DIFFRACTION OF LIGHT BEAMS BY DAMPED ULTRASONIC WAVES IN OPTICALLY ISOTROPIC MEDIA V. N. Belyi, I. G. Voitenko, and G. V. Kulak

UDC 535.42:534

For the purpose of optimizing acoustooptic (AO) instruments (modulators, deflectors, processors) it is necessary to know the effect of the incident light polarization, as well as that of ultrasound (US) damping, on the characteristics of diffracting light beams. The polarization and energy characteristics of diffracted light beams can be used for the experimental measurement of photoelastic constants, US dampingicoefficients, and elasticity moduli. Several studies are presently known [1-5], devoted to the study of this problem. The treatments are restricted, however, to low US intensity or to the plane-wave approximation [i, 2], without accounting for US damping [3, 4] and without using polarization effects during diffraction of light beams by damped US waves [5]. In the present paper the effect of the incident light polarization on the Bragg diffraction of finite light beams by damped US waves in optically isotropic media (including crystals of cubic systems) is investigated. The maximum energy shift of a light beam diffracted by a US is determined. Consider a sound conductor, filled with an optically isotropic material, in which a US wave propagates along th X axis and occupies the region between the planes z = 0 and z = ~. A light beam with a certain amplitude distribution in the scattering plane XZ propagates with a Bragg angle

q ) = a r c s i n 2%-2-~ A n (%0 is the wavelength of light in vacuum, A is the wavelength

of sound, and n is the refractive index of the sound-conducting material). It is assumed that the incident light beam is linearly polarized, while the electric field intensity vector E spans an angle ~ with the Y axis. A US plane wave with vector displacement U = U o exp (- ~x) exp i (.Hx -- ~t) ( J E - - - - ~ -,

and ~,

v,

~ are,

respectively,

the

circular

frequency,

phase

velocity,

a n d US

v

damping coefficient) creates spatially and temporally periodic variations in the dielectric permeability, related to the elastic deformations

Uhz = - 71~ f 0Uk a& ~ 0UI ] and the photoelastic -

,,

~x~ I

constants Pijk/ by the relation: 2

Asu = - - s o p ~ z U ~ l ,

(1)

were s o is the dielectric permeability of the unperturbed medium. From the Maxwell and the material equations we obtain, with account of (1), the following wave equation for the electric field intensity E in the region occupied by the US: V~E

eo "~ c 2 0t 2

- ~ 2c"

{As*E -i- AsE) = 0, ~g2 '

(2)

where ~, c are the frequency and speed of light, and the asterisk denotes the complex conjugate. According to the Floquet theorem, the solution of the wave equation (2) is sought in the form of a sum of two coupled waves with slowly varying amplitudes

Mogilev Branch of the B. I. Stepanov Institute of Physics, Belarus Academy of Sciences. Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 56, Nos. 5-6, pp. 831-836, May-June, 1992. Original article submitted November 29, 1991.

0021-9037/92/0506-0513512.50

9 1992 Plenum Publishing Corporation

513

E = [e,Aop (x, z) + e.,Ao~ (x,z)l exp i (kor - - o~t) § -k [ e l A , ; (x, z) -}- e~,A~ (x, z)l exp i ( k ~ r .

where

m o_+f2 ko_.= ~ n~ cosg, kix -.C C

kox = --n0 sin % C

(3)

c%t),

m~-Q_, nisin 9, kiz = - n i cos 9 C

are

the

wave

,

vectors of the incident and diffracted waves, n o = //0(m); n~ = /go(~ +_ ~); and el, e~ are unit polarization vectors, lying in the scattering plane and being perpendicular to the propagation direction of the zeroth and first mode, respectively, and e~ is a unit polarization vector, perpendicular to the interaction plane AO. Following substitution of (3) into the wave equation (2) and transition to the coordinate system r = z sin 9i- x cos % 9 = z sin qi+ x cos ~ [6], a system of equations is obtained for the amplitudes of the interacting waves

OAos & -~- i exp [[~ (r - ~)1 (x11Als @ xl~All~) = 0,

--

0Aop + i exp [} ( r - - ~)1 (• 8r

-b •

= 0, (4)

