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Diffraction Theory by Means of Singular Integral Equations VI Diffraction of Plane Waves by an Infinite Strip Grating.

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Annalen der Phvsik. 7. Folae, Band 27, Heft 3, 1071, S. 257--288
.J. 21. Barth, 1,cipzig
Diffraction Theory by Means of Singular Integral Equaiions VI’)
Diffraction of Plane Waves by an Infinite Strip Grating2)
By E. LCTERIJRQand K. WESTPBAHL
With 20 Figures
Abstract
For an infinitc plane grating formed by strips antl gaps of equal width, a DIRICIII.ET
boundary value problem for the HEuIloLTz equation is solved rigorously by function-throretic techniques. Plane wave excitation with an arbitrary angle of incidence is considered.
The high frequency asymptotics of the solution is complctfily evaluated and compared with
KIRCIIH~FF’S
diffraction theory as well as with the asymptotics for a single strip. Extpnsivc
numerical data arc laid down graphically.
1. Introduction
This paper adds a further sample t o the small number of rigorously solvable
boundary value problems in diffraction theory. I t may be considered as a
heurist,ic method for the half plane to a
systematic extension of SOMMERFBT.D’S
non separable diffraction problem.
We consider the diffraction of plane electromagnetic waves striking a strip
grating in the plane z - = 0. The grating consists of an infinite set of identical
perfectly conducting antl infinitely thin strips with edges parallel t o the y-axis.
The periodically spaced strips have the width 2a in the direction of the x-axis
and the spacing between any two adjacent &ips is also 2a (Fig. 1).The angle of
incidence is 8,, and the electric vector is polarized parallel t o the edges of the
strips.
The case of normal incidence (ao = cos 8, = 0) has b e m treated rigorously
b y AIDWI WIN and HICI\S111, whereas the independent treatment of WEISSTEIN
[ 2 , 31 includes also thc discretr directions a. = zn/’.’ak ( k waw: n u m b e r ,
n integer). These authors consider the grating as an equivalent infinite waveguide of width 2a formed by tbc planes TC = 0, x = ’La. and a transverse &aphragm z :0 , 0
x
n, a problem which is aintmable to the \VIE:SER-HOI>E’
technique (cf. also [ 1, 51).
An approximate solution for a grating with the distances between the strips
not equal t o their widths has been obtained by AGRAXOVJCH
et al. [ 61
< <
I ) This work is n seqiicl of the series publishetl in Ann. l’liys. 4 (1959) 283; 20 (1967) L 1 ;
21 (1968) 12; 25 ( l ! J i O ) 3 7 5 ; 26 (1971) 103.
2) Detlicatcd t o Prof. H. 110s~
on his retirement.
17
Ann. Yh,sih. 7 . E’olgr. l M . 27
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and by MALIN [7]. The only other rigorous solution of a grating problem, known
to us, is concerned with an infinite set of staggered equally spaced semi-infinite
plates of zero thickness [9-171.
The purpose of the present paper is twofold :
(a) To produce a closed-form solution for a r b i t r a r y angles of incidence, by
reducing the boundary value problem to a singular integral equation of the
CAUCHY type, adapted by one of us to a class of two-dimensional diffraction
problems [18].
'T
'I
Fig. 1.
Cross section of the strip grating
(b) To provide extensive numerical data for the rigorous solution as well as
a thorough discussion of the high frequency properties of the solution including
diffraction theory.
its relation to KIRCHHOFF'S
I n Sec. 2 the reduction of the boundary value problem to an appropriate
transform of the induced surface
singular integral equation for the FOURIER
current density on the central strip is accomplished by means of a conformal
mapping. The solution of the integral equation by means of a factorization
theorem is provided in See. 3, and
procedure in conjunction with LIOUVILLE'S
supplemented by a complete evaluation of the high frequency asymptotics in
See. 4. Most of our numerical results are laid down in See. 5 where the amplitudes
of the first five propagating modes of the RAYLEIGH
representation are graphically displayed for various angles of incidence. The comparison with KIRCHHOFF'S theory is accomplished via the asymptotic modes and via a properly
defined form factor which is numerically evaluated. Finally, in Sec. 6 we
use a modified optical theorem to evaluate analytically and numerically the
transmissivity in comparison with its high- and low-frequency approximation.
The appendices are concerned with infinite product decompositions and boundedness considerations, and give a table of phase functions.
2. Reduction of the boundary value problem to a singular integral equation
Let (z, x ) and (r, 8) be Cartesian and polar coordinates, respectively, in a
plane perpendicular to the infinite set of strips forming the grating and let
m
S
=
u
n=
-m
S,,
S, = (-a
+ 4an 5 z 5 a + 4an, z = 0)
(1)
be its cross section (Fig. 1).Let v be the total electric field polarized parallel to
the edges of the st,rips (omitting the time factor exp (-id)). Then, according to
[l], v is uniquely determined by the following data,:
LUNEBURG
and WESTPFAHL:
Diffraction of Plane Waves by an lnfinite Strip Grating
259
(a) Field equation
(V2
+ k2)v(z, z ) = 0
(k wave number).
(b) Boundary conditions
w(z,S O ) = 0 ( x E 8).
(c) Radiation condition
(d) Edge condition
O w = O(p-l'z) (e = distance from the edges --f 0).
wo is the incident field for which we take the plane wave
wo(x,z )
+
(2d)
~
(ikjaox 1/1 - at z ) ]
(ao = cos 6,),
(3)
propagating perpendicular t o the edges with an angle of incidence Go (cf.
