Abstract

Spatial and temporal effects arising in photorefractive crystals during the process of double phase conjugation are analyzed numerically with a novel beam-propagation method. Slowly varying envelope wave equations in the paraxial approximation are solved under the appropriate boundary conditions. Our analysis includes dynamical effects caused by the buildup of diffraction gratings in the crystal and the turn-on of phase-conjugate beams as well as spatial effects caused by the finite transverse spread of beams and by the propagation directions of the beams. Various phenomena are observed, such as self-bending of phase-conjugate beams, convective flow of energy out of the interaction region, mode oscillations, critical slowing down at the oscillation threshold, and irregular spatial pattern formation. For a real beam-coupling constant and constructive interaction of interference fringes in the crystal we find steady or periodic behavior. For a complex coupling constant and/or induced phase mismatch in the grating a transition to spatiotemporal chaos is observed. We believe that under stable operating conditions the transverse double phase-conjugate mirror in the paraxial approximation is a convective oscillator, rather than an amplifier. Improved agreement with experimental results is obtained.

© 1995 Optical Society of America

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References

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  1. M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
    [CrossRef]
  2. B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550–569 (1989); S. Sternklar, S. Weiss, M. Segev, and B. Fischer, “Beam coupling and locking of lasers using photorefractive four-wave mixing,” Opt. Lett. 11, 528–530 (1986); S. Weiss, S. Sternklar, and B. Fischer, “Double phase-conjugation: analysis, demonstration, and applications,” Opt. Lett. 12, 114–116 (1987).
    [CrossRef] [PubMed]
  3. A. A. Zozulya, “Double phase-conjugate mirror is not an oscillator,” Opt. Lett. 16, 545–547 (1991); V. V. Eliseev, V. T. Tikhonchuk, and A. A. Zozulya, “Double phase-conjugate mirror: two-dimensional analysis,” J. Opt. Soc. Am. B 8, 2497–2504 (1991).
    [CrossRef] [PubMed]
  4. N. V. Bogodaev, V. V. Eliseev, L. I. Ivleva, A. S. Korshunov, S. S. Orlov, N. M. Polozkov, and A. A. Zozulya, “Double phase-conjugate mirror: experimental investigation and comparison with theory,” J. Opt. Soc. Am. B 9, 1493–1498 (1992).
    [CrossRef]
  5. A. A. Zozulya, M. Saffman, and D. Z. Anderson, “Propagation of light beams in photorefractive media: fanning, self-bending, and formation of self-pumped four-wave-mixing phase conjugation geometries,” Phys. Rev. Lett. 73, 818–821 (1994); “Double phase-conjugate mirror: convection and diffraction,” J. Opt. Soc. Am. B 12, 255–264 (1995).
    [CrossRef] [PubMed]
  6. M. Segev, D. Engin, A. Yariv, and G. C. Valley, “Temporal evolution of photorefractive double phase-conjugate mirrors,” Opt. Lett. 18, 1828–1830 (1993); S. Orlov, M. Segev, A. Yariv, and G. C. Valley, “Conjugation fidelity and reflectivity in photorefractive double phase-conjugate mirrors,” Opt. Lett. 19, 578–580 (1994); D. Engin, M. Segev, S. Orlov, and A. Yariv, “Double phase conjugation,” J. Opt. Soc. Am. B 11, 1708–1717 (1994).
    [CrossRef] [PubMed]
  7. K. D. Shaw, “The double phase conjugate mirror is an oscillator,” Opt. Commun. 90, 133–138 (1992).
    [CrossRef]
  8. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York), Chap. 6, p. 112.
  9. The situation has changed considerably since this paper was submitted for publication. A number of treatments have appeared and are listed in Refs. 5, 6, and 10.
  10. M. Cronin-Golomb, “Whole beam method for photorefractive nonlinear optics,” Opt. Commun. 89, 276–282 (1992); K. Ratnam and P. P. Banerjee, “Nonlinear theory of two-beam coupling in a photorefractive material,” Opt. Commun. 107, 522–530 (1994).
    [CrossRef]
  11. S. R. Liu and G. Indebetouw, “Spatiotemporal patterns and vortices dynamics in phase conjugate resonators,” Opt. Commun. 101, 442–455 (1993).
    [CrossRef]
  12. W. Krolikowski, M. R. Belić, M. Cronin-Golomb, and A. Bledowski, “Chaos in photorefractive four-wave mixing with a single interaction region,” J. Opt. Soc. Am. B 7, 1204–1209 (1990).
    [CrossRef]
  13. J. V. Moloney, M. R. Belić, and H. M. Gibbs, “Calculation of transverse effects in optical bistability using fast Fourier transform techniques,” Opt. Commun. 41, 379–382 (1982); M. Lax, G. P. Agrawal, M. R. Belić, B. J. Coffey, and W. L. Louisell, “Electromagnetic field distribution in loaded unstable resonators,” J. Opt. Soc. Am. A 2, 731–742 (1985).
    [CrossRef]
  14. M. R. Belić and M. Petrović, “Unified method for solution of wave equations in photorefractive media,” J. Opt. Soc. Am. B 11, 481–485 (1994).
    [CrossRef]
  15. M. R. Belić, J. Leonardy, D. Timotijević, and F. Kaiser, “Transverse effects in double phase conjugation,” Opt. Commun. 111, 99–104 (1994).
    [CrossRef]
  16. N. Wolffer, P. Gravey, J. V. Moisan, C. Laulan, and J. C. Launay, “Analysis of double phase conjugate mirror interaction in absorbing photorefractive crystals: application to BGO:Cu,” Opt. Commun. 73, 351–356 (1989); N. Wolffer and P. Gravey, “High quality phase conjugation in a double phase conjugate mirror using InP:Fe at 1.3 μ m,” Opt. Commun. 107, 115–119 (1994).
    [CrossRef]
  17. F. T. Arecchi, “Space-time complexity in nonlinear optics,” Physica D 51, 450–464 (1991); F. T. Arecchi, S. Boccaletti, G. Giacomelli, G. P. Puccioni, P. L. Ramazza, and S. Residori, “Space–time chaos and topological defects in nonlinear optics,” Physica D 61, 25–39 (1992).
    [CrossRef]
  18. W. Krolikowski and B. Luther-Davies, “The effect of a high external electric field on a photorefractive ring phase conjugator,” Appl. Phys. B 55, 180–182 (1992).
    [CrossRef]

