere,” Appl, Phys. 10, pp.129-160 (1976).

12. A. A. Vedenov and O. A. Markin, “Propagation of Intense Laser Radiation in an Absorbing 

Medium,” Sov. Phys. JETP 49 (4), pp. 608-615 (1979).

13. P. Sprangle, J. Penano, and B. Hafizi, “Propagation of High energy Laser Beams in Various 

Environments,” Report prepared for High-Energy-Laser Joint Technology Office, May, 2007. Also, 

appears as NRL Report, NRL/MR/6790-07-9032, June 2007.

14. R. Fischer, et al., “Absorption and Scattering of 1.06 μm Laser Radiation from Oceanic Aerosols,” 

Appl. Opt. 48 (36), pp. 6990-6999 (2009).

15. S. L. Glickler, “Propagation of a 10.6 μ Laser Through Cloud Including Droplet Vaporization,” Appl. 

Opt. 10 (3), pp. 644-650 (1971). 

16. V. E. Zuev and A. V. Kuzikovskii, “Thermal Bleaching of Aqueous Aerosols by Laser Radiation,” 

Russ. Phys. J. 20 (11), pp.1480-1500 (1977).

17. B. S. Agrovskii, et al., “Thermal Blooming of Laser beams in Turbulent Media,” J. Opt. Soc. Am. A2

(12), pp. 2304-2312 (1985). 

18. L. C. Bradley and J. Herrmann, “Phase Compensation for Thermal Blooming,” Appl. Opt. 13 (2), pp. 

331-334 (1974).

19. D. E. Novoseller, “Zernike-Ordered Adaptive-Optics Correction of Thermal Blooming,” J. Opt. Sic. 

Am. A5 (11), pp, 1937-1942 (1988). 

20. NIST Handbook of Mathematical Functions, (Cambridge University Press, 2010). Chapter 18. 

21. J. R. Penano, P. Sprangle, and B. Hafizi, “Propagation of High Energy Laser Beams Through 

Atmospheric Stagnation Zones,” NRL Report NRL/MR/6790-06-8925, January, 2006.

22. J. W. Hardy, “Active Optics: A New Technology for the Control of Light,” Proc. IEEE 66, pp. 651-

697 (1978). 

23. M. A. Vorontsov, et al., “Adaptive Imaging System for Phase-Distorted Extended Source/Multiple 

Distance Objects,” Appl. Opt. 36, pp. 3319-3328 (1997).

NASA/TM—2015-218726 38
24. M. A. Vorontsov, et al., “Adaptive Optics Based on Analog Parallel Stochastic Optimization: 

Analysis and Experimental Demonstration,” J. Opt. Soc. Am. A17 (8), pp. 1440-1453 (2000).

25. M. A. Vorontsov, “Decoupled Stochastic Parallel Gradient Descent Optimization for Adaptive 

Optics: Integrated Approach for Wave-Front Sensor Information Fusion,” J. Opt. Soc. Am. A19 (2), 

pp. 356-368 (2002).

26. M. A. Vorontsov and V. Kolosov, “Target-in-the-Loop Beam Control: Basic Considerations for 

Analysis and Wave-Front Sensing,” J. Opt. Soc. Am. A22 (1), pp. 126-141 (2005).

27. P. Piatrou and M. Roggemann, “Beaconless Stochastic Parallel Gradient Descent Laser Beam 

Control: Numerical Experiments,” Appl. Opt. 46 (27), pp. 6831-6842 (2007). 

28. M. A. Vorontsov, V. V. Kolosov, and A. Kohnle, “Adaptive Laser Beam Projection on an Extended 

Target: Phase- and Field-Conjugate Precompensation,” J. Opt. Soc. Am. A24 (7), pp. 1975-1993 

(2007). 

29. C. A. Primmerman and D. G. Fouche, “Thermal-Blooming Compensation: Experimental 

Observations Using a Deformable-Mirror System,” Appl. Opt. 15 (4), pp. 990-995 (1976).

30. C. A. Primmerman, F. B. Johnson, and I. Wigdor, “Thermal Blooming Compensation Using the 

CLASP System,” Appl. Opt. 17 (18), pp. 2909-2912 (1978). 

