Abstract

A semiderivative method, which consists of illumination of a photorefractive medium with a weak beam that is incoherent with the interacting beams, is described for stabilization of unstable steady states in photorefractive phase conjugators, such as an externally pumped four-wave mixer, a semilinear self-pumped phase conjugator, and a double phase conjugator.

© 2000 Optical Society of America

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References

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  1. P. Gunter, E. Voit, M. Z. Zha, and J. Albers, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214 (1985); A. M. C. Smout, R. W. Eason, and M. C. Gower, “Regular oscillations and self-pulsating in self-pumped BaTiO3,” Opt. Commun. 59, 77–82 (1986); D. J. Gauthier, P. Narum, and R. W. Boyd, “Observation of deterministic chaos in a phase-conjugate mirror,” Phys. Rev. Lett. PRLTAO 58, 1640–1643 (1987); A. V. Nowak, T. R. Moor, and R. A. Fisher, “Observations of internal beam production in barium titanate phase conjugators,” J. Opt. Soc. Am. B JOBPDE 5, 1864–1878 (1988); T. Rauch, C. Denz, and T. Tschudi, “Analysis of irregular fluctuations in a self-pumped BaTiO3 phase-conjugate mirror,” Opt. Commun. OPCOB8 88, 160–166 (1992); P. M. Jeffrey and R. W. Eason, “Lyapunov exponent analysis of irregular fluctuations in a self-pumped BaTiO3 phase-conjugate mirror, establishing transition to chaotic behavior,” J. Opt. Soc. Am. B JOBPDE 11, 476–480 (1994).
    [CrossRef] [PubMed]
  2. Y. Braiman and I. Goldhirsh, “Taming chaotic dynamics with weak periodic perturbations,” Phys. Rev. Lett. 66, 2545–2548 (1991); M. Ciofini, R. Meucci, and F. T. Arecchi, “Experimental control of chaos in a laser,” Phys. Rev. E 52, 94–97 (1995); P.-Y. Wang and P. Xie, “Eliminating spatiotemporal chaos and spiral waves by weak spatial perturbations,” Phys. Rev. E (to be published).
    [CrossRef] [PubMed]
  3. E. Ott, C. Grebogi, and J. A. York, “Controlling chaos,” Phys. Rev. Lett. 64, 1196–1199 (1990).
    [CrossRef] [PubMed]
  4. K. Pyragas, “Continuous control of chaos by self-controlling feedback,” Phys. Lett. A 170, 421–428 (1992).
    [CrossRef]
  5. S. Bielawski, D. Derozier, and P. Glorieux, “Controlling unstable periodic orbits by a delayed continuous feedback,” Phys. Rev. E 49, R971–R974 (1994).
    [CrossRef]
  6. S. Bielawski, M. Bouazaoui, D. Derozier, and P. Glorieux, “Stabilization and characterization of unstable steady states in a laser,” Phys. Rev. A 47, 3276–3279 (1993); D. J. Gauthier, “Controlling lasers by use of extended time-delay autosynchronization,” Opt. Lett. 23, 703–705 (1998).
    [CrossRef] [PubMed]
  7. A. Bledowski, W. Krolikowski, and A. Kujawski, “Temporal instabilities in single-grating photorefractive four-wave mixing,” J. Opt. Soc. Am. B 6, 1544–1547 (1989); W. Krolikowski, K. D. Shaw, M. Cronin-Golomb, and A. Bledowski, “Stability analysis and temporal behavior of four-wave mixing in photorefractive crystals,” J. Opt. Soc. Am. B 6, 1828–1833 (1989).
    [CrossRef]
  8. J. Limeres and M. Carrascosa, “Influence of multigrating operation on the generation of phase-conjugate beams by four-wave mixing,” J. Opt. Soc. Am. B 15, 2037–2044 (1998).
    [CrossRef]
  9. P. Xie, P.-Y. Wang, and J.-H. Dai, “Origin of frequency shift and temporal instability in a photorefractive self-pumped phase conjugator,” J.Opt. Soc. Am. B (to be published).
  10. H. Risken, The Fokker-Plank Equation: Method of Solution and Application (Springer-Verlag, Berlin, 1984), pp. 60–62.

