polarization properties of light-induced scattering in bi12tio20 crystals: theory and experiment for...

Filippov et al. Vol. 20, No. 4 /April 2003 /J. Opt. Soc. Am. B 677

Polarization properties of light-induced scatteringin Bi12TiO20 crystals:

theory and experiment for diagonal geometry

O. Filippov and K. H. Ringhofer*

Department of Physics, University of Osnabruck, D-49069 Osnabruck, Germany

M. Shamonin

Department of Electrical Engineering, University of Applied Sciences, D-93025 Regensburg, Germany

E. Shamonina

Department of Physics, Imperial College, London SW7 2BZ, UK, and Department of Physics, University ofOsnabruck, D-49069 Osnabruck, Germany

A. A. Kamshilin and E. Nippolainen

Department of Physics, University of Joensuu, P.O. Box 111, Joensuu, Finland

B. I. Sturman

International Institute for Nonlinear Studies, Koptyg Ave 1, 630090, Novosibirsk, Russia

Received June 6, 2002; revised manuscript received October 18, 2002

Illumination of ac-biased photorefractive Bi12TiO20 crystals with a coherent light beam results in the develop-ment of strong nonlinear scattering. Theoretically and experimentally we investigate the angular and polar-ization characteristics of the scattered light for the diagonal (@ 111#) optical configuration and different polar-ization states of the pump. A satisfactory understanding of the observed scattering properties is achieved formost of the cases investigated. © 2003 Optical Society of America

OCIS codes: 160.5320, 190.5330.

1. INTRODUCTIONCubic crystals of the sillenite family (Bi12SiO20 ,Bi12TiO20 , and Bi12GeO20) as well as cubic semiconduc-tors such as GaAs, CdTe, InP are the fastest photorefrac-tive materials, which makes them attractive for numer-ous applications.1–3 Two techniques, one dc and one ac,have been proposed to enhance the insufficiently highnonlinear response of these materials.1,2,4–6 The dc tech-nique requires application of a dc electric field and the in-troduction of proper frequency detuning between the in-teracting light beams. The ac technique, which hasproved to be the more appropriate for practical purposes,is based on the use of an alternating field, preferably of asquare-wave shape.7 The characteristic rates of spatialamplification in the ac technique approach 102 cm21,8,9

which is comparable with the rates attainable with slowferroelectrics such as LiNbO3 and BaTiO3 .1

Strong spatial amplification achieved in sillenite crys-tals manifests itself in pronounced light-induced (nonlin-ear) scattering.10,11 The underlying mechanism of thisphenomenon is not much different from that known forphotorefractive ferroelectrics1: Weak seed waves, whicharise because of surface and bulk crystal imperfections,

0740-3224/2003/040677-08$15.00 ©

experience a strong spatial amplification at the expense ofthe pump.

The observable properties of the light-induced scatter-ing in cubic crystals are, however, essentially differentfrom those known for ferroelectrics. The basic reason forthis difference is the vectorial character of beam couplingin cubic (optically isotropic) photorefractive materials.12,13

In contrast to those of anisotropic ferroelectrics, the spa-tial changes of light energy and polarization cannot beseparated here from each other. Furthermore, the ab-sence of a polar axis makes cubic materials highly sensi-tive to the orientation of the electric field vector about theprincipal crystal directions.1,2

Considerable efforts made during the past decade tounderstand and improve the nonlinear photorefractiveproperties of cubic materials stimulated the developmentof the vectorial theory of beam coupling.14–20 The latestversion of this theory19,20 incorporates the properties im-posed by the spatial symmetry and the enhancementmechanisms described above as well as by the presence ofoptical activity and elasto-optic contributions to the cou-pling parameters. This theory recently permitted the de-scription of some angular distributions of scattered light

2003 Optical Society of America

678 J. Opt. Soc. Am. B/Vol. 20, No. 4 /April 2003 Filippov et al.

in ac-biased Bi12TiO20 (BTO) crystals.19,21 Here we applyit to describe the polarization properties of small-anglelight-induced scattering in BTO crystals. To the best ofour knowledge, polarization states of scattered waves incubic crystals were never previously analyzed theoreti-cally; this problem is actually more difficult and challeng-ing than the description of the angular intensity distribu-tions.

