DIFFRACTION THEORY A CRITIQUE OF SOME RECENT DEVELOPMENTS

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NEW YORK UNIVERSITY
WASHINGTON SQUARE COLLEGE OF ARTS AND SCIENCE
MATHEMATICS RESEARCH GROUP
RESEARCH REPORT No. EM-50
DIFFRACTION THEORY
A CRITIQUE OF SOME RECENT DEVELOPMENTS
by
C. J. BOUWICAMP
CONTRACT No. AF-19(122)42
50X1
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New York University
Washington Square College of Arts and Science
Mathematics Research Group
Research Report No. EM-50
DIFFRACTION THEORY
A CRITIQUE OF SOME RECENT DEVELOPMENTS
bY
C. J. Bouwkamp
The research reported in this document has been made possible through
support and sponsorship extended by the Geophysics Research Directorate
of the Air Force Cambridge Research Center, under Contract No.AF-19(122)42.
It is published for technical information only, and does not necessarily
represent recommendations or conclusions of the sponsoring agency.
April 1953
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?
_ABSTRACT
A number of recent developments in the theory of diffraction
of electromagnetic waves, particularly hose dealing with apertures
in plane conducting screens, are review d. The subjects treated
include modifications of Kirchhoff's th-?ry, the theory of small
apertures, Babinet s principle for plan obstacles, variational
principles, and singularities at sharp dges.
For completeness, a discussion f m an alternative view-
point of the problem of diffraction by ? aperture by Professor
N. Marcuvitz has been included in this eport.
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iii ?
Table of Contents
Section rage
I Introduction 1
I/ Kirchhofes Theory of Diffraction 3
III Modified Kirchhoff Theory 6
rr Braunbek's Modification of the Kirchhoff Theory 15
V Variational Formulation of Scalar Diffraction Problems 23
VI Rigorous Form of Babinet's Principle on Electromagnetic
Diffraction Theory 39
VII Diffraction by a Small Aperture in a Perfectly Conducting
Plane Screen 44
VIII On Copsonts Theory of Diffraction 59
IX Diffraction by Narrow Slits 67
X Diffraction by an Aperture ins. Planar Screen
by
N. Marcuvitz
References
77
89
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I. Introduction
The theory of diffraction has three major f
(2) radio-wave propagation, and (3) acoustics.
sidered are usually of the same order of magnitu
while in the case (1) the wavelengths are umuall
A further difference between these three fields
sent tally vectorial problems, while the problems
However, in most applications of diffraction the
is considered as a scalar wave phenomenon (polar
ror example, calculations on diffraction by aper
Kirchhoff's mathematical formulation of Huygens'
shown that this is justifiable when the waveleng
size of the aperture. Polarization cannot be ig
where the wavelength is of the same order of mag
to cover this is by using an electromagnetic equ
Kirchhoff formula. This scalar formula may be a
tangular components of the electric and magnetic
six wave functions so obtained should satisfy Ma
introduce certain contour integrals along the ri
The theory of Kirchhoff and Kottler are poo
diffraction theory (wave equation plus boundary
range because they do not correctly describe the
aperture and the edge.
elds of application: (1) optics,
(2) and (3) the wavelengths con-
e as the diffracting obstacle,
small compared to the obstacle.
s that (1) and (2) involve es-
involved in (3) are mainly sealer.
ry to classical optics, light
zation effects are ignored).
urea are usually based on
principle. Experiments have
Ii is small in comparison to the
ored in radio-wave propagation,
itude as the aperture. One way
Talent of the scalar Huygens-
!I ?
lied to any of the six rec-
vectors. In order that the
ell's equations we have to
of the aperture (Rattler).
substitutes for rigorous
onditions) in the quasi-optical
field in the vicinity of the
In the extreme ease of :.ry long waves they entirely
the field far from the aperture
fail to predict the correct order of magnitude o
(Rayleigh).
The purpose of this report is to comment on some of the new developments
in diffraction theory. Various modifications of the Kirchhoff theory have re-
cently been proposed. Rayleigh's potential-appr
oh has been extended to
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higher-order approximations. The integral-equation technique has been developed
extensively, and variational methods have shown their usefulness in a great number
of problems. Also, the rigorous form of Babinet's principle in plane obstacle
diffraction theory has been obtained. New insight into the character of singu-
larities at sharp edges has profoundly influenced many aspects of diffraction
theory.
A number of important recent developments have not been treated in these
lectures.
Among these we can mention the exact solutions recently obtained for
diffraction by circular apertures and disks, and the Wiener-Ropf technique which
has proved its power in the solution of certain waveguide problems.
Only steady-state problems will be discussed. The time factor is under-
stood to be exp(-imit). For a general introduction into diffraction theory,
which includes descriptions of the early work by Kirchhoff, Kottler and
Rayleigh, see:
Baker and Copson, The Mathematical Theory of Huygens' Principle, Oxford,
Clarendon Press, 1950.
Sommerfeld, Vorleeungen uber theoretieche Physik, vol. 4, Pptik
Wiesbaden, Dieterich Verlag, 1950.
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II. KArchhoff's Theory of Diffraction
let E be a screen of vanishing thickne
plane s = 0. Consider a system of sources in
the screen were absent, these sources would p
point P. The actual field u(P), produced whe
is the sum of u(P) and ua(P), where ua(P) is
secondary sources on E. By Green's theorem
ikr eikr
(P \ te ) _
d? ' Alr as r
where the integration is over both faces of
d Vurther, afatt denotes differentiation
point coordinates in the direction of the no
Kirchhoff made the following assumptions
au
(i)
o as at,
L.111 = n
9 ay ?0
on S (illuminat
(on dark s
The total field then becomes, in Kirchhoff 's
(2.1) x5(P) = u(P)
eikr ano
r an
4.n
s covering a finite part of the
the left half-space (z < 0). If
?duce a wave field u(P) at the
the diffracting screen is present,
the diffracted field due to the
, and r is the distance from P to
ith respect to the integration,-
1 to E drawn into free space.
d side of screen)
de of screen).
pproximat ion,
a ieikr, I d r
on r /
where now the integration is only over the ill inated part S of , while n
refers to the normal of S drawn into the shado region (z >0).
Serious objections can be raised against irchhoff's theory-1:13. In fact, as
we let P approach a point q on the screen E, quation (1) fails to reproduce the
assumed values u and auo/an. This can be sho n with the use of the following
theorem:
1 a
Lvs uo an
ikp 8
S
d = ? LITT z
u
2
ikr
P
?
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Then the limiting values of KS(P) are
(2.2)
4( ikr au
ou + ..1.- 6 .....9.
, o 4r r an
where C = 1 or 3. according as P is on the
2 2
Consequently, only-if
z .
1 u 4. 1 .1sikr auo
2 6 4n r an
d E =
dark or on the illuminated side of
will the limiting values of KS be identiea with the assumed values U01.However,
this condition cannot be fulfilled for arb trary S and uo, as can easily be seen
If we take two screens, one inside the 0th
Then, using the above condition, we obtain
ikr 5u
1 v? e o
4r r Bn
d E
where (the "difference" between the two
au
It would then follow that --II= 0 on 3. B
an
au
fore --2 would not necessarily be equal t
an
This shows that Kirchhoff's procedure is no
follows from a consideration of the limiti
au
(2.1) C
an
(k2+ + a2 1 1"-
2c2 ar2 / 4r
Generally it can be said that the reason f
theory is that u and aufay cannot simultane
equation Au + k2u = 0 is elliptic.
If Kirchhoff's boundary conditions on
world jump by the amounts uo and auo/an res
jumps are produced by Ks, as may at once be
(1) for the limiting values on S. This is
pretation of Kirchhoff's formula (1) in tha
of a boundary-value problem, but of a saltu
r, and subtract their field equations:
riginal screens) and .126 are arbitrary.
t U0 was arbitrarily chosen and there-
zero. Hence we have a contradiction.
self-consistent. The same conclusion
values of aKs/an, which are
eikr
?
the inconsistency of Kirehhoffle
usly be prescribed on E since the
were exact, then u and au/an
ectively across S. In fact, these
verified from expressions (2) and
n accordance with Kottler's inter-
Ks is the rigorous solution, not
problem[2].
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We shall now discuss the Complementary problem of diffraction by an infinite
plane screen with a finite aperture A. Aastming the same primary field uo, we
apply equation (1) where now S means the complement of A. To avoid the slight
difficulty that S is no longer finite, it may be necessary to assume that the
imaginary part of k is positive. Now the integral over S is equal to the integral
over S + A minus the integral over A. The integral over S + A equals -u(P)
if z >0 and equals zero if z 0 and assuming that u and?au/8n are zero
on the dark face of the screen and that in the aperture they are equal to the
unperturbed values. However, Kirchhoffia original method is preferable since
it avoids the difficulty that (h) does not reproduce the values assumed in the
aperture but rather the values (6).
For complementary problems (A = S), it follows from (1), (4) and (5)
that
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Ka + Ks = no
K + K = 2uo
a
This is Babinet's principle in the sense
It has often been suggested[h] that
term of an accurate solution of a bounda
mations. This was disproved by Franz
Es]
Franz derived equations (24) and. (5) as f
given by Schelkunoff.) A zero-order app
(z >0). This agrees with the boundary
wave equation is violated in the apertur
by a correction term arising from the se
term equals the right-hand side of equat
the new interpretation, Franz[7] devised
is applicable for all wavelengths. In t
an open question whether Franz's theory
III. Modified Kirchhoff Theory
11.1.1111?1110M?01??????????????
Modifications of Kirchhoff's theory
diffraction problems*. One aim of this
a means for distinguishing between the tw
(I) scalar wave function vanishing on a s
scalar wave function vanishing on a ri id
The modified theory makes use of the two
space, which are known explicitly.
or z >0
or z 0) then becomes
(3,3)
(3.10
(3.5)
( 3.6)
duo
11.(p) . 12.0(p) + j.... ./(e
sl ikr d Z ,
27, s r 8n
, feikr au_
al
R(P) = l' / v dE
- 2n A r an ?
ikr
Rs 2 (p) uo (P) - 2:n7 uo dan (2?r ) d E
,L.
ikr
R62(P) = 1 fu d E
7n A on r
?
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Unlike Kirchhoff's theory, the mod fied theory is self-consistent[81. The
reeson for this is that in the modified theory it was sufficient to assume
boundary values for either u (in the ca. e of R2) or au/an (in the ease of 111)-.
In fact, all values assumed to exist at z = +0, whether behind the screen or
in the aperture, are exactly reproduced by the Rayleigh solutions when P approaches
the plane z = 0 from the right.
The analytic continuation of the yleigh solutions into the illuminated half-
space are easily obtained. Equations (3) and (5) remain valid for points P to the
left of the plane z = O. On the other nd,.equations (4) and (6) are to be re-
placed by
(3.7)
and
(3.8)
R]?(P) = u0(P) u(-P)
eikro
r an
ikr
Ra (P) = u0(P) + u (-P) + d
o an r
respectively. Here u0(-?) means the value assumed by uo at the reflection of P
in the plane z = 0; u(P) u0(-P) is tiae zero-order reflected field in the sense
of geometrical optics.
Like Kirchhoff's solution, the Ra leigh solutions are exact solutions of
saltus problems. The functions R92 an Ra2 jump from 2a0 on the illuminated Dice
of the screen to zero on the dark face. Similarly the normal derivatives of Ftel
and Ral -jump across the screen from 2 /an to zero. Further, the Kirchhoff
solution is just the average of the twe corresponding Rayleigh solutions, viz.)
K= (R 1 ) - (Ral Re2)1
a 2 al a2 -
s 2
while Robinette principle now assumes ither the form
B1 + R1 u (z > 0), o Ral Bel = 2ao (P) uo (-P) (2: < o) ,
=
or the form
= 20.a2 s2 0(P) + 1.1.0(.4) ?: o)
a2 Bs2 = uo (2 o),
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As we mentioned before, one aim of the modified Kirchhoff theory was to furnish
a method for distinguishing between the two principal boundary-value problems pre-
viously noted. Accordingly, since aRai/an = 0 on the dark face of the screen, Rai
of equation (4) was proposed [9] as an approximate expression for the diffracted
field behind the aperture in the acoustically rigid screen for the incident wave
110 (i.e., Raiz 02). (01 and 02 are defined in equations (9) and (10)0 Similarly,
Ra2 was suggested for the diffracted field behind the acoustically soft screen
(i.e., Ra2 ? It is obvious that these approximations will be accurate immediate-
ly behind the screen but poor in the vicinity of the aperture. The approximation
is a complete failure if it is extended to the respective analytic continuations
through the aperture into the illuminated space, because on the lit face of the
screen we have aRai/an = 2 auo/an and Ra2 = 2u0. In fact, the reflected-field
terms in equations (7) and (8) suggest the opposite correspondence between the
Rayleigh solutions and the solutions of the boundary-value problems. We shall
now discuss this correspondence in more detail.
