Propagation of Electromagnetic

Fields Over Flat Earth

ARL-TR-2352 February 2001

Joseph R. Miletta 

Approved for public release; distribution unlimited.
The findings in this report are not to be construed as an

official Department of the Army position unless so

designated by other authorized documents.

Citation of manufacturer’s or trade names does not

constitute an official endorsement or approval of the use

thereof.

Destroy this report when it is no longer needed. Do not

return it to the originator.
ARL-TR-2352 February 2001

Army Research Laboratory

Adelphi, MD 20783-1197

Propagation of Electromagnetic

Fields Over Flat Earth

Joseph R. Miletta 

Sensors and Electron Devices Directorate

Approved for public release; distribution unlimited.
Abstract

This report looks at the interaction of radiated electromagnetic fields with

earth ground in military or law-enforcement applications of high-power

microwave (HPM) systems. For such systems to be effective, the microwave

power density on target must be maximized. The destructive and constructive scattering of the fields as they propagate to the target will determine

the power density at the target for a given source. The question of field

polarization arises in designing an antenna for an HPM system. Should

the transmitting antenna produce vertically, horizontally, or circularly polarized fields? Which polarization maximizes the power density on target?

This report provides a partial answer to these questions. The problems of

calculating the reflection of uniform plane wave fields from a homogeneous

boundary and calculating the fields from a finite source local to a perfectly

conducting boundary are relatively straightforward. However, when the

source is local to a general homogeneous plane boundary, the solution cannot be expressed in closed form. An approximation usually of the form

of an asymptotic expansion results. Calculations of the fields are provided

for various source and target locations for the frequencies of interest. The

conclusion is drawn that the resultant vertical field from an appropriately

oriented source antenna located near and above the ground can be significantly larger than a horizontally polarized field radiated from the same

location at a 1.3 GHz frequency at observer locations near and above the

ground.

ii
Contents

1 Introduction 1

2 Problem Formulation 2

2.1 Vertical Dipole Over Earth .................... 3

2.2 Horizontal Dipole Over Earth .................. 4

2.3 Comparisons ............................ 5

2.3.1 Comparison of Field Components Near the Earth . . 6

3 Conclusion 14

References 15

Appendix. MATLAB m-Files 17

Distribution 27

Report Documentation Page 29

Figures

1 Geometry for vertical and horizontal dipole formulations .... 2

2 Comparison of main electric field components from a vertical

(Ez) and from a horizontal (Eφ) broadside ideal dipole over a

perfectly conducting ground ..................... 7

3 Reflection coefficients for vertically and horizontally polarized

plane waves of 1.3-GHz frequency incident on a flat ground for

five earth-parameter cases listed in table 1 ............. 8

4 Fresnel reflection coefficient for a vertically polarized plane wave

field compared to “reflection coefficient” as calculated by complete formulation for a vertical dipole over homogeneous earth

as observed at 1-m height 1000 m down range at a frequency of

1.3 GHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Fresnel reflection coefficient for a horizontally polarized plane

wave field compared to “reflection coefficient” as calculated by

complete formulation for a horizontal dipole over homogeneous

earth as observed at 1-m height, broadside 1000 m down range

at a frequency of 1.3 GHz . . . . . . . . . . . . . . . . . . . . . . 10

iii
6 Comparison of principal fields from an ideal dipole oriented

perpendicular and horizontal to a homogeneous flat earth . . . 11

7 Comparison of principal fields from an ideal dipole oriented

perpendicular and horizontal to a homogeneous flat earth . . . 12

8 Effect of ground reflection on primary field components near

ground for typical earth parameters . . . . . . . . . . . . . . . . 13

Tables

1 Earth parameters ........................... 6

iv
1. Introduction

Effective military or law-enforcement applications of high-power microwave (HPM) systems in which the HPM system and the target system

are on or near the ground or water require that the microwave power density on target be maximized. The power density at the target for a given

source will depend on the destructive and constructive scattering of the

fields as they propagate to the target. Antenna design for an HPM system includes addressing the following questions about field polarization:

Should the fields the transmitting antenna produces be vertically, horizontally, or circularly polarized? Which polarization maximizes the power

density on target? (The question of which polarization best couples to the

target is beyond the scope of this report.) While this report does not completely answer these questions, it addresses the interaction of the radiated

electromagnetic fields with earth ground. It is assumed that the transmitting antenna and the target (or receiver) are located above, but near the

surface of a flat idealized earth (constant permittivity, ε, and conductivity,

σ) ground. First an ideal vertical dipole (oriented along the z-axis perpendicular to the ground plane) is addressed. The horizontal dipole (parallel

to the ground plane) follows.

1
2. Problem Formulation

The problems of calculating the reflection of uniform plane wave fields

from a homogeneous boundary and calculating the fields from a finite

source local to a perfectly conducting boundary are relatively straightforward. However, when the source is local to a general homogeneous plane

boundary, it is found that the solution cannot be expressed in closed form.

An approximation usually of the form of an asymptotic expansion results.

The problem of an ideal dipole over a homogeneous half-space has been the

topic of a number of studies starting at the turn of the last century with the

solution provided by Sommerfeld (1949) and leading to the more contemporary work of Banos (1966) and King et al (1994, 1992). References such

as Maclean and Wu (1993) address in detail the many approaches to solving the problem. Considerable controversy has surrounded these studies.

We will not attempt to derive the solution or otherwise discuss the solution of the problem in this report. We will rely on the work of King et al

for a complete and concise formulation of the problem. Figure 1 depicts the

problem geometry. The King expressions have been encoded and solved in

MATLAB (1984–1999). Comparisons of the field structures are provided

for various source and target locations for frequencies of interest.

The expressions that follow are constrained by the magnitude of the wave

numbers (k = ω(μ)1/2, ω = 2π × frequency) in each region:

|k1| ≥ 3 |k2| , (1)

where, after King, the subscript 2 denotes the upper half-space (air) and the

subscript 1 denotes the lower half-space (earth). Also, μ is the permeability

in henries per meter, ε is the permittivity in farads per meter, and σ is the

conductivity in siemens per meter. The subscripts 0 and r represent free

space and relative to free space, respectively.

