1004 J. Opt. Soc. Am. B/Vol. 17, No. 6 /June 2000 Xie et al.

Stabilization of unstable steady states inphotorefractive phase conjugators

Ping Xie, Peng-Ye Wang, and Jian-Hua Dai

Laboratory of Optical Physics, Institute of Physics and Center for Condensed Matter Physics,Chinese Academy of Sciences, P.O. Box 603, Beijing 100080, China

Received October 4, 1999; revised manuscript received February 10, 2000

A semiderivative method, which consists of illumination of a photorefractive medium with a weak beam that isincoherent with the interacting beams, is described for stabilization of unstable steady states in photorefrac-tive phase conjugators, such as an externally pumped four-wave mixer, a semilinear self-pumped phase con-jugator, and a double phase conjugator. © 2000 Optical Society of America [S0740-3224(00)01906-8]

OCIS codes: 190.5330, 190.5040, 190.3100, 190.4380.

1. INTRODUCTIONIt is well known that photorefractive phase conjugatorshave many promising applications in optical signal pro-cessing, wave-front correction, optical neural networks,etc. However, these applications are degraded when theoutput phase-conjugate intensity becomes unstable.1

Therefore it is imperative to eliminate these instabilitiesand stabilize the conjugator at steady states.

Recently the control of chaos has been an interestingtopic. Generally, the control schemes fall into two cat-egories: nonfeedback and feedback schemes. The non-feedback scheme2 modulates a system parameter with aweak periodic signal to stabilize the system at a periodicorbit. For the feedback scheme, after the pioneeringwork of Ott et al.,3 a continuous time-delayed feedbackmethod to control the chaotic system at a periodic orbitwas proposed.4,5 More recently, several researchers havesuggested what is called a derivative control method tostabilize the unstable steady state in lasers.6 In this pa-per, we present a semiderivative method, based on the de-rivative method, to stabilize the unstable steady state in aphotorefractive phase conjugator.

Consider the standard four-wave mixing geometry witha single transmission grating. For an externally pumpedphase conjugator [Fig. 1(a)], beams 1, 2, and 4 are the ex-ternal inputs; beams 1 and 2 are the pump beams, beam 4is the signal beam, and beam 3 is the phase-conjugatebeam. For a semilinear self-pumped phase conjugator[SPPC; Fig. 1(b)], only beam 4 is an external input, beam2 is the reflection of self-induced beam 1 by mirror M, andbeam 3 is the phase-conjugate beam. For a double phaseconjugator [DPC; Fig. 1(c)], beams 4 and 2 are the exter-nal inputs and beams 3 and 1 are the correspondingphase-conjugate beams. All beams are assumed to beplane waves. The time-dependent coupled-wave equa-tions in the slowly varying envelope approximation can bewritten as follows:

]A1 /]z 5 QA4 , (1a)

]A2* /]z 5 QA3* , (1b)

0740-3224/2000/061004-04$15.00 ©

]A3 /]z 5 2QA2 , (1c)

]A4* /]z 5 2QA1* , (1d)

t]Q

]t1 Q 5

g

I0~A1A4* 1 A2* A3! (1e)

where Aj ( j 5 1, 2, 3, 4) is the slowly varying complexamplitude of the jth beam, Q is the complex amplitude ofthe index grating, I0 5 ( j51

4 uAju2 is the total light inten-sity, t is the time constant, and g is the photorefractivecoupling constant (which depends on the optical param-eters of the media and the geometrical factors).

Equations (1) are solved numerically by the samemethod as that used in Refs. 7 and 8. To be sure aboutaccuracy of the numerical procedure, we reduced the spa-tial and time integration steps until the solution re-mained invariant.

For externally pumped four-wave mixing, a number ofprevious papers have predicted unstable dynamics for ahigh real coupling constant gL (where L is the interactionlength) and specific regions of the physical parameters.7,8

For example, for pump ratio p 5 I2(L)/I1(0) 5 20 andsignal–pump ratio q 5 I4(0)/@I1(0) 1 I2(L)# 5 1024, thenumerical simulation predicts that, when gL > 6.28, thesystem will become unstable.

For a semilinear SPPC our numerical results9 haveshown that, for a real coupling constant gL, the con-jugator yields stable phase-conjugate output. For a com-plex coupling constant gL (for convenience, we writegL 5 gRL 1 igIL), when either gRL or gIL becomeslarge, the steady-state solution becomes unstable. Forexample, when gRL 5 8.5 the solution is unstable atugILu > 7.06 for a reflectivity of mirror M of I2(L)/I1(L)5 0.98.