Let u (x22 y z) denote the wave function for the diffraction of the primary
al
field u (x2y2z), incident from the left (z < 0), through a finite aperture in a
[10]
perfectly soft plane screen. Then
uo(x,y,z) - uo(x,y,-z) + V1(x?y2-z) (z < 0)
(3.9) ual
01(x,Y2z),
where V12 defined for z > 0 only, has the following
(z > 0)
properties: (i) V1 is a solution
of the wave equation; (ii) Vi = 0 on the dark face of the screen; (iii) 01 is
regular at infinity (Sommerfeldts radiation condition); (iv) a01/az = 3u0/8z in
the aperture; (v) 01 is uniformly bounded, and 'grad 911is integrable over any
finite part of three-dimensional space, including the rim of the aperture. ?
Let ua2(xyz) denote the corresponding wave function for an aperture in a
perfectly rigid screen. Then
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(1.10
2 =
liuo(x,y,z) + uo(x
02(x9Y,z),
3r10,z) -
02(x,y,-z) (2 0)
(z o)
where 02 is also defined :or z >0 and 6 similar properties as 01 except that
(ii) and (iv) should be -replaced by: (ii') 802/8z = 0 on the screen, and (iv!)
f62 uo in the aperture. Existence theo ems and questions of uniqueness will
not be discussed for the time being. lie it suffice to mention that the pro-
perty (v) ensures that no energy is radi ted by the singularities at the rim.
It is difficult, if not impossible, to determine the flanctions 01 and 02
for an aperture of arbitrary shape. The trouble is that either 01 or 02 solves
a mixed boundary-value problem: 0 and 8 /8n are given on mutually complementary
parts of the plane z = 0 (see (ii) and ( )). However, by virtue of (ii) and (ii')
and Rayleigh's formula, we have for any i.erture A the following relations:
ikr 80 ikr
(7'11) 01 = 2n r
4 lied 1 A_ (e 1 d r".
A
2n an r d
A
rol,r if we assume that the unknown values f 01 and802 /an in the aperture may be
replaced by the respective unperturbed va ues of the incident wave, we find that
01? Ra2' 02z, Ral (z > 0). If this is s bstituted into equations (9) and (10)
we obtain
ual Ra2 (z >0),
-- - Ra2 ' (z 0)
(3.12)
uel2re?Ral (z >o), 11a2- Hal ' (z 0 the approximation (12) is ident cal with that discussed previously.
It should be noted that the approximate s lutions do satisfy the correct boundary
conditions at the screen. However, they a e not analytic functions: either the
normal derivative or the approximate solution itself is discontinuous in the
aperture.
In deriving the approximations (12) t e properties (ii) and have been
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11
-used. Yet almost equally characteristic for planar boundary-value problems
the properties (iv) and (ivi). The latter express that for problem I (
the screen) duNI is unperturbed in the aperture.
are
= 0 on
Starting from this point of
view we can derive a different set of approxiMationst1. From Bayleighls
formula (1) and property (iv) it follows that
Vhfortunetely,
ikr u
2r r
A 2r
ikr bco
e s?
ots r "
?
the values of 01An are not known on the infinite screen S. If
we assume that they are approximately equal to zero,
we arrive at 01 Ra1. By
a similar reasoning in the case of problem TI, we get 027/... Ra2. It is not diffi-
cult to verify that this ultimately gives
everywhere
produce the
R
nal'e Re1 a2 a2
in free space. The approximate solutions are now analytic and they
correct (unperturbed) values of au41/bn and uin the aperture, but
they violate the correct boundary conditions at the screen. Insofar as an accurate
approximation to the field is more important in the aperture than in the vicinity
of the
(12).
screen, the approximations (14) seem preferable to the opposite relations
An alternative way of showing the close relation between the original and
modified Kirchhoff solutions is the following. Consider, for example, the
diffracted field behind an aperture. Noting that 8/bn = a0z1 = for any
function of r, we hsve from equation (4) that
2Ka(P) = f(P) + ?R-z- g(P),
where f(P) and g(P) are both even functions of z, viz.)
(1.16)
f(p) = ifeikr
2rt
A r
ilcr
?
d E , g(P) =
an 2r A r
Comparison with (4) and (6) shows that
nod E.
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(3.17)
-12-
Ral(P) = f(P)'a2(P)
g(p).
Consequently, the functions R1(P) and Rs, CP) are simply twice the even and odd
parts respectively of
In addition, the function f(P) is eq 1 to the velocity potential of a mem-
brane vibrating in a, rigid baffle with velocity distribution -8110/8n; the same
holds for g(P) and -ub. Methods and numer cal results of the theory of acoustic
radiation are therefore valuable in diffra tion theory also. Various authors
have used the modified Kirchhoff theory in one way or another. Bremmerl).2] applied
Rayleighls second formula to the diffracti n theory of Gaussian optics. Various
mathematical aspects of equation (2) were iscussed by LunebergM, and Scheffers
emphasized the usefulness of this equation in the Fourier form)forthe calculation of
Fraunhofer natterns. DurandE1 applied t e Fourier equivalent of (6) to a circular
aperture and to a. half-plane for the case f a plane wave at normal incidence.
Spencer:161 compared Ral for a circular ape ture (plane wave at normal incidence)
with the corresponding exact solution of t e boundary-value problem II. Experi-
mental results on the diffraction of sound around a circulardisk were discussed
in connection with the Kirchhoff approxima ions by Primakoff, Klein, Keller and
[17]
Cars tensen ?
In concluding this section, we shall riefly discuss the Kirchhoff solutions
for the diffraction of a plane wave normal y incident on a circular aperture. Let
a denote the radius of the aperture, and it the incident wave ub = eikz impinge
from the left. Choosing the origin of coo dinates at the center of the aperture,
we have in the. shadow region
1 8U
Ka m - 2 (iklg+ 5-.7) -ikU,
a
(1.18)
ikr
U = /12-- d3:,r
where U is the velocity potential of Bayle ghls piston for unit velocity dis-
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tribution.
The integral is easy to evaluate if P lies on. the z axis. The result is
(1.19)
B1 177-
= eikz eik z +a
a
?
kz 177
eik z +a
(1.20) Ra2 = i
e
177
z
?
(:21) K = eikm 1 ik1:2----F.a2 1 m eip
k z +a--i
a 2 e - 2
It should be noticed that thee() expressions are equally valid if P is on the
negative z axis. If a tends to infinity, Bel and Ke do not reduce to the inci-
dent field eikz, unless the medium is assumed to be slightly absorbing (i.e.,
Imaginary part of k is positive).
The respective Fraunhofer patterns are also easy to calculate. Let p, e
denote spherical coordinates with the positive z axis as polar axis. Then at
large distances from the aperture
1 ikp
", A (0) e ?
where A(0) is the amplitude of the spherical wave.
J (kasinG)
(1.22) (i/a)Ael =1
(.2g)
Sine
J (kasinG)
(1/0Aa2 1
tan?
?
?
31(kasin0)
0.21-1) (i/a)AK 2tan(0/2) ?
We find
where we 1180 an obvious notation for the amplitude, and where 0 e
J is a Pessel function.
1
and
The amount of energy transmitted through the aperture can be computed by
integrating IA12 over half of the unit sphere. It is convenient to introduce
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-1
the transmission coefficient, which is the ratio of the transmitted energy to
the energy incident on the aperture in the sense of geometrical optics. In
the problem under discussion this coef icient is
(3.25)
1
jj 1Al2
2
Tr 2 2
- 2- A( )1
2 j 1 4) sin0d8 .
a o
The relevant expressions for the three ifferent cases mentioned above are
(3.26)
(.27)
1 (2ka)
T1 = 1 -
ka
J1(2ka)
T2 = 1 +
ka
Jo(t)dt,
2ka
= 1 - EJ2(ka + J2 (ka) + J (1,)di] .
2 o 1 2ka o -
(1.28) 1
Equation (26) is a classical resul obtained by Lord Rayleigh.
:For very small values of ka we he
(1. 29)
T1 1 ' .(ka)2 T2 1(
2 6
T (ka)2.
K 24
?
These values of T are in complete disagr?ement with the results for the exact
boundary-value problems. This is not s rising, since the NIrchhoff approxima-
tion holds for small wavelengths. For v ry large values of ka we have
sin 4
2
(3.30) T^, 1
+
2 km
-
1 1 os(2ka-7/4)
2FT (ka)5 2 '
(-17- .?,1)
sin(2)-TO(ica)3/2
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IT. Braunbek's Mcdificatign of the Kirchhoff Theory.,
An attempt to improve the modified Kirchhoff solution was made by
581
Braunbek - who observed that the Kirchhoff solution does not constitute
the main term (in a series of powers of lika) of the exact solution. How-
ever, before entering into a discussion of Braunbek's theory, we shall
first discuss the solution constructed by Sommerfeld for the diffraction
of a plane wave by a half-plane:
-iktocos(0-80) 2kpcosg8-0J/2) ilc2
Aee dc
+ Be
-ikccos(4144)o
2kpcosi(O+02/2)
e d-C ,
on which Braunbek's theory is based.
A simple derivation of Sommerfeld's formulas is implicit in an inter-
esting paper by 3. Brillouint191; this proof we now present.
Consider a plane wave incident on a screen which cuts the plane of
drawing in the upper half of the y axis (fig. 1, direction of incidence
normal to edge of screen).
Screen
Fig, 1
(The restriction to normal incidence causes no lack of generality since
the proof can be generalized to arbitrary oblique incidence.) Introduce
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- 16
new 4semi-parabolic" coordinates u, such that
(h.1)
V IR
u
7 2g If
z g--ju24-2uv
(i.e., one of the parabolic coordinat s is rejected in favor of a rectangular
coordinate), which can be written in erms of polar coordinates as well:
(4.2)
V = - p cos 0
U = 2p cos2
2 '
where 0 < p co and 0 5".: Tr.
Now, from (1) we see that
(4.3)
With the use
a z a a a u a
az - ir+v au ; ay av u+v au
of (3), A=;+ a22
aY az
ecomes
(4.4) b. a2 1. 2a a2
av2 v+u ava
au2 1 1 a
- -
v+u au ?
The wave equation LIE+ k2 I= 0 is t separable in these coordinates;
however, we follow the same procedure s we would if it were separable and
assume that i:= F(y)Cr(),, The wave eq tion then becomes
(4.5) Ggitt+kailj + EIG11-1-
1.1:11r 2u
which can be satisfied if we take F and
(b.6a)
(h. 6b)
are satisfied.
F. , 170
71 on+ fa. oil
From (16a) it follows that
( 4 . )
ikv
P = cc e
G such that the relations
'=0
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and when this result is substituted into (6b), the latter becomes
(4.8)
P' d
+ ik =?LLog .
G 211. 7 dn.
It is seen from (8) that
and finally that
( 4 . 9 )
GI . l'e"n
u ikt
G = 0j( dt .
Irt
Using the results of (7) and (9) the solution I= P(v)G(u) becomes
ikt
9? at.
(4.10) PG = (const.) e jr ? fter,
\
Transforming to polar coordinates, we obtain
2pcos2(0/2),
ikt
(4.11) PG = (const.)
e-ikpcosy
and moreover, if we let kt =z2, (11) becomes
2kpcos(Q/2) i.r2
(4.12) PG = (const. )e-ikpcosA clic ?
dt
?
If we had started with 8 + 490 instead of A (this just means a rotation
of the coordinate system) we would have obtained two other solutions, and
by taking a linear combination of these two, we would have as our final
result
_ikpcos(0_00) '2kpcos(i2.0-1 ir2
(4.13) 7G = As 11-
(040 )
Viii7cos ,o
-ikpcos(o+o0)./ ei ,c 2
+ Be dt'
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which is exactly the form of Sommerfel Is solution and hence verifies it.
(The boundary conditiont can be met by an adjustment of the constants A
and B.)
We shall now treat the two specie cases referred to in Section III
(i.e., 01 = 0 on the screen, and 802M= 0 on the screen)-. However, we
shall use henceforth a slightly differ nt system of coordinates; see fig.
2.
Defining 01 and 02 to be
Inc! ent Wave
?
\A
?
?
Fig.
2
ei-c2
ikpcos (0-00 11 e-171/14f e dr ikpcos(04-0o )e -
irr/4
Vq7 J d-r. co
-c? ; e =
41 -Go t( 44.0 ikpcos(0-0)
where el =12Frce. sin( -r-), s2 = -1217r i ( 2 0) , and where uo= e o
is the incident wave, it can be verifie directly that 0, and 2102Pri both
vanish on the screen.
There are other functions which sa isfy the wave equation and, at the
same time, the bdundary conditions. T for example, the functions
8201/az2, a401/Bz, etc., which obvious y satisfy the wave equation, also
vanish on the screen
[20].
However, the :e functions are too singular at
the edge and hence are not admissible s lutions; 01 and 02 of equation
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(114) are admieeible[201.
