1-R179 285 SCAITTERING BY A TRUNCATED PERIODIC RRARY(U) ILLINOIS 1/1UNIV AT URBANA COORDINATED SCIENCE LRB Wi L KO0 ET AL.MAR 87 UILU-ENG-87-22i7 N99914-94-C-S149

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; 'The scattering characteristics of a doubly periodic array of metallic patches isinvestigated in this paper from the point of view of detemining the effect of truncatingsuch a screen. The induced surface current on each patch of the array as well as thescattered far-field pattern of the truncated array are computed using the spectral-Galerkin method. The usefulness of the results for the truncated screen problem forthe design of frequency selective surfaces (FSS) is discussed.

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SCATTERING BY A TRUNCATED PERIODIC ARRAY

W. L. Ko and R. Mittra

'WITq

.4

ABSTRACT

The scattering characteristics of a doubly periodic array-of metallic*1

.. - patches is investigated in this paper from the point of view of determining

the effect of truncating such a screen. The induced surface current on

each patch of the array as well as the scattered far-field pattern of

_4' the truncated array are computed using the spectral-Galerkin method. The

usefulness of the results for the truncated screen problem for the design

* ..~of frequency selective surfaces (FSS) is discussed.

Acccioii For

NTIS CRA&I

DrIC TAB [U :, - oCJ.;c cd

J1,,t1!Cj j4() ILJ :,:, l~iC b(, . ... . ..............Ix, B ... ... ... ..... .......Byi N "\ .l~ibtto:,I

":~Dr .JD tif -

J . . . . . . . . . _ . .. .

%'. ',,% " .' . , -. '" ,," - " % % % % ,. ." .""."%" ,,, ",."'%" - ,,,-%" %, % , . - . %. . - % ,.,

1. INTRODUCTION

In this paper, the problem of scattering by a frequency selective surface

(FSS), comprising an array of metallic patches in the x-y plane, is addressed,

with the primary objective of investigating the effect of truncation of the FSS

on its scattering characteristics. The geometry of the truncated array is shown

in Fig. 1. For simplicity of analysis, the screen is truncated in the x- direc-

tion only, and it is still infinitely periodic in the y- direction. As is well

known, the frequency selective surface finds numerous applications in telecom-

munications. In the past, this type of array has been treated as an infinite,

doubly periodic structure with the resulting simplification that only a single

cell of the periodic structure need be considered [1]. However, it is often

desirable to assess the edge effects introduced by the truncation of the array.

Recently, the finite array problem has been addressed in the literature in the

context of patch antennas (2]. In addition, scattering by a finite number of

periodic strips has been investigated [3]. In some of these cases, advantage is

taken of the long and thin geometry of the structure, comprising strips or

-" dipoles; in others, certain simplifications are employed to make the problem

, ~ manageable by reducing the number of basis functions for the representation of

the current distribution to just one.

In the following sections, we treat the singly truncated periodic array

with the spectral-Galerkin method (4], using a set of entirp domain basis

functions to represent the surface current. The choice of these basis functions

is based on the following two criteria: (1) they be analytically Fourier trans-

P formable; and, (2) they have the proper edge condition incorporated into their

expressions. Section 2 details the analysis of the finite screen problem.

. Section 3 presents the computed data for the surface current as well as for the

A 0

2 3

far field. Section 4 discusses the usefulness of this type of structure for

frequency selective purposes. Concluding remarks and recommendations forj

further investigations are found in Section 5.

I32. ANALYSIS

I For the sake of simplicity, we consider only the case of TM-polarized

plane-wave incidence in this work, although the approach applies equally well to

the TE case. The array, which is located in the z-0 plane, has a finite number

Sof metallic patches in the x-direction and is infinitely periodic in the y-

direction (see Fig. 1). The period in the x-direction is A, and the period in

4the y-direction is C. The size of the patch is B in the x-direction and D in

the y-direction. Since the array is periodic in y, we can solve this problem by

considering a single, finite array along the x-axis with appropriate periodic

boundary conditions in the y-direction. The problem of scattering is solved in

two steps. First, we find the surface current induced on each patch; second, we

"p compute the scattered far-field pattern from the surface currents.

