WAVE MOTION 1 (1979) 287-298.
©' NORTH-HOLLAND PUBLISHING COMPANY
ON THE D I F F R A C T I O N
ULTRASONIC WAVE
W. H E R E M A N
OF LIGHT
BY
AN
AMPLITUDE-MODULATED
and R. M E R T E N S
Instituut voor Theoretische Mechanica, Ri]ksuniversiteit Gent, Belgium
Received 1 February 1979, Revised 10 June 1979
The partial differential equation for the wave function in the problem of the diffraction of light by an ultrasonic wave is
derived as the result of transformations and related approximations of the electrical field equation itself. It is shown that this
way of treating the diffraction problem is equivalent to the generating function method applied to the Raman-Nath system of
difference-differential equations for the amplitudes of the diffracted waves. This general method is worked out here for
diffraction due to an amplitude-modulated ultrasonic wave, for which the symmetry properties of the diffraction pattern are
also investigated with the help of the transformed wave equation. Afterwards the wave function is considered as a generating
function for the amplitudes. The corresponding partial differential equation for this generator is solved exactly in the case of
large ultrasonic wave lengths at oblique incidence of the light.
1. Introduction
T h e p r o b l e m of diffraction of light in a liquid m e d i u m , disturbed by the passage of ultrasonics waves,
predicted by Brillouin [1] in 1921 was i n d e p e n d e n t l y observed in 1932 by D e b y e and Sears [2] and by Lucas
and B i q u a r d [3]. T h e first theories were established by Brillouin [4] and D e b y e [5], w h o used the m e t h o d of
r e t a r d e d potentials and o b t a i n e d in this way only first-order lines in the diffraction spectrum. In o r d e r to
a c c o u n t for the higher o r d e r lines, Brillouin [4] derived a m o r e general theory, based on the expansion of
the solution of the wave e q u a t i o n into a series of M a t h i e u functions (in the case of standing ultrasonic
waves); the expressions for the intensities; however, were limited to the first three terms of a series
expansion and the spectral character of each diffracted light b e a m was not taken into account. O n the o t h e r
h a n d R a m a n and N a g e n d r a N a t h established a simplified t h e o r y [6, 7, 8] based on geometrical optics and a
general t h e o r y [9, 10, 11] based on Maxwell's equations, in which a system of difference--differential
equations for the amplitudes of the diffracted light waves was set up. A s u m m a r y of those theories m a y be
f o u n d in B o r n and Wolf's b o o k 'Principles of Optics' [12, C h a p t e r 12]. T h e latter reference also contains an
extensive exposition of Bhatia and N o b l e ' s t h e o r y based on the integral e q u a t i o n m e t h o d [13, 14]. Until
1968 m o s t theoretical investigations were limited to a p p r o x i m a t e solutions, a survey of which m a y be f o u n d
in [15] for n o r m a l incidence of the light and in [16] for oblique incidence. Exact solutions of the system of
difference-differential equations of R a m a n and N a g e n d r a N a t h were studied by M e r t e n s and Kuliasko [ 17]
in the case of n o r m a l incidence of the light, by P l a n c k e - S c h u y t e n and M e r t e n s [18, 19] in the case of oblique
incidence. H e r e b y the solution of the R a m a n - N a t h system was r e d u c e d to the integration of a partial
differential e q u a t i o n for an u n k n o w n generating function. A f t e r solving this e q u a t i o n the generating
function was e x p a n d e d into a L a u r e n t series, the coefficients of which yield the expressions for the
amplitudes of the diffracted light waves.
287
W. Hereman and R. Mertens / Diffraction of light by A M ultrasonics
288
The purpose of the present p a p e r is threefold:
(1) We shall develop here a modification of the generating function method, applicable to calculate the
intensities of the optical spectra produced when light traverses a liquid disturbed by ultrasound.
T h e r e b y we have tried to maintain in our approach the advantages of the exact scalar wave theory (as
treated by Berry [20]) and to include the approximations of the general R a m a n - N a t h theory [11], yet
without using their system of difference-differential equations.