?or the purpose of discussing Sommerfeld's theory, two electromagnetic
fields can be constructed from the solutions of the scalar wave equation
previously discussed. The first field is
(4.15a)
which vanishes on the screen, and for which H, determined by Maxwell's
equations, is given by
, -01
ik ? ?
1 ?
az 9 by '
and the second field is
(11.15b)
the normal derivstive of which vanishes on the screen, and for which
ad - 0
4
(o, __2., -2
5z 8y
The explicit expressions for all field components for the Sommerfeld
half-plane problem are
Ex = 01
Hy = 02sine_o-cos
Hz = 01cos00-cos
1.0 co. 2Y ei(kp+r/4)
2 o rkp
1. 0 sin 1.0 ei(kp+r/4)
2 o 2 41;
in the ease of indident field polarized parallel to the edge, and
Hx=02
1 _ ei(kp+r
/4)
E = -01singo-sin - sin m
2 o 2 YTTkp
1
Ez = 02005.00 + sin- 0
2 o
ei(kp+r/4)
cos
2 Ti'kp
for the incident magnetic vector parallel to the edge. For the defini-
tion of coordinates, see Fig. 2, not Fig. 1.
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Brambek's theory makes use of the val es of the scalar fields and their normal
derivptives on the x.7 plane (i.e.. th plane determined by the screen).
1 8" necessary to compute the field 0
The non-vanishing derivatives of 0
quantities are
0,
= (iksin00)02
+ e
,p1P ?ir/4
cos
8?2
= kiksin0o)01
az e-17/4sin
(4.16)
? = - e
ay
8?1 tA ira" -17/4 sin
COS-1
2
{cosi 0
2
ao2
ay
= (ikcos00)p2 _f ? e
A a -ir/4sin
eikp 1
eikP
eikP
eikp 1
1
cos-0
2
VI.;0"
To calculate the fields on the x,y pla e, we let 0 = 0 and 0 = n. We
confine ourselves to the ease of norma incidence and hence take 0 = -
o 2 ?
This gives us
(F.17)
and
(.18)
0=0
?1 = 0
80 2
832
0
2 k e-ir
r = act e-in/h"
e (It
8z
Vc7
02 =4.- e?i7TiVet dz
e?,
(CP'
= ik
az
?2 = 1
= ?rT 01 = 1 - e -47/4 oo
ei z2d.r.
\I:7T Ircf-;
ikp
? ni, ikn
4, Zia e-i /4je ciT
) .4";;e
2 k 4
1/77
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- 21 -
We shall now discuss Braunbek's attempt to improve the modified
Kirchhoff solution for a circular aperture. As was mentioned in the be-
ginning of this section Braunbek observed that Kirchhoff's solution does
not constitute the main term of the exact solution. To obtain the correct
main term, we must estimate the effect of the second integral of (3.13)
on 01. (The problem for 02 is analogous.) Braunbek replaced an on the
screen by the value derived from Sommerfeld's theory for the half-plane,
as if S had a locally straight edge. This is a plausible assumption, be-
cause a$1/an is expected to be rapidly decreasing from the edge over a
distance of a few wavelengths(confirmed experimentally in the case of 02
by Severin and Starke[211.,), and the wavelengths here considered are small
with respect to the radius of the aperture. For the circular aperture
and a normally incident plane wave, Braunbekts approximation becomes
ikr ikr
(14.19) ?.B1 = - fe clE ap---
2n r 2rt r
A
(k0.-tif(q a. E ,
where is the distance to the rim and
Ti
(h.20) I:(x) = IL e-in/Ve.3ei T2dr ; w(x) = eix
v7T-Tx
The evalurtion of B1 on the axis of the aperture is comparatively simple.
Integration by parts and some trivial transformations yield
(1-.21) 131(0,0,z)
e ?11
ikz -17/4eikf27:7-71 T4-ka+de+e'4 el
Tr ti 2 21
:11/2
tr.. el:.
z+2 z +a
The integral is elementary if z = 0. At the center of the aperture Bramnbek's
function is exactly equal to
(4.22)
B1 (o,o,o) = 1 - elka.
When z is greater than zero, the integral in (21) can be expaoled -0ymptotically.
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Ina first approximation, the square ro t in the integrand may be replaced
by its value of the integrand at the lo er limit of integration. Then ex-
cept for terms of order l/ka,
2
(4.23) B1(0,0.0:: eiks ei E 1 + a -11/2
11 27 2
V 2 +a
which is Braunbek's result (obtained so ewhat differently). His computations
showed that (22) and (23) are in excell nt agreement with Meixner and Pritzes
exact values. The corresponding diffra tion pattern was also evaluated by
Braunbek.
In studying Braunbekle paper, the uthor encounters some difficulties
in connection with a second type of app oximation to Or In starting from
the equation
ilcr
1 t d a re -
/51 = 2r7wi 371 r
A
given in Section III, Brauribek replaces
01 in the aperture by the value
1 - Pkg) derived from Sommerfeldls the ry. Then
(4.24) ?lz Bt = I; [ i7rj{i i(ks) er '
A ikr
l dE
Braunbek claims that, exciept for terms f order 1/ka, the functions Bl and
B* are identical on the axis of the ape ture. The present author believes
1
this to be incorrect, because from the lues at the center of the aperture
we can see, without any calculation at 11, that
eil7/4 ika
e
B* (0" 0 0) = 1 -
1
which differs from (22).
V711?le.
[221
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- 4)
V. Variational Formulation o. Scalar Diffraction Problems.
A variational formulation of planar diffraction problems, which permits
accurate numerical evaluation of the diffracted amplitude and the trans-
mission cross-section fora wide range of frequencies, was given by Levine
and 5chwingerE231. They illustrated the utility of the variational method
by applying it to the circular aperture for a normally incident plane wave.
[24]
The analysis was criticized by Copson
since many of the integrals in-
volved appeared to diverge. Copson, however, in deriving what he calls
"Levine and Schwingeres variational principle" in a mathematically sound
way, confined himself to the problem for 02, while his criticism concerns
that for ? 7ortunately the divergent integrals that occur in Levine
and Schwingerls paper are easy to eliminate without affecting the numerical
results. However before we treat these problems let us first discuss the
integral formulation of scalar diffraction problems with application to
small apertures.
As is seen from the second equation of (3.11), the wave function 02
is uniquely determined in space by the values of its normal derivative in
the aperture. Let the unknown values of ?f an in the aperture be denoted
by f(x,y).. Recalling that 02 = u in the aperture, we find the integral
equation
(5.1a) ff(xl,y1)0.(x,20,y,y9dx1dyl =
A
[251
where the kernel G is symmetric and singular, viz.
= (1/2ns)elks
2 = (x-x') (Y-30)2
s
and where (x,y,o) is any point of the aperture.
Similarly, the first equation of (3.11) shows that 01 is uniquely
determined by its value in the aperture:
ikr
(5.1) ? a [
1 j ?e E
1
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- 214
In this case we have the further con ition that 801/8z = 8110/8z in the
aperture. Since in equation (1) the term in brackets is a solution W,
say, of the wave equation, we have
82
01 . la. . _ oc2 + a2 a2 8x2 az ,
3z2 j
+ 20y2 w'
so that the following relation is ob ined[26]:
duo(x,y,o)
r-2 a2
,
(5.2) a -1 + + ploci,y1,0)G(x,x14r.30)axlaye,
aZ
8x2 a? A A
where (x,y,o) is in the aperture. T is relation is not a pure integral
equation; it is a. differential-integ 1 equation. It should be noted that
the differential operator may not be :.pplied to G under the integration sign
[]since the resulting kernel would not 27
integrable. Mane gave an equi-
valent form of equation (2), namely,
(5.1) k2 jid Gd E ? (g
az
an
P51 ? grad G)d
where both gradients are to be taken ith respect to the coordinates of
integration x', yl. Equation (3) fol owe from (2) by a process of differ-
entiation and integration by parts, a d use of the condition 01 = 0 at
the edge of the aperture. The second integral in (3) is a Cauchy's -
principle value (small circle around ,y,o of radius e 0). A second
integration by parts is impossible bes.ue of the singularities, at the
edge.
Only in a few simple cases can t e differential-integral equation (2)
be transformed into a pure integral e. ation. If the incident field is a
plane wave, uo = exp[ik(cm+Sy+Yzni
au0(x,y,o) 2
- Ek2 + +
az
ax2
so that (2) becomes
(1-m2-02)1/2,
we have
(i/e) eik(axi-Py)
9
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(5.4) /01(xtor'.0)G(x,xt,y,y1) -L eik(ax4*) +
A
where 2( is a solution of the twb-dimensionnl wave equation in the aperture.
The function A/. can be uniquely determined (except for a constant multiplier)
for a normally incident wave (m = 0 = o) in the oases of the circular aper-
ttrel:281 and the infinite slit[2 ? 9] The resulting equation can be trans-
formed into a pure integral equation with a non-symmetric kernel.
Let us now return to the discussion of the variational formulation of
planar diffraction problems as given by Levine and Schwinger. As was mention-
ed before, the divergent integrals that occur in their paper are easy to
eliminate without affecting the numerical results. In what follows, the
notation is suggested by that of Levine and Schwinger.
Let A1(nn1) denote the amplitude of the diffracted wave, where n and
nt are unit vectors in the direction of observation and of propagation of
the incident plane wave respectively. Further, let Q and Al be the angles
between the positive z axis and these unit vectors. From equation (1) it
follows that
(5.5) Al(n,nt) = (ik/2n)cos OfOn, (poe-iknptdsl,
where 0211(p9 is the value of 01 in the aperture. The integral equation
in Levine and Schwingerts paper [their equation (2.9D contains a non,-
integrable kernel and should be interpreted in the sense of one of the
differential-integral equations (2) or (3). Let us choose Mauets equation
(1). Then
(5.6) 2rr ik cos e1 = 01fin1(o9G(p,p9dSt -.1rVIOni(p1)171G(pool)dSI,
where p is in the aperture and
eikip-Pil
(5.7) G(p.ps)
110-P ?
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If we mdltiply througl in equation (
solution for a plane wave in the dir
aperture, there results
(5.8) 2Trikcosa (p )elicniPdS
The last integral appears after an i
term drops out because 011", like On
The right-hand side of equation (8)
sequently so is the left-hand side.
side and by a similar term in which
use (5) and invert, we obtain
(5.9) A1(n",n1) = Al(-nt.-nn)
cosOlcose"
It can be proved that the equation (
independent variations of 010 and 0
variations which do not violate the c
admissible.
The stationary character of the
approximate calculations. In fact, a
butl-ms in equation (90 may result in
without the necessity of solving the
Scale factors are of no account since
0. The same remarks apply to the pla
of any aperture in a plane screen (ID
[3?1
ing to Levine and Schwinger , is r
) by On"(p)(where 021"(p) is the
ction n"), and integrate over the
.2i6nu(p)G(e,p1)0n1(p9dSdS1
v0n"(p)VIOnt(p1)0(0,p9dSdSt.
tegration by parts; the integrated
vanishes at the edge of the aperture.
s symmetrical in nl and n" and con-
If we divide (8) by this left-hand
and 0 are interchanged, and then
)-k
dS./0_nol(p )eiknip dS
ikn"p
gemmilm.p...????????
0nI(P)710-ns(POJG(e,p9dSdSt ?
) is stationary with respect to small
about their correct values; those
ndition 01 = 0 at the edge are
xpress ion (9) is of importance for
judicious choice of aperture distri-
a reasonably correct value for AI
rig inal differential-integral equation.
(9) is homogeneous of degree zero in
ef-wave transmission cross-section Cr'
rfectly rigid or soft) whinh)accord-
lated to the amplitude A of the
spherical wave at large distances beh nd the aperture in the direction of
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the incident wave by
(5.1o)
Cr' = 22-7 Im A.
rrom eqvPtiOn (10) it follows that
.10.(p)e-iknPdsjo_.(p)eikniDds
(5.11) cr (n) = - cos2,g Im
hn(P)O-n(P ')-k-2VOn(P) Vt0,(p)G(p?p I )(1SciS I
1
Levine and Schwinger discussed the limiting form of this equation for
low and high frequencies. In the static case as k approaches zero
3
(9.12) G _ e 1 k2
"k- --1P 111-1 12
JP-p 2 --P - 6 P-P' O(k4)
Using equation (12), we obtain the following for the denominator of (11):
(5.13))(I "nN7k-n
? - ikVOnc710...n k2("Ii. 1 P-P '1 V OnVI0 1-13:11-95 )
-n ip_pli
3(0 0 11 121V0 VIO
+ ik n -n+ ?6 p".p n -n)
At this point we shall prove the theorem
/ 14%
40vk ) dSdSf.
