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Diffraction of light by a three-dimensional system of ultrasonics
Phariseau, P. 1958
Physica X X I V 985-995
D I F F R A C T I O N OF L I G H T BY A T H R E E - D I M E N S I O N A L SYSTEM OF U L T R A S O N I C S by P P H A R I S E A U Laboratorium voor Kristalkunde, Rozier, 6, Gent, Belgie.
Synopsis We establish a system of linear and homogeneous equations describing the diffraction of light by a three-dimensional system of ultrasonics. In order to find solutions of this system we lollow a well-known method of first approximation (l). This enables to explain diffraction patterns of the first order. To obtain explicit expressions for the intensities we are obliged to distinguish carefully between the Laue-case and Braggcase. In some circumstances, total reflection is possible. Neglecting the absorption we have verified the conservation of energy. The reflections in the disturbed slab satisfy nearly the Bragg-law.
1. Position o/the problem. In order to treat mathematically the diffraction of light b y a three-dimensional system of ultrasonics, which are mutually independent and non-orthogonal, we consider an isotropic homogeneous medium with refractive index/~0. In it we consider a plane parallel layer in which the supersonic disturbance is created. We call d the thickness of this slab, laterally it is indefinitely extended. The orientation of the boundaryplanes will be described by means of a unit vector u, normal to these planes and pointing into this slab of liquid. Choosing the origin in one of the boundary-planes, their equations become u.r=d
u.r =0,
(1)
r being the position-vector. Inside this layer, the medium is disturbed by three independent progressive sound-waves, travelling in three mutually arbitrary directions. As a result of these longitudinal harmonic compression waves, we get fluctuations of the density of the medium. If we assume that the refractive index # is a linear function of the density, i.e. the maximum variation of the refractive index is directly proportional to the maximum variation of the density, we write the following expression for the refractive index in the disturbed region :
/~ = / ~ 0 + / . 1 cos(o~l*t -- k l * . r + 61) + / ~ 2 cos(co2*t -- k2*. r + 62) + + #3 cos(aJa*t -- k3*. r + 63)
(2)
where /*1, #2,/*3 are the maximum variations of the refractive-index due --
Physiea XXIV
9 8 5
--
986
P. P H A R I S E A U
respectively to the waves (Wl*, kl*), (cos*, ks*) and (0~3", ks*) only, o~1", ~2", o~s* are the angular frequencies of the ultrasonics, hi*, ks*, k3* are the wave-vectors of the three sound-waves, &l, ~2, &s are the phase-constants of the three sound-waves. It is immediately seen from the expression (2) that the refractive-index # is periodic in three directions with periods specified by KI*, Ks* and K3* which are defined by the relations:
k~*. ~j* = 2~,j
(i, i = l, 2, 3)
(3)
Hence,
+ lKl* + .*Ks* + nK3* t) =
t)
(4)
where l, m and n are three arbitrary integers. We already notice the analogy with crystal-physics. Indeed, we m a y consider the three vectors KI*,/£2* and Ks* as the ordinary lattice-vectors, whilst the vectors (kl*/2a), (k2"/2~) and (k3"/2~) form the basis-vectors of the reciprocal lattice. Concerning the light-waves in this slab, it can be easily deduced from the equations of Maxwell, that they must satisfy the following partial differential equation (2). / I E = (/~/C) s (cqSE/cqt 2)
(5)
where E is the electric field of the light-waves in the disturbed region, c is the light-velocity in vacuum. It is obvious that the electric field E' of the light-waves outside the disturbed layer satisfies the ordinary wave-equation:
./1E' = (#o/c) s (c92E'/cgts)
(6)
The solutions of the differential equations (5) and (6) must agree at the boundary planes in order that the mathematical problem would have a physical meaning. The integration of these equations and the elaboration of the boundary-conditions are the subjects of the following sections.
2. The electric/ield inside the plane parallel region. We shall now confine ourselves to the case of ultrasonics with nearly the same intensities. So we m a y neglect the terms in #12, #2 ~', #82, /~1#2, #1#3, #2#3 with respect to #0 s,/~0#1, #0#9. and #0#3 in the expression of/~9., which is found in the righthand side of the equation (5). Hence /~2 = / , 0 s + 2/~0/~1cos(wl*t -- kl*. r + ~1) + 2/,0/,2 cos(cos*t -- ks*. r + c52) + 2#0#3 cos(o~3*t -- k3*. r + &3) (7) Treating the problem of wave-propagation in a three-dimensional periodic medium, B r i l l o u i n 3) gives solutions of the corresponding wave-equation in the form of a triple Fourier sum with constant coefficients. In order to find solutions for the partial differential equation (5) we shall follow about
DIFFRACTION
OF LIGHT BY ULTRASONICS
987
the same method. We consider an oblique incidence of the light-wave, which m a y be represented by A~ exp[i(o~t -- K. r)] (8) where ~o is the angular frequency of the incident light, K is the wave-vector, A~ is the amplitude of the incident beam with frequency oJ. As the expression (8) represents a plane-wave, we get the well-known relation between co and K:
lI-l"l =/,o lc
(9)
Inside the disturbed slab, the incident light-wave will no longer be represented by the expression (8) as a result of the interaction between incident and scattered radiation. Hence, we have to distinguish between the external incident wave given by (8) and the incident wave once in the disturbed region (further called internal incident beam), which is then represented by Nooo exp[i(o~t -- k. r)]
(I 0)
The amplitude N0oo and the wave-vector k have to be determined from the boundary conditions and the conditions of self-consistence of the incident and diffracted waves. Since the disturbance of the refractive-index is periodic as expressed by the relation (4), the disturbed light-wave will also present the same periodicity. Hence, we put as a solution of the equation (5) a so-called Bloch-wave: E = ¢(r, t) exp[i(cot -- k. r]
(11)
where ¢(r, t) has the same periodicity as the refractive index/~, i.e.