(2) Applying this modified method to the diffraction due to an amplitude-modulated ultrasonic wave, we
m a k e some progress in the theoretical explanation of the diffraction pattern, a problem for which until now
only approximations were obtained by Aggarwal et al. [21, 22], Phariseau [23, 24] and Mertens [25].
(3) In the last p a p e r correct intensities were calculated in the case of large ultrasonic wave lengths, at
normal incidence of the light. Besides, the s y m m e t r y properties of principal as well as of satellite lines were
investigated. A closer inspection of that p a p e r brings to light some invalidities of the extended (three
dimensional) generating function m e t h o d used there. Here, we remove these difficulties using a m o r e
refined technique to calculate the intensities in the case of oblique incidence of the light.
2. The wave equation with boundary conditions
For the description of the p h e n o m e n o n of diffraction of light by ultrasonic waves, consider the physical
situation illustrated in Fig. 1 and suppose that the incident m o n o c h r o m a t i c plane light wave (with wave
length in vacuum A, frequency v and circular wave n u m b e r in vacuum k = 2"rr/)t) has a direction of
X
DIRECTION
ULTRASONICWAVE
(-1.-1)
(-1.0)
f (-1.,1)
II/
II/
~
~-/z~
~
¢o.-i)
(O,O)
(0.÷1) DIFFRACTION
WAVES
----~(.,. o)
(+1.+1)
FLUID
O~YSTAL
Z
Fig. 1. Geometry of the phenomenon of diffraction of light by ultrasonic waves in the case,of oblique incidence. Representation of
principal (thick arrows) and satellite (thin arrows) diffraction waves (l = O, ± 1; m = ± 1).
propagation making an angle ~ with the positive z-axis. Let further the light wave be linearly polarized with
its electrical vector E normal to the plane of incidence Ozx, so that E is along the y-axis. The wave equation,
resulting from Maxwell's equations, may be written, after some simplifications and some approximations
justified by R a m a n and Nagendra Nath [9] and by Born and Wolf [12], as a o n e - c o m p o n e n t equation
az(p
02qb+ - Ox 2
8z 2 -
-k2e(x,
t)~,
(1)
W. Hereman and R. Mertens / Diffraction of light by A M ultrasonics
289
where the function qb(x, z, t) is related to Ey(x, z, t) (Ex and Ez being negligible) 1 by
Ey(x, z, t) = crp(x, z, t) exp(i tot).
(2)
Here, to = 2,try and e(x, t) is the relative permittivity of the fluid, periodic in x and t. According to the
physical situation sketched above and as a consequence of the approximations, the three-dimensional
vectorial problem is reduced to a two-dimensional one, described by a scalar wave equation.
As the incident light beam, propagating through the undisturbed medium (with dielectric constant eo)
may be written as
~o(X, z) = exp[-ik4~o(X sin q~+ z cos ~o)],
(3)
the boundary conditions are
• (x, 0, t) = exp(-ikx/~oX sin ~0),
(4)
(~-~
(5)
= - i k X/~o cos ~o e x p ( - i k X/~oX sin ~o).
Z=0
Furthermore it is worthwhile mentioning that, the R a m a n - N a t h theory being a scalar wave theory, all
exponentially decaying disturbances near the walls of the column are ruled out. Also, in general, evanescent
effects due to reflected waves inside and outside the non-magnetic liquid are omitted.
3. The wave equation in the case of amplitude-modulated ultrasonics
The relative permittivity of a liquid medium disturbed by an ultrasonic wave can be written as
(6)
e = eo+e' sin(to*t-k'x),
where to* is the pulsatance of the ultrasonic wave, k* its circular wave number, A* its wave length.