31. D. G. Fouche, C. Higgs, and C. F. Pearson, “Scaled Atmospheric Blooming Experiments (SABLE),” 

Lincoln Lab. J. 5 (2), pp. 273-293 (1992). 

32. C.-H. Qiao, et al., “Simulation Experiment of High Energy Laser Propagation in the Atmosphere,” 

Proc. SPIE 7382, Art # 73822I (2009).

33. M. A. Molchan, E. V. Doktorov, and R. A. Vlasov, “Propagation of Vector Fractional Charge 

Laguerre-Gaussian Light Beams in the Thermally Nonlinear Moving Atmosphere,” Opt. Lett. 35 (5), 

pp. 670-672 (2010). 

NASA/TM—2015-218726 39

Chapter 2.—An Analytical Assessment of the Thermal Blooming Effects on 

the Propagation of Optical And Millimeter-Wave Focused Beam

Waves For Power Beaming Applications*

2.1 Introduction

When delivering large amounts of power through the Earth’s atmosphere via millimeter or infrared 

‘beams’ (i.e., laser beams or beams formed at the output of a millimeter wave antenna system), many 

propagation mechanisms must be addressed that can be potentially deleterious such power transmission.

The most obvious one is the ever-present random variation of the atmospheric refractive index due to 

local temperature variations known as ‘turbulence’. This naturally occurring phenomena is driven by 

thermal convection of heat from the Earth’s surface; once the resulting air motion exceeds a critical value 

of velocity, laminar flow essentially evolves into turbulent flow and fluctuations in the temperature 

distribution becomes statistically random (Ref. 1). These temperature fluctuations then act directly on the 

prevailing refractive index, thus rendering the refractive index a random quantity. These refractive index 

variations randomly focus and defocus the intervening electromagnetic wave field. Thus, the atmosphere 

can be considered to be composed of ‘lenses’ of random focusing and defocusing characteristics that, due 

to the gross atmospheric motion due to wind, move across the beam. This gives rise to many beam quality 

variations; the major ones being beam broadening and beam steering. The statistical analysis and 

modeling of this type of atmospheric propagation as a long and rich history and has resulted in analytical 

descriptions for the impact of turbulence on the operation of systems relying on such beam propagation.

Many models and descriptions exist for the ‘engineering analysis’ of the operation of transmission 

systems that rely on the propagation of electromagnetic beam propagation in the atmosphere (for a good 

recent treatment, see Ref. 2 and the reference therein).

The scenario discussed above may be considered as ‘passive’ electromagnetic wave propagation, i.e., 

the wave field moves through an atmosphere the refractive index of which is determined by other sources, 

not the field itself. However, as the energy density of the beam increases, absorption of the beam energy 

by atmospheric gas components results in local heating of the atmospheric which does indeed act directly 

on the refractive index causing it to decrease in value. This thermal change of the refractive index field 

then acts on the electromagnetic wave field causing it to also change, and so on. The propagation scenario 

now becomes an ‘active’ one, whereby the propagating field modifies the very medium it which it exists.

This heating process is called ‘thermal blooming’ and substantially differs from that of the passive 

propagation discussed earlier. Here, a ‘thermal lens’ is created within the atmosphere by the heating due 

to the energy density of the beam. This ‘self-action’ of the beam will not only bend the beam into regions 

of higher refractive index (beam steering), but convection within the atmospheric fluid will also arise 

which is the source of self-induced turbulent flow of the medium. The situation is further complicated 

when one includes the effects of atmospheric wind and the abovementioned passive propagation effects.

Defocusing and other such associated nonlinear thermal blooming distortions of the beam cross-section 

will then result. In extreme cases of very large energy densities, the propagating beam will essentially 

break up into smaller beams, or filaments, which severely constrains the amount of energy density that the 

beam will be able to possess as it travels through the atmosphere. Unlike the situation of passive 

propagation, the thermal blooming mechanism introduces nonlinearities into the analysis of the 

phenomena that substantially complicates a complete mathematical description. Complete analyses of 

*

See also the reference by the author, “High Energy Laser Beam Propagation in the Atmosphere: The Integral 

Invariants of the Nonlinear Parabolic Equation and the Method of Moments,” NASA/TM—2012-217634, July 2012.