¢(r + lKl* + inK2* + nK3*, t) = ¢(r. t)
(12)
Consequently we m a y expand the function ¢(r, t) in a triple Fourier series with constant coefficients. So we finally put as a solution of the equation (5) : E = exp[i(cot -- k. r)]. Y, zmn ~O~mnexp[i(lkl* + ink2* + nk3*), r] Putting now with
k~mn = k - - k~mn*
(k0oo ------k)
k~mn* = lkl* + ink2* + nk3*
(13) (14) (15)
and substituting the expression (13) into the equation (5), taking into account that wj* <~ o~ ( / = 1, 2, 3) i.e. taking into consideration the quasistationary character of the phenomenon, we get after equating the corresponding terms : [Ikzm~12/IKI 2 -
13 ~zm. = ( m / m ) {W.+l,m~ exp [i(o~l*t + (~1)] + ~(l--1)mn exp[-- i(col*t + 61)]} + (#2//zo){~V~(m+l~nexp[i(w2*t + 62)] + ~/z(m-l~n exp[--i(wg.*t + 68)]} + (/~a/#o) {~m(~+l~ exp[i(a~a*t + 63)] + ~vlm(n-i~ exp[--i(co3*t + 63)]} (16)
988
P. PHARISEAU
Putting V:zmn =- Nzmn exp[-- i(CO*tmnt + ~zmn)]
(17)
co*tmn =/COl* + mCO2* + nCOs*
(18)
8lmn = lS1 + mc52 + m58
(19)
where
and substituting the expression (17) into the set of equations (16), we get
rlkz,,,,,12/lHl 2,- 1]Nzmn = = (ffl/#o)[N(z+l)mn + N{z-1)mn] + (#21#o)[Nl(m+l). + Nl(m-l,n] + (#3/ffo)[Nzm(n+l, + Nzm(.-1)] (20) The exact solution of this infinite set of linear and homogeneous equations leads to insurmountable mathematical difficulties. So we are obliged to use approximating methods. 3. Approximate solution o / t h e system (20). In order to find approximate solutions of the homogeneous set of equations (20), we shall use a method which is carefully exposed in Zachariasen's well-known book 1). Considering the equations (20), it is obvious that only these amplitudes will have a non negligible value for which the following relation applies: Ikz~.l
~
IKI
(21)
Hence, for convenience we put Ikzmnl = IKI(1 + ~lmn)
(22)
where ~llmn are very small quantities with respect to unity. On the other hand, at the boundary plane u . r = O, the exponential functions of the incident wave (8) and of the non diffracted wave in the disturbed region given by (10), must agree. In order to fulfil this requirement we put kooo = k = K + (ru
(, ~ 1)
(23)
From the relation (22) we get immediately = ~7ooo [K]2/K.u = ~/ooo ]K[/cos ~o
(24)
where ~o is the angle between the direction of propagation of external incident light-wave and the normal to the boundary plane. The relation (14) can now be written: kunn = K + ~u - - k*tmn
(25)
An expression for ~lmn is found, combining the relations (22) and (25). We get where
~7~mn = a~ooo + b
(26)
a = 1 -- u.k*~mn/u.K
(27)
b = ½[Ik*z,,,,,I 2 -
2 K . k*z,,,,]/IKI 2
(28)
It is to be noted that the quantity a has an important signification for the
DIFFRACTION OF LIGHT BY ULTRASONICS
989
diffraction phenomenon. Indeed, calling k~O~mn the vector for which the equality (21) is exactly satisfied, i.e. [kzm~c°~I = !'KI we get a = u . k~m~(°~/u. I f ~ cos ~zmn/cos ~o
(29)
where =tmn is the angle between the diffracted wave and the normal to the boundary plane. Now it is immediately seen from equation (29) that when a is positive the diffracted waves emerge through the boundary plane u. r = d, which we shall call the "Laue-case" as in the diffraction theory of Xrays by crystals. When a is negative, the diffracted waves leave the slab through the boundary u. r = 0, which we shall indicate henceforth as the "Bragg-case". Case 1 : Let us assume that ]kzmn[ ~ ]KI for each vector kzmn*. In that case, the incident wave will produce diffracted waves of negligible amplitudes. Hence, we m a y put in the system (20) that all the amplitudes Nzran vanish except Nooo and the equations (20) lead to
lk0ooI =-Ikl = IKI So we have determined ]hi and it is obvious that the light-waves with such frequencies traverse the disturbed region without any change. Consequently we shall drop this non interesting case in future. Case 2: We are now particularly interested in the case for which the incident wave produces only one diffracted wave of appreciable amplitudes, i.e. among all the different values of kzmn* there is only one for which the relation (21) is valid. Now it is evident that for this fixed value of integers l, m, n, all the amplitudes become negligible except Nooo and N~mn. From the system of equations (20) it is obvious that we have to distinguish two cases according as ]lI + [m[ + Inl = 1 or Ill + Imi + Inl > 1. W e shall treat them now sucessively. a) Ill + ]m] + In t = 1. - In order to treat simultaneously the six cases contained in this possibility, we put /A100 =
/A(--1)00 =
/A1, /~010 =
/~0t--1}0 =
/¢2, /~001 =
/~00(--1) =
/~3
(31)
while all other values of #~mn a priori vanish. The system (20) becomes now, taking into account the relations (22)"
(m~,d~o)Nz,,~
(32a)
2~lzmnNzmn = (#zmn//ao)Nooo
(32b)
2~000Noo0 =
This set of linear and homogeneous equations has a nontrivial solution only if the determinant of the coefficients vanish, i.e. if no0o.nzmn
=
¼(#zm./#o) 2
(33)
Considering (26) this equation becomes: a~ooo2 + b~ooo = ¼(m~/~o). 2
(34)
990
P. PHARISEAU
The solutions of this quadratic equation determine two possible values for 7000, ViZ.,
7ooo (1' = (1/2a){-- b + [b 2 + a(/zzmn/#o)2] ~}
(35a)
7000 (2) = (1/2a){-- b --[32 + a (/~mn//~o)2] ½}
(35b)
W i t h each value of 7ooo there corresponds an amplitude ratio x given b y x = Nzmn/Nooo = 27ooo(/*o/#~mn)
(36)
Since there are two possible values for 7ooo and consequently also for 7~mn and the amplitude ratio x, we find two internal incident waves and two internal diffracted waves. Hence, we get for the electric field inside the disturbed region: E = {Nooo(1~ exp (--ip(l~z) + Nooo (~) exp(--ipcg"~z)} exp[i(~ot --/1[. r)] + {x(l~Nooo el) exp (-- ipCl)z) + xlg~Nooo (9"' e x p ( - - ip(9)z)} exp[i(cot -- k~mn (o) . r)]. e x p [ - - i(cozmn*t + ~mn)]
(37)
p(l~ -- 7ooo<1~ iKi/co s ~o, p = 7ooo~Z~ iKi/co s ~o, z - - . r
(38)
where b) Ill + Ira[ + Jnl > 1. - In this case, the homogeneous set of equations (20) leads to
IkJ = IK[, Ikzmn[ = [KJ
(39)
and the amplitudes No00 and N~mn remain undetermined. So we obtain for the electric field in the disturbed layer the simple expression: E = No00 exp Ei(o~t -- K. r)] + N~mn exp [i(o~t -- kzmn ~°) . r)] exp [-- i(co~mn*t + 6zmn)]
(40)
R e m a r k : It is of course, possible t h a t there are several values of k~ran* for which the relation (21) is valid at the same time so t h a t two or more diffracted waves are p r o d u c e d siinultaneously. It is obvious t h a t such cases are rarely encountered in practice and therefore t h e y will not be discussed in this paper. 4. The boundary-conditions.
