A Fast Algorithm forComputing Scattered Fields

Using Physical OpticsEquivalent 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 anofficial Department of the Army position unless sodesignated by other authorized documents.

Citation of manufacturer’s or trade names does notconstitute an official endorsement or approval of the usethereof.

Destroy this report when it is no longer needed. Do notreturn it to the originator.

ARL-TR-2132 July 2000

Army Research LaboratoryAdelphi, MD 20783-1197

A Fast Algorithm forComputing Scattered FieldsUsing Physical OpticsEquivalent Approximation inHalf-SpaceT. Raju Damarla Sensors and Electron Devices Directorate

Approved for public release; distribution unlimited.

In this report, a fast algorithm for computing scattered fields with theuse of physical optics (PO) equivalent approximation in half-space ispresented. The theoretical basis for the algorithm and the derivationof formulas used in the algorithm is presented. The algorithm is usedto compute the radar cross sections (RCSs) of several objects. The RCSof the objects computed by the algorithm is compared with those thatwere computed with the method of moments (MOM). The resultspresented are found to be accurate when the target dimensions aregreater than or equal to 2λ, where λ denotes the wavelength. It isconcluded that the PO algorithm presented in this report can be usedfor 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 ................................................................... 22.1 Physical Optics Equivalent ............................................................................................................. 32.2 Half-Space Approximations ............................................................................................................ 62.3 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 ............................................................................... 22. Field geometry for derivation of currents on a surface of an object.......................................... 33. Field geometry for physical optics approximation ...................................................................... 44. Geometry used for layered medium .............................................................................................. 75. Ray dynamics .................................................................................................................................... 86. A 1- × 1-m2 plate ............................................................................................................................. 137. RCS VV characteristics of a plate in a half-space ....................................................................... 138. RCS HH characteristics of a plate in a half-space ...................................................................... 149. Cylindrical target used................................................................................................................... 14

10. RCS VV of a cylinder, θ = 60°, φ = 0° (broadside) ...................................................................... 1511. RCS HH of a cylinder, θ = 60°, φ = 0° (broadside) ...................................................................... 1512. RCS VV of a cylinder, θ = 60°, φ = 45° (off axis) ......................................................................... 1613. RCS VV of a cylinder, θ = 60°, φ = 45° (off axis) ......................................................................... 1614. RCS VV of a cylinder, θ = 60°, φ = –90° (axial) ............................................................................ 1715. RCS HH of a cylinder, θ = 60°, φ = –90° (axial) ........................................................................... 1716. RCS VV of a cylinder above Eglin soil, θ = 60°, φ = 0° (broadside) ......................................... 1817. RCS HH of a cylinder above Eglin soil, θ = 60°, φ = 0° (broadside) ........................................ 1818. RCS VV of a cylinder above Eglin soil, θ = 60°, φ = 45° (off axis) ............................................ 1919. RCS HH of a cylinder above Eglin soil, θ = 60°, φ = 45° (off axis) ........................................... 1920. RCS VV of a cylinder above Eglin soil, θ = 60°, φ = –90° (axial) .............................................. 2021. RCS HH of a cylinder above Eglin soil, θ = 60°, φ = –90° (axial) ............................................. 2022. Left column MOM, right column PO........................................................................................... 2123. UXO used for measurements and simulations........................................................................... 2224. Measured versus simulated SAR images using PO................................................................... 22

1

1. IntroductionIn general, scattered fields from complex objects are computed in thefrequency domain with the use of the method of moments (MOM).Although this method computes the scattered fields accurately, thecomputational complexity is shown1 to be O(n3), where n is the numberof elements for which currents need to be computed. Because of theexponential nature of the computational complexity, as n increases, theuse of MOM for computing scattered fields becomes less and less attrac-tive. Other techniques for computing frequency domain scattered fieldsexist, namely, the fast multipole method (FMM), multilevel FMM, etc. Themultilevel FMM techniques have the complexity O(n log n).1 Within therealm of MOM, other techniques exist that take advantage of targetgeometry, such as exploiting body of revolution (BOR) symmetry toreduce the size of matrixes, which must be inverted. In spite of greatadvances in computational electromagnetics, computing the scatteredfields from an arbitrary target can be prohibitive, especially as the targetis electrically large. However, at higher frequencies for targets withdimensions greater than 2λ, where λ denotes the wavelength, computingscattered fields by physical optics (PO) approximations yield relativelyaccurate results. Some of the commercially available high-frequencymodeling packages, such as XPATCH, model targets in free space with orwithout conducting ground planes. The PO code that was developedcomputes the scattered fields in a lossy and dispersive half-space via thelayered medium dyadic Green’s function specialized for a half-space. Thiscapability is of interest for simulating targets in real soil. The PO codeallows one to compute the radar cross section (RCS) and scattered fieldsof an arbitrary target that is underground, above ground, or partiallyburied.