Figure 1. Geometry for

vertical and horizontal

dipole formulations.

z

z′

0

r0

r1

r2 Region 2

μ0, ε0, σ = 0

Region 1

μ0, ε2 = εr ε0,σ ≠ 0

y

x

Looking down

θd

θ

d

φ

ρ

Ψ

2
2.1 Vertical Dipole Over Earth

The electromagnetic fields from a dipole with dipole moment I
 oriented

perpendicular to and at a height z = d above the ground plane have three

components in cylindrical coordinates. The magnetic field is symmetric

about the z-axis, perpendicular to the direction of propagation. For that

reason the fields will be termed transverse magnetic (TM) fields. (We will

find that the horizontal dipole has TM and transverse electric (TE) components.) The formulation as provided in King et al (1994) is, referring to

figure 1,

Hφ(ρ, z) = − I


2π







eik2r1

2

 ρ

r1

 ik2

r1 − 1

r2

1



+ eik2r2

2

 ρ

r2

 ik2

r2 − 1

r2

2



−eik2r2

k3

2

k1

  π

k2r2

 1

2 e−iP F(P)





 , (2)

Eρ(ρ, z) = −ωμ0I


2πk2







eik2r1

2

 ρ

r1

 z−d

r1

 ik2

r1 − 3

r2

1

− 3i

k2r3

1



+eik2r2

2

 ρ

r2

 z+d

r2

 ik2

r2 − 3

r2

2

− 3i

k2r3

2



−k2

k1 eik2r2

 ρ

r2

 ik2

r2 − 1

r2

2



−

k3

2

k1

  π

k2r2

 1

2 e−iP F(P)









, and (3)

Ez(ρ, z) = ωμ0I


2πk2







eik2r1

2

ik2

r1 − 1

r2

1

− i

k2r3

1



−

z−d

r1

2 ik2

r1 − 3

r2

1

− 3i

k2r3

1



+eik2r2

2

ik2

r2 − 1

r2

2

− i

k2r3

2



−

z+d

r2

2 ik2

r2 − 3

r2

2

− 3i

k2r3

2



−eik2r2 k3

2

k1

 ρ

r2

  π

k2r2

 1

2 e−iP F(P)







, (4)

where

r1 =



ρ2 + (z − d)

2

 1

2 ,

r2 =



ρ2 + (z + d)

2

 1

2 , (5)

P = k3

2r2

2k2

1

k2r2 + k1 (z + d)

k2ρ

2

, (6)

F(P) =  ∞

P

eit

(2πt)

1

2

dt = 1

2 (1 + i) − C2 (P) − iS2 (P) , (7)

and C2(P), S2(P) are the Fresnel integrals as defined in Abramowitz and

Stegun (1970). Since

C2 (P) + iS2 (P) = 1

2 (1 + i) −



i

2

eiP w

√

iP, (8)

we have

F (P) = 

i

2

eiP w

√

iP, (9)

where w(z), called the plasma dispersion function with complex argument,

is a form of the error function defined in Abramowitz and Stegun (1970). A

3
convenient MATLAB m-file is available for solving the function with complex argument (Chase) and is provided in the appendix. Note that √i can

be written as (i+1)

2 and that F(P) always appears with e−iP , thus canceling the exponential term in equation (9). This is reflected in the MATLAB

m-files that calculate the fields (provided in the appendix).

The equations have been written such that the direct and reflected components appear first. The last term is referred to as the surface or lateral wave.

Often it is called the Norton surface wave from the engineering models he

developed in the mid-1930s. The equations reduce to the fields above a perfectly conducting ground when k1 → ∞; the surface or lateral wave term

then goes to zero.

2.2 Horizontal Dipole Over Earth

The electromagnetic fields produced by an ideal dipole oriented at a height

z = d above and parallel to the ground plane consist in general of both TM

and TE components. The formulation as provided in King et al (1992) for

the TM wave components in cylindrical coordinates, referring to figure 1

and the above definitions, is

Hφ(ρ, φ, z) = I


4π

cos φ













eik2r1

z−d

r1

 ik2

r1 − 1

r2

1



− eik2r2

z+d

r2

 ik2

r2 − 1

r2

2



+2k2

k1 eik2r2







ik2

r2 − 1

r2

2

− i

k2r3

2

−

k3

2

k1

 r2

ρ

  π

k2r2

 1

2 e−iP F(P)

















 , (10)

Eρ(ρ, φ, z) = ωμ0I


4πk2

cos φ







eik2r1



2

r2

1

+ 2i

k2r3

1

+

z−d

r1

2 ik2

r1 − 3

r2

1

− 3i

k2r3

1



−eik2r2







2

r2

2

+ 2i

k2r3

2

+

z+d

r2

2 ik2

r2 − 3

r2

2

− 3i

k2r3

2



−2k2

k1

z+d

r2

 ik2

r2 − 1

r2

2



+2k2

2

k2

1







ik2

r2 − 1

r2

2

− i

k2r3

2

−

k3

2

k1

 r2

ρ

  π

k2r2

 1

2 e−iP F(P)



















, and (11)

Ez(ρ, φ, z) = ωμ0I


4πk2

cos φ







−eik2r1

 ρ

r1

 z−d

r1

 ik2

r1 − 3

r2

1

− 3i

k2r3

1



+eik2r2

 ρ

r2

 z+d

r2

 ik2

r2 − 3

r2

2

− 3i

k2r3

2



−2k2

k1 eik2r2

 ρ

r2

 ik2

r2 − 1

r2

2



− k3

2

k1

 π

k2r2

 1

2 e−iP F(P)









. (12)

The TM fields are zero broadside to the dipole orientation, φ = π

2 . The TE

components are

4
Hρ(ρ, φ, z) = I


4π

sin φ













eik2r1

z−d

r1

 ik2

r1 − 1

r2

1



− eik2r2

z+d

r2

 ik2

r2 − 1

r2

2



+2k2

k1 eik2r2







2

r2

2

+ 2i

k2r3

2

+

 ik2

2

k1ρ

 r2

2

ρ2

  π

k2r2

 1

2 e−iP F(P)

+

z+d

r2

2 ik2

r2 − 3

r2

2

− 3i

k2r3

2



















 , (13)

Hz(ρ, φ, z) = I


4π

sin φ





















eik2r1

 ρ

r1

 ik2

r1 − 1

r2

1



− eik2r2

 ρ

r2

 ik2

r2 − 1

r2

2



+2  ρ

r2



eik2r2







k2

k1

 z+d

r2

 ik2

r2 − 3

r2

2

− 3i

k2r3

2



−

k2

k1

2







1

r2

2

+ 3i

k2r3

2

− 3

k2

2r42

+

z+d

r2

2 ik2

r2 − 6

r2

2

− 15i

k2r3

2



































, and (14)

Eφ(ρ, φ, z) = −ωμ0I


4πk2

sin φ







eik2r1

ik2

r1 + 1

r2

1

− i

k2r3

1



− eik2r2

ik2

r2 − 1

r2

2

− i

k2r3

2



−eik2r2







−2k2

k1

z+d

r2

 ik2

r2 − 1

r2

2



+2k2

2

k2

1







2

r2

2

+ 2i

k2r3

2

+

z+d

r2

2 ik2

r2 − 3

r2

2

− 3i

k2r3

2









+

2ik4

2

k3

1ρ

 r2

ρ

2  π

k2r2

 1

2 e−iP F(P)













. (15)

These are the predominant fields broadside to the dipole orientation. Here

Eφ is often termed the horizontal electric field. Again, the equations reduce

to the fields above a perfectly conducting ground when k1 → ∞; the surface

or lateral wave term goes to zero.

