772 OPTICS LETTERS / Vol. 18, No. 10 / May 15, 1993

Fanning effects in photorefractive crystals

Yan-Hua Hong, Ping Xie, Jian-Hua Dai, Yong Zhu, Hua-Guang Yang, and Hong-Jun Zhang

Institute of Physics, Academia Sinica, Beijing 100080, China

Received October 14, 1992

A model to describe the fanning effect based on the beam-coupling mechanism in photorefractive crystals is pre-sented. The intensity distribution of the fanning beam in a 45'-cut BaTiO3 crystal is measured experimentally.The results show good quantitative agreement with theory.

In photorefractive crystals with a large two-beam-coupling coefficient, for example, BaTiO3 and stron-tium barium niobate crystals, beam fanning is strong.It is a decisive factor in permitting self-pumpedphase conjugation1'2 and has applications such asin optical limiters.Y4 However, in some cases beamfanning is undesirable; for example, it may preventhigh-gain signal amplification in two-wave mixing inthick photorefractive crystals.5 6 As a consequence,various methods to suppress beam fanning have beenproposed.7 -9 The existence of beam fanning canthus greatly influence the optical nonlinear couplingin photorefractive crystals. Various studies of themechanism of beam fanning have been reported.10-'3Feinberg proposed a self-defocusing mechanism forthe fanning effect in which the Gaussian shapeof the beam causes the fan.1' Another mechanismis that beam fanning is the amplification of thescattered noise by two-beam coupling.'12"3 However,the calculations are restricted to the couplingbetween only one direction of the scattered beamand the incident beam. In this Letter we presentan improved model to describe the fanning effectbased on the beam-coupling mechanism, in whichvarious directions of the fanning beam are takeninto consideration. We also present, for the firsttime to our knowledge, experimental results of theintensity distribution of the fanning beam in 45°-cut BaTiO3 crystal. The experimental results are ingood agreement with the theoretical results.

As shown in Fig. 1, the extraordinary-polarizedplane wave, I, is incident upon the crystal surfaceat an incident angle 0' with respect to the normal(along the z axis) to the crystal surface. The incidentlight I will produce light scattered from defects orimperfections within the photorefractive crystal inevery direction. The scattered light interferes withincident beam I, and, because of beam coupling inthe photorefractive crystal, the scattered light obtainsenergy from incident light I, finally forming a broadfan-shaped beam. Assuming that the intensity dis-tribution density per radian of the fanning beamis i(a), where a is the angle of i(a)da elementwith respect to the z axis inside crystal, consideringthat beam overlap of different i(a)da elements isvery small, i.e., the effective interaction length isvery small, and, for simplicity, neglecting the beam-coupling between different i(a)da elements of the

fanning beams, we can obtain the following coupled-wave equations:

U = - I ' y(t) Ii(a) ddz _-f7/2 Io

- f/2f(a)Id ,-v/l2

di(a) cos 0 Ii(a)~~ ~~y (a)' + f(a)I,dz cos a Io

7r/2

Io = I(z=0) = I + 2 i(a)da.J-,l

(1)

Here Io is the total light intensity in the crystal, 0 isthe angle of beam I with respect to the z axis insidethe crystal, and f (a) is a phenomenological fanningscattering coefficient describing the scattered lightfrom incident beam I into the fanning beam in the adirection. We assume f (a) to be constant here, i.e.,f (a) = f = constant [the calculated results show thatdifferent forms of function f (a) have little effect onthe intensity distribution of the fanning beam]. y(a)is the two-beam-coupling coefficient between incidentbeam T and the i(a)da element of the fanning beam,with a value

y ) 2 ec cos 0 ESC(a)reff(a),

z

(2)

D

C(+45")

C(-45-)

Fig. 1. Scheme for illustrating beam fanning.

0146-9592/93/100772-03$5.00/0 © 1993 Optical Society of America

May 15, 1993 / Vol. 18, No. 10 / OPTICS LETTERS 773

where a) is the optical frequency, c is the speed oflight, and ne is the extraordinary index of refraction.In the absence of applied or intrinsic electric fields,the space-charge electric field is given by

(3)Esc.(a) = kBT Kg(a) -e * ea,q 1I+ Kg2 (a)/Ko 2

where kBT is the thermal energy, q is the charge ofthe mobile charge carriers, and Kg(a) is the magni-tude of the grating wave vector formed by beams Iand i(a)da. e and ea are unit polarization vectors ofbeams I and i(a)da, respectively, and Ko is a constantof the material that depends on the number densityN of available mobile charge carriers:

K° [ e(a)eokBT ]

laser is incident upon the 450-cut BaTiO3 crystal.The output fanning intensities are measured by aNewport Laser powermeter (Model 835), the detectorlocated on a precise rotating plate, the acceptanceangle of the detector receiving area being -0.01 rad.The results are shown in Figs. 2 and 3; the Fresnel

(a)

10

8

6

4(4)

