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A General Solution to the Scattering

of Electromagnetic Waves from a

Strip Grating

S. Sohail H. Naqvia

& N.C. Gallagherb

aDepartment of Electrical and Computer Engineering,

University of New Mexico, Albuquerque, NM, 87131, U.S.A.b

School of Electrical Engineering, Purdue University, West

Lafayette, IN, 47907, U.S.A.

Available online: 01 Mar 2007

To cite this article: S. Sohail H. Naqvi & N.C. Gallagher (1990): A General Solution to theScattering of Electromagnetic Waves from a Strip Grating, Journal of Modern Optics, 37:10,

1629-1643

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JOURNAL OF MODERN OPTICS, 1 9 9 0 , VOL . 3 7 , NO . 1 0 , 1 6 2 9 -1643

A general solution to the scattering of

electromagnetic waves from a strip grating

S . SOHAIL H . NAQVI

D e p a r t m e n t o f E l e c t r i c a l a n d C o m p u t e r E n g i n e e r i n g ,

University of New Mexico, Albuquerque, NM 8713 1, U . S . A .

and N . C . GALLAGHERS c h o o l o f E l e c t r i c a l E n g i n e e r i n g , P u r d u e U n i v e r s i t y ,

W e s t L a f a y e t t e , I N 4 7 9 0 7 , U . S .A .

(Received 10 November 1989 ; revision received and accepted

13 February 1990)

A b s t r a c t . W e d e s c r i b e a n e w r o b u s t a p p r o a c h f o r t h e a n a l y s i s o f s t r i p g r a t i n g s ,

b o t h o f f i ni t e a n d in f i n it e c o n du c t i v it y , f o r t h e T E a n d T M c a s e s . T h e f i el d

d i s t r i b u t i o n s i n t h e p l a n e o f t h e g r a t i n g a r e e x p a n d e d i n a F o u r i e r s e r i e s , w h o s e

c o e f f i c i e n t s a r e d e r i v e d a s t h e s o l u t i o n t o a n i n f i n i t e - d i m e n s i o n a l s y s t e m o f l i n e a r

e q u a t i o n s . V a r i o u s c o n f i g u r a t i o n s o f t h e s c a t t e r e r a r e c o n s i d e r e d a n d i t i s s h o w n

t h a t e v e n i n c a s e s w h e r e t h e T s a o - M i t t r a S I T p r o c e d ur e s f a i l s t o c o n v e r ge a n d

t h e moment met h od re q uire s a large matri x to arri v e at a s olution, our met h od

y i e l d s r e a s o n a b l e r e s u l t s e v e n f o r s m a l l m a t r i x s i z e s . T h e a c c u r a c y o f t h e s o l u t i o n

p r o c e d u r e i s a n a l y s e d b y c o n s i d e r i n g t h e m e a n - s q u a r e e r r o r i n t h e f i e l d

magnitudes as a function of the truncation size of the infinite system of linear

e q u a t i o n s .

1 . Introduction

T h e s c a t t e r i n g o f e l e c t r o m a g n e t i c w a v e s f r o m p e r i o d i c m e t a l l i c s t r u c t u r e s i s a

c l a s s i c p r o b l e m t h a t h a s b e e n l o o k e d a t b y n u m e r o u s r e s e a r c h e r s o v e r t h e p a s t f e w

d e c a d e s [ 1 - 8 ] . T h e g e o m e t r y o f t h e p r o b l e m u n d e r c o n s i d e r a t i o n i n t h i s p a p e r i s

given in figure 1 . T h e b a s i c a s s u m p t i o n s a r e t h a t w e h a v e i n f i n t e l y t h i n p e r i o d i c s t r i p s

o f m e t a l o f f i n i t e o r i n f i n i t e c o n d u c t i v i t y . T h e s e s t r i p s a r e o f i n f i n i t e l e n g t h a n d a r e

l o c a t e d i n t h e x y p l a n e . A p l a n e w a v e i s o b l i q u e l y i n c i d e n t a t a n a n g l e 0 f r o m t h e

n o r m a l a n d w e w i s h t o d e t e r m i n e t h e e l e c t r i c f i e l d d i s t r i b u t i o n e v e r y w h e r e . T h e

s t a n d a r d a p p r o a c h t o t h e s o l u t i o n o f t h i s p r o b l e m i s t o f o r m u l a t e t h e E - f i e l d o r