a)
[l] + Jm[ + in] = 1.
Since the b o u n d a r y conditions are different according as we are in the Lauecase or in the Bragg-case, it is necessary to distinguish b e t w e e n them. 1) L a u e - c a s e . At the plane u. r = 0, the external and internal incident waves m u s t agree. Hence, considering the expressions (8) and (37), we get A K = Nooo (1) + Nooo (2)
(41)
No diffracted waves leave the m e d i u m through this b o u n d a r y , so t h a t 0 = x(1)Nooo (1) + x(2~Nooo (~
(42)
The equations (41) and (42) form an ordinary s y s t e m of two equations, the
991
D I F F R A C T I O N OF L I G H T B Y U L T R A S O N I C S
solution of which is given b y NoooCll = AKx¢2~/x~2~ - - x¢l~, Nooo~9.~ = -- AKx¢l~/x~2~ -- x~l~
(43)
At the boundary plane u. r = d there must be agreement between the internal incident and transmitted light as well as between the internal and external diffracted wave. Since outside the disturbed region the electric field is a solution of the ordinary wave-equation (6), we write E ' = Tooo exp[i(cot -- K. r)] + Tzm• exp[i (o~t -- kzm~~°' . r)~]
(44)
Comparing the expressions (37) and (44), we get taking into account the formulae (43): Tooo = ( A n / x c2' -- xCl~)[x ~9"'exp(-- ip~l~d)
-
-
x {1}
exp (-- ip~2~d)]
(45)
Tz,nn = (Anx~l~xC~/X c2~ -- x!10[exp (-- ipClld) -- exp(-- ipC2~d)] exp[-- i(¢o~mn*t + ~lmn)]
(46)
2) B r a g g - c a s e . Since a is negative now, the diffracted wave emerges through the boundary plane u. r = 0, while it must vanish at the plane u. r = d. Before the slab the electric field is of the form AK exp [i(wt -- K. r)] + R~m~ exp[i(oJt -- kzmn ~o~. r)]
(47)
Comparing the expressions (47) and (37) we get in the plane u. r = 0 AK = Nooocl~ + Nooo~
(48)
Rzmn = (x~l~N0oo~1' + x~2'No0o~9'~)exp[-- i(co~mn*t + ¢Szmn)]
(49)
Beyond the disturbed layer we get only the transmitted incident beam. Hence, To0o = Nooocl~ exp (-- iprl'd) + Nooo ~'~ exp (--ipC2'd) (50) 0 = x~llNooo~l~ exp(--ipll~d) + x l2~Nooo~ exp (-- ip~Z~d)
(51)
The amplitudes Nooo ~1~ and Nooo ~z~ are determinated b y the system of equations formed b y (48) and (51), the solutions of which are: N o o o ~ l ~ = x ~ A K [ x ~z~ exp(--ip~Z~d)--x ~1~ exp(-- ip~l~d)] - 1 . e x p ( Nooo~2~=--xll,AK[x~,exp(--
ip~d)
i p ~ d ) - - x ~ l ~ exp(--ip~lld)] - 1 . e x p ( - i p ~ l ~ d )
(52) (53)
b) ] / [ + I m i + [ n [ > l As opposed to the preceding case, it is immediately seen from the expression (40) of the electric field inside the disturbed region that w e never can satisfy the boundary-conditions, unless Nlmn vanish. That means we don't find any diffracted wave. Henceforth we shall omit this non interesting case. 5. Intensity-expressions o/ the di//racted beams. From the preceding sections it is evident that we are only able to explain diffracted waves characterized b y the integers l, m, n provided Ill + [ m I + [hi = !~ We. con: Sider again the Lane- and the Bragg-case in succession.
992
P. P H A R I S E A U
1) L a u e - c a s e . From the expression (44) it is seen that the diffracted beam of the order (1, m, n) travels in the direction defined b y k~mn(°), while from the relation (46) it is evident that we have a frequency-change -- ~ozmn* and that the phase-constants don't have any influence. For the intensities of the transmitted incident beam, viz., Ioo0 r" = T00oTo0o* and of the diffracted beam of the order (l, m, n), viz., I z m . r- = T z m , T z m n * we get after effectuating the calculations: Iooo L = AKg"[1 + 4xCl)x(2)(x c2~ - - x(10-9 sin s ½(p(1) -- O(~))d]
(54)
I~.r.
(55)
= AKg.(2x(1)x(202(x(Z) _ x(1))-2 sin s ½(O(1) __ p(2))d
It is obvious that between these two intensities the following relation exists, calling IK the intensity of the incident beam of light with frequency o~: Iooo L = I K -- aIzmn z"
(56)
Defining the power P of a beam of light, the product of its intensity b y its cross-section S, the relation (56) can be written P00oL + P~mnL = PK
(57)
since a ~ cos ~mn/cos c~o'~ S~mn/So and where So and S~mn are respectively the cross-sections of the incident and diffracted beams, PK, Pooo L, Pzmn L are respectively the power of the external incident, the transmitted incident and the diffracted beams. This result was normally to be expected, since we have asssumed zero absorption so that there is no energy dissipation within the disturbed slab. The power ratio PzmnL/PK takes the following simple form, taking into consideration the formula (55): P~mnL/PK = (1 + ~ ) - 1 sin 2[D(1 + ~)t]
(59)
where = b2a-l(l~m~/tzo)-2,
D = (/zzmn/#0)(IK[ d/2a½ cos a0)
(60)
2) B r a g g - c a s e . Working along similar lines as in the Laue-case, we get for the intensity of the transmitted incident beam Iooo B = A K ~ [1 + 4x(1)x (2) sin 9"½(p(1) _ p(2))d]-i
(61)
The reflected wave has again a frequency-change defined b y --OJ~mn*. The intensity is given b y
I~,~,~B =
(2x(1)x(~)AK)~. [(x(Z) _ x(~))~ + 4x(1)x(~) sin2½(p(1) _ p(~))d]-l.