Fig. 1).
By means of HUYCEN'S
principle (HELMHOLTZ'S
formula) we obtain an integral representation for v(x,z ) which satisfies (2a, c), viz.
w(z, z )
= exp
= vo(x,z )
where 3,
Y(+,)
1
+- sJ;y(x'la0) H:l'(k
= i, rv,(x,
-0)
- w,(x, +ON
V(T
-
x')2
+z2)ax',
E s)
(4)
(5)
is the induced surface current density on the strips.
Due t o the periodic structure of t h e grating the current densities on different
strips are related by the FLOQUET
condition
y(x
4anla0) = y(z/ao)e42en0n(1x1 < a ; n = 0, &l,& 2 , ...)
E = ka
and subject t o the edge condition (2d),viz.
(6a)
(Gb)
N x o n : fi= Ef/&.
1/1- a2> 0 for a real and la/< 1. The branch cuts are taken along (1,icu) and
(-1, -iw).
cf. Fig.%.
17*
3,
4,
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I n the last equation use has been made of the FOURIER
representation of the
HANKEL-function H t ’ , and we have introduced the FOURIER
transform of the
current density on the single strip So:
U
ji(cyIcyo)
=
/ y(xlao)e--ikmx dx.
--a
For reasons of convergence the infinite sum has to be ident,ified with
m
2 e--4i€(n-aa,)n = 1
cot 2 4 3 - ,I).
2
n=
(9)
-m
where the hooks indicate that the path of integration in (4“) has to pess above
( ), respectively below ( ), the poles of the cot-function. Inserting (9) in (4”) we
thus obtain
v
n
(10)
The path of integration is sketched in Fig. 2. (In order to simplify our analysis
n integer, corresponding
we temporarily exclude the values a,, = & 1 nn/2&,
+
I
-
I
I
-7
”
111
111
L
I
I
I
I
I
t
1
I
I
I
Fig. 2. Integration path for Eq. (10)
t o the coincidence of a pole of the cot-function with the branch points m = & 1.
I n the final results, however, these values may be included by a limiting process.)
Applying the boundary condition ( 2 b) for IC t So to (10) yields the dual integral
equations
inversion of (8).
where the second equation results from FOURIER
I n order t o indicate the method of getting singular integral equations of the
CAUCHYtype we begin with the (rather trivial) Eq. ( l l b ) , where we replace cy
b y a’, multiply by exp ( - ikmz) and integrate over the 2-intervals (-00, -a) and
LuNEsuRa
and WESTPFAHL:Diffraction of Plane Waves by an Infinite Strip Grating
261
( a , co),
respectively. The results are the singular integral equations5)
-m
f y(&jag)
-m
m
da'
a' - a = 0,
$ ij(a+x0)e-iaa'
--m
da'
___ -0
a' - a
(12a, b)
or, equivalently,
The equations (12 a) and (12 b) are the necessary and sufficient conditions for
their integrands being holomorphic functions in the upper and lower &-half
plane, respectively, and vanishing there a t infinit;y (cf. [19] p. 27). Thus
@(also) is an entire function
(13)
or, more precisely,
1
holomorphic
i
(13a)
(13b)
whereby the holomorphy, of course, is already implied by the representation (8)
and the exact behaviour a t infinity is implied by the edge condition (7).
Proceeding in a similar manner with ( l l a ) we find the integral equation
y(alao)e " i E 1
=
0(a-'l2) (la1+oo)
in the upper/lower
a-half plane,
Our goal is now t o convert (14) into such a singular integral equation which
admits only an entire function as solution. To this end we remark that (14) is
equivalent t o
sin &(a- a )
+-,--sin &(a- a , ) } ~~da'
- STC--------'-,
sln &(a - a,) a' - a
a - a,
(15)
where t'he point a lies outside the integration oontour. This may be seen by
subtracting from (14) the term
vanishing b y virtue of (13).
Bearing in mind (13b) we can deform the path of integration into the loops
C and 6 encircling the branch cuts (1, ioo) and (-1, -ioo), respectively,
~.
5)
We use the notation
1
+
1 - ni S(n' - a),
1
1
97
= 9 _ _ - ni a(&' - a )
a-a
a' - QL
a' - a
or, in integrated form,
a'-F
-
CAUCHYprincipal values being taken for the respective first terms on the r.h.s. (cf. [MI).
262
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(cf. Fig. 3) :
+=
(
or for values of a on the bra.nch cuts6)
c-L‘
+(a’la,)
1-
I
-
cos &(a - no) + sin &(a - cl,) -~
dn‘ sin ~ ( n x,) a’ - a
cos &(a’- a,)
~~
sin &(a- a,)
a - 01,
-
.
(l6b)
The solution of this singular integral equation is indeed an entire function as
may be seen by regarding (16a) as an integral representation for
which
satisfies (12) by means of (16b).
Now, substituting’)
we conformally map the cut a-plane onto the region
from above by the contour D defined as
R- of the !-plane, bounded
D passes from ca exp (37ci/4) through the points - 1, 0, 1t o 00 exp (- ni/4)and
is symmetrical t o the origin (cf. Fig. 4).
Equation (16 b) now reads
1
where a and 00 are rational functions of
48
sin &(a - a,)
/?and / I r ,respectively, cf. (17b).