(5.10 Idxdyirdx'dyli701(x,y).N7102(xi,y017(x,xs,y,y1) =
S S
-idxidy'02(x',y0idxdy03.(x,y) IP,
which is necessary for the further calculation of expression (13). We
assume that 01 and 02 are arbitrary functions defined in the region S and
eonal to zero on the boundary of S. The left-hand side of equation (13)
can be written in the form
(5.15) fd-xdY "1(x'7) ../dx1c1Y1 { { ?21}
- 02v '3' 1
However, it follows from partial integration and the condition that 02 = 0
on the boundary of S that
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jdx fdy1 V'
-28
021 4 112
Bounder
Hence, expression (15) becomes
-irdady V$ (x,y) Irdx1dy
which can be written as
(5.16) idxidy102(xl,y1)Id
(direction cosine) = 0.
I yl) Wry
V 01(x,y)
Using a similar procedu:re on the inte ral of v01.(x.7) VT, the left side
of equation (14) can be shown to be eq 1 to
IA001 02(xl,y1))(dxdy $ (x,y) V VT,
If we assume that F is a function of a argument of the form (x-xl )24.(30,..y92
then VV 1 = - 71, and thus the theorem is proved. Therefore if r
1
is of the form lp-p'12 then we find, king use of the theorem (14), that
.101 ? V1021p-p1i2dSdS1 = -9(01(p 0 ( )
2 p dSdS
Now, to continue our discussion of the limiting form of equation (11),
we can see that the expression for the eAominator becomen
r yO n .4 -n 4 k2 .295
(5.17) + o(k4)3
-- 1 P-P 11 P-ro
EP + k2 + k3R]
The numerator of equation (11) becomes
(5.18) [fin?-n ika ./(?-P 1) C6n9/-n- dfin8-n En (p-p' 2] d5cIS'
However, for small apertures (with respe t to the wavelength) On= = 0
since in this case 0 is not dependent on a special n. Therefore the leading
term of the transmission cross-section given by
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2 ab(P)d]4
(5?19) (5. (n)^,2-1 ki4cos-e as k --> 0
3
Ejr7f6P)7*13(P1) 00132
which is an illustration of Rayleighls X-4 law for small scatterers.
The analogous problem for V2, discussed by CopsonN1 and by Levine[32],
is easier since it is based on the pure integral equation (1a). From this
equation it follows that
(5.20)
(pt)G(pIpt)dS1 = -2nei
kntp
nt
where Yni(pl) is the value of a912/az in the aperture due to a plane incident
wave traveling in the direction nt. The amplitude can be found from the second
equation (3.11):
dS
(5.21) A2(n,nt) = - 41ii/Yni(pl)e-iknpl 1
and the analogue of (9) becomes
(5.22)
A2(n!'ni) = A1(-ny1-:11)
j/ini(P)e (p)eikn'PdS
(P)G(P,P911.n5 (p )dSdSe
which expression, again, is stationary with respect to small variations of Yril
and Y about their correct values. In this case the variations are not res-
tricted by a condition at the edge; the correct aperture fields Y are infinite
there.
The corresponding formula for the transmission cross-section is
21c7 imn(p)e-iknodsji-n(p)eiknods
(5.23) aVn) _
fin(p)G(p,p1)Y_n(pi)d8dS1
while the leading term in the static case is given by
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(5.24) ?2(n) 2n
- 30 -
[fi(p)dS]4
1r("139 dSdS
Ip-p'i
(k 0).
This leading term, it may be noted, is eq al to 2nC2 where C is the electro-
static capacity of a metal disk of the s shape and size as the aperture.
The variational principle for 01 was uccessfully applied by Levine and
Schwinger (333 for the diffraction of a nor ally incident plane wave through
a circular aperture. The correct aperture distribution is assumed in the
4
form D1
(5.25)
op
Vi =
n
n=1
where the coefficients an have yet to be determined. The denominator of
equation ( 9 ) 1
(5.26) F .../POn(p)Oni(p1)-V9n(p)Vici
can be written in the form
a a 2n
(5.27) F = pdp fpldp ir 1 fk2Vn( )
2n
sb 0
eik Vp2-2ppicos(Y-
p,p9dSdSTI
aPn(OWni(W)
PIap ---cos(Y-Y1)3
apt
VV:72pplcos(T-Y1)+p
where we used the relationships VOn(p)?Vik pi
eikc?f:7:27:ptcos(Y-r)+PI
hP:2;ptco:s(Y-Tt )+pt2
Now G can be written in the form
(5.28)
eik?i7:2;Pico8(Y-1" )410'2
n 2 , - 1)..2
cn ....cPP-COS\x..x1)+
a9n,
ar--5T- COS (T-11) and
OD XIX
a J 2
(XVP -2Pplcos(T-Y9+1112)
0 1,2.777 ?
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- 31 -
When we substitute equation (28) into equation (27) and use the addition theorem
for Bessel functions, the-denominator becomes
OD a
(5.29) F s 2nitXdX
-------/pdpfpldpl[k2On(p)nt(pl)J0(Xp)J0(kpf)
o a-T-7 0 0
vx -k
an 89/ni
ap' 1 (XP)J1 (XPI)].
ap
a Kin
However, the expression./ pdp Ji(kp) when integrated by parts gives us
a 89n
fpdp Tp?
o
From this we see that
la r a f
j1(AP) = P9 j1(AP)lo 9n P
a a (
j()`P) dP
a
=
./Vn f (XP) XPj1(110 (X01
0
= .'19nOtTO(XP)411.
a a 89 ap 1
_fop! apn pldp1 J1 (Xp)119m.
(Xp
ap t
o o
dp
1
Vpdpf()Ido on(p)onipi)Jo(xp)J0(xpf)
l
Hence equation (29) becomes
oax(k2x2 a a
(5.30) F= 2n/_ \ %0 / pdpipidolJo(A.W0(xpOcifn(p)9n,(p1),
which can be written finally as
(5.31) F =-271/X X -k dXfpdpiplcip195n(p)reni(p0J0(Xp)J0(Xpi),
o o
Equation (31) is the same as that obtained by Levine and Schwinger. However,
they used a non-integrable kernel in the integral equation, and hence the valid-
ity of their procedure is somewhat doubtful. Although the present method is more
difficult than Levine and Schwingerfs, it is a valid procedure.
Now, if the series expansion (25), where the individual terms are of the form
(1 _ p2/a2)n-1/2
0
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- 32 -
is substituted in (31), we get a double s of terms:
(5.32) F = -2n4( -k:dX /pdp(1 p2/a2)n-1/2,10(4)]
nm 41) t 177--7
'I a
4` PdP(1
However
p /a2)m-1/2 jo(xp)]
2
a n/2
a ? /
f,xal2n-1 r-
/2n+1/2)
ficiP(1-P2/a2)n-1/20.0(X0u a2jr30(Xasi sinOcos2 jn+1/2 1(
119dG=
(xa)n+1/2
Substituting this into equation (32) we h ve
(5.33) Fnm = -n2n+9rCn4) gm4)a3
and then writing X = kv we obtain finally
(la) J OA)
1 2 m+1 2
x2.47 dx,
(Xa)114111
(5.34) Fnm= -2na(2/ka)n+IT'-2 ri-11+1/2)Fm 1/2)
1/2v- ( n+m) Jn+1/2(kav) JM+1/2( kav) dv.
n-1
Moreover, (i_p /a2)/2
using the equation con(p) p we have
jr a
0n(p)dS = 2ndfpdp(1 -p2/a -1/2
'o
= 2n P:inAcos nQ na2 rzi)frnia/2)
Fn+3/2)
a2
n+177
and we can now write A (the amplitude i the forward direction) as
1 r ?41)
_Al
(5.35) 2a03 CO
cmnam n
m=1 n=1
where the coefficients c are defined
mn
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(5.36)
and
(5.37)
-33-
cmn = (2/ka)m+n [Tm+1/2) n+1/2)f (ka)
oo
f (ka)=./(v2-1)1/2v-(m+n)JM+1/2(kav)Jn+1/2(kav)dv.
mn
Let us now apply small variations 6a to the true values of an. In view
of the stationary character of (35) we then find
1 oo an -Al oo
= c a (m=1,2, ...).
n=1
2m+1. 2n+]. 2a mn n
n=1
On the other hand, it follows from equation (5) that
OD a
(5.38) A, = -ika2 n
rig.T. ?
n=1
Elimination of Ai thus gives the following infinite System of linear equations
for the unknown at:
(5.39)
OD
cninan v.
n=1
1
ika(m+1/2)
(m = 1,21...).
An approximate solution of (39) was obtained by Levine and Schwinger by assuming
an = 0 if n > N and solving from the first N equations of (39). The
corresponding approximate value of the transmission coefficient becomes
(5.)40) t(N) = Re co
2 (N)
1 717177-1 an
It seems worth noting that the integrals (37) can be expressed in terms of
Fn = Jn+44n where tin is Watson's notation for the Struve function. The symmetry
between the real and imaginary parts of fmn were not recognized by Levine and
Schwinger, although their expressions are easily transformed into the symmetrical
form by a partial integration.
11. Levine and Schwinger apparently overlooked the fact that their coefficients An and
Dn are simply related by An = ikDn. The factor C of Magnus[35] is thus equal
to ik, so that his Table I provides at once the first terms of the power series
for An.
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34
One has
f11(a) = - 2.Tiq - .41.-c-c ) + TIL71.(1 4* -17
4a 0
f12(a) = i(a2 - 1 3 1
_
8a 8a
(5,141) i a3 a 4 3 _ 45
f12(a) = - 7/(7 - 12 32a 64a3
+ -2-7 (1 - -)F0(2a)
32a 4o
a
1 1
Fo(t)dt- F, (2a)-
(2a)-
8a
32
2a VFo(t)dt - -2-- F (2a) a3 o'
Lta
1 3 45 2a
Ica.(1 + --y ?Of F? (t)dt
2a 16a' o
3
- F1(2a),
8a
The infinite system of linear equat
Magnus[)6]. He showed that for suffici
an is unique and can be expanded in a co
also shown that in the N--th
approximation
cients of the power series for a11 ...,
formulae were given. Special attention
for ka -oco. Owing to an error of sign i
part of c the conjugate values should
mn
same error (and others) occurred in the p
the wrong definition of (X2 -k2)1/2 when 0
table of coefficients and found complete
except for the last column. The foilowin
paper [381; we give one term in addition t
2ika
al = n
a + t(ka)2
(5(3)L110 81
a2
a3 =
45) F (2a).
4a2 1
ons (39) was thoroughly studied by
ntly small values of ka, the solution
vergent power series in ka. It was
he exact values of the first N coeffi-
are obtained. Explicit recurrence
paid to the limiting form of (39)
the definition of the imaginary
taken in Magnus,s table. The
per by Sommerfeld 1373, who took
o),
mn n ika mo oo mo
n=o
where
(5.46)
d = (6/ka)2 14+3/2)(7n+3/2) (ka);
mn ml ni bran
ginn(a) .1* (v2..1)1/21,-2 j2m+3/2(av)J2n+3/2(av)dv.
In this case the various approximations to the transmission coefficients are
given by the ratio of two determinants, viz.,
gll
?
?
? glN
?
?
?
gN1
?
?
? gNN
4ka
N+1) . Imagin. of
(5.47) part -----------------
goo
?
?
? goN
gNo
?
?
? gn
in which for N = 0 the upper determinant should be interpreted as unity. It
may be verified that equation (47) gives exactly the same approximations as
equation (39). The advantage of (47) is in its explicit analytical form, which
invites a detailed study along the lines of Magnus's paper.
tt is also simpler than the Legendre-function expansion of Levine and Schwinger [p]
btained by direct integration of the differential-integral equation (5.4).
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- 36 -
Only slight changes in the preceding analysis are necessary in order to
cover the second boundary-value problem. First, the series analogous to that
of (25) represents the aperture values of aV /az; in this series we include
2
a term with n = 0. Secondly, +A2/2a is g ven by equation (26) modified to
include the terms mon = O. Thirdly, fm a) should be replaced by
(5.48) hinn(a) .. fc?(v2-1)-1/2v-(
m+1/2( av) Jn+1/2(av) dv,
o
which function is related to f by (440
mn
(5.49)d
h (a) = (m+n-2)f - a
Mil Mri da mn
For example,
(.5.50)
2a
h00(a) = Tu.( Fo(t)dt,
2a
1 1
1'1i(a)+ --7.1F0(t)dt ..271 F0(2a),
4a 0
3 1 3 3
h11(a) = w + + -11a( 4. 2 )/Fo(t)dt- -,-7F0(2a)- rizt F1(2 a).
4a o 8a
The corresponding first-order approxi ation of the transmission coefficient,
t(1) was calculated by Miles1411 although this was obtained by a less powerful
2 '
variational principle. Miles introduced impedance parameter Z = R-IX, and
the admittance Y = Z-1 = G-iB, which were valuated for constant and static-
field aperture values of apf,/az and compar d with the rigorous and Kirchhoff
values of the transmission coefficient (t2 = ReZ). His curves are represented
by
(p2 42)y p2 F1(2ka)
ka
where
2ka
I Q2I Fo(t)dts
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-37-
(P,Q) = (1,0), (0,1) or (1,2/kam).