2. BASIC RELATIONSAssuming that the scattered waves are weak compared tothe pump beam propagating along the @ 110# crystal axis,we neglect pump depletion and consider the spatial am-plification of a scattered wave traveling at a small angleto @ 110#. We denote the vectorial complex amplitudes ofthe pump and scattered waves e and a, respectively; theygenerally are functions of propagation coordinate z. Thepump intensity is normalized such that ueu2 5 1. Thismeans that e can be treated as a unit polarization vector.Within the paraxial approximation, the light amplitudespossess only x and y components. Amplitude a 5 a(z) isgoverned by the ordinary first-order differentialequation19

da

dz2 i~k – s!a 5 QuE0 cos cu~a – e* !~n0 1 n – s!e.

(1)

Here uE0u is the amplitude of a square-wave ac field, c isthe azimuth angle that characterizes the propagation di-rection (see Fig. 1), k 5 (k1 , k2 , k3) and n5 (n1 , n2 , n3) are two known real three-dimensional(3D) vectors that characterize the linear and the nonlin-ear properties of the medium, respectively, s5 (s1 , s2 , s3) is the standard set of two-dimensional(2D) Hermitian sigma (Pauli) matrices,22,23 and, lastly, n0and Q are two known scalar parameters. The structureof Eq. (1) is adequate to the vectorial problem understudy. Note immediately the difference between the com-plex 2D amplitudes, such as a 5 (ax , ay), and the real3D vectors such as k. The 3D vectors are introduced forthe sake of mathematical convenience; they enter intoonly the scalar products, such as k – s [ k1s1 1 k2s21 k3s3 , which represent certain 2D matrices acting onthe vectorial 2D amplitudes.

Vector k is responsible for the linear optical properties.Its components k1,3 characterize the optical anisotropy in-duced by field E0 , whereas the component k2 5 2r (r isthe rotatory power) characterizes optical activity. In theabsence of the field the crystal is optically isotropic. Fur-ther, scalar n0 and vector n are responsible for the scalarand vectorial parts, respectively, of nonlinear coupling.These parameters depend on the orientation of scattering(grating) vector K (the difference between the wave vec-tors of the scattered and pump waves) about the crystalaxes [see Fig. 1(b)]. They are macroscopic in nature; i.e.,they have nothing to do with the processes of photoexci-tation, recombination, and charge transport. On the con-trary, real factor Q depends essentially on the transportproperties. To be more precise, it is the quality factor fora space-charge wave with wave vector K and, at the same

time, the enhancement factor for both ac and dc enhance-ment techniques.5,6,24 This factor is considerably largerthan one within a certain range of the grating vector, i.e.,for a certain range of the propagation directions of scat-tered waves. These directions, which usually correspondto small-scattering angles, are of interest chiefly for stud-ies of light-induced scattering. The presence of weaklydamped space-charge waves correlates strongly with therapidity of the photorefractive response of cubic crystals.

As we have neglected pump depletion, the governingequation for the pump amplitude, e 5 e(z), differs fromEq. (1) for a only in the absence of the right-hand side.This means that e changes only because of optical activityand the ac-field-induced optical anisotropy.

From now on we restrict ourselves to the so-called di-agonal configuration (E0 i @ 111#' @ 110#),25 distin-guished by the strongest nonlinear coupling.26,27 Fur-thermore, the relevant experimental characteristics areobtained just for this geometry. Here we have for k1,3

k1 5A2

A3sE0 , k3 5 2

1

2A3sE0 , (2)

Fig. 1. (a) Schematic of an experiment on light-induced scatter-ing. (b) Geometrical diagram. For diagonal geometry, c is thepolar angle measured from the @ 111# axis, z0 is the angle be-tween @ 111# and [001], and the propagation direction is @ 110#.


where s 5 2pn03r41 /l, n0 is the nonperturbed refractive

index, r41 is the only nonzero electro-optic coefficient, andl is the length of the light wave. Obviously, the compo-nents k1,3 alternate in time with E0 .