2

where e(a)eo is the dielectric constant at low fre-quency (1 kHz) along the direction of the gratingwave vector Kg(a). For extraordinary polarizationthe effective electro-optic coefficient reff (a) is

reff (a) = 24 { r1 3[cos(da- 0) - cos 2,/] + 4ne2 no2 r42

x sin2 /3 + ne4r33[cos 2,8 + cos(a - 0)]}cos /,B (5)

where n. is the ordinary index of refraction, rij arethe elements of the linear electro-optic tensor, and,8 is the angle between the c axis and the gratingwave vector Kg(a), which is a function of a. ForBaTiO3 crystal at A = 0.514 ,gm with n0 = 2.488and ne = 2.424, the number density N of availablecharge carriers is assumed to be 8 x 1022 m-3 andthe charge is assumed to be q = + 1.6 x i0-'9 C. Thethermal energy kBT = 4.14 x 10-21 J. The nonzero,linear electro-optic coefficients are r13 = 8, r33 = 28,and r42 = 820, in units of picometers per volt. Therelative dielectric constants parallel and perpendic-ular to the c axis are ell = 135 and e1 = 3600,respectively. With the equations and parametersgiven above, the intensity distribution of the outputfanning beams i(a')/Io, in units of inverse radians,in a ±450-orientation BaTiO3 crystal are calculatedby the combination of the Runge-Kutta method andthe trapezoidal integration method, as seen in Fig. 1,where a' is the angle of the fanning beams outside thecrystal. Some of the results are presented in Figs. 2and 3. The fanning scattering coefficient f in the fig-ures is assumed to be 1 x 10-5 m'1; for this choice off value the calculated output intensity of the incidentbeam is approximately the same as the one obtainedfrom experiment. For different choices of f, the half-widths and the maximum positions of the fanningbeam distribution are identical, and only the maxi-mum intensities are different (the larger the valueof f, the larger the maximum intensity, and viceversa). To confirm the theory, we measured the out-put fanning beam intensities for various angels of a'in a 450 -cut BaTiO3 crystal of dimensions 6.00 mm X6.15 mm X 5.81 mm (L = 5.81 mm). The experi-mental setup is shown in Fig. 1. The extraordinary-polarized beam I, which has a beam diameter of-2 mm, from a single-frequency (A = 0.5145 um) Ar'

0

(b) 12

10

10 0 10 20 30 40 50 60 70 80 90a'(degree)

:o ofj v off~

-10 0 10 20 30 40 50 60 70 80 90

a'(degree)

Fig. 2. Intensity distribution of the fanning beam in+450 -orientation BaTiO3 crystal. The solid curves corre-spond to the theory; the circles and triangles correspondto experiment. (a) Curve 1, 0' = -10°; curve 2, 0' = 10°.(b) Curve 1, 0' = 00; curve 2, 0' = 20°.

12 I '

10 08~~~6~~~~

2 2

4V V

2

-10 0 10 20 30 40 50 60 70 80 90a'(degree)

Fig. 3. Intensity distribution of the fanning beam in+450 -orientation (curve 1 and circles) and -450 -orienta-tion (curve 2 and triangles) BaTiO3 crystals. The solidcurves correspond to theory; the circles and trianglescorrespond to experiment, with 0' = 00.

I

774 OPTICS LETTERS / Vol. 18, No. 10 / May 15, 1993

losses have been considered. From the figures wesee that near the incident angle, the experimentalresults are a little larger than that from theory.This is because in the neighborhood of the incidentdirection the measured intensities include not onlythe fanning beam but also the Rayleigh forwardscattering of the incident beam. Except for this,the theoretical curve matches the experiment well.This indicates that neglecting the coupling betweendifferent i(a)da elements of the fanning beam ispermissible in a practical situation. From the figurewe can conclude that the intensity distribution ofthe fanning beam is not too wide, with its half-width being approximately 100 for a +450 -orientationBaTiO3 crystal and approximately 150 for the -45°orientation. When the incident angle 0' increases,the direction of the maximum intensity of the fanningdistribution tends toward a larger a', whereas the an-gle between the incident direction and the maximumintensity direction of the fanning distribution staysroughly the same, i.e., approximately 30°. For the-450 orientation, the maximum intensity directionof the fanning distribution tends toward larger a',and its half-width is larger than that for the +45°orientation, and the incident beam depletion causedby beam fanning is a little larger, too.

It should be pointed out that, owing to the finitediameter of the incident beam and for thick photore-fractive crystals, the effective interaction length willbe reduced, but this has no effect on the half-widthand the maximum position of the fanning intensitydistribution, and only the maximum intensity will bereduced.

In conclusion, we have presented a more rigorouslytheoretical model to calculate beam fanning basedon the beam-coupling mechanism in photorefractive

crystals, in which various directions of the fanningbeam are considered. The output intensity distri-bution of the fanning beam has been measured fora 450-cut BaTiO3 crystal. The calculated results forthe BaTiO3 crystal are in good agreement with theexperiment. The fanning effect on two-wave mixingis under investigation now.

This research was supported by the Chinese Na-tional Science Foundation and the Nonlinear ScienceProject of China.

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