1998

1994

S. Bielawski, D. Derozier, and P. Glorieux, “Controlling unstable periodic orbits by a delayed continuous feedback,” Phys. Rev. E 49, R971–R974 (1994).
[CrossRef]

1992

K. Pyragas, “Continuous control of chaos by self-controlling feedback,” Phys. Lett. A 170, 421–428 (1992).
[CrossRef]

1990

E. Ott, C. Grebogi, and J. A. York, “Controlling chaos,” Phys. Rev. Lett. 64, 1196–1199 (1990).
[CrossRef] [PubMed]

Bielawski, S.

S. Bielawski, D. Derozier, and P. Glorieux, “Controlling unstable periodic orbits by a delayed continuous feedback,” Phys. Rev. E 49, R971–R974 (1994).
[CrossRef]

Carrascosa, M.

Derozier, D.

S. Bielawski, D. Derozier, and P. Glorieux, “Controlling unstable periodic orbits by a delayed continuous feedback,” Phys. Rev. E 49, R971–R974 (1994).
[CrossRef]

Glorieux, P.

S. Bielawski, D. Derozier, and P. Glorieux, “Controlling unstable periodic orbits by a delayed continuous feedback,” Phys. Rev. E 49, R971–R974 (1994).
[CrossRef]

Grebogi, C.

E. Ott, C. Grebogi, and J. A. York, “Controlling chaos,” Phys. Rev. Lett. 64, 1196–1199 (1990).
[CrossRef] [PubMed]

Limeres, J.

Ott, E.

E. Ott, C. Grebogi, and J. A. York, “Controlling chaos,” Phys. Rev. Lett. 64, 1196–1199 (1990).
[CrossRef] [PubMed]

Pyragas, K.

K. Pyragas, “Continuous control of chaos by self-controlling feedback,” Phys. Lett. A 170, 421–428 (1992).
[CrossRef]

York, J. A.

E. Ott, C. Grebogi, and J. A. York, “Controlling chaos,” Phys. Rev. Lett. 64, 1196–1199 (1990).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B

Phys. Lett. A

K. Pyragas, “Continuous control of chaos by self-controlling feedback,” Phys. Lett. A 170, 421–428 (1992).
[CrossRef]

Phys. Rev. E

S. Bielawski, D. Derozier, and P. Glorieux, “Controlling unstable periodic orbits by a delayed continuous feedback,” Phys. Rev. E 49, R971–R974 (1994).
[CrossRef]

Phys. Rev. Lett.

E. Ott, C. Grebogi, and J. A. York, “Controlling chaos,” Phys. Rev. Lett. 64, 1196–1199 (1990).
[CrossRef] [PubMed]

Other

S. Bielawski, M. Bouazaoui, D. Derozier, and P. Glorieux, “Stabilization and characterization of unstable steady states in a laser,” Phys. Rev. A 47, 3276–3279 (1993); D. J. Gauthier, “Controlling lasers by use of extended time-delay autosynchronization,” Opt. Lett. 23, 703–705 (1998).
[CrossRef] [PubMed]

A. Bledowski, W. Krolikowski, and A. Kujawski, “Temporal instabilities in single-grating photorefractive four-wave mixing,” J. Opt. Soc. Am. B 6, 1544–1547 (1989); W. Krolikowski, K. D. Shaw, M. Cronin-Golomb, and A. Bledowski, “Stability analysis and temporal behavior of four-wave mixing in photorefractive crystals,” J. Opt. Soc. Am. B 6, 1828–1833 (1989).
[CrossRef]

P. Xie, P.-Y. Wang, and J.-H. Dai, “Origin of frequency shift and temporal instability in a photorefractive self-pumped phase conjugator,” J.Opt. Soc. Am. B (to be published).

