Ii. 12. 13. 14. 15. 16. 17o

A. A. Oliner~ S. T. Peng, T. I. Hsu, and A. Saucher, Trans. IEEE, MTT-29, 855 (1981). K. Ogusu, J. Opt. Soc. Am., 7_33, No. 3, 353 (1983). A. I. Gipsman, I. S. Nefedov, and R. A. Silin, Electronic Engineering, Microwave Electronics Series, No. 8, 12 (1979). E. V. Avdeev and V. I. Potapova, Electronic Eng., Set. 3, No. 2, 36 (1972). M. A. Evgrafov, Analytical Functions [in Russian]~ Nauka, Moscow (1968). G. I. Veselov, V. M. Temnov, and S. V. Ruzhitskii, Summary of Lectures on "Calculation and Design of Two-Dimensional Antennas," Sverdlovsk (1982), p. 79. G. I. Veselov and V. M. Temnov, Summary of Lectures at the 39th All-Union Scientific Session, Radio i Svyaz', Moscow, Pt. 2 (1984), p. 21.

DIFFRACTION OF ELECTROMAGNETIC WAVES AT PERIODIC BOUNDARIES BETWEEN TWO MEDIA I. V. Borovskii and N. A. Khizhnyak

UDC 537.226.2

The key problem of diffraction of a plane H-polarized wave at the periodic interface of two nonmagnetic isotropic dielectrics is solved by the method of integral equations of macroscopic electrodynamics. Rigorous analytical expressions are obtained for the components of the scattered field. The solution of this key problem is used as the basis for solving the problem of diffraction of a plane wave at a dielectric comb and at an array of rectangular bars.

One of the most effective methods of computing the diffraction fields at periodic dielectric structures is the nt~erical-analytical method [I, 2], which permits one to obtain analytical expressions for the scattered fields only in the long-wave approximation. The object of the present article is to derive exact expressions for the fields scattered at such structures for an arbitrary ratio of the wavelength and the period of the structure. i. The key aspect of the problem under investigation is the diffraction of a plane electromagnetic H-polarized wave at the periodic interface of two isotropic nonmagnetic media. This interface is shown in Fig. i. In the coordinate system chosen here it is described by the function z = ~ (x). This interface does not exclude the presence of a comb (Fig. 2). The dielectric constant of the medium below the boundary is equal to unity, that of the medium above is g. We divide the entire space into three two-dimensional regions homogeneous along y: region 1 (z S 0), region 2 (z ~ h), region 3 (0 S z ~ h) - strictly an array. We introtuce the inverse function x = ~ - Z ( z ) and specify it by the following conditions: if x ~ [ 0 ,

b], then x = ~ - I z ( z )

if x ~ [b, L], then x = @ - 1 2 ( z ) f(z), defined by the expression

/ (z)

for z ~ [0, h] and x = 0 for z ~

[h, ~);

for z ~ [0, h] and x = L for z E [h, ~);

/ ~ ~ ( z ) - ~V~(z) ~, ! L ~r

Then function

O~,h

(1) '

describes the width of its element within the region of the array proper; above the array it is equal to the period of the diffraction structure. The fields in all the above-mentioned =

+

regions satisfy the integral equations

(g ad

+ k')

l

')gClr - -

[3] (2)

1 v

Institute of Radiophysics and Electronics, Academy of Sciences of the Ukrainian SSR. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 29, No. 5, pp. 586596, May, 1986. Original article submitted October 17, 1984.

0033-8443/86/2905-0437512.50

9 1986 Plenum Publishing Corporation

437

[

L

2L

*

Reg. 1

F Fig. i

I-I (r)

=

11, (r)

~

+ ~

E (r') g (:r - r'l) dr',

(3)

V

where E0 and H0 are the incident electric and magnetic fields and g is Green's function for free space. The integration is carried out over the entire volume V of the dielectric half space. The incident wave vector k has two components kx and kz. Equations (2) and (3) are integral equations for the field components inside the scattering medium, but also expressions for external scattered fields that can be described by the integral terms. We now turn to the scattered fields. Considering the geometry of the scattering medium, the plane wave nature of the incident field, and Flok theorem, the condition of quasiperiodicity of the fields has the form