(19)
f we mean t h a t principal values are t o be taken on either side of the branch cuts.
‘j)
By
7)
1/1.11> 0 for a real and > 1.
LijNEBURG
and WESTPFAHL:
Diffraction of Plane Waves by a n Infinite Strip Grating
263
I
Fig. 4. Integration path in the B-plane
The more general case of distances 2b between the edges of two adjacent
strips (not equal t o their width 2a) leads t o ihe singular integral equation
(7 == kb)
which, for the special cases 7 = 1 E (I
-
4p
p
--
(19a)
1, 2, . . .) can be written in the form
sin c(z - __
no)
-
1
=
z - xo
.
Our method, however, works only for 1 = 1, i. e. a
=
b.
3. Solution of the singular integral equation
Let us define t,wo sectionally holomorphic functions
(20b)
and denote by C“(Blno) and S*(Pla,) their boundary values, if /Iapproaches D
from the semi-infinite regionss) R’ (as well as their holomorphic branchesin R*).
From (IS) we obtain the two equivalent functional equations
1
x - a,
- 1 (Y
--
xo
(B E D )
( 2 1 % b)
(upper and lower signs, respectively) with
L(ocJa,)= (a- a,) cot &(a- no).
8)
R+ is the complement of R- in the B-plane.
(22)
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Using the fact that L(a[ao)as well as D is symmetrical with respect to /3 we
conclude for the increment of arg ( L )along the path D
arg -wajao)lD = 0,
(23)
which is the well known condition for the unicrueness of the solution of the
RIEMANN-HILBERT
problems (21) (cf. [I91 p. 85 [20] p. 1 2 7 ) .
The essential step towards their solution is the factorization
-&la,)
L-(Pla,)
= L+(Pi.o)
with
L*(P1a0)being holomorphic and without zeros in R*
or, more explicitly,
L+(Bla,) = L.-(-Pla,).
Infinite product expansions of L*(PIao)are given in App. A. Using
we rewrite equation ( 2 l a ) in the form
and decompose the term on the r.h.s. of (27) into:
Defining the holomorphic functions
-
11- a!
2i
~
1
+
"0
L-(Bol"o)
1
~
B-
a
in R', we arrive a t the homogeneous RIEMANN-HILBERT
problem
With
(cf. Appendix B)
and WESTPBAHL:
Diffraction of Plane iVaves by an Infinite Strip Grating
1,LXE:BI*RG
'k -
(where the r.h.s. applies t o both L+ and L-) and L G bounded for
whereas C' (B[ao)and X-(plao) are bounded everywhere, we obtain
265
-+ 00,
(32)
The sectionally holomorphie function
m p a01 = Q=(B/no) (p E R ' 1
(31%)
vanishes for )3 --f 00 and is holomorphic in the p-plane punctured a t /3 = f1.
A t these points s2(plao)cannot have a pole nor a branch point due to (32) and the
holomorphic character of f2-'(/3iao)in conjunction with the continuity condition
(30). There remains only the possibility of a reinoveablc singularity. By LrouVILLE'S theorem we conclude
SZ-L(p:KO) =
a-(p'a,)= 0 ,
(34)
and find from (29)
and
llerein C-(Plao) and SJ (B/ao)e m be derived either in a similar manner from
(21b), or simply by using the oddness of C(p/ao)and X(p'ao)with respect t o p
and property (2513) of L i ( P a , ) .
Sow, the difference of the boundary values of C*(p do)or S ' ( ~ ' K according
~)
t o (20) yields
(30)
which by inserting (35) can be written as:
(37)
From (26a) we obtain, replacing p' by -/3'
and adding :
(38)
Returning to the a-plane by means of (17) and writing
L+(P,a,)= LIJI(aln,)
(39)
we find
LIJI(Cxja,)
(xE sheet I, 11). (39')
= esp
C'fC
Lr(ala0)and LI1(a'a0)
are analytic functions of ix on the two-sheeted RIEMANX
surface of 1/m2 - 1. L' is holomorphic and without zeros on sheet I (which we
266
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identify with the originally introduced cut a-plane) and is meromorphic on
sheet I1 while LI1 has analogous properties with sheet I and sheet I1 interchanged. These functions are related by
LI(nja,)LII(ajol,)= ( a - a,) cot &(a- a,).
(Infinite product expansions of
are given in App. A.)
It's not hard to conclude from (39') that the quantities
+
1
L1(ala,)
LI1(a/ao) are holomorphic
(a2- 1)-1/2 [L'(ala,) - Lrl(a/ao)]
within C and
Thus, the solution (37) in the a-pIane reads
and
(40)
(41a)
(41b)
or, by virtue of (40),
This solution, indeed, is an entire function: Eq. (42b) tells us that y(aIao)is
holomorphic in the cut &-plane, but according to (42a) and (41a) there are no
outs a t all. The behaviour (13b) a t infinity is demonstrated by means of the
~ / ~la1
) -+ 00, proved in App. B.
relatiorl L1(ala,)= O ( O L for
For completeness we finally indicate the verification of the dual integral equations ( l l a , b). Denoting the 1.h.s. of ( l l a ) by J [ y ] ,viz.
-m
--m
we insert y(alao)from (42b) and obtain by shifting the path of integration from
the real axis to C and C, respectively, by virtue of (40) :
CtC
The contour integral vanishes since by means of ( 4 1b) the integrand is holomorphic within the loops C and C.