Coupling parameters n0 and n, in contrast to param-eters k1,3 , generally include not only electro-optic but alsoelasto-optic contributions16–18; the latter result from thedeformations produced by the space-charge field bymeans of the piezo effect. Furthermore, in specifyingthese parameters we should keep in mind that gratingvector K is not parallel to diagonal @ 111#. In otherwords, these parameters are functions of angle c betweenK and @ 111#. Lastly, component n2 is 0 in the generalcase, which means that the space-charge field does not af-fect optical activity. If we provisionally neglect theelasto-optic contributions, the nonzero coupling param-eters are

n0,3 5 6~s/2!cos~z0 1 c!, n1 5 s sin~z0 1 c!, (3)

where z0 5 arctan(A2) . 54.7° [see Fig. 1(b)]. For c5 0 we have k1 /n1 5 k3 /n3 5 E0 . The elasto-optic con-tributions modify the dependences n0,1,3( c). A relevantexample of such modification for BTO crystals is pre-sented in Fig. 2. One can see that components n0 and n3are subject to the strongest changes. Anyhow, the rela-tive changes do not exceed 40%. In the subsequent cal-culations of the scattering characteristics (Section 4 be-low) the elasto-optic contributions are taken intoconsideration.

Fig. 2. Dependences n0,1,3( c) for the diagonal geometry; thedashed curves are plotted for zero elasto-optic contributions.

The last parameter to be specified is enhancement(quality) factor Q. Within the conventional one-speciesmodel of charge transport,1,2 it includes mobility–lifetimeproduct for photoelectrons mt and effective trap concen-tration Nt . Otherwise it depends on the absolute valueof the ac field uE0u and on the K vector. For BTO crystalswe chose the following representative values of the micro-scopic parameters: mt 5 1027 cm2/V and Nt 5 33 1016 cm23. These values are within the range ofvariations that is typical of BTO crystals; they fit well theangular dependences of the light-induced scattering inour sample. Figure 3 exhibits the angular dependence ofthe factor Q 5 Q(K)ucos cu that enters into Eq. (1) foruE0u 5 17 kV/cm and the wavelength of light, l5 633 nm. For convenience we use azimuth angle c andpolar angle u instead of the components Kx and Ky . Po-lar scattering angle u is recalculated for air. One can seethat the maximum value of Q(u, c) occurs at umax' 5°, cmax 5 0,p. With increasing uE0u the value of umaxdecreases while the value of Qmax remains almost con-stant. Increasing ratio mt/Nt makes angle umax smallerbut does not change Qmax .24 The value of Qmax is deter-mined by the product mtNt . The features mentionedabove are important for understanding many features oflight-induced scattering.

To complete the background information we commentbriefly on the status of our theory. The conventional one-species model1 is used for the description of the photore-fractive ac response. The amplitudes of the scatteredwaves are assumed to be small compared with the pumpwave’s amplitude. With respect to other relevant as-pects, our basic equations are quite general. They incor-porate the electro- and elasto-optic contributions to opti-cal permittivity and optical activity. They include thevectorial character of beam coupling in cubic photorefrac-tive crystals, i.e., the possibility of a change in the polar-ization of light. In this sense the vectorial theory is ageneralization of the scalar theory that is applicable to bi-refrigent media. An application of the scalar theory tothe description of light-induced scattering can be found inRef. 28.

Fig. 3. Contour plots Q(u, c) 5 constant for uE0u 5 17 kV/cm,mt 5 1027 cm2/V, and Nt 5 3 3 1016 cm23. The filled circlesmark the positions of two symmetric maxima.