H - f i e l d i n t e g r a l e q u a t i o n s . A m a t r i x e q u a t i o n i s t h e n o b t a i n e d b y u s i n g t h e m e t h o d

of moments [ 9 ] . T h e i n f i n i t e s y s t e m o f l i n e a r e q u a t i o n s o b t a i n e d c a n t h e n b e s o l v e d

u s i n g a n y o f t h e t e c h n i q u e s d e s c r i b e d i n [ 1 0 ] . The C-G method [11 ] was proposed

b y H e s t e n e s a n d S t i e f e l n e a r l y 3 0 y e a r s a g o f o r t h e s o l u t i o n o f a s y s t e m o f l i n e a r

e q u a t i o n s . H o w e v e r , i t i s o n l y i n t h e l a s t d e c a d e t h a t t h i s m e t h o d h a s b e e n a p p l i e d t o

t h e e l e c t r o m a g n e t i c s c a t t e r i n g p r o b l e m [ 1 2 , 1 3 ] . A d i f f e r e n t a p p ro a c h t o t h e s o l u t i o n

o f t h i s p r o b l e m w a s p r o p o s e d b y T s a o a n d M i t t r a [ 5 ] . F o r m u l a t i n g t h e p r o b l e m i n

t h e s p e c t r a l d o m a i n t h e y o b t a i n e d a s e t o f a l g e b r a i c e q u a t i o n s f o r t h e c o e f f i c i e n t s o f

t h e E l e c t r i c f i e l d a n d c u r r e n t d i s t r i b u t i o n s . T h e s e e q u a t i o n s a r e s o l v e d u s i ng t h e

s p e c t r a l i t e r a t i o n t e c h n i q u e d e v e l o p e d b y t h e a u t h o r s .

W e f i n d t h a t b o t h t h e m o m e n t m e t h o d a n d t h e s p e c t r a l i t e r a t i o n m e t h o d h a v e

` r e g i o n s o f o p e r a t i o n ' a s s o c i a t e d w i t h t h e m w h e n a p p l i e d t o t h e p e r f e c t l y c o n d u c t i n g

0 9 5 0 - 0 3 4 0 / 9 0 $ 3 - 0 0 © 1 9 9 0 T a y l o r & F r a n c i s L t d .

D o w

n l o a d e d b y [ I N A S P - P a k i s t a

n ] a t 0 6 : 1 4 0 8 D e c e m b e r 2 0 1 1



1 6 3 0 S . S . H . Naqvi and N . C . G a l l ag h e r

Region 3

Figure 1 . Geometry of scatterer .

strip-grating problem . These regions are defined b y the particular method of

solution, the scattering geometry and the angle of incidence of the incoming wave .

Various researchers have addressed these problems by proposing modifications to

these solution procedures which make a particular method work b etter under certain

circumstances . These include the addit ion of an edge mode to improve convergence

of the electric field expansion [ 1 4 - 1 6 ] . Techniques to improve the convergence of the

spectral iteration routine have been proposed in [17 , 18 ] . Kas and Yip [19] show that

the addition of preconditioners c an improve the convergence rate of the co njugate

gradient method .

In this paper, we introduc e a new method for the so lution of scat tering of

electromagnetic waves from a strip grating . The initial formulation of the problem is

carried out as described in [ 5 ] . The iteration procedure is, however, applied by

representing the truncation of the electric and current fields as a matrix multipli-

cation . Assuming that the initial guess is the solution to the problem a sy stem of

linear equatio ns for the field coefficients is derived . T h e s i g n i f i c a n t r e s u l t i s t h a t t h e

solution of this system of linear equations is actually the closed form solution of the

iteration procedure . The system of equations derived is equivalent to that arrived at

b y using the moment method solution . A solution is obtained by truncati ng the

infinite-dimensional matrix and using any of the procedures described in[ 1 0 ] , to

solve the system of linear equations . In this paper, we use Gauss elimination for the

solution of the system of equations . Although a proof of convergence is not given, a

solution is obtained for all `regions of operation' and for any matrix size . Thus the

accuracy of the solution is directly related to the size of the truncated matrix . I t i s

observed that the error decreases monotonically with increasing matrix size in all the

cases considered .

Region 1

Yr.

D o w


n ] a t 0 6 : 1 4 0 8 D e c e m b e r 2 0 1 1



S t r i p g r a t i n g g e n e r a l s o l u t i o n 1631

2 . Problem formulationMoharram and Gay lord in [20] discuss the formulation of the general problem of

the scattering of a plane wave from a periodic surface . T h e t o t a l f i e l d i n r e g i o n 1 c a n

be written in the most general form as the sum of an incident plane wave and multiple

backward diffracted orders .