1994 (3)

A. A. Zozulya, M. Saffman, and D. Z. Anderson, “Propagation of light beams in photorefractive media: fanning, self-bending, and formation of self-pumped four-wave-mixing phase conjugation geometries,” Phys. Rev. Lett. 73, 818–821 (1994); “Double phase-conjugate mirror: convection and diffraction,” J. Opt. Soc. Am. B 12, 255–264 (1995).
[CrossRef] [PubMed]

M. R. Belić and M. Petrović, “Unified method for solution of wave equations in photorefractive media,” J. Opt. Soc. Am. B 11, 481–485 (1994).
[CrossRef]

M. R. Belić, J. Leonardy, D. Timotijević, and F. Kaiser, “Transverse effects in double phase conjugation,” Opt. Commun. 111, 99–104 (1994).
[CrossRef]

1993 (2)

1992 (4)

K. D. Shaw, “The double phase conjugate mirror is an oscillator,” Opt. Commun. 90, 133–138 (1992).
[CrossRef]

M. Cronin-Golomb, “Whole beam method for photorefractive nonlinear optics,” Opt. Commun. 89, 276–282 (1992); K. Ratnam and P. P. Banerjee, “Nonlinear theory of two-beam coupling in a photorefractive material,” Opt. Commun. 107, 522–530 (1994).
[CrossRef]

N. V. Bogodaev, V. V. Eliseev, L. I. Ivleva, A. S. Korshunov, S. S. Orlov, N. M. Polozkov, and A. A. Zozulya, “Double phase-conjugate mirror: experimental investigation and comparison with theory,” J. Opt. Soc. Am. B 9, 1493–1498 (1992).
[CrossRef]

W. Krolikowski and B. Luther-Davies, “The effect of a high external electric field on a photorefractive ring phase conjugator,” Appl. Phys. B 55, 180–182 (1992).
[CrossRef]

1991 (2)

F. T. Arecchi, “Space-time complexity in nonlinear optics,” Physica D 51, 450–464 (1991); F. T. Arecchi, S. Boccaletti, G. Giacomelli, G. P. Puccioni, P. L. Ramazza, and S. Residori, “Space–time chaos and topological defects in nonlinear optics,” Physica D 61, 25–39 (1992).
[CrossRef]