3. LINEAR PROPERTIES OF LIGHT WAVESThe linear optical properties determined by 3D vector kare of importance for what follows. Within the linear ap-proximation we have for pump amplitude e

de

dz5 i~k – s!e. (4)

The same linearized equation for a(z) follows from Eq.(1). To find the optical eigenmodes we search the solu-tion of Eq. (4) in the form e } exp(id kz), where dk is thewave vector measured from its nonperturbed value k05 2pn0 /l. Then we arrive at the 2D eigenproblem:

~k – s!e 5 dke. (5)

From this equation we can find two eigenvalues for (dk)6

and two corresponding eigenpolarization vectors e6 . Be-cause matrix k – s is Hermitian, its eigenvalues are real.Using the known properties of the s matrices,22,23 wehave immediately (dk)6 5 6k, where k 5 uku. The dis-tance between the wave surfaces is therefore 2k. For thediagonal configuration we have 2k 5 (3s2E0

2 1 4r2)1/2.The eigenpolarizations that correspond to eigenvalues

6k are generally elliptical; one of the ellipses extendsalong @ 111# (parallel to E0), another, along [112] (perpen-dicular to E0). The degree of ellipticity (i.e., the ratioshort axis/long axis) in both cases is the same, uru/(k1 A3usE0u/2). In the limit uE0u @ Ec [ 2uru/A3usu theeigenmodes are polarized linearly; in the laboratory coor-dinate system the corresponding polarization vectors aree1 5 (1, 0) and e2 5 (0, 1); in experiment they are usu-ally horizontal and vertical, respectively. This experi-mental configuration is of practical interest for BTO crys-tals at l . 630 nm. The rotatory power is fairly smallhere, uru . 6.5 deg/mm, so the characteristic field is Ec. 3.2 kV/cm. The values of the ac fields used in the ex-periment are typically much higher. In other words, theapplied field strongly suppresses optical activity, whichcan actually be neglected.29

Vectorial equation (4) admits of the general solution19

e~z ! 5 exp@i~k – s!z#e0

5 @cos kz 1 ik21~k – s!sin kz#e0 , (6)

where e0 5 e(0) is the input pump polarization vector.If it is chosen to be an eigenvector, e0 5 e6 , no polariza-tion changes occur during propagation. This means, inparticular, that neither the vertical nor the horizontal in-put polarization is subject to any strong changes owing tolinear effects for uE0u @ 3.2 kV/cm.

4. CALCULATION OF SCATTERINGCHARACTERISTICSAs we have seen, the vectorial pump amplitude e that en-ters into Eq. (1) is not constant in the general case; itchanges because of the linear optical effects. This gener-ally happens to probe amplitude a. The linear effects donot directly influence the beam intensities. They changethe polarization state, however, and in this way affect thepolarization-sensitive nonlinear coupling.

Within the vectorial theory we can get rid of the linearterms by switching from vector a to the new vectorial am-plitude b:

a 5 exp@i~k – s!z#b. (7)

This procedure is similar to the transition above frome(z) to e0 [see Eq. (6)]. As matrix k – s is Hermitian,transformation (7) is unitary; it changes neither the val-ues of the 2D vectors nor their scalar products. It isequivalent to the so-called interaction representation inquantum mechanics.22

After the unitary transformation we have instead ofEq. (1)

db

dz5 ~b – e0* !~q0 1 q – s!e0 , (8)

where q0 5 QuE0 cos cun0 is a real constant and q is a real3D vector:

q 5 QuE0 cos cuFk~n – k!

k2 1 Fn 2k~n – k!

k2 G3 cos 2kz 2

~n 3 k!

ksin 2kzG . (9)

This vector depends on propagation coordinate z, whereinthe oscillating terms (presented in the second line) aredue to the fact that k and n are not parallel to each other,i.e., the eigenvectors of the operators k – s and n – s aredifferent. The spatial frequency of the oscillations is 2k.

The problem of calculation of scattering characteristicsis reduced now to two different tasks: calculation of in-tensity distributions for different pump polarizations andcalculation of polarization properties of scattered light.To solve the first task it is sufficient to find b(z, u, c).Then the value ub(z0)u2 [ ua(z0)u2, where z0 is the crystalthickness, defines the angular distribution of the light-induced scattering. The second task is more difficult.The direction of vector b(z) is more sensitive to the modelassumptions than to its absolute value. Furthermore, itis necessary to perform unitary transformation (7) fromb(z0) to a(z0) to find the output polarization.

