Electric-field dependence of phase-conjugate wave-frontreflectivity in reduced KNbO3 and Bi12GeO20
P. N. Gunter
Laboratory of Solid State Physics, Swiss Federal Institute of Technology, ETH Honggerberg, CH-8093 Zurich, Switzerland
Received July 21, 1981
The electric-field dependence of the wave-front reflectivity p in degenerate four-wave mixing experiments with re-duced KNbO3 and Bi12GeO20 has been measured and compared with the theoretical expressions [N. Kukhtarevet al., Ferroelectrics 22, 949, 961 (1979)] describing the influence of the different photoinduced space-charge fieldsin photorefractive media. Peak reflectivities of p = 1.4% for Bi12 GeO2 0 and p = 10% for KNbO3 :Fe2 + have beenreached for field strengths of 12 and 8 kV/cm, respectively. It is shown that complex-conjugate wave fronts canbe generated and that distorted optical wave fronts can be effectively corrected by degenerate four-wave mixingexperiments in KNbO3:Fe2 +.
Phase conjugation by degenerate four-wave mixing(FWM) in photorefractive materials can be used for therestoration of phase-distorted images (adaptive optics).In FWM two counterpropagating pump waves withcomplex amplitudes R1 and R2 and a weak signal waveS3 interact in the volume of a nonlinear medium toproduce a fourth wave S 4 that is a complex conjugateto the signal wave (Fig. 1).1,2 First observations ofphase conjugation by four-wave mixing in photore-fractive materials were reported for Bi12SiO2 0 in Ref.3 and for LiNbO 3 and LiTaO 3 in Ref. 4. Recent ex-periments with BaTiO3 (Ref. 5) show that, in materialswith large electro-optic coefficients and sufficient pumppower R2 cw phase-conjugate wave generation with si-multaneous amplification can be achieved.
In this Letter we present for the first time to ourknowledge measurements of the electric-field depen-dence of the phase-conjugate wave-front reflectivity ofBi12GeO2 0 and KNbO3:Fe2+. These materials, togetherwith BiU2SiO2 0, have been shown to be the most at-tractive ones for dynamic holography experiments sincethey have highest photosensitivities at the argon-laserlines6-8 and optimized energy transfer between re-cording beams because of a 7r/2 phase shift of the re-fractive-index grating with respect to the intensitygrating observed in these materials.9 Both the highphotorefractive sensitivity and intensity transfer aredue to an efficient charge transport of photoexcitedcarriers, which, after being trapped at other locations,lead to a space-charge field modulating the refractiveindices through the electro-optic effect.
Measurements of the electric field dependence of thewave-front reflectivity and the comparison of this de-pendence with theoretical expectations based on Ku-khtarev's'0 nonlinear theory of photorefractive crystalswith long photocarrier drift length Id = AurEo (where /iis the charge mobility, -r is the lifetime before trapping,and E0 is the applied electric field) allow one to deter-mine the optimum recording parameters (fringe spac-ing, applied electric field, recording wavelength, etc.)and to check the limits of available theories. InKNbO3:Fe2 , which in addition to optimized photo-
conductivity also shows unusually large electro-opticcoefficients (r3 3 = 64 pm/V, r 42 = 380 pm/V),"1" 2
wave-front reflectivities similar to the ones observed inBaTiO 3 (Ref. 5) are reported.
Our experimental results are interpreted on the basisof the theories of Kukhtarev et al., which are the onlyones valid for (1) arbitrary transport lengths of photo-carriers, (2) the initial (transient) or steady-state stageof recording process, and (3) dynamic changes of thefringe pattern contrast along the crystal length that aredue to beam-coupling effects of writing beams.
In our previously published detailed two-wave mixingexperiments,9"13 we verified that this formalism mostaccurately describes the volume hologram formation inphotoconductive materials. The intensity changes ofinteracting beams for the transmission grating can bederived by solving the wave equation, including thenonlinear dependence of the refractive index on thelight-field amplitude or intensity.' 0 For a 7r/2-shiftedgrating, neglecting light absorption, Kukhtarev andOdulov4 obtain for the most interesting case of a singlesignal wave 13(x = 0) 5 0 and I4(x = d) = 0
dx1 d = - a, =rF/Io[I13 - (II2II4)1/2I,dx ax
a,= - a12= -F/I[124 - (1112i314)i/21,ax O9x
(1)
(2)
where Io is the total intensity of all interacting beamsand r is the exponential gain characterizing the effi-ciency of the FWM process determined by the electro-optic properties and the 7r/2 phase-shifted componentof the photoinduced space-charge field.9 10 In theweak-field approximation (I3 << I, and I4 << I2), one getsfor the wave-front reflectivity4
response in which I4 rapidly decreases if the pump-beam intensities differ.14 In our experiments, I2 wassimply the retroreflected pump wave I,, which on beingtransmitted through the recording crystal was atten-uated because of absorption and reflection losses,yielding I2(x = d)/II(x = 0) - 0.6 only. According toEq. (3), an optimum reflectivity of p - 50% can be ex-pected. Increased values for p according to Eq. (3),particularly p > 1, as in BaTiO3 ,5 can be expected onlyfor 12/I(x = 0) > 1.