Ex,~(x+nL, z) --~-exp (i~s nL) E~,z(X, z),

x ~ [0, L],

(4)

where ~s = ~ + (2~/L)s, ~ = k s i n ~ , ~ is the angle of incidence of the wave, and s is a given element of a set of integers. Using (4) and carrying out transformations in (2) taking account of the discrete spectrum of the scattered field, we obtain the expressions for the components of the scattered field:

& {x,z)

exp (ix%) =

(1/2)

(~ -

i)i 8~--c0

X (/~-- q']) e x p ( i l z - z ' l V k ~ - 9]~C~x@~,z')dz'+i%d 2- X 0

(s)

oo

E~(x,z) = (1/2) (~ -- 1) i E s~

X

Iu+;/2

0

H,(x,z) =~

438

-0o

exp(ll~ - z'l-V ~i-'~'2~cv~j ~ (%,z')dz' + i% ~d i exp (i I z -, z'l V~ ~ - ~,~) c,~ (% z') az' ] ;

0

where

exp (ix%)

k'T7--

o~ ] '

(6)

~-~(z) E~,, (~', z') e

c,.~,, ( % z) = L-

(7)

"a~ .

~F~(z) The following representations out the integration:

of Hankel function

{

H(Z)o(k]p

- ~'I) were used in carrying

oQ

M~o" (k It, - ~"1) ==-~ [§

I (~k~--wD~/~}, x ~ x ' in~%. 3.

Thus, in order to obtain the expressions for the fields scattered by a periodic interface of two media, it is sufficient to find linear functionals C1x,z(~s , z) of these fields relating the corresponding components of the internal fields and the properties of the media interface over its period L. For this purpose we consider the system of integral equations for Ex and Ez obtained from (2):

Ex(x,z)=Eox(x,z)+~--I, --t 4

( k~+ 0~) T x ( x , z ) + -,--t--1- .T z ( xO= ,z), ~ 4 Ox~ (8)

E~(x, z) = Eo~(x, z) +

where

-

-

i k~

4

T,(x,z)+

Tx.z(X.Z): .IEx'z(x"z')H~l~(ik ~ (x---x')2+(z--z')2)ds"

~

4

i -Oxaz L ( x , z),

The integration

is carried out in X'OZ'

S

plane over normal section S' of the entire scatterer. Integral TE,Z converges at all points of region S', which is ensured by the weak divergence of function H(1)0(kl9 -- P'l) in the neighborhood of the point 9' = 9- We enclose point 9' = 9 by contour ZI covering a v e r y small region Sz, such that H(1)0(klp - P'I) ~ 2 i v - l i n k p -- P' I in this region. Then in the remaining region of integration S' S I function H(1)0(k O - P'l) exists at each point. In system of equations (8) both differentiation and integration operators can interchange places. By direct computations it can be verified that in region S I

a T o, (x, z) = ~ Or ~.,

a~ T ? ~ x , 2 ) , Or z .

e ~.. ( x ' , z') - da;

i./o, ) (k ]P -- P'[) dx'dz';

(9)

S,

a' ii~ (~ IP -- p'! ) dx'dz', e ; , ( x ' , ~') Yr'

(i0)

.

I Ex'zH~)dss"

where r is either x or z and T(Z)xz denotes

The right-hand side of (10) d i -

S~

verges; S l9

therefore,

rearrangement

of the above-mentioned

operators

is not possible

in region

However, if the integration in region S I is taken in the sense of the principal value [4], the rearrangement of the operators is possible. For this purpose we make use of Green's theorem and change over from integration over region S I to that over closed contour Z I containing no singularities. We let contour Z z shrink to point p' = 9 such that S I ~ O. As a result, we get

02 f Or~ . F,.=(x'.~')H'r S,

2i[

=--~

e~.~

~ /~'(~)sln~cos~ ] - ~-[~ d~+ ~e~.~(x.z) .

