A Fast Algorithm for

Computing Scattered Fields

Using Physical Optics

Equivalent Approximation in

Half-Space

ARL-TR-2132 July 2000

T. Raju Damarla 

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

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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-2132 July 2000

Army Research Laboratory

Adelphi, MD 20783-1197

A Fast Algorithm for

Computing Scattered Fields

Using Physical Optics

Equivalent Approximation in

Half-Space

T. Raju Damarla 

Sensors and Electron Devices Directorate

Approved for public release; distribution unlimited.
In this report, a fast algorithm for computing scattered fields with the

use of physical optics (PO) equivalent approximation in half-space is

presented. The theoretical basis for the algorithm and the derivation

of formulas used in the algorithm is presented. The algorithm is used

to compute the radar cross sections (RCSs) of several objects. The RCS

of the objects computed by the algorithm is compared with those that

were computed with the method of moments (MOM). The results

presented are found to be accurate when the target dimensions are

greater than or equal to 2λ, where λ denotes the wavelength. It is

concluded that the PO algorithm presented in this report can be used

for majority of applications as it captured all the salient features of the

Abstract

targets.

ii
iii

Contents

1. Introduction ....................................................................................................................................... 1

2. Scattering of Radiation by an Object in Free Space ................................................................... 2

2.1 Physical Optics Equivalent ............................................................................................................. 3

2.2 Half-Space Approximations ............................................................................................................ 6

2.3 Implementation of PO Approximations in Half-Space for Computation of Scattered Fields ......... 7

3. Validation of PO Code ................................................................................................................... 12

4. Conclusions ...................................................................................................................................... 23

Acknowledgments ................................................................................................................................. 24

Distribution ............................................................................................................................................ 25

Report Documentation Page ................................................................................................................ 27

Figures

1. Coordinate system for radiation by an object............................................................................... 2

2. Field geometry for derivation of currents on a surface of an object.......................................... 3

3. Field geometry for physical optics approximation ...................................................................... 4

4. Geometry used for layered medium .............................................................................................. 7

5. Ray dynamics .................................................................................................................................... 8

6. A 1- × 1-m2 plate ............................................................................................................................. 13

7. RCS VV characteristics of a plate in a half-space ....................................................................... 13

8. RCS HH characteristics of a plate in a half-space ...................................................................... 14

9. Cylindrical target used................................................................................................................... 14

10. RCS VV of a cylinder, θ = 60°, φ = 0° (broadside) ...................................................................... 15

11. RCS HH of a cylinder, θ = 60°, φ = 0° (broadside)...................................................................... 15

12. RCS VV of a cylinder, θ = 60°, φ = 45° (off axis) ......................................................................... 16

13. RCS VV of a cylinder, θ = 60°, φ = 45° (off axis) ......................................................................... 16

14. RCS VV of a cylinder, θ = 60°, φ = –90° (axial) ............................................................................ 17

15. RCS HH of a cylinder, θ = 60°, φ = –90° (axial) ........................................................................... 17

16. RCS VV of a cylinder above Eglin soil, θ = 60°, φ = 0° (broadside) ......................................... 18

17. RCS HH of a cylinder above Eglin soil, θ = 60°, φ = 0° (broadside) ........................................ 18

18. RCS VV of a cylinder above Eglin soil, θ = 60°, φ = 45° (off axis) ............................................ 19

19. RCS HH of a cylinder above Eglin soil, θ = 60°, φ = 45° (off axis) ........................................... 19

20. RCS VV of a cylinder above Eglin soil, θ = 60°, φ = –90° (axial) .............................................. 20

21. RCS HH of a cylinder above Eglin soil, θ = 60°, φ = –90° (axial) ............................................. 20

22. Left column MOM, right column PO........................................................................................... 21

23. UXO used for measurements and simulations........................................................................... 22

24. Measured versus simulated SAR images using PO................................................................... 22
1

1. Introduction

In general, scattered fields from complex objects are computed in the

frequency domain with the use of the method of moments (MOM).