Section 2 presents the basic formulation of the PO approximations andthe Green’s functions used. Section 3 presents some of the RCS character-istics of various targets, namely, a sphere, a vertical plate, a mine, and anunexploded ordnance (UXO), that were computed with both MOM andPO for comparison. Section 3 also presents some of the actual syntheticaperture radar (SAR) images of UXOs along with the images generatedby the simulated data by PO. These images show that most of the salientfeatures of the targets are captured, which in turn, help develop auto-matic 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 situatedin 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 SpaceOne 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 andpermittivity of free space, respectively; β is the wave number; and rdenotes the distance from the target to the observation point. Figure 1shows the coordinate system used for computing scattered fields in thefar field. I approximate the distance R from any point on the object to theobservation 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 bybold letters). Then the electric and magnetic fields radiated by the objectin the far field are given by

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

where

A =

µ4π Js

e–jβRR ds ∼ µe–jβr

4πr NS

, and (3)

N = Jsexp jβr′ cos ψ dsS

, (4)

z

y

x

φ

θ

rψ

r´

(x´, y´, z´)

R

Figure 1. Coordinatesystem for radiationby an object.

3

where Js and Ms denote the electric and magnetic current densities in-duced on the object. Clearly, scattered field E can be computed if Js isknown. To compute Js in an object, one assumes a known plane wave isimpinging on the object. PO approximation gives a relation between theinduced magnetic field and the current density Js. The following subsec-tion presents the relevant concepts.

2.1 Physical Optics Equivalent

For the sake of completeness, I present in this subsection a brief descrip-tion of PO approximation theory. Detailed explanation and derivation ofrelevant formulation can be found in literature by Balanis.2

Let us consider an unbounded medium with constituting parameters ε1and µ1. Let us also assume that a current source J1 radiating E1 and amagnetic current source M1 radiating H1 are everywhere in the medium.Moreover, consider an imaginary region V1 enclosed by a surface S1

inthis medium as shown in figure 2. Assume that the surface S1 is replacedby another medium with parameters ε2 and µ2. Let us also assume thatthe sources J1 and M1 are allowed to radiate in the presence of the newmedium in V1. The total field outside the region V1, produced by J1 andM1, 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 andH1) 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 bysolving Maxwell’s equations. To compute Es and Hs, I introduce bound-ary 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 theregion V1 as follows:

Jp = n × Hs – Ht ,

Mp = – n × Es – E t ,(6)

Figure 2. Fieldgeometry forderivation of currentson a surface of anobject. J1

ε1, µ1

E = E1 + Es

H = H1 + Hs

Et, Ht

ε2, µ2

V1

S1

M1

J1

E1, H1

ε1, µ1

V1

S1

M1

n̂ n̂

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

E1 tan + Estan = E t

tan ⇒ n × E1 + Es = n × E t ,

H1 tan + Hstan = Ht

tan ⇒ n × H1 + Hs = n × Ht ,(7)

or

Estan – E t

tan = –E1 tan ⇒ n × Es – E t = –n × E1 ,

Hstan – 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 perfectelectric conductor (PEC) with conductivity σ = ∞. Then the fields in theregion Et and Ht equal zero as shown in figure 3. Over the boundary S1 ofthe conductor, the total tangential components of the E field are equal tozero, and the total tangential components of the H field are equal to theinduced current density Jp. Hence,

Mp = –n × E – E t = –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 thatH1|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

