tions are present in the distribution of the oxygen in the combination: vacancy-interstitial site and, hence, that the energy level of an electron captured by an insulated oxygen vacancy changes. The decrease in the energy of thermal ionization which was observed for the traps at increasing gamma radiation doses (deductions made from the results of calculations; see Table i) can obviously be explained by a development which takes place at high levels of crystal lattice perturbations and which resembles the radiation-induced bleaching of smokey quartz crystals irradiated with neutrons [7]. Defects of this type also develop when quartz is irradiated with gamma doses in excess of 8.10 ~ R [2]. Despite the higher concentration of electrons trapped in samples irradiated with a gamma photon dose of 109 R in comparison with the samples irradiated with doses of 107 and 108 R, the thermoluminescence intensity of these samples is much lower. The weak luminescence can be explained by concentration quenching and also by quenching resulting from the presence of several types of electron--hole traps having characteristic radiationless transitions [8]. The measurements have shown that energy is stored in the irradiated samples at impuritytype defects (AI 3+, Li+, Li, and Na) as well as at defects formed by the radiation, i.e., at oxygen vacancies. The oxygen vacancies are the deepest traps. The thermoluminescence spectra differ not only in the various samples but also in samples obtained from one single crystal. These fine details of the energy spectrum so far cannot be explained. The interpretation of energy maxima with similar E i values, the determination of the concentration of electron-hole traps, and absolute measurements of the intensity of thermoluminescence spectra will make it possible to simplify the evaluation and the analysis of the results of the measurements and to deduce substantially more information from the thermo!uminescence technique. The author thanks M. I. Samoilovich for discussing the present work and I. A. Kudin for his help in adjusting the measuring setup. LITERATURE CITED i. 2. 3. 4. 5. 6. 7. 8.

A . V . Ipatov and A. B. Berlin, Prib. Tekh. Eksp., No. i, 118 (1973). J. Xchikawa, Jpn. Jo Appl. Phys., 7, No. 3, 220 (1968). M . I . Samoilovich and A. A. Fotchenkov, in: Physics Research on Quartz [in Russian], Nedra, Moscow (1967), pp. 12-20. A . G . Smagin and F. A. Zaitov, in: Radiation Phenomena in Optical Materials Having Broad Energy Bands [in Russian], FAN, Tashkent (1979), pp. 29-31. Ch. B. Lushchik, Tr. Inst. Fiz. Astron. Akad. Nauk Est. SSR, Izd-vo Akad. Nauk Est. SSR, Tallin, No. 31, 19-83 (1966). A . S . Marfunin, Spectroscopy, Luminescence, and Radiation-Induced Centers in Minerals [in Russian], Nedra, Moscow (1975). E . W . Mitchell and E. G. Page, Philos. Meg., Set. B, ~, No. 12, 1085 (1956). V . V . Antonov-Romanovskii, The Kinetics of the Photoluminescence of Crystal Phosphors [in Russian], Nauka, Moscow (1966).

DIFFRACTION OF LIGHT AT ULTRASOUND IN GYROTROPIC MEDIA UDC 535.42

G. V. Kulak

Diffraction of light at ultrasound is widely employed in various acoustooptica! devices (deflectors, modulators, filters, etc.) [I, 2] and for displaying acoustic fields in various media. It is well known that certain materials used in acoustooptics are characterized by optical activity [3, 4]. The author of [5] has considered the diffraction of light in the approximation of a given field in a weak acoustooptical interaction.

1985.

930

Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 43, No, 2, pp, 299-302, August, Original article submitted April 29, 1984.

0021-9037/85/4302-0930509.50

9 1986 Plenum Publishing Corporation

#

Fig. i. Dependence of the intensity of light diffracted at ultrasound (KIi[001]) on the angle ~ of incidence in a bismuth germanate crystal, v = 3.65.10 s cm/sec, f = 200 MHz, and %o = 0.632 ~m.