0 0F1 (x,y,z)=exp(-ik 1 • r )+ Y R„exp(-ik l i • r ) , ( 1 )

with harmonic exp ( i w t ) assumed and suppressed . Similarly for region 3 we have

only the forward diffracted orders and

F3 ( x , Y , z )= Y - T„exp (-ik 3 i • r ) , ( 2 )

where r=xil+yk'+z4 and k, 9 and 4 are unit vectors in the x , y and z directions

respectively . Thus the scattered field can b e treated as a sum of plane waves with

wave vectors k l , , . Where the incident electr ic field lies parallel to the metallic strips

(TE case), the field F ( x , y , z ) represents the electric field expansion . For the dual case

o f t h e i n c i d e n t m a g n e t i c f i e l d l y i n g p a r a l l e l t o t h e s t r i p s ( T M c a s e ) , t h e f i e l d F ( x , y , z )

represents the magnetic field . In either case the only component of the field present is

in the y direction . F 1 and F 3 represent these y components .

In the limiting case of zero grating modulation, we have an infinite metallic sheet

and only the n=0 diffracted mode is present and has to be phase matched to the

incident field at the z=0 boundary . Thus

k 1 • k= kio • k .

Since the scattering surfac e is periodic, each diffracted mode must satisfy the

'Floquet condition' whereby the scattered field is also periodic in the x direction with

period d. We th en have

Simplifying we obtain

k l „ •k=k . 4- nd .

sing„=sing-dwhich is nothing other than the usu al grating formula .

The problem formulation is continued by Moharram and Gaylord by noting that

each nth diffracted mode in regions 1 and 3 must be phase matched at the z=0

boundary. Thus the x component of the wavevector of the nth diffracted mode in

region I and the nth diffracted mode in region 3 must be the same, that is

k 1 „ • k = k3. - ' -

(4 )

Since the waves in regions 1 and 3 are travelling in opposite directions, for the z

components of t he wave vector we have

k 1 .4=-k10 . =k30 .~ ~

and

k1=-k3 i '4 .

( 3 )

( 5 )

D o w


n ] a t 0 6 : 1 4 0 8 D e c e m b e r 2 0 1 1



1 6 3 2 S . S . H . Naqvi and N . C . G a l l ag h e r

k

k

Figure 2 . D e c o m p o s i t i o n o f w a v e v e c t o r s , o f t h e i n c i d e n t w a v e a n d t h e z e r o o r d e r r e f l e c t e d

and transmitted waves, into rectangular components .

Figure 2 displays graphically the result of applying the formalism of [20] to the strip

graing prob lem . In regions 1 and 3 the wave vectors have magnitudes

I k 1 1 = l k t . l , I k 3 1 = I k 3 n 1 •

Since these regions are filled with the same material

z=0Yr.

z

I k i 1 = 1 k 3 1 . ( 6 )

We now wish to find the coefficients R„ and T„ such that the total wave satisfies the

boundary conditions .

In the TE polarized case only the following field components are present

Ey , Hx, H Z , J, .

For the TM polarized case we have

H , E x , E 2 , J x ,

where H i s the magnetic field and J is the induced surface current on the metallic

strips .

2 . 1 . A s s u m p t i o n s

(a) We have an infinitely thin metallic grating .

(b) The grating is periodic and of infinite dimension .

(c) The incident wave is TM or T E polarized with wave vector in the x z plane .

The procedure developed in this paper can be conveniently extended to

consider the case of an arbitrarily polarized incident field [21] .

(d) Regions 1 and 3 separated by the metallic strips are filled with the same

homogeneous material . We assume the material is air in this problem . An

identical procedure can be developed for the case where regions 1 and 3

contain different materials [ 2 2 ] . The procedure has also been extended to

consider a strip grating placed on a dielectric slab of some thickness h

[22,231 .

