April 1, 1992 / Vol. 17, No. 7 / OPTICS LETTERS 475

Large thermal nonlinearities and spatial self-phasemodulation in SrxBai-xNb2O6 and BaTiO3 crystals

Moshe Horowitz, Ron Daisy, Ofer Werner, and Baruch Fischer

Advanced Opto-Electronics Center, Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel

Received August 15, 1991

We find strong optical nonlinearities in SrxBal1 xNb2O6 (x = 0.61 and x = 0.75) and in BaTiO3 crystals causedby thermally induced changes of the refractive indices. This gives strong spatial self-phase modulation andself-focusing of Gaussian light beams and light-induced birefringence. The light intensities are of the order of102_103 mW/mm2, and the time response of the effect is of the order of milliseconds.

Sr.Ba,-..Nb 2O6 (SBN) and BaTiO3 crystals areprobably the most intensively studied nonlinear pho-torefractive crystals."2 The standard photorefrac-tive mechanism, based on space-charge fields thatinduce index of refraction changes, benefited fromthe large electro-optics coefficients of these crystalsat room temperatures. The strongly enhanced dcsusceptibility arises from being close to the phasetransition temperature in the crystals.3 This isalso the reason for the strong dependence on tem-perature of the susceptibility and the refractive in-dices of SBN3 '4 and BaTiO3.5

Here we report on a strong nonlinear effect inSBN and BaTiO3 crystals that has a thermal origin.It is shown to give strong spatial self-phase modula-tion and self-focusing with a Gaussian light beamtraversing the crystals along a path of 2-5 mm witha power of the order of 102_103 mW/mm2. This ef-fect has not been reported yet in SBN and BaTiO3crystals, although its mechanism has a stronger ca-pability of changing the refractive indices comparedwith the usual photorefractive effect and might bean important factor in wave-mixing experiments.Strong dependence of the refractive indices on tem-perature has been reported for these crystals.4'5Thermal lensing has been observed recently6 in an-other crystal (Ba2NaNb5O15), in liquid crystals,7 andin liquids.8 We also demonstrate light-induced bire-fringence that is due to polarization dependence ofthe thermal effect.

In the experiment we used several SBN (x = 0.61and 0.75 and various nominal doping) and BaTiO3crystals. Most of them were strongly photorefrac-tive, exhibiting large beam-coupling capability. Alight beam from an argon-ion laser was shined di-rectly or through a focusing lens onto the crystal.The beam direction was normal to the input surface,and the c axis of the crystal was along one of thesesurface edges. The self-focusing effect, shown inFig. 1, is obtained with SBN (x = 0.61, nominallydoped with 0.05 wt. % Ce, and with an intensity ab-sorption coefficient of 1.1 cm-'). The effect isstrong for an extraordinary polarization and weakerfor an ordinary polarization.

We believe that the origin of the self-focusingeffect is thermal. We rule out regular photorefrac-tive,"2 photovoltaic, or photoabsorptive9 origins forour self-focusing effect: the symmetrical rings ofthe diffraction in the present experimental geometryof the crystal and the beam cannot be explained bythem. The fact that the self-focusing process is sostrongly dependent on the input intensity, as dis-cussed below (see Fig. 2), is also not in accordancewith these mechanisms. Moreover we observed thesame self-focusing effect even in samples that werenonphotorefractive. On the other hand, we knowthat SBN and BaTiO3 crystals can give large changesof the index of refraction4'5 as they are heated or illu-minated by a laser beam. We were also able to af-fect strongly the diffraction pattern by changing thethermal environment and boundary conditions.When we put a heat sink on the crystal surfaceswe observed a decrease in the ring number andstructure.