-OAi~ + i exp [~ (r ---- -c)] (•

-}- u,~Aov) = O,

OAiv ---k i exp [} (r - ~:)1 (• Or

-k•

where B = a/2 cos 9, and the quantity

~n3A• ~ 410sin

•

= 0,

2[~ p / pv:~

(p is the crystal density,

and

f~, is the US intensity) is expressed in terms of a convolution of the dielectric permeability variation tensor with the polarization vectors. We write down expressions for the nonvanishing coefficients f~!eij for the various cases of AO interactions. For diffraction by longitudinal elastic waves, propagating (Pzl + PI2 + P21 - 2P44);

in the [ IIi ] direction,

in the [001] direction - dz22 = P11,

I

direction -

A•

== ~- (2pn -~ P18 @ P1~ + 4p~), A• .

by transverse elastic modes the [iii] direction,

1~ I Ax~o.----~-:~pll@ PI~ @ P21 @ 4p~)~ A X n = --3 X

A z ~_ = Ax.21 ~- ~

and iXi2 = A•

Az~l =

Pl2;

and in the [ii0]

I

= - ~ (Pz~ n- P21).

(2pii

l

I

Pl'~-"P2i

In the case of diffraction

'. . . ~. . (Pi~ - 4Rsvp), [iXii

P'2i)

for

= P44 for the [001] and [ii0] directions.

In most cases of AO interactions considered the system of equations (4) is decomposed into two independent systems of equations, which can be solved in closed form. In the case of diffraction by US shear wave propagating in the [iii] direction, system (4) can be solved only by numerical methods. In the most important case of diffraction by longitudinal US waves the system of equations (4) for perpendicular (s) and parallel (p) diffraction planes of field components is

-}- i~,p exp [~ (r--"c)] A1~,~, = O, 8~

(5)

OAI~,; +i~,;exp[~(r-.-T)]Ao~,;=O, Or where ~':~s= x!Iz; ~p = • The Cauchy problem for Eq. (5) was solved by Riemann's method [7] with the use of the boundary conditions A0s p(T = -r) = A(r), A1s,p(~ = -r) = 0, assuming the absence of Fresnel reflection of incident light at z 0. 514

7

1

o,4

o# o,z i

2

3

#

5

6

"% O

I/A

,,55% I

2

5

~,

5

6

~s

Fig. I. The diffraction yield N as a function of the modulation index v s for beams of Gaussian (i-5), Lorentzian (6), and rectangular (7) profiles a = 0 (i), 1 (2), 5 (3) and 7dB/cm (4). For the zeroth and first modes the Riemann functions Rs'P(r, T; r0, c 0) = Vs, p (r, ~; r0, ~0)exp(-~r), Rs'P( r, ~; ro, ~a) = Vs,p( r, ~; r0, ~0)exp(-~ ~) are found from the equation

O~"V'~'~

o~g~' > + V< p = 0

( ~,,p = •

exp [-- ~ Q...... Q l i c o s h

following

conditions

(6)

,-,82

9

~ (r + z), ~ = •

exp [-- c~ (r--- ~:)1 s i n h

" + ~) ) w i t h a c c o u n t ~}(r

of the

on t h e c h a r a c t e r i s t i c s

V,.r,(ro, z; to, %)=V
V~ :~ = $o J- "

[(exp (-- 2o~%) --, exp (- - 2o~T)) (exp 2s~r - exp 2~ro)l'/'-"[

where J0 is the Bessel function of the first kind, and r 0, c 0 are the coordinates of points on the intersection of characteristics (see [7]). By means of (6) one can obtain exact solutions of Eq. (5). In this case the amplitude distribution at the outlet boundary z = ~ of the sound.conductor is written in the following form for an s-polarization wave in the zeroth A0s and first Ais diffraction orders: q-i

Ao,(r)=A(r ) cos g - - - ~ j cos {~sin q: sh (2cz/sin q~)exp(Tr) S A [r - - t sin q0(q@ 1)] x --1