D o w


n ] a t 0 6 : 1 4 0 8 D e c e m b e r 2 0 1 1




2 . 2 . A p p l i c a t i o n o f M a x w e l l ' s e q u a t i o n s a n d b o u n d a r y c o n d i t i o n s

I n t h e T E c a s e w e w i s h t o o b t a i n a n e x p r e s s i o n f o r t h e s u r f a c e c u r r e n t d e n s i t y o n

t h e m e ta l li c s t ri p s . F o r t h e d u a l T M c a s e w e r e q u i r e a n e x p r e s s i o n f o r t h e e l e c t r i c

f i e l d d i s t r i b u t i o n i n t h e z = 0 p l a n e . W e a p p l y t h e f o l l o w i n g b o u n d a r y c o n d i t i o n s :

( i ) T h e t a n g e n t i a l e l e c t r i c f i e l d i s c o n t i n u o u s a c r o s s t h e z = 0 b o u n d a r y . I n t h e

T E c a s e w e g e t t h e n

R + b o = T n . ( 7 a )

W h i l e f o r t h e T M c a s e w e o b t a i n

T n =-Rn +S n 0 . ( 7 b )

( i i ) I n t h e p r e s e n c e o f s u r f a c e c u r r e n t d e n s i t y J , t h e H - f i e l d i s a b r u p t l y

d i s c o n t i n u o u s b y J [ 2 4 ] s o

4 x ( H 3 -H 1 )=J . ( 8 )

I n t h e T E c a s e u s i n g M a x w e l l ' s e q u a t i o n s w e o b t a i n a n e x p r e s s i o n f o r t h e

m a g n e t i c f i e l d a n d s u b s t i t u t e i n t o t h e a b o v e e q u a t i o n t o g e t

a oJy = Y Cn exp(-ik t n • x k ) ,

n=-ao

( 9 )

where

Cn=BnRn,Bn=

u ( k 1) .

( 1 0 )

( 1 1 )

F o r t h e T M c a s e b y s i m i l a r a r g u me n t s w e o b t a i n

. 1 1 x =x _ ( R I . + a n o ) e x p ( - i k l n • x A ) ,

2 n=-co

where

R , , +Sn o =Rn .

T h e e l e c t r i c f i e l d c a n t h e n b e r e p r e s e n t e d a s ,

a o

E1x(x)= E Cn exp(-ik l n • xk ) ,

( 9 a )

n=-co

where

Cn =1k l n • 4Rln,CUE (10a)

=BnR1n,and

Bn= I

k 1 n • 4 . ( 1 1 a )

C o e

D o w


n ] a t 0 6 : 1 4 0 8 D e c e m b e r 2 0 1 1



1634 S . S . H. Naqvi and N . C . G a l l a g h e r

In both cases we have

_ [ k i -(kin't)2]

1 / 2 ,k 2 > ( kl

.1)2 ,

kin

•

Q

{i[(kln • 31 ) 2 -kl] 1 1 2 , k 2 <(k 1 • 8)2 .

( 1 2 )

The strip grating is made either from an infinitely conducting material or

from a resistive sheet material, whereby the losses in the material can be

approximated by a surface boundary cond ition [25] . This boundarycondition is also known as the Leontovitch boundary condition [ 2 6 ] . Theelectric field on the metallic strips satisfies the relationship

E ( x ) = R , J ( x ) , on metallic surface .

where Rg is the resistivity of the metallic stri ps .

3 . Solution

In order to solve for the reflected and diffracted waves we use the following two

equations at the z = 0 plane .

F or the T E case

c c

E l y l .=O = Y (Rn+an0)exp(-i k 1 n . ' x k ) ,

n= - a o

a o

Jylz=o- ` Cn exp(-lk l n • x k ) .

n=-oo

For the TM case

XE11=0=Y Cnexp(-ik l n • x* ) ,n=-m

J1x IZ=o = (Rln+5no)exp( -ikln • x k) .

n= - oo

As a first step, let us describe the iterative procedure based on the T sao-Mittra SIT

approach . The procedure is described in detail for the TE case . The solution in the

TM case can b e obtained in a similar manner .

We utilize the periodicity of the grating and use equation (1) to write the above

equations for the TE case as

° ° 2n \

E l y l z = o = Y (Rn+B n o )exp(-ik l • x k)exp i n-X/ ,

n=-ao

J yl z = 0 = Y

0 0 C exp(-ik l • x k)exp/ in d x ' .

n=-ao

These equati ons can be recognized as being in the form of a Fourier Series . The

iteration procedure is as follows :

( a ) Make any initial guess E 1 ° j ( x ) fo r E l y ( x ) I Z = o .

(b) Calculate the coefficients R n using this E-field representation by applying the

Fourier series coefficient fomula .

(c) Using equation 10, calculate C,l o ) , the current-density Fourier-series

c o e f f i c i e n t s .

D o w


n ] a t 0 6 : 1 4 0 8 D e c e m b e r 2 0 1 1



Strip grating general solution 1 6 3 5

° ° 2 i t

Jy ° )(x)= Y C,° ) exp (-ik l • xk) exp idn= - 00

Now make a new estimat e JY

' ) ( x) of JY( ' ) (x) by applying the constraint that the

surface current density is zero where there is no metal . So

J y l ) ( x ) =J; , ° ) ( x ) S ( x ) ,

where

S(x) = (0, in gap,

j

1

1 , else .