I n order to verify ( l l b ) for 2 > a and x < - a , we insert y(ocIa,) from
(42a) and shift the path of integration to C and 6, respectively:
The contour integrals vanish by virtue of (41a).
LUNEBURG
and WESTPFAHL:
Diffraction of Plane Wavee, by an Infinite Strip Grating
267
4. High frequency asymptotics
The purpose of this section is to obtain a n asymptotic representation of our
solution (42) useful for the high frequency case E 9 1. Writing Eq. (40) in the
form
(43)
where the upper signs refer to n E C, the lower ones to
(39‘)
In L1(clan)=
In [-;(a’
-
%,,)I
01
E 6 , we obtain from
dx‘
-
C
(44)
with
By means of the indentities
we obtain
(Y3
(7
This result has been achieved by deforming the contours C, C of the integrals
containing
~n [(1/v2i)(k1/1Tir
p‘1-
& F 1/ K 7 vi+pLyo)]
11_-
into the contours -C, -6 on the other sheet of tlhe RIEMANN
surface of
- a2
and redefining the radicals, i. e. replacing V1-y n by - 1’1
a. With the aid
of (47) we find
C-t
c
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The last form has been arrived a t by residue calculation since there is no contribution from the arcs closing the path C
C a t infinity. With (48) equation
(44) reads
+
___
_ _
(1/1+ a 1
1- a, + 1/1_-_
a 1/1+ a,)
-~
LI(a]a,)= (I@)
(49)
( a € sheet I).
We now expand the logarithm in the integrand of (45) into a n infinite series
converging on C:
x exp [-
with
7
= 2&(2n
nl(alao)- nl(-a/-ao)]
+ 1)
and
We decompose the last integrand according to
ewa
__
-..--.
v>;
eirp'
_ _ _ _ _am'
___
1
_
- 2ni
a' - or
C
1
2ni
$1-
_ eZqa'
_ _ _ _dor'_ _
a' - a
(53)
C
9
where and f mean that the loop C in the &'-plane is distorted by the pole
a' = a approaching C from the interior and exterior, respectively.
The first term on the r.h.s. of (53) is holomorphic within C and the second
term can be expressed as a FRESNEL
integral
Inserting (53) and (54) in (52) we obtain
(55)
1 B 1)
The asymptotic expansion') ( q la - 1
then yields (this term by term integrated series being again asymptotic)
9)
Representations by means,of asymptotic expansions are denoted by the symbol &.
LUNEBURG
and WESTPFAHL:
Diffraction of Plane Wevee by a n Infinite Strip Grating
269
or, by residue calculation, finally
where the phase functions a t ( % are
) given by
a (x)
ezn(ex+l/l-z/!)
1
-
cc
In
elin%%
c -;7pi
-___--
((%a)
?L=O(,n
with
((iOb)
x = (&/7C)(1 - ao)
or (cf. [21], p. 27)
((iOc)
From (GO) it follows
a l ( x )= a l ( x mod 1)
(tila)
at(. -k 112) = - a&)
I m a J x ) = (-1)1Re a,(l - x ) .
(61b)
(61c)
Therefore, in order to calculate a,(%), i t is sufficient t o know R e a l ( % )for
0 5 x 5 112. Tables for the cases 1 = 0 and 1 = 1 are given in App. C.
It should be noted that the derivative dao/dx does not exist for the discrete
values x = n/2 ( n integer, 20). These points characterize the change of the
n-th order right hand modes from the evanescent to the propagating form
(cf. See. 5).
More explicitly we obtain for e l l
- 011
~-
nI(aIa,)L+&- I + %Ix [a,(.)
&-I/%
$ 1 the following approximation
+-
i(l:+ I - %
L)al(z)s-3/2+
--
0(&-"2)]
(62)
and finally, taking into account (49) lo),
~-~_ _ _
____
LI(a~a,)-=(tiV2i)(1/1+a1/1-Mo+~~1-011/l+OIo)
>< exp
1")
to
{
-
1
2
-- [bo(oljao) &-1/2
+ b,(alao)&-3/q +
I
(63)
0(&-j/2)
Expanding the exponential we obtain the correct asymptotic series for L1(r]a,)u p
O(E-2).
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with1I)
(64b)
If a. can be written as a,";
= nn/2&( n = 0, 1, 2, ...) (cf. [3]) the asymptotic
expression simplifies considerably. Indeed, due to (45) and (60) we have12)
nI(a1at) = ( - 1 ) T k
a,(%,)
= (-1)n
nI(n/O)
(65a)
a,.
(65 b)
For the corresponding discrete angles of incidence the current densities on all
strips become identical (amplitude a n d phase, cf. (6)).
We are now in a position t o write down the (asymptotic) high frequency
development of our solution (42) in terms of elementary functions and the phase
factors a l ( x )for ~ lf
l a1 9 1 and 811 f a0l 9 1:
+ +(-a/-.,)
Y(ala0)= i ( + O )
with
+(also) + 2 ?jn(ala0)
&-n/2
n = O , I , ...
where the first five Gn are explicitly given by11)
.~
?j0(a/ao)
= - v1
___
e-i&(%-%o)
+ a1/1 - aoP-
01
- LYo
e-i&(a-a,)
?j1(aIao)=
11)
12)
-
IT+; IF-;,a(7
Notation: a: = a L ( x * ) ,xi = (&in)
(1'f: -ao).