Supposing the "maximum variation" e' of the relative permittivity to be modulated, we write,
e' = el + e2 cos(to l*t - k * x + 6),
(7)
where to1* is the modulation pulsatance, kl* the circular wave number of the modulating wave, A* its wave
length and 8 a phase constant, m = e2/el is called the depth of modulation, and since e2 is always much
smaller than el it is a small parameter of the problem. The relative permittivity of the perturbed medium
hence becomes
e(x, t) = eO+ el s i n ( t o * t - k * x ) + le2 sin[(to * +to *l ) t - ( k * + k *l )X +B]
1
+~e2 sin[(to* - t o * ) t -
( k * - k * ) x - 8].
(8)
In order to integrate the scalar Helmholtz equation (1) we split off the incident light beam (3) by the
substitution
• (x, z, t) = ~o(X, z ) ~ ' ( x , z, t).
x W h e n E is not polarized, Brillouin [4] has shown that the three components each satisfy a wave equation of the type (1).
(9)
W. Herernan and R. Mertens / Diffraction of light by A M ultrasonics
290
W e r e m a r k that this is always the first step in the establishment of a R a m a n - N a t h system of d i f f e r e n c e differential equations too. N o w we introduce the c o o r d i n a t e t r a n s f o r m a t i o n (x, z, t) + (~r, xl, x2), defined by
kexz
(10)
~"= 2~/7-~ocos ¢ '
3
Xl = ½(k*x - w*t + 5rr),
(11)
x2 =½(k*x - t o * t - S ) ,
(12)
so that we shall only consider solutions ~ ' ( x , z, t) d e p e n d i n g on w ' t - k ' x ,
w * t - k * x +8 and ~r.
Expressing the relative permittivity (8) in Xl and x2, we obtain
E = eO - - E 1 COS
2x 1 - - ½g 2
COS
2 (x 1 + x 2 ) - ~ e1 2 cos 2 ( X l - x 2 ) ;
(13)
since it is a rr-periodic function in b o t h variables, we require the wave function
gt(Xl, x2, ~') = @'(x(xl, x2), z (~'), t(Xl, x2)),
(14)
to satisfy the periodicity relations
gt(Xl +~r, x2, () = qt(xa, x2, ~'),
(15)
gt(xl, x2 +'rr, 5) = gt(xl, x2, ~').
(16)
O n the hand of eqs. (1), (3), (8), (9), (10), (11), (12) and (14), it is straightforward to verify that gt(Xl, x2, 5)
has to be solution of the following partial differential e q u a t i o n
2 00~5 - 2i[cos
t
2 X l + ~ 82
e l C O S 2 ( X l + X z ) + 2 e8_.I..~2COS 2(X~ - x 2 ) ] ~ =
=-¼i0
(~XT1+ c ~2 )2 ' / ' - a
"
sm
-OlIl
-~l'I)"
~t~OX
1 al sin q~Ox2'
(17)
where
c = A*/A*,
(18)
a = 2h~/eo/h*el,
(19)
al :
(20)
ca = 2A ~/eo/h*ex,
p = 2A 2/A " 2 E 1 ,
(21)
are four i m p o r t a n t dimensionless p a r a m e t e r s .
O b s e r v e that in eq. (17) the t e r m containing the second o r d e r derivative with respect to ~" is neglected,
which was also the case for a R a m a n - N a t h system. C o n s e q u e n t l y o n e of the b o u n d a r y conditions must be
d r o p p e d . K e e p i n g in mind the calculation of intensities, it is well k n o w n that only the b o u n d a r y condition
(4) should be retained. With the help of eqs. (9) and (14) it b e c o m e s
qt(Xh X2, 0) = 1.
(22)
T h e foregoing a p p r o x i m a t i o n for obtaining eq. (17) with b o u n d a r y condition (22) is k n o w n in the optical
literature as the " p a r a b o l i c a p p r o x i m a t i o n " .
W. Hereman and R. Mertens / Diffraction of light by A M ultrasonics
291
4. T h e w a v e function as a g e n e r a t o r for the amplitudes of the diffracted w a v e s
The periodicity conditions (15), (16) enable us to look for a solution in the form of a double Fourier
expansion,
q-OC~
lt¢(x1, X2, ~') =
t~lm(~)i l+m exp[2i(lxl + rex2)],
~
(23)
I,m ~ --00
where 4~t,~ are the unknown amplitudes of the emerging waves.