NASA/TM—2015-218726 41

 
these types of propagation scenarios can only be done numerically, which was a major activity within the 

United States and Russia in the late 1980s. Other than the usual ‘order-of-magnitude’ estimates using the 

equations of fluid mechanics and wave propagation, analytical treatments appropriate for an engineering 

analysis of atmospheric propagation systems encountering thermal blooming have been lacking, making 

difficult a comparative assessment of the operation of through-the-atmosphere power transmission.

In an effort to quickly assess the operation of power beaming methodologies proposed for application 

in continuous wave (CW) beamed energy propulsion, but at the same time, transcend the ‘order-ofmagnitude’ analysis approach that can be employed (Ref. 3), an analytical model is needed to describe 

two very important beam parameters, i.e., beam broadening and beam steering, due to thermal blooming 

that may occur within the atmosphere in the presence of wind. (The use of the CW case represents the 

‘worst case’ scenario for power beaming since much larger power densities can be supported using pulsed 

source operation (Ref. 3).) Of course, beam broadening dictates the severity of the spread of the energy 

distribution of the beam at the target and the steering will capture an assessment of the variations of 

pointing of the beam at the target. Although such effects can be dealt with by an application of adaptive 

optics at the transmitter, an estimation of these parameters will provide a good comparative analysis of 

systems operating at different frequencies and power density levels as well as an idea of the overall 

impact of thermal blooming on system operation. (Only brief descriptions will be given concerning the 

derivation of the equations given here; the detailed derivations will be the subject of a forthcoming 

paper.)

2.1.1 A Simple Analytical Model for Beam Broadening and Displacement 

To this end, consider the fundamental equations for a scalar electric field E(x,y,z) propagating along 

the z-axis of a coordinate system within the atmosphere of permittivity (refractive index) ε(x,y,z),

( ( ) ) 2 2 2 ,, 1 0 E ik E k x y z E

x

⊥

∂ +∇ + ε − =

∂ (1)

where 2 ∇⊥ is the transverse Laplacian and k = 2π/λ is the wave number of the radiation of wavelength λ.

In general (Refs. 1 and 3), the permittivity ε(x,y,z) is a function of the temperature T as well as a 

wavelength-dependent imaginary component ε′(x,y,z) accounting for absorption and δε which accounts 

for the permittivity fluctuations due to turbulence, i.e.,

(, ,) 1 (, ,) () xyz i xyz T T ε = + ε + ε + δε ′ (2)

The temperature contribution can be written as 

T (T T ) T

∂ε

ε =

∂ (3)

where, for the gasses within the Earth’s atmosphere, ∂T/∂t < 0. For the purposes of this analysis, the 

turbulent fluctuations δε will be neglected since only thermal blooming effects are being considered. (The 

combined effects of turbulence and thermal blooming should be considered in a future treatment.) Hence, 

using Equations (2) and (3) into Equation (1) gives

( ( ) ) 2 2 2 , , T 0 E ik E k i x y z T E

x

⊥

∂ +∇ + ε +ε = ′ ∂ (4)

NASA/TM—2015-218726 42
As discussed above, for fields possessing a large energy density, the temperature T will be a function of 

the value of the field E, i.e., T = T(E) due to the absorption ε′. This propagation equation must thus now 

be augmented by an equation from fluid dynamics that describes the creation of temperature within the 

atmosphere due to absorption and its spatial and temporal evolution within the atmosphere. The 

appropriate equation is (Ref. 3)

2 2

8 0 p

T VT c T E

t xc ⊥

∂ ∂α −∇ + =

∂ χ ∂ πχρ (5)

where V is the atmospheric wind velocity, taken to occur along the x-axis, α = kε′ is the wavelength 

dependent absorption coefficient, c is the velocity of light, χ is the atmospheric molecular diffusivity, cp is 

the specific heat at constant volume, and ρ0 is the nominal air density. (The term on the right side of 

Equation (5) is the heat source due to wave absorption and, of course, is proportional to the energy 

density |E|

2 within the electric field.) Finally, the boundary condition giving the source of the radiation 

must be specified. In general, a good model for the field source is given by a collimated Gaussian beam 

wave at the entrance z = 0 to the atmosphere,

( ) ( ) ( )

2 2

2 2 0 0 2

0

, ,0 exp 2 2

x y ik E xy E x y a F

  +

= − −+  

 

(6)

where a0 is the radius of the output aperture and F is the focal length of the beam.