Although this method computes the scattered fields accurately, the

computational complexity is shown1 to be O(n3), where n is the number

of elements for which currents need to be computed. Because of the

exponential nature of the computational complexity, as n increases, the

use of MOM for computing scattered fields becomes less and less attractive. Other techniques for computing frequency domain scattered fields

exist, namely, the fast multipole method (FMM), multilevel FMM, etc. The

multilevel FMM techniques have the complexity O(n log n).1 Within the

realm of MOM, other techniques exist that take advantage of target

geometry, such as exploiting body of revolution (BOR) symmetry to

reduce the size of matrixes, which must be inverted. In spite of great

advances in computational electromagnetics, computing the scattered

fields from an arbitrary target can be prohibitive, especially as the target

is electrically large. However, at higher frequencies for targets with

dimensions greater than 2λ, where λ denotes the wavelength, computing

scattered fields by physical optics (PO) approximations yield relatively

accurate results. Some of the commercially available high-frequency

modeling packages, such as XPATCH, model targets in free space with or

without conducting ground planes. The PO code that was developed

computes the scattered fields in a lossy and dispersive half-space via the

layered medium dyadic Green’s function specialized for a half-space. This

capability is of interest for simulating targets in real soil. The PO code

allows one to compute the radar cross section (RCS) and scattered fields

of an arbitrary target that is underground, above ground, or partially

buried.

Section 2 presents the basic formulation of the PO approximations and

the Green’s functions used. Section 3 presents some of the RCS characteristics of various targets, namely, a sphere, a vertical plate, a mine, and an

unexploded ordnance (UXO), that were computed with both MOM and

PO for comparison. Section 3 also presents some of the actual synthetic

aperture radar (SAR) images of UXOs along with the images generated

by the simulated data by PO. These images show that most of the salient

features of the targets are captured, which in turn, help develop automatic target detection (ATD) and automatic target recognition (ATR)

algorithms.

1N. Geng, A. Sullivan, and L. Carin, “Fast multipole method for scattering from 3-D PEC targets situated

in a half-space environment,” Microwave Opt. Tech. Let. 21, No. 6 (June 1999), 399–405.
2

2. Scattering of Radiation by an Object in Free Space

One assumes that the observations are made in the far field; that is,

(βr >> 1), where β = ω με and μ and ε represent the permeability and

permittivity of free space, respectively; β is the wave number; and r

denotes the distance from the target to the observation point. Figure 1

shows the coordinate system used for computing scattered fields in the

far field. I approximate the distance R from any point on the object to the

observation point by

R = r – r′ cos ψ for phase variations ,

r for amplitude variations , (1)

where ψ is the angle between the vectors r and r ́ (vectors are denoted by

bold letters). Then the electric and magnetic fields radiated by the object

in the far field are given by

EA = –jωA (θ and φ components only) ,

HF = –jωF (θ and φ components only) , (2)

where

 A = μ

4π Js

e –jβR

R ds ∼ μe–jβr

4πr

N

S

, and (3)

N = Jsexp jβr′ cos ψ ds

S , (4)

z

y

x

φ

θ

ψ r

r ́

(x ́, y ́, z ́) 

R

Figure 1. Coordinate

system for radiation

by an object.
3

where Js and Ms denote the electric and magnetic current densities induced on the object. Clearly, scattered field E can be computed if Js is

known. To compute Js

 in an object, one assumes a known plane wave is

impinging on the object. PO approximation gives a relation between the

induced magnetic field and the current density Js

. The following subsection presents the relevant concepts.

2.1 Physical Optics Equivalent

For the sake of completeness, I present in this subsection a brief description of PO approximation theory. Detailed explanation and derivation of

relevant formulation can be found in literature by Balanis.2

Let us consider an unbounded medium with constituting parameters ε1

and μ1. Let us also assume that a current source J1 radiating E1 and a

magnetic current source M1 radiating H1 are everywhere in the medium.

Moreover, consider an imaginary region V1 enclosed by a surface S1 in

this medium as shown in figure 2. Assume that the surface S1 is replaced

by another medium with parameters ε2 and μ2. Let us also assume that

the sources J1 and M1 are allowed to radiate in the presence of the new

medium in V1. The total field outside the region V1, produced by J1 and

M1, is E and H and inside V1 is Et

 and Ht.

