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)

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]

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

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.

Feinberg, J.

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.

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]

MacDonald, K. R.

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.

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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