Notation: x, = (&in)
(1 - a:), az = -aL(&/n).
271
LUNEBURG
and ~VESTPFAHL:Diffraction of Plane Waves by an Infinite Strip Grating
@,a)l,.(
1
,-ze(.-r.)
~~
~
p
- -
1
16
V
T
;
-
-
-
~
a l / l - a.
(67 e)
1
Note that only yo(aiao)is peaked a t a = ao.
It is instructive t o compare this result with the high frequency asymptotics
of the corresponding solution for a single strip (width Za, cf. [ZL']) :
-
pe-n/r)
~-
-~
dl
s\/';
G 11 +
a0
,-i&+n,)E-3/2
+ O(&-6/0-).
(68)
(1- a ) (1- ao)
There is agreement of the dominant term of both solutions, this term referring
t o the two (noninteracting) half planes complementary to the strip. The higher
order terms for the single strip describe the interaction of these two half planes,
whereas for the grating there is a predominant interaction with the other strips.
5. Node representation and comparison with Kirchhoff's theory
5.1. Exact modes
The only singularities of the integrand of the field representatiorl (10) are
the poles of the cot-function. We thus obtain by residue calculation the well
expansion
known RAYLEIGH
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Fig. 5
Fig.5--12. Amplitudes ITn\of propagating RAYLEICH
modes versus &in= 2 a / l for 8 different angles of incidence t?o (2a is the common width of strips and gaps, I the wavelength)
Fig. 6
-
LUNEBURQ
and ~VESTPFAHL:
Diffraction of Plane ITaves bp an Infinite Strip GratJing 273
25
2
2,s
3
3,5
E,&,
n h
Fig. 7
f
lT,i
Fig. 8
with
(an,,= 1 for 72 = 0 and is zero otherwise)
4
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Fig. 9
t
IT,I
0.6 0.5 -
0.4 -
0,3-
These representations tell us that through any point ( T , 6) travels a finite
number of propagating modes (undamped plane waves) corresponding to
Ia,&l5 1, i.e. t o n- 5 n 5 n+ with l 3 )
f [ ( 2 & / 4 (1 TaCx0)l.
the modes (la,J > 1) is damped and
721 =
(72)
The rest of
can be neglected for kz
1.
I n the high frequency domain E & 1 (where, by virtue of (72), these sums become
practically useless) the approximation of Sec. 5.3 applies.
13)
[a] means the integer part of a
L~NEBUR
and
C WESTPFAHL:
Diffraction of Plane Waves by an Infinite Strip Grating 275
0
'
0.5
15
1
2
2.5
3
3;5
€_la,
5-A
4
Fig. 11
/
4',
=ZOO
-7---'-'
2
2.5
3
4
3.j
L=&r h
Fig. 1 2
The solution for the complementary excitation (viz. the case in which the
magnetic field G is parallel t o the edges of the strips) is obtained in a similar
boundary value problem) or by applying the
manner (solution of a NEUMANN
rigorous version of BABINET'S
principle to the above solution. The result is
m
G(r, 8)=
11=
2
+n(Lyo) e ? k r c o s ( e - 8 , )
-_
- 1 -
eakreos(rP-rP,)
with
(z
> 0)
(73 a)
m
+2
n=
-m
n
(a
O
ezlircos(iY+rP,,)
( z < 0)
= ,a
, - %(a,) = - ( E l l n R n ( 4 .
(73b)
0 this agrees with the results of BALDWIN
and HEINS[l]and WEINSTEIN [2, 31 (of. App. A).
The amplitudes IY',l, lT.tll and IT*.zl of the first five propagating modes are
shown in Fig. 5-12 for 8 different angles of incidence as functions of E ( E 5 4n).
TnbO)
For a.
18*
=
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By virtue of (6Yb) the propagating forms of themodesT,,,begin a t E = nlnl/
2 ( 1 T a0). With increasing E the amplitudes lTol, ITzn]and IT2n+l/approach
values l / 2 , 0 and l/nl2n+ll, respecbively
asymptotically the KIRCHHOFP
(cf. Sec. 5.2.). It should be noted that for ( n / s ,a,J E T , where T is the triangle { ( l , O ) , (4/3, l/3), (2,O)) in the ( n / s ,&,)-plane, the amplitudes of TI and
Y'-l are exactly equal, cf. Fig. 6. This statement, which may be verified from
(71b) and App. A, corresponds physically to the situation where for ( n / s ,ao)
E T the modes T , and T-,are (apart from T o )the only propagating mode forms.
5.3. KIRCHHOFF
modes and high frequency modes
For the sake of simplicity we confine ourselves t o normal incidence (ao = 0)
and consider KIRCHHOFF'S
formula ( Lis the union of the grating slits)14)
wK(z,Z )
=
i
O ( Z )v,,(x, Z ) - 4 J (u0 8,
L
+ uo2)HI:) (k l ( z - x')~+ z 2 )dx'
or
We have to compare these coefficients with the asymptotics of Sec. 4 applied
t o the rigorous coefficients (71) :
As expected, KIRCHHOFF'S
formula yields only the dominant term of the correct
high frequency asymptotics.
5.3. Form factor
I n this section we point out that our result may also be interpreted in
terms of a finite N-strip grating in the E N % 1approximation. To begin with we
formula for a finite gratJing (consisting of N strips
write down KIRCHHOFF'S
11)
O(2) = (1/2) (1- 2/12]).