The total field outside the region V1 is equal to the original field (E1 and

H1) in the absence of the obstacle plus a perturbation (scattered) field (Es

,

Hs

), introduced by the obstacle. Hence,

E = E1 + Es ,

H = H1 + Hs . (5)

In the above equation, one can compute the original fields E1 and H1 by

solving Maxwell’s equations. To compute Es

 and Hs, I introduce boundary currents Jp and Mp on the surface.

These currents can be expressed in terms of the difference in fields (Es

, Hs)

that are outside the region V1 and the fields (Et

, Ht) that are inside the

region V1 as follows:

Jp = n × Hs – Ht ,

Mp = – n × Es – Et , (6)

Figure 2. Field

geometry for

derivation of currents

on a surface of an

object. J1

ε1, μ1

E = E1 + Es

H = H1 + Hs

Et

, Ht

ε2, μ2

V1

S1

M1

J1

E1, H1

ε1, μ1

V1

S1

M1

n^ n^

2C. A. Balanis, Advanced Engineering Electromagnetics, John Wiley and Sons, Inc., New York, NY (1989).
4

where “×” denotes the cross product. Since the tangential components of

E and H must be continuous across the boundary, I get

E1 tan + Es tan = Et

tan ⇒ n × E1 + Es = n × Et ,

H1 tan + Hs tan = Ht

tan ⇒ n × H1 + Hs = n × Ht ,

(7)

or

Es tan – Et

tan = –E1 tan ⇒ n × Es – Et = –n × E1 ,

Hs tan – Ht

tan = –H1 tan ⇒ n × Hs – Ht = –n × H1 .

(8)

Substituting equations (8) in (6), I get

Jp = – n × H1

Mp = n × E1 .

(9)

Let us now assume that the obstacle occupying the region V1 is a perfect

electric conductor (PEC) with conductivity σ = ∞. Then the fields in the

region Et

 and Ht equal zero as shown in figure 3. Over the boundary S1 of

the conductor, the total tangential components of the E field are equal to

zero, and the total tangential components of the H field are equal to the

induced current density Jp. Hence,

 Mp = –n × E – Et = –n × E = – n × E1 + Es =0 ,

or

– n × E1 = n × Es , and (10)

 Jp = n × H – Ht = n × H = n × H1 + Hs ,

or for an infinitely large object, it can be shown by image theory2 that

H1|tan = Hs

|tan. Hence, the PO for equation (10) becomes

Jp = 2n × H1 in the lit region and

Jp = 0 in the shadow region .

(11)

J1

ε1, μ1

ε1, μ1

S1 S1

–E1, –H1

ε1, μ1

V1

Et = Ht = 0

σ = ∞

V M1 1

E = E1 + Es

H = H1 + Hs

n^ n

^

Jp

Figure 3. Field

geometry for physical

optics approximation.

2C. A. Balanis, Advanced Engineering Electromagnetics, John Wiley and Sons, Inc., New York, NY (1989).
5

Equation (11) allows us to compute the current density Jp on the target

because of H1, which one assumes can be computed. Once we have

determined Jp, we can use equations (2) to (4) to compute the field radiated by the target with current density Jp. In particular, the scattered field

can be computed once N in equation (4) is determined; namely,

N = J exp jβr′ cos ψ ds

S = J exp jβr ⋅ r′ dsS , (12)

where “⋅” denotes the dot product. Note that J in equation (12) is computed by equation (11). Expanding equation (12) in terms of its x, y, and z

components of the current J produces

N = axJx + ayJy + azJz exp jβr ⋅ r′ ds

S . (13)

Using the rectangular-to-spherical component transformation, where

ax

ay

az

=

sinθ cosφ cosθ cosφ – sinφ

sinθ sinφ cosθ sinφ cosφ

cosθ – sinθ 0

ar

aθ

aφ

, (14)

we can write the θ (vertical polarization) and φ (horizontal polarization)

components of N as

Nθ = Jx cosθ cosφ + Jy cosθ sinφ – Jz sinθ exp jβr ⋅ r′ ds

S , (15)

Nφ = Jx sin φ + Jy cos φ exp –jβr ⋅ r′ ds

S . (16)