The thermal nonlinearity is not a local Kerr-likeeffect since it involves heat diffusion. For an exactanalysis of the effect, we have to find the space- andtime-dependent temperature distribution in thecrystal driven by the heat source of the Gaussianlight beam. This can be done by solving the heatdiffusion equation. An explicit solution depends onthe specific boundary conditions and can be compli-cated. An analysis of a simplified case is given inRef. 8. In our experiment, the crystals were slabswith a width of 2-4 mm and surfaces with sides of5-8 mm. The heat relaxation time is roughly givenby6810 7 _ pCl 2/(4K), where I is a characteristicdimension, p is the mass density, K the thermal con-ductivity, and C the specific heat at constant pres-sure. Thus the time scale is composed from theflow in the z direction (along the light path), with thetwo surfaces (input and output) as quasi-isotherms,and in the plane that is transverse to the beam direc-tion, with the relatively large crystal cross sectioncompared with the small beam cross section. Infact, the nonperfect circular structure of the rings(seen in Fig. 1) is a result of the nonsymmetricalstructure of the crystal (rectangular with one pair of

0146-9592/92/070475-03$5.00/0 ©0 1992 Optical Society of America

476 OPTICS LETTERS / Vol. 17, No. 7 / April 1, 1992

Fig. 1. Output beam from the SBN crystal for the follow-ing input-beam powers and conditions: (a) 3.5 mW andextraordinary polarization, which give a spot diameter of7 mm at the detection screen; (b) 224 mW and extraordi-nary polarization, which give a spot diameter of 135 mmat the detection screen; (c) 318 mW and extraordinary po-larization, which give a spot diameter of 360 and 290 mm(two axes of the elliptical shape); (d) 318 mW and ordinarypolarization, which give a spot diameter of 35 mm at thedetection screen. The input beam was focused onto thesample with a lens of 44-cm focal length, and the crystallocation was 47.5 cm from the lens, which give a spot sizeof approximately 0.2 mm at the crystal. The detectionscreen, from where the pictures were taken, was 118 cmfrom the crystal.

z

x

35 x

30 x

25

20 x15~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~15 x

10x x

5L x

0 0 50 1 00 150 200

P [mW]250 300 350

Fig. 2. Number of rings versus input-beam power (withthe conditions of Fig. 1).

its edges coated with metallic electrodes that radiateheat and change the thermal boundary conditions).Another complication may arise from the expectedchange of K owing to photoionization of carriers tothe valence band.

The ring pattern is caused by spatial self-phasemodulation owing to the nonlinear index change in-duced by the nonuniform Gaussian profile of thelight beam. The ring pattern in the far field can becompared with the theory in Ref. 8, which results ina good agreement. A rough estimate of Ref. 7, forthe change of the nonlinear index in the beam centerby the number of rings N is given by (Anel) = NA.For the specific example of SBN shown in Figs. 1(c)and 1(d), we have for the extraordinary polarization

N 40. Then with a wavelength A = 514.5 nmand a crystal width of I = 2 mm, we obtainAnle = 0.01. For an ordinary polarization with thesame conditions, we have N = 3 and An, 0.0006.These changes in the indices are similar to reporteddata4 for the temperature dependence in SBN. Weobserved the effect in several SBN crystals. Thestrongest self-focusing was achieved for SBN withx = 0.61, nominally doped with 0.1 wt. % of Ce,1' butthe pictures in Fig. 1 were taken for 0.05 wt. % ofCe.11 In our BaTiO3 samples the self-focusing ef-fect was weaker. Typically we obtained N = 2.5with direct illumination (without any lens) of -1 Wfrom an argon laser beam.

The number of rings was strongly dependent onthe input-beam intensity, as shown in Figs. 1 and 2.The expected8 linear dependence on intensity is en-hanced and becomes nonlinear when the intensityincreases because the index of refraction changesfaster 4 as the crystal heats up and the temperatureapproaches the phase transition. When the intensitywas raised beyond a certain value (of approximately350 mW in this experiment), we saw a breakdown ofthe focusing. We attribute this behavior to an in-crease of the local temperature to values that arenear (or above) the phase transition, such that thecrystal partially looses its uniform, single-domainstructure. The phase-transition temperature de-pends3'4 on x and on the doping and is in the range of50-90TC. Sometimes, at the high-input-intensityregime, we saw a dramatic continuous increase inthe rings number with an increasing self-focusing inthe crystal (although the input was held constant),until the crystal was damaged and changed its uni-formity. We also note that the self-focusing effect isstronger as the crystal doping and absorption werehigher, as expected from the thermal origin ofthe effect.