LUNEBURG
and WEESTPFAHL:
Diffraction of Plane JVaves by a n Infinite Strip Grating
277
placed symmetrically to the origin, N odd)
vF(x,
= eik(wA$izqz)
(78%)
which by the method of steepest descent reduces to (x t i z = r
0 < 6 < n):
eiA,
r $ as,
I<
iki cos(4--it0)
vlv+
e
8
+ i(sin 6 + sin 8,) sin [c(cos
;c
1ioy79,
-
Y
-
-
On the other hcnd our rigorous representation (4") for an infinite grating may by means of
(in the sence of distributional convergence) - be written in the form
-__
v(z, z )
eik(w-Lv1-4?:z)
The comparison with (78a) suggests t o consider the expression
Vlv(z,
z ) = eih(lo7t dl-niz)
as a n approximation for a finite grating (at least for N $ 1)taking into account
the interaction of its individual elements. Thus, we obtain by the method of
steepest descent ( r 9 a N , 0 < 8 < n):
vAT(r,8)--f
ezkrcos(-it--4,)
Hence, the form factors multiplying the common grating pattern
are to be defined as follows:
KIRCHHOPF'S
theory :
F"(BJ6,)= i(sin 6
+ sin 8,) sin cos
--
8
6 )]
8,
[ ~ ( C O S - cos
--~79 -- cos
278
Annalen der Physik
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*
1971
our theory:
P(616,)= ,(cos 6lcos 79,).
(83b)
I n KIRCHHOFF’S
theory the form factor is given by the FRAUNHOFER
diffraction
pattern of a single element (strip). However, in view of the high frequency
Fig. 13-16. Grating form factor IF1 for 4 different angles
of incidence 8, and 4 different values of E versus the diffraction angle 8
Fig. 13
Fig. 1 4
L~NEBIJRG
and WESTPFAHL:
Diffraction of Plane Waves by an Infinite Strip Grating
279
Fig. 15
relations (66), (67) and (tjS), in our theory besides the dominant term
8
po($j]$jo)
= 2 i sin
(
+ 8, sin [e(cos 8 - cos So)]
1
-
~~
cos 6 - cos8,
2
6 - 6, cos [E(COS 6 -- cos .a,)]
sin (
7
cos )
6 - cos 8,
(8312)
+
of the form factor, wich is also given by the diffraction pattern of a single strip
(and agrees with KIRCHHOFF’s expression in the vicinity of the direction of incide nee), there are’ higher order terms
etc. which take into account the mutual interaction of the other strips. (These
remarks indicate that our approximation yields the dominant term of the
EN $ 1 approximation even for N = 1.) The interaction is described by the
phase factors a , which involve a summation over the grating elements, cf. (60 a).
280
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*
1971
Fig. 17. Comparison of form factors IF1 and jFKl versus 6 for F = 47c and;? different angles
of incidence 8, = 90" (--)
and 8, = 30" ( - - -). The arrows indicate the diffraction
orders n
The absolute value of the form factor P(B16,) is shown in Fig.13-16 for
4 different angles of incidence and for various values of c as function of the dif0 ) ~KIRCHHOFF'S
value lFK(6[6,)1
fraction angle 0.Fig. 1 7 compares ~ F ( 8 ~ 8with
for Oo = 90" and 30" in the frequency domain E = 4n. The agreement decreases
with increasing @-distance from the direction of incidence and mith increasing
6,-deviation from normal incidence. The peaks of the common grating pattern G
(viz. the orders of diffraction) are marked by arrows.
The most striking fact: in contrast t o KIRCHHOFF'S
theory the spectra of
order 2% ( n $. 0) do not vanish in our approximation, judged both from the
(asymptotic)mode and from the form factor aspect.
6. Transmissivity and reflectivity
The transmissivity t and reflectivity
Q
of the grating are defined by
L ~ V E B U Rand
G WESTPFAHL:
Diffraction of Plane Waves by a n Infinite Strip Grating
281
where the summation extends over all propagating modes. The individual terms
of the sunis represent the relative power of the trainsmitted and reflected modes,
of order n (i. e. the power leaving a unit area of the grating in thedirection 6,,
dividcd by the power arriving a t a unit area due t o the incident wave with the
angle of incidence 8,).
We now proceed to derive a modified optical theorem appropriate t o grating
problems. From (G9a) we obtain for z = 0, 1x1 5 a the identity
(85)
which me substitute in (8) :15)
Since y n is real for n- 5 TL 5 n and purely imaginary otherwise we obtain
I m V(+O)
=
1
?)+
1
2 r /Y("%laO)l2.
~ = n -Jn
(87)
Taking into account the definition (701 ) of Tn(c~o),
we rewrite ( M a ) with the
aid of (87) in two equivalent forms (mod fied optical theorem):
1
n+
C
1
- iv(~n/.o)12.
16E2Y, n==%-Y n
=I-
~
The reflectivity @(ao)
follows from the law of conservation of energy
Tho) 3- e b o ) = 1
(89)
which is easily verified by means of (84), (87) and (70b).
