1462 OPTICS LETTERS / Vol. 17, No. 20 / October 15, 1992

Self-pumped forward phase conjugator

Nery Strasman, Ron Daisy, and Baruch Fischer

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

Received June 11, 1992

A self-pumped forward phase-conjugate mirror based on nonlinear wave mixing in photorefractive media isdemonstrated. A single input beam to a BaTiO3 crystal excites an oscillation of pump beams between surfacesof the crystal and a forward phase-conjugate beam. The phase conjugation is carried on the first-orderdiffraction from the grating induced by the input and the self-generated pump beams.

One of the most common methods for obtainingphase conjugation is based on nonlinear four-wavemixing." The built-in solution for the phase-matching requirement made this technique superiorto an earlier suggestion for phase conjugation byusing three-wave mixing.5 The attractiveness andsimplicity of the four-wave-mixing scheme becameeven more evident when the self-pumped phase con-jugators in photorefractive media were developed'and the need for external pump beams was relaxed.In these devices, the pump beams are self-generatedwhen feedback is added by external mirrors (in thelinear, semilinear, and ring conjugators3), by thecrystal surfaces (in the cat4 and linear devices3), orwithout any mirrors (in the double phase-conjugatemirror6). In all these four-wave-mixing techniquesa backward phase conjugation is generated. Thereis another way to obtain phase conjugation byusing the first-order diffraction of the gratingsin two-beam coupling.7 -9 This method gives aforward-propagating phase conjugation and needsthin gratings (Raman-Nath regime) to permit thefirst-order diffraction. In this Letter we demonstratesuch a forward phase conjugator that does not needthe external second beam for two-beam coupling.The first-order diffraction by the grating that givesthe forward phase conjugation is induced by theinterference of the input beam and a self-generatedpump beam.

The experimental configuration is schematicallyshown in Fig. 1. Beam 1, carrying spatial informa-tion and having an extraordinary polarization, entersa photorefractive crystal that has a 450 orientationof the c axis with respect to the input facets (a 45°-cut crystal). Amplified light in a broad angle (theFanning effect) occurs, and eventually an oscillationbuilds up between the opposite crystal facets. Thefeedback for the oscillation is supplied by Fresnel re-flections of the facets. In fact, this part of the processprovides a compact self-pumped linear (or semilinear)phase conjugator2'3 in which the feedback is providedby the crystal surfaces instead of by external mirrors.A similar oscillation was demonstrated with LiNbO3by Odoulov and Soskin.'0 The four-wave mixing ofthe beams produces a backward-propagating phaseconjugation (beam 1') of input beam 1. The new re-

sult in the present research is the second stage inthe beam-coupling dynamics. The grating, inducedby the input beam 1 and the self-generated pumpbeam 2, produces a first-order diffracted beam 3. Inthis three-beam coupling process beam 3 is a for-ward phase conjugation of the input beam.7 "9 Aphotograph of the beam patterns inside the crystalis shown in Fig. 2, and the forward phase-conjugateimage carried on beam 3 is shown in Fig. 3. Theangle between beams 2 and 3, which has to be ap-proximately the same as the angle between beams 1and 2, seems in Fig. 2 to be slightly smaller notonly owing to illusion that results from the shiftedbeams' intersection (the buildup of beam 3 is at theright side of the grating zone), but there is also areal angular reduction of approximately 3% owing tothe crystal anisotropy, for which the extraordinaryrefractive index is higher as the beam direction movestoward the c axis.

We now turn to a more detailed description of theeffect through the wave-mixing processes. As men-tioned above, we have two stages in the dynamics.In the first part, the pump beams 2 and 2' build up(see Fig. 4) between the two crystal surfaces, andthe linear phase conjugator starts to operate. Infact, in the analysis below we consider it a semilin-ear self-pumped conjugator having only one feedbackmirror since the Fresnel reflectivity of the surfacesis relatively low (M = 0.17), so that the front surfacecontributes only -M 2 to the pump's feedback. Nev-ertheless, this small feedback, together with the factthat it is generated by a flat surface, causes signifi-cant fluctuations in the intensities of the backwardand forward phase-conjugate beams. The inducedgrating in the buildup of the semilinear conjugator

BaTiO3 , 3

Fig. 1. Schematic of the self-pumped forward phaseconjugator.

0146-9592/92/201462-03$5.00/0 © 1992 Optical Society of America

October 15, 1992 / Vol. 17, No. 20 / OPTICS LETTERS 1463

Fig. 2. Photograph of the beams inside the crystal whenthe self-pumped forward phase conjugator operates, show-ing the input beam 1 (bottom pattern), the pump beams 2and 2' (middle pattern), and the forward phase-conjugatebeam 3 (top pattern).

beams 2 and 3 adds a new component (1/IO)A3A2 +c.c. to the grating g in relation (1), with the samegrating wave number, and another grating, which isdue to the coupling of beams 1 and 3, with a wavenumber that is larger by a factor of 2. Nevertheless,we can neglect these two contributions since theintensity of beams 3 is much smaller than that ofbeams 1 and 2, as we show below.