2.3 Comparisons

The peak power radiated by a unit dipole is given in Collin and Zucker

(1969):

P = k2

2ζ0

12π , (16)

where ζ0 is the free-space impedance μ0

ε0



. The fields and power density

comparisons that follow are normalized to one watt radiated peak power.

To obtain the unnormalized quantities, simply multiply the power results

by P and the field results by P 1

2 . The outward component of the complex

Poynting vector over a closed surface is

1/2



S

E × H ∗ · dS = −Pcomplex , (17)

where the negative sign indicates power flow away from the surface. Our

interest is in the real part of the power density at the observer (or target)

location. In our cylindrical coordinate system for the TM components, this

becomes

1/2 Re (E × H∗) = 1/2 Re 

ρ0EzHφ ∗ + z0EρHφ ∗



, (18)

5
where ρ0 and z0 are the unit vectors in cylindrical coordinates. Near the

ground, the power flow is predominantly radial with a small z component.

And, for the TE components, we have

1/2 Re (E × H∗) = 1/2 Re 

ρ0EφHz ∗ − z0EφHρ ∗



. (19)

For the vertical dipole, the power density at the observer (on target) will be

Pv = 1/2 (Re (EφHz ∗) + Re (EφHρ ∗)) . (20)

The horizontal dipole in general will produce a target power density of

Ph = 1/2 ({Re (EφHz ∗) − Re (EzHφ ∗)} + {Re (EρHφ ∗) − Re (EφHρ ∗)}) . (21)

For broadside calculations this becomes

Ph = 1/2 (Re (EφHz ∗) + Re (EφHρ ∗)) . (22)

The calculations that follow are limited to a frequency of 1.3 GHz (wavelength, λ0, is 0.23 m) and dipole heights of 1 to 3 m. Observer (target)

heights range from 0 to 5 m. The frequency of 1.3 GHz is chosen, since most

of the HPM source and antenna design work at ARL is centered around that

frequency. The height ranges are chosen to be consistent with ground vehicle source and target applications. Five classes of ground parameters that

are representative of distinctly different terrain will be addressed. These

five classes are those discussed in King et al (1994) and are given in table 1.

2.3.1 Comparison of Field Components Near the Earth

When thinking about the interaction of electromagnetic waves with the

earth’s surface, we often view the earth as a perfect conductor. The horizontally polarized field is reflected with the opposite sign of the incident

field, leading to a difference or destructively interfered-with field. The vertically polarized field is reflected with the same sign as the incident field,

leading to a sum or constructively interfered-with field. Figure 2 compares

the resultant primary field components produced from a horizontal and a

vertical ideal dipole located at the same point in space over a perfectly conducting ground plane. From these calculations, one might conclude that

Table 1. Earth

parameters.

Case σ,* S/m εr*

1 Sea water 4 80

2 Wet earth .4 12

3 Dry earth .04 8

4 Lake water .004 80

5 Dry sand .000 2

*The variable name representing σ

in the MATLAB m-files is SIGMA

and for εr it is EPSREL. These variable names appear on many of the

figures.

6
Figure 2. Comparison of

main electric field

components from a

vertical (Ez) and from a

horizontal (Eφ)

broadside ideal dipole

over a perfectly

conducting ground. Ez

is a constructively

interfered-with field,

while Eφ is a

destructively

interfered-with field.

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

0

0.5

1

1.5

2

2.5

3

Dipole height = 2 m, frequency = 1.3 GHz,

range = 1000 m

Normalized field magnitude (V/m)

Horizontal E-field, horizontal dipole

z-directed E-field, vertical dipole 

Observer height (m)

an antenna that radiated fields polarized such that the electric field was

essentially perpendicular to the ground plane would produce the largest

fields and, consequently, the greater power densities on a target or near

the ground. This is generally not the case for real earth grounds. While

the perfectly conducting ground plane model may provide some insight

into the interaction with horizontally polarized fields, it clearly is an inadequate model for vertical polarization. The complex reflection coefficients

for a plane wave polarized with the electric field in the plane of incidence

(vertical polarization) and with a plane wave polarized with the electric

field perpendicular to the plane of incidence (horizontal polarization) are

given by the following Fresnel expressions. For vertical polarization, it is

(Collin, 1985)

ρcomplex =



εr − i σ

ωε0



sin ψ −



εr − i σ

ωε0



− cos2 ψ



εr − i σ

ωε0



sin ψ +



εr − i σ

ωε0



− cos2 ψ

, (23)

and for horizontal polarization, it is

ρcomplex =

sin ψ −



εr − i σ

ωε0



− cos2 ψ

sin ψ +



εr − i σ

ωε0



− cos2 ψ

, (24)

where ψ is the angle of incidence as measured between the ray path and

the surface of the earth (see fig. 1). These reflection coefficients for the five

cases of table 1 are presented in figure 3. One can clearly see that for shallow grazing angles, a Brewster angle effect exists (often called a pseudoBrewster angle). The result is destructive interference of the vertically polarized field for incident angles that are less than this pseudo-Brewster

7
Figure 3. Reflection

coefficients for

vertically and

horizontally polarized

plane waves of 1.3-GHz

frequency incident on a

flat ground for five

earth-parameter cases

listed in table 1. Note

that case 5 provides a

true Brewster angle

(since conductivity is

zero) and case 4 is very

near to producing a true

Brewster angle at

1.3-GHz frequency:

(a) vertical polarization

and (b) horizontal

polarization.