Using equations (15) and (16) in equations (2) and (3) produces expressions for scattered field in vertical and horizontal polarizations:

vertical polarization

Eθ = – jωμ

4πr

e– jβr Jx cos θ cos φ + Jy cos θ sin φ – Jz sin θ

S

× exp jβ x′ sin θ cos φ + y′ sin θ sin φ + z′ cos θ ds

(17)

horizontal polarization

Eφ = – jωμ

4πr

e–jβr –Jx sinφ + Jy cosφ S× exp jβ x′ sinθ cos φ + y′ sinθ sinφ + z′ cos θ ds (18)

Note that equations (17) and (18) correspond to the scattered fields in free

space. The following section presents the half-space approximations. We

can obtain these approximations by incorporating the half-space dyadic

Green’s functions.*

*T. Dogaru, private notes, Electrical Engineering Dept., Duke University, Durham, NC.
6

2.2 Half-Space Approximations

Consider two layers with an xy-plane with z = 0 as the boundary between

the two as shown in figure 4. The medium in layer 1 has the permittivity

ε1 and permeability μ1 while the medium in layer 2 has ε2 and μ2. Often

the medium in layer 1 is air; thus, μ1 = μ0 = 4π10–7 H/m and ε1 = ε 0 =

10–9/36π F/m. Define the following functions:

 β 1 = ω μ 1ε 1 ,

 β 2 = ω μ 2ε 2 ,

exp 1 = exp jβ 1 x′ sinθ cosφ + y′ sinθ sinφ + z′ cosθ ,

exp 2 = exp jβ 1 x′ sinθ cosφ + y′ sinθ sinφ – z′ cosθ ,

exp 3 = exp j β 1 x′ sin θ cosφ + y′ sinθ sinφ + z′ β 2

2 – β 1sinθ 2 ,

 Txx = Tyy = 2β 1cosθ

β 1cosθ + β 2

2 – β 1sinθ 2 ,

 Tzz = 2β 1cos θ

β 2

2

β 1

cos θ + β 2

2 – β 1sinθ 2

,

 Tzx =

β 2

2

β 1

cosθ – β 2

2 – β 1sinθ 2

β 2

2

β 1

cosθ + β 2

2 – β 1sinθ 2

+ β 1cosθ – β 2

2 – β 1sinθ 2

β 1cosθ + β 2

2 – β 1sinθ 2

cosθ cos φ

sinθ ,

 Tzy = Tzx

sin φ

cos φ ,

 Rxx = Ryy = β 1cosθ – β 2

2 – β 1sinθ 2

β 1cosθ + β 2

2 – β 1sinθ 2 ,

 Rxx = Ryy = β 1cosθ – β 2

2 – β 1sinθ 2

β 1cosθ + β 2

2 – β 1sinθ 2 ,

 Rzx = Tzx ,

 Rzy = Rzx

sin φ

cos φ .
7

The following are the expressions for Eθ and Eφ for the target in layer 1:

Eθ = –jωμ 1

4πr

e– jβ 1r Jx cos θ cos φ exp1 + Rxx exp2 – sin θRzx exp2 S

+ Jy cos θ sin φ exp1 + Ryy exp2 – sin θRzy exp2 – Jz sin θ exp1 + Rzz exp2 ds , (19)

Eφ = – jωμ 1

4πr

e–jβ 1r –Jx sinφ exp1 + Rxx exp2 S+ Jy cosφ exp1 + Ryy exp2 ds . (20)

Expressions for Eθ and Eφ for the target in layer 2 are given by

Eθ = – jωμ 1

4πr

e–jβ 1r Jx cosθ cosφ Txx – sinθ Tzx exp3 ds

S

+ Jy cosθ sinφ Tyy – sinθ Tzy exp3 – Jz sinθ Tzz exp3 ds ,

(21)

Eφ = – jωμ 1

4πr

e–jβ 1r –Jx sin φ Txx exp3 + Jy cosφ Tyy exp3 ds

S . (22)

Equations (19) to (22) can be used for computing the scattered fields by a

target.