It has been shown in two-wave mixing experiments 1 0
that the gain r is determined by the component of theholographic grating that is 7r/2 phase shifted with re-spect to the intensity pattern. Expressing the pho-toinduced space-charge field in terms of the appliedelectric field E0 , the diffusion field ED = (2 7r/e) (kT/A)= A/A, and the maximum possible field allowed by acomplete trap center filling Eq = (eNA/2-7rE3E0)A = BA(where A is the holographic fringe spacing, NA is theconcentration of trapping centers, and E3 is the dielectricconstant), one gets for the linear recombination case andfor weak beam coupling the steady-state wave-frontreflectivity
P I2 EO 2 + ED 2 4n32 A 2
I1 I +ED2 + E02 4n3
2 A2 - X2
steady-state photoinduced refractive-index changes.Second, the photoconductivity is high enough that evenat wavelengths near 600 nm these steady-state refrac-tive changes are reached in less than a second at mod-erate optical power densities, i.e., 2 orders of magnitudeless than Bi12GeO20. Whereas beam coupling inBi12SiO20 and Bi12GeO2 0 is substantially reduced be-cause of the large optical activity of these materials, theoptical polarization remains the same along the lightpath for polarization directions parallel to the crystal-lographic directions.
Photorefractive recording is much more efficient inKNbO3:Fe2+ than in BaTiO3, another material withlarge electro-optic effects, with typical response timeT of the order of 10 msec at recording wavelengths of X= 488 nm and T of the order of 100 msec at X = 592 nmin KNbO3 :Fe2 + for I = 1 W/cm 2 . These values comparefavorably with T - 1 sec measured in BaTiO3 (Ref. 5)for X = 514 nm, which until now showed the best per-formance.
The electric-field dependence of the wave-front re-flectivity for different values of the beam ratio 13o = I1/I3is shown in Fig. 2 for Bi12GeO2 0 and in Fig. 3 forKNbO3:Fe2+. The other parameters used in these ex-periments are indicated in the Figs. 2 and 3 and repre-
(4)
where X is the wavelength of light, n 3 is the refractiveindex, and R is a proportionality constant.
Experimental results of wave-front reflectivity p havebeen obtained with the setup shown in Fig. 1. The lightsource for recording holograms was a tunable dye laser(Spectra-Physics Model 375, rhodamine 6G) pumpedby an Ar laser (Lexel Model 95), which permittedwavelength tuning for the best reflectivity.15
Experiments have been performed both with a crystalof photoconductive electro-optic and paraelectricBi12GeO20 (d = 9.15 mm) and with reduced Fe-dopedKNbO 3 (d = 3.33 mm). The orientation of the KNbO 3crystal is shown in Fig. 1. The Bi12GeO2 0 crystal wasoriented to give optimized coupling between interferringbeams. 13
Reduced KNbO3 as a ferroelectric material withphotoconductivity comparable with or larger than thatin Bi12 Si020 and Bi12GeO2 0 (Refs. 7 and 8) offers manyadvantages compared with the other more commonlyused photoconductors Bi12Si020 and Bi12GeO20. First,the rather large value of the spontaneous polarizationgives large electro-optical effects leading to large
0 5 E 0kV/cm 15 20
Fig. 2. Wave-front reflectivity versus applied electric fieldfor different intensity beam ratios for Bi12GeO20.
20%
15%
a 10%
5%
0%0 4 8 12 16
E0 [kV/cm]20
Fig. 3. Wave-front reflectivity versus applied electric fieldfor different intensity beam ratios for KNbO3 :Fe2 +.
