zotero-db/storage/QLPUR8Y3/.zotero-ft-cache

Propagation of Electromagnetic Fields Over Flat Earth

ARL-TR-2352

Joseph R. Miletta

February 2001

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.

Army Research Laboratory
Adelphi, MD 20783-1197

ARL-TR-2352

February 2001

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

Figures

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

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

Tables

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
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 θd

d θ
0

r1
r0 r2
Ψ

z′

Region 2 µ0, ε0, σ = 0

Region 1

ρ

µ0, ε2 = εr ε0,σ ≠ 0

y

φ

x

Looking down

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)

=

eik2r1 ρ

−

I 2π



2
−eik2r2 

r1
k23 k1

− + ik2 1

r1

r121

eik2 r2 2

ρ r2

π k2 r2

2 e−iP F (P )

ik2 r2

−

1 r22

 ,



(2)

Eρ(ρ, z)

=

− ωµ0I 2πk2

 

eik2r1 ρ

2

r1

+ eik2r2 ρ

2

r2

−

k2 k1

eik2

r2

z−d r1

ik2 r1

−

3 r12

−

3i k2 r13

z+d r2

ik2 r2

−

3 r22

−

3i k2 r23

ρ r2

ik2 r2

−

1 r22

−

k23 k1

π k2 r2

1
2 e−iP F (P )

  , and

(3)

Ez(ρ, z)

=

ωµ0I 2πk2

  

eik2 r1 2

ik2 r1

−

1 r12

−

i k2 r13

−

z−d 2 r1

+ eik2r2
2

ik2 r2

−

1 r22

−

i k2 r23

−

z+d r2

−eik2

r2

k23 k1

ρ r2

1

π k2 r2

2 e−iP F (P )

ik2 r1

−

3 r12

−

3i k2 r13

2

ik2 r2

−

3 r22

−

3i k2 r23

   ,

(4)

where

1
r1 = ρ2 + (z − d)2 2 ,

1

r2 = ρ2 + (z + d)2 2 ,

(5)

P

=

k23r2 2k12

k2r2 + k1 (z + d)

2
,

k2ρ

(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 eiP w

√ iP

2

,

(8)

we have

F (P ) =

i eiP w

√ iP

,

2

(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 wit√h complex argument (Chase) and is provided in the appendix. Note that i can

be

written

as

(i+1) 2

ing the exponential

and that F (P ) always appears with e−iP , thus cancelterm 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 

+

2k2 k1

eik2r2





ik2 r1

−

1 r12

− eik2r2

ik2 r2

−

1 r22

−

i k2 r23

−

k23 k1

r2 ρ

π k2 r2

z+d r2

ik2 r2

−r122

1



2 e−iP F (P )

 ,



Eρ(ρ, φ, z)

=

ωµ0I 4πk2

cos

φ


 

eik2r1 −eik2

r2

2
r12  

+
2 r22
−
+

2i k2 r13

+

z−d 2 r1

ik2 r1

−

3 r12

−

3i k2 r13

+

2i k2 r23

+

z+d 2 r2

ik2 r2

−

3 r22

−

3i k2 r23

2k2 z+d
k1  r2

ik2 r2

−

1 r22

2k22 k12



ik2 r2
−

−

1 r22

k23

k1

−

i k2 r23

r2

π

ρ

k2 r2

1
2 e−iP F (P

)

 

  

 



, and

Ez(ρ, φ, z)

=

ωµ0I 4πk2

 −eik2r1

ρ r1

cos

φ



+eik2r2

ρ r2

−

2k2 k1

eik2

r2

z−d r1

ik2 r1

−

3 r12

−

3i k2 r13



z+d r2

ik2 r2

−

3 r22

−

3i k2 r23

ρ r2

ik2 r2

−

1 r22

−

k23 k1

π k2 r2

1
2 e−iP F (P )

 .