It must be remarked that representations of the type eq. (23) do not always converge. Cassedy and Oliner
[26, 27] have determined the region of the relative phase velocities of the disturbing wave, where the
expansion (23) is not valid. This corresponds in our case to a very small region (of width 10 -s) in the
neighbourhood of unity, far away from the values of the relative phase velocities of ultrasonic waves in
liquids, which are of the order 10 -s.
From this double Fourier series (written in the original variables x and t with the help of (11) and (12)), it
is clear that the light wave is split into different subwaves. Each subwave represents a diffracted beam of
light, making the angle 0t,, with the z-axis, given by
sin 01,~= sin • - (A/',/-~o)(I/A * + m/A *1),
(24)
and having the frequency-shift,
Avl,. = --(lu* + rnv*).
(25)
The intensities lt,. (of order (l, m)) of the diffraction lines are then given by
I,m (~') = ~b~m(()~b*,. (~'),
(26)
where the asterisk stands for complex conjugation. Although the amplitudes ~blm can be calculated
according to the formula
~(Xl, x2, ¢) exp[-2i(lxl + rex2)] dxl dx2,
~tm(sr) = ~-~
(27)
once the solution 1F(Xl, x2, ~r) is known, we shall deduce them in an easier way.
It is our aim to transform the wave function in the so-called "generator form" for the amplitudes,
affirming at the same time the form (23) of the requested solution. Therefore we introduce the complex
transformation
~: = i exp(2ixl),
r / = iexp(2ix2),
(28)
and if we denote ~(xl(~), x2(n), () by G(~:, n, ~'), we obtain for (23),
+CO
G(~, r/, br) =
E
l,m
¢~,.(sr)sCtrlm.
(29)
= --o0
From eqs. (17) and (22), it is easy to verify that this generating function must satisfy
2 0 G - (~ - ~¢-1)G + (ieE/2e 1)(s¢ - ¢ - l ) ( r / - r/-l)G =
.[
2
_2 0 G
=l[p~ -~+Pln
2
2 0 G,,~
~
2
0 G
, z
~*~:02¢'O0--~*tp-2a
0G
0G']
sin ~o)~:--~--+( 0 1 - 2 a l sin ~o)rl-~--~--J,
(30)
W. Hereman and R. Mertens / Diffraction of light by A M ultrasonics
292
G(~, r/, 0) = 1,
(31)
with the abbreviations
Px = 2A2/A*2e1 =c2p,
P2=2A2/A*A*el =co.
(32)
Eq. (30) (up to some trivial modifications) in precisely the partial differential equation one would obtain for
the generating function introduced by Kuliasko, Mertens and Leroy [15, 17] for the system of differencedifferential equations of Raman-Nath type established by Phariseau [24].
Remark that once the function G(sc, r/, st) is known, the amplitudes of the diffracted light waves simply
may be found as the coefficients in the Laurent series (29) of G(~:, r/, ~r).
On the other hand, from complex function theory, it is well-known that they also can be calculated by the
integral
~b,,,(() = (1/2~ri) 2 ~ ~ G(~:, r/, ¢)~ 't+"r/-("+1~ dsc dr/,
(33)
where each contour is any closed path within an annular region encircling the origin O once, in a
counterclockwise sense, respectively in the complex sc- and r/-plane. Naturally the generating function G is
supposed to be holomorphic in the respective regions.
5. Intensities in the case of large ultrasonic wave lengths
We want to solve eq. (30) under the assumptions that A* >>A and A* >>h. With regard to the expressions
(21) and (32) we may write
(34)
/9 -~ P l -----P 2 = 0 .
Hence, eq. (30) reduces to a linear first order partial differential equation,
0G
2 - - + 2ia~: sin , p ~ - + 2ial r/sin ~
a~"
0¢
OG _(
= [ ~ - sc-1) - (ie2/2e 1)(~ - ~c-l)(r/- r/-~)]G.