Substituting from (42) we obtainI6)
This yields the high frequency asymptotics (with the aid of the develop1 a,) % 1:
ments of Sec. 4) for ~ ( &
t(aO)
with
II
. 1
2
~
+ 4&(11 -T I m exp (bo~-lI2+ bl&--3'2) + O ( E - ~ ' ~ )
~
- 010)
(91)
= a. in (G4).
a* denotes the complex conjugate of a.
From (Tla) follow simple relations betwcen the total transmitted and reflected
power and the amplitude of the dominant mode :
16)
4 4 = Re To(oc,), e b o ) = - Re Ro(oco).
(90')
Annalen der Physik
09
~
t
T
08 07 -
*
7. Folge
*
Band 27, Heft 3
*
1971
4-900
j11._
80°
!/
60O
Fig. 18. Transmissivity t versus c / n = 2a/A for 8 different angles of incidence 6,.(2aIA =
strip widthlwavelength)
The low frequency approximation follows from (90) and (A9) by an &-power
expansion for E < 1:
2(ao)F
y -cos (C1&+ C,&3 + C3&5)+ 0(&8)
2
( 92)
and
m
((n)= 2
k= 1
(94)
LUNEBURG
and WESTPFAHL:
Diffraction of Plane Waves by an Infinite Strip Grating
283
I n Fig. 18 we have plotted t versus e l n _< 4 for 8 different angles of incidence.
The comparison with the high frequency asymptotics is shown in Fig. 19: There
is excellent agreement for (e/n)(1 - ao) 0,8. The also shown low frequency
approximation improves a n old result due t o LAMB[23].
2
t
&=goo
5
/
0.6
0.6
0.4
0,2
///
I
Fig. 19. Test of the quality of the high and low frequency approximations (91) and (92),
respectively, for the transmissivity t (dashed curves) for 3 different angles of incidence @,,
Appendix A . Infinite product factorization
I n order t o decompose L(alao)= (a - ao)cot &(a- ao)according t o (24a),
we use the infinite product expansions
284
Annalrn der Physilr
*
7. Folge
*
Band 27, Heft 3
*
1971
and
with
In ( A l ) and (A2) the last product is to be extended over the four possible comand - signs. Hence
binations of the
+
2
+ (4.).(
x%o
X%(%
-
l/Wx % b o - ( 4 4 ( n - l i 2 ) )
+ nni4
x
3
.
0
- nn/4
and
or, by making use of (A3), and passing t o the a-plane
~
-t
~
_
_
1/1 - 2 4 1 q= a O ) / n n ]
(where sum and product extend over the two terms with upper and lower sign,,
”)
+(a),X ~ ( C U>
) 0 for 01
respectively.
> 1;the branch cuts are taken along (1,ioo) and (-1,
-ice)>
L ~ K E B U and
R G WESTPFAHL:
Diffraction of Plane IVaues by an Infinite Strip Grating
285
respectively) and
~
~
_
~
~
~
~~
_
~
_
_
I m !,'I - %(I
a,)/nn: 5 0.
+ a,)/;d
we find simply
For n 2 2 4 1
&,(&,)
E
_
(AGb)
11- [a&(&
5 n/2(l + ao)there is a furt,her simplificatjion:
I&,(al&o)i =
For a = a. and
_
- ao)/n;d]2.
(L47)
(A8a)
= e-i'%(ao)
with
and
with
For
=
0 t>hequantity
LI acquires the explicit form
(AlOa)
(AlOb)
Finally, using a somewhat more explicit notation suggested by (AG) the
equivalence (2.5 b) in the &-planereads as follows
= LI (01, -
v1-
azla,).
(All)
Appendix B. Houndednoss considerations
The investigation of the behaviour of L-(/ljoc,'l in the vicinity of /l = 00,0 and
0 == 1,which are boundary points of the coniour D ,requires by (17) the examination of LI(alix,) in the vicinity
of a = & 1and m , respectively. To this end
-~
we represent (50) as (0 5 arg l/afz - 15n/a)
Obviously we have
%I(
-
1I a,) = 0.
286
Annalen der Physik
*
7. Folge
*
Band 27, Heft 3
*
1971
h
Now, if p(t) E H ( l ) l s )on the closed arc ab, the CAUCHYtype integral
h
behaves like
where Qo(z) is bounded a t z = a for L > 1 / 2 (cf. [19] p. 53ff. or [20] p. 73ff).
Since (Bl)is of the form of (B3) with p(z) E H(1)we conclude
nI(a/ao)
= O(1) (a --f 1).
I n order t o examine the behaviour of nl(alao)in the vicinity of
stitute a’= l/y‘, LY = l / y and consider the behaviour of
c*3,
(B5 )
we sub-
in the vicinity of y = 0. This integral is of the type
h
a
h
where p(z)is HOLDERcontinuous on the closed arc ab. For z + a it behaves like
@ ( z ) = - v(a)
---;In ( z - a )
2na
+ Q0(z)
(B8)
with Qo(z) bounded a t z = a (cf. [19, 201 1. c . ) . Since the logarithm in (B6)
vanishes for y -+ 0 it follows
nI(alao)= O(1)
(la/-+ 0 0 ) .
H ( 1 ) means the class of HOLDERcontinuous function with the index 1.
(B9)
LUNEBURQ
and WESTPFAHL:
Diffraction of Plane Waves by en Infinite Strip Grating
287
Collecting results we obtain taking into account (49)
which, in the b-plane, correspond t o
I n view of (25b) this behaviour is also valid for L-f-(/?lao).