A. A. Zozulya, “Double phase-conjugate mirror is not an oscillator,” Opt. Lett. 16, 545–547 (1991); V. V. Eliseev, V. T. Tikhonchuk, and A. A. Zozulya, “Double phase-conjugate mirror: two-dimensional analysis,” J. Opt. Soc. Am. B 8, 2497–2504 (1991).
[CrossRef] [PubMed]

1990 (1)

1989 (2)

N. Wolffer, P. Gravey, J. V. Moisan, C. Laulan, and J. C. Launay, “Analysis of double phase conjugate mirror interaction in absorbing photorefractive crystals: application to BGO:Cu,” Opt. Commun. 73, 351–356 (1989); N. Wolffer and P. Gravey, “High quality phase conjugation in a double phase conjugate mirror using InP:Fe at 1.3 μ m,” Opt. Commun. 107, 115–119 (1994).
[CrossRef]

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550–569 (1989); S. Sternklar, S. Weiss, M. Segev, and B. Fischer, “Beam coupling and locking of lasers using photorefractive four-wave mixing,” Opt. Lett. 11, 528–530 (1986); S. Weiss, S. Sternklar, and B. Fischer, “Double phase-conjugation: analysis, demonstration, and applications,” Opt. Lett. 12, 114–116 (1987).
[CrossRef] [PubMed]

1984 (1)

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

1982 (1)

J. V. Moloney, M. R. Belić, and H. M. Gibbs, “Calculation of transverse effects in optical bistability using fast Fourier transform techniques,” Opt. Commun. 41, 379–382 (1982); M. Lax, G. P. Agrawal, M. R. Belić, B. J. Coffey, and W. L. Louisell, “Electromagnetic field distribution in loaded unstable resonators,” J. Opt. Soc. Am. A 2, 731–742 (1985).
[CrossRef]

Anderson, D. Z.

A. A. Zozulya, M. Saffman, and D. Z. Anderson, “Propagation of light beams in photorefractive media: fanning, self-bending, and formation of self-pumped four-wave-mixing phase conjugation geometries,” Phys. Rev. Lett. 73, 818–821 (1994); “Double phase-conjugate mirror: convection and diffraction,” J. Opt. Soc. Am. B 12, 255–264 (1995).
[CrossRef] [PubMed]

Arecchi, F. T.

F. T. Arecchi, “Space-time complexity in nonlinear optics,” Physica D 51, 450–464 (1991); F. T. Arecchi, S. Boccaletti, G. Giacomelli, G. P. Puccioni, P. L. Ramazza, and S. Residori, “Space–time chaos and topological defects in nonlinear optics,” Physica D 61, 25–39 (1992).
[CrossRef]

Belic, M. R.

M. R. Belić, J. Leonardy, D. Timotijević, and F. Kaiser, “Transverse effects in double phase conjugation,” Opt. Commun. 111, 99–104 (1994).
[CrossRef]

M. R. Belić and M. Petrović, “Unified method for solution of wave equations in photorefractive media,” J. Opt. Soc. Am. B 11, 481–485 (1994).
[CrossRef]

W. Krolikowski, M. R. Belić, M. Cronin-Golomb, and A. Bledowski, “Chaos in photorefractive four-wave mixing with a single interaction region,” J. Opt. Soc. Am. B 7, 1204–1209 (1990).
[CrossRef]

J. V. Moloney, M. R. Belić, and H. M. Gibbs, “Calculation of transverse effects in optical bistability using fast Fourier transform techniques,” Opt. Commun. 41, 379–382 (1982); M. Lax, G. P. Agrawal, M. R. Belić, B. J. Coffey, and W. L. Louisell, “Electromagnetic field distribution in loaded unstable resonators,” J. Opt. Soc. Am. A 2, 731–742 (1985).
[CrossRef]

Bledowski, A.

Bogodaev, N. V.

Cronin-Golomb, M.

M. Cronin-Golomb, “Whole beam method for photorefractive nonlinear optics,” Opt. Commun. 89, 276–282 (1992); K. Ratnam and P. P. Banerjee, “Nonlinear theory of two-beam coupling in a photorefractive material,” Opt. Commun. 107, 522–530 (1994).
[CrossRef]

W. Krolikowski, M. R. Belić, M. Cronin-Golomb, and A. Bledowski, “Chaos in photorefractive four-wave mixing with a single interaction region,” J. Opt. Soc. Am. B 7, 1204–1209 (1990).
[CrossRef]

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

Eliseev, V. V.