0 20 40 60 80

0

0.2

0.4

0.6

0.8

1

Incidence angle (°)

–150

–100

–50

0

170

175

180

185

190

195

200

Case 1

2 

5 3 

4 

4,5

0 20 40 60 80

Incidence angle (°)

0 20 40 60 80

Incidence angle (°)

0 20 40 60 80

Incidence angle (°)

0.5

0.6

0.7

0.8

0.9

1

Reflection coefficient

Phase Phase

Reflection coefficient

(a)

(b)

Case 1

5

2

4

3

5

3

2 4

Case 1

3

Case 1

2

angle. This would lead one to conclude that horizontally and vertically polarized field levels above a ground would be comparable at shallow angles

of incidence. The Fresnel equations (22) and (23) do not take into account

the surface or lateral wave produced by finite sources above the ground.

They are also not applicable to near-field problems. We can compare the

reflection coefficients as determined from the Fresnel equations with the

results of our complete solution. To do this with the complete equations

(eqs (2) to (4) for the vertical dipole and eqs (10) to (15) for the horizontal

dipole), we fix the radial (or range) distance to 1000 m. The dipole height is

varied to provide a range of incident angles. The observer will be fixed at a

1-m height. The incident and ”reflected” fields are

Evtotal =



E2

ρ + E2z , (25)

for the vertical dipole and

Ehtotal =



E2

φ + E2

ρ + E2z , (26)

for the horizontal dipole. For broadside calculations, this simply becomes

Eφ. The terms involving r1 in equations (2) to (4) and (10) to (15) represent the incident field and the remaining terms represent the reflected and

surface wave terms. The reflection coefficient will be calculated as the total incident electric field divided into the total reflected and bound (surface wave) fields. Figures 4 and 5 provide the comparison of the magnitude of the reflection coefficient for the five cases of table 1. For the most

part, the Fresnel expressions are a good approximation for the calculation

of horizontally polarized field values (fig. 5) above realistic earth ground at

8
Figure 4. Fresnel

reflection coefficient for

a vertically polarized

plane wave field

compared to “reflection

coefficient” as

calculated by complete

formulation for a

vertical dipole over

homogeneous earth as

observed at 1-m height

1000 m down range at a

frequency of 1.3 GHz:

(a) sea water, (b) wet

earth, (c) dry earth,

(d) lake water, and

(e) dry sand.

0 10 20 30 40 50 60 70 80 90

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Incidence angle (°)

Reflection coefficient

0 10 20 30 40 50 60 70 80 90

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Incidence angle (°)

Reflection coefficient

0 10 20 30 40 50 60 70 80 90 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Incidence angle (°)

Reflection coefficient

0 10 20 30 40 50 60 70 80 90 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Incidence angle (°)

Reflection coefficient

0 10 20 30 40 50 60 70 80 90

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Incidence angle (°)

Reflection coefficient

(a)

(c)

(e)

(d)

(b)

SIGMA = 4 S/m, EPSREL= 80

Fresnel Eq.

Complete Eq.

SIGMA = 0.4 S/m, EPSREL= 12

SIGMA = 0.04 S/m, EPSREL= 8 SIGMA = 0.004 S/m, EPSREL= 80

SIGMA = 0 S/m, EPSREL= 2

Fresnel Eq.

Complete Eq.

Fresnel Eq.

Complete Eq.

Fresnel Eq.

Complete Eq.

Fresnel Eq.

Complete Eq.

1.3 GHz. The horizontally polarized fields diverge from the Fresnel expressions for low conductivities and relative dielectric constants. The resulting

total fields in such cases predicted by the Fresnel expressions will be larger

than the complete solution prediction. The results for vertical polarization

(fig. 4) show significant divergence for shallow incident angles. In this case

the resulting total fields predicted by the complete solution can be significantly higher than that predicted from employing the Fresnel expressions.

9
Figure 5. Fresnel

reflection coefficient for

a horizontally polarized

plane wave field

compared to “reflection

coefficient” as

calculated by complete

formulation for a

horizontal dipole over

homogeneous earth as

observed at 1-m height,

broadside 1000 m down

range at a frequency of

1.3 GHz: (a) sea water,

(b) wet earth, (c) dry

earth, (d) lake water,

and (e) dry sand.

0 10 20 30 40 50 60 70 80 90

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Incidence angle (°)

SIGMA = 4 S/m, EPSREL = 80

Fresnel eq. 

Complete eq.

Reflection coefficient

0 10 20 30 40 50 60 70 80 90

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Incidence angle (°)

SIGMA = 0.4 S/m, EPSREL = 12

Reflection coefficient

0 10 20 30 40 50 60 70 80 90

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Incidence angle (°)

SIGMA = 0.04 S/m, EPSREL = 8

Reflection coefficient

0 10 20 30 40 50 60 70 80 90

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Incidence angle (°)

SIGMA = 0.004 S/m, EPSREL = 80

Reflection coefficient

0 10 20 30 40 50 60 70 80 90

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Incidence angle (°)

SIGMA = 0 S/m, EPSREL = 2

Reflection coefficient

(a)

(c)

(e)

(d)

(b)

Fresnel eq. 

Complete eq.

Fresnel eq. 

Complete eq.

Fresnel eq. 

Complete eq.

Fresnel eq. 

Complete eq.

10
Figures 6 and 7 are plots of the fields produced by the two-dipole orientations; these plots compare the predominant field components for each

dipole orientation at 1000 and 100 m, respectively. The plots for the 100-m

case (fig. 7) show the beginning of the lobe effect produced by the phase

difference between the incident and reflected wave. This is shown more

dramatically in figure 8.

Figure 6. Comparison of

principal fields from an

ideal dipole oriented

perpendicular and

horizontal to a

homogeneous flat earth.

In each case, dipole is

placed 2 m above

ground plane and

observer or target is

1000 m down range:

(a) sea water, (b) wet

earth, (c) dry earth,

(d) lake water, and

(e) dry sand.