The Fourier coefficients for the truncated J field can thus be obtained by

convolving the Fourier coefficients of the square wave S(x) with the Fourier

c o e f f i c i e n t s o f t h e f i e l d Jy° )(x ) . Thus if we write

00

J y l ) ( x ) = E C, 1 ) exp (-ik l • x*) exp imdM= - .0

then

1Cm)-00

C"° ex p [i27t(n-m)]i21t n = _ W (n-m)n#m-exp Ci dc(n-m) +d d

c C „ ° ) .

(d) To o bt ai n E iy° ) ( x ) , we use equation (10) to write

R1 a ) =1CMM B mBThis electric field, however, does not satisfy the boundary condition for the

electric field on the metallic surface .

( e ) We require that

Eili, )=R.Jy 1 ) ( x ) , on strips .

Obtain a better approximation for the electric field as follows .

EiV(x) = Ei li°)[1- S(x)] +RJ , 1)(x ) .

The Fourier coefficients of the new electric field can thus be evaluated by

convolving the Fourier coefficients of the field E i y ° ° ( x ) with the squar e wave

[ 1 -S(x)] and adding to it the Fourier coefficients of the field R , J 3 , 1 ) ( x ) .

ThusICnl)+ano=d m y~(bmo+Rlm°)) e x p C icd(m-n) -1 i d(m-n)

m#nC

+ a ( S no+R"l a ))+RB„Rk l a ).

(f) Repeat until desired accuracy is obtained .

D o w


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1636 S . S . H . Naqvi and N. C . G a l l ag h e r

3 . 1 . I t e r a t i o n p r o ce d u r e u s i n g m a t r i x n o t a t i o n

Again the procedure is described in detail for the TE case .

( a ) Let

R[m] =Rm , - 00<m<00,

C[m] = Cm , - ° o < m < 0 0 ,

=B(m, n ) _

Bm , m n , - c o < m , n < 0 C ) .

0 , min,

(b) Obtain R ( ° using the Fourier series coefficient integration formula on the

i n i t i a l g u e s s E i°y ) ( x ) .

(c) Using equation (10) we obtain the Fourier series coefficients for the current

density

00 ) = BR ( ( ) ) ,

and

C ( 1 ) = A 1 C ( 0 ) ,

where

i2n(n-m)(1-expIic d ( n - m ) 1 ) , min,

d-cd '

( d ) We now obtain a better approximation of the electric field from the current

density

R (1 a ) =B - 1

C ( 1 ) =B - 1 A 1 BR( 0 ) .

( e ) Applying the boundary conditions, the Fourier series coefficients of the

electric field after one iteration are given by

RM=A2 R(1a ) + p ,

where-A [ m , n ] , m#n,

A 2 [ m , n ] = cd+R B n , m=n .

and A2 [ m , 0 ] , m :A 0 ,

P[m ]= (c-d)d , M=O

R" )=QR ° + P,

m = n .

Consequently the electric-field Fourier coefficients, after one iterative step, are

given b y

D o w


n ] a t 0 6 : 1 4 0 8 D e c e m b e r 2 0 1 1



where

Q=A2B - 'A 1 B . ( 1 5 a )

I n t h e T M c a s e , w e b e g i n w i t h a g u e s s R ( ° f o r t h e c u r r e n t d e n s i t y a n d o b t a i n t h e

c u r r e n t d e n s i t y R " l ) a f t e r o n e i t e r a t i o n a s

R c 1 > = Q ' R c ° + P ' ,

where

S t r i p g r a t i n g g e n e r a l s o l u t i o n 1 6 3 7

and

and

P[M] =

1

Q'=AZB - ' A ' l B .

2R 8A+ B.

f

A 1 [ m , n ] , min,AZ[m,n]= d-c

d ' m = n '

C e x p I i c d ( n - m ) I - 1 ) , min,

[ I - Q]R = P .

M: A 0 ,

M=0 .

m = n ,

( 1 5 b )

d B0 d'

T h e m a t r i c e s Q a n d Q , a r e f i x e d o n c e t h e g e o m e t r y o f t h e s c a t t e r e r a n d t h e a n g l e o f

i n c i d e n c e o f t h e i n c o m i n g w a v e i s d e f i n e d .