sin ~. ½(p(1) __ p(fl))d
(62)
J u s t as in the preceding case there is not any energy dissipation within the
D I F F R A C T I O N OF L I G H T BY ULTRASONICS
993
disturbed region, there exist once-more the relations Io00 B
=
IK --
lal
Ilmn B
(63)
PK
(64)
or
Pooo B + PzmnB
=
The power ratio P~mnB/PK can again be written in the following w a y : (a) if b 2 + a(#~mn/#0) 2 > 0 we get
PlmnB/PK -- {~2 __ 1 + sin 2 [D(~ 2 -- 1)-~]}-l.sin 2 [D(~ 2 -- I)~.] =
{~2 +
(~2 __ 1) cotg2[D(~ 2 - -
1)½]} - 1
(65) (66)
(b) if Y" + a(#zmn/#o) 2 < 0 we get
PlmnB/PK = {1 -- ~2 + sh 2 [D(I -- ~2)l]}-l.shg'[D(l -- ~2)~] = {~2 +
(1 - - ~2) coth 2 [D(1 -- ~2)½]}-1
(67) (68)
where
-=
lal-X
D = (#~mnll~o)(lKld/2 lal cos
(69)
6. Investigation o~ the intensity-expressions. 1) D < 1. In this case, the power ratio in the Laue-case as well as in the Bragg case m a y be written plmn/1%K ~. ~-9. sin ~.D~ For such small values of D, the power ratio is small with respect to u n i t y and we note i m m e d i a t e l y the periodicity of the i n t e n s i t y with respect to the width of the disturbed slab of liquid. 2) D >~ 1. Now we have to distinguish the Laue- and Bragg-cases. a) Laue-case: At a given value of ~ and when D is going to infinity the power ratio P~rnn/PK will oscillate between zero and (1 + ~g)-i b) Bragg-case : In the range [~[ > 1 in which the relation (65) is valid, the power-ratio Pzm,B/PK approaches no definite limit as D increases just as in the Laue-case because of the presence of the trigonometric functions. Considering however the range [~1 < 1 in which the relation (67) is valid, we get for the power-ratio when D becomes v e r y great:
PZmnB/PK ~, 1
(70)
It is obvious from equation (70) t h a t the inward energyflux across the b o u n d a r y plane u. r = 0 is equal to the o u t w a r d flux. So we m a y conclude t h a t there is total reflection in the range I~l < 1 and D has a sufficient order of magnitude. 3) I n t e r m e d i a t e case (D ~ 1). In this case neither a p p r o x i m a t i o n is valid a n d so we have to use the expressions (59), (65) and (67) for the power-ratio in the different cases. I t is seen from these equations t h a t the intensities are s y m m e t r i c a l with respect to the values ~ = 0. a) L a u e - c a s e : The value of the power ratio for ~ = 0 is sin s D. More-
994
P. PHARISEAU
over, when D is in the neighbourhood of ½a the power ratio will approaches the value 1, corresponding to total reflection. When D < ½a we also get a maximum for ~ -----0. Beyond the value ½~ for D, the power ratio begins to oscillate between zero and 1 at ~ = 0 (fig. 1).
-4
-3
-2
-I
0
I
2
3
4
Fig. 1. D i f f r a c t i o n p a t t e r n s i n t h e L a u e - c a s e for a s l a b of l i q u i d of i n t e r m e d i a t e t h i c k n e s s . T h e t h r e e c u r v e s s h o w n in t h e figure c o r r e s p o n d t o D = ¼n, ½• a n d n.
b) B r a g g - c a s e : From fig. 2 it is evident that in the range I~] < 1 the value PzmnB/PK m 1 is rapidly attained as D increases.
7. On the Bragg-reflection in the disturbed layer. We shall now show that the condition ]k!°)zmn[ = lk[ is equivalent with exact Bragg-reflection. Writing the expression for k(°)zmn: k l m ~ (0) - -
K
=
--
k*lmn
(71)
and squaring this expression, taking into account that [k~,nn~0)] -----]KJ and introducing the angle 2Cplmn between the direction of propagation of the external incident light-wave and the direction'of the diffracted wave characterized by klmn (0) w e get 2 sin 9zm. - - Ik*~m~l/IHl,
(72)
Since kzmn*12a is a vector of the reciprocal lattice it is well-known that
DIFFRACTION OF LIGHT BY ULTRASONICS
995
L = 2~/Jk~mn[ is the distance between two planes of the family kzmn*, r = q
(73)
where q is an arbitrary integer. 2 sin 9~r.n = ~/#oL
Hence
(7 4)
k being the wavelength of the light-wave in vacuum. Since the condition is however Jkzmn] ~ [K] reflections which satisfy a law slightly different from the Bragg-law will occur in the disturbed slab. pB
tmn /PK
1.0
.o.9
~,n
• 0,7
.0.6 .0.5
0.3 0,2
0.1
-4
-3
-2
1
2
'3
4
Fig. 2. D i f f r a c t i o n p a t t e r n s in t h e B r a g g - c a s e for a slab of liquid of i n t e r m e d i a t e t h i c k n e s s . T h e t h r e e c u r v e s s h o w n in t h e figure c o r r e s p o n d to D = ¼r~, ½~ a n d ~.
A c k n o w l e d g m e n t s . This work was made possible through a grant from the Centre national Belge de Chimie physique mol~culaire to which the author is greatly indebted. He also expresses his thanks to Prof. W. D e k e y s e r for his continuous interest and to Prof. M. N u y e n s and Dr R. M e r t e n s for fruitful discussions. Received I-8-58. REFERENCES
I) Z ach a r i a s e n , W. H., Theory of X-ray diffraction in crystals. Wiley, New York, 1945, p. 115. 2) B r i l l o u i n , L., La diffraction de la lumi~re par des ultrasons. Act. Sc. et Ind. 59 (1933). 3) B r i l l o u i n , L., Wave propagation in periodic structures, Dover Publications, 1953, p. 143.