2.1 Computation of Surface Current

.To obtain the surface current, we follow the spectral-Galerkin method,

' which entails the representation of the surface currents in terms of a set of

Fourier-transformable basis functions with unknown coefficients. These unknown

coefficients are solved for by using the Galerkin method, which generates a

system of linear equations by means of a suitably defined inner product that

uses the same set of functions for basis as well as testing. It is well known

that by Fourier transforming a set of equations involving differential and

integral operators, a set of algebraic equations involving only multiplications

and divisions can be obtained. To take advantage of this fact, Galerkin's

method of analysis is carried out in the Fourier transform or the spectral

le domain; hence, the name spectral-Galerkin method.

An alternative approach to solving this problem would be to use an itera-

p.. ~,tivP approach, e.g., the conjugate-gradient method, which has found extensive

UI

o '-3o . . . ..

4

application to the infinite FSS problem [5]. Recently, an attempt has also been

made to apply the iterative procedure to the solution of the truncated FSS

problem [6]. However, it has been found that the iterative procedure is fraught

with difficulties associated with convergence and accuracy, due primarily t 4

very large number of unknowns required to adequately describe the finite FSS

problem using a FFT-based iteration procedure.

Returning to the spectral-Galerkin procedure, the first step is to write the

integral equation for the surface current in the transform domain. The Fourier

transform J of the surface current '(x,y)mxiX(X,y)+y (xy) satisfies the

equation .

2001):.

in whichG G yx,

rk2 - 02 21 -:

(2)

- k2 _2 2 if k > (a2 +82) -

and 2 2 2 Pad 2 k2 if k < (a + B2) (3) N

The symbol - on top of a quantity designates the Fourier transform of that quan-

tity. The conventions of exp(jwt) and exp(yz) are used in this paper. The . ,

Fourier transform pair, as employed herein, is defined as

Ni

-v

.'%'% " % % "- . % " % • "," • .% " "" "% " " q " ""%""@

"" • '.k

%" % " ' " "" ' "'"" """ '"-""" "

"''" % "S

51.N.F (fs)f (x,y) exp H(- + By)] dxdy

F (x,y) ._. f f F (a,s) exp[j(ax + By)] dadB (4)(2,)

2

The right-hand side of (1) contains the su9mation of the Fourier transform of

the incident field E ,t on the individual patches. The boundary condition thatx

the tangential E-field vanish on the patches has been used in writing (1). .

Consider the current on the t-th patch:

i x j L + t (5) %x y

where t t (6)

s ,drsx,rsI. Jx "(7)

a .?

L J,q cpq #;,Pq(7

tL Land ds and c are unknown coefficients. The basis functions xrand **rs pq x ,rs y~pq.* are chosen such that they incorporate the correct singularities for the current ' -

at the edges of the patch. These basis functions can be written as

* sin - t[(x IA) +-1]) (8)2r [_2 2 (8)

where

r 1, 2, . .. , IRR (8a)

s 0, 1, . •., ISS-i (8b) "."

N-1 (.. (8c)

0' " .

6

T [(x-LA)sand '2 -U.+Z2ypq (x A 1- D 29

where p - 0, 1, . .. , IPP-! (9a)

q - 1, 2, . ., IQQ (9b)

The Ti in (9) is the i-th order Chebyshev polynomial of the first kind. There

are a total of N patches. The index t designates the 1-th patch, and it is com-

puted according to (8c). The number of basis functions in the x- direction is

given by (IRR x ISS) and in the y- direction (IPP x IQQ). Note that the running

indices r and q start from I and the running indices p and s start from 0. The

basis functions given in (8) and (9) are then Fourier transformed to obtain

whose x- and y-components are given by

: M D Dt t: (10)

r,s r x,rs

]'t~ ~ %6.t t(ll

Y p q pq *ypq

where Dt . d t C" (12)rs rs rs %

I-

C1 a c t C1 (13)pq pq Pq

The quantities dt and ct are defined in (6) and (7). From (6), (7), (10), "rs pq(11), and (4), we can write -

rs xrs Oxs_ ,rs (x,y) exp[-J(ax + By)] dxdy (14)