2.3 Implementation of PO Approximations in Half-Space

for Computation of Scattered Fields

To compute the scattered fields near a target, one must first partition the

target into small triangles with each side of the triangle approximately

0.1λ, where λ denotes the wavelength. For each triangle patch, an outward normal and its centroid are computed. This normal is used to

compute the current induced by a magnetic field in the patch. The centroid becomes the reference point in the patch for computing electric and

magnetic fields. The magnetic field in a patch is then used to determine

the current with the use of PO approximation. Once the current in a patch

is computed, it is then used to compute the electrical field radiated (scattered field) by the patch. The sum total of all the fields radiated by all the

patches constitutes the scattered field by the target.

ε2, μ2

z

Source at (x ́, y ́, z ́)

Observer at (r, θ, φ)

y

x

φ

θ

r

Half-space

2

x-y plane 1

r

θ ε1, μ1

Layer

Layer

Figure 4. Geometry

used for layered

medium.
8

Now to compute the induced current caused by a magnetic field, let us

assume that an incident plane wave where the electric field has a magnitude E0 = 1 is impinging on the target. Let the direction of the incident

plane wave be given by θ and φ; then the unit-transmitted vector in

rectangular-coordinate system is given by

layer 1: tx = – [sin θ cos φ, sin θ sin φ, cos θ]

 r = r tx ,

and the reflected vector is rl

 = –[sin θ cos φ, sin θ sin φ, –cos θ], and

layer 2: t

x = – [sin θt

 cos φ, sin θt sin φ, cos θt],

where θt

 denotes the angle of refraction in medium 2 and is determined

by Snell’s law: sin θt

 = β1 sin θ/β2 .

Figure 5 shows the transmitted, reflected, and refracted rays (vectors) and

the corresponding angles.

Field computations in layer 1:

The electric fields in vertical and horizontal polarizations impinging on a

patch are given by

transmitted field

EV = E0 cosθ cosφ, cosθ sinφ, – sinθ exp –jβ 1 tx ⋅ Pi ,

EH = E0 – sin φ, cosφ, 0 exp –jβ 1 tx ⋅ Pi ,

(23)

where Pi

 = [xi, yi, zi] denotes the coordinates of the centroid of an ith

triangular patch of the target.

Transmitted vector

Reflected vector

Refracted vector

z

θt

θ θ

Layer 1

Layer 2

Figure 5. Ray

dynamics.
9

reflected field

EV

r = E0rv cosθ cosφ, cosθ sinφ, sinθ exp –jβ 1 rl ⋅ Pi ,

EH

r = E0rh – sin φ, cosφ, 0 exp –jβ 1 rl ⋅ Pi ,

 (24)

where the reflection coefficients rv and rh are given by

rv = η 2 cosθ t – η 1 cosθ

η 2 cosθ t + η 1 cosθ ,

rh = η 2 cosθ – η 1 cosθ t

η 2 cosθ + η 1 cosθ t

and

η 1 = μ 1

ε 1 intrinsic impedance of medium 1 ,

η 2 = η 1

ε 2 intrinsic impedance of medium 2 .

Then the magnetic fields can be expressed in terms of the electric fields as

HV = 1

η 1

tx × EV ,

HH = 1

η 1

tx × EH ,

HV

r = 1

η 1

rl × EV

r ,

HH

r = 1

η 1

rl × EH

r ,

(25)

where a × b represents the cross product between two vectors a and b.

Computation of currents in a triangular patch:

Having determined the magnetic fields in a triangular patch, I can now

compute the currents using the PO approximation given by equation (11)

as follows:

JV = 2 nP × HV

JH = 2 nP × HH

JV

r = 2 nP × HVr

JH

r = 2 nP × HHr ,

(26)

where nP denotes the outward normal to the triangular patch P. Note

that the normal nP and the HV should correspond to the same triangular
10

patch. Now I can compute the scattered fields using equations (19) and

(20). Note that equation (19) gives the scattered field in vertical polarization. By using JV, the current caused by the vertically polarized transmitted field in equations (19) and (20), I can obtain scattered fields in vertical

transmit, vertical receive (VV) and vertical transmit, horizontal receive

(VH) polarizations. Similarly, using JH in equations (19) and (20), I can

obtain scattered fields in HV and HH polarizations:

EVV = – jωμ 1

4πR e–jβ 1R JVx cos θ cos φ exp1 + Rxx exp2 – sin θRzx exp2 S

+ JVy cos θ sin φ exp1 + Ryy exp2 – sin θRzy exp2 – JVz sin θ exp1 + Rzz exp2 ds ,