Physica X X I V 985-995
D I F F R A C T I O N OF L I G H T BY A T H R E E - D I M E N S I O N A L SYSTEM OF U L T R A S O N I C S by P P H A R I S E A U Laboratorium voor Kristalkunde, Rozier, 6, Gent, Belgie.
Synopsis We establish a system of linear and homogeneous equations describing the diffraction of light by a three-dimensional system of ultrasonics. In order to find solutions of this system we lollow a well-known method of first approximation (l). This enables to explain diffraction patterns of the first order. To obtain explicit expressions for the intensities we are obliged to distinguish carefully between the Laue-case and Braggcase. In some circumstances, total reflection is possible. Neglecting the absorption we have verified the conservation of energy. The reflections in the disturbed slab satisfy nearly the Bragg-law.
1. Position o/the problem. In order to treat mathematically the diffraction of light b y a three-dimensional system of ultrasonics, which are mutually independent and non-orthogonal, we consider an isotropic homogeneous medium with refractive index/~0. In it we consider a plane parallel layer in which the supersonic disturbance is created. We call d the thickness of this slab, laterally it is indefinitely extended. The orientation of the boundaryplanes will be described by means of a unit vector u, normal to these planes and pointing into this slab of liquid. Choosing the origin in one of the boundary-planes, their equations become u.r=d
u.r =0,
(1)
r being the position-vector. Inside this layer, the medium is disturbed by three independent progressive sound-waves, travelling in three mutually arbitrary directions. As a result of these longitudinal harmonic compression waves, we get fluctuations of the density of the medium. If we assume that the refractive index # is a linear function of the density, i.e. the maximum variation of the refractive index is directly proportional to the maximum variation of the density, we write the following expression for the refractive index in the disturbed region :
/~ = / ~ 0 + / . 1 cos(o~l*t -- k l * . r + 61) + / ~ 2 cos(co2*t -- k2*. r + 62) + + #3 cos(aJa*t -- k3*. r + 63)
(2)
where /*1, #2,/*3 are the maximum variations of the refractive-index due --
Physiea XXIV
9 8 5
--
986
P. P H A R I S E A U
respectively to the waves (Wl*, kl*), (cos*, ks*) and (0~3", ks*) only, o~1", ~2", o~s* are the angular frequencies of the ultrasonics, hi*, ks*, k3* are the wave-vectors of the three sound-waves, &l, ~2, &s are the phase-constants of the three sound-waves. It is immediately seen from the expression (2) that the refractive-index # is periodic in three directions with periods specified by KI*, Ks* and K3* which are defined by the relations:
k~*. ~j* = 2~,j
(i, i = l, 2, 3)
(3)
Hence,
+ lKl* + .*Ks* + nK3* t) =
t)
(4)
where l, m and n are three arbitrary integers. We already notice the analogy with crystal-physics. Indeed, we m a y consider the three vectors KI*,/£2* and Ks* as the ordinary lattice-vectors, whilst the vectors (kl*/2a), (k2"/2~) and (k3"/2~) form the basis-vectors of the reciprocal lattice. Concerning the light-waves in this slab, it can be easily deduced from the equations of Maxwell, that they must satisfy the following partial differential equation (2). / I E = (/~/C) s (cqSE/cqt 2)
(5)
where E is the electric field of the light-waves in the disturbed region, c is the light-velocity in vacuum. It is obvious that the electric field E' of the light-waves outside the disturbed layer satisfies the ordinary wave-equation:
./1E' = (#o/c) s (c92E'/cgts)
(6)
The solutions of the differential equations (5) and (6) must agree at the boundary planes in order that the mathematical problem would have a physical meaning. The integration of these equations and the elaboration of the boundary-conditions are the subjects of the following sections.
2. The electric/ield inside the plane parallel region. We shall now confine ourselves to the case of ultrasonics with nearly the same intensities. So we m a y neglect the terms in #12, #2 ~', #82, /~1#2, #1#3, #2#3 with respect to #0 s,/~0#1, #0#9. and #0#3 in the expression of/~9., which is found in the righthand side of the equation (5). Hence /~2 = / , 0 s + 2/~0/~1cos(wl*t -- kl*. r + ~1) + 2/,0/,2 cos(cos*t -- ks*. r + c52) + 2#0#3 cos(o~3*t -- k3*. r + &3) (7) Treating the problem of wave-propagation in a three-dimensional periodic medium, B r i l l o u i n 3) gives solutions of the corresponding wave-equation in the form of a triple Fourier sum with constant coefficients. In order to find solutions for the partial differential equation (5) we shall follow about
DIFFRACTION
OF LIGHT BY ULTRASONICS
987
the same method. We consider an oblique incidence of the light-wave, which m a y be represented by A~ exp[i(o~t -- K. r)] (8) where ~o is the angular frequency of the incident light, K is the wave-vector, A~ is the amplitude of the incident beam with frequency oJ. As the expression (8) represents a plane-wave, we get the well-known relation between co and K:
lI-l"l =/,o lc
(9)
Inside the disturbed slab, the incident light-wave will no longer be represented by the expression (8) as a result of the interaction between incident and scattered radiation. Hence, we have to distinguish between the external incident wave given by (8) and the incident wave once in the disturbed region (further called internal incident beam), which is then represented by Nooo exp[i(o~t -- k. r)]
(I 0)
The amplitude N0oo and the wave-vector k have to be determined from the boundary conditions and the conditions of self-consistence of the incident and diffracted waves. Since the disturbance of the refractive-index is periodic as expressed by the relation (4), the disturbed light-wave will also present the same periodicity. Hence, we put as a solution of the equation (5) a so-called Bloch-wave: E = ¢(r, t) exp[i(cot -- k. r]
(11)
where ¢(r, t) has the same periodicity as the refractive index/~, i.e.