Equations (4) to (6) form the basis of most investigations of the effects of thermal blooming. As 

mentioned above, the investigations can only be performed numerically for specific geometrical and 

atmospheric scenarios. In an effort to yield analytical expressions for only the most important 

performance characteristics, the following approximation procedure will be applied to these model 

equations. Assuming that a steady state condition exists for the induced temperature within the 

atmosphere, i.e., ∂T/∂t = 0 (such an assumption cannot be made for the analysis of a pulsed laser 

situation), and taking the wind velocity to essentially move the parcel of atmosphere from the volume of 

the beam before diffusion takes effect, i.e., ( )( ) 2T V Tx ⊥ −∇ << χ ∂ ∂ , Equation (5) reduces to

2

8 0 p

T c V E

x c

∂ α = ∂ πρ (7)

The coupled equations represented by Equations (4) and (7) will now be solved using successive 

approximations, viz, for the first approximation E1(x,y,z) for the electric field of the beam wave, 

Equation (4) will be written in which the term εTT is ignored,

1 2 1 1 0

2 2

E i E E

x k ⊥

∂ α −∇ + =

∂ (8)

where the relation α = kε′ which defines the absorption coefficient in terms of the imaginary part of the 

permittivity was used as noted earlier. Equation (7) becomes, within this approximation, 

1 2

1 8 0 p

T c V E

x c

∂ α = ∂ πρ (9)

NASA/TM—2015-218726 43
Now the solution of Equation (8) is given by

( ) ( ) ( ) 2 1 0 0 , , , ,0 , ; , ; E x y z e E x y G x y x y z dx dy z

∞ ∞

−α

−∞ −∞

= ′′ ′′ ′′ ∫ ∫ (10)

where 

( ) (( ) ( ) ) 2 2

0 , ; , , exp 2 2

ik ik G xyx y z x x y y

z z

  ′ ′ ≡ − − + − ′ ′   π   (11)

is the Green function of the differential equation, Equation (8). Substituting Equations (6) and (11) into 

Equation (10) yields

( ) ( ) ( ) ( )

2

2 22 2 1 0 , , exp 1

22 2

z ik R z ik ik E xyz Ee x y R z

z zz

−α        = −   + +              

(12)

where

( ) ( ) 1 2 2 1 R I

z R z i ziz

k

  − = −γ + γ     (13)

with

2

0

1 1

R I , ka F γ ≡ γ≡ (14)

The solution of Equation (9) is written

( ) ( )

2

1 1 , , , , 8

x

p

c T xyz E x y z dx c V −∞α = ′ ′ πρ ∫ (15)

Using Equation (12) in this relation yields, after some significant manipulation (as mentioned above, 

all the derivations that give the results displayed here will be the subject of a separate paper),

( ) ( )

1/2

1/2 2 2 1 0

1 , , ( ) exp 1 erf

8 ( ) ( )

z I I

p I

c k k T xyz Ee z y x

c V k z z

− −α   α     γ γ    

= Λ −+         

πρ     γ   Λ   Λ    

(16)

where

( ) ( )

2 2 2 1 R I Λ ≡ −γ +γ z zz (17)

This ‘first approximation’ result can now be used in the second approximation of Equation (4) which 

includes Equation (16) as the thermal source, viz,

NASA/TM—2015-218726 44
2 2 2 2 12 2 22 T

E i k E E i TE

x k ⊥

∂ α −∇ + =ε

∂ (18)

the solution of which can be written

( ) ( ) ( ) ( ) 2 2 0 0 1

0

, , , ,0 , ; , ; exp , , 2

z

z T

ik E x y z e E x y G x y x y z T x y z dz

∞ ∞

−α

−∞ −∞

 

= ′′ ′′  ε ′ ′     ∫ ∫ ∫ (19)

Using this second approximation expression, one can now obtain the behavior of the intensity of the 

propagating beam wave, of arbitrary initial radius and focal length, which includes the effects of thermal 

blooming within the atmosphere. From this, one can then immediately find expressions for the beam 

broadening and beam bending. Substituting Equations (6), (11), and (16) into Equation (19) gives for the 

beam wave intensity I(x,y,L) at a point (x,y) in a transverse plane at distance L downrange from the 

transmitter

( ) ( ( ) ( ))