A simple quantitative derivation of the beams' in-tensities Ii = IA 12 can be done by using our assump-tions of treating the wave-mixing configuration asa semilinear phase conjugator with the buildup ofpumps and thin gratings from which the first-orderbeam 3 is diffracted. Then we can use the knownsolution3 for the beam intensities in the semilinearself-pumped phase conjugator and the double phase-conjugate mirror. In particular, we want to findthe grating's diffraction transmissivity that is thetransmissivity of a double phase-conjugate mirror(Eq. 4.17 of Ref. 3),

I2(1) _ II,(O)MlO) I21(1)

(3)

This can be obtained by using Eq. 4.10 in Ref. 3for the reflectivity of the semilinear phase-conjugatemirror, defined by R = I (O)/II(O) (where 0 and 1denote the input and output sides, respectively, ofthe coupling zone whose length is 1) and the surfacefeedback reflectivity M (it is defined as M2 in Ref. 3),

image carried on

is proportional to

g = I (Al*A2 + A1,A2 1) + c.c. x I Al*A2 + c.c.,

(1)

where Ai is the complex amplitude of beam i andIo = ZIAi 12. We also used the relations Al, cc Al"' andA21 °c A *. The grating turns out to be thin, in theRaman-Nath regime. The reason is the strong de-pletion of the input beam that confines the beaminteraction zone and the grating near the input sur-face. Then the Bragg condition can be violated andthe first-order diffraction of beam 3 is allowed. Itsk vector is k3 = 2k2 - k, + Ak, where Ak is the smallphase mismatch involved in such high-order diffrac-tion processes. The effective width of the grating,which is measured in terms of its period, can be re-duced by taking a small angle between the input andthe oscillating beams. This improves the diffractionefficiency of beam 3, while still maintaining a highenough coupling constant, owing to the 45° cut of thecrystal. Then

(2)

and if the self-induced pump A2 is spatially uniform,A3 is the forward phase conjugation of the inputAl. After beam 3 is generated (its experimental timeevolution is shown in Fig. 4) the coupling between

T = (R ) 2 =1 - a2

M(1 + a2) - 2M"12[a2(M + 1) - 1]12

(4)

Here a is a constant derived from the photorefractivecoupling strength (yl) by3 tanh[-(yl/2)a] = a. Thethreshold of the semilinear conjugator is given by(Eqs. 4.12 and 4.13 in Ref. 3)

NI+ )1"2 - 1](ylX (1± M)"2 In (1 + M)1"2 + 1]'

L1004

Ce S-

i a

Time [sec]

Fig. 4. Buildup behavior of the output intensities 11(curve 1), I2 (curve 2), and I3 (curve 3). In this ex-periment, the input was a 5-mW He-Ne laser beam (at633 nm) that had an angle of 10° (inside the crystal)with respect to the input surface normal.

Fig. 3.beam 3.

Forward phase-conjugate

A 3 c gA 2 ih IA 22Al*XIo

1464 OPTICS LETTERS / Vol. 17, No. 20 / October 15, 1992

which in our experiment gives (yl)t = -3.5 for M =0.17 [compared with (yI)t = -2.49 for maximum feed-back with M = 1 that can be obtained with an exter-nal mirror or by high-reflection coating of the crystal].It is important to emphasize that all the above rela-tions are derived for the steady-state behavior andneglect absorption in the crystal.

At this point, the experimental data of the outputintensities in Fig. 4 can be examined. The figuregives II(t = 0), which can be taken as I,(l = 0)-II(0), and the steady-state values of the beams inten-sities, from which we obtain T = I2(1)1I1(0) 0.18.Then we can also have a good estimate of the forwardphase-conjugate beam intensity, relying on our aboveexplanations that the same grating responsible forthe diffraction I, - I2 gives the diffraction I2 - I3,if we assume that the coupling constant does notvary significantly for the slightly different beam di-rections. Consequently,

I3(1) : T(I2) _ T12(l) (5)2

where the angle brackets indicate an averageover the interaction length. Thus we have thediffraction efficiency from the previously obtained T(for the diffraction I, -- 12), I3(1)/I2(l) T/2 - 0.09.This is in reasonable agreement with the directexperimental steady-state value from Fig. 4, which

gives 13(1)/I2(1) 0.08. More experimental data forthe beams' intensities with slightly different inputangles, and therefore different coupling constants,showed similar agreement with the expected values.

In conclusion, we have demonstrated a self-pumpedforward phase-conjugate mirror. The demonstrationwas done with a photorefractive BaTiO3 crystal. Ananalysis was shown to give a good understanding ofthe wave-mixing process.

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