V-

N . -- ''

1- 7

q ~ ff ~,p (y) exp(-J(ax + By)] dxdy (15)CUq ;ypq *yq

U, ' Substituting (8) and (9) into (14) and (15), we obtain

-t i - -LQ + (_1)r- 1sn ' +w -LOOx rs 2~ic- 2~ 2ic~ 2 7)

'U' • exp[-j a (1A)1 • J s)8 (16)

B (r-)D (.' r. c s 2 _~ (17)

•T,.~p ~t p (T a) exp[-Ja(L.A)]

" -,. y, pq p

U.,

-sn 2 fI) + (-l)q-1 sinc (29 + ) (18)

C , B P D (q-1) (19)U. pq 2(i i~

.-

-? In the above equations, sinc(.)-sin(.)/(.) and J is the i-th order Bessel func-i

tion of the first kind.

With all of the quantities defined above, we can write (1) in its x- and y-

• .components as

ZO x

rG.!-x( .J) +y(G.J)-.4. ; (G

~mx J k.. (-E ',') -' y (0) (20)z° KL

-4

.4* .

4." d" .4 - . - - - . m . - ,..

*,y..yo'.X .. ,..: ,: . : . .% -..',...,.'. ;.'";j."','':"" -""..".."..' " .. .. ..". : . ..'-" . " J-

8

Defining an inner product

- a

<A • B> - ff X •T dad$ (21)-e

and applying the spectral-Galerkin method to (20), we obtain the following

(L x R x S) x-component equations:

f-L -

rsx,S xx Ox,rs dadS

+ C f G dodopq pq - xRSxy y,pq

Lm

S-h [ I o (-- i't*) 1 dad$ (22)_f -.

*mx,RS o E

and the following (L x P x Q) y-component equations:

D G* f f * dadBrs rs -.i ny,PQ YX x,rs

.- *-

+~ C~~ fG * dd

pq pq ~ y,PQ Yyp

-0 (23)

4.-

19The above equations can be rearranged into a matrix form:

Xpq columns Lrs columns

LRS < LRS, tpq > < LRS, Irs > c ) Pq Ip'qt LRsrows p unknowns rows

I -

JP < xQ pq > PIa> DIaLPQ, rows rs unknowns 0 rows

(24)

where the matrix elements are given as

W". ( LRS, Ipq >

S.

- x,RS x X yPq

-4x

I- f -j) + (_I) R - snc (- + a)]'-" "n 2 2 2[ln 2.

exp(-jJ ( ) a 1ne J

.1

m - ) exp(j a I A)Esinc( - Dp 2 2 T Bn

+ (1 qIsinec 9 + X- + n] do (25)

2. 2 n

5% *.

• .*S.

- 5'.-,,,, ''' ''''o,,,,,, '''' ,- , "". . . , "•" -. ." ' ',"- , . '.' , ; ''" .2 ""' - ' . '.' .. .

-- .r 'r w - VC rTrflrrf wrrMrwWWW'n-.M

10

< LPQ, Irs > f f ;L G do-m y,PQ GX *x,ru s d

I f J (f a) exp(-j a LA)

n *

[ wn(.. - B rI r

[sic(- 3) + (-I) sinc(L- + I a)]

*exp(j a t A) * J (-)do (26)s 2 n

LRS, tIr > -ff 8 a-. ,RSxx #x,ru

- l B R-1 Ri B .If Isn( ~ -- y ) +(-l) sincC- + -a))

" exp(-j aLA) - i (k2- 2)M 1

" [sinc(L!. - BaQ + (1 -I ic +B

" exp(j a LA) *J~ (R.B) do (27)

powp

1

< LPQ, Lpq > -ff f ;" pQ G' dodBGyy dd

f- [J Ia exp(-j aL1A)4w 2

2 " 0.) + (-0) 1sinc( + B )

(2 02) 1• n 2y(a,BnT]

J (~.~a) exp(j a L A)p

_D + ((I)q - i +

F, 2 7 On 2 T~) nic..