(35)
It is easy to write down the differential system of the characteristics [28, Ch. 1, § 4], which is equivalent to
three ordinary differential equations
d_{= bd~',
(36)
d__~_~= b~dsr,
r/
(37)
d__G= ½(~ _ s¢_1)[1 _ D ( r / - n-l)] d~',
G
(38)
b = ia sin ~,
(39)
where
bl = ial sin ~0,
D = ie2/2e1.
Integration of the first two equations yields
:
C1 exp(b~r),
77 = C2 exp(bl~r),
(40)
293
w.. Hereman and R. Mertens/ Diffraction of light by AM ultrasonics
wherein C1 and C2 are arbitrary integration constants. Introducing these relations into (38) we get after
integration,
G(~, 71, ~) = C3(, exp(-b~), ~7 exp(-bl~'))
D
1
exp 1(~:+~)
exp{~(~-~)}.
(41,
The arbitrary function C3(~ exp(-b~'), rt exp(-bl~')) now follows from the boundary condition (31), so that
we obtain
G(~¢, r/, st)= exp{-~b[S¢ exp(-bsr)+ ~ exp(b~')]} exp{~b(S¢+ 1)}
x exp{ 2(bD+bl)[ ' r l e x p ( - ( b + b l ) ' ) - l e x p ( ( b + b x ) ( ) ] } e x p { - ~
× exp{~ [
2(bD+bl)((r/- ~)
]
~ exp(-(b- bl)sr)-~ exp((b- hi)()]} exp{ 2(bD bl)(~-~)}"
(42)
Rewriting the right-hand side as
G(~:, r/, st)= exp{~b [~: exp(-½bsr) -~ exp(½b~')]}[ exp(lbsr) -exp(-½b~')]
×
ex.
+
+
× [exp(½(b+ bl)~')-exp(-½(b + b 1)()]
× exp{~ [ ~
exp(-½(b- b 1)~')+~ exp(l(b- bl)sr)]}
× [exp(½(b- bl)()- exp(-½(b - bl)sr)],
(43)
it is possible to apply the generating function formulae respectively for the Bessel functions J, (x) and the
modified Bessel functions I, (x),
q-oo
exp[½(t-1/t)x]=
~
J~(x)t",
(44)
In(x)t".
(45)
n = --oo
and
4-OO
exp[~(t+ l / t ) x ] =
Z
n =-¢x3
So, we obtain a product of three absolutely convergent Laurent series, which, taking into account the
relation
I,, (x) = i-"Jn (ix),
(46)
294
W. Hereman and R. Mertens / Diffraction of light by A M ultrasonics
may be written as the following triple series
G(~:, 7, ~') =
•
i-(r+')J,
sinh½b~')Jr - - -
b+bl
p ......
x J,
_
sinh½(b + bl)
exp[-5(p+r+s)b(]exp[-5(r-S)bl(]~
sinhs(b-bl)
"0
(47)
It is meaningful to put
l=p+r+s,
m =r-s,
(48)
from which
r=½(l+m -p),
s =½(l-m -p).
(49)
r and s being integers, it is clear that l+m (or l - m ) and p must be both even or both odd.
Regarding the abbreviations (39) and using the properties
J_.(x) -- ( - 1)"J. (x) = J. (-x),
(50)
we finally arrive at
a(~, 7, ~) =
~
(-i)mJp 2 sin(~a~" sin ~0)
a sin
q~
2
l,m =--eop
even
or odd
x J~(t+,,,-p)
[
]
e2 sin(½(a +al)~" sin q~). J~(l-m-p)
el(a + al) sin ~o
el(a -al) sin ~
"
x exp(- ½ila~ sin q~) exp(- ~lrna1~ sm q~)~:lrlm.