For a DPC, the numerical results are similar to thosefor the semilinear phase conjugator. When gRL 5 8 andinput ratio I2(L)/I4(0) 5 1, the steady-state solution be-comes unstable at gIL ' 67.17.

Our purpose is to stabilize these unstable steadystates. To this end we adjust coupling constant g by il-luminating the medium with a beam that is incoherent

2000 Optical Society of America

Xie et al. Vol. 17, No. 6 /June 2000/J. Opt. Soc. Am. B 1005

with the interacting beams. When a spatially uniformincoherent intensity I in(t) is illuminating the photorefrac-tive medium, the coupling constant can be written as

g 5g0

1 1 I in~t !/I0~t !, (2)

where g0 is the coupling constant without illumination ofI in(t) and I0(t) is the total intensity of the interactingbeams, as defined in Eqs. (1). For a small illuminatingintensity, i.e., I in(t)/I0 ! 1, Eq. (2) can be rewritten as

g 5g0

1 1 I in~t !/I0' g0F1 2

I in~t !

I0G . (3)

In a way similar that for the derivative control methodused to stabilize the unstable steady state in lasers,6 wetake the time-dependent illuminating intensity as

I in~t !

I05 2b

dI3~z 5 0, t !

dt,

dI3~0, t !

dt, 0, (4a)

I in~t ! 5 0,dI3~0, t !

dt> 0, (4b)

Fig. 1. Geometrical configurations for (a) an externally pumpedfour-wave mixer, (b) a semilinear self-pumped phase conjugator,and (c) a double phase conjugator.

where b . 0 is the feedback gain parameter and I3(z5 0, t) 5 uA3(z 5 0, t)u2. Considering that the illumi-nating intensity can only be larger than or equal to zero,we use a semiderivative scheme here (i.e., the derivativehas a particular sign). Equations (4) mean that whenthe unstable steady state is stabilized the feedback illu-mination vanishes.

2. EXTERNALLY PUMPED FOUR-WAVEMIXINGIt is numerically verified that the semiderivative method,which is described by Eqs. (2) and (4), is an efficient onefor stabilizing unstable steady states. Figure 2 shows anexample of the numerical results, where gL 5 6.8 andthe feedback gain parameter b is taken as 249.4 (normal-ized by t and I0). The incoherent intensity is illuminat-ing the photorefractive medium for t > 500t. We seethat after the illuminating intensity is turned on the un-stable state gradually changes into a stabilized stablesteady state, and correspondingly the incoherent illumi-nating intensity gradually vanishes and eventuallyequals zero. In real experiments, noise is inevitable, es-pecially the fluctuation owing to the thermal excitation ofchange carriers. To investigate the influence of this ther-mal noise, we consider the coupling constant g0@11 f(z, t)# instead of g0 , where f(z, t) is assumed to be aGaussian random variable with a zero mean and a d cor-relation function:

^ f~z, t !& 5 0, (5a)

^ f~z, t !f~z8, t8!& 5 2Dd ~z 2 z8!d ~t 2 t8!. (5b)

Figure 3 is the numerical result for D 5 1026, with otherparameters the same as those in Fig. 2. Q(t) is numeri-cally integrated by use of the molecular dynamics

Fig. 2. Temporal evolution of (a) phase-conjugate reflectivityR 5 I3(0)/I4(0) and (b) feedback illuminating intensityI in(t)/I0 . D 5 0. Insets in (a) and (b) are enlargements of (a)and (b), respectively.

1006 J. Opt. Soc. Am. B/Vol. 17, No. 6 /June 2000 Xie et al.

method.10 It can be seen that the stabilization method isstill efficient in the presence of noise. The small fluctua-tion of the incoherent illumination after the stabilizationis used to cancel the small fluctuation that is due to ther-

Fig. 3. Temporal evolution of (a) phase-conjugate reflectivityR 5 I3(0)/I4(0) and (b) feedback illuminating intensityI in(t)/I0 . D 5 1026. Insets in (a) and (b) are enlargements of(a) and (b), respectively.

Fig. 4. Temporal evolution of phase-conjugate reflectivity R5 I3(0)/I4(0). D 5 1028. The inset is an enlargement of thefigure.