(ll)

It

439

where R(9) = IP - P'J in the polar coordinate system with its center at the point p' = 9, R'(g) = (8/SN)R(~). The first integral term in (ii) depends on the shape of the contour s If we choose the representation

H~~ (k D -- 0"1) = ~-~ j" (~kZ--w~) -~/z exp { i w ( x - - x ' l - F i l z - - z ' I f k e - - w Z I d w ,

zv~z',

(12)

for the internal, points of tile scattering medium, then it is necessary to let contour s shrink to point p' = O in such a way that z ~ z'. Then expression (ii) equals zero if r = z and 4iEx(x~ z) if r = x. It is shown in [5] also that the problem of transposition of linear differential and integral operators in the source region has a definitive solution only for a correct choice of the shape of contour s However, the rules for choosing this shape are not indicated. In the present article the problem of determining the shape of the contour has a natural solution. Replacing operator k 2 + 32/8x a by -82/3z 2, the equivalence formula

(k* + ~we w r i t e system of i n t e g r a l

equations

of which is obvious from the

-)

+ o7

Hg ~ (klo - P'l) = o,

(8) in t h e form

eEl(x, z) = Eel(X, z)-]- (e--l) (4~)-iLX

p~ (~, z) ] U --

U-~-

(13)

w= az

r~O

Ez (x, z) = Eoz(x, e) + (e-- 1) (4z)-~LX

X

i

dwe ~x

[

iw2 -- 2C, (w, z) 4- V ~

P~ (w, z)

w a ]/~---~_ w ~. de

p. (w,z)],

(14)

--oo

where

Px,. (w, z) = i C~:,. (w, z')

e'l*-''lr

dz',

o

C~,~

mL+~Fz(z) 1 ; dx,e_,~X, Exz(X,,Z, )' (w, z) = ~ "-L ' m=-,~ mL+~f~-l(z)

C~ (w'z) -- E~

E

y

dx'e-tx'(~x--W)"

ra mL+~ll(Z) L+ Applying operator

I

-L- ~ dxexp(--ixv) m mL+~l(z)

to both Eqso

(13) and (14), we obtain the system

of integral equations

c~(v, z ) [ 8 - - ( ~ - - l ) f ( z ) , L - q

440

= Co~(V, z) +

(15)

+ ,-- ! f(z) 2 L

Cz(v, z)

[l+(~--l)t(z)L-q

V~--v~

iPx(v,z)

= Co,(v, ~) + ~ -

2

~ /(2)

v ]/r~-f~

0 p,(v,z) a2

v

i v G (v, z) -

V~_~

]; a

&_.v,~

(15)

,

(16)

using algebraic transformations, this system is reduced to a system of two first-order linear differential equations for Cx and ~z [Cx,z(v, z) = Cx,z(V, z)f(z)L-l]:

rt Gx;

(17)

~ x l -d-2

a~ dz

(18)

d"z

where

a ~ l = fiv[e--(e--l)f(z)L-1], azl=iv-l[lq-(e--1)

a~z~-----iv(e--l)ff(z)L-',

f (z)L-i],

azz = f w - l ( e - - 1 ) f f ( , z ) L -I ,

ax3 ~--- - - (e-- 1) kzf (z) L - ' - - k z + v ~ . Solving system (17) and (18) for ~x(v, z) we get

a4 +

az-T + G~ ~

(19)

&~c~ = &~(v, z),

where Fax =

az'axtaxa +aztax2ax* aztaxta'~s + az2axtax' ($xaaztaxt -

a~,a'ia,3

--

-

(20)

a~,axiax3 + a,zaxia~ - - ,a~a

Flr x (Lxt(L zI(L x l

ax3~zl~xt H~ :

in (20) t h e d i f f e r e n t i a t i o n Eq. (19) i s of t h e form

- - Co~(k~--vD + i v G ~ ,

H~ = - - C o ~ + i v - ~ C ~ , ,

w i t h r e s p e c t t o z i s d e n o t e d by prime. s

z) = A i w , (o, z) + A z w z ( v , z) - - ~ ( v ,

The g e n e r a l s o l u t i o n o f

z) ,

(21)

where -wI 2(v, z) are the fundamental solutions of the corresponding homogeneous equation and ~x(V, z)'is its particular solution: ~]x(,V, z) = w~(v,,z)

Fo~(v,z')wz(v,z')w~ 1 (v,z')dz'~-wz(v,.z) 0

Fox(V,Z') wi ( v , z ' ) w ~ 1 ( v , z ' ) d z ' .