X exp [7/sip. rp (q + 1)] J~ { ~* exp (%,:r--6)[(exp ( = 5 ) - - e x p ( . S q ) ) (exp(--Sq)---exp6)] 1/2}[(exp(-

6)--exp(

-&7))(exp(-8$)--expS)]-i/2dq,

(7) AI~ (~) . . . . . i 1 as I sin ~ cos ~ exp [3[5 (l sin {p -- T)] S A [I sin (p (q & 1) --2 i -I ..... T] exp ([3l sin q)q) Jo/. •

exp [2o~(l sin go- - ~)1 [(exp 6 - - exp (6q)) X

• (exp 6 -- exp (-- 6q))] 1/2 } dq,

where 6 = 2 ~ s i n ~; ~ = $ - 2~, and J0, Ji are Bessel functions of the first kind. In the case of interactions of p-polarization waves one must replace cos ~ by sin ~, and the subscript s by p, in Eqs. (2), (3). Expressions (6), (7) describe strong AO interaction, and are valid for arbitrary damping coefficients and Bragg angles, as well as for a large class of light distributions A(r') at the inlet boundary of the sound conductor. In the absence of sound attentuation expressions (7) transform to the equations for two-dimensional diffraction [3, 6], while for A(r) = A 0 = const they lead to the well-known expressions for the amplitudes.of diffracted waves A0s,p = A0 costs,p, Als,p = - i A 0 sin Vs,p, where vs,p = ~• s i n ~ are the modulation indices for diffracted waves of s- and p-polarizations. 515

The diffraction yield N was calculated by the equation [6]:

A1A'~d'c

(8)

for collimated beams with Gaussian

A (r')= A0exp

I--(\ 2r' W ]'12:i J' Lorentzian

A(r') = A0/[l + (4~r'/

3W)2], and rectangular A(r') = A0(Ir' I ~ W) profiles of the transverse amplitude distribution of the light field, where W is the radius of the transverse cross section of the incident bealll. At the outlet boundary of the sound conductor the diffracted beams are linearly polarized. The electric field vector for light in the zeroth order of diffraction spans with the ing plane XZ an angle #0, and the'first order - ~i. The polarization azimuthal-,values of tHe incideht and diffracted beams are found from the expressions

tg ~ r

--

Iov + llv los -+- Ils

where lo,r = ~ [Ao~,plZdr, I~,p = j' IA~.~! ~ dw

lop Io~

, tg ~ % = - ,

tg z~l=

are the diffracted

lip 11~

,

(9)

beam i n t e n s i t i e s .

Using expressions (9), it is not difficult to show that the diffracted beam intensity for an arbitrary azimtuh ~ of the incident polarization, normalized to the diffracted light intensity at ~ = 90 ~ , can be found from the equation

I1(*) _ 1 _ _ ( 1 _ & (90 ~)

\

~_s

(10)

~ /

where qs,p is the diffraction yield of the s-, p-components of the incident light. For light beam diffraction by US shear waves, propagating in the [i00] and [Ii0] directions, AO mode interaction takes place of the zeroth and first orders with orthogonal polarizations. In this case the diffracted wave amplitudes are identical (• = x12) and are given by expressions (7), in which one must put • = • = ~12. The diffraction yields are Ks = Dp, and therefore the diffracted light intensity' is independent of the incident polarization azimuth. Figure 1 shows the diffraction yield N as a function of the modulation index v s = 4~ss sin ~ for s-polarization Gaussian beams for different values of the US damping coefficient ~. In this case s = 1 cm, ~ = 5 ~ , W = 0.5 cm. It is necessary to note that for strong, as opposed to weak (Vs << I), interations the modulation index can acquire arbitrary values within the limits of achievable US intensities (~s,p ~ ~f=--~ and fa is the US wave intensity). As c o u l d be expected, even for a = 0 and exact satisfaction of the Bragg condition, 100% diffraction yield is not achieved for the beams (~max = 97% for 9s = 1.57). In the absence of damping the curve N(Vs) has an oscillatory nature, while with increasing v s the amplitude of the maxima decreases gradually. With increasing damping the oscillation amplitude also decreases, and the separation between neighboring maxima of ~max increases. Finally, for high damping (~ > 7 dB/cm) D(~s) practically becomes a linearly decreasing function of the modulation index. The diffraction yield for the various profiles of transverse cross section of incident beams is represented by curves 5-7 (Fig. i), while ~ = 1 dB/cm, W = 0.5 cm and s = 1 cm. It is seen that in the presence of damping the effectiveness of AO interactions depends substantially on the slope of the amplitude distribution. The maximum yield (for ~ z 1.57) is reached for a rectangular, and the minimum - for a Lorentzian profile. This behavior of q(~s) implies an insignifcant contribution of the lateral portions of beams to the effectiveness of AO interactions. It follows from Fig. 1 and Eq. (i0) that the attentuation of high-frequency sound worsens substantially the polarizing properties of AO cells. An ideal polarizer is a cell for which 516