0 1 2 3 4 5 67

× 10–3

0

0.5

1

1.5

2

2.5

3

Normalized field magnitude (V/m)

Observer height (m)

(a)

0

0

0.5

1

1.5

2

2.5

3

Normalized field magnitude (V/m)

Observer height (m)

(d)

0

0

0.5

1

1.5

2

2.5

3

Normalized field magnitude (V/m)

Observer height (m)

(e)

0

0

0.5

1

1.5

2

2.5

3

Normalized field magnitude (V/m)

Observer height (m)

(c)

0 0

0.5

1

1.5

2

2.5

3

Normalized field magnitude (V/m)

Observer height (m)

(b)

SIGMA = 4 S/m, EPSREL = 80

dipole height = 2 m, frequency = 1.3 GHz

range = 1000 m

Horizontal E-field, horizontal dipole

z-directed E-field, vertical dipole

SIGMA = 0.4 S/m, EPSREL= 12

dipole height = 2 m, frequency = 1.3 GHz

range = 1000 m

SIGMA = 0.04 S/m, EPSREL= 8

dipole height = 2 m, frequency = 1.3 GHz

range = 1000 m

SIGMA = 0.004 S/m, EPSREL= 80

dipole height = 2 m, frequency = 1.3 GHz

range = 1000 m

SIGMA = 0 S/m, EPSREL= 0.2

dipole height = 2 m, frequency = 1.3 GHz

range = 1000 m

1 2 3 4 5 67

× 10–3

1 2 3 4 5 67

× 10–3

1 2 3 4 5 67

× 10–3

1 23 4 5 6

× 10–3

Horizontal E-field, horizontal dipole

z-directed E-field, vertical dipole

Horizontal E-field, horizontal dipole

z-directed E-field, vertical dipole

Horizontal E-field, horizontal dipole

z-directed E-field, vertical dipole

Horizontal E-field, horizontal dipole

z-directed E-field, vertical dipole

11
Figure 7. Comparison of

principal fields from an

ideal dipole oriented

perpendicular and

horizontal to a

homogeneous flat earth.

In each case, dipole is

placed 2 m above

ground plane and

observer or target is

100 m down range:

(a) sea water, (b) wet

earth, (c) dry earth,

(d) lake water, and

(e) dry sand.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.16 0.18 0.2

0

0.5

1

1.5

2

2.5

3

Normalized field magnitude (V/m)

Observer height (m)

(a)

0

0.5

1

1.5

2

2.5

3

Normalized field magnitude (V/m)

Observer height (m)

(d)

0

0.5

1

1.5

2

2.5

3

Normalized field magnitude (V/m)

Observer height (m)

(e)

0

0.5

1

1.5

2

2.5

3

Normalized field magnitude (V/m)

Observer height (m)

(c)

0

0.5

1

1.5

2

2.5

3

Normalized field magnitude (V/m)

Observer height (m)

(b) SIGMA = 4 S/m, EPSREL = 80

dipole height = 2 m, frequency = 1.3 GHz

range = 100 m

Horizontal E-field, horizontal dipole

z-directed E-field, vertical dipole

SIGMA = 0.4 S/m, EPSREL = 12

dipole height = 2 m, frequency = 1.3 GHz

range = 100 m

SIGMA = 0.04 S/m, EPSREL = 8

dipole height = 2 m, frequency = 1.3 GHz

range = 100 m

SIGMA = 0 S/m, EPSREL = 0.2

dipole height = 2 m, frequency = 1.3 GHz

range = 100 m

SIGMA = 0.004 S/m, EPSREL = 80

dipole height = 2 m, frequency = 1.3 GHz

range = 100 m

0.16 0.18 0.2 0.2

Horizontal E-field, horizontal dipole

z-directed E-field, vertical dipole

Horizontal E-field, horizontal dipole

z-directed E-field, vertical dipole

Horizontal E-field, horizontal dipole

z-directed E-field, vertical dipole

Horizontal E-field, horizontal dipole

z-directed E-field, vertical dipole

12
Figure 8. Effect of

ground reflection on

primary field

components near

ground for typical earth

parameters. Phase

difference between

incident and reflected

waves results in

development of field

lobes that are more

pronounced as

radiating antenna is

approached: (a) 100 m

down range and (b)

1000 m down range.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0

5

10

15

20

25

30

Normalized field magnitude (V/m)

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

0

5

10

15

20

25

30

Normalized field magnitude (V/m)

Observer height (m) Observer height (m)

(a)

(b)

SIGMA = 0.4 S/m, EPSREL = 12

dipole height = 2 m, frequency = 1.3 GHz

range = 1000 m

SIGMA = 0.4 S/m, EPSREL = 12

dipole height = 2 m, frequency = 1.3 GHz

range = 100 m

Horizontal E-field, horizontal dipole

z-directed E-field, vertical dipole

Horizontal E-field, horizontal dipole

z-directed E-field, vertical dipole

13
3. Conclusion

A suite of MATLAB m-files have been developed to calculate the electromagnetic fields produced by a vertical and a horizontal infinitesimal unit

dipole over a homogeneous flat (ground) plane. Calculations for the fields

above the ground plane have been made for various ground-plane conductivities and relative dielectric constants. The calculations bound the practical range of parameters representative of natural earth terrain.

For a frequency of 1.3 GHz, where the dipole and the observer are close to

the ground plane (<3 m), significant difference is seen in the magnitude of

the fields from either dipole orientation. The power density on target will

be much larger for vertical dipole orientation. The effects of rough terrain,

foliage, or scattering from manmade or natural objects in the path from the

dipole to target may alter this conclusion. How these other scatterers might

affect the field structure at a target at 1.3 GHz is not known at this time. If

the effects are random in nature, the present conclusion is most likely still

valid. From the simple, smooth, flat ground model, then, one must conclude that vertical polarization (antenna radiating a vertically polarized

field) will deliver the most energy to the target. Unless the target’s preference for field orientation for maximum pickup is known, a vertically polarized antenna may in fact be the best choice for a ground weapon system.

14
References

Abramowitz, M., and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Mathematics Series–55, U.S. Department of Commerce (1970).

Banos, A., Dipole Radiation in the Presence of a Conducting Half-Space, Pergamon, Oxford (1966).

Chase, R., MATLAB m-file, Plasma Dispersion Function with Complex Argument (unpublished).

Collin, R. E., Antenna and Radiowave Propagation, McGraw-Hill, New York

(1985).

Collin, R. E., and F. J. Zucker, Antenna Theory, Part 1, McGraw-Hill, New

York (1969).

King, R.W.P., and S. S. Sandler, “The electromagnetic field of a vertical electric dipole over the earth or sea,” IEEE Trans. Antennas Propag. 42, No. 3

(March 1994), pp 382–389.

King, R.W.P., M. Owens, and T. T. Wu, Lateral Electromagnetic Waves,

Springer-Verlag, New York (1992).

Maclean, T.S.M., and Z. Wu, Radiowave Propagation Over Ground, Chapman

and Hall, London (1993).