I

l

6"

We have used in our work the theory of coupled waves to investigate the Bragg diffraction in optically isotropic gyrotropic media and in cubic crystals in the case of strong acoustooptical interaction. We introduce an XYZ coordinate system so that the ultrasonic wave propagates in the direction of the X axis, whereas a plane light wave is incident under the angle ~ relative to the Z axis. When the wave is linearly polarized, four types of interaction of circular modes are possible in the medium and a specific Bragg angle exists for each of the modes. In the case of the interaction of waves with identical polarization (isotropic diffraction), the Bragg condition is simultaneously satisfied at a particular angle of incidence,

~B= arcsJn--

2n

A

,

where ~o and A denote the wavelength of the light wave and of the acoustic wave, respectively, and n denotes the index of refraction of the medium when the gyrotropy is disregarded, In the case of interaction of waves with various types of polarization (anisotropic diffraction), the Bragg angles differ from ~:B by the quantity gyrotropy parameter.

2y , where y denotes the A~-----_--+_-nsin2~B

Therefore, when a diverging, linearly polarized light beam is incident under the angle ~B on a gyrotropic medium, in the case of a sufficient thickness of the ultrasonic region Z ~ ~o/y, we will observe in the zeroth and first orders of diffraction a complicated threepeak structure of discernible diffractionmaxima (see Fig. i). The authors of [6, 7] have studied the fine structure of the Bragg maximum in the diffraction of light at static phase lattices in gyrotropic media. By contrast to the work of [6, 7], the two outermost peaks are characterized by elliptic polarization in the case of diffraction of light at ultrasound. When the condition y >> ~n holds, where ~n denotes the degree of modulation of the index of refraction, all four diffraction processes in gyrotropic media can be independently considered [6]. This condition is satisfied in the majority of gyrotropic media which are used in acoustooptics. A plane ultrasonic wave

U = Uoexpi(Kx~QO with its displacement vector (where K = ~/V and ~ denotes the cycle frequency of the ultrasound) creates periodic changes in the dielectric constant Aeij in both space and time; the

1

changes are associated with elastic deformations Ue~=--~

Oue

Oxm

8x~

,J

and with the photo-

elastic constants Pijem by the following expression: Asij =

(!)

-- ~oPue~Uem,

where co denotes the dielectric constant of the medium when gyrotropy is disregarded. When we start from the Maxwell equations and the material equations of a gyrotropic medium [8], we obtain with Eq, (i) the following wave equation for the electric field strength E in the region of the ultrasound V2E

eo

0~E

2~z

O2

c~

Ot~

c~

8t2

rot E

1

0~

2c ~

Or2

(Ae~Eh +

Aei~Eh) = O,

( 2)

931

where in the case of monochromatic waves a is associated with the gyration parameter 7 by the formula y =(m/c)~; w denotes the frequency and c the velocity of light. We try to find a solution to Eq. (2) in the form of a sum of two coupled elliptically polarized waves with slowly changing amplitudes:

E = [e~Ao• where

k~=

(z) 4- i%Bo• (z)] exp i ik~r- ~t) + [e;Al• (z) ::E ie2Sl• (z) exp i ( k ~ r - - ~ , t ) ,

~(n+Y)(sin%

O, cos~); k ~ =

C

~ + fl

(n 4- y)( - - sin %

O, c o s q 0 )

denote

the

(3) wave vectors

C

of the incident and the diffracted waves, respectively. In Eq. (3), e, and e,' denote the unit vectors of polarization which are disposed in the plane of scattering and perpendicular to the direction of propagation of the zeroth and first modes, respectively; e~ denotes the unit vector of polarization perpendicular to the plane of the acoustooptical interaction. The four types of interaction which can exist in the particular medium between the ellipticallyy polarized waves have been taken into account in Eq. (3). By substituting Eq. (3) into Eq. (2), we obtain the following equation system for the amplitudes of the diffracted waves:

dz

~z

2

,A 2

L

+ng 2iak~A,~:+

-J

(e,a,,~e,) Ao• 4- i -~- (e~a,,je~) Bo•

= O,

where
(elAelj%)----- (elAei~e2) = O, (elAe~]et)9 --

2

eo Peff H "/-T, [7~-~ ~

where plleff = p**, p elf = p,a; I denotes the flux density of the ultrasonic energy; and p denotes the density of the medium. In the case of diffraction at the transverse elastic modes, we have ?