¢(r + lKl* + inK2* + nK3*, t) = ¢(r. t)
(12)
Consequently we m a y expand the function ¢(r, t) in a triple Fourier series with constant coefficients. So we finally put as a solution of the equation (5) : E = exp[i(cot -- k. r)]. Y, zmn ~O~mnexp[i(lkl* + ink2* + nk3*), r] Putting now with
k~mn = k - - k~mn*
(k0oo ------k)
k~mn* = lkl* + ink2* + nk3*
(13) (14) (15)
and substituting the expression (13) into the equation (5), taking into account that wj* <~ o~ ( / = 1, 2, 3) i.e. taking into consideration the quasistationary character of the phenomenon, we get after equating the corresponding terms : [Ikzm~12/IKI 2 -
13 ~zm. = ( m / m ) {W.+l,m~ exp [i(o~l*t + (~1)] + ~(l--1)mn exp[-- i(col*t + 61)]} + (#2//zo){~V~(m+l~nexp[i(w2*t + 62)] + ~/z(m-l~n exp[--i(wg.*t + 68)]} + (/~a/#o) {~m(~+l~ exp[i(a~a*t + 63)] + ~vlm(n-i~ exp[--i(co3*t + 63)]} (16)
988
P. PHARISEAU
Putting V:zmn =- Nzmn exp[-- i(CO*tmnt + ~zmn)]
(17)
co*tmn =/COl* + mCO2* + nCOs*
(18)
8lmn = lS1 + mc52 + m58
(19)
where
and substituting the expression (17) into the set of equations (16), we get
rlkz,,,,,12/lHl 2,- 1]Nzmn = = (ffl/#o)[N(z+l)mn + N{z-1)mn] + (#21#o)[Nl(m+l). + Nl(m-l,n] + (#3/ffo)[Nzm(n+l, + Nzm(.-1)] (20) The exact solution of this infinite set of linear and homogeneous equations leads to insurmountable mathematical difficulties. So we are obliged to use approximating methods. 3. Approximate solution o / t h e system (20). In order to find approximate solutions of the homogeneous set of equations (20), we shall use a method which is carefully exposed in Zachariasen's well-known book 1). Considering the equations (20), it is obvious that only these amplitudes will have a non negligible value for which the following relation applies: Ikz~.l
~
IKI
(21)
Hence, for convenience we put Ikzmnl = IKI(1 + ~lmn)
(22)
where ~llmn are very small quantities with respect to unity. On the other hand, at the boundary plane u . r = O, the exponential functions of the incident wave (8) and of the non diffracted wave in the disturbed region given by (10), must agree. In order to fulfil this requirement we put kooo = k = K + (ru
(, ~ 1)
(23)
From the relation (22) we get immediately = ~7ooo [K]2/K.u = ~/ooo ]K[/cos ~o
(24)
where ~o is the angle between the direction of propagation of external incident light-wave and the normal to the boundary plane. The relation (14) can now be written: kunn = K + ~u - - k*tmn
(25)
An expression for ~lmn is found, combining the relations (22) and (25). We get where
~7~mn = a~ooo + b
(26)
a = 1 -- u.k*~mn/u.K
(27)
b = ½[Ik*z,,,,,I 2 -
2 K . k*z,,,,]/IKI 2
(28)
It is to be noted that the quantity a has an important signification for the
DIFFRACTION OF LIGHT BY ULTRASONICS
989
diffraction phenomenon. Indeed, calling k~O~mn the vector for which the equality (21) is exactly satisfied, i.e. [kzm~c°~I = !'KI we get a = u . k~m~(°~/u. I f ~ cos ~zmn/cos ~o
(29)
where =tmn is the angle between the diffracted wave and the normal to the boundary plane. Now it is immediately seen from equation (29) that when a is positive the diffracted waves emerge through the boundary plane u. r = d, which we shall call the "Laue-case" as in the diffraction theory of Xrays by crystals. When a is negative, the diffracted waves leave the slab through the boundary u. r = 0, which we shall indicate henceforth as the "Bragg-case". Case 1 : Let us assume that ]kzmn[ ~ ]KI for each vector kzmn*. In that case, the incident wave will produce diffracted waves of negligible amplitudes. Hence, we m a y put in the system (20) that all the amplitudes Nzran vanish except Nooo and the equations (20) lead to
lk0ooI =-Ikl = IKI So we have determined ]hi and it is obvious that the light-waves with such frequencies traverse the disturbed region without any change. Consequently we shall drop this non interesting case in future. Case 2: We are now particularly interested in the case for which the incident wave produces only one diffracted wave of appreciable amplitudes, i.e. among all the different values of kzmn* there is only one for which the relation (21) is valid. Now it is evident that for this fixed value of integers l, m, n, all the amplitudes become negligible except Nooo and N~mn. From the system of equations (20) it is obvious that we have to distinguish two cases according as ]lI + [m[ + Inl = 1 or Ill + Imi + Inl > 1. W e shall treat them now sucessively. a) Ill + ]m] + In t = 1. - In order to treat simultaneously the six cases contained in this possibility, we put /A100 =
/A(--1)00 =
/A1, /~010 =
/~0t--1}0 =
/¢2, /~001 =
/~00(--1) =
/~3
(31)
while all other values of #~mn a priori vanish. The system (20) becomes now, taking into account the relations (22)"
(m~,d~o)Nz,,~
(32a)
2~lzmnNzmn = (#zmn//ao)Nooo
(32b)
2~000Noo0 =
This set of linear and homogeneous equations has a nontrivial solution only if the determinant of the coefficients vanish, i.e. if no0o.nzmn
=
¼(#zm./#o) 2
(33)
Considering (26) this equation becomes: a~ooo2 + b~ooo = ¼(m~/~o). 2
(34)
990
P. PHARISEAU
The solutions of this quadratic equation determine two possible values for 7000, ViZ.,
7ooo (1' = (1/2a){-- b + [b 2 + a(/zzmn/#o)2] ~}
(35a)
7000 (2) = (1/2a){-- b --[32 + a (/~mn//~o)2] ½}
(35b)
W i t h each value of 7ooo there corresponds an amplitude ratio x given b y x = Nzmn/Nooo = 27ooo(/*o/#~mn)
(36)
Since there are two possible values for 7ooo and consequently also for 7~mn and the amplitude ratio x, we find two internal incident waves and two internal diffracted waves. Hence, we get for the electric field inside the disturbed region: E = {Nooo(1~ exp (--ip(l~z) + Nooo (~) exp(--ipcg"~z)} exp[i(~ot --/1[. r)] + {x(l~Nooo el) exp (-- ipCl)z) + xlg~Nooo (9"' e x p ( - - ip(9)z)} exp[i(cot -- k~mn (o) . r)]. e x p [ - - i(cozmn*t + ~mn)]
(37)
p(l~ -- 7ooo<1~ iKi/co s ~o, p
(38)
where b) Ill + Ira[ + Jnl > 1. - In this case, the homogeneous set of equations (20) leads to
IkJ = IK[, Ikzmn[ = [KJ
(39)
and the amplitudes No00 and N~mn remain undetermined. So we obtain for the electric field in the disturbed layer the simple expression: E = No00 exp Ei(o~t -- K. r)] + N~mn exp [i(o~t -- kzmn ~°) . r)] exp [-- i(co~mn*t + 6zmn)]
(40)
R e m a r k : It is of course, possible t h a t there are several values of k~ran* for which the relation (21) is valid at the same time so t h a t two or more diffracted waves are p r o d u c e d siinultaneously. It is obvious t h a t such cases are rarely encountered in practice and therefore t h e y will not be discussed in this paper. 4. The boundary-conditions.
a)
[l] + Jm[ + in] = 1.