(22)

0

The expression for ~z(V, z) can be obtained by substituting (17) into (21): C,(v, ,z) = Ai (v)w~i (v, z) + A z ( v ) w,z(v, z) - - ~ ( v , z) ,

(23)

where ~xa

~x3

(24) ax3

ax3

ax3 441

In region 3, where the fields get formed, fundamental solutions wl and w2 depend only on the shape of the boundary. The particular solutions depend also on the type of incident field. In region 2 (above the boundary), f(z)L -~ = i and for any boundary between two media fundamental solutions w~,~ are exp [ • ~ - v2( z - 11)]- However, for constructing general solution (21) of Eq. (19) and also expression (2 3) f o r Cz(v, z) it is necessary to consider only w~ = exp [ i ~ ( z - h)] ensuring in this region waves having physically realizable phase velocity along the z axis~ Therefore, in region 2 we have

C~(v, z) = B~ exp [if 8kz -- vz (z--h) ] + B a (v) exp (iGz) ; C~ (v, z) . . . .

( 25 )

v (f 8k~--v z)-~B* exp [i~/ 8,kZ--v '~ (,z--h) ] q-B~ (v) exp (ik~z),

(26)

where

B~ (v) =

[Co~ (v, z) ( s G - - v2) + co~ (v, z).k~v (~--1 ) ] / 8 ( 8 ~ - - v ~ - - k ~ ) ;

(27) ~(v)

Here Be, , exp ( i k z z ) tions

=

[c0~{v, z) (sk~-- 8v~--,k~) +co~(~,.z) (8--1)k~v]/8(sk~--v~--k~).

is the particular

solution

of s y s t e m o f e q u a t i o n s

(28)

(17) and (28) i n r e g i o n 2.

System of i n t e g r a l e q u a t i o n s (1) c o n t a i n s in i t a l l t h e b o u n d a r y c o n d i t i o n s [ 3 ] . Equa(15) and (16) f o r C x , z ( v , z) c o n t a i n t h e c o n d i t i o n s a t t h e b o u n d a r i e s b e t w e e n r e g i o n s 1,

2, 3. Therefore, for determining unknown coefficients At, A2, BI it is sufficient to substitute into integral equation (15) solutions (21), (23), (25), (26) taking into consideration the relationship of Eqs. (15) and (16). Then even at the boundaries of region 3 (at z = 0 and z = h) we obtain two linearly independent equations for A I, A 2, B I. We obtain the third equation by considering (15) at the lower boundary of region 2 (at z = h). Thus, we obtain a closed linear inhomogeneous system of equations for AI, A a, BI:

At{wt(v, O) 8~(0) -- U,(v, 0)} + A2{w~(v, O) 8~(0) -- U2(v, 0)} + +B~ exp (ih f -k-.2--- - ~v) = C0~ (v,, 0) q- P~ (v, 0) ;

( 29 )

A~{w~(v, h)et(h) -- U,(v, h)} -+-A2{wa(v, h)e~(h) -- Uz(v, h)} + B,f]~ ~- C-o~(v, h)+P~(v, h);

(30)

-- A~U,(v, h) -- A2U2(v, h) + Bd~2=Co~(v, h) +P2(v, h},

(31)

where

81(z) == 8 - - ( 8 - - 1 ) [ ( z ) L - ' ,

u,,2 (v, z)

~ --2 i i .

h t= f(2") I

--if- | / k ~ -

~2= (J/ e.k2--v 2 + 8f ,k2--v2) / 2 f J 82-v2,

( ~ - - l } 2'U2 .t'(2 t)

v ~ ~*~,2 (v, 29 + V - ~ - v ~

L

Led1,2(U, 23')

(8--1)k~[(z)L-~+~--v~

(32)

Oz

0

v~

w l ~ (z') 8, (z')

0 ]

h fi,(v, z) = ~ , ( z ) ~ . ( v .

z) - - - 2

t

.