(10) (11) (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 

+

2k2 k1

eik2r2



ik2 r1

−

1 r12

− eik2r2

2 r22
+

+z+kd22ri232
r2

+

ik22 k1ρ

ik2 r2

−

3 r22

z+d ik2

r2

r2

r22

π

ρ2 k2r2

−

3i k2 r23

−
1

1 r22

2 e−iP F (P )

 

 

,

(13)

Hz(ρ, φ, z)

=

I 4π

sin φ 

eik2r1

+2

ρ r2



ρ r1

irk12−

1 r12



k2 k1

eik2r2  −

− eik2r2

ρ r2

ik2 r2

−

1 r22

z+d
r2

ik2 r2

−

3 r22

−

3i k2 r23

k2 k1

2 

1 r22

+

z+d

r2

3i k22 r23

−
ik2

r2

3 k22 r24

+

−

6 r22

−

15i k2 r23

 

  



, and

(14)



Eφ(ρ, φ, z)

=

− ωµ0I 4πk2

sin

φ

 

eik2r1 −eik2r2

ik2
r1 


+

1 r12

−

i k2 r13

− eik2r2

−

2k2 k1

z+d
 r2

ik2 r2

−

1 r22

+

2k22 k12



2 r22
+

+z+kd22ri232
r2

ik2 r2

+

2ik24 k13 ρ

r2 2 π

ρ

k2 r2

ik2 r2

−

1 r22

−

−

3 r22

−

3i k2 r23

1

2 e−iP F (P )

i

k2r23

 

 

 

.

(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 = k22ζ0 , 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

1
by P and the ﬁeld results by P 2 . The outward component of the complex

Poynting vector over a closed surface is

1/2 E × H ∗ · dS = −Pcomplex ,

(17)

S

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 3

main electric ﬁeld

components from a

vertical (Ez) and from a 2.5

horizontal (Eφ)

broadside ideal dipole

over a perfectly

2

Observer height (m)

conducting ground. Ez

is a constructively

interfered-with ﬁeld,

1.5

while Eφ is a

destructively

1

interfered-with ﬁeld.

0.5

Dipole height = 2 m, frequency = 1.3 GHz, range = 1000 m
Horizontal E-field, horizontal dipole z-directed E-field, vertical dipole

0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Normalized field magnitude (V/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)

εr

−

i

σ ωε0

sin ψ −

εr

−

i

σ ωε0

− cos2 ψ

ρcomplex =

,

(23)

εr

−

i

σ ωε0

sin ψ +

εr

−

i

σ ωε0

− cos2 ψ

and for horizontal polarization, it is

sin ψ −

εr

−

i

σ ωε0

− cos2 ψ

ρcomplex =

,

(24)

sin ψ +

εr

−

i

σ ωε0

− cos2 ψ

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

(a) 1

coefﬁcients for

Reflection coefficient

vertically and

0.8

horizontally polarized plane waves of 1.3-GHz 0.6

frequency incident on a 0.4

ﬂat ground for ﬁve

earth-parameter cases

0.2

listed in table 1. Note

that case 5 provides a

0 0

true Brewster angle

(since conductivity is

zero) and case 4 is very (b) 1

near to producing a true 0.9 Brewster angle at

1.3-GHz frequency:

0.8

(a) vertical polarization

and (b) horizontal

0.7

polarization.

0.6

Reflection coefficient

0.5 0

Case 1 2

4

3

5

20

40

60

80

Incidence angle (°)

Case 1

42

3 5

20

40

60

80

Incidence angle (°)

Phase

Phase

0
–50
–100
–150
0
200 195 190 185 180 175 170
0

Case 1

2

5 4
3

20

40

60

80

Incidence angle (°)

4,5 3 Case 1 2

20

40

60

80

Incidence angle (°)

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 = Eρ2 + Ez2 ,

(25)

for the vertical dipole and

Ehtotal = Eφ2 + Eρ2 + Ez2 ,

(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

(a) 1

reﬂection coefﬁcient for 0.9

a vertically polarized

SIGMA = 4 S/m, EPSREL= 80 0.8

plane wave ﬁeld

0.7

Fresnel Eq. Complete Eq.