(elAeijel) -=

(%aelj%)=

O, (elAeije.~) =

(e,,Aeije~)

8o

| . /~o ~~

p!~ e n lrT,

2

9

where

p,~ff= p~f= p~. By solving equation system (4) with the matrix technique and the boundary conditions Ao+(0) = Bo+(0) = A_+, At• = B~+(0) = 0, and by disregarding second-order terms of the gyration parameter y, we obtain the following equations for the field intensities in the zeroth and first order:

lo___= - ~1 l+_ (COS~• [i 1 -[- cos,~• 932

l) '

9 ~.

I1+ = -!-I I§ (sin'e• j'erIf ~- sin •

i

(5)

where I• = A• 2 denotes the intensities of the waves with circular polarization which are incident on the region of the interaction. The quantity ~ is defined by the formula <

~sn ~ ., / T 4co8~_____27'z F ~ '

in which the plus sign is selected for isotropic diffraction of

waves with right-hand polarization, whereas the minus sign is used for waves with left-hand polarization, One must set y = 0 when the anisotropic diffraction of the two types is considered. It follows from the solution of equation system (4) that the diffracted light has elliptic polarization in both orders of diffraction; the ellipticity (ratio of the minor to the major axis of the ellipse of polarization) T~ of the first and To of the zeroth mode is given by % =

2sin • sin • sin2• ~ @ sin~•

!

(6)

The corresponding equation for To is obtained from Eq. (6) by replacing sin x by cos x. In the case Pleff < plleff the major axis of the ellipse of polarization is perpendicular to the plane of scattering in the zeroth order and is located in the plane of scattering in the first order. It follows from formulas (5) that the intensities of the diffracted modes of the four interactions are identical except for the gyrotropy parameters and are the sum of the intensities of the components which are perpendicular to (s) and parallel with (p) the plane of diffraction. We infer from Eq. (6) that the ellipticity of the light is given by the anisotropy of the photoelasticity (pIIeff ~ P• and by the intensity of the ultrasound, In the case of diffraction at transverse elastic modes, we have pileff = P• and the scattered light waves preserve their circular polarization in both orders. The author thanks in conclusion V. N. Belyi and V. V. Shepelevich for discussing the results of the present work. LITERATURE CITED i. 2. 3. 4. 5. 6. 7.

8.

R. Damon, V. Maloney, and D. MacMahon, Physical Acoustics [Russian translation], Vol. 7, Mir, Moscow (1974). Yu. V. Gulyaev, V. V. Proklov, and G. N. Shkerdin, Usp. Fiz. Nauk, 124, No. i, 61 (1978). A . W . Warner, D. L. White, and W. A. Bonner, J. App!. Phys., 43, No. ii, 4489 (1972). M . F . Brizhina and S. Kh. Esayan, Zh. Tekh. Fiz., 47, No. 9, 1937 (1977). V . V . Soroka, Fiz. Tverd, Tela, 19, No. !i, 3327 (T977). T . G . Pensheva, M. P. Petrov, and S. J. Stepanov, Opt. Commun., 40, No. 3, 175 (1981). M . P . Petrov, S. I. Stepanov, and A. V. Khomenko, Photosensitive Electrooptical Media in Holography and Optical Processing of Information [in Russian], Nauka, Leningrad (1983). V . N . Belyi and V. V. Shepelevich, Opt. Spektrosk., 52, No. 6, 842 (1982).

933