(27)

EVH = – jωμ 1

4πR e–jβ 1R –JVx sinφ exp1 + Rxx exp2 S

+ JVy cosφ exp1 + Ryy exp2 ds , (28)

EHV = – jωμ 1

4πR e–jβ 1R JHx cosθ cos φ exp1 + Rxx exp2 – sinθRzx exp2 S

+ JHy cos θ sin φ exp1 + Ryy exp2 – sinθRzy exp2 – JHz sinθ exp1 + Rzz exp2 ds ,

(29)

EHH = – jωμ 1

4πR e–jβ 1R –JHx sin φ exp1 + Rxx exp2 S

+ JHy cosφ exp1 + Ryy exp2 ds , (30)

where JV = [J

Vx, JVy, JVz] and R is the distance between the observation

point and the target. Similarly, EVV

r , EVHr , EHVr , and EHHr can be computed with similar equations (eq. (27) to (30)), where JV and JH are replaced by JV

r and JHr , respectively. These scattered electric fields are

computed for only those triangle patches illuminated by either tx or rl

.

The sum of EVV + EVV

r for all the illuminated patches gives the scattered

electric field in VV polarization. Similarly, other summations give the

field in VH, HV, and HH polarizations.

Computations in layer 2:

Field computations in layer 2 are as follows:

EV = E0 ⋅ TV cosθ t cosφ, cos θ t sin φ, – sin θ t exp –jβ 1 tx ⋅ Pi ,

EH = E0TH – sinφ, cosφ, 0 exp –jβ 1 tx ⋅ Pi ,

(31)

where

tx = –[sin θt cos φ, sin θt sin φ, cos θt

] ,

and the transmission coefficients TV and TH are given by
11

TV = 2η 2 cosθ

η 2 cosθ t + η 1 cosθ

TH = 2η 2 cosθ

η 2 cosθ + η 1 cos θ t

.

Notice that no reflected fields are in layer 2. As in layer 1, magnetic fields

and currents in each triangular patch are computed, and the scattered

fields are computed with equations (27) to (30) and appropriate currents.
12

3. Validation of PO Code

A PO code was implemented in a matrix laboratory (MATLAB) environment. To verify the accuracy of the code, I created several targets and

computed their RCSs using both PO code and MOM code. The MOM

code is developed by Duke University. The comparison results are presented below.

Vertical plate: A 1-m2 plate was created as shown in figure 6 and placed

0.15 m above the Eglin soil with permittivity of εr

 = 5.0 – j0.0 and conductivity of σ = 0.003. For the computation of the scattered fields at various

frequencies, an incident field at θ = 45° and φ = 0° was used. The plate

was oriented to get the dihedral effect from the ground. I computed the

RCS characteristics of the plate from 25 to 2000 MHz. The results are

shown in figures 7 and 8 for VV and HH polarizations, respectively.

Figures 7 and 8 show that PO output is fairly accurate compared to that of

the actual output computed by MOM at frequencies 600 MHz and higher.

In other words, when the target was greater than or equal to 2λ, I obtained good correlation with the MOM simulations. Why the PO output

for HH polarization had better correlation at even lower frequencies with

MOM output compared to VV polarization is not clear.

The next object I used for simulations and comparison was a horizontal

cylinder with a spherical head on one side as shown in figure 9. The

dimensions of the cylinder were length = 0.58 m and diameter = 0.078 m.

I performed two simulations, namely, a cylinder buried 1 in. below Yuma

soil and a cylinder placed 1 in. above Eglin soil. The results of the simulations are shown in figures 10 to 15 for a target buried 1 in. below the

surface in Yuma soil. PO output is plotted by a solid line and MOM

output is plotted by a dotted line. Notice from these figures that the RCS

computed for the cylinder using the PO code is accurate. The largest

difference between the MOM and PO code is observed in figure 15, where

φ = –90°; at that angle, it is known2 that PO does not do well. Figures 16 to

21 show the RCS characteristics of a cylinder placed 1 in. above the

surface of Eglin soil.