(( ) ( ) ) (( ) ( ) )

( )( ) ( ) ( )

2

2 2 22 22 * 2 0 11 22

22 2 2

11 2 2

2 2 1 1 2 2 1 2 112 2

(, , ) , , exp 2 2

2 2

z k ik IxyL E xyL Ee x y x y

z

ik ik

xx yy xx yy

z z

i z x x z y y dx dy dx dy

∞∞∞∞

−α

−∞ −∞ −∞ −∞

   ≡ = − Γ − −Γ − ′′ ′′      π 

+ − +− − − +− ′′ ′ ′

 + Φ − −Φ − ′ ′ ′ ′ ′′′ ′ 



∫∫∫∫

(20)

where

( ) ( ) ( ) ( ) ( )

5/2 2 2 2 2

1 2 2 9/2 , , 2

R I R I

R I

I I

i z z z zk γ +γ π γ +γ Γ≡γ − γ Φ ≡Ξ Φ ≡Ξ

γ γ

(21)

and

( )

2

0

2 2 , 2

T R

p R I

k E

z ze

c V

−αη

  ε γ Ξ ≡ α η≡  

ρ γ +γ  

(22)

Again, leaving the details to a future publication, Equation (20) reduces to

( ) ( ) ( )

( ( ))

( ) ( )

2 2 2 2 2 0

2 0 2 2 (, , ) , , exp

L

x y x y

a e x xL y IxyL E xyL E a La L a L a L

−α   − ∆ ≡= −−  

 

(23)

where

aL a L x ( ) ≡ Λ 0 ( ) (24)

is the radius along the x-axis of the beam at a distance z from the transmitter,

NASA/TM—2015-218726 45
( ) ( ) ( ) ( ) ( )

1/2 2 2

2 2

0 2

4 4

y R 1

zz zz

az a z z

k k

  Φ Φ ≡   − −γ +Λ

 

(25)

is that along the y-axis of the beam and finally,

( ) 0 1 ( ) 2

I xz a zz

k

γ ∆ ≡Φ (26)

is the displacement of the initial axis of the beam along the x-direction at the target. Equation (23) can be 

used to easily calculate the energy that is transmitted to a target within the atmosphere. These relations 

indicate that in the presence of an atmospheric wind taken to be directed along the x-axis, the beam radius 

along the x-axis does not deviate from that of just pure diffractive broadening (given by Eq. (24)) but that 

along the y-axis will appreciably broaden. However, the beam will also be steered along the x-axis as 

described by Equation (26). These results will now be applied to two CW power beaming scenarios to 

ascertain the comparative behavior of their overall characteristics at two disparate frequencies.

2.1.2 Application of the Foregoing to Millimeter and Optical Wavelength Power Beaming

Equations (24) to (26) will now be applied to assess the behavior of high power electromagnetic wave 

beams within the atmosphere at two wavelengths λ of 1.06×10–6 m and 2.0×10–3 m. The nominal wind 

velocity within the atmosphere is taken to be V = 4.5 m/sec (≈ 10 mi/hr) along the x-axis. For the Earth’s 

atmosphere, εT ≈ 2.3×10–6 K–1

, cp ≈ 1.005 J (g K)–1, and ρ = 1.225×103 g/m3. The initial beam radii are 

taken to be a0 = 1.0 m. Ideally, the beams should be focused at the target, that is taken to be at a distance 

of L = 1.0×104 m, i.e., F = L. In the millimeter wave case, this is realistic. That is, a millimeter wave beam 

with λ = 2.0×10–3 m, a0 = 1.0 m at a distance of L = 1.0×104 m in the absence of thermal blooming will 

possess a beam radius of a(L) ≈ 3.2 m (using Eqs. (17) and (24)). However, at the wavelength of 

