Abstract

The physical origin of frequency shift and dynamic instability in a photorefractive self-pumped phase conjugator is analyzed by use of four-wave mixing and multi-four-wave mixing. We show that the frequency shift and the instability result from the requirement of the maximal energy transfer between the interacting beams. Two examples of numerical results in a cat self-pumped phase conjugator are given, which are in good agreement with previous experimental results. A simple method is proposed to change an unstable phase-conjugate output to a stable one.

© 2000 Optical Society of America

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References

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  1. J. Ramsey and W. Whitten, “High-resolution self-scanning continuous wave dye laser,” Anal. Chem. 56, 2979–2981 (1984).
    [CrossRef]
  2. M. C. Gower, “Photoinduced voltages and frequency shifts in a self-pumped phase-conjugating BaTiO3 crystal,” Opt. Lett. 11, 458–460 (1986).
    [CrossRef] [PubMed]
  3. 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).
    [CrossRef]
  4. A. V. Nowak, T. R. Moore, and R. A. Fisher, “Observations of internal beam production in barium titanate phase conjugators,” J. Opt. Soc. Am. B 5, 1864–1878 (1988).
    [CrossRef]
  5. D. J. Gauthier, P. Narum, and R. W. Boyd, “Observation of deterministic chaos in a phase-conjugate mirror,” Phys. Rev. Lett. 58, 1640–1643 (1987).
    [CrossRef] [PubMed]
  6. 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 11, 476–480 (1994).
    [CrossRef]
  7. See, for example, J. Feinberg and K. R. MacDonald, in Photorefractive Materials and Their Application, P. Gunter and J. P. Huignard, eds. (Springer-Verlag, New York, 1989), pp. 176–179 and reference therein.
  8. M. D. Ewbank and P. Yeh, “Frequency shift and cavity length in photorefractive resonators,” Opt. Lett. 10, 496–499 (1985).
    [CrossRef] [PubMed]
  9. B. Fischer, “Theory of self-frequency detuning of oscillations by wave mixing in photorefractive crystals,” Opt. Lett. 11, 236–239 (1986).
    [CrossRef] [PubMed]
  10. P. Xie, P. Y. Wang, J. H. Dai, and H. J. Zhang, “Frequency shifts and dynamic instabilities in photorefractive self-pumped and mutually pumped phase conjugation,” J. Opt. Soc. Am. B 16, 420–427 (1999).
    [CrossRef]
  11. W. Krolikowski, M. R. Belic, M. Cronin-Golomb, and A. Bledowski, “Chaos in photorefractive four-wave mixing with a single grating and a single interaction region,” J. Opt. Soc. Am. B 7, 1204–1209 (1990).
    [CrossRef]
  12. W. Krolikowski, M. R. Belic, and A. Bledowski, “Phase transfer in optical phase conjugation,” Phys. Rev. A 37, 2224–2226 (1988).
    [CrossRef] [PubMed]
  13. P. Xie, J. H. Dai, and H. J. Zhang, “Multigrating optical phase conjugation with considerations of phase effects,” J. Opt. Soc. Am. B 9, 2240–2247 (1992).
    [CrossRef]
  14. P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Self-pumped phase conjugation in photorefractive crystals: reflectivity and spatial fidelity,” Phys. Rev. A 55, 3092–3100 (1997).
    [CrossRef]
  15. J. F. Lam, “Origin of phase conjugate waves in self-pumped photorefractive mirrors,” Appl. Phys. Lett. 46, 909–911 (1985).
    [CrossRef]

1999 (1)

1997 (1)

P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Self-pumped phase conjugation in photorefractive crystals: reflectivity and spatial fidelity,” Phys. Rev. A 55, 3092–3100 (1997).
[CrossRef]

1994 (1)

1992 (1)

1990 (1)

1988 (2)

1987 (1)

D. J. Gauthier, P. Narum, and R. W. Boyd, “Observation of deterministic chaos in a phase-conjugate mirror,” Phys. Rev. Lett. 58, 1640–1643 (1987).
[CrossRef] [PubMed]

1986 (3)

1985 (2)

M. D. Ewbank and P. Yeh, “Frequency shift and cavity length in photorefractive resonators,” Opt. Lett. 10, 496–499 (1985).
[CrossRef] [PubMed]

J. F. Lam, “Origin of phase conjugate waves in self-pumped photorefractive mirrors,” Appl. Phys. Lett. 46, 909–911 (1985).
[CrossRef]

1984 (1)

J. Ramsey and W. Whitten, “High-resolution self-scanning continuous wave dye laser,” Anal. Chem. 56, 2979–2981 (1984).
[CrossRef]

Belic, M. R.