S1S1

–E1, –H1

ε1, µ1

V1

Et = Ht = 0

σ = ∞

V1M1

E = E1 + Es

H = H1 + Hs

n̂ n̂

Jp

Figure 3. Fieldgeometry for physicaloptics 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 targetbecause of H1, which one assumes can be computed. Once we havedetermined Jp, we can use equations (2) to (4) to compute the field radi-ated by the target with current density Jp. In particular, the scattered fieldcan be computed once N in equation (4) is determined; namely,

N = J exp jβr′ cos ψ dsS

= J exp jβr ⋅ r′ dsS

, (12)

where “⋅” denotes the dot product. Note that J in equation (12) is com-puted by equation (11). Expanding equation (12) in terms of its x, y, and zcomponents of the current J produces

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

. (13)

Using the rectangular-to-spherical component transformation, where

ax

ayaz

=sinθ cosφ cosθ cosφ – sinφsinθ sinφ cosθ sinφ cosφ

cosθ – sinθ 0

araθaφ

, (14)

we can write the θ (vertical polarization) and φ (horizontal polarization)components of N as

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

, (15)

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

. (16)

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

vertical polarization

Eθ = –jωµ4πr 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 = –jωµ4πr 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 freespace. The following section presents the half-space approximations. Wecan obtain these approximations by incorporating the half-space dyadicGreen’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 betweenthe two as shown in figure 4. The medium in layer 1 has the permittivityε1 and permeability µ1 while the medium in layer 2 has ε2 and µ2. Oftenthe 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 µ1ε1 ,

β 2 = ω µ2ε2 ,

exp1 = exp jβ 1 x′ sinθ cosφ + y′ sinθ sinφ + z′ cosθ ,

exp2 = exp jβ 1 x′ sinθ cosφ + y′ sinθ sinφ – z′ cosθ ,

exp3 = exp j β 1 x′ sin θ cosφ + y′ sinθ sinφ + z′ β 22 – β 1sinθ 2 ,

Txx = Tyy =2β 1cosθ

β 1cosθ + β22 – β 1sinθ 2

,

Tzz =2β 1cos θ

β 22

β 1cos θ + β 2

2 – β 1sinθ 2,

Tzx =

β 22

β 1cosθ – β 2

2 – β 1sinθ 2

β 22

β 1cosθ + β 2

2 – β 1sinθ 2+

β 1cosθ – β 22 – β 1sinθ 2

β 1cosθ + β 22 – β 1sinθ 2

cosθ cos φsinθ ,

Tzy = Tzxsin φcos φ ,

Rxx = Ryy =β 1cosθ – β 2

2 – β 1sinθ 2

β 1cosθ + β 22 – β 1sinθ 2

,

Rxx = Ryy =β 1cosθ – β 2

2 – β 1sinθ 2

β 1cosθ + β 22 – β 1sinθ 2

,

Rzx = Tzx ,

Rzy = Rzxsin φcos φ .

7

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

Eθ =–jωµ1

4πr e– jβ1r Jx cos θ cos φ exp1 + Rxx exp2 – sin θRzx exp2S+ Jy cos θ sin φ exp1 + Ryy exp2 – sin θRzy exp2 – Jz sin θ exp1 + Rzz exp2 ds , (19)

Eφ = –jωµ14πr e–jβ1r –Jx sinφ exp1 + Rxx exp2S

+ Jy cosφ exp1 + Ryy exp2 ds . (20)

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

Eθ = –jωµ14πr e–jβ1r Jx cosθ cosφ Txx – sinθ Tzx exp3 ds

S

+ Jy cosθ sinφ Tyy – sinθ Tzy exp3 – Jz sinθ Tzz exp3 ds ,(21)

Eφ = –

jωµ14πr e–jβ1r –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 atarget.

2.3 Implementation of PO Approximations in Half-Spacefor Computation of Scattered Fields

To compute the scattered fields near a target, one must first partition thetarget into small triangles with each side of the triangle approximately0.1λ, where λ denotes the wavelength. For each triangle patch, an out-ward normal and its centroid are computed. This normal is used tocompute the current induced by a magnetic field in the patch. The cen-troid becomes the reference point in the patch for computing electric andmagnetic fields. The magnetic field in a patch is then used to determinethe current with the use of PO approximation. Once the current in a patchis computed, it is then used to compute the electrical field radiated (scat-tered field) by the patch. The sum total of all the fields radiated by all thepatches 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

1x-y plane

rθε1, µ1

Layer

Layer

Figure 4. Geometryused for layeredmedium.