Reflection coefficient

compared to “reﬂection 0.6

coefﬁcient” as

0.5

calculated by complete 0.4

formulation for a

0.3

vertical dipole over

0.2

homogeneous earth as 0.1

observed at 1-m height 1000 m down range at a

0 0 10 20 30 40 50 60 70 80 90 Incidence angle (°)

frequency of 1.3 GHz: (a) sea water, (b) wet earth, (c) dry earth, (d) lake water, and (e) dry sand.

(c) 1 0.9 0.8 0.7 0.6

SIGMA = 0.04 S/m, EPSREL= 8

Fresnel Eq. Complete Eq.

Reflection coefficient

0.5

0.4

0.3

0.2

0.1

0 0 10 20 30 40 50 60 70 80 90 Incidence angle (°)

(b) 1 0.9

Fresnel Eq. Complete Eq.

0.8 SIGMA = 0.4 S/m, EPSREL= 12

Reflection coefficient

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0 10 20 30 40 50 60 70 80 90 Incidence angle (°)

(d) 1 0.9 0.8 SIGMA = 0.004 S/m, EPSREL= 80

Fresnel Eq. Complete Eq.

Reflection coefficient

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0 10 20 30 40 50 60 70 80 90 Incidence angle (°)

(e) 1 0.9 0.8

SIGMA = 0 S/m, EPSREL= 2

Fresnel Eq. Complete Eq.

Reflection coefficient

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0 10 20 30 40 50 60 70 80 90 Incidence angle (°)

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

Reflection coefficient Reflection coefficient

Reflection coefficient Reflection coefficient

Figure 5. Fresnel

(a)

1

reﬂection coefﬁcient for

0.9

a horizontally polarized

0.8

plane wave ﬁeld

0.7

compared to “reﬂection 0.6

SIGMA = 4 S/m, EPSREL = 80

Fresnel eq. Complete eq.

coefﬁcient” as

0.5

calculated by complete 0.4

formulation for a

0.3

horizontal dipole over

0.2

homogeneous earth as 0.1

observed at 1-m height, broadside 1000 m down

0 0 10 20 30 40 50 60 70 80 90 Incidence angle (°)

range at a frequency of

1.3 GHz: (a) sea water, (c) 1

(b) wet earth, (c) dry

0.9

earth, (d) lake water,

0.8

Fresnel eq. Complete eq.

and (e) dry sand.

0.7

0.6

0.5 SIGMA = 0.04 S/m, EPSREL = 8
0.4

0.3

0.2

0.1

0 0 10 20 30 40 50 60 70 80 90 Incidence angle (°)

(b) 1 0.9 0.8 0.7 0.6 SIGMA = 0.4 S/m, EPSREL = 12 0.5 0.4 0.3 0.2 0.1 0 0 10 20 30 40 50 60 Incidence angle (°)

Fresnel eq. Complete eq. 70 80 90

(d) 1 0.9

Fresnel eq. Complete eq.

0.8 SIGMA = 0.004 S/m, EPSREL = 80 0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0 10 20 30 40 50 60 70 80 90 Incidence angle (°)

Reflection coefficient

(e) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

SIGMA = 0 S/m, EPSREL = 2

Fresnel eq. Complete eq.

10 20 30 40 50 60 70 80 90 Incidence angle (°)

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.

Observer height (m) Observer height (m)

Figure 6. Comparison of (a)

3

principal ﬁelds from an

SIGMA = 4 S/m, EPSREL = 80

dipole height = 2 m, frequency = 1.3 GHz

ideal dipole oriented

2.5 range = 1000 m

perpendicular and

horizontal to a

2

homogeneous ﬂat earth. 1.5

In each case, dipole is

placed 2 m above

1

ground plane and observer or target is

0.5

Horizontal E-field, horizontal dipole

z-directed E-field, vertical dipole

1000 m down range: (a) sea water, (b) wet

0

0

1

2

3

4

5

6

7

Normalized field magnitude (V/m)

× 10–3

earth, (c) dry earth, (d) lake water, and (e) dry sand.