Again, the simplest analytical approximations of the transmission coeffi-
cient t2 (circular aperture, plane wave at normal incidence) are obtained if
we start from an expansion in Legendre functions. We now assume that in the
aperture
(5.51)
462 %--??
= P
2
(11-P2/a2)
Bn
n=o \A_p2/a2
Then by equation (21), the scattered amplitude becomes A2 = -a2B0, while
insertion of (51) in (22) gives
A2 B2
.
(5.52)
2a = Co oz
E z DmnBmBn
m=o n=o
where
(5.53)
1
2 -7
mi ra 0 (ka); G (a).1(v -1) J2m+1/2(av)J2n+1/2(av)dv.
mn Inn 0
Application of the variational principle for P2 gives the following infinite system
of'equations for the unknowns Bn:
oo 2
(5.54) E DmnBn = 3. 6m0 (m = o1112,...).
n=o
The successive approximations or the transmission coefficient are then simply
given in close analogy with equation (47), by
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- 38
(5.55) .,.(14+1)
L,2 Tin
nka ?
G11 ? ? GIN
?
GN1 ?? ? GNN
?
oo ? ? ? GoN
0 ? ? ? GNN.
It should be noted that in the integral (37), (46), (48), and (53) the roots
are Understood in the seme
(v2-1)1/2. .4(1..v2)1/2,(v2- 2
+i(1-v2)-1/2 when 0 < v < 1.
-1/
All these integrals can be expressed in terms of fmn and, therefore, in terms
of Fn and the indefinite integral of F?
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- -
VI. Rigorous Form of Babinet's Principle in Electromagnetic Diffractien
Theory.
On several occasions we have discussed Babinet's principle in one form
or another. Only recently has it been possible to extend this principle so
as to be applicable to rigorous electromagnetic diffraction theory. Our
representation is essentially that of 0opsonf*23433 . In what follows,
the time factor exp(iwt) is omitted; as before, k denotes the wave number.
Let 6,70 denote any arbitrary incident field, where f stands for the
electric field vector and 1 for the magnetic field vector. It is assumed,
therefore, that f and g satisfy Maxwell's equations. Tater on we shall use
the term "complementary" incident field. This is the field defined by
(4), in the order (electric, magnetic) vector. As is well known, this
complementary field also satisfies Maxwell's equations. The complementary
field of the complementary field is identical with the original field ex-
cept for sign.
First of all we consider the diffraction of the field (f,-;) by a per-
fectly conducting plane screen (finite or infinite) of zero thickness.
Secondly, we consider the diffraction of the complementary field (-Vi')
by an aperture in a perfectly conducting plane screen of zero thickness;
the aperture in the second problem is of the same size and shape as
the screen in the first problem. For simplicity we call these two diffraction
problems complementary diffraction problems. The rigorous form of Babinet's
principle asserts that the solution of one of these apparently different
problems gives, at once, the solution of the remaining problem. We now
turn to the proof of this statement, and to its precise form.
In the first problem the total field everywhere in space is given by
(f + E + H ), where the scattered field (E , H-) can be derived from the
vector potential? of the currents induced in the screen by the incident
flow. Let I denote the surface current density vector. Then
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- 40
(6.1)
/ ikr
Ae= e
ci r dS,
-s -s "13
He = curl As -ike 2As + grad div A .
The unknown two-component vector I, defined only on the screen, has
to satisfy certain integro-differenti 1 equations in order that the well-
known boundary eonditioas shall be sa isfied at the surface of the screen.
The superscript s will he omitted for all quantities evaluated at the
screen's surface. Por example, the n rmal component of the total magnetic
field must vanish at the screen. Thi requires that ( z = 0 in the plane
of the screen)
(6.2)
aA aA
= - gz for 11 points (x,y,o) on the, screen.
8x
Similarly the x-componeat of the tota electric field must vanish at the
screen. Thus
ikfx(x,y,o)
_ 2- 4. 3_1_
= A
x ?Ix
62A
= k2A + +.L
ay
[Ai
2 a
- gzi
ax
yj
al'gz
=kA x ay
= k + A -
=1c2Ax + AA + ikf (x ,yo),
0)
"
where A is the two-dimensional Lapl ce operator and where we have used
the facts that le has a Tem E-compon nt and (f;0) is a solution of Maxwell's
equations. The last equation can be implified to
8g
(6.1) k2Ax + = --Z or all points (x,y,o on the screen.
x az
Also, from the condition that the y-c mponent of the total electric field
vanish at the screen, we have
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-41 -
(6.4)
ag
k2A + ZSA =_-
I - az
for all points (x,y,o) on the screen.
Finally, bearing in mind that on the screen we have
ikr ikr
(6.5) A =111 1--- dS, A = 1 I '419 dS,
x c x r 7 oj y r
where r2 - (x-30)2 + (.7-Y02, dS = dx/dYlt 1 =7(xtat), we see that
equations (2) through (5) constitute a set of integro-differential relations
for the unknown current density I.
By physical intuition we expect these relations to have at least one
"admissible" solution I = (Ix, Iy) satisfying all physical requirements as
to singularities possibly occurring at the edge of the screen. It is not
known whether the assumption of absolute integrability of I over the screen
would entail a unique admissible solution. The integrals in (5) cannot be
proper Riemann integrals; they are improper because of singularities of Ix
E41/1
and I at the edge. Recent work of Meixner , Mane , and others makes it
plausible that the component of I tangential to the edge becomes infinitely
-1/2
1srge as D snd that the component of I normal to the edge vanishes as D1/2,
where D is the distance to the edge. Similar properties hold for the field
vectors themselves, although it is not clear at present what conditions are
necessary and/or sufficient for a unique physically acceptable solution.
We now consider the complementary diffraction problem. There is need
for a proper distinction between the fields in front of and behind the aper-
ture, Let (F, ) denote the total field in the illuminated space if there
oo
is no hole in the screen. For example
r(x.Y.) gx(x,Y,-z)
ox x
z 0.
7
oy fy(x.Y,z) fy(x,y,-z)
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- 142 -
Met (' IT1) denote the total field for z S*0 in the presence of the hole,
l
and let (Yr' it) denote the total field for z >0 (that le, behind. the
2 2
aperture). Further let rbe any two-co onent vector defined in the open-
ing.
(6.6)
-3c1.
Define the vector It by
ikr
113 = 1 dS.
c, r
Then we will show that when = (K , K
x y
the fields can be expressed as follows:
I1 = 10 - curl I'd
1
+ike2
(6.7)
= + k5a + gra
= curl cd
k2P + grad divld
First of ell, so long ge f is integrebl
equations in terms of K vie, Fd setisfie
spaces z < 0 and z > 0, end it satisfie
at the screen (1.e., that the tangentia
vanish). Furthermore, ric matter what
magnetic fields are automatically conti
The only conditions that are not a
normal E component and the tangential H
ture. Tn fact these components are eas
corresponding components of the undistu
the superscript d when we refer to valu
conditions are
8F 8F
(6.8)- _ - -''
for a 1 points (x,y,o) on the opening.
8x 8y z
satisfies the proper conditions,
(z 0)
(z >o).
div r
the field defined by the preceding
Maxwell's equations in the half-
the appropriate boundary conditions
electric and normal magnetic fields
s, the tangential electric and normal
unus in the aperture.
tomatically satisfied are that the
component be continuous in the aper-
ly seen to be identical with the
bed incident field,. Again dropping
S on the aperture, we find that these
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- 43-
(6.9 )
(6.10
ag
+ = -
y az
for all points (x,y,o) on the
opening.
It is remprImIlle that these equations are precisely the same as those for
A' A in the first problem. Assuming that either system has a unique solu-
tion, we see that the two mutually complementary problems are simply identical:
F = 7I, K= henceld =18. Moreover, if we introduce the notation
$2' -02) = (7e, 0d) for the diffracted field behind the aperture, we have
(6.11) = and = (z>0)
and this is Babinetis principle in its rip!.orous form for diffraction by
plane perfectly conducting screens and apertures.
By analogy to the first problem, is called the magnetic current
density. Its components show the same behavior at the edge as does the
electric current density in the complementary problem. The vector is
called the magnetic vector potential.
In concluding this section, it may be noted that taking the comple-
mentary field Eiransformation (it,) --> (=OE of a plane wave is equi-
valent to rotating the plane of polarization through a right angle counter-
clockwise, looking in the direction of propagation.
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- 44
VII. Diffraction by a Small A erture 1. a Perfectly Condulalm_Vane Screen
In this section we shall comment u on Pethels [446] theory of electro-
magnetic diffraction by small apertures. It appears[4] that Bethels first-
order approximation is fundamentally incorrect with respect to the field
near the aperture.
-1 --i
Let E , H denote the incident fie
in the illuminated half-space
(z = 0). Then
and let! ,H denote the field
0 0
S' 0) t there Is no eperture in the screen
= Ow -
ox x x
E m Ki(z?; -
0Y Y Y
E = Ei(z) -I- E:(-z)
O5 z ,
F = H!(7) 4- H1(-z)
ox
+
oy y ?
F = Hi(z) -
02 2 Z )
where z ,5: 0 and *here we have omitted licit reference to the x,y-coordin-
ates.
As we have seen in the preceding se tion, the diffracted field can be
derived from fictitious magnetic current (and charges) in the aperture:
El = Eo curll (z o)
11 .1 + a+ (i/k) grad div F (in front of aperture)
(7.1)
= curl F
= (i/k) grad div?
(z > 0)
(behind aperture))
where F is the magnetic vector potential given in terms of the currents K
by means of
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ikr
(7.2) p _ e
f
aperture
The magnetic cherge density, 7 , is found from
MC bK
(7.1) div i = --2-; + ?1 = ikepi,
.51?
and the scalar magnetic potential Iv satisfies the equation
eikT
(7.1) W =
r dS = div
aperture
In order that these formulae hold it is necessary that the component
of K normal to the edge of the screen vanishes at the edgeC cf,Bouwkamp 1
For arbitrary lc the tangential electric and normal magnetic components
are automatically continuous in the aperture. We have for these components
(7.5)
c
K = Ec .50 , _21:4 (i?r-i)
where script letters refer to values in the aperture, and 71 is the unit
vector in the positive 2-direction.
Conditions for K are obtained by requiring that the norMal electric
and tangential magnetic components are continuous in the aperture.
t = lim rlz = lim E =
22 2 -oz
= lim Rix = lim Tr2x = 1 R
2 ox (and same for y-comp.).
In the opening, therefore,
5;2' = Eo ? r
ge X r) r
Consequently, as a by-product we get the theorem: in the aperture the values
of the tangential magnetic and normal electric field components are exactly
equal to the values of the corresponding components of the undisturbed inci-
dent field.
Thus we have
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As we have seen in the preceding se tion, the above requirements lead
to the system of integral-differential e uations
gr - lIK
e
c x -- dS
r T. lix,y)
ikr
K *2 K(x1,30)
iji( CIY
k2 F + gi? = =
az
.i
x c ' y
x - az
r
a ey,
a
i
r2 = (x,-x')2 + (y-y1)
X
y 7 2
(77) i
dS dxidyl
(x,y,o) any point in aperture
e .... _.
acr a F
ax ay
So far our formulation is general.
(xl,yi,o) same
ti
y
az
e".., 0
e next consider the ease of long
waves Olt -30), and assume that K can be xpanded in a power series of k.
-'o
let K end K denote the parts of K of relative order zero and one respect-
ively. It arpears that relative order is the same as absolute order. Thus
assume
- Ko 4:11 + 0(k2),
wherel? = 0(1),7111 = 0(k),, Then, expan ng the exponential function in
powers of k, we have...
- ds 1p1
oj r c r
kK dS + 0(k2).
The third integral, of order k, apparentl is independent of x and y. The re-
sult of differentiation, of this term with respect to x or y is identically zero.
We need not retain this term in the expre sions for the fields in our order
1
of approximation. Therefore T= 9r6 constak + 0(k2), where
?4,-"b3co
Cf ? ? ? :!-- ? dS (order 1)
C r
(7.8a)
fl
ET ? if F-- as
c r ' (order k)
with the same convention. pa to the sunere ripts zero and one as before.
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We have dr= + .01 + 0(k2). Therefore,except for terms of order k2,
we may replace k2qr+g- by 4g. Equations (7) then reduce to (8a) and the
following
(7.8b
A7?
cr 0 = 8elx.1
-o
Acrx
P--1" - -1?1.
k =o
p_i+
a 7 k az )11c.0
a5:D51-1 ; 6c5 a5'1.
ax ay
8x ?y = [14 Ek=
!
2-1k=o o.
Let us evaluate the right-hand sides of equations (8b) for a plane-
polpriyee nbliollelv incident electromagnetic wave., We choose the xz-plane
as plane of incidence and call Go the angle between the z-axis and direction
of incidence. The phase function of the incident wave therefore is
expak(xein004-zdos001]. Further, let 00 denote the angle between E andthe
xz-plane. Then, in an obvious component notation,
Ea' = (cos00cos00,sin$0,-cosco0sinG0)expt3.k(xsin80+zcosG0D
Hi
(-sinO0coseecos$0,sinO0sine0)expEk(xeinG0+zcos8on .