(51)
The intensity of order (l, m) is then given by
Ii,,,((; ~o) =
Jp[2sin(½a~sin~)]Jq[ 2sin(½a~sin~°)]',
[e2sin(½(a+al)~sin~o)]
asm~
as,n;
jJ~(t+,.p)[ ~l(a~-a~m-~
~
p,q both
even or odd
×J~(t+,n-q)
[ e2 sin(½(a + al)sr sin q~).] J',, ,,,-~,)[ezsin(½(a-al)_~sin
e~(a -~)a) sin-~ ~o,]
el(a + a l ) sin q~
×J~l-,.-q)
[62 sin(½(a- al)~" sin ~o)]
el(a - al) sin ~ • •
(52)
At normal incidence of the light (~0 = 0), eq. (35) can be solved immediately. The solution satisfying (31)
reads
1
1
1.
rl.[ e+n e2Cl
±)~Cl
G(sC'rt'~r):exP[2(~:-~) ~r] e x p [ - ~ , ( ~ : r / +scr/J2exjeXpL2'k;
~]2--~elJ'
/53/
and effectuating analogous calculations as in the case of oblique incidence, it transforms to
G(~, rl, ~) = Z
Y (-0 Jo(~)J,<t+m-p)
l, r n = - - o o p e v e n
or odd
J~(~-,,,-p)
~ 71 •
(54)
W. Hereman and R. Mertens / Diffraction of light by A M ultrasonics
295
This solution determines the amplitude and the intensity of order (l, m),
~btm(~') =
Y~ (-1) Jo(~)Ji{t+,,_p)
J~t-,,-p)
(55)
,
p even
or odd
II,,, (~') =
Y.
Jp(~)Jq(~)J'o+=-p)
J~.+,.-q)
J~<t-.,-p)
J~{l-,,,-q)
•
(56)
p,q both
even or odd
It is also possible to deduce these expressions from the general ones as a limit for ~ tending to zero.
Anyhow we get expressions for the amplitudes different from those obtained by Aggarwal et al. [21].
1
1
These authors considered the products Jp(~)J,(~i~eZ/el)Js(~e2/el) as the amplitudes and ruled out the
possibility of taking combinations of the integers p, r and s. They considered any line of the diffraction
pattern as the consequence of one single subwave (part of (8)) passing through the liquid and excluded
possible interference. Our results (52) and (56) however are in correspondence with those derived from the
elementary R a m a n - N a t h theory (cf. Mertens [29] and Hereman [30]). To end this paragraph, we remark
that, using elementary properties of the Bessel functions, it is easy to control the following symmetry
properties of the diffraction pattern: At normal incidence of the light we have
/--,.--m (~') = I,--,. (~') = L,.,, (~') = I,,. (~').
(57)
At oblique incidence of the light we find
I_,._,~(~'; ~o)= k.,(~'; ¢).
(58)
In the next section we investigate and justify such symmetry properties from a general point of view.
6. Symmetries in the diffraction pattern
Aggarwal et al. [21, 22], Phariseau [23, 24] and Mertens [25] have tried to interpret the spectrum of the
considered optical diffraction. They came to the more or less uniform conclusion that the diffraction pattern
consists of 'principal lines' for l = 0, +1, +2 . . . . with m = 0, having the same directions and frequencies as
in the case where the ultrasonic wave is not modulated, and 'satellite lines' for each value of l with
m = +1, +2 . . . . . This interpretation follows from the formulae (24) and (25). In Fig. 1 principal and
satellite lines are indicated. We first consider symmetries of the principal lines with respect to the zero-order
central line; afterwards the symmetry of the side bands surrounding each principal line will be discussed.