A

lO -z cm

fb

z~ @

Fig. 2. function (i); 0.3 0 . 6 3 pm,

I

[

~

I

...........

The shift of a Gaussian beam A as a of the parameter W/I 0 for v = 0.i ( 2 ) ; 0 . 6 ( 3 ~ and 0 . 8 (4). 10 = f = 200 MHz, a = 0 . 0 0 1 d B / c m .

Iz(~)/Iz(90 ~ = sin 2 ~, i.e., when qp = 0, qs ~ 0. A real cell, however, polarizes the incident radiation only partially. For cubical system crystals the best polarization properties are acquired by a cell in which the longitudinal US wave propages in the [iii] direction, since in this case the value of A•215 is maximum. Expressions are given in [8] for the shift values of light beams with a sufficiently narrow spectral width by an expansion of the light beam in plane waves. In the opposite case of a beam with a wide spectrum, along with the shift there occurs a deformation of its envelope, and the expressions earlier obtainted are not valid. It is assumed that the center of the incident light beam is located at the point x = 0, when the energy shift of the center of the first order diffracted beam is determined by the relation [9] i'~-~; =

A~:~

-o~

(ll) ,

' I

" :~ (-r) 4 ~ "

O

The shift value of the zeroth order diffracted beam is determined by a similar expression with the replacement of A l by A 0 and of % by r. Figure 2 shows the shift in the diffracted light beam (A) as a function of the radius of the incident Gaussian beam W/I 0 for various values of the modulation index v ~ ~-a 9 It follows from the figure that the magnitude of the shift decreases with increasing radius of the incident beam and modulation index v. The modulation indices for the parallel Vp and perpendicular v s components of the linearly polarized components of the incident beam differ in magnitude. Hence follows the possibility of spatial separation of nonpolarized light into s- and p-polarized beams. LITERATURE CITED i.

2. 3. 4.

,

6. 7. 8.

9.

H. Ekund, A. Roos, and S. T. Eng, Opt. Quant. Electron., Q E-7, 73-79 (1975). S. V. Bogdanov, Opt. Spektrosk., 49, No. i, 146-150 (1980). V. N. Belyi and G. V. Kulak, in: Applications of Acoustooptic Methods and Devices in Industry [in Russian], Leningrad (1984), pp. 84-89. Ao S. Zadorin and S. N. Sharangovich, "Polarization characteristics of acoustooptic interactions of wave beams in optically isotropic media," Moscow (1985); Deposited in VINITI, No. 6219 (July 8, 1985). V. A. Pilipovich and Yu. M. Shcherbak, Izv. Akad. Nauk Belarus SSR, Ser. Fiz., No. 4, 100-104 (1975). M. G. Moharam, T, K. Goylord, and R. Magnusson, J. Opt. Soc. Am., 70, No. 3, 300-304 (1980). A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics [in Russian], Moscow (1981). V. N. Belyi, I. G. Voitenko, and G. V. Kulak, Opt. Spektrosk., 62, No. 5, 1161-1164 (1987). O. A. Godin, Zh. Tekh. Fiz., 54, No. ii, 2094-2104 (1984).

517