We measured the buildup behavior of the self-focusing by plotting the intensity of the central partof the far-field diffraction of the output beam (itpassed through an aperture). This is shown inFig. 3. The time constant is of the order of millisec-onds, and it depends approximately on the inputpower P as r OC p-2 3 . We observed, however, a muchslower change in the ring pattern in the time scaleof seconds to minutes. This is in accordance withthe formula for r given above. This is also seen anddiscussed in the next experiment, in which we ex-amine light-induced birefringence. We therefore

n30 0''''''6o ' 3b0

P [mW]

Fig. 3. Dependence of the time response on the inputbeam power (with the conditions of Fig. 1 with an aperturein front of the detector).

2o10 x

xx

- x

~~ ~xxx

1: xx

x

,3 ....

I

1

April 1, 1992 / Vol. 17, No. 7 / OPTICS LETTERS 477

d=,

U 20 40 60

0 t[sec] 0.480 100 120 140 160 180

t Isec I

Fig. 4. Time dependence of the output intensity after thebeam passes through an analyzer (the input beam was 45°to the crystal axes). The inset shows the initial responsein Fig. 4 (curve a), which is the time dependence of theoutput intensity after the beam passes through an aper-ture and an analyzer, the response without the output po-larizer (curve b), and the response without the outputpolarizer with extraordinary input polarization (curve c).

think that there are two time scales in the buildupof the temperature and indices changes. A part ofthe index change occurs in the short period of sev-eral milliseconds, and we can view it as local heatingby the light beam. Then a heat flow with a muchslower time scale causes a further change. Thispart strongly depends on the bulk geometry.

In another experiment, we examined the differ-ence in the nonlinearity of the ordinary and extra-ordinary polarizations with refractive indices n0and n,, respectively. This gives a light-inducedbirefringence through a thermal effect. The input-beam polarization was at 45° with the c axis. Theintensity-dependent retardation caused a change ofthe polarization state. We were able easily to ob-serve retardations of several periods (multiples of270). This can be nicely shown in the buildup of theretardation for a Gaussian beam when we measurethe time dependence of the output beam after itpassed an analyzer (a polarizer with a perpendicularaxis with respect to the input polarization) and anaperture (to detect only the central part of the dif-fracted beam). We found an expected oscillatorybehavior, shown in Fig. 4, as a result of the rotationof polarization (in time) of the beam that passes thecrystal. The input power was 135 mW. The oscilla-tion period increased in time, corresponding to aslowing down in the buildup of the retardation. Theoscillations can be compared to the changes of thering pattern in the former experiment. This re-veals that after the initial quick buildup of the self-focusing pattern there is a further slow change inthe ring pattern that is related to the oscillations inFig. 4. The initial rapid response, shown by curve ain the inset of Fig. 4, corresponds to the formationof the ring pattern of the former experiments. Thefast decrease in the intensity is a result of the aper-ture that blocks diffracted light that is due to theself-focusing. In curves b and c of Fig. 4, we see thecorresponding fast response in which the output

polarizer was removed and when the input had anextraordinary polarization, respectively. In thesetraces we again see the two time scales discussedabove. A time dependence of the refractive indexand also the oscillations in the reflectivity of a self-pumped phase-conjugate mirror that resemblesthe above slow response was recently observed inBaTiO3.5 There was no report on any self-focusingeffect, and only the slow part of the index changewas recorded. The reason why the intensity in theoscillation in Fig. 4 does not reach zero, as expectedfrom a continuous change of the retardation, is thenonuniformity of the retardation in the beam crosssection owing to its Gaussian profile. By taking theaverage value, we obtained the dependence of Fig. 4.

We have also performed a first study of inducedgratings and beam coupling by the thermal effect.Except for the usefulness of this process in wavemixing, its dynamics can provide information aboutcrystal parameters, the heat flow, and more. How-ever, this process has been found to be ineffecientbecause of diffusion and grating washout.

In conclusion, we have demonstrated a strong non-linear behavior in SBN and BaTiO3 crystals thatoriginates from thermally induced index changes.This effect can be used as a tool to investigate crys-tal parameters and heat diffusion processes. Thefunction of the light is twofold: to induce the ther-mal energy and to probe and visualize the changesin real time, for the whole medium in a parallel way.

We thank K. Megumi of Hitachi Research Labora-tory, Tokyo, Japan, for providing one of the SBNcrystals.

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