8

Now to compute the induced current caused by a magnetic field, let usassume that an incident plane wave where the electric field has a magni-tude E0 = 1 is impinging on the target. Let the direction of the incidentplane wave be given by θ and φ; then the unit-transmitted vector inrectangular-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: tx = – [sin θt cos φ, sin θt sin φ, cos θt],

where θt denotes the angle of refraction in medium 2 and is determinedby Snell’s law: sin θt = β1 sin θ/β2 .

Figure 5 shows the transmitted, reflected, and refracted rays (vectors) andthe corresponding angles.

Field computations in layer 1:

The electric fields in vertical and horizontal polarizations impinging on apatch 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 ithtriangular patch of the target.

Transmitted vector

Reflected vector

Refracted vector

z

θt

θ θ

Layer 1

Layer 2

Figure 5. Raydynamics.

9

reflected field

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

EHr = 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 cosθη 2 cosθ t + η 1 cosθ ,

rh =η 2 cosθ – η 1 cosθ tη 2 cosθ + η 1 cosθ t

and

η 1 =µ1ε1

intrinsic impedance of medium 1 ,

η 2 =η 1ε2

intrinsic impedance of medium 2 .

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

HV = 1η 1

tx × EV ,

HH = 1η 1

tx × EH ,

HVr = 1

η 1rl × EV

r ,

HHr = 1

η 1rl × 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 nowcompute the currents using the PO approximation given by equation (11)as follows:

JV = 2 nP × HV

JH = 2 nP × HH

JVr = 2 nP × HV

r

JHr = 2 nP × HH

r ,

(26)

where nP denotes the outward normal to the triangular patch P. Notethat 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 polariza-tion. By using JV, the current caused by the vertically polarized transmit-ted field in equations (19) and (20), I can obtain scattered fields in verticaltransmit, vertical receive (VV) and vertical transmit, horizontal receive(VH) polarizations. Similarly, using JH in equations (19) and (20), I canobtain scattered fields in HV and HH polarizations:

EVV = –jωµ14πR e–jβ1R JVx cos θ cos φ exp1 + Rxx exp2 – sin θRzx exp2S

+ JVy cos θ sin φ exp1 + Ryy exp2 – sin θRzy exp2 – JVz sin θ exp1 + Rzz exp2 ds ,(27)

EVH = –jωµ14πR e–jβ1R –JVx sinφ exp1 + Rxx exp2S

+ JVy cosφ exp1 + Ryy exp2 ds , (28)

EHV = –jωµ14πR e–jβ1R JHx cosθ cos φ exp1 + Rxx exp2 – sinθRzx exp2S

+ JHy cos θ sin φ exp1 + Ryy exp2 – sinθRzy exp2 – JHz sinθ exp1 + Rzz exp2 ds ,(29)

EHH = –jωµ14πR e–jβ1R –JHx sin φ exp1 + Rxx exp2S

+ JHy cosφ exp1 + Ryy exp2 ds , (30)

where JV = [JVx, JVy, JVz] and R is the distance between the observationpoint and the target. Similarly, EVV

r , EVHr , EHV

r , and EHHr can be com-

puted with similar equations (eq. (27) to (30)), where JV and JH are re-placed 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 scatteredelectric field in VV polarization. Similarly, other summations give thefield 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 =2η2 cos θ

η 2 cos θ t + η 1 cos θ

TH =2η2 cos θ

η 2 cosθ + η 1 cos θ t.

Notice that no reflected fields are in layer 2. As in layer 1, magnetic fieldsand currents in each triangular patch are computed, and the scatteredfields are computed with equations (27) to (30) and appropriate currents.

12

3. Validation of PO CodeA PO code was implemented in a matrix laboratory (MATLAB) environ-ment. To verify the accuracy of the code, I created several targets andcomputed their RCSs using both PO code and MOM code. The MOMcode is developed by Duke University. The comparison results are pre-sented below.