0

-2--

exp (ilz--z" I ]/k2--.v2) dz' -[oo

+

-- 1 i exp ( - - s 2

i) f exp (iz']/ k 2 - - v2) [ f~2__02 B, (v, z' ) + v B ~ ( v , z ' ) ] d z ' ,

h ?, =

(~--1) ( f 8k2"v~' f k2--v2--- v2) 2 f ~k2 _ v2 (fsk2 - - v~ + 1/k~ - - v 2)

The l i n e a r indep_~ndenee o f Eqs. ( 2 9 ) - ( 3 1 ) becomes o b v i o u s i f i t i s r e c a l l e d t h a t U l , a ( v , h) r e x p ( i h ~ k a C v-2)U1,2(v, 0 ) , P z ( v , h) r exp (ihvrk a - v 2 ) P l ( v , 0). Therefore, the p r i n c i p a l d e t e r m i n a n t o f t h i s s y s t e m a 0 ( v ) ~ 0, and we find its solution by Kramer's rule:

442

A,(v)--

A,(v)

6o(V)'

Aa(v)=A~(v__..) B,(v)=ha(v.~} 6o(V)' ~o(v)"

Functionals Cx,z(v, z) are thereby determined at any internal point of the dielectric. Determining the relationship between Cx,z(V, z) and ~Ix,z(V, z) from their definitions (7) and (14) in the form

C7:qz(v, z) ~-- ~ exp [ira L (%---v) ] ~,,= (v, z) and keeping in mind that

exp [imL (kx--v) ] A,,,,2.,a (v), A, == )1,~ ~ exp [imL (te,~v) ]

Co,~,~(v, z) ~- Co,~,z(v, e) ~ exp [imL (kx .-- v)], and, h e n c e , A,.2,a(v) = m ~,~(v, z) --= ~ e x p [imL(k~ --- v)]~lx,~; Fo~(V, z) -~- ~.~ exp[imL(~k~ - - v)]Fio~(v, z),

(i=~ 1,2,3), we g e t t h e d e s i r e d

functionais:

C~.(~,~ , z) = t (z) L-~[A**w, ( % , z) +. A,~w~.( ,., .z}

~ ~,.(~b.,, z)];

(33)

C ~ ( % , z) = f(z)L-*[A.w~l(%~, z) +. ~-Aiz~za(~s,Z) --~iz(~,z)]-- in region 3;

C,x (q),, z)----~S,~(~.~) exp [ i ( z - - h ) 1/e.k z --. I[,~] -{-B,a(%) exp (ik~z); C~ (.q~,, z) :

(34)

(35)

--~, (]/ek z - - ~ ) - ~ B , , (%) exp [i ( z - - h ) ~ eleZ - - ~ ] q-k B1~(%) exp (ik~z) - - i n r e g i o n 2.

(36)

Expressions (5) and (6) together with (33)-(36) determine the field reflected by the periodic boundary of the two media. The homogeneous diffracted waves originating from the boundary are defined on a limited set of integers S o given by the condition ek 2 ,~2s ~ 0. If f(z) is not given by a specific expression, it is possible to get only an estimate of the behavior of the field amplitudes for large values of s, which permit one to outline only very broad limits of their acceptable behavior. The condition of bounded energy of the scattered field in any finite volume of the space ensures that field amplitudes belong to space E2 of infinite sequences {Xn}, given by the condition [i]

k

I x , 1 2 ( l + l s l ) ' < co.