Appendix C. Table of phase functions
I n Table 1 and Fig. 20 the real part of the first two phase functions a o ( x )
and a,(%),cf. (GO), is given for 0 5 ~t l / 2 .
continuation
Table 1
x
0,00
0,01
0,02
0,03
0,04
0,05
0,OG
0,07
0,08
0,09
0,10
0,11
0,12
0,13
0J.4
0.185
Oil6
0,17
0,18
0,19
0.20
-0,6736
-0,4422
-0,3390
-0,2708
-0,2144
-0,1651
-0,1218
-0,0828
-0,0471
-0,0137
0,0177
0,0471
0,0745
0,1003
0,1247
0,1481
0,1707
0,1927
0,2142
0,2350
0,2552
0,2746
0,2932
0,3110
0,3281
0,3446
-0,441
-0,474
-0,496
-0,516
-0,531
-0,543
-0,551
-0,557
-0,562
-0,564
-0,564
-0,562
- 0,558
-0,552
-0,544
-0,534
-0,524
-0,512
-0,500
-0,487
-0,472
-0,457
-0,439
-0,421
- 0,401
-0,379
0,26
0,27
0,28
0,29
0,30
0,3i
0,32
0,33
01,34
0,35
0,36
0,37
0,38
0,39
0,40
0,41
0,42
0,43
0,44
0,45
0,46
0,47
0,48
0,49
0,3608
0,3765
0,3921
0,4075
0,4228
0,4378
0,4526
0,4671
0,4812
0,4949
0,5081
0,5210
0,5336
0,5460
0,5583
0,5707
0,5830
0,5950
0,6068
0,6183
0,6297
0,6411
0,6524
0,6624
0,50
0,6736
-0,356
-0,332
-0,306
-0,280
-0,254
-0,227
-0,199
-0,171
-0,142
-0,112
-0,081
-0,049
-0,016
0,018
0,053
0,088
0,124
0,161
0,199
0,238
0.277
0:317
0.357
0;400
0,441
References
[l] BALDWIN,
G. L., and A. E. HEINS,Math. Scand. 2 (1954) 103.
[2] WEINSTEIN,
L. A., Zh. Tekn. Fiz. 25 (1965) 841, 847.
[3] WEINSTEIN.
L. A., The Theory of Diffraction and the Factorization Method. Boulder:
The Golem Press 1969.
[4] GNATO OW SKI, fir. S.. Dokl. Akad. Nauk SSSR 22 (1934) 21.
[5] KARP,S. N., and W. E. WILLIAMS,
Proc. Camb. phil. Soc. 63 (1957) 683.
[6] AORANOVICH,
2. S., V. A. MARCHENKO
and V. P. SHESTOPALOV,
Zh. Tekn. Fiz. 32
(1962) 381.
288
Annalen der Physik
*
7. Folge
*
Band 27, Heft 3
*
1971
[7] MALIN,V. V., Radiotekhn. Electr. 8 (1963) 211.
[8] CARLSON,
J. F., and A. E. HEINS,Quart. Appl. Math. 4 (1947) 313.
[9] HEINS,A. W., and J. F. CARLSON,
Quart. Appl. Math. 5 (1947) 82.
[lo] HEINS, A. E., Quart. Appl. Math. 8 (1950) 281.
[ll] BERZ,I?., Proc. ZEE 98 (1951) 47.
[12] WHITEHEAD,E. A., Proc. I E E 98 (1951) 133.
[13] HEINS, A. E., J. Math. Mech. 6 (1957) 401, 629.
[14] MEISTER,E., Arch. rat. Mech. Anal. 10 (1962) 67, 127.
[15] BAKLANOV,
E. V., Dokl. Akad. Nauk SSSR 163 (1963) 570.
[16] IGARASHI,.A.,J. phys. Soc. Japan 19 (1964) 1213.
[17] IGARASHI,
A., J. phys. SOC.Japan 26 (1968) 260, 607.
[IS] WESTPFAHL,K., Ann. Physik, Leipzig 4 (1959) 283.
[19] GAKHOV,
F. D., Boundary Value Problems. Oxford: Pergamon Press 1966.
[20] MUSCHELISCHWILI,
N. I., Singuliire Integralgleichungen. Berlin : Akademie-Verlag
1965.
[21] ERDELYI,A., W. MAGNUS,F. OBERHETTINGER
and F. G. TRICOMI,Higher Transcendental Functions Vol. I. New York: McGraw-Hilll953.
[22] LUNEBURG,
E., u. K. WESTPFAEL,Ann. Physik, Leipzig 21 (1968) 12.
1231 LAMB,H., Hydrodynamics. Cambridge: University Press 1932.
Die numerischen Rechnungen wurden auf der IBM 7040-Anlage des Rechenzentrums der Wniversitat Freiburg durchgefiihrt.
F r e i b u r g / B r ., Institut fur Theoretische Physik der Albert-Ludwigs-Universit at.
Bei der Redaktion eingegangen am 5. Februar 1971.
Anschr. d. Verf.: Dr. E. LUNEBURG,
Deutsche Forschungs- und Versuchsanstalt fur Luftund Raumfahrt (DFVLR), BRD-8031 Oberpfaffenhofen,
Post WesslinglObb.
Prof. Dr. K. WESTPBAHL,
Inst. f. theoret. Physik d. Univ. Freiburg
BRD-78 FreiburgIBr., Hermann-Herder-Str. 3
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