Accordingly, in this case we have
8e
- vt? = -iksin$ cos0 exp(ikxsin00)
o
ikcos0
0 cos% exp(ikxsinGo)
Vif ' 0
= --cos06 sinGo exp(ikxsinGo),
and egnations (Rh) become
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(7.9)
= 0
65 = -isin0 cos()
o o
biefv? 0 icoso0cos280
ace al?qi
x -cos0 sin4
8x - 8y o o = -ixcosO0sin2G0o
Note that if we had replaced
a
ei
-'- az 11 Cz in equations
(7)
by their corresponding cohstant valu s at the origin of coordinates(which
may be the center of a circular apert e, for example) we would have obtained
the approximate equations
= -iksinkcos0
(7.10) mriccos?os2o
Y 0
2.&
- ?= -cos00sin?o
ax ay
and consequently, beariag in mind thai 17. I.', we would, have obtained
ecs.
(9) except for the last one:
n1 41
= 0 instead of = -
ax ay
xcos0osin2o
Fquations (10) are essentially those o BetheL+l and Copoon . They can,
therefore, only lend to a correct cr erm; the term-) is necessarily wrong
in their approximation.
It is possible to eliminate deriv tives with respect to k. by introducing
derivatives with respect to x,y:
ri,
-cos00sine0 j+ikxsi n9 k 0, 0, 0
aZi
z ,
koix4o).
ax
We then have expressed all constants i terms of center values. More generally,
we assume
8E1 8E1
8i(x,Y,o)=.1E1(0,0,0) + x -'Co 0,0) + y
2 aX
Our final equations thus become, for an plane-wave excitation,
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LQ
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(7.11) AT; = 0;
(7.12) L9 = 0;
a/0
1 f a0
.41
- =
ax ay
(7.13)
(7.14) &II.D.
=
(7.15) All ;
arra all
(7.16) ? x pac
ax ay
ali
aR
aE am
where P - --4t
+ Sy,
All these equations are evaluated at the center of coordinates.
hav,. the following orders of magnitude:
The constants
= 0(1); = 0(k).
This theory will now be applied to the case of a circular aperture of
smell rndius a NUR b) means "contribution to a is bu.
4(1)
= 21)(7.2)12+2x24-y1
c x
3n2)42-x2-Y2
--> (20.2...9x2-3,2)
P
3
T., a
8x ?y.
Second Part:
k(2)_ 2Q.
c x
2 Va
2 Va2-x2-72
?> P
Ty- 4 xy ;
67. -> 0
1 k(2) = -
ti"(-2a24-x24-2z!1
y
3,2 142..x2..y2
Tft (20a2-3x2-9y2)
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- 52
Third Part:
1 k0) 411.21E.
c x
ITT281/11;2..y2
--> ? 1 R xy
2
_
ay
rourth Part:
?>R x .
-x
32 v-2. .y
2 2
rr a
a 0 on
a perfectly conducting screen in the pl e z = 0, the aperture in the screen being
denoted by S. Then the total field in t e half-space z < 0 is
(8.1)
whereat.
(8.2)
and
(8.3)
Ex 18 E
au av
az y az z
aw
Hx = ikv +?-- H
y
(u,v,w) =
ay au
ikw
ax ay 4'
The functions e se sh which are connecte by the relation
x y z
e sh
aw
aw
?
Hz az $
dxrdyt
( 8 .14.
ikh
ae ae
- --Y x
axl ay' 2
satisfy the differential-integral equatio
iodxidyl + ik?
(8.5) a ir
hz 0d:c1 dy - 1k /f
fiek locb"d3r1 2-av ffe
S
dxtclyt 2Itklii(x y o)
x 1.1
Iodx"Y'
dxt dyt
2nifi(x,ylo),
= -2nEitx y o)
z. s 7
when (x,y,o) is a point of S. If there we ?no aperture in the screens the total
field would be null in z < 0, but would be E?, -ri?, say, in z > 0. In the presence
of the aperture, the total, field in z >
0 auay o au av
E =E --,E -E? --,E -E +?+-.,
x x az y y aza z ax :
(8.6)
H -H? - ikv - , a aw Hy Hy + iku
x x aw
oy z z az e
-ikR
=e /R,
1111/11.1011111111111111111111iNIONSURISWINSINIMMIleillitalliallili
(x-xf)2+(y-yl + z ; when z = 0.
* Copson terms them "integral equatio an
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My comment upon this theorem was as follows[5];I
"There remains only one question not properly accounted for by this
analysis, namely whether or not line charges along the rim of the screen are
necessary*. For instance the proof of theorem 4 is incomplete. suppose
the equations (5) are solved rigorously, under the side-condition (4). It is
not at once evident whether the lialre functions 1.1,V,W defined by (2) fulfill (3).
Now it can be shown that
(8.7)
a au 1 fArr_i
-ikw -, + Tit-4 lax? (ey ) - 41 (ex )idxidyi ?
Maxwell's equations are satisfied if and only if the right-hand integral van-
ishes. Because ex=E?e =EY in the hole, the integral overScan be trans-
formed
into an integral along the rim of the hole
(8.8)
-1-1-1,Esds
2n
where Es denotes the electric field tangential to the rim of the hole. Therefore
the condition for Maxwell's equations to be satisfied is
(8.9)
Es = 0.
This is an extra condition with regard to the solution e,,h. Foracircular
x z
hole the condition is equivalent to x'e - ylex = 0 at the rim. This condition is
satisfied in Copson's (and Bethe's) theory for the small circular hole, and Copson
states that his approximate solution does not violate (3)."
The validity of Copson's assertions and his theorem has not been questioned;
however, I claim that a necessary condition for Conson's theorem 4 to be self -
condistent is
(8.10) es = 0 on the rim of S,
where es denotes the projection of the vector e- ? = (exle ) at the rim upon the
tangent to the rim.
The condition (10) does not imply that either ex or e or both are finite
on the rim. In fact, they generally become infinite on the rim of order D-1/2
where D denotes the distance to the rim. Thus (10) does not exclude that en is
1,m4aari by Amryin in a review of Bethe's paper (Bether.46]
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infinitely large on the rim, where en me
upon the normal of the rim.
Incidentally, es and en should
the corresponding components of when t
the interior of the aperture. The vecto
following Copson, -we may take it to be z
over the whole plane z = 0. Of course,
when the field point is tending to the r
states that es is continuous on the rim.
discontinuous there.
To prove that (10) is a necess
is a finite part of the plane z = 0, with
over, it will be supposed that e 2e and
x y
and have continuous first-order derivativ
interior of S.
Let So be any sutdomain of S wi
so have no points in common. Then Stokes
the projection of at the rim
be considered as limiting values of
e field point tends to the rim from
-1.
e is not defined outside S although,
ro there: in this case (e ,e 1E )
x y x y
he limiting values of ex and e are zero
from outside St In this sense, (10)
On the other hand, en is in general
y condition it will be assumed that S
a simple closed boundary curve S. More-
hz are absolutely integrable over S
s with respect to xl and y! in the
h boundary curve so such that 8 and
s theorem may be applied to the vector
tin the domain So when the field point ( ly,z) is outside S. Thus
ko i 4-ct ( 41 ey)- 4-0- ( ) cbci dyt I esds,
so
where the integrand of the surface integr
Now using (4) and the fact that (a/ax.,
that the integrandis equal to
-
ey +
It thus follows that
(8.11) - -17/1 eAdx1c1?ya //e
So. So.
The change in the order of integration an
all integrals involved are uniformly conve
is the z component of curl ( 44).
lay!) -(a/axlaoy) we see
dx'dyt-ik)/(hyjI dxidyl ends.=1.4?
So ao
differentiation is justified, since
gent with respect to xly,z and ab-
solutely convergent with respect to ex, e hzis
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The same remark applied if in (11) the limiting case is taken where So becomes
S, though it must be implicitly understood that certain regularity conditions
as to the boundary curves so and s are satisfied. If so, then (11) leads to
(8.12)
av au
+ -ikw 2-1-14/ 5
eeds
and then (10) follow from (12) and (3).
As emphasized before, in the proof of (10) it has not been required that
ex and e are finite on 8. Yet an explicit example may be useful.
If S is a circular aperture of radius a, the following functions are
admissible (so far as integrability is concerned):
(8.13)
2a2-act2-2302
ex = A
e -A
1042-x124'2
xty"
CXI
B Va.2-x12-y12 CYIA
where A, B, and C are constants. It is easy to verify that the corresponding
function h is given by
(8.14)
ikhz = -(3A+B)
Y1 +2C.
2 2 2
Va -xf -Y1
Clearly these functions are absolutely integrable when x'2 +y'2 < a2) and are
continuously differentiable when x12+y12 < a2. Unless A = 0, they are all
singular on the rim. Notwithstanding this, es is finite on the rim, as is
readily verified if polar coordinates are introduced. Letting x' = r' cos
y1 = r1 sin 91 we have
er
(8.15)
2a-r'
=062-1.'2
I. k/a2-r12
cos
= -Cr' - (2A+B) P--7:12'a sin ?
and so in this case es = -Ca on the rim. Consequently, the condition (10)
would rule out the case C # 0, but it does not require A =0.
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-64-
In view of this, I shall now quote*
"Bouwkampts argument depends on different'
and the use of Greenrs theorem. If e is
whereas, if it is discontinuous, it becom
himself has point out. And similarly for
on the rim, and (3) is a consequence of (
the functions ex and e is infinite on the
As a matter of fact, my proof of (10)
simply incorrect, as is also borne out by
ie that es should not. be singular on the r
case which has been overlooked by Copson.
rom Copsonts answer:[]
ation under the sign of integration
continuous it vanishes on the rim,
s infinite on the rim, as Bouwkamp
e. Thus either e and e vatish
Y
) and (4); or else at least one of
rim, and Bouwkamp's Argument Mild'.
shows that Copson's assertion is
he explicit example (13), What matters
m;. this constitutes an intermediate
Consequently, I fully maintain my "cr ticise of Copson vs paper. That is
to say, if we intend to solve Copsonts dif erential-integral equations (4), (5),
we should look for those solutions e 1 e ? h satisfy the auxiliary condition
x y
(10) because then and only then will the e ectromagnetic field specified by (1)
and (6) solve Maxwell's equations. This ' formation concerning the solution of the
differential-integral equations (5), which was omitted in Copson's paper and
which I therefore added in my review, does not weaken the value of Copson's
theorem. On the contrary, it was meant to be and is in fact a further step
towards the practical application of the t orem, especially in the construction
of approximate solutions.
In addition, it is now evident that a rigorous formulation of plane
diffraction problems there is never need o additional line integrals along
the rim. This settles an old question con erning the "rigorous" extension of
the Huygens--Kirchhoff principle to electr magnetic diffraction problems. Where-
as fictitious boundary values of the field vectors on an open surface in general
require these line integrals in order that axwellts equations be satisfied,
the correct boundary values automatically ake these equations vanish identically.
At the time of writing my review of C son's paper, his approximate solutions
for the small circular disk and aperture se med to be correct, since the condi-
* Actually, Copson discusses the complementary problem, his theorem 5. I pre-
fer to keep the argument in accordance mi h what I criticized in Copson's
work.
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tion (10) happened to be fulfilled and Bethe's earlier results were confirmed.
However, mainly because of Meixner's investigations, I have since come to the
conclusion that Bethe's as well as Copson's first4-order solutions are correct
in the wave zone, but that they fail in and near the aperture or disk.
The fact that Copson's approximate solution for the small circular aperture
is incorrect will be shown by the simplest possible example, namely the diffrac-
tion of a plane-polarized wave impinging in the normal direction. In this case
Copson's equations (6.4) - (6.6)04'21 reduce to
14k(8.16) k I) 2 12 .
= -W- Va -r 3 e = 0; hz - 4
n'VraT];-17
if it is assumed that the incident wave is polarized parallel to the x axis
and of unit amplitude (Ei =1 Hi = 1). In the limiting case ka 0, Copson's
x y
equations (5) then reduce to
h
dx1dy, dx1dyl ah + = 0,
ax z p J1?P
(8.17) # dx1pyl ikJje
dx1cptyl -211,
aY"S x
where
a ire dx'dy,
ax x p
a 11
ay/is ey
. 0,
2 2 2 2
P2 = (x-x, )2 + (y-y') x.+37. < a ?
If the expressions (16) are substituted in (17), the right-hand members become,
in the same order,
0; -2n + nk2(2a2-x2-y2) = -2n+O(k2a2); -2nikx = 0(ka);
thus, as in Copson's paper, (16) is an approximate solution of (17).