(c) 3
2.5

SIGMA = 0.04 S/m, EPSREL= 8 dipole height = 2 m, frequency = 1.3 GHz range = 1000 m

(b) 3 SIGMA = 0.4 S/m, EPSREL= 12
dipole height = 2 m, frequency = 1.3 GHz 2.5 range = 1000 m

2

1.5

1

0.5 0 0

Horizontal E-field, horizontal dipole z-directed E-field, vertical dipole

1

2

3

4

5

6

7

Normalized field magnitude (V/m)

× 10–3

(d) 3
2.5

SIGMA = 0.004 S/m, EPSREL= 80 dipole height = 2 m, frequency = 1.3 GHz range = 1000 m

Observer height (m) Observer height (m)

2

2

1.5

1.5

1

1

0.5
0 0

Horizontal E-field, horizontal dipole z-directed E-field, vertical dipole

1

2

3

4

5

6

7

Normalized field magnitude (V/m)

× 10–3

0.5
0 0

Horizontal E-field, horizontal dipole z-directed E-field, vertical dipole

1

2

3

4

5

6

7

Normalized field magnitude (V/m)

× 10–3

(e) 3 2.5

SIGMA = 0 S/m, EPSREL= 0.2 dipole height = 2 m, frequency = 1.3 GHz range = 1000 m

Observer height (m)

2

1.5

1

0.5
0 0

Horizontal E-field, horizontal dipole z-directed E-field, vertical dipole

1

2

3

4

5

Normalized field magnitude (V/m)

6 × 10–3

11

Observer height (m) Observer height (m)

Figure 7. Comparison of (a)

3 SIGMA = 4 S/m, EPSREL = 80

principal ﬁelds from an

dipole height = 2 m, frequency = 1.3 GHz

ideal dipole oriented

2.5 range = 100 m

perpendicular and

2

horizontal to a homogeneous ﬂat earth. 1.5

In each case, dipole is

1

placed 2 m above

ground plane and observer or target is 100 m down range: (a) sea water, (b) wet

0.5

Horizontal E-field, horizontal dipole

z-directed E-field, vertical dipole

0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Normalized field magnitude (V/m)

earth, (c) dry earth, (d) lake water, and (e) dry sand.

(c) 3 SIGMA = 0.04 S/m, EPSREL = 8 dipole height = 2 m, frequency = 1.3 GHz
2.5 range = 100 m

(b) 3 SIGMA = 0.4 S/m, EPSREL = 12 dipole height = 2 m, frequency = 1.3 GHz
2.5 range = 100 m

2

1.5

1

0.5 00

Horizontal E-field, horizontal dipole z-directed E-field, vertical dipole
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Normalized field magnitude (V/m)

(d) 3 SIGMA = 0.004 S/m, EPSREL = 80 dipole height = 2 m, frequency = 1.3 GHz
2.5 range = 100 m

Observer height (m) Observer height (m)

2

2

1.5

1.5

1

1

0.5

Horizontal E-field, horizontal dipole

z-directed E-field, vertical dipole

0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Normalized field magnitude (V/m)

0.5

Horizontal E-field, horizontal dipole

z-directed E-field, vertical dipole

0 0

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Normalized field magnitude (V/m)

(e) 3 SIGMA = 0 S/m, EPSREL = 0.2 dipole height = 2 m, frequency = 1.3 GHz
2.5 range = 100 m

Observer height (m)

2

1.5

1

0.5
0 0

Horizontal E-field, horizontal dipole z-directed E-field, vertical dipole
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Normalized field magnitude (V/m)

12

Figure 8. Effect of ground reﬂection on primary ﬁeld components near

(a) 30 SIGMA = 0.4 S/m, EPSREL = 12
dipole height = 2 m, frequency = 1.3 GHz
range = 100 m 25

ground for typical earth

parameters. Phase

Observer height (m)

difference between

20

incident and reﬂected

waves results in

development of ﬁeld

15

lobes that are more

pronounced as radiating antenna is approached: (a) 100 m

10

Horizontal E-field, horizontal dipole

z-directed E-field, vertical dipole

down range and (b)

5

1000 m down range.