MATLAB, release 5.3, The Math Works, Inc., Natick, MA (1984–1999).

Sommerfeld, A., Partial Differential Equations in Physics, Academic Press,

New York (1949).

15
16
Appendix. MATLAB m-Files

This appendix documents the MATLAB m-files that implemented the field

equations for the vertical and horizontal dipole and array cases. Slight modification may be required to obtain all the calculations presented.

WERF Function

function w = werf(z,N)

% WERF(Z,N) Plasma Dispersion Function with complex argument.

%

% Computes the function w(z)=exp(-z^2)*erfc(-iz) using a rational

% series with N terms. N should be a power of 2 or it gets SLOW.

% Default value of N is 64. z can be a matrix of values.

% Taken from Siam Journal on Numerical Analysis, Oct 1994, V31,#5.

% Modified by R. Chase to work for all z (4/3/95).

%

% N=32 gives approx 14 place accuracy, N=64 is better and

% it seems as fast.

%

% See wfn.m -- Draws graph in Abramowitz & Stegun,

% Handbook of Mathematical Functions, p. 298

if nargin == 1, N=64; end % Default value for N

M=2*N; M2=2*M; k=[-M+1:1:M-1]'; % M2=no. of sampling points

L=sqrt(N/sqrt(2)); % Optimal choice of L

theta=k*pi/M; t=L*tan(theta/2); % Variables thetaand t

f=exp(-t.^2).*(L^2+t.^2); f=[0;f]; % function to be transformed

a=real(fft(fftshift(f)))/M2; % Coefficients of transform

a=flipud(a(2:N+1)); % Reorder coefficients

nz=imag(z) <= 0; % Find im(z) <=0

z(nz)=conj(z(nz)); % Use conj for above

Z=(L+i*z)./(L-i*z); p=polyval(a,Z); % Polynomial evaluation

w=2*p./(L-i*z).^2+(1/sqrt(pi))./(L-i*z); % Evaluate w(z)

w(nz)=2*exp(-(conj(z(nz)).^2)) - conj(w(nz)); % Handle im(z) <= 0

%if all(imag(z)==0), w=real(w); end % Rtn real if real

17
Constants

function [w,cv,epso,u0,zo,k0,k1,kappap,lo]=cnstdg(f)

% This m-file contains the basic constants to be used by various

% m-files and functions

%

% f is input in GHz

% global EPSREL SIGMA

w=2*pi*f*1e9;

cv=2.99792458e8;

epso=(1/(4*pi))*1e7*(1/cv^2);

eps1=EPSREL;

u0=4*pi*1e-7;

zo=sqrt(u0/epso);

k0=w*sqrt(epso*u0);

epsr=(eps1-j*SIGMA./(w*epso));

k1=k0*sqrt(epsr);

kappap=-k0/sqrt(epsr+1);

lo=cv/(f*1e9);

Vertical Dipole

function [ez,er,hp]=dipg(f,d,rho,z)

%

% Electromagnetic fields from a vertical z-directed current element

% at a height, d, over a conducting dielectric plane (ground) calculated

% by the King/Sandler model. This formulation is that developed

% as equations 6, 7, and 8 of "The Electromagnetic Field of a Vertical Electric

% Dipole over the Earth or Sea", IEEE Trans. on Antennas and Propagation,

% March 1994, Vol 42 No. 3, page 383. It is the same as the King/Owens/Wu

% formulation. This formulation is that developed as equations 4.2.30-32 of

% "Lateral Electromagnetic Waves", Springer-Verlag, 1992, pp 293-297.

%

% > f- frequency; rho- radial distance from the z-axis

% < z- height of observer above ground plane (input as an array)

% > ez,er,hp- field components at the observer, where the second letter

% < designates the component -- z, r (rho), p (phi)

%

% The formulation is subject to |k1(ground)|>3|k2(air)|

%

[w,c,eps0,u0,zo,k2,k1,kappap,lo]=cnstdg(f);

%

18
% King, et. al assume an exp(-i*omega*time)dependency thus we convert k1

%

k1=conj(k1);

%N2=(k1/k2)^2;

k21=k2/k1;

%

% We assume a unit dipole

%

il=1;

r1=(rho^2+(z-d).^2).^.5;

r2=(rho^2+(z+d).^2).^.5;

sd=rho./r1;

sr=rho./r2;

cd=(z-d)./r1;

cr=(d+z)./r2;

p=(r2*k2^3/(2*k1^2)).*((k2*r2+k1*(z+d))./(k2*rho)).^2;

%

% The attenuation function less the exp(i*p) term - related to the

% error function is

%

Ferf=((1+i)/4)*werf(sqrt(i*p));

%

% Develop the terms for the field expressions

%

t1=w*u0*il/(2*pi*k2);

t11=-il/(2*pi);

t2=exp(i*k2*r1)/2;

t3=(i*k2./r1)-(1./r1.^2)-i./(k2*r1.^3);

t31=(i*k2./r1)-(1./r1.^2);

t4=(cd.^2).*((i*k2./r1)-(3./r1.^2)-(3*i)./(k2*r1.^3));

t41=((i*k2./r1)-(3./r1.^2)-(3*i)./(k2*r1.^3));

t5=exp(i*k2*r2)/2;

t6=(i*k2./r2)-(1./r2.^2)-i./(k2*r2.^3);

t61=(i*k2./r2)-(1./r2.^2);

t7=(cr.^2).*((i*k2./r2)-(3./r2.^2)-(3*i)./(k2*r2.^3));

t71=((i*k2./r2)-(3./r2.^2)-(3*i)./(k2*r2.^3));

t8=(2*t5*(k2^3)/k1).*((pi./(k2*r2)).^.5).*sr.*Ferf;

t81=((k2^3)/k1)*((pi./(k2*r2)).^.5).*Ferf;

t82=(2*t5.*(k2^3)/k1).*((pi./(k2*r2)).^.5).*Ferf;

%

% Calculate the fields

% ez=t1*(t2.*(t3-t4)+t5.*(t6-t7)-t8);

er=-t1*(t2.*sd.*cd.*t41+t5.*sr.*cr.*t71-k21*2*t5.*(sr.*t61-t81));

hp=t11*(t2.*sd.*t31+t5.*sr.*t61-t82);

19
Horizontal Dipole

function [ez,ep,er,hz,hp,hr]=dipgh(f,d,phi,rho,z)

%

% Electromagnetic fields from a horizontally directed (perpendicular to z)

% current element located in the phi=0 plane

% at a height, d, over a conducting dielectric plane (ground) calculated

% by the King/Owens/Wu formulation. This formulation is that developed

% as equations 7.10.75, 7.10.80, 7.10.84, 7.10.92, 7.10.93 and 7.10.94 of

% "Lateral Electromagnetic Waves", Springer-Verlag, 1992, pp 293-297.