However, a second solution is provided by
(8.18)_ 14k 2a2-x12-2302
14ke = k hz
x 77-yra7=;:12 Y 3n
I I
a -r
14 y'
va -rt
which, if substituted in (17), will make the right-hand members equal to
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- 7 nk xy 0(k2a2); -2n + 1
12
1 2
k2(20a2-3x2-9y2)
-2n+0.(k2a2); 0.
This clearly demonstrates that equatio s (4) and (5) of 0opson's theory
must be handled very carefully in order to etermine an approximate solution
e ey h. In addition it is to be noted hat the approximate solution (18)
x z
is better than Copson's (16) since in the f ret case equation (17) is satisfied
up to terms of relative order (ka)2 and in he second case only up to terms
of relative order ka.
The condition (10) is not decisive as
(18) is the physical solution, since both (
(10). Thus (10) is not a sufficient condit*
of finding the long-wave approximation.
The solution of this difficulty is si
the electromagnetic field calculated on the
approximate solution (16) has revealed (B
electric field, which is throughout of ord
is discontinuous in the aperture. There i
and ignoring Ez, as Copson did in his appr
of the same order of magnituie.
As was shown elsewhere (Bouwkamp54)
not lead to a discontinuity in the electri
the question whether (16) or
6) and (18) are consistent with
on, at least not for the purpose
le. A detailed investigation of
basis of Copson's (that is, Bethe's)
kamp[471) that the corresponding
ka compared to the magnetic field,
no sense in retaining Ex in the aperture
ximation, since these quantities are
the approximate solution (18) does
field in the aperture.
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- 07 -
IX. Diffraction by Narrow Slits
In this section we shall be concerned with approximate solutions for
the diffraction of plane waves by narrow slits. The x-axis is assumed to
be parallel to the edges, and we shall consider only problems that are in-
dependent of x. That is the direction of incidence of the plane wave is
in a plane normal to the edge. The integral-equation formulation of these
two-dimensional problems follows at once from the corresponding formulation
in three dimensionst if we integrate with respect to the x-coordinate. If
in three dimensions we have r2= (x-x')2 + (y-y1)2 + (z-z02 and in two
dimensions p2
' (Y-30)2 (t-e)2, then
co Jim
(9.1) r
ext ni H(1)(kp).
-co
The two principal boundary-value problems for the slit are to be for-
mulated as follows:
Problem I, 9 = 0 on the screen:
V = 00(Ilt) - 00(y,-a) 01(Y)-z) (z < 0)
V ' Vi(Y,z), (t > 0).
Problem II, Mg/an = 0 On the screen:
0 = 00(Y)z) 00(Y,-t) - 02(Y0-0
V 0
02.(Y,z) 0)1
where the wave functions rgi and V2 (defined for z.> Olonly) can be represented
in the form of integrals extended over the aperture, the integrands contain-
ing the aperture values of V, and a2/an respectively. The two-dimensional
analogue of Rayleigh's formulas are
t See chapter III,
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i
(9.2) 'Vi(y,z)
V2(y,z)
iaa ( )
az
-a
. a
r a
' az
-a.
V2(3?3?
H
where 2a is the width of the slit.
As in the case of the circular apert
relations for V, and V2 by requiring that
the case of protlem I-, and
o is the incident field.
(9.3)
(9.4)
r a
-2i
az ro'Y'4=o
21 Vo(y$0)
where -a < y < a.
- -
in the
These differe
2 d2 a
= (k + f
47 -a
-a
Z912(y',z') 16z.?
azt
Henceforth we consider only the case
choose
o = exp(ikz). We also introduce
y = a sin 9, yl = a sin GI, Ica
1
(0) = --- cos 9 01 (a sin A,
1 ika
2(0) = a cos a2L 0 (a sin
?z 2
(7 iH7-41 ' _:9I 0 )
1
? in the aperture,
of the aperture. As Bouwkamp first
e not sufficient to insure a unique
singularity of the electric field at
h a) and b), does insure such a solu-
n of the far fields due to current
distance r. As Bouwkamp discussed
id region into two parts, z < 0 and
n, with sources prescribed in the
ion z < 0 is equivalent to that pro-
ductor at z 0, both by the pre-
etic current source n x E
* i.e., parallel to the plane of the scr
** M.K.S. system but normalized so that i
unity,
en.
trinsic impedance vi.77i of vacuum is
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17
in the aperture (i2, . unit normal vector in positive z - direction). Similarly
the fields in z > 0 may be regarded as produced solely by an induced magnetic
current density -n x E in the aperture.
It is convenient to introduce a uhalf-spacen dyadic Green's function
Y(rir,l) defined by (cf.[40])
(10.3) V x (V x Y) = k2Y =
subject to boundary conditions
a) Y -> 0 as 1r-r,1 oo (Im k > 0)
b) n x (V x T) = 0 at z st Os
where e is the unit dyadic defined by e? A = A, and 6(r -r1) is the three-
dimensional delta function. Physically -Y'r,r1) ? e is the magnetic field
produced at r by a delta-function magnetic current flowing in the direction
e at rl in the half-space. The half-space Green's function can be defined
in terms of a ?free -spaces' okyadic Green's function Y1(E1r1) which obeys
Eq. (3) with the omission of condition b). The free-space Green's function
is given by (cf. 1401)
(10.4) Yi(E,E!) = -ik(e + 2;) g(r,r1)
Ic
where g(r1t9 is the scalar Green's function defined by
(10.5)
Although
(10.6)
(V2 k2)g
g -> 0 as Ir.-x.11
eiklE-r11
(un k > 0).
is a simple closed form, a more convenient representation for subsequent appli-
cations will be considered below. The half-space dyadic Green's function Y(r,r1)
is obtained by additive or subtractive superposition of two free-space dyadic
Green's functions Y (r r1) one corresponding to the source at r' and the
f
other to its image at r' - 2n n ? r'. In particular, for a transverse magnetic
source on the z = 0 plane,
V
(10.7) Y(r,r1) V
-2ik(e + -T) g(r1r,),
k
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where rl= (xlly1,0).
The magnetic field due to both
can be expressed in terms of Y(r1r1), th
the prescribed sources are assumed to li
z < 0 can be represented by means of su
of the vector Green's theorem) as
(10.8) H(r) = g (r) Y(r.
o ap
where the first term H0 is the magnetic
ev
sources in the absence of the aperture,
produced by the induced sourcesnxEi
magnetic field in z > 0 produced by the
aperture is
(10.9)
H(r) = +
ap
It is evident from Eq. (2) that the ft
the field equations (1) in z 0 and t
Moreover, since Li x (V x Y) has a jump
at z = = 0, it follows that Eqs. (8
that are continuous as z -> t 0 and eq
aperture. The requirement of continu
and (9) in the aperture region (z =
(10.10) -H (r) x n = 2/7 n x Y(r rt
-0 - sj ap
on the transverse electric field E
-t
11xYxnat z= 0 should be noted; h
= 0 since the integral in (10) bec
of this latter fact, as Bouwkamp has
integral equation but may be called a
In view of the representati
(10.11) -Ho(r) x n= 4ik(et
V
the prescribed and induced sources
field of a 'point" source. Since
in z < 0, the magnetic field in
rposition (or equivalently by use
)? n x E(r?OdS1, z < 0
-
field produced by the prescribed
and the second term is the field
the aperture. Similarly, the
induced sources -n x E in the
OdS1, z > 0.
ld representations (8) and (9) satisfy
e boundary conditions (2a) and (2b).
discontinuity of value -iketo(p-pl) t
and (9) yield values of 2 x ?E?(r)
al to the value of n x E in the
'ty of the H(r) x rolgiven by (8)
) imposes the condition
x n ? Et(ri)dS11 r -? (x,y1t0)
"in the aperture%
the aperture. The continuity of
wever z is not permitted to equal
nes divergent at r rl. Because
mphasized, Eq. (10) is not a true
pseudo-integral eciapAlm, for
n (7), Eq. (10) may be rewritten as
74 g(rIri)E (r1)dS,, r in the aperture,
ap ,t
In a rectangular x y coordinate
st = + zoy,0 = tran
stem, 45(p-p') =
verse unit dyadic.
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ol
where the interchange of differentiation and integration is permissible since
the integral exists before and after interchange. Since g(r,rt) is integrable
even in the limit z = zl = 0, Eq. (11) is a true integro-differential equation.
To emphasize this fact we may rewrite (11) in the form*:
2
(10.12a) curlt k2F = -ikH (p) x n
-o
(10.12b) F(P) = 4J:p g(pppi)Et(EI)dS1
where we have defined
(10.13) curl2F=nxV9 xn? 2
? ?F,?
t t t t t
and where p = (x,y10) is the coordinate vector in the aperture. It is
necessary first to obtain the general solution of the vector differential
equation (12a) for F, and then to solve the integral equation (12b) for
)Fh. The arbitrary constants in the resulting solution for Et are determined
by imposing the remaining boundary condition (2c).
A general solution to Eq. (12a) can be obtained ma variety of
ways, depending on the nature of the excitation Ho and the shape of the
aperture. It thus appears desirable to specialize at this pent.
Diffraction of a Plane Wave by a Circular Aperture
Since an arbitrary source distribution can be resolved into plane
wave constituents, it is basic to consider a plane wave incident on the aper-
ture. There are two independent types of vector plane waves: the E- and the
H- waves**, distinguished by their polarization. If a rectangular coordinate
system with origin at the center of the aperture is oriented so that the plane
wave is incident in the xz-plane, the transverse field distribution of an E-mode
[551 wave is given by
* Bouwkamp writes Eq. (12a) in a somewhat different form. From (12a) one notes
that
(1)
-k2V ? F --ikV ? H x n = -k2Eon
whence on expansion of curl2 of (12a)
t
2 2k2 a E
(ii) (Vt + k) F = ik(et + VtVt) H() x ? n =
,... .... az -ot ?
Eqs, (i) and (ii) which together are equivalent to Eq. (12a), are employed by
Bouwkamp but with,E replaced by F x n.
** Cf. M; Section 26 contains a description of a complete orthogonal set
of vector plane waves.
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- 82
(10.14a) el(p) = 11,(p) x n =
"...? 'to
and that of an H-mode wave is given by
(10.14b)
eft(p) = h"(P) x n = -Vt
where e and h denote, respectively, the
distributions of the mode in question.
excitation) the unperturbed transverse
is a superposition of an E- and H- mode*
ikxx
=-e
-0
ikxx
ikxx
ikx e ?To '
transverse electric and magnetic field
With Bouwkamp's choice of plane wave
agne tic field Ho(p) x n in the aperture
(10.15) H (p) x n Ithqp) x n Iuh"(p) x ns
where I' and I' are twice the incident plitudes of the transverse magnetic
fields of the plans waves defined in Eqs. (14). For Bouwkamp's choice of plane
wave excitation,*
(10.16) It = -2cos0'
gf Iu = 2sinO0cosA0
kx = k sinGo .
The solution of Eqs. (12) for composite wave of the form (15) is un-
necessarily complicated since the symmet y of the excitation is concealed.
Accordingly it is desirable to represent F(p) as a superposition of an E- and
H- mode component, viz:
(10.17)
.E(P) = ITt(p) + PIP
then (12a) decomposes Into
2
(10.17a) curltEl - kEt
2
(10.17b) dur1t7 - k-Fu =
and correspondingly (12b) decomposes int
(10.18a)
Ft(P) = /41( g(p,p')Eqp )dS1
j ap
g(p,p9,01(pi)dS,,
ap
x n
xn,
(10.18b)
where
(10.18c) !:t(P) = PET(P) + TuEu(p)
* See Bouwkamp's definition of the an
Go' p on p.47.
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Solutions of the vector partial differential equations (17) can be
obtained readily. In fact the general solution of Eqs. (17) may be expressed
in the form:
(10,19a) LIShip_ WWI% ,22 ,Jm(kip)
m t k
= + 2_ 2a V x -
k m=1
co J (kp)
E 2atvt x n( k
m cos Mg)
M
m=o
(10.19b)r(a) br(OxIt,
ik
oo J.,(kp)
2 2
sin MV)
E 21m7tx 4( 4"k cos
k -kx M=0
OD J (kp)
+ E 2131VtX 3.4,0L,( mk sin m0) .
m=1 m
From Eqs. (13) and (14)
mcil)
we see that the first term in Eq. (19a or b) is a parti-
cular solution, while the remaining "H-mode terms", which alone can satisfy the
homogeneous equations (17), represent the complementary solution. The 0-polar-
izationt of the complementary solution is of the most general. form; however, it
may be delimited by using the symmetry properties of the field. In view of the
rotational symmetry of the structure about the center of the circular aperture,
the 0-dependence of the field is determined by the nature of the excitation. The
symmetry of the excitation is evident when a solution in powers of ik is considered.
As Bouwkamp has shown, only the E,umode component (19a) contributes to zero-order
in ik whereas both the E- and H- components of F contribute to first-order in ik.