The first half of the Tight-hand side in (24) is given by

< LRS, -x >

Lx

k= f f *1 Hj I (_' tx, dd

",R Ba,-I..R

a) +) (--1s- I I f - ( 1 )RI2 _ iL + -B a)]

e exp(-j a LA) J 3S ("2n) (-j ho) (-coa)

exp (n ao ) A] 0 L D B • sinc (a a - 2=1

•sinc (Sn -) d3 (29)-. =2

• . . ' " . .' . '"". ' "". . -" " . " " " . " . - " . -"""""". ' . "" ", . -"" " .""" "". """ . . """ " .""""'

12

wherea k sin -

k is the wave number, and

Si is the incident angle.

The second half of the right-hand side is equal to zero because of the

assumed polarization of the incident field in this study. It can be readily

modified to handle other polarizations as well.

Because the array is infinitely periodic in y, we can sample the

corresponding transformed variable 0 according to Floquet's theorem. In the

present case, where incidence is parallel to the 61= 0 plane, we have

S--- n-0, ±1, ±2, ... (30)n C '

where C is the spatial period in the y-direction. On the other hand, since the

periodic array is truncated along x, the number of patches along the x-direction

is finite. Therefore, we integrate along this direction over the finite length

of the array. The integration can be performed conveniently by using the fast

Fourier transform (FFT) algorithm. Notice that the integrals in (25), (26),

(27), (28), and (29) are of the following generic forma

f f(a,$n) explj[(Z-L)AQ} do (31)

Therefore, only one FFT of the function f(,on ) is needed to compute the above

integral for various combinations of the indices I and L, whose values are

given in (8c). In other words, we choose the FFT samples where x - (Z - L) A.

After all of the entries in the matrix and the right-hand side in (24) have been

computed, we can solve for the unknown coefficients C's and D's by standard

*% l .- ,% % *'. ". ,-.* " *" • .% - ' • . . ...- ,_. ' .

3 13

methods available for the solution of simultaneous algebraic equations. For

example, Cholesky's method is used to generate results in this paper. The C's

and D's are then substituted into (12) and (13) and the c's and d's are com-

puted. These are, in turn, substituted into (6) and (7) to evaluate the surface

current on each patch. Numerical results based on this procedure are presented

in Section 3.

2.2 Computation of Scattered Far Field

To complete the scattering analysis, we compute the scattered far field

from the induced surface currents on the patches. In the * - w half-plane, we

desire to compute the 8- and *- components of the far E-field, viz., E8 and

E,, for 8 between 0 and 90 degrees. It is found (71 that IE0I a IN81 and

IE I a IN, , where

No - cose f f J exp[-J(k sinS)x] dxdy (32)-e All patches x

VaIdV alLd

N -- ffJ exp(-J(k sinO)xl dxdy (33)All patches y

Using (4),(6),(7),(10), and (11), we can revrite the above two equations in

the following manner:

N o - cose i (a - k sin, - 0) (34)p.t

and N4 - - J (a"k sin, a 0) (35)Y

_1A

14

It is now seen that the far field can be computed readily by a weighted sum-

mation of the transformed basis functions evaluated at a - k sin 6 and 0 - 0.

The weights of the terms in the above summation are simply the appropriate coef-

ficients C's and D's given in (12) and (13). Thus, we can write

L-

andjLF ~IX C.* (a ksine, 60 )1

- p q pq y,pq (37)

The far-field patterns computed using the above equatio' s are presented in

the next section along with the surface current distribut;ons.

pi

..1m

U

* .. . .. . .. . . . .

~,- ".

3 15

3. NUMERICAL RESULTSUIn this section, we present the results for the surface currents and the

far-field patterns of a truncated FSS screen computed at various frequencies.