I n t h e T E c a s e , i f t h e i n i t i a l g u e s s i s t h e s o l u t i o n t o t h e p r o b l e m , t h e n

R" = R ( O ) =QR( 0 ) +P .

O r w e h a v e t h e s y s t e m o f l i n e a r e q u a t i o n s

( 1 6 )

w h e r e I i s t h e i d e n t i t y m a t r i x . A s i m i l a r s y s t e m o f e q u a t i o n s c a n b e d e r i v e d f o r t h e

T M case .

T h u s a n y s ol u t i o n t o t h e p r o b l em m u s t s a t i s f y e q u a t i o n ( 1 6 ) . S i n c e w e k n o w t h e

s o l u t i o n t o b e u n i q u e [ 2 4 , 2 7 ] , w e c a n s o l v e e q u a t i o n ( 1 6 ) f o r t h e F o u r i e r s e r i e s

c o e f f i c i e n t s o f t h e r e q u i r e d f i e l d c o m p o n e n t s .

4 . Results

T o i m p l e m e n t t h i s p r o c e d u r e o n c o m p u t e r , w e n e e d t o t r u n c a t e t h e i n f i n i t e

d i m e n s i o n a l m a t r i c e s . T h e e l e c t r i c f i e l d is n o w r e p r e s e n t e d b y 2N+ 1 F o u r i e r s e r i e s

c o e f f i c i e n t s s i g n i f y i n g t h e i = - N, . . . , N d i f fr a c te d mo d e s i n e q u a t i on ( 1 3 ) . Thus,

e a c h o f t h e m a t r i c e s A 1 , A 2 , B i s t r u n c a t e d t o a 2N+ I b y 2N+ 1 m a t r i x a n d t h e Q

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n ] a t 0 6 : 1 4 0 8 D e c e m b e r 2 0 1 1



1 6 3 8 S. S . H . N a q v i a n d N . C . G a l l a g h e r

F i g u r e 3 . M a g n i t u d e o f e le c t r i c f i e l d i n th e z = 0 p l a n e . T h e i n c i d e n t p la n e w a v e i s T E

p o l a r iz e d w i t h 0 = 6 0 ° , d=1. 1 2 a n d c=0-9d. R,=O . (a) N=5, (b) N=20 .

x x

F i g u r e 4 . M a g n i t u d e o f s u r f a c e c u r r en t d e n s i t y d i s t r i b u t i on i n t h e z = 0 p l a n e . T h e i n c i de n t

p l a n e w a v e i s T M p o l a r i z e d w i t h 0=45 °, d=5 . 022 a n d c=0-75 d. R,=O . (a) N=5,

(b) N=20 .

m a t r i x , g i v e n b y e q u a t i o n ( 1 5 a ) , i s a l s o o f d i m e n s i o n 2N+1 by 2N+1 . A s o l u t i o n t o

t h e p r o b l e m i s o b t a i n e d b y s o l v i n g t h e s y s t e m o f l i n e a r e q u a t i o n s

[ I - Q] R = P .

u s i n g t h e G a u s s e l i m i n a t i o n p r o c e d u r e [ 9 ] .

I n g e n e r a l , f o r t h e p e r f e c t l y c o n d u c t i n g g r a t i n g c a s e , t h e e l e c t r i c a n d c u r r e n t f i e l d

d i s t r i b u t i o n f o r t h e T M c a s e b e h a v e d i n a s i m i l a r m a n n e r t o t h e r e s p e c t i v e d u a l

c u r r e n t a n d e l e c t r i c fi e l d d i s t r i b u t i o n s c a l c u l a t e d in t h e c a s e o f T E i n c i d e n c e . F o r

c a s e s w h e r e t h e p e r i o d o f t h e g r a t i n g w a s n e a r t h e w a v e l e n g t h o f i n c i d e n t w a v e

(figure 3 ) , only a few terms in the expansion were needed to obtain a good

D o w


n ] a t 0 6 : 1 4 0 8 D e c e m b e r 2 0 1 1



2 .4-

_ s a -0 .8 -

2 .4-

0

- 0 . 2 d

B = 8 00

d = 1 .5X

c = O . 6 d

x

Strip grating general solution

2 .4 -

1 .6-

0.8 -

0I 1 1 1 I I I

- 0 . 2 d 0 0.2d 0 . 4d c 0 . 8 d 0 0 .2d 0.4d Old 0. 8 d c

2 .4 -

1 .6 -

0 .8-

0 =450

d = 5 .02X

c = 0 . 7 5d

x

1639

- - -I I 1 0 1 J I I I I ('