Bledowski, A.

Boyd, R. W.

D. J. Gauthier, P. Narum, and R. W. Boyd, “Observation of deterministic chaos in a phase-conjugate mirror,” Phys. Rev. Lett. 58, 1640–1643 (1987).
[CrossRef] [PubMed]

Cronin-Golomb, M.

Dai, J. H.

Eason, R. W.

Ewbank, M. D.

Fischer, B.

Fisher, R. A.

Gauthier, D. J.

D. J. Gauthier, P. Narum, and R. W. Boyd, “Observation of deterministic chaos in a phase-conjugate mirror,” Phys. Rev. Lett. 58, 1640–1643 (1987).
[CrossRef] [PubMed]

Gower, M. C.

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).
[CrossRef]

M. C. Gower, “Photoinduced voltages and frequency shifts in a self-pumped phase-conjugating BaTiO3 crystal,” Opt. Lett. 11, 458–460 (1986).
[CrossRef] [PubMed]

Jeffrey, P. M.

Krolikowski, W.

Lam, J. F.

J. F. Lam, “Origin of phase conjugate waves in self-pumped photorefractive mirrors,” Appl. Phys. Lett. 46, 909–911 (1985).
[CrossRef]

Moore, T. R.

Narum, P.

D. J. Gauthier, P. Narum, and R. W. Boyd, “Observation of deterministic chaos in a phase-conjugate mirror,” Phys. Rev. Lett. 58, 1640–1643 (1987).
[CrossRef] [PubMed]

Nowak, A. V.

Ramsey, J.

J. Ramsey and W. Whitten, “High-resolution self-scanning continuous wave dye laser,” Anal. Chem. 56, 2979–2981 (1984).
[CrossRef]

Smout, A. M. C.

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).
[CrossRef]

Wang, P. Y.

P. Xie, P. Y. Wang, J. H. Dai, and H. J. Zhang, “Frequency shifts and dynamic instabilities in photorefractive self-pumped and mutually pumped phase conjugation,” J. Opt. Soc. Am. B 16, 420–427 (1999).
[CrossRef]

P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Self-pumped phase conjugation in photorefractive crystals: reflectivity and spatial fidelity,” Phys. Rev. A 55, 3092–3100 (1997).
[CrossRef]

Whitten, W.

J. Ramsey and W. Whitten, “High-resolution self-scanning continuous wave dye laser,” Anal. Chem. 56, 2979–2981 (1984).
[CrossRef]

Xie, P.

Yeh, P.

Zhang, H. J.

Anal. Chem. (1)

J. Ramsey and W. Whitten, “High-resolution self-scanning continuous wave dye laser,” Anal. Chem. 56, 2979–2981 (1984).
[CrossRef]

Appl. Phys. Lett. (1)

J. F. Lam, “Origin of phase conjugate waves in self-pumped photorefractive mirrors,” Appl. Phys. Lett. 46, 909–911 (1985).
[CrossRef]

J. Opt. Soc. Am. B (5)

Opt. Commun. (1)

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).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (2)

P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Self-pumped phase conjugation in photorefractive crystals: reflectivity and spatial fidelity,” Phys. Rev. A 55, 3092–3100 (1997).
[CrossRef]

W. Krolikowski, M. R. Belic, and A. Bledowski, “Phase transfer in optical phase conjugation,” Phys. Rev. A 37, 2224–2226 (1988).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

D. J. Gauthier, P. Narum, and R. W. Boyd, “Observation of deterministic chaos in a phase-conjugate mirror,” Phys. Rev. Lett. 58, 1640–1643 (1987).
[CrossRef] [PubMed]

Other (1)

See, for example, J. Feinberg and K. R. MacDonald, in Photorefractive Materials and Their Application, P. Gunter and J. P. Huignard, eds. (Springer-Verlag, New York, 1989), pp. 176–179 and reference therein.