%

% > f- frequency(GHz); phi- angle about the z-axis, rho- radial distance from

% < the z-axis, z- height of observer above ground plane (input as an array)

% > ez,ep,er,hz,hp,hr- field components at the observer, where the second

% < letter designates the component -- z, r (rho), p (phi)

%

%

% The formulation is subject to |k1(ground)|>3|k2(air)|

%

[w,c,eps0,u0,zo,k2,k1,kappap,lo]=cnstdg(f);

%

% King, et. al assume an exp(-i*omega*time)dependency thus we convert k1

%

k1=conj(k1);

%N2=(k1/k2)^2;

k21=k2/k1;

k2p=k2/rho;

%

% We assume a unit dipole

%

il=1;

cp=cos(phi);

sp=sin(phi);

r1=(rho^2+(z-d).^2).^.5;

r2=(rho^2+(z+d).^2).^.5;

sd=r1/rho;

sdo=1./sd;

sr=r2/rho;

sro=1./sr;

cd=(z-d)./r1;

cr=(d+z)./r2;

p=(r2*k2^3/(2*k1^2)).*((k2*r2+k1*(z+d))./(k2*rho)).^2;

20
%

% The attenuation function less the exp(i*p) term - related to the

% error function is

% Ferf=((1+i)/4)*werf(sqrt(i*p));

%

% Develop the terms for the field expressions

%

t1=w*u0*il*cp/(4*pi*k2);

t11=-w*u0*il*sp/(4*pi*k2);

t12=il*sp/(4*pi);

t13=il*cp/(4*pi);

t2=exp(i*k2*r1);

t32=(i*k2./r1)-(1./r1.^2)-i./(k2*r1.^3);

t3=(2./r1.^2)+2*i./(k2*r1.^3);

t31=(i*k2./r1)-(1./r1.^2);

t4=(cd.^2).*((i*k2./r1)-(3./r1.^2)-(3*i)./(k2*r1.^3));

t41=(cd).*((i*k2./r1)-(3./r1.^2)-(3*i)./(k2*r1.^3));

t5=exp(i*k2*r2);

t62=(i*k2./r2)-(1./r2.^2)-i./(k2*r2.^3);

t6=(2./r2.^2)+2*i./(k2*r2.^3);

t61=(i*k2./r2)-(1./r2.^2);

t7=(cr.^2).*((i*k2./r2)-(3./r2.^2)-(3*i)./(k2*r2.^3));

t71=(cr).*((i*k2./r2)-(3./r2.^2)-(3*i)./(k2*r2.^3));

t72=((1./r2.^2)+(3*i)./(k2*r2.^3)-3./((k2^2)*r2.^4));

t73=(cr.^2).*((i*k2./r2)-(6./r2.^2)-(15*i)./(k2*r2.^3));

t8=((k2^3)/k1)*((pi./(k2*r2)).^.5).*sr.*Ferf;

t81=((k2^3)/k1)*((pi./(k2*r2)).^.5).*Ferf;

t82=(2*t5*(k2^3)/k1).*((pi./(k2*r2)).^.5).*Ferf;

t83=((pi./(k2*r2)).^.5).*Ferf;

%

% Calculate the fields

%

ez=t1*(-t2.*sdo.*t41+t5.*sro.*t71-2*k21*t5.*(sro.*t61-t81));

er=t1*(t2.*(t3+t4)-t5.*(t6+t7-2*k21*t61+2*(k21^2)*(t62-t8)));

ep=t11*(t2.*t32-t5.*t62-t5.*(-2*k21*cr.*t61+2*(k21^2)*(t6+t7)...

+2*i*k21^3*k2p*(sr.^2).*t83));

hr=t12*(t2.*cd.*t31-t5.*cr.*t61+2*k21*t5.*(t6+i*k21*k2p*(sr.^2).*t83+t7));

hp=t13*(t2.*cd.*t31-t5.*cr.*t61+2*k21*t5.*(t62-k21*(k2^2)*(sr).*t83));

hz=t12*(t2.*sdo.*t31-t5.*sro.*t61+2*sro.*t5.*(k21*t71(k21^2)*(t72+t73)));

21
Plot m-File for Fields

%

% This m-file plots the fields over a conductive flat earth produced by an ideal

% dipole placed a distance d above the earth. It compares the results from

% a vertical and horizontal dipole.

%

%

% Establish the problem conditions

%

%

% EPSREL- Relative dielectric constant; SIGMA- Earth conductivity (S/m)

%

EPSREL=80;SIGMA=4;

global EPSREL SIGMA

%EPSREL=12;SIGMA=.4;

%EPSREL=8;SIGMA=.04;

%EPSREL=80;SIGMA=.004;

%EPSREL=2;SIGMA=.000;

%

% Location of dipole (m)

%

d=2;

%

% Location of observer, rho (m); phi (radians); z (m) ---> an array

%

rho=1000;

phi=pi/2;

z=.1*[1:1:30];

%

% f- frequency in GHz

%

f=1.3;

%

% Field normalization factor - one Watt radiated

%

[w,c,eps0,u0,zo,k2,k1,kappap,lo]=cnstdg(f);

Fn=sqrt(12*pi/((k2^2)*zo));

%

% Horizontal dipole fields

%

[ez,ep,er,hz,hp,hr]=dipgh(f,d,phi,rho,z);

plot(Fn*abs(ep),z,'-b')

hold

22
%

% Vertical dipole fields

%

[ezv,erv,hpv]=dipg(f,d,rho,z);

plot(Fn*abs(ezv),z,'-.r')

ts=Fn*abs(erv(1));

title('Vertical and Horizontal Ideal Dipole fields over Ground')

text(ts,z(19),['SIGMA = ',num2str(SIGMA),' S/m EPSREL=

',num2str(EPSREL)])

text(ts,z(18),['Dipole height = ',num2str(d),' meters; Frequency =

',num2str(f),' GHz'])

text(ts,z(17),['Range = ',num2str(rho),' meters'])

ylabel('Observer height - meters')

xlabel('Normalized field magnitude - V/m')

legend('Horizontal E-field, horizontal dipole','z-directed E-field,

vertical dipole')

hold

Fresnel Reflection Coefficients and Plots

function [rcomv,rcomh]=Fresnel1(f,psi)

%

% This routine calculates the Fresnel reflection coefficients for

% vertical and horizontal incident fields

%

% after Collin, "Antenna and Radiowave Propagation", page 345

%

% f-frequency in GHz; psi- array of incident angles

%

% Code returns the arrays rcomv and rcomh, the vertical and horizontal

% complex reflection coefficients, respectively.