0 0 .2d 0 . 4 d c 0. 8 d - 0 . 1 d O 0.2d 0 .4d 0 . 6 d c 0. 9 d

x x

Figure 5 . M a g n i t u d e o f e l e c t r i c f i e l d i n t h e z = 0 p l a n e c a l c u l a t ed f o r t h e d i f f e r e n t c a s e s u s i n g

matrix s ize N=20 . The incident wave is TM polarized . R,=0. (a) 0=0°, d=1 . 5 . 1 ,c=0 6d . ( b ) 0 = 0 ° , d = 1 . 5 . 1 , c = 0 .9d . ( c ) 0 = 6 0 ° , d = 1 . 5,, c=0 . 6d . ( d ) 0 = 4 5 ° , d = 5 . 02 . 1 ,

c=0-75 d .

approximation to the field . N o t e t h a t i n a l l c a s e s t h e i n c i d e n t f i e l d w a s a u n i t e l e c t r i c -

field amplitude TM- or TE -polarized plane wave . Since the conservation of energy

criterion does not guarantee the solution to b e correct [ 2 8 ] , our criterion for

convergence here was the satisfication of the boundary conditions by the electric and

current fields . If only the reflection coefficient is desired, a good approximation can

be obtained using N= 5 . As the period of the grating increased with respect to the

wavelength (figure 4), more terms were required in the expansion to obtain an

accurate description of the field .

In figure 5 we display the electric fields calculated for different scatterer

configurations in the case of a T M polarized incident plane wave . To eliminate the

ripples due to Gibb's phenomenon, the electric field Fourier coefficients are first

multiplied with a hamming window of length 2 N + 1 .

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1 6 40 S . S . H . Naqvi and N . C . G a l l ag h e r

For comparison purposes we implemented the Ts ao-Mittra spectral iteration

procedure [ 5 ] , and applied it to calculate the electric field distribution, in the TE

case, for various scatterers . We found that for small grating periods with respect to

wavelength and/or large angles of oblique incidence, it becomes impossible to supply

a good initial guess to the iteration procedure . Co nseque ntly the method fails to

converge to the correct solution .

Richmond in [ 1 4 ] presents two moment method solutions to the perfectly

conducti ng scatteri ng problem. The second method presented, using an edge mode,

is applicable only in the normally incident case . I n f i g u r e 2 o f [ 1 4 ] the magnitudes o f

the reflection coefficients are presented, calculated using the second method, for

various configurations o f the scatterer . We obtain the same results using ou r

procedure . Since we consider oblique incidence, for comparison purpose weimplement the first moment method (equations ( 1 2 ) , ( 1 3 ) ) given in [ 1 4 ] . I t i s a p p l i e d

to calculate the electric field distribution in various cases . We find that, using this

method, the matrix size req uired for the Gauss elimination procedure to arrive at a

solution becomes ext remely large when the strip width becomes small compared to

the period o f the grating . No such convergence problems are observed using our

method .

In figure 6 we consider the convergence o f the electric-field magnitude for

various cases using a mean-square error criterion . S i n c e t h e a c t u a l f i e l d d i s t r i b u t i o n

is not available, the electric field calculated using various values o f N i s comparedwith the electric field obtained using N=100 . We have

where

NP-1Error= EM=0a n d

E1 00 (NP) EN(NP)I } 2 ,

N2 7 c

EN(x)

n

E N (R„+8 f o )exp(-ik l •x4)exp i n

dx .

NP represents the total number o f sampling points for the region 0 < x < d . For a

particular matrix size, the error increases upon decreasing the strip width, increasing

Figure 6 . M e a n s q u a r e d e r r o r i n t h e e l e c t r i c f i e l d m a g n i t u d e f o r di f f e r e n t s t r i p w i d t h s a n d

angles of oblique incidence of TE polarized plane wave . R S =O . : 0 = 0 ° , d = 1 - 5 A ,

c=0-6d ,0=0°, d=1 . 5 . 1 , c = 0 - 9 d , -- - : 0=60°, d=1 . 5 . 1 , c = 0 - 6 d .