Engin, D.

Fischer, B.

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550–569 (1989); S. Sternklar, S. Weiss, M. Segev, and B. Fischer, “Beam coupling and locking of lasers using photorefractive four-wave mixing,” Opt. Lett. 11, 528–530 (1986); S. Weiss, S. Sternklar, and B. Fischer, “Double phase-conjugation: analysis, demonstration, and applications,” Opt. Lett. 12, 114–116 (1987).
[CrossRef] [PubMed]

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

Gibbs, H. M.

J. V. Moloney, M. R. Belić, and H. M. Gibbs, “Calculation of transverse effects in optical bistability using fast Fourier transform techniques,” Opt. Commun. 41, 379–382 (1982); M. Lax, G. P. Agrawal, M. R. Belić, B. J. Coffey, and W. L. Louisell, “Electromagnetic field distribution in loaded unstable resonators,” J. Opt. Soc. Am. A 2, 731–742 (1985).
[CrossRef]

Gravey, P.

N. Wolffer, P. Gravey, J. V. Moisan, C. Laulan, and J. C. Launay, “Analysis of double phase conjugate mirror interaction in absorbing photorefractive crystals: application to BGO:Cu,” Opt. Commun. 73, 351–356 (1989); N. Wolffer and P. Gravey, “High quality phase conjugation in a double phase conjugate mirror using InP:Fe at 1.3 μ m,” Opt. Commun. 107, 115–119 (1994).
[CrossRef]

Indebetouw, G.

S. R. Liu and G. Indebetouw, “Spatiotemporal patterns and vortices dynamics in phase conjugate resonators,” Opt. Commun. 101, 442–455 (1993).
[CrossRef]

Ivleva, L. I.

Kaiser, F.

M. R. Belić, J. Leonardy, D. Timotijević, and F. Kaiser, “Transverse effects in double phase conjugation,” Opt. Commun. 111, 99–104 (1994).
[CrossRef]

Korshunov, A. S.

Krolikowski, W.

W. Krolikowski and B. Luther-Davies, “The effect of a high external electric field on a photorefractive ring phase conjugator,” Appl. Phys. B 55, 180–182 (1992).
[CrossRef]

W. Krolikowski, M. R. Belić, M. Cronin-Golomb, and A. Bledowski, “Chaos in photorefractive four-wave mixing with a single interaction region,” J. Opt. Soc. Am. B 7, 1204–1209 (1990).
[CrossRef]

Laulan, C.

N. Wolffer, P. Gravey, J. V. Moisan, C. Laulan, and J. C. Launay, “Analysis of double phase conjugate mirror interaction in absorbing photorefractive crystals: application to BGO:Cu,” Opt. Commun. 73, 351–356 (1989); N. Wolffer and P. Gravey, “High quality phase conjugation in a double phase conjugate mirror using InP:Fe at 1.3 μ m,” Opt. Commun. 107, 115–119 (1994).
[CrossRef]

Launay, J. C.

N. Wolffer, P. Gravey, J. V. Moisan, C. Laulan, and J. C. Launay, “Analysis of double phase conjugate mirror interaction in absorbing photorefractive crystals: application to BGO:Cu,” Opt. Commun. 73, 351–356 (1989); N. Wolffer and P. Gravey, “High quality phase conjugation in a double phase conjugate mirror using InP:Fe at 1.3 μ m,” Opt. Commun. 107, 115–119 (1994).
[CrossRef]

Leonardy, J.

M. R. Belić, J. Leonardy, D. Timotijević, and F. Kaiser, “Transverse effects in double phase conjugation,” Opt. Commun. 111, 99–104 (1994).
[CrossRef]

Liu, S. R.

S. R. Liu and G. Indebetouw, “Spatiotemporal patterns and vortices dynamics in phase conjugate resonators,” Opt. Commun. 101, 442–455 (1993).
[CrossRef]

Luther-Davies, B.

W. Krolikowski and B. Luther-Davies, “The effect of a high external electric field on a photorefractive ring phase conjugator,” Appl. Phys. B 55, 180–182 (1992).
[CrossRef]

Moisan, J. V.