(37)

In the final analysis this condition determines the choice of the fundamental solutions w 1 and w 2, We determine the asymptotic forms of wl(s , z) and w2(s , z) from (19) and (20): w1(s, z) - exp(-azs), w=(s, z) - exp [-~(z - h)s], where ~ is a function connecting E with the geometry of the boundary and wavelength %. Its specific form is determined from (19) and (20). Omitting unwieldy computations we note that the convergence of the series of the electric field components is not worse than exp (-~zs)s -l and exp(-~zs)s -z for the magnetic field components. Condition (37) is satisfied in this case. The investigation of the field in the neighborhood of the comb is equivalent to the investigation of the field for large values of s. Therefore, in the neighborhood of the comb the behavior of the field amplitudes also conforms to condition (37)~ An improved convergence of the field is possible only for a specific expression of f(z). Since it is not feasible to carry out the subsequent analysis of the scattered field in the general case, we shall discuss two specific problems. 2. The dielectric comb (Fig. 2) consists of the array proper (region 3) and the half space of the isotropic dielectric (region 2). In the case when the dielectric constant of region 2 is equal to unity, the dielectric comb degenerates into an array of rectangular dielectric bars. Therefore, the dielectric comb may be regarded as a matching of the array with the dielectric half space. For both investigated structures f(z) = L - b = const in region 3. Therefore, in this region the fundamental solutions w1(v, z) and w2(v , z) of Eq. (19) for the array and the comb are the same:

w~(:v, z)---e•

wz(v, z) -~ exp[--ika(v) (.z--h) ] ,

(38)

443

0

2L

L

Fig. 2 and the coefficients of Eq. (19) have the form

(39)

It is convenient to determine the particular solution of Eq. (19) for fundamental solutions (38) directly from the form of ~0x(V, z) ~ ~0x(V)exp (ikzz):

Therefore, in this case expression (21) has the form

C.(v,z) ~-- A,(v) exp (ikg) + A2(v)exp[-- ~ 3 ( z - - h) ] +F~.(v)cxp(ik.z)/ (k] -- k~).

(40)

From (23) and (24) we get

C. (v, z) =: - - (vefk3e~) A, (v) exp (ik~z) + (ve/k~ex) A2 (v) exp [--ik3 (,z--h) ] - - ~ (v. z),

(41)

where

For the array it is sufficient to determine coefficients A l and A 2 from the system of two linear inhomogeneous equations (29) and (30). For the comb it is necessary to determine also expressions (35) and (36) of functionals C1x(~s, z) and C1z(~s, z) in region 2. We discuss in detail the array of rectangular dielectric bars. We find coefficients A I and A= from the system of two linear equations (29) and (30). Expressions (33) and (34) together with (39)-(41) determine the desired functionals Clx(~s, z) and ~iz(~s, z) within an element of the array proper. For determining the field transmitted through the array it is necessary to put Iz - z' I = z - z' in (5) and (6) and to limit the integration over z' to the thickness of the array h. Then, according to the magnetic component of the transmitted field has the form

Hy(X,Z)

"x -- 1 L - - b

2 X'

+

etksh

1 --

L

2 $~--OD

exp (ix@s + i z V ~ f - - ~/~) X

V k,

-

(42)

exp [ih(k~ -- V ~ - ~ ) ]

I--e•

-

[--ih (k3-5 ] / M -

.

,

#~)] (_~/~7.,_ # 2 +

@ ] * ) A , (#s)}

__

- ....

Vk - % We do not present the expressions for All/A 0 and Azl/A 0 because of their unwieldiness. The method of determining them, as also of each element of the determinant, is obvious from the system of equations (29), (30). 444

IIH~/H~

1,O

-

: I/lli!

0.2 0

! ._.a._..).

_ ,

.

0,2

t

._,.1.

0,4

~._[

.L

0,6

.~.,~

0.8

.~, ..........

i,O L/~.