Vertical plate: A 1-m2 plate was created as shown in figure 6 and placed0.15 m above the Eglin soil with permittivity of εr = 5.0 – j0.0 and conduc-tivity of σ = 0.003. For the computation of the scattered fields at variousfrequencies, an incident field at θ = 45° and φ = 0° was used. The platewas oriented to get the dihedral effect from the ground. I computed theRCS characteristics of the plate from 25 to 2000 MHz. The results areshown 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 ofthe 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 ob-tained good correlation with the MOM simulations. Why the PO outputfor HH polarization had better correlation at even lower frequencies withMOM output compared to VV polarization is not clear.

The next object I used for simulations and comparison was a horizontalcylinder with a spherical head on one side as shown in figure 9. Thedimensions of the cylinder were length = 0.58 m and diameter = 0.078 m.I performed two simulations, namely, a cylinder buried 1 in. below Yumasoil and a cylinder placed 1 in. above Eglin soil. The results of the simula-tions are shown in figures 10 to 15 for a target buried 1 in. below thesurface in Yuma soil. PO output is plotted by a solid line and MOMoutput is plotted by a dotted line. Notice from these figures that the RCScomputed for the cylinder using the PO code is accurate. The largestdifference between the MOM and PO code is observed in figure 15, whereφ = –90°; at that angle, it is known2 that PO does not do well. Figures 16 to21 show the RCS characteristics of a cylinder placed 1 in. above thesurface of Eglin soil.

First, I simulated the scattering fields for the cylinder shown in figure 9for all azimuthal angles φ = 0° to 360° and generated SAR images usingthe back-projection algorithm. The SAR images generated using MOMand 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 anexcellent correlation exists between the MOM and PO simulated SARimages in terms of resolution and that both the images have similarmagnitudes. Next, I used a UXO (shown in fig. 23), which was buried inYuma soil. I made actual measurements of the UXO using a radar. Its SARimages and the PO simulated images for the same UXO are presented infigure 24. It shows that the SAR images using PO simulated data cancapture all the features of the UXO. This helps in developing ATD andATR 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

POMOM

Frequency (MHz)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

RC

S V

V (

dBsm

)

Figure 7. RCS VVcharacteristics of aplate in a half-space.

1

0.8

0.6

0.4

0.2

0.5

xy

1

0.5

0

–0.5

–1–0.5

0

z

Figure 6. A 1- × 1-m2

plate.

14

30

20

10

0

–10

–20

–30

POMOM

Frequency (MHz)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

RC

S H

H (

dBsm

)

Figure 8. RCS HHcharacteristics of aplate in a half-space.

0.05

0

–0.05

x

y

z

0.050

–0.05–0.25

–0.2–0.15

–0.1–0.05

00.05

0.10.15

0.20.25

Figure 9. Cylindricaltarget used.

15

POMOM

–5

–10

–15

–20

–25

–30

–35

–40

–45

–50

–55

Frequency (MHz)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

RC

S V

V (

dBsm

)

Figure 10. RCS VV ofa cylinder, θ = 60°,φ = 0° (broadside).

–10

–15

–20

–25

–30

–35

–40

–45

POMOM

Frequency (MHz)0 200 400 600 800 1000 1200 1400 1600 1800 2000

RC

S H

H (

dBsm

)

Figure 11. RCS HH ofa cylinder, θ = 60°,φ = 0° (broadside).

16

–15

–20

–25

–30

–35

–40

–45

RC

S V

V (

dBsm

)

0 200 400 600 800 1000 1200 1400 1600 1800 2000Frequency (MHz)

POMOM

Figure 12. RCS VV ofa cylinder, θ = 60°,φ = 45° (off axis).

–20

–25

–30

–35

–40

–45

–50

POMOM

Frequency (MHz)0 200 400 600 800 1000 1200 1400 1600 1800 2000

RC

S H

H (

dBsm

)

Figure 13. RCS VV ofa cylinder, θ = 60°,φ = 45° (off axis).

17

Figure 14. RCS VV ofa cylinder, θ = 60°,φ = –90° (axial).

–15

–20

–25

–30

–35

–40

–45

–50

–55

POMOM

Frequency (MHz)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

RC

S V

V (

dBsm

)

–25

–30

–35

–40

–45

–50

–55

POMOM

Frequency (MHz)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

RC

S H

H (

dBsm

)

Figure 15. RCS HH ofa cylinder, θ = 60°,φ = –90° (axial).