H. Risken, The Fokker-Plank Equation: Method of Solution and Application (Springer-Verlag, Berlin, 1984), pp. 60–62.

P. Gunter, E. Voit, M. Z. Zha, and J. Albers, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214 (1985); A. M. C. Smout, R. W. Eason, and M. C. Gower, “Regular oscillations and self-pulsating in self-pumped BaTiO3,” Opt. Commun. 59, 77–82 (1986); D. J. Gauthier, P. Narum, and R. W. Boyd, “Observation of deterministic chaos in a phase-conjugate mirror,” Phys. Rev. Lett. PRLTAO 58, 1640–1643 (1987); A. V. Nowak, T. R. Moor, and R. A. Fisher, “Observations of internal beam production in barium titanate phase conjugators,” J. Opt. Soc. Am. B JOBPDE 5, 1864–1878 (1988); T. Rauch, C. Denz, and T. Tschudi, “Analysis of irregular fluctuations in a self-pumped BaTiO3 phase-conjugate mirror,” Opt. Commun. OPCOB8 88, 160–166 (1992); P. M. Jeffrey and R. W. Eason, “Lyapunov exponent analysis of irregular fluctuations in a self-pumped BaTiO3 phase-conjugate mirror, establishing transition to chaotic behavior,” J. Opt. Soc. Am. B JOBPDE 11, 476–480 (1994).
[CrossRef] [PubMed]

Y. Braiman and I. Goldhirsh, “Taming chaotic dynamics with weak periodic perturbations,” Phys. Rev. Lett. 66, 2545–2548 (1991); M. Ciofini, R. Meucci, and F. T. Arecchi, “Experimental control of chaos in a laser,” Phys. Rev. E 52, 94–97 (1995); P.-Y. Wang and P. Xie, “Eliminating spatiotemporal chaos and spiral waves by weak spatial perturbations,” Phys. Rev. E (to be published).
[CrossRef] [PubMed]

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Figures (7)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Temporal evolution of (a) phase-conjugate reflectivity R=I3(0)/I4(0) and (b) feedback illuminating intensity Iin(t)/I0. D=0. Insets in (a) and (b) are enlargements of (a) and (b), respectively.

Fig. 3
Fig. 3

Temporal evolution of (a) phase-conjugate reflectivity R=I3(0)/I4(0) and (b) feedback illuminating intensity Iin(t)/I0. D=10-6. Insets in (a) and (b) are enlargements of (a) and (b), respectively.

Fig. 4
Fig. 4

Temporal evolution of phase-conjugate reflectivity R=I3(0)/I4(0). D=10-8. The inset is an enlargement of the figure.

Fig. 5
Fig. 5

Stability domain of β versus γL for externally pumped four-wave mixing. β is normalized by τ and I0.

Fig. 6
Fig. 6

Temporal evolution of (a) phase-conjugate reflectivity R=I3(0)/I4(0) and (b) frequency shift Δf for a semilinear SPPC. β=10.6 [normalized by τ and I4(0)] for a reflectivity of mirror M of I2(L)/I1(L)=0.98. The inset in (a) is an enlargement of (a).

Fig. 7
Fig. 7

Stability domain of βth versus γIL for a semilinear SPCC at γRL=8.5 (solid curve) and for a DPC at γRL=8 (dashed curve). βth is normalized by τ and I4(0).

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

A1/z=QA4,
A2*/z=QA3*,
A3/z=-QA2,
A4*/z=-QA1*,
τ Qt+Q=γI0(A1A4*+A2*A3)
γ=γ01+Iin(t)/I0(t),
γ=γ01+Iin(t)/I0γ01-Iin(t)I0.
Iin(t)I0=-β dI3(z=0, t)dt,dI3(0, t)dt<0,
Iin(t)=0,dI3(0, t)dt0,
f(z, t)=0,
f(z, t)f(z, t)=2Dδ(z-z)δ(t-t).
Iin(t)=0,-β dI3(0, t)dt0,
Iin(t)I0=-β dI3(0, t)dt,
0<-β dI3(0, t)dtIin0,
Iin(t)I0=Iin0,-β dI3(0, t)dt>Iin0,
δp(t)=-βn·dX(t)dt,-βn·dX(t)dt>(or<) 0,
δp(t)=0,otherwise,

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