To make explicit the perturbation solution in powers of ik one employs
the scheme:
(10.20) h(p) = 120(p) +ik 1!1(p) + (ik)2ht(p) +
F(p) F0(e) +ik ti(p) + (ik)2F2(p);
then Eqs. (17) decompose into
(10.21a) cur12F =
(10.21b) curl2F = -h n
t-1 -o
(10.21c) curl2F2 + Fo = -h x n
t- - Ar
(10.21d) curl2F + F = -h x n
t-3 -1
? ? ?
t Note that x = p cos V y = p sin 0.
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- 84
(the superscripts I and" are omitted).
first-order solutions F and F are not
ev0
to determine the transverse divergence
respectively are required. Use of Eqs.
in zero-order both Fo and. the correspon
-
ferred dependence, i.e., they are radi
the electric field (or h x n) of an inc
the x: or y directions respectively.
(21V and d) indicate that in the first
fields must have the following polar co
(10.22a) E/.,cos g
V
From (22a) the rectangular components fo
(10.22b) Ex = cos V Ep - sin V Eiif
Ey = sin V Ep + cos V Ecl,
It is convenient to deal with rectangul
that the angular symmetry of the rectan
F are similar. Since the symmetry of th
we can give a more detailed characterize
given in (2c). Equations (22a) state th
(2c) may be decomposed into
(10.23)
E t cos 91 Et
It should be noted that the zero- and
determined solely by Eqs. (21a and b);
f these two solutions, Eqs. (21c and d)
(14) in Eqs. (21a and c) indicates that
ng aperture field E0(p) have no pre-
. Furthermore, Eqs* (14) reveal that
dent E- or H- mode is polarized along
n accordance with this symmetry Eqs.
rder the corresponding aperture electric
onentst:
EH ~sin V
q?
low by the transformation
field components since Eqs. (18) imply*.
lar (but not polar) components of E and
first-order aperture field is known,
ion of the rim singularity than that
in the first order the rim condition
yri ?%, sin V
? co. ,
Bouwkamp has pointed out that t
the literatures** have usually been corre
the first-order solutions. Since the dif
and that of Bouwkamp is most evident in t
tion will be considered below.
e zero-order solutions obtained in
t, but that this is not the case for
erence between the method employed here
e first order, only this order of solu-
lh See note on previous page.
* Since g(p pi) is independent of the V
* *
See section VII. For a derivation empl
authors HCoupling of Waveguides by Sma
PI3-106, Polytechnic Institute of Brook
ientation of the coordinate as.
ying the methods herein , cf. the
1 Apertures", p.68, report R17_47,
yn.
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- -
First order solution in (ik).
The general solutions (19) permit a ready determination of the first-
order form* of F(p). It is relevant therefore to consider at this point the
,???
first-order form Of the integral equations (18). Let
g(p,p1) g (plpt) + (ik)g1 (p,o) + =
h? ? ,/
1 ik
Ect ???
4n1O -Oil
E(p) = Eje) + (ik)El(p) + ***;
Then in view of the corresponding expansion of F(p) in (20) and the stated
radial nature of ,4(e), one has the first-order integral equation (for either
the ' or component):
Fl(P) = 4)7 go(p,p')Al(pl)dSts
ap
where g0(E,r1) is the static form of the Green's function in (5), defined to
within a constant by
V2 go (r,r9 = -6(r -r1).
In an oblate spheroidal coordinate system Q, 0 associated with an aperture of
radius a,
(10.26)
x = a cos 9 cos 0
y = a sin Q sin 0,
where 0 < 9 < 2n and 0 < Q < m/2. The static Green's function go can be represented
diagonally in this coordinate system in terms of a complete set of orthogonal
functions P (cosQ) e?8714 which are periodic in 0 and whose derivative with respect
sinm0
to Q vanishes at 0 2, rr - The latter property implies that n Pnd m are either both
*'s-
even or both odd integers, and that m < n. The desired diagonal representation5'15
OD n2 _
1
go(P/Pf) = m E E e (2n+1)
m n n
(11-m)1 P111(0] Pw
(cosQ)1411(cosW)cosm(95-09
)! n J
(10.27) no m=o (n+m
where em is the Neumann number and equals 1 or 2 depending on whether m = 0 or
>0. On substitution of (26) and (27) one obtains as the "diagonal" form of the
integral equation (25):
* The first-order solution can likewise be obtained by means of Eqs. (21).
** Note that Eq. (27) represents a convenient form for some of Bouwkamp's integral
theorems.
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2n n/2
2:1(13)= Vd004/"dglsingtE(gt) Em
0 0 113M
(10.28)
where
E(A) = cos A ;1(2).
It is evident that Eq. (26) can be solve
a series of Pm(cosg)m9 cos functions. Tw
sinm9
which F = Ft and the other in which F
.v1 -.1
E-mode Solution
In view of Eqs. (14a) and (22a
general E-mode solution (19a) that yield
ik are
(10.29)
. 2 2
1
;:c
N.1 -
Fl(p) =- -2(1 - - xP ----cos )x
2n+1) Pm(0)12Pm(cosg)P:(cos60)
(n+m)1 n n
cosm(V-9./))
immediately on representation of Ii(p)in
cases will be distinguished, one in
F".
we see that the
contributions to
2aivtx
2m317tx
only terms of the
the first order in
3.1(kp)
n( stn0)
n (34!_cp) silos)
where the omitted am , al terms have the cng symmetry and the omitted ikxx?so
m
term has been considered in the zero-orde approximation. To the desired order
Eq.. (29) becomes
(10.30)
Y1(P) =
,1
rar 5in2Q0-'1k2)
L k
C
2 ,2
+
where we have put kx = ksingo. Substitut
(10.31) sin2g = p.-P2(cosg)] =
one obtains
(10.32)
r 1 2
El(P)E1 [11- ?7 +(sin2 go-alk2
k
2 2 ,a2 _2
+((a1+a3)k +2sin 610)74 F2(
The integral equation for the first-order
with (32) as the left-hand member, On e
terms, one finds that
2 2
(10.11) Cos0 N.(p) o +A2 1),(cosg)+A2P2
c
where
o2
+ ((a1+a3)k2+2s1n2 go)t-cos2dx0
29 yo
ng p = a sing and noting that
2,
2cosg),
2
(a1 k2-sin2g )aP (cos)g 0
2
sg)cos2)501+(a1 a3)k2a2-74-P2(cosg)sin2ko.
aperture field E{(2) is given by (26)
ating coefficients of corresponding
CO
cos2dx +B.2P2(cosg)sin29y $
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1 4. (sin290_a1k2) a2
al- --
naAo = 2-
k
1 2 2 a2
nal2 = (alk -sin Ceo) -b-
3
2
_
-8. RaJAA2 2 = ((al+a3)k2 + 2sin2 Go) h
.2 2
ii. naB22 = (al-a3) Ic2a .
The application of the rim boundary condition (23) to (33) yields
A2 2
= B2
A2 2
Ao- 0;
-3A2 =
from this the two arbitrary constants al and a3 follow:
a3k2 = - sin2Q0
a1k2 = 1 (to first order).
Equation (33) then yields for the first-order E-mode component of the electric
field in the aperture:
(10.34) ace) 72 [1/77 (2-sin20G )xo + fe.x -? (l+sin2Ao)
al77
H-mode solution
As in the case of Eq. (29) the only terms in the general H-mode
solution (19b) which are of interest for the first-order solution ars:
(1-k
2 2
J (kp) (kp)
(10.3 F
5) l(P)=
2xx2/2) ,31.20+2ply(),s(W--- coe0)+20317tx cos*,
k -kx
where the coefficients Po and p2 have been employed to remove in first order
the ikyo term of the particular solution, and where all pm and plin terms with
the -wrong symmetry (cf. Eqs. 22) have been omitted. Using the relations (31),
we can rewrite Eq. (36) to the first order as follows:
k2a2 22 2 a2
21.t(R) . _03113)74- P2(cos)sin2v2% -p1+(pik -tan Go) -6-
2 2 a2 2 a2 2
-(pik -tan G0).7 P2(cosG)+((pl-3)k2-2tan 80)74 P2
(10.37) (cosQ)cos2dy0.
As before, if we substitute (37) into the integral equation (28) for the aperture
field and equate the corresponding coefficients)we obtain
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8 -
cose (co rd,sin2,.0. [no
(10.38)
where 3 2
staC2
naDo
1
staD2
3 2
6 rtaD2
(cos0)+D22P22(cosA)cos29/ ,
k2a2
= 1 2b
1 a (131k2.4a22?30)
k -kx
= (tan2plk2)
2
)k2-2tan2 ) a
o
(031
?
2
Furthermore, the imposition of the im conditions (23) on (38) leads to
2 2
C2 = -D2
2 2
D- -A?-?+ 31) 32
o z 2
0
and hence the arbitrary constants p and 133 in (38) become
k2 = -tan2o
3
sec2g0 (to the first order).
Equation (38) then yields for the -mode component of aperture field in the
first order:
(10.39) Et; (p) = - ? 2 a22
2
Pe.Y0
't? I. "., 3n P -Yo +
In view of Eqs. (16) and (18c) we find by superposition that the
total first-order aperture field ?r the Bouwkamp choice of incident wave is
ika(P)= -FE Vs -p ( Ilx -Illy -Iisin2A x )
2.ik [ / 2 2
-o o o-o
(io.Loa)
or, as Bouwkamp obtained,
n 2 II /72-7 T (_ik
I Va -P
(10.40b)
(I ix + I'sin28 x )
o -o o-o
tra72.7-p
x - Von)
? (-ikH xrs + V,E )
T, on
where H and VtEon are the wiper rbed values evaluated at the center of the
aperture.
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References
1. Raker and Copson, The Mathematical Theory of Huygens' Principle Oxford'
Clarendon Press)11950, p. 71.
2. rbid. p. 98.
3. Sommerfeld, Vorlespngen i;ber theoretische Physik, vol. 4, Optik (Wiesbaden,
Dieterich Verlag) 1950, P. 208 If.
4. Born, Optik (Berlin, Springer Verlag) 1933, P. 152.
5. Franz, Z. Physik 121 (1949) 563.
6. Sehelkunoff, Comm. Pure Appl. Math. 4 (1951) 43.
7. Franz, Z. Physik j (1950) 432; Proc. Roy. Soc. AL (1950) 925.
8. Sommerfeld, p. 203.
9. Bonwkamp, Thesis (Groningen, Wolters) 1941.
10. mid.
11. Braunbek, Z. Physik 127 (1950) 381.
Veinier and Fritze, Z. Angew. Physik 1 (1949) 535.
Severin and Starke, Acoustioa (Akust. Bethel t, 2) 1952, p. 59.
12. Bremer, Comm. Pure Appl. Math. 4 (1951) 61.
13. Luneberg, Mathematical Theory of Optics (Brown University) 1944.
14. Scheffers, Ann. Physik 42 (1942) 211.
15. Durand, C.R. Acad. Sci. Paris 226 (1948) 1440, 1593.
16. Spence, J. Acoust. Soc. Amer. 21 (1949) 98.
17. Primakoff et al., J. AC011At. Soc. Amer. la (1947) 132.
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- 90 -
18. Braunbek, Z. Physik 127 (1950) 81; Ibid. 127 (1950 405.
19. Brillaain, C. R. Acad. Sci, Paris 229 (1949) 513.
20. Bouvkamp, Physica 12 (1947) 467
21. Severin and Starke., Acoustica- t St. Beiheft 2 (1952) 59.
22. Meixner and Fritze, Z. Angew. Ph sik 3 (1951) 171.
23. Levine and Schwinger, Phys. Rev. 74 (1948) 958, 1212.
24. Baker and Copson, ap.cit., p. i6.
25. Ibid., p. 160.
26. Ibid., p. 184.
27. Maue, Z. Physik 126. (1949) 601, equation 60.
28. Levine and Schwinger, Phys. Rev.
4
(1948)
958.
29. Sommerfeld, op p. 283.
30. Levine and Schwinger, Phys. Rev,
4
(1948)
958.
31. Baker and Copson, op .cit" p. 18.
32. Levine, Comm. Pure Appl. Math. 3 1950) 355.
33. Levine and Schwinge:?, Phys. Rev. 4 (1948) 958.
34. Sommerfeld, Ann. Physik 42 (1942) 89.
35. Magnus, Res. Rep. No. EM-321 N.Y.0 Washington Sq. College, Math. Res.
Group (1951).
36. Idem.
37. See ref. 34, also An. Physik 2 (19 8) 85.
38. Bouwkamp, Physica 16 (1950) 1.
39. Levine and Schwinger, Phys. Rev. 7 (1948) 958; appendix 1.
40. Levine and Schwinger? Comm. Pure Ap 1. Math. 3 (1950) 355.
41. Miles, J. Acoust. Soc. Amer. 21 (19 9) 140, 434.
42. Copson, Proc. Roy. Scc. A 186 (1946 110.
43. Copson, Ibid, A 202 (1950) 277.
44. Meixner, Ann. Physik .5 (1949) 2.
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- 91 -
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