λ = 1.06×10–6 m, one has a(L) ≈ 3.1×10–3 m (the optical wavelength beam can achieve a much sharper 

focus than its millimeter wave counterpart). At this small radii, the power density would not only surpass 

that which the theory can treat within the approximations made, but is not representative of a condition 

for optimal power transfer to the target. Thus, in the optical wave case, the focus must be backed-off, e.g., 

to achieve a beam radius at the target that is representative of that of the target size (which, for purposed 

of this preset calculation, is ~ 1.0 m). This dictates that for the optical case, the focal length should not be 

smaller than F = 1.0×108 m. Finally, the absorption coefficients α for each if the wavelengths must be 

found. For λ = 1.06×10–6 m, α ≈ 4.6×10–6 m-1 (Ref. 4). Similarly, for λ = 2.0×10–3 m, α ≈ 2.2×10–4 m–1

(Ref. 5). Finally, the three power levels of P0 = 1.0×106 W, P0 = 10.0×106 W and P0 = 100×106 W will be 

considered. (The relationship between the total beam power P0 and the radiant intensity 2

0 E that appears 

in Equations (22) and (23) is 2 2 0 0 0 P Ea = π .) The operational results for such beamed transmission systems 

follows:

3 2.0 10− λ= × m, 4 L = × 1.0 10 m, a0 =1.0 m, F L =

6 P0 = × 1.0 10 W

a L x ( ) = 3.18 m

a L y ( ) = 4.15 m

NASA/TM—2015-218726 46
∆ =− x L( ) 1.3 m

6 P0 = × 10.0 10 W

a L x ( ) = 3.18 m

a L y ( ) = 26.9 m

∆ =− x L( ) 13.1 m

6 P0 = × 100 10 W

a L x ( ) = 3.18 m 

a L y ( ) = 267.6 m

∆ − x L( ) 131.1 m

6 1.06 10− λ= × m, 4 L = × 1.0 10 m, a0 =1.0 m, 8 F = × 1.0 10 m

6 P0 = × 1.0 10 W

a L x ( ) =1.0m

a L y ( ) =1.03 m

∆ =− x L( ) 0.02 m

6 P0 = × 10 10 W

a L x ( ) =1.0m

a L y ( ) =1.34 m

∆ =− x L( ) 0.19 m

6 P0 = × 100 10

a L x ( ) =1.0 m

a L y ( ) = 4.43 m

∆ =− x L( ) 1.94 m

Clearly, the use of optical wavelengths will minimize thermal blooming effects as compared to those 

of millimeter waves. This is due simply to the fact that the absorption coefficient is two orders of 

magnitude less at optical wavelengths than at millimeter wavelengths and, of course, diffraction effects 

dominate the larger wavelengths. The negative numbers representing the beam displacements along the xaxis indicate that the beam is deflected in a direction opposite to that of the atmospheric wind, i.e., the 

deflection is toward the cooler side of the beam, i.e., toward regions of larger refractive index.

NASA/TM—2015-218726 47
The results of the analytical expressions derived here can also be displayed in various graphical 

formats. These will be included in the forthcoming publication that will detail the derivations of the 

equations. Equations (24) to (26) are, however, sufficient for use in the overall assessment of high-energy 

beam wave propagation within the atmosphere.

It also remains to subject a pulsed laser scenario to the same type of straightforward analysis as given 

above for the CW case. This will form the subject of another forthcoming paper. Also, atmospheric 

conditions have been assumed in which aerosols and such do not exist that may give rise to explosive 

detonation upon heating by the radiation field. 

References

1. R. M. Manning, Stochastic Electromagnetic Image Propagation Through the Atmosphere, McGrawHill Book Co., New York, NY. 1993, Chapter 2.

2. A. D. Wheelon, Electromagnetic Scintillation: Volume 2, Weak Scattering, Cambridge University 

Press, 2003.

3. R. M. Manning, “High Energy Laser Beam Propagation and Thermal Blooming in the Atmosphere,” 

notes from an SPIE short course, Los Angeles, CA (1987).

4. E. P. Shettle and R. W. Fenn, “Models for the Aerosols of the Lower Atmosphere and Effects of 

Humidity Variations on Their Optical Properties,” Environmental Research Papers, No. 676, AFGLTR-79-0214, Sept. 20, 1979.

5. H. J. Liebe and D. H. Layton, “Millimeter-Wave Properties of the Atmosphere: Laboratory Studies 

and Propagation Modeling,” NTIA Report 87-224, October, 1987.

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