A. Intensity DistributionsThe spatially oscillating terms in Eq. (9) are of minor im-portance when there is strong spatial amplification. Thepoint is that the presence of the spatially uniform contri-bution to q and of the spatially uniform parameter q0gives rise to exponential growth of b with increasingpropagation distance z, whereas the oscillating contribu-tions to q cannot produce any permanent spatial growth.Furthermore, the amplitudes of the oscillating terms inEq. (9) are often relatively small for the actual angles uand c. For these reasons we neglect these terms in theleading approximation. Then we have, from Eq. (8),

b ' QuE0u~e0* • b0!exp~Gz!

GFn0 1

~n – k!~k – s!

k2 Ge0 ,

(10)

where b0 5 b(0) is the seed amplitude and G is the rateof spatial amplification, also called the increment. Thegeneral expression for G can be presented as follows:


G 5 QuE0 cos cu@n0 1 k22~n – k!~k – j0!#, (11)

where j0 5 e0* :s :e0 [ ^e0usue0& is the real 3D vectorthat characterizes the pump state and is often called theStokes vector.30 The descriptions of the pump polariza-tion by j0 and e0 are equivalent; they differ only in degreeof convenience. The absolute value of Stokes vector j0for a totally polarized pump wave equals unity.

The dependence of the increment on polar scatteringangle u is defined by the factor Q (Fig. 3). The depen-dence of G on azimuth angle c comes from factor Q andalso from parameters n0 and n (Fig. 2). Owing to thegeneral properties of parameters n and n0, we haveG( c) 5 2G( c 6 p); in other words, the increment is anodd function of scattering (grating) vector K. This fea-ture is indeed a generalization of the known property ofthe scalar spatial amplification, which is caused by a non-local photorefractive response, to the vectorial case.1 In-crement G does not depend on the polarization of the testbeam. That polarization influences only the preexponent(e0* • b0); most probably, e0ib0 , so ue0* • b0u 5 ub0u.

Let us consider several representative cases of pumppolarization. For horizontal and vertical polarization wehave k – j0 5 6k, respectively, so the bracketed notationin Eq. (11) is simply @n0 6 (n – k)k21#. Figures 4a and4b show the corresponding angular dependences of the in-crement [in the region of its positive values] for uE0u5 17 kV/cm and the accepted BTO parameters. Forhorizontal (1), polarization the dependence G5 G1(u, c) is characterized by a pronounced horizontalright-hand lobe. The maximum rate of spatial amplifica-tion (G1

max ' 48 cm21) takes place at c 5 0, u . 5°. Thelight-induced scattering is strongest in this case.19 Forthe vertical (2) polarization the distribution G2(u, c) isquite different; it possesses one tilted left-hand lobe, atc ' 150°. The maximum value of the increment is no-ticeably smaller here, G2

max ' 27 cm21.Next we calculated the gain factor, G

5 (2z0)21 ln@ua(z0)u2/ua(0)u2#, by solving numerically theinitial vectorial equation (1) for the thickness z0 5 1 cmand the same material and experimental parameters.We found that the angular distributions of increment Gand of gain G differ from each other only in fine details.This proves that the oscillating terms in q are of minorimportance and that increment G is really a useful char-acteristic of the intensity distribution for light-inducedscattering.

Figures 5a and 5b show intensity distributions forlight-induced scattering, recorded from an observationscreen placed in the far field, for e0i@ 111# and e0'@ 111#.In our experiment we employ a 9.94-mm-thick sample,the amplitude of the ac field [applied along the diagonal@ 111#] is '17 kV/cm, the typical pump intensity is '0.8W/cm2, and the ac frequency is '70 Hz. The setup usedis similar to that described in Ref. 19. One can see thatexperiment gives similar angular distributions of scat-tered light. The azimuth positions of the lobes and of po-lar angle umax are in good agreement with theory.