Fig. 3 We present the analysis of expression (42), which represents a set of homogeneous and inhomogeneous fields in the form of series expansion in space harmonics. Let us investigate the convergence of this series. For s + ~ the orders of magnitudes of convergence of the corresponding quantities are as follows: k3(~s) ~ s, &zl(@s) ~ s-l, A=1(~s) - s-l, fi0(@s) ~ so Then the order of magnitude of convergence of expression (42) is exp (-hs)s -2. The electric components

of the transmitted

,~---1 L--_.b 2 2 L $~

field are:

exp(txq,+iz]/~ i-9])R,, --co

(43)

E~ = sx -- 1 L -- b Z exp (ixgs + iz l/']z" 2

qD q~Rs (t/'k ~ _

_

%~ ) -' ,

L s

where Rs denotes the expression within curly brackets in (42). It is obvious that the orders of magnitude of convergence of the expressions for Ex and Ez are equal to exp (-hs)s -I. Similarly, putting [z - z' I = z' - z for the reflected field and limiting the integration over z' to the thickness of the array h, from (5), (6), and (3) we get the following expressions for the reflected field:

Ex __

exp(ixg~-izV"~--c.~)N,,

~ -- 1 Z -- b 2 2 L

(44) - --- I L -- b- ~ ] exp (ixOs "x

e.=

2

-

tzV-~

--

,~1 ,~N~ ( V ~

--

%) =

-'

,

L

k ( ~ - 11 z. - o ~ 2

ixp

(ix,,

-

-

izVk' a 4~)N, ( V ~ - ,~)-',

L

where

explih(k3+V~--~)]--I

+

~. (%)

exp tih (k,~ + V'k 2 -- 9~ )1 --1 AI_3~/

+

.

On the basis of the estimates of the order of magnitude of convergence already made above, we determine the order of magnitude of convergence of the components of the reflected field:

445

Ex,z ~ exp(-zs)s -l, Hy ~ exp(-zs)s -2. In the neighborhood of the comb at z = h and z = 0 expressions (44) satisfy the condition of boundedness of the energy of the electromagnetic field:

llrn I(51E[ 2+ ]Hl~)d~e< oo. V~O o ~

(45)

Substituting into (45) the expressions for the transmitted field and then the expressions for the reflected field, and evaluating the integrals in (45) by maximum modulus, we find that the convergence of the obtained series is not worse than s -2. Therefore, condition (45) is satisfied. For L/X << 1 (long-wave approximation) condition (37) leads to a unique value s = 0. Therefore, in this case we have a single transmitted plane wave and a single reflected plane wave. The modulus of the transmission coefficient IHy/H0yl has the form Inv/n~

=

2ezek3ok~ ~ ~2 = E cos(k*oh)k3og:ek~(1 + r e--) __ isin(,kaoh)(ekao+ ,k~V ).

(46)

This expression coincides with the expression derived in [2, 6] by other methods. A detailed analysis of expression (46) is given in [6]. The dependence of the modulus of the transmission coefficient of the wave through the array on the ratio L/A, obtained from formula (42) for normal incidence of the field, is shown in Fig. 3. The parameters of the array are: s = 5, h/L = 0.5, b/L = 0.i for the continuous curve and 0.5 for the broken curve. Certain characteristic features of this figure are worth noting: the resonance region gets wider with the decrease of the distance between the array elements. New resonances appear that are obviously caused by the emergence of new waveguide modes in the array interferring with each other. The total transmission and total reflection resonances come together. This means that for a sufficiently small change in the frequency the array is transformed from totally reflecting to totally transmitting, in the longwave part of the range the cosine dependence of IHy/H0yl on L/X becomes more pronounced with the decrease of b/L. A graphical comparison o of the results discussed here with those presented in [i] shows good agreement. LITERATURE CITED i~ 2. 3. 4. 5. 6.

446

V . P . Shestopalov, L. N. Litvinenko, S. A. Masalov, and V. P. Sologub, Diffraction of Waves at Arrays [in Russian], Kharkov State Univ. (1973). S . A . Masalov and Yu. T. Repa, in: Radio Engineering, Resp~ Mexhved. Temat. NauchnoTekhn. Cb., Kharkov State Univ., No. 20 (1972), p. 1592. N . A . Zhizhnyak, Zh~ Tekh. Fiz., 28, No. 7, 1592 (1958)o A . N . Tikhonov and L. A. Samarskii, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1972)o A . D . Iadzhian, TIIER, 68, No. 2, 62 (1980). I . V . Borovskii and N. A. Zhizhnyak, Izv. Vyssh. Uchebn. Zavedo, Radiofiz., No. 2, 231 (1983).