3.1 Surface Current

The x- and y-components of the surface current normalized to the incident

H-field, i.e., J /H and J /H, have been computed along the center line of eachx y

,. patch in the x- and y- directions, with the exception of J /H vs. x, which has

been plotted for y - -d/4, since it is essentially negligible along the center

line (i.e. y - 0). The x- and y-dimensions of the plot have been normalized to

the respective cell sizes in these dimensions. Figures 2,3,4,5, and 6 show the

- surface currents of a 9-patch array illuminated by a TM-polarized plane wave

incident from 8 -45 degrees, and # - 0*, for five frequencies, viz., 3, 15,

24, 36, and 45 GHz. These currents have been computed using sixteen basis func-

', tions (4 in x and 4 in y) to ensure convergence, although typically 9 basis

functions are found to be adequate. The corresponding results for the surface

currents of a 5- and a 3-patch array are shown in Figs. 7 and 8, respectively,

.,. for the frequency of 24 Gliz. Note that the starting frequency of computation is

well below the resonant frequency of the FSS, which occurs at around 24 GHz,

P. whereas the highest frequency of computation is well above the resonance. This

allows us to investigate the out-of-band performance of the FSS.

3.2 Far-Field Patterns

,, The co-polarized and the cross-polarized far-field components, E and

E are displayed in Figs. 10, 11, 12, 13, and 14 for the 9-patch array illumi-

nated by a TM-polarized plane wave incident from 9 - 45 degrees and * - 0,

*- for five frequencies spanning the range 3 to 45 GHz. The corresponding surface

.currents of these patterns were shown in Figs. 2, 3, 4, 5, and 6. Figures 15

U

16

and 16 display the corresponding patterns for a 5- and a 3-patch array, respec-

tively. Finally, the pattern of a 3-patch array with normal incidence is shown

in Fig. 17.

A discussion of the above results is presented in the following section.

C-

C.,

I

C..

.- ,

'p

L',.," , " ., , ' ; . , ,, ,..," , -v ; . ; , ,. , -, . ..- , , . " "., , - -. . .'. .., ' - . "..• , - . . - .". ,.", . , ". . " ." . . - . - " .,. , , ,.' - -" -,.' ." , .

17

4. SCATTERING CHARACTERISTICS

From the computed data presented in the previous section, we can observe 'N

that the array scatters strongly in the specular direction with a veil-defined

main beam of the scattered field pointing in this direction. As expected, the

frequency selective array shows a strong dependency on the frequency of the

incident plane wave. For example, the magnitude of the patch currents at 3 G~z,

which is veil below the resonant frequency, and the corresponding radiated

field are negligible compared with their counterparts at higher frequencies.

The frequency selective property of the screen is evident from the behavior-of

the currents and the far field at the other frequencies. Parametric investiga-

tion of these characteristics enables one to design the FSS screen to meet the

desired specifications.

Another important observation is that the surface currents are almost iden-

tical for all of the patches in the array except for those near the truncated

edges of the array. This edge effect is clearly noticeable from the computed

data and leads one to conclude that the simple approximation based on the '.~

assumption that the perturbation from the doubly infinite periodic case is

negligible is not a good one. However, since the modification of the surface *''

current is seen to be confined to the edge patches only, it appears reasonable %

to assume that for the finite, rectangular patch FSS problem, all of the inner

patches have the same current distribution as that existing on the center patch

and extend the size of the array in the finite dimension to a viery large number,

without significantly increasing the number of unknowns to be solved for in the

application of the spectral- Galerkin method.

.0* %

5.*CONCLUS IONS

In this paper, we have solved the protklem of scattering by a truncated

periodic planar array of metallic rectangular patches with TM-polarized plane- "

wave illumination at an oblique incidence angle. Since the array is infinitelyW,

periodic in one dimension and finitely periodic in the other, the problem can be

solved by considering one row of patches in the finite dimension. Therefore, it

is equivalent to treating the problem of scattering by a linear array of a

finite number of patches. The induced surface currents on the patches have been

computed using the spectral-Galerkin method. The scattered far-field pattern

of the finite array has also been obtained. ~*.