0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Normalized field magnitude (V/m)

(b) 30 25

SIGMA = 0.4 S/m, EPSREL = 12 dipole height = 2 m, frequency = 1.3 GHz range = 1000 m

Observer height (m)

20

15

10

5

Horizontal E-field, horizontal dipole

z-directed E-field, vertical dipole

0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Normalized field magnitude (V/m)

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

WERF Function

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.

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 M=2*N M2=2*M k= -M+1:1:M-1]' L=sqrt(N/sqrt(2)) theta=k*pi/M t=L*tan(theta/2) f=exp(-t.^2).*(L^2+t.^2) f= 0 f] a=real(fft(fftshift(f)))/M2 a=flipud(a(2:N+1)) nz=imag(z) <= 0 z(nz)=conj(z(nz)) Z=(L+i*z)./(L-i*z) p=polyval(a,Z) w=2*p./(L-i*z).^2+(1/sqrt(pi))./(L-i*z) w(nz)=2*exp(-(conj(z(nz)).^2)) - conj(w(nz)) %if all(imag(z)==0), w=real(w) end

% Default value for N % M2=no. of sampling points % Optimal choice of L % Variables thetaand t % function to be transformed % Coefficients of transform % Reorder coefficients % Find im(z) <=0 % Use conj for above % Polynomial evaluation % Evaluate w(z) % Handle im(z) <= 0 % 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))
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
%

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% 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')

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
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Mgmt ATTN AMSRL-CI-AP Techl Pub (2 copies) ATTN AMSRL-CI-LL Techl Lib (2 copies) ATTN AMSRL-SE-DP M Litz ATTN AMSRL-SE-DP R A Kehs (3 copies) ATTN AMSRL-SE-DS J Miletta (10 copies) ATTN AMSRL-SE-DS J Tatum (5 copies) ATTN AMSRL-SE-DS M Berry ATTN AMSRL-SE-DS W O Coburn Adelphi MD 20783-1197

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REPORT DOCUMENTATION PAGE

Form Approved OMB No. 0704-0188

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503.

1. AGENCY USE ONLY (Leave blank)

2. REPORT DATE
February 2001

3. REPORT TYPE AND DATES COVERED
Summary, Oct 99 to Sept 00

4. TITLE AND SUBTITLE Propagation of Electromagnetic Fields Over Flat Earth 5. FUNDING NUMBERS

6. AUTHOR(S) Joseph R. Miletta

DA PR: AH94 PE: 62705A

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
U.S. Army Research Laboratory

Attn: AMSRL-SE-DS

email: jmiletta@arl.army.mil

2800 Powder Mill Road

Adelphi, MD 20783-1197

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)
U.S. Army Research Laboratory
2800 Powder Mill Road Adelphi, MD 20783-1197

8. PERFORMING ORGANIZATION REPORT NUMBER
ARL-TR-2352
10. SPONSORING/MONITORING AGENCY REPORT NUMBER

11. SUPPLEMENTARY NOTES
ARL PR: 0NE6YY AMS code: 622705.H94
12a. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution 12b. DISTRIBUTION CODE unlimited.

13. ABSTRACT (Maximum 200 words)
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.

14. SUBJECT TERMS Antenna, ground interaction

15. NUMBER OF PAGES
33
16. PRICE CODE

17. SECURITY CLASSIFICATION OF REPORT
Unclassified
NSN 7540-01-280-5500

18. SECURITY CLASSIFICATION OF THIS PAGE
Unclassified

19. SECURITY CLASSIFICATION OF ABSTRACT
Unclassified

20. LIMITATION OF ABSTRACT
UL
Standard Form 298 (Rev. 2-89) Prescribed by ANSI Std. Z39-18 298-102
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