First, I simulated the scattering fields for the cylinder shown in figure 9

for all azimuthal angles φ = 0° to 360° and generated SAR images using

the back-projection algorithm. The SAR images generated using MOM

and PO simulated data are shown in figure 22 for a cylinder buried 1 in.

below the surface of Yuma soil. From the SAR images, I found that an

excellent correlation exists between the MOM and PO simulated SAR

images in terms of resolution and that both the images have similar

magnitudes. Next, I used a UXO (shown in fig. 23), which was buried in

Yuma soil. I made actual measurements of the UXO using a radar. Its SAR

images and the PO simulated images for the same UXO are presented in

figure 24. It shows that the SAR images using PO simulated data can

capture all the features of the UXO. This helps in developing ATD and

ATR algorithms for various targets.

2C. A. Balanis, Advanced Engineering Electromagnetics, John Wiley and Sons, Inc., New York, NY (1989).
13

20

15

10

5

0

–5

–10

–15

–20

–25

PO

MOM

Frequency (MHz)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

RCS VV (dBsm)

Figure 7. RCS VV

characteristics of a

plate in a half-space.

1

0.8

0.6

0.4

0.2

0.5

y x

1

0.5

0

–0.5

–0.5 –1

0

z

Figure 6. A 1- × 1-m2

plate.
14

30

20

10

0

–10

–20

–30

PO

MOM

Frequency (MHz)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

RCS HH (dBsm)

Figure 8. RCS HH

characteristics of a

plate in a half-space.

0.05

0

–0.05

x

y

z

0.05 0

–0.05 –0.25

–0.2 –0.15

–0.1 –0.05

0

0.05

0.1

0.15

0.2

0.25

Figure 9. Cylindrical

target used.
15

PO

MOM

–5

–10

–15

–20

–25

–30

–35

–40

–45

–50

–55

Frequency (MHz)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

RCS VV (dBsm)

Figure 10. RCS VV of

a cylinder, θ = 60°,

φ = 0° (broadside).

–10

–15

–20

–25

–30

–35

–40

–45

PO

MOM

Frequency (MHz)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

RCS HH (dBsm)

Figure 11. RCS HH of

a cylinder, θ = 60°,

φ = 0° (broadside).
16

–15

–20

–25

–30

–35

–40

–45

RCS VV (dBsm)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (MHz)

PO

MOM

Figure 12. RCS VV of

a cylinder, θ = 60°,

φ = 45° (off axis).

–20

–25

–30

–35

–40

–45

–50

PO

MOM

Frequency (MHz)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

RCS HH (dBsm)

Figure 13. RCS VV of

a cylinder, θ = 60°,

φ = 45° (off axis).
17

Figure 14. RCS VV of

a cylinder, θ = 60°,

φ = –90° (axial).

–15

–20

–25

–30

–35

–40

–45

–50

–55

PO

MOM

Frequency (MHz)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

RCS VV (dBsm) –25

–30

–35

–40

–45

–50

–55

PO

MOM

Frequency (MHz)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

RCS HH (dBsm)

Figure 15. RCS HH of

a cylinder, θ = 60°,

φ = –90° (axial).
18

10

0

–10

–20

–30

–40

–50

PO

MOM

Frequency (MHz)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

RCS VV (dBsm)

Figure 16. RCS VV of

a cylinder above

Eglin soil, θ = 60°,

φ = 0° (broadside).

10

0

–10

–20

–30

–40

–50

–60

PO

MOM

Frequency (MHz)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

RCS HH (dBsm)

Figure 17. RCS HH of

a cylinder above

Eglin soil, θ = 60°,

φ = 0° (broadside).
19

–10

–15

–20

–25

–30

–35

–40

–45

–50

PO

MOM

Frequency (MHz)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

RCS VV (dBsm)

Figure 18. RCS VV of

a cylinder above

Eglin soil, θ = 60°,

φ = 45° (off axis).

Figure 19. RCS HH of

a cylinder above

Eglin soil, θ = 60°,

φ = 45° (off axis).

5

10

15

20

25

30

35

40

45

50

55

RCS HH (dBsm)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (MHz)

PO

MOM
20

–5

–10

–15

–20

–25

–30

–35

–40

–45

PO

MOM

Frequency (MHz)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

RCS VV (dBsm)

Figure 20. RCS VV of

a cylinder above

Eglin soil, θ = 60°,

φ = –90° (axial).