The important findings of this work are summarized as follows: (1) The

edge effect is clearly observable and should not be ignored; (2) The array scat-

ters strongly in the specular reflection direction; and, (3) This type of sur-

face scatters with a strong dependency on the frequency of the incident wave and,

hence, can be used as a frequency selective surface.

Because the truncated screen problem requires a rather large number of

unknowns to derive an accurate solution, the computation can be quite time con-

suming. It takes about 2 hours of CPU time (about 4 hours of elapsed time) on a

VAX-8600 to generate a set of data at a single frequency for a 9-patch array

with (4 x 4) basis functions along each dimension. Although not discussed here,

recent efforts have been successfully directed toward reducing this computation

time by increasing the efficiency of the computation of the matrix element. .'4

Additional insight gained from the present work is helpful in formulating other

approaches by which the computer time and storage problems could be alleviated

even further. One could attack the large finite array problems without ignoring

the edge effects by assuming that the inner patches have identical current

*i.* ,.~* '. ' . *'~ % % ~ %~~~ ~ ' -*'' *

19

distributions as the center patch and by treating the currents on the edge

patches and possibly on one or two inside patches next to the edge patches as

unknowns in the spectral-Galerkin's method. In this manner, we can increase the

number of patches in the finite array without increasing the number of unknowns.

We also plan to extend this work to a conformal FSS printed on a cylindrical

surface, by using an approach similar to what has been described in this paper

for planar arrays. The curved array is inherently more complicated because for

each patch on the curved surface, the incident angle of the plane-wave illumina-

tion may look different.

-4

*1

RI

20

REFERENCES

[1] C.C. Chen, "Scattering by a two-dimensional periodic array of conducting

plates," IEEE Trans. Antennas Propagat., vol. AP-18, pp. 660-665,

September 1970.

(2) D.M. Pozar, "Finite phased arrays of rectangular microstrip patches," IEEE

Trans. Antennas Propagat., vol. AP-34, pp.658- 665, May 1986. "

[31 T. Cwik and R. ittra, "The effects of the truncation and curvature of W

periodic surfaces: A strip grating," to be published in AP-S.

(4] C. H. Tsao and R. Mittra, "Spectral-domain analysis of frequency selective

surfaces comprised of periodic arrays of cross dipoles and Jerusalem.

crosses," IEEE Trans. on Antennas and Propagation, vol. AP-32, no. 5,

pp. 478-486, May 1984. %

(5 T. C-dik and R. Mittra, "Scattering from general periodic screens,"

Electromagnetics, vol. 5, no. 4, pp. 263-283, 1985.

[61 R. Kastner and R. Mittra, "Iterative analysis of finite-sized planar fre-

quency selective surfaces with rectangular patches or perforations," sub-

mitted to AP-S.

[71 C.A. Balanis, Antenna Theory Analysis and Design, New York: Harper & Row,

1982.

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22

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Figure 2. Current distribution on 9 patih finite FIS (A =1 cm, C - 1 cm, B =0.8 cm, D - 0.8 cm) for 8 - 45°ando - 00 and TM polarization.Frequency f - 3 GHz.

(a) J /IH'I vs. x sampled at y - -d/4, and J /IH'I vs. y sampledat center line of each patch. Y

(b) J /IH'J vs. x and y sampled at center lines of each patch inthe hohizonal and vertical directions.

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p.€. Figure 3. Current distribution on 9 patch finite F S (A - 1 cm, C - I cm, B

0.8 cm, D 0.8 cm) for 8- =45°and - 00 and TM polarization.Frequency f - 15 GHz.

(a) J /IH'i vs. x sampled at y - -d/. and J /IH'i vs. y sampledat center line of each patch. y

(b) J x/IHi" vs. x and y sampled at center lines of each patch inthe horizonal and vertical directions.

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28

Figure 4. Current distribution on 9 patch finite FJS (A - 1 cm, C -1 cm, B j0.8 cm, D 0.8 cm) tor e - 454and = 0o and TM polarization.Frequency t -24 GHz.

(a) J /IHI vs. x sampled at y - -d/4, and J'IHI vs. y sampledat center line of each patch. y

(b) J / jH' vs. x and y sampled at center lines of each patch inthe hohizonal and vertical directions.