18

10

0

–10

–20

–30

–40

–50

POMOM

Frequency (MHz)0 200 400 600 800 1000 1200 1400 1600 1800 2000

RC

S V

V (

dBsm

)

Figure 16. RCS VV ofa cylinder aboveEglin soil, θ = 60°,φ = 0° (broadside).

10

0

–10

–20

–30

–40

–50

–60

POMOM

Frequency (MHz)0 200 400 600 800 1000 1200 1400 1600 1800 2000

RC

S H

H (

dBsm

)

Figure 17. RCS HH ofa cylinder aboveEglin soil, θ = 60°,φ = 0° (broadside).

19

–10

–15

–20

–25

–30

–35

–40

–45

–50

POMOM

Frequency (MHz)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

RC

S V

V (

dBsm

)

Figure 18. RCS VV ofa cylinder aboveEglin soil, θ = 60°,φ = 45° (off axis).

Figure 19. RCS HH ofa cylinder aboveEglin soil, θ = 60°,φ = 45° (off axis).

5

10

15

20

25

30

35

40

45

50

55

RC

S H

H (

dBsm

)

0 200 400 600 800 1000 1200 1400 1600 1800 2000Frequency (MHz)

POMOM

20

–5

–10

–15

–20

–25

–30

–35

–40

–45

POMOM

Frequency (MHz)0 200 400 600 800 1000 1200 1400 1600 1800 2000

RC

S V

V (

dBsm

)

Figure 20. RCS VV ofa cylinder aboveEglin soil, θ = 60°,φ = –90° (axial).

0

–10

–20

–30

–40

–50

–60

POMOM

Frequency (MHz)0 200 400 600 800 1000 1200 1400 1600 1800 2000

RC

S H

H (

dBsm

)

Figure 21. RCS HH ofa cylinder aboveEglin 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 (

dow

n ra

nge)

–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 (

dow

n ra

nge)

–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 (

dow

n ra

nge)

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 (

dow

n ra

nge)

–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 (

dow

n ra

nge)

–1 –0.5 0 0.5 1X (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 (

dow

n ra

nge)

–1 –0.5 0 0.5 1X (cross range)

Figure 22. Leftcolumn MOM, rightcolumn PO. Top rowφ = 0°, middle rowφ = 135°, bottom rowφ = –90°.

22

0.05

0

–0.05

x

y

z

–0.05

0.050

0.25

–0.25–0.2

–0.15–0.1

–0.050

0.050.1

0.150.2

Figure 23. UXO usedfor measurementsand simulations.

Figure 24. (a) Meas-ured versus (b) simu-lated SAR imagesusing PO.

φ = 0°

φ = 45°

φ = 90°

(a) (b)

23

4. RemarksOne of the drawbacks in the current PO code is that normals of variouspatches are being used to determine whether a patch is illuminated by thetransmitted/reflected ray. However, it does not determine whether apatch is blocked by some part of the target. For some simple targets, thisis not a problem. If the target is complex and has many corners andprojections, the PO code may not give accurate results unless the above-mentioned problem is solved.

24

AcknowledgmentsI would like to acknowledge my sincere appreciation and gratitude toProfessor L. Carin for giving me an opportunity to spend six months atDuke University. I also would like to thank Yanting Dong, Duke Univer-sity, and A. Sullivan, ARL, for their help and discussions.

I would like to thank Jeffrey Sichina, Microwave Branch, and the SEDDmanagement for allowing me to go to Duke University.

25

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NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)Prescribed by ANSI Std. Z39-18298-102

A Fast Algorithm for Computing Scattered FieldsUsing 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 andthe derivation of formulas used in the algorithm is presented. The algorithm is used to computethe radar cross sections (RCSs) of several objects. The RCS of the objects computed by thealgorithm is compared with those that were computed with the method of moments (MOM). Theresults presented are found to be accurate when the target dimensions are greater than or equal to2λ, where λ denotes the wavelength. It is concluded that the PO algorithm presented in thisreport can be used for majority of applications as it captured all the salient features of the targets.

RCS, MOM

Unclassified

ARL-TR-2132

0NE4I4622120.H16

AH1662120A

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

27

T. Raju Damarla

email:2800 Powder Mill RoadAdelphi, MD 20783-1197