To analyze the effect of pump polarization on the scat-tering characteristics in more detail, we considered rightand left circular pump polarization and also two kinds of

linear polarization for polarization angle w0 5 /e0 ,@ 111# 5 645°. The main theoretical prediction is thatincrement G is almost the same for these four cases:

G . ~G1 1 G2!/2 5 QuE0 cos cun0 . (12)

To prove this assertion we mention that for all four casesthe pump polarization vector can be represented as e05 c1e1 1 c2e2 , with uc6u . 1/A2. This is true becauseeigenvectors e6 are parallel and perpendicular to the@ 111# axis for uE0u @ 3.2 kV/cm. Therefore we have(k – s)e0 5 k(c1e12c2e2), which is perpendicular to e0 ,and from Eq. (11) we immediately get the necessary re-sult. Note that the suppression of optical activity by theapplied field is crucial for the validity of expression (12).

Figure 4c shows the angular distribution G(u, c) givenby expression (12) for uE0u 5 17 kV/cm. It is the sum ofthe distributions presented in Figs. 4a and 4b. Here wehave two scattering lobes. The strongest one is tilted by'15° to the horizontal. It originates indeed from thehorizontal lobe of increment G1(u, c); the tilt is causedby the influence of the negative lobe of G2(u, c), whichcan be obtained by a p rotation of the positive lobe shownin Fig. 4b. The weakest lobe in Fig. 4c originates fromthe tilted lobe shown in Fig. 4b; its increasing tilt to thehorizontal is caused by the influence of the negative hori-

Fig. 4. Angular dependences of the increment for uE0u5 17 kV/cm and the accepted parameters of BTO crystals: (a),(b), (c) correspond to horizontal, vertical, and circular pump po-larization, respectively. Polar angle u is recalculated for air.


zontal lobe of G1(u, c) (situated symmetrically to the lobeshown in Fig. 4a). The maximum values of G for thequasi-horizontal and the tilted left lobes in Fig. 4c are.30 and .8 cm21, respectively; they are considerably

Fig. 5. Experimental distributions of scattered light for uE0u5 17 kV/cm for the diagonal geometry: (a), (b), (c), (d) corre-spond to horizontal, vertical, 145°, and (1) circular pump polar-ization, respectively. The scattering distributions for 245° and(2) circular polarization are not visually different from thoseshown for (c) and (d).

smaller than the corresponding values for Figs. 4a and 4b.This is, indeed, due to the partial compensation mecha-nism described above.

Direct simulation of vectorial equations (7) and (8) hasshown that the spatial distributions of G5 (2z0)21 ln@ua(z0)u2/ua(0)u2# for the (6) circular and the645° pump polarization are in good qualitative agree-ment with the distribution presented in Fig. 4c. At thesame time, quantitative distinctions become noticeablehere. Figures 6a and 6b show the angular dependence ofgain factor G for right and left circular polarization, re-spectively. These distributions are slightly different, en-tirely because of the influence of optical activity. Thegreater influence of optical activity than for eigenpumppolarization is caused by partial cancellation of the big-gest contributions to G that come from G1 and G2 .

Experiment does not show any noticeable difference be-tween the intensity distributions for the four choices ofpump polarization described above. Figure 5c shows arepresentative distribution obtained for the left circularpolarization at uE0u 5 17 kV/cm. In accordance withtheory, there is a slightly tilted main right lobe and a con-siderably weaker, strongly tilted left lobe. The angularpositions of these lobes are c ' 5°, 135°, respectively.These numbers are slightly different from the model pre-dictions. The intensity of the scattered wave is, as ex-pected, considerably lower than that for the cases shownin Figs. 5a and 5b. It should be understood that thepump is essentially depleted in the available 9.94-mmsample; i.e., the spatial growth experiences saturation.This makes quantitative estimates of the total scatteringintensity unreliable but can hardly strongly affect the an-gular and polarization scattering properties.

Fig. 6. Dependence G(u, c) for (a) (1) and (b) (2) circular pumppolarization. The maximum values for the main lobe are .18.2and 22.9 cm21, whereas for the secondary lobe they are .9.0 and.5.9 cm21.


B. Scattering PolarizationThe polarization properties of scattered light seem to bemore sensitive to the choice of experimental and materialparameters than the intensity distributions are. This isespecially true with respect to the weakest lobes consid-ered above.