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Figures (10)

Fig. 1
Fig. 1

Geometrical configuration for four-wave mixing.

Fig. 2
Fig. 2

Curve represents boundary between the stable state and the unstable state. The reflectivity of mirror M is taken as I2(L)/I1(L)=0.98.

Fig. 3
Fig. 3

Temporal evolution of (a) the phase-conjugate reflectivity R and (b) the frequency shift Δf for four-wave mixing. At t=0 the phase conjugator resides in the unstable steady state.

Fig. 4
Fig. 4

Geometrical configuration for multi-four-wave mixing.

Fig. 5
Fig. 5

Temporal evolution of phases of A31 (curve 1), A32 (curve 2), and A33 (curve 3) for multi-four-wave mixing. Temporal evolution of the phase of A3 for four-wave mixing for γ=8+2i (dashed line), 8-2i (dotted line), and 8-6i (dashed and dotted line). Plotted in the inset are input phases of A31, A32, A33 (hollow dots) and the output phases (filled dots) at an instant.

Fig. 6
Fig. 6

Temporal evolution of (a) the phase-conjugate reflectivity R and (b) the frequency shift Δf for multi-four-wave mixing.

Fig. 7
Fig. 7

Geometrical configuration for a cat self-pumped phase conjugator in a BaTiO3 crystal.

Fig. 8
Fig. 8

Temporal evolution of (a) the phase-conjugate reflectivity R, (b) the intensity of the totally reflecting beam at z=L, and (c) the frequency shift Δf for model (I). E0=72 V/mm.

Fig. 9
Fig. 9

Temporal evolution of (a) the phase-conjugate reflectivity R and (b) the frequency shift Δf for model (II). E0=70 V/mm.

Fig. 10
Fig. 10

Temporal evolution of the phase-conjugate reflectivity R for model (I). E0=68 V/mm. For t200τ the illuminating intensity is turned on.

Equations (26)

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A1/z=QA4,
A2*/z=QA3*,
A3/z=-QA2,
A4*/z=-QA1*,
τQt+Q=γI0(A1A4*+A2*A3),
I0dI1dz=2γR(I1I4+I1I2I3I4 cos Φ)-2γII1I2I3I4 sin Φ,
I0dI2dz=2γR(I2I3+I1I2I3I4 cos Φ)+2γII1I2I3I4 sin Φ,
I0dI3dz=-2γR(I3I2+I1I2I3I4 cos Φ)-2γII1I2I3I4 sin Φ,
I0dI4dz=-2γR(I4I1+I1I2I3I4 cos Φ)+2γII1I2I3I4 sin Φ,
I0dϕ1dz=γRI2I3I4I1 sin Φ+γI(I4+I2I3I4I1 cos Φ),
I0dϕ2dz=γRI1I3I4I2 sin Φ-γI(I3+I1I3I4I2 cos Φ),
I0dϕ3dz=γRI1I2I4I3 sin Φ-γII2+I1I2I4I3 cos Φ,
I0dϕ4dz=γRI1I2I3I4 sin Φ+γII1+I1I2I3I4 cos Φ,
Φ=ϕ3+ϕ4-ϕ1-ϕ2.
A1/z=j=13QjA4j(j=1, 2, 3),
A2*/z=j=13QjA3j*,
A3j/z=-QjA2,
A4j*/z=-QjA1*,
τQjt+Qj=γjI0(A1A4j*+A2*A3j).
fF(θ, z, t)z=1cos θθ[Q(θ, θ, z, t)fF(θ, z, t)]-αL2fF(θ, z, t),
fB*(θ, z, t)z=1cos θθ[Q(θ, θ, z, t)fB*(θ, z, t)]+αL2fB*(θ, z, t),
τ(θ, θ)Q(θ, θ, z, t)t+Q(θ, θ, z, t)
=γ(θ, θ)I0(z, t)[fF(θ, z, t)fF*(θ, z, t)+fB*(θ, z, t)fB(θ, z, t)],
γ(θ, θ)=-iωno32creff(θ, θ)Esc(θ, θ)cos(θ-θ),
Esc(θ, θ)=Eq(Ed-iE0)E0+i(Eq+Ed),
γ=γ1+Iin/I0,

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