The Solution of the Raman-Nath Equations Ii

DIFFRACTION OF LIGHT BY SUPERSONIC WAVES: THE SOLUTION OF THE RAMAN-NATH EQUATIONS II. The Exact Solution

BY ROBERT MERTENS (Rijksuniversiteit Gent. Seminarie voor Analytische Mechanica, Gent, Beigium) AND FREDERIK KULIASKO (RijksuniverMteit Gent. Seminarie voor Wiskundige Natuurkunde, Gent, Belglum) Received January 1, 1968 (Communicated by Sir C. V. Raman, V.R.S., Nobel Laureate)

A~STRACT The partial differential equation associated with the system of differ- ence-differential equations of Raman-Nath for the amplitudes of the diffracted light-waves is solved exactly by the method of the separation of the variables. The solution is presented asa double in¡ series con- taining the Fourier coefficients of the even periodir Mathieu functions with period zr and the corresponding eigenvalues. Considering this solution asa Laurent series in one of the variables, the Laurent coetª immediately give the exact expressions for the amplitudes of the diffrac- red light-waves, from which the formulae for the intensities are caleu- late& The connection between the Raman-Nath method and Brillouin's Mathieu function method has thus been achieved.

1. II'~TRODUCTION

ThE problem of the diffra•tion of light by a supersonie wave has given ¡ to two important methods of solution: (1)the method of Brillouin1 based on the expansion of the solution of the wave equation into a series of Mathieu functions; (2) the method of Raman-Nath 2 starting from a Fourier expansion of the solution of the wave equation and leading to a system of diffcrence-differential cquations for the amplitudes of the diffracted light- waves. B¡ exposed his method in the case of a standing ultrasonic wave. It was extended by Mertens 3 to the case of a progressive sound wave from which the problem of a standing supersonic wave r be treated in

303 304 ROBERT MERTENS AND FREDERIK KULIASKO

a more detailed way,4 leading to the spectral character of each diffracted light beato. Recently Berry~ establishzd a theory in the sense of BriUouin taldng into account th, effect of the reflected diffracted waves. At the first sight the theories of Brillouin and Raman-Nath are eom- pletely different. A eonnection between both theo¡ was established by Berry6 using a rather involved solution of the truncated Raman-Nath system in terms of a continued fraction and its relationship with the Mathieu furtcfion solution. In the present study we derive the conneetion between both theo¡ in another way, employing the method that was described in Part I of this paper. 6 There it was shown that the amplitudes of the diffraeted light- waves eould be obtained as the Laurent expansion eoetticients of a generating function whieh satisfies a partial differential equation. It was seen that the approximate solutions of the latter equation lead rather quickly to known approximate expressions for the amplitudes of the diffracted light beatos. In w we derive an exact solution of the paxtial differential equation by the method of the separation of the variables. By a ehange of one of the independent variables one of the obtained ordinary differential equations is transformed into a Mathieu equation, the solutions of whieh are shown to be even and periodic with period ~r. The solution of the partial differential equation is presented as an infinite series eontaining the eonsidered Mathieu funetions. The Fourier expansion of those Mathieu funetions leads to a solution having the forro of a double series from whieh, in w exaet expressions for the amplitudes and the intensities of the diffraeted light-waves are obtained in terms of the Fou¡ eoetfieients of the Mathieu funetions and the eorresponding eharaetefistie numbers. The intensities of zero and first order are also derived from the general formulae when p >~ 1.

2. EXACT SOLUTION OF THE PARTIAL DIFFERENTIAL EQUATION* It has been shown by Kuliasko, Mertens and Leroy e that the amplitudes Sn (0 of the diffraoted light beams, whieh ate solutions of the system of Raman and Nath," 2 d_~~_ _ (4~-x -- 4n+x) in~lM?n (n .... , --2, --1, 0, 1, 2, .-.), (1) with the boundary eonditions 4~ (0) = ~.0, (2)

* Tho notations arc the sarao as in Part L Diffraction of Light by Supersonic Waves--H 305 may be obtained as the Laurent expansion coefficients of a generating function

G (~, 7) = S ~. (0 ~~, (3) nl=.-~O satisfying the partial differential equation b'G ;~G 2 -~-~- with the boundary eondition G (0, 7/) = 1. (5)

We shaU now try to find a solution of the partial differential equation (4) satisfying the condition (5) by the method of separation of the variables. Putting G ( r/) = Z( H (r/), (6)

Z (O being a function of ~ only and H 01) representing a funetion of 7/only, into the Eq. (4), we obtain the following differential equations for Z and H: dZ~~= 1 iaZ, (7)

~3 d~H[l_~-z q_ ~ ~ dH.d~l.-- ~ (~2 _ iay -- 1) H ---~0, (8) ia being the separation eonstant.

The general solution of Eq. (7) is easily obtaincd as Z ( = Ce t i'r (9) whereby C is an arbitrary constant. We shall further transform the Eq. (8) by changing the independent variable 7. In order to do this, we remark that ~ is a complex variable, 6 whose image point in the complex w-plane must lie on a elosed contottr encircling the o¡ once in a countercloekwise sense. The elosed path itself is situated within ala annular region with centre 0 in the complex ~-plane, 306 ROBERT MERTENS AND FREDERIK KULIASKO

where the generating function G (~, r/), considered as a function of the eomplex variable ~7 alone, is holomorphic. Taking for the closed path a circle with radius unity we may write ~1 -~- ieniX, (10) where x is the new independent rem variable. Putting y (x) -----H (,/), Eq. (8) be,comes

ate+~d~y (4~ ,o8e~ 2x)y =0, (11)

whieh is nothing else than the weU-known Mathieu equation. If we set 4a a = - , (12a) P 4 q = ~, (12 b)

we obtain the canonical forro ~, s

daY + (a -- 2q cos 2x) y = 0. dx 2 (13)

The function G (~, ~7), being holomorphic within the annular region of the eomplex rrplane , has to be one-valued in each point of the circle with radius unity. Hence, H(~I) must also be a one-valued function in each point of this circle, so that, with respect to (10), this stipulation yields y (x + ,0 = y (x). (14)

This condition signifies that the allowed solutions of Eq. (13) ate the even Mathieu functions ce~ (x, q) or the odd Mathieu functions ~e~ (x, q) periodic in x with period ,r. 7, s A general solution of Eq. (13) belonging to the characteristic number a = a~ may be written as (ref. 7, p. 145) y = Ace~n (x, q) + Bfem (x, q), (15)

A and B being arbitrar), eonstants and f%n (x, q) signifying the non- periodic second solution of Eq. (13) reducing to sin 2nx when q = 0 [ce2n (x, q) reduces to cos 2nx for q : 0]. According to the periodicity condition (14), we must put B = 0. In respect of Eq. (12 a) we see that the value of the separation constant a eorresponding with the charaeteristic number azn is given by a~ = (16) Diffraction of Light by Supersonic Waves--H 307

A solution of Eq. (4) belonging to a = a2n is thus e Li*"~. ce2n (x, q)

anda sum of those solutions is still a solution:

oo G(~, 7) = E C2neti*"~ce2n(x, q), (17) limo

where C2n are arbitrary constants, which we shall now determine, by using the boundary condition (5). For ~ -----0, (17) becomes

oo 1 : S C2ncez~(x, q). (18) i1~-0

Multiplying both sides of this relation by ce,.r (x, q), integrating over x between 0 and 2~r, and taking into aceount the orthogonality relation (ReŸ 7, p. 23)"

S ce.,n (x, q) ce2r (x, q) dx - 0 (n :~ r), (19) O

we find

S ce2r (x, q) dx -- 0 C2r -- 2r (20) I ce%.r (x, q) dx O

Using the Fourier expansion (Ref. 7, p. 21),

Cezr (x, q) : Z" A~ '~r~ cos 2px, (21) pm0

one has,

271" ce2r (x, q) dx = 2~r Ao ~~r) O

2W" r S ce2ze (x, q) dx = 21r [A0~2rl]2 -'1- 7r S [A2p~2r~]2 ~- 7r 0 l~s=l according to the stipulation of normalization, generally accepted (Ref. 7, p. 24). The expression (20) for the coefficients C2r then gives C~. ~- 2Ao r (22) 308 RoaFatT MERIENS AND FREDERIK KULIASKO so that the solution of the partiat differential equation (4), safisfying the boundary condition (5), becomes G (~, 7) = 2 27 A0t~r) e ti`''t ce~r (x, q)*. (23)

We shaU now return to the original independent variable ~/in the r.h.s. of Eq. (23). Writing the Fourier expansion (21) as

ce~r (x, q) = 27 A2n (zr) (e i2= + e -ignx) 11----0 and taking into account the transformation formula (10), the solution (23) finally becomes

oo oo G (~, 7/) = S 27 A0 (zr) A~n(2r) e (i-n~n + q (24) n~O f----O

3. AMPLITUDES AND INTENSITIES OF THE DIFFRAC~ED LIGBT WAVES From the series (24) and in view of Eq. (3), we can immediately deduce the expression for the amplitude of the diffracted light-wave of order n, ~ (~), as the coeflficient of 7/n. So we have

$n ( = (-- l)n4-n ( = i-n 27 Ao(2r) Ain (~r) e i.,,r (n >0) (25 a) fino

O0 ~0(~) = 2 27 [Ao(2r)]2e ti*'~. (25 b) rmO

Analogous formulae have been obtained by Berry5 from a Brillouin- like theory, neglecting reflected waves. We remark here that the application of the boundary conditions (2) to the expressions (25) leads to known

* Instoad of using the general solution (15) of Eq. (13), wo could also b-ave chosen anothel general solution, be]onging to the characteristic numbcr a = bs~ ~ at, (ReŸ 7, p. 145), y = A' sean (x, q) + B' gell (x, q), whorr A' and B' are arbitrary constants and ges (x, q) is the non-pr sr solution o! (Eq. 13), reducing to cos 2nx for q = O[se2n (x, q) reduces to sin 2nx whon q = 0]. Here again wo must put B' = 0. The solution of Eq. (4) then br

oo G (~, ~1) = "Y' C'tr e ~'tt'~r:; sezr (x, q). fa: 0 However, taking into account the boundary condition (5) it is seen that suela a series expansion is impossible, the r.h.s, being an odd, the l.h.s, an oyen function with respect to x, if one puts ~--0. DiffractŸ of Light by Supersonic Waves--H 309 identities betwcen the Fourier coefficients of the Mathieu functions Ce2r (x, q) [Ref. 7, p. 210, Eqs. (6) and (7)],

O0 2 S Ao (2r) A2n (zr) = (26) rw-O

For the n-th order intensity we have

In = ~n~n*

O0 O0 ~--- Z, Ao (~') A2n (9r) e S Ao (2s) A2n (2s) e-tia"~ r~0 l~O

co ~r' Ao(2r) Ao(2S) A2n(Zr) Azn(zs) eti (,,.~-a,.)~ r4/:0

X [A0 (2r) A2n(2r)] 2 -~- 2 X Ao (2r~ Ao ("s) A2n (2r) A2,~ (2s) r~O f, 1=0 r~e

• cos (=~, - ,~=) g.

Taking the square of the identity (26) for n 4: 0, we find that

S [Ao (2r) A2n(Zr)] 2 + 2 27 Ao (9r) Ao (~ A~n (re) Azn (~s) = 0, r_~~O y, II:=O so that In = -- 4 S A0 (zr) A0 (zs) A2n (zr) A2n (=s) sin 2 (r -- azs) t<~s or introdudng the relation (16), between azr and a~.r,

In = -- 4 X Ao (2r) Ao t2s) Ag.nt~r) A2u (2s) sin z 1 (azr --azs) p~. r<~l (n >0). (27)

From (25 b) we obtain

lo = 'ko'/'o*

= 4 S [Ao(mP + 8 27 [Ao ~m Ao(2~)] ~ cos (,~~. -- '~~s) 5 tino r ~e

Taking the square of the identity (26) for n = 0, we see tlaat 4 Z [Ao(2r)] 4 +8 ~' [Ao (2r~ Ao(2S)] ~= 1 r r~$ 310 ROBERT MERTENS AND FREDERIK KULIA$KO

so that I0 = 1 -- 16 27 [Aor (*.s~]~.sin* (a~r -- a2s) r

or employing again (16) we obtain finally

q = 1 -- 16 27 [Ao r Aor ~ sin z -~~ (a~r - a2s) p[. (28)

With regard to the relation between ']'n (D and 6-n ( D, given by Eq. (25 a), we may conclude that, Ln = In, (29)

confirming the experimental result, that in the case of normal incidence of the light the diffraction spectrum is always symmet¡ with respect to the zero order. The expressions for the intensities (27) and (28) are presented here in a forro, which allows their numerical computation with the help of tables giving the values of the Fourier coefficients of the periodic Mathieu functions with period ~r and the corresponding characteristic values. Such tables have been established by the U.S. National Bureau of Standards. a We shall limit ourselves, here, to the approximate calculation of lo and I• from the general formulae (27) and (28) for values of p >~ 1 or, taking into account (12 b), q ~ 1. From Ref. 8, p. 123, we find that

A0 t~ = ~/2[1 Ao'" -V6q20(q,)][(~._[_O(qa))] + AoC~ Ao r = O (q2),

Ao ~~ Ao ~s) = O (q3), ... Ao ~~) Ao ~~~ = O (q3) ..., Ao~4~ AoC,) = O (q5), ...

so that neglecting all terms of an order higher than the second one, the double series in (28) is reduced to only one term, having as coefficient q~ [A~176 A~ ~ ~2"

Further, using the expressions for the characteristic values in Ref. 8, p. 120, we see that ao -- a2 = --4 + O(q2). Diffraclion of Light by Supersonic Waves--H 311

Introducing the explicit values of q given by (12 b), we obtain for the zero order intensity 8 I 8 ~rAz lo = 1 -- ~ sin~ ~ V~ = 1 -- ~ sin 2 2/z0A..-- ~ (30)

Neglecting all the terms of an order higher than the second one, the double series in (27) is also reduced to only one tcrm with coefficient, 1 q. q . 1 A~176Ao(2)A~~~ ~ -- x/2 " 4 2V'2 1 = -- ~ q'' so that the intensities of orders q-1 become 4 1 4 . ~ ~rh I+1 = ~ sin~~ pE =2 sin 2tL~-~,~ z. (31)

The approximate formulae (30) and (31) have already been obtained by Mertens ~ as the result of a direct calculation, up to the second order, in the case of a progressive ultrasonic wave. Analogous expressions were found by Bri/louin I in the case of a standing supersonic wave. Eqs. (30) and (31) slightly different from the expressions found by Nagendra Nath, ~o using the NOA-method, with N----1. We may conclude that the method of solution of the Raman-Nath equations exposed in this paper is a quite general one. Besides most approximate solutions it gives also the exact solution, once the integration, approximately or exactly, of the partial differential cquation (4)with boundary condition (5) has been performed.

4. REFERENCES 1. Brillouin, L. .. "La diffraction do la lumi6re par des ultrasons," Actual. Sci. Industr., 1933, Nr. 59. 2. Raman, C. V. and Proc. lnd. Acad. Sci., 1936, 3 A, 119. Nagendra Nath, N. S. 3. Mertens, R. .. Simon Stevin, 1949/50, 27, 212. 4. .. Ibid., 1950/51, 28, 1. 5. Berry, M. V. .. The Diffraction of Light by Ultrasound, Academic Pre,,s, 1966. 6. Kuliasko, F., Mertens, R. Proc. Ind. Acad. Sci., 1968. and Leroy, O. 7. Me Lachlan, N.W. Theory and Application of Mathieu Functions, Oxford Clarendon Press, 1947. 8. Meixner, J. and Mathieusche Funktionen und Sphiiroidfunktionen, Springer, Sch~fke, F. W. 1954. 9. Tables Relating to Mathieu Functions, Columbia University Press, 1951. 10. Nagendra Nath, N.S. .. Proc. lnd. Acad. Sci., 1938, 8A, 499. A2