N. Wolffer, P. Gravey, J. V. Moisan, C. Laulan, and J. C. Launay, “Analysis of double phase conjugate mirror interaction in absorbing photorefractive crystals: application to BGO:Cu,” Opt. Commun. 73, 351–356 (1989); N. Wolffer and P. Gravey, “High quality phase conjugation in a double phase conjugate mirror using InP:Fe at 1.3 μ m,” Opt. Commun. 107, 115–119 (1994).
[CrossRef]

Moloney, J. V.

J. V. Moloney, M. R. Belić, and H. M. Gibbs, “Calculation of transverse effects in optical bistability using fast Fourier transform techniques,” Opt. Commun. 41, 379–382 (1982); M. Lax, G. P. Agrawal, M. R. Belić, B. J. Coffey, and W. L. Louisell, “Electromagnetic field distribution in loaded unstable resonators,” J. Opt. Soc. Am. A 2, 731–742 (1985).
[CrossRef]

Orlov, S. S.

Petrovic, M.

Polozkov, N. M.

Saffman, M.

A. A. Zozulya, M. Saffman, and D. Z. Anderson, “Propagation of light beams in photorefractive media: fanning, self-bending, and formation of self-pumped four-wave-mixing phase conjugation geometries,” Phys. Rev. Lett. 73, 818–821 (1994); “Double phase-conjugate mirror: convection and diffraction,” J. Opt. Soc. Am. B 12, 255–264 (1995).
[CrossRef] [PubMed]

Segev, M.

Shaw, K. D.

K. D. Shaw, “The double phase conjugate mirror is an oscillator,” Opt. Commun. 90, 133–138 (1992).
[CrossRef]

Sternklar, S.

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550–569 (1989); S. Sternklar, S. Weiss, M. Segev, and B. Fischer, “Beam coupling and locking of lasers using photorefractive four-wave mixing,” Opt. Lett. 11, 528–530 (1986); S. Weiss, S. Sternklar, and B. Fischer, “Double phase-conjugation: analysis, demonstration, and applications,” Opt. Lett. 12, 114–116 (1987).
[CrossRef] [PubMed]

Timotijevic, D.

M. R. Belić, J. Leonardy, D. Timotijević, and F. Kaiser, “Transverse effects in double phase conjugation,” Opt. Commun. 111, 99–104 (1994).
[CrossRef]

Valley, G. C.

Weiss, S.

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550–569 (1989); S. Sternklar, S. Weiss, M. Segev, and B. Fischer, “Beam coupling and locking of lasers using photorefractive four-wave mixing,” Opt. Lett. 11, 528–530 (1986); S. Weiss, S. Sternklar, and B. Fischer, “Double phase-conjugation: analysis, demonstration, and applications,” Opt. Lett. 12, 114–116 (1987).
[CrossRef] [PubMed]

White, J. O.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

Wolffer, N.

N. Wolffer, P. Gravey, J. V. Moisan, C. Laulan, and J. C. Launay, “Analysis of double phase conjugate mirror interaction in absorbing photorefractive crystals: application to BGO:Cu,” Opt. Commun. 73, 351–356 (1989); N. Wolffer and P. Gravey, “High quality phase conjugation in a double phase conjugate mirror using InP:Fe at 1.3 μ m,” Opt. Commun. 107, 115–119 (1994).
[CrossRef]

Yariv, A.

Zozulya, A. A.

Appl. Phys. B (1)

W. Krolikowski and B. Luther-Davies, “The effect of a high external electric field on a photorefractive ring phase conjugator,” Appl. Phys. B 55, 180–182 (1992).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550–569 (1989); S. Sternklar, S. Weiss, M. Segev, and B. Fischer, “Beam coupling and locking of lasers using photorefractive four-wave mixing,” Opt. Lett. 11, 528–530 (1986); S. Weiss, S. Sternklar, and B. Fischer, “Double phase-conjugation: analysis, demonstration, and applications,” Opt. Lett. 12, 114–116 (1987).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B (3)

Opt. Commun. (6)

M. R. Belić, J. Leonardy, D. Timotijević, and F. Kaiser, “Transverse effects in double phase conjugation,” Opt. Commun. 111, 99–104 (1994).
[CrossRef]