Fig. 5. Stability domain of b versus gL for externally pumpedfour-wave mixing. b is normalized by t and I0 .

mal noise. The zero magnitude of the feedback illumi-nating intensity for the ideal case (without noise) and thesmall magnitude for the real case (with noise) imply thatthe feedback illuminating intensity does not alter thevalue of the solution of the conjugator and that only thestability of the solution is changed. One can clearly seethis by turning off the feedback illumination well afterthe unstable steady state is stabilized, as shown in Fig. 4,where for t , 800t incoherent illumination is turned onand for t > 800t incoherent illumination is turned off.We see that when the incoherent illumination is turnedoff the stabilized steady state gradually turns into an un-stable state. For a given coupling constant gL, there ex-ists a region for b within which the unstable steady statecan be stabilized. Figure 5 shows the stability domain(b versus gL) for p 5 I2(L)/I1(0) 5 20 and q 5 I4(0)/@I1(0) 1 I2(L)# 5 1024. It can be seen that for a verylarge gL(gL > 7.6) there exists no region within whichthe unstable steady state can be stabilized.

3. SEMILINEAR SELF-PUMPED PHASECONJUGATOR AND DOUBLE PHASECONJUGATORFor the SPPC and the DFC, pump beam 1 results fromcoupling between the scattering light and input signalbeam 4. A great decrease of coupling constant gL willlead to a great decrease of the output phase-conjugate in-tensity (the conjugator does not even work for a smallcoupling constant). Therefore for these conjugators weshould also introduce a restriction on the maximum valueof the incoherent illuminating intensity. Then Eqs. (4)become

I in~t ! 5 0, 2 bdI3~0, t !

dt< 0, (6a)

I in~t !

I05 2b

dI3~0, t !

dt,

0 , 2bdI3~0, t !

dt< I in0 , (6b)

I in~t !

I05 I in0 , 2 b

dI3~0, t !

dt. I in0 , (6c)

where I in0 is the maximum value of the illuminating in-tensity, which we take as 0.01I0 in this paper. We findthat the method described by Eqs. (2) and (6) can stabilizeunstable steady states in the SSPC and the DPC. Figure6 shows an example of the numerical results for the SPPCat gL 5 8.5 2 7.8i. In the figure we take D 5 1026; theincoherent intensity is illuminating the photorefractivemedium for t , 3000t, and for t > 3000t the illuminat-ing intensity is turned off. From the figure we see thatonce the illuminating intensity is turned off the stabilizedphase-conjugate reflectivity and frequency shift Df be-tween the phase-conjugate beam and the input signalbeam are destabilized and then jump to their periodic os-cillations. For a given coupling constant gL 5 gRL1 igIL, there exists a value of bth larger than which theunstable steady state can be stabilized. Figure 7 shows

Xie et al. Vol. 17, No. 6 /June 2000/J. Opt. Soc. Am. B 1007

the stability domain (bth versus gIL) for the SPPC atgRL 5 8.5 @I2(L)/I1(L) 5 0.98# and for the DPC at gRL5 8 @I2(L)/I4(0) 5 1#. It has been verified that whenwe increased b to 100 we did not detect an upper limitleading to destabilization.

Finally, it should be pointed out that the semideriva-tive method used in this paper, i.e., an adjustment of thesystem control parameter by an amount

dp~t ! 5 2bn •

dX~t !

dt, 2 bn •

dX~t !

dt. ~or , ! 0,

dp~t ! 5 0, otherwise,

where n is the measurement direction and X(t) is the vec-tor of the system output variables, can also stabilize theunstable steady state in other chaotic systems.

Fig. 6. Temporal evolution of (a) phase-conjugate reflectivityR 5 I3(0)/I4(0) and (b) frequency shift Df for a semilinearSPPC. b 5 10.6 [normalized by t and I4(0)] for a reflectivity ofmirror M of I2(L)/I1(L) 5 0.98. The inset in (a) is an enlarge-ment of (a).

Fig. 7. Stability domain of bth versus gIL for a semilinear SPCCat gRL 5 8.5 (solid curve) and for a DPC at gRL 5 8 (dashedcurve). bth is normalized by t and I4(0).

In summary, the unstable steady state in photorefrac-tive conjugators, such as the externally pumped four-wave mixer, the semilinear self-pumped phase conjuga-tor, and the double phase conjugator, can be stabilized byillumination on the photorefractive medium of a weakbeam that is incoherent with the interacting beams.

ACKNOWLEDGMENTThis research was supported by the National ScienceFoundation of China.

P. Xie’s e-mail address is pxie@aphy.iphy.ac.cn.

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