12 OPTICS LETTERS / Vol. 7, No. 1 / January 1982
sent the optimum data determined in two-wave mixingexperiments.1 2
In the limit of E0 < Eq, a quadratic-field dependenceof the wave-front reflectivity has been observed, inagreement with Eq. (4). For very large electric fieldsEo > Eq, the wave-front reflectivity should saturatesince, according to Eq. (4), p a Eq 2 for E 0 >> Eq, inde-pendently of E0. The expected field dependence [Eq.(4)] with Eq = 7 kV/cm for KNbO 3 and Eq = 16 kV/cmfor Bi12GeO2 0 (determined in two-wave mixing exper-iments) has been plotted in Figs. 2 and 3. Peakreflectivities of 19 and 4% are expected for KNbO3 andBi12GeO20 , respectively.
For large electric fields, however, the experimentalresults of Figs. 2 and 3 show a saturation of p(Eo) for E0- 12 kV/cm, with a peak value of p _ 1.4% at E0 - 12kV/cm in Bi12 GeO2o and of p = 10% for E0 near 8 kV/cmin KNbO3 . For larger field strengths p shows a decreasethat cannot be explained by the theoretical relation [Eq.(4)] since this theory is valid only for weak beam cou-pling and small photoinduced phase changes, and thisassumption seems to be violated at large electric fields.Also, self-interference inside the crystal volume betweenthe phase-conjugate generated wave and retroreflectedreference beam R, may not be neglectable if wave-frontreflectivity becomes larger.
The main difference between the two materials is themore than sixfold increase in values of reflectivity in thed = 3.328-mm-thick crystal in KNbO3 compared withthat of Bi12 GeO20 with d = 9.15 mm, peaking at p = 10%for E0 near 8 kV/cm. This illustrates the increasedelectro-optic activity of KNbO3. Using Eq. (3), one getsa peak gain of r = 2.6 cm-' for the experimental data:p = 0.1, I1 = 100 mW/cm 2 , I2 = 60 mW/cm 2 , d = 3328mm, and M 13 = 0.625. This peak-gain value should becompared with I' = 2 cm-' obtained for the diffusion-only case by using ED = 271 V/cm and with r = 20 cm-'for the drift-only case assuming a 7r/2-shifted compo-nent of the space-charge amplitude of B = 2 kV/cm.
The peak value of p = 10% represents one of thelargest phase-conjugate reflectivities measured inphotorefractive materials. It compares favorably withthe peak reflectivity of 0.2% measured in Bi12Si020 .3
We believe that, with more-sophisticated experimentalarrangements such as the ones used in BaTiO3,5 phaseconjugation with simultaneous amplification could beachieved at much shorter response times and for thevisible wavelength range X = 400-600 nm with the fol-lowing methodology: (1) Use of a strong read-out signalwave I2 (12/I1 = 100 in Ref. 5, I2/I1 = 0.6 in this work).(2) Changed mixing geometry to create a space-chargefield component along the b axis and use of increasedelectro-optic coefficient r4 2 = 380 pm/V.1 1' 12 (3) Sup-pressed reflection losses at crystal surfaces by antire-flection coatings or index-matching fluids. (4) In-creased value of space-charge fields by increasing therecording intensity [p - 2 for I, = 2 W/cm2 , X = 514 nm,and I2/1I1 = 1 in BaTiO3 (Ref. 5); p = 0.1 for I, = 0.1W/cm2 at X = 592 nm and I2/11 = 0.6 in KNbO 3 ].
In order to confirm that the backward wave S4 is acomplex conjugate replica of the signal wave 83, we in-troduced a diverging lens into the signal beam. Thebackward wave then indeed became convergent andfocused to a point at a certain distance from the lens.
BS Beom splitters Photorefi(KNbO,
Spot size of distortedSpot size of beamoutpUt beam
tratisoersal Reflected beam
distorting plateRefled aedbetm dfter plase diortr.nsvers.1 of dist.rtmgq plot. iegigln
Fig. 4. Experimental setup for phase-conjugate wave gen-eration and for correction of phase distortions by four-wavemixing in photorefractive materials.
Also, more-complicated images placed in the signalbeam in front of the diverging lens were reconstructedas real images, the resolution being limited by thecrystal quality.
The effect of wave-front restoration of the laser spotdistorted by a roughened glass plate instead of by thediverging lens by four-wave mixing in reduced KNbO 3is illustrated in Fig. 4. Besides static phase distortions,dynamic phase inhomogeneities occurring, e.g., in thephotorefractive medium itself, can be compensated for15since the recording time is of the order of 100 msec atthe power levels used above and at X = 600 nm. Thisresponse time can be further decreased by increasingthe light intensity or using light sources with shorterwavelengths.
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