N. Wolffer, P. Gravey, J. V. Moisan, C. Laulan, and J. C. Launay, “Analysis of double phase conjugate mirror interaction in absorbing photorefractive crystals: application to BGO:Cu,” Opt. Commun. 73, 351–356 (1989); N. Wolffer and P. Gravey, “High quality phase conjugation in a double phase conjugate mirror using InP:Fe at 1.3 μ m,” Opt. Commun. 107, 115–119 (1994).
[CrossRef]

J. V. Moloney, M. R. Belić, and H. M. Gibbs, “Calculation of transverse effects in optical bistability using fast Fourier transform techniques,” Opt. Commun. 41, 379–382 (1982); M. Lax, G. P. Agrawal, M. R. Belić, B. J. Coffey, and W. L. Louisell, “Electromagnetic field distribution in loaded unstable resonators,” J. Opt. Soc. Am. A 2, 731–742 (1985).
[CrossRef]

K. D. Shaw, “The double phase conjugate mirror is an oscillator,” Opt. Commun. 90, 133–138 (1992).
[CrossRef]

M. Cronin-Golomb, “Whole beam method for photorefractive nonlinear optics,” Opt. Commun. 89, 276–282 (1992); K. Ratnam and P. P. Banerjee, “Nonlinear theory of two-beam coupling in a photorefractive material,” Opt. Commun. 107, 522–530 (1994).
[CrossRef]

S. R. Liu and G. Indebetouw, “Spatiotemporal patterns and vortices dynamics in phase conjugate resonators,” Opt. Commun. 101, 442–455 (1993).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (1)

A. A. Zozulya, M. Saffman, and D. Z. Anderson, “Propagation of light beams in photorefractive media: fanning, self-bending, and formation of self-pumped four-wave-mixing phase conjugation geometries,” Phys. Rev. Lett. 73, 818–821 (1994); “Double phase-conjugate mirror: convection and diffraction,” J. Opt. Soc. Am. B 12, 255–264 (1995).
[CrossRef] [PubMed]

Physica D (1)

F. T. Arecchi, “Space-time complexity in nonlinear optics,” Physica D 51, 450–464 (1991); F. T. Arecchi, S. Boccaletti, G. Giacomelli, G. P. Puccioni, P. L. Ramazza, and S. Residori, “Space–time chaos and topological defects in nonlinear optics,” Physica D 61, 25–39 (1992).
[CrossRef]

Other (2)

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York), Chap. 6, p. 112.

The situation has changed considerably since this paper was submitted for publication. A number of treatments have appeared and are listed in Refs. 5, 6, and 10.

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Figures (17)

Fig. 1
Fig. 1

DPCM. Pump beams A2 and A4 enter the crystal from opposite sides. A1 is the PC of A2, and A3 the PC of A4. z is the propagation direction and x is one of the transverse directions, the other, y, being perpendicular to the xz plane. Q represents the amplitude of the transmission grating. V is the high-voltage source of the electric field E0 (see Section 5).

Fig. 2
Fig. 2

Self-consistency parameter a as a function of the coupling strength Γ. Solid curves are the numerical solutions of Eq. (B6); dashed curves are the approximate solutions, Eq. (B7). Curves AA and BB are the simple PC curves, with the boundary conditions |C1|2 = |C2|2 = |1, |C3|2 = 0, and |C4|2 = 0.7 for the AA and |C4|2 = 0.4 for the BB curves. Curve CC is the 4WM curve, with the conditions |C1|2 = 0.4, |C2|2 = 1, |C3|2 = 0.1, and |C4|2 = 1. Curve D is the DPC threshold curve, Eq. (7), with the conditions |C2|2 = 1, |C4|2 = 0.1 (arbitrary units).

Fig. 3
Fig. 3

Checking the numerics: comparison with PW solutions (curves, analytical results; squares, numerical results). Four fields inside the crystal are shown as functions of the longitudinal spatial variable z. (a) δz = 0.01, (b) δz = 0.005. In the following computations δz is kept fixed at 0.005. The other parameters are β = 0, Γ = 5, η = 1, |C2|2 = 0.1, |C4|2 = 1, |C1|2 = |C4|2, |C3|2 = |C2|2, and = 10−9. Linear dependence of the numerical error on δz is evident.

Fig. 4
Fig. 4

(a) Limit cycle oscillation of the total transmissivity. Solid curve, T0; dashed curve, Td. (b) Contour plot of the output profile I3 versus transverse coordinates at t = 150τ. The profile is shifted from the center at