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~, Figure 5. Current distribution on 9 patch tinite FS (A - 1 cm, C =1 cm, B -

0.8 cm, D - 0.8 cm) for e - 50and* - 00 and TM polarization.Frequency t 36 G-iz.

(a) J /1101 vs. x sampled at y - -d/4, and J /1101 vs. y sampledat center line of each patch. y

(b) J /IHI vs. x and y sampled at center lines ot each patch Inthe hohizonal and vertical directions.

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Figure 6. Current distribution on 9 patch tinite F S (A 1 1 cm, C 1 1 cm, B -

0.8 cm, D 0.8 cm) for e - 450ando - 00 and TM polarization.Frequency f 4 45 GHz.

(a) J /IJHI vs. x sampled at y - -d/4, and J /1H1 vs. y sampledat center line of each patch. Y

(b) J /IHI vs. x and y sampled at center lines of each patch inthe hohzonal and vertical directions.

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Figure 7. Current distritition on 5 patch finite FS (A - 1 cm, C - 1 cm, B -

0.8 cm, D 0.8 cm) for e - 450andi 00 and TM polarization.Frequency f - 24 GHz.

(a) J /IH'I vs. x sampled at y - -d/4, and J /IHiI vs. y sampledat center line of each patch.

(b) J /IHiI vs. x and y sampled at center lines of each patch in

the horizonal and vertical directions.

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40

Figure 8.Current distribution on 3 patih finite FS( m m0.8 cm, D -0.8 cm) for 8 456ando 00 and TM polarization.Frequency f 24 Q~z.

(a) J /JH iIvs. x sampled at y - -d/4, and J y/jHi IVS. y sampledat center line of each patch.

(b) J x/IH'I vs. x and y sampled at center lines of each patch inthe horizona. and vertical directions.

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41

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F~ 13 u

42

Figure 9. Current distribution on 3 patch finite FSS (A - 1 cm, C - 1 cm, B -

0.8 cm, D - 0.8 cm) for 8- Oand.' - 00 and TM polarization.Frequency f - 24 GHz.

(a) J /IHij vs. x sampled at y - -d/4, and J /IHil vs. y sampledat center line of each patch. Y

(b) J /IHil vs. x and y sampled at center lines of each patch inthe hoizonal and vertical directions.

.

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Figure 10. Co-polarized and cross-polarized far-freld components, Ea and .Parameters same as in Fig. 2.

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Fgure 11. Co-polarized and cross-polarized far-field components, Ee and E0 iParameters same as In Fig. 3.

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[ Figure 13. Co-polarized and rosqs-polarized ,far-field components, E0 and EParameters same as in Fig. 5.Q

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" i.,'~ ; . 'igure 13. Co :,.,'-..-.,'.'.;-..,---.---poaieadcos-oaie .-.-.-..-.- components:., .....and . ......-"----" Parmeters sam as - in Fig. 5. 8, '' -- ,", .. ' ':-.-.'. .",'",,

48

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-gure 14. Co-polarizea and cross-polarized tar-field components, E and ,Parameters same as in Fig. 6.

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k*0

. r

p%

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49

.5j

i Xt0-6

1-.

";'p

- Ir

X10 - 14 J .E U I 0 Lim

TE_________' 'p

7.1 Fgure 15. Co-polarizedL andl cross-polarizedl far-f'ield componentsE ancd E ..: Parameters same as in Fig. 7.

%'p;J. .--

F%,p

?i~ue 15 Co-ola~zedand rosspo~r~ze tartiel coponets, an

Parametrs sam as inFig. 7

,'k % % ,j% % % % , % k " % % %' " " ',". " , , . %". " . •• " U

50

.,

I

II -

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TNC

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7igure 16. Co-polarized and cross-polarized far-field components, E, and E.

Parameters same as in Fig. 8.

..

51

I- s2

II

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S.0

'-p5,"

Figure 17. Co-polarized and cross-polarized far-field components, Ee and E

Parameters same as in Fig. 9. .

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