In order to investigate a symmetry of the first kind we replace in a/t(xl, x2, ~r), xl by -x~, x2 by - x 2 and
verify whether the equation for gt(Xl, x2, ~r) (and the corresponding boundary condition) remains invariant
for this operation. For the sake of clarity we denote t/e(Xl, x2, ~') by ~+(Xl, x2, ~r) and gZ(-x~, -x2, ~r) by
t/t-(x1, x2, ~r). We find that
+~
'~'-(xl, x2, ~)=
Z
~bl,,(~')it÷" exp[--2i(/xl + mx2)]
l,m =--~
+oo
=
~
/,trl = - - o 0
(-1)t+m~b_z_,,,i I÷m exp[2i(/xl+mx2)],
(59)
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W. Hereman and R. Mertens / Diffraction of light by A M ultrasonics
satisfies the following partial differential equation
2 Ofi- - 2i[cos 2xl + (½e2/el) cos 2(Xl + x2) + (½e2/el) cos 2(xl - x2)] f i - =
a(
1-
[ t~
a \2
= - ( ~ l p ) / - - + c - - - - - I f i - + a sin ~
\OXl
OX2]
afi-
" OXl + a l
al/~-
sin ~p
(60)
ax2
while the boundary condition (22) transforms to
f i - ( x l , x2, O) = 1.
(61)
So, at normal incidence of the light (~ = O) we see from (17) and (60) that fi+ and f i - are solutions of the
same partial differential'equation with the same boundary condition, which leads to the identity of the
(periodic) solutions: fit+ = f i - . Consequently the comparison of the series expansions (23) and (59) yields
then
&-t,-,.(sr) = (-1)~+"&t.,(sr).
(62)
Hence, we conclude from (26) that
I-~.-m(() =Ii,.(().
(63)
For oblique incidence of the light (q~ # 0) eqs. (17) and (60) in general could never be identical, so the
s y m m e t r y (63) of the intensities with respect to the zero-order line (l = m = 0) will no longer occur.
However it is easy to check that
~btm(~'; tO) = (--1)Z+m&-l.-m((; --~),
(64)
which implies that
(65)
I-l,-m(~;--~P) = I~m((; frO.
With the help of this last property it is easy to prove symmetry relations in special cases. So, for instance,
from (52) it is clear that all Bessel functions remain invariant for the transformation ~0'~-~o, so that,
It,,,((; -~o)= It,,(~; ~o). Then from (65), the symmetry relation (58) follows immediately.
Now we investigate under which conditions the satellite lines (given by m and - m ) exhibit a symmetry
with respect to a corresponding principal line (determined by l). To control this we only have to replace x2
by - x 2 in fi(Xl, x2, ~') and verify whether fi(xa, - x z , () is solution of eq. (17). We find that fi(xl, -x2, ()
satisfies
20fi(xl, -x2, ~)
2i cos 2Xl + 2e:1 cos 2(xt - x2) + 2e----~cos 2(Xl + x2) fi(xl, - x2, ~) =
1./a
aV
ofi(xl, -xz, ~)
afi(xl, -x2, ~)
~-a 1 sin ¢
,
= - 4 - a p ~ x l - C ~ x 2 ) fi(xl, -x2, ~ ) - a sin ~0
aX1
aX2
(66)
together with
fi(xl, -x2, ~')= 1.
(67)
From comparison of (17) and (66) it follows then that: At normal incidence of the light (¢ = 0) the desired
symmetry occurs if p = pl =/92 = 0 , which corresponds with the particular case of Section 5.
W. Hereman and R. Mertens / Diffraction of light by A M ultrasonics
297
A t o b l i q u e i n c i d e n c e of the light, n o s y m m e t r y p r o p e r t i e s of satellite lines with respect to a principal line
holds.
A f u r t h e r i n v e s t i g a t i o n of the case p # 0 (eq. (17)) will be the s u b j e c t of later study.
Acknowledgements
T h e a u t h o r s are very grateful to Professor A . T . de H o o p of the T e c h n o l o g i c a l U n i v e r s i t y of D e l f t (The
N e t h e r l a n d s ) for the very positive a n d v a l u a b l e suggestions c o n c e r n i n g the final v e r s i o n of the paper. O n e of
the a u t h o r s (R.M.) also wishes to t h a n k the " N a t i o n a a l F o n d s voor W e t e n s c h a p p e l i j k O n d e r z o e k "
(Belgium) for a grant.
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