%

global EPSREL SIGMA

w=2*pi*f*1e9;

cv=2.99792458e8;

epso=(1/(4*pi))*1e7*(1/cv^2);

epsr=(EPSREL-j*SIGMA./(w*epso));

rcomv=(epsr*sin(psi)-sqrt(epsr-cos(psi).^2))./(epsr*sin(psi)+sqrt(epsr-cos(psi).^2));

rcomh=(sin(psi)-sqrt(epsr-cos(psi).^2))./(sin(psi)+sqrt(epsr-cos(psi).^2));

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

%

23
% This m-file plots the reflection coefficient for vertical and horizontal

% fields over a homogeneous ground plane

%

%

% EPSREL- Relative dielectric constant; SIGMA- Earth conductivity (S/m)

%

EPSREL=80;SIGMA=4;

f=1.3;

global EPSREL SIGMA

psi=[pi/1000:pi/1000:pi/2];

[rcomv,rcomh]=Fresnel1(f,psi);

EPSREL=12;SIGMA=.4;

[rcomv1,rcomh1]=Fresnel1(f,psi);

EPSREL=8;SIGMA=.04;

[rcomv2,rcomh2]=Fresnel1(f,psi);

EPSREL=80;SIGMA=.004;

[rcomv3,rcomh3]=Fresnel1(f,psi);

EPSREL=2;SIGMA=.000;

[rcomv4,rcomh4]=Fresnel1(f,psi);

subplot(2,2,1),plot((180/pi)*psi,abs(rcomv),'k')

hold

axis([0,90,0,1])

xlabel('Incidence angle (degrees)')

ylabel('Reflection Coefficient')

%text(5,.8,['SIGMA = ',num2str(SIGMA),' S/m EPSREL= ',num2str(EPSREL)])

subplot(2,2,1),plot((180/pi)*psi,abs(rcomv1),'r')

subplot(2,2,1),plot((180/pi)*psi,abs(rcomv2),'b')

subplot(2,2,1),plot((180/pi)*psi,abs(rcomv3),'g')

subplot(2,2,1),plot((180/pi)*psi,abs(rcomv4),'c')

hold

subplot(2,2,2),plot((180/pi)*psi,(180/pi)*angle(rcomv),'k')

hold

axis([0,90,-190,5])

xlabel('Incidence angle (degrees)')

ylabel('Phase')

subplot(2,2,2),plot((180/pi)*psi,(180/pi)*angle(rcomv1),'r')

subplot(2,2,2),plot((180/pi)*psi,(180/pi)*angle(rcomv2),'b')

subplot(2,2,2),plot((180/pi)*psi,(180/pi)*angle(rcomv3),'g')

subplot(2,2,2),plot((180/pi)*psi,-(180/pi)*angle(rcomv4),'c')

hold

subplot(2,2,3),plot((180/pi)*psi,abs(rcomh),'k')

24
hold

axis([0,90,.5,1])

xlabel('Incidence angle (degrees)')

ylabel('Reflection Coefficient')

subplot(2,2,3),plot((180/pi)*psi,abs(rcomh1),'r')

subplot(2,2,3),plot((180/pi)*psi,abs(rcomh2),'b')

subplot(2,2,3),plot((180/pi)*psi,abs(rcomh3),'g')

subplot(2,2,3),plot((180/pi)*psi,abs(rcomh4),'c')

hold

subplot(2,2,4),plot((180/pi)*psi,(180/pi)*angle(rcomh),'k')

hold

axis([0,90,170,200])

xlabel('Incidence angle (degrees)')

ylabel('Phase')

subplot(2,2,4),plot((180/pi)*psi,(180/pi)*angle(rcomh1),'r')

subplot(2,2,4),plot((180/pi)*psi,(180/pi)*angle(rcomh2),'b')

subplot(2,2,4),plot((180/pi)*psi,(180/pi)*angle(rcomh3),'g')

subplot(2,2,4),plot((180/pi)*psi,(180/pi)*angle(rcomh4),'c')

hold

25
26
27

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NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. Z39-18

298-102

Propagation of Electromagnetic Fields Over Flat Earth

February 2001 Summary, Oct 99 to Sept 00

This report looks at the interaction of radiated electromagnetic fields with earth ground in military or lawenforcement applications of high-power microwave (HPM) systems. For such systems to be effective, the

microwave power density on target must be maximized. The destructive and constructive scattering of the 

fields as they propagate to the target will determine the power density at the target for a given source. The 

question of field polarization arises in designing an antenna for an HPM system. Should the transmitting 

antenna produce vertically, horizontally, or circularly polarized fields? Which polarization maximizes the 

power density on target? This report provides a partial answer to these questions. The problems of calculating 

the reflection of uniform plane wave fields from a homogeneous boundary and calculating the fields from 

a finite source local to a perfectly conducting boundary are relatively straightforward. However, when the 

source is local to a general homogeneous plane boundary, the solution cannot be expressed in closed form. An 

approximation usually of the form of an asymptotic expansion results. Calculations of the fields are provided 

for various source and target locations for the frequencies of interest. The conclusion is drawn that the resultant 

vertical field from an appropriately oriented source antenna located near and above the ground can be 

significantly larger than a horizontally polarized field radiated from the same location at a 1.3 GHz frequency 

at observer locations near and above the ground.

Antenna, ground interaction 

Unclassified

ARL-TR-2352

0NE6YY

622705.H94

AH94

62705A

ARL PR:

AMS code:

DA PR:

PE:

Approved for public release; distribution

unlimited.

Unclassified Unclassified

2800 Powder Mill Road

Adelphi, MD 20783-1197

U.S. Army Research Laboratory

Attn: AMSRL-SE-DS jmiletta@arl.army.mil

U.S. Army Research Laboratory

Joseph R. Miletta

email:

2800 Powder Mill Road

Adelphi, MD 20783-1197

33

UL

29