0

–10

–20

–30

–40

–50

–60

PO

MOM

Frequency (MHz)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

RCS HH (dBsm)

Figure 21. RCS HH of

a cylinder above

Eglin soil, θ = 60°,

φ = –90° (axial).
21

–1.00 –0.80 –0.60 –0.40 –0.20 0.00 0.20 0.40 0.60 0.80 1.00

1

0.5

0

–0.5

–1

X (cross range)

–1 –0.5 0 0.5 1

Y (down range)

–1.00 –0.80 –0.60 –0.40 –0.20 0.00 0.20 0.40 0.60 0.80 1.00

X (cross range)

–1 –0.5 0 0.5 1

1

0.5

0

–0.5

–1

Y (down range)

–0.40 –0.32 0.24 –0.16 –0.08 0.00 0.08 0.16 0.24 0.32 0.40

1

0.5

0

–0.5

–1

–1 –0.5 0 0.5 1

Y (down range)

X (cross range)

–0.40 –0.32 –0.24 –0.16 –0.08 0.00 0.08 0.16 0.24 0.32 0.40

X (cross range)

–1 –0.5 0 0.5 1

1

0.5

0

–0.5

–1

Y (down range)

–0.30 –0.24 –0.18 –0.12 –0.06 0.00 0.06 0.12 0.18 0.24 0.30

1

0.5

0

–0.5

–1

Y (down range)

–1 –0.5 0 0.5 1

X (cross range)

–0.30 –0.24 –0.18 –0.12 –0.06 0.00 0.06 0.12 0.18 0.24 0.30

1

0.5

0

–0.5

–1

Y (down range)

–1 –0.5 0 0.5 1

X (cross range)

Figure 22. Left

column MOM, right

column PO. Top row

φ = 0°, middle row

φ = 135°, bottom row

φ = –90°.
22

0.05

0

–0.05

x

y

z

–0.05

0.05

0

0.25

–0.25

–0.2

–0.15

–0.1

–0.05

0

0.05

0.1

0.15

0.2

Figure 23. UXO used

for measurements

and simulations.

Figure 24. (a) Measured versus (b) simulated SAR images

using PO.

φ = 0°

φ = 45°

φ = 90°

(a) (b)
23

4. Remarks

One of the drawbacks in the current PO code is that normals of various

patches are being used to determine whether a patch is illuminated by the

transmitted/reflected ray. However, it does not determine whether a

patch is blocked by some part of the target. For some simple targets, this

is not a problem. If the target is complex and has many corners and

projections, the PO code may not give accurate results unless the abovementioned problem is solved.
24

Acknowledgments

I would like to acknowledge my sincere appreciation and gratitude to

Professor L. Carin for giving me an opportunity to spend six months at

Duke University. I also would like to thank Yanting Dong, Duke University, and A. Sullivan, ARL, for their help and discussions.

I would like to thank Jeffrey Sichina, Microwave Branch, and the SEDD

management for allowing me to go to Duke University.
25

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298-102

A Fast Algorithm for Computing Scattered Fields

Using Physical Optics Equivalent Approximation in Half-Space

July 2000 Final,

In this report, a fast algorithm for computing scattered fields with the use of physical optics (PO)

equivalent approximation in half-space is presented. The theoretical basis for the algorithm and

the derivation of formulas used in the algorithm is presented. The algorithm is used to compute

the radar cross sections (RCSs) of several objects. The RCS of the objects computed by the

algorithm is compared with those that were computed with the method of moments (MOM). The

results presented are found to be accurate when the target dimensions are greater than or equal to

2λ, where λ denotes the wavelength. It is concluded that the PO algorithm presented in this

report can be used for majority of applications as it captured all the salient features of the targets.

RCS, MOM

Unclassified

ARL-TR-2132

0NE4I4

622120.H16

AH16

62120A

ARL PR:

AMS code:

DA PR:

PE:

Approved for public release;

distribution unlimited.

33

Unclassified Unclassified UL

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U.S. Army Research Laboratory

Attn: AMSRL-SE-RU rdamarla@arl.mil

U.S. Army Research Laboratory

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T. Raju Damarla

email:

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