Because the simplified model that ignores the spatiallyoscillating contributions to q but includes the elasto-opticcontributions to G gives satisfactory results for the inten-sity distributions, it is natural to apply the model to thedescription of the output polarizations. Therefore wehave to calculate b(z0) from expression (10) and performunitary transformation (7) to find a(z0) as a function of uand c. The most important propagation directions corre-spond indeed to the maxima of increment G.

Let us revisit first the cases of horizontal and verticalinput pump polarization. Here e0 is the eigenvector ofoperator k – s with good accuracy (we assume again thatuE0u @ Ec . 3.2 kV/cm). Therefore b(z0) is almost par-allel to e0 . The unitary transformation does not changethe eigenvector directions. We come therefore to the con-clusion that for the main horizontal lobe presented in Fig.4a the scattering polarization has to be horizontal and forthe tilted left lobe of Fig. 4b it must be vertical. Experi-mental polarization measurements confirm this predic-tion with high accuracy.

In the case of mixed pump polarization it is useful torepresent the amplitude b in the form b 5 b1(z)e1

1 b2(z)e2 . Then for all four cases considered above theratio of horizontal to vertical intensity components is thesame:

ub1 /b2u2 5 ~G1 /G2!2. (13)

This ratio depends on azimuth angle c and does not de-pend on polar angle u. Let us estimate it first for themain scattering lobe of Fig. 4c. At the maximum of G(u,c) we have ub1 /b2u2 ' 102. This means that vectorb(z0) at this maximum is almost horizontal, as is vectora(z0). Simulation of vectorial equation (8) with the os-cillating terms and optical activity included followed byunitary transformation (7) from b(z0) to a(z0) has giventhe range (3 –23) 3 102 for the ratio of horizontally tovertically polarized intensity components for the cases ofmixed pump polarization described above. In this waywe expect from theory that the polarization of the quasi-horizontal (strongest) lobe will be almost horizontal$a(z0)i@ 111#% for the (6) circular and 645° pump polar-ization. Experimental measurements confirm this resultwith high accuracy.

We turn last to the weakest lobe of Fig. 4c. Here thesimplified model (we neglect oscillating terms and opticalactivity) gives the estimate ub2 /b1u2 ' 3 for the corre-sponding maximum of the increment. This number can-not be considered big enough that we can expect quasi-vertical polarization of the scattering lobe. More-accurate numerical calculations based on Eqs. (7)–(9)have shown that the intensity ratio of vertical/horizontalcomponents ranges from '1.6 to '12.5 for mixed pumppolarization. This theoretical prediction has found only aqualitative experimental confirmation. Experiment

shows that the polarization of the weakest (tilted) lobe isvertical with good accuracy for (6) circular and 645°pump polarization.

5. CONCLUDING REMARKSWe have applied vectorial theory to describe the angulardistributions and polarization properties of light-inducedscattering in cubic ac-biased BTO crystals for several po-larization states of the incident pump beam. Diagonalgeometry, distinguished by the strongest vectorial cou-pling, was chosen for our comparison of theory and ex-periment.

We have found that a large variety of angular intensitydistributions can be described uniformly within a rela-tively simple and transparent vectorial model of spatialamplification without extensive use of numerical simula-tions. The measured scattering distributions are in goodquantitative agreement with theory. The photoelasticcontributions to the coupling constants affect mostly theabsolute values of the rates of spatial amplification.

For the strongest lobes of light-induced scattering, thetheory also allows for a simple and satisfactory descrip-tion of their main polarization properties. The polariza-tion properties of the weakest scattering lobe are found tobe sensitive to the model assumption made. At thispoint, there is only qualitative agreement between experi-ment and theory (including numerical simulations).

The results obtained are of interest for the use of fastphotorefractive materials for various interferometricapplications31–34 and also for characterization ofscattering.35,36

ACKNOWLEDGMENTSFinancial support by the German Research Council(Graduate College 695 ‘‘Nonlinearities of optical materi-als’’ and the Emmy-Noether program) is gratefully ac-knowledged. M. Shamonin acknowledges financial sup-port from the Scheubeck-Jansen Stiftung.

*K. H. Ringhofer died December 26, 2002.

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