Since the b o u n d a r y conditions are different according as we are in the Lauecase or in the Bragg-case, it is necessary to distinguish b e t w e e n them. 1) L a u e - c a s e . At the plane u. r = 0, the external and internal incident waves m u s t agree. Hence, considering the expressions (8) and (37), we get A K = Nooo (1) + Nooo (2)
(41)
No diffracted waves leave the m e d i u m through this b o u n d a r y , so t h a t 0 = x(1)Nooo (1) + x(2~Nooo (~
(42)
The equations (41) and (42) form an ordinary s y s t e m of two equations, the
991
D I F F R A C T I O N OF L I G H T B Y U L T R A S O N I C S
solution of which is given b y NoooCll = AKx¢2~/x~2~ - - x¢l~, Nooo~9.~ = -- AKx¢l~/x~2~ -- x~l~
(43)
At the boundary plane u. r = d there must be agreement between the internal incident and transmitted light as well as between the internal and external diffracted wave. Since outside the disturbed region the electric field is a solution of the ordinary wave-equation (6), we write E ' = Tooo exp[i(cot -- K. r)] + Tzm• exp[i (o~t -- kzm~~°' . r)~]
(44)
Comparing the expressions (37) and (44), we get taking into account the formulae (43): Tooo = ( A n / x c2' -- xCl~)[x ~9"'exp(-- ip~l~d)
-
-
x {1}
exp (-- ip~2~d)]
(45)
Tz,nn = (Anx~l~xC~/X c2~ -- x!10[exp (-- ipClld) -- exp(-- ipC2~d)] exp[-- i(¢o~mn*t + ~lmn)]
(46)
2) B r a g g - c a s e . Since a is negative now, the diffracted wave emerges through the boundary plane u. r = 0, while it must vanish at the plane u. r = d. Before the slab the electric field is of the form AK exp [i(wt -- K. r)] + R~m~ exp[i(oJt -- kzmn ~o~. r)]
(47)
Comparing the expressions (47) and (37) we get in the plane u. r = 0 AK = Nooocl~ + Nooo~
(48)
Rzmn = (x~l~N0oo~1' + x~2'No0o~9'~)exp[-- i(co~mn*t + ¢Szmn)]
(49)
Beyond the disturbed layer we get only the transmitted incident beam. Hence, To0o = Nooocl~ exp (-- iprl'd) + Nooo ~'~ exp (--ipC2'd) (50) 0 = x~llNooo~l~ exp(--ipll~d) + x l2~Nooo~ exp (-- ip~Z~d)
(51)
The amplitudes Nooo ~1~ and Nooo ~z~ are determinated b y the system of equations formed b y (48) and (51), the solutions of which are: N o o o ~ l ~ = x ~ A K [ x ~z~ exp(--ip~Z~d)--x ~1~ exp(-- ip~l~d)] - 1 . e x p ( Nooo~2~=--xll,AK[x~,exp(--
ip~d)
i p ~ d ) - - x ~ l ~ exp(--ip~lld)] - 1 . e x p ( - i p ~ l ~ d )
(52) (53)
b) ] / [ + I m i + [ n [ > l As opposed to the preceding case, it is immediately seen from the expression (40) of the electric field inside the disturbed region that w e never can satisfy the boundary-conditions, unless Nlmn vanish. That means we don't find any diffracted wave. Henceforth we shall omit this non interesting case. 5. Intensity-expressions o/ the di//racted beams. From the preceding sections it is evident that we are only able to explain diffracted waves characterized b y the integers l, m, n provided Ill + [ m I + [hi = !~ We. con: Sider again the Lane- and the Bragg-case in succession.
992
P. P H A R I S E A U
1) L a u e - c a s e . From the expression (44) it is seen that the diffracted beam of the order (1, m, n) travels in the direction defined b y k~mn(°), while from the relation (46) it is evident that we have a frequency-change -- ~ozmn* and that the phase-constants don't have any influence. For the intensities of the transmitted incident beam, viz., Ioo0 r" = T00oTo0o* and of the diffracted beam of the order (l, m, n), viz., I z m . r- = T z m , T z m n * we get after effectuating the calculations: Iooo L = AKg"[1 + 4xCl)x(2)(x c2~ - - x(10-9 sin s ½(p(1) -- O(~))d]
(54)
I~.r.
(55)
= AKg.(2x(1)x(202(x(Z) _ x(1))-2 sin s ½(O(1) __ p(2))d
It is obvious that between these two intensities the following relation exists, calling IK the intensity of the incident beam of light with frequency o~: Iooo L = I K -- aIzmn z"
(56)
Defining the power P of a beam of light, the product of its intensity b y its cross-section S, the relation (56) can be written P00oL + P~mnL = PK
(57)
since a ~ cos ~mn/cos c~o'~ S~mn/So and where So and S~mn are respectively the cross-sections of the incident and diffracted beams, PK, Pooo L, Pzmn L are respectively the power of the external incident, the transmitted incident and the diffracted beams. This result was normally to be expected, since we have asssumed zero absorption so that there is no energy dissipation within the disturbed slab. The power ratio PzmnL/PK takes the following simple form, taking into consideration the formula (55): P~mnL/PK = (1 + ~ ) - 1 sin 2[D(1 + ~)t]
(59)
where = b2a-l(l~m~/tzo)-2,
D = (/zzmn/#0)(IK[ d/2a½ cos a0)
(60)
2) B r a g g - c a s e . Working along similar lines as in the Laue-case, we get for the intensity of the transmitted incident beam Iooo B = A K ~ [1 + 4x(1)x (2) sin 9"½(p(1) _ p(2))d]-i
(61)
The reflected wave has again a frequency-change defined b y --OJ~mn*. The intensity is given b y
I~,~,~B =
(2x(1)x(~)AK)~. [(x(Z) _ x(~))~ + 4x(1)x(~) sin2½(p(1) _ p(~))d]-l.