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n ] a t 0 6 : 1 4 0 8 D e c e m b e r 2 0 1 1



F i g u r e 7 . M a g n i t u d e s o f e l e c t r i c a n d R R x c u r r e n t f i e l d s i n t h e z = 0 p l a n e c a l c u l a t e d u s i n g

m a t ri x s i z e N=20 . W e h a v e n o r m a l i n c i d e n c e , d=0. 9 7 . 1 , c=0-5d, R,=5000 a n d t h e

i n c i d e n t w a v e i s T E p o l a r i z e d .


x

AI (I i 1

0 0 . 2 d 0 . 4 d c 0 .6d 0. 8 d d

x

F i g u r e 8 . M a g n i t u d e s o f e l e c t r i c a n d R a x c u r r e n t f i e l d s i n t h e z = 0 p l a n e c a l c u l a t e d u s i n g

m a t r ix s i z e N=20 W e h a v e n o r m a l i n c i d e n c e , ( . i .-d)=3 x 10 - ' , c=0 - 5 d, R g =500Ia n d t h e i n c i d e n t w a v e i s T E p o l a r i z e d .

t h e p e r i o d o f t h e g r a t i n g r e l a t i v e t o t h e w a v e l e n g t h , o r i n c r e a s i n g t h e a n g l e o f o b l i q u e

i n c i d e n c e . I r r e s p e c t i v e o f t h e s c a t t e r e r o r t h e a n g l e o f i n c i d e n c e o f t h e p l a n e w a v e , i t

c a n b e o b s e r v e d t h a t t h i s e r r o r d e c r e a s e s m o n o t o n i c a l l y w i t h i n c r e a s i n g m a t r i x s i z e .

T h e t r a d e - o f f i n v o l v e d h o w e v e r i s t h a t t h e c o m p u t e r t i m e i n c r e a s e s c o r r e s p o n d i n g l y

w i t h i n c r e a s i n g m a t ri x s i z e .

H a l l a n d M i t t r a i n [ 8 ] c o n s i d e r i m p e r f e c t l y c o n d u c t i n g s t r i p s . We ran our

p r o g r a m f o r t h e c a s e s c o n s i d e r e d a n d o b t a i n e d s i m i l a r r e s u l t s f o r t h e r e f l e c t i o n

c o e f f i c i e n t i n a l l c a s e s e x c e p t f o r t h e c a s e s w h e r e w e h a d n o r m a l i n c i d e n c e a n d t h e

p e r i o d o f t h e g r a t i n g w a s n e a r l y e q u a l t o t h e w a v e l e n g t h o f t h e i n c i d e n t w a v e . When

2 . 5

P

2 .0-

K 1 . 0

0 . 5 -

0

1 . 5

1 .25-

1 .0-

k

0.75-

-.t - 0 .5-

0 .25 -

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1 6 42 S . S . H . Naqvi and N . C . G a l l ag h e r

the wavelength of incident wave is equal to the period of the grating, one of the terms

in the diagonal matrix B is equal to 0 . Thus B - 1 can not be computed . To avoid this

problem we considered 2-d= + 3 x 1 0 - ' . As expected, almost identical results were

obtained for the two cases . I n f i g u r e s 7 a n d 8 w e c o n s i d e r t h e e l e c t r i c a n d c u r r e n t f i e l d

distributions in the case of a T E-polarized plane wave normally incident on a strip

grating with resistivity R S = 500 . Again to reduce ripples due to G ibb's phenomenon,

the current-fi eld Fourier coefficients are f irst multiplied by a hamming window of

lengt h 2N+ 1 . Although the period of the grating used in figure 8 was only 3% larger

than the one used in figure 7, the fields changed significantly and the magnitude of

the reflection coefficient also decreased sharply . It can be ob served that in both cases

the boundary condition for the tangential electric field on the conducting strips is

well satisfied by the field R,J 5 , ( x ) .

5 . ConclusionsWe have introduced a new robust method for the solution of the scattered field

distributions in the case of a plane wave obliquely incident on a metallic strip grating .

An infinite-dimensional system of linear eq uations is derived which is equivalent to

that arrived at using the moment method soluti on . A solution is obtained b y

truncating the infinite-dimensional matrix and using G auss elimination to solve this

system of linear equations . In this paper we consider both T M and T E polarized

incident plane waves . The solution procedure is general enough to treat gratings

both of finite and infinite conductivity .

We have shown how, using our method, an arbitrarily accurate description of the

electric and current fields can be obtained for any configuration of the strip grating

and for any angle of oblique incidence of the plane wave (figure 1) . D i f f e r e n t s c a t t e r e r

configurations are considered . In each case it is shown that the error in the magnitude

of the fields in the plane of the strip grating depends on the size of the truncated

matrix . It is shown that this error decr eases monotonica lly as the matrix size

increases .

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n ] a t 0 6 : 1 4 0 8 D e c e m b e r 2 0 1 1

a general solution

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