sin ~. ½(p(1) __ p(fl))d
(62)
J u s t as in the preceding case there is not any energy dissipation within the
D I F F R A C T I O N OF L I G H T BY ULTRASONICS
993
disturbed region, there exist once-more the relations Io00 B
=
IK --
lal
Ilmn B
(63)
PK
(64)
or
Pooo B + PzmnB
=
The power ratio P~mnB/PK can again be written in the following w a y : (a) if b 2 + a(#~mn/#0) 2 > 0 we get
PlmnB/PK -- {~2 __ 1 + sin 2 [D(~ 2 -- 1)-~]}-l.sin 2 [D(~ 2 -- I)~.] =
{~2 +
(~2 __ 1) cotg2[D(~ 2 - -
1)½]} - 1
(65) (66)
(b) if Y" + a(#zmn/#o) 2 < 0 we get
PlmnB/PK = {1 -- ~2 + sh 2 [D(I -- ~2)l]}-l.shg'[D(l -- ~2)~] = {~2 +
(1 - - ~2) coth 2 [D(1 -- ~2)½]}-1
(67) (68)
where
-=
lal-X
D = (#~mnll~o)(lKld/2 lal cos
(69)
6. Investigation o~ the intensity-expressions. 1) D < 1. In this case, the power ratio in the Laue-case as well as in the Bragg case m a y be written plmn/1%K ~. ~-9. sin ~.D~ For such small values of D, the power ratio is small with respect to u n i t y and we note i m m e d i a t e l y the periodicity of the i n t e n s i t y with respect to the width of the disturbed slab of liquid. 2) D >~ 1. Now we have to distinguish the Laue- and Bragg-cases. a) Laue-case: At a given value of ~ and when D is going to infinity the power ratio P~rnn/PK will oscillate between zero and (1 + ~g)-i b) Bragg-case : In the range [~[ > 1 in which the relation (65) is valid, the power-ratio Pzm,B/PK approaches no definite limit as D increases just as in the Laue-case because of the presence of the trigonometric functions. Considering however the range [~1 < 1 in which the relation (67) is valid, we get for the power-ratio when D becomes v e r y great:
PZmnB/PK ~, 1
(70)
It is obvious from equation (70) t h a t the inward energyflux across the b o u n d a r y plane u. r = 0 is equal to the o u t w a r d flux. So we m a y conclude t h a t there is total reflection in the range I~l < 1 and D has a sufficient order of magnitude. 3) I n t e r m e d i a t e case (D ~ 1). In this case neither a p p r o x i m a t i o n is valid a n d so we have to use the expressions (59), (65) and (67) for the power-ratio in the different cases. I t is seen from these equations t h a t the intensities are s y m m e t r i c a l with respect to the values ~ = 0. a) L a u e - c a s e : The value of the power ratio for ~ = 0 is sin s D. More-
994
P. PHARISEAU
over, when D is in the neighbourhood of ½a the power ratio will approaches the value 1, corresponding to total reflection. When D < ½a we also get a maximum for ~ -----0. Beyond the value ½~ for D, the power ratio begins to oscillate between zero and 1 at ~ = 0 (fig. 1).
-4
-3
-2
-I
0
I
2
3
4
Fig. 1. D i f f r a c t i o n p a t t e r n s i n t h e L a u e - c a s e for a s l a b of l i q u i d of i n t e r m e d i a t e t h i c k n e s s . T h e t h r e e c u r v e s s h o w n in t h e figure c o r r e s p o n d t o D = ¼n, ½• a n d n.
b) B r a g g - c a s e : From fig. 2 it is evident that in the range I~] < 1 the value PzmnB/PK m 1 is rapidly attained as D increases.
7. On the Bragg-reflection in the disturbed layer. We shall now show that the condition ]k!°)zmn[ = lk[ is equivalent with exact Bragg-reflection. Writing the expression for k(°)zmn: k l m ~ (0) - -
K
=
--
k*lmn
(71)
and squaring this expression, taking into account that [k~,nn~0)] -----]KJ and introducing the angle 2Cplmn between the direction of propagation of the external incident light-wave and the direction'of the diffracted wave characterized by klmn (0) w e get 2 sin 9zm. - - Ik*~m~l/IHl,
(72)
Since kzmn*12a is a vector of the reciprocal lattice it is well-known that
DIFFRACTION OF LIGHT BY ULTRASONICS
995
L = 2~/Jk~mn[ is the distance between two planes of the family kzmn*, r = q
(73)
where q is an arbitrary integer. 2 sin 9~r.n = ~/#oL
Hence
(7 4)
k being the wavelength of the light-wave in vacuum. Since the condition is however Jkzmn] ~ [K] reflections which satisfy a law slightly different from the Bragg-law will occur in the disturbed slab. pB
tmn /PK
1.0
.o.9
~,n
• 0,7
.0.6 .0.5
0.3 0,2
0.1
-4
-3
-2
1
2
'3
4
Fig. 2. D i f f r a c t i o n p a t t e r n s in t h e B r a g g - c a s e for a slab of liquid of i n t e r m e d i a t e t h i c k n e s s . T h e t h r e e c u r v e s s h o w n in t h e figure c o r r e s p o n d to D = ¼r~, ½~ a n d ~.
A c k n o w l e d g m e n t s . This work was made possible through a grant from the Centre national Belge de Chimie physique mol~culaire to which the author is greatly indebted. He also expresses his thanks to Prof. W. D e k e y s e r for his continuous interest and to Prof. M. N u y e n s and Dr R. M e r t e n s for fruitful discussions. Received I-8-58. REFERENCES
I) Z ach a r i a s e n , W. H., Theory of X-ray diffraction in crystals. Wiley, New York, 1945, p. 115. 2) B r i l l o u i n , L., La diffraction de la lumi~re par des ultrasons. Act. Sc. et Ind. 59 (1933). 3) B r i l l o u i n , L., Wave propagation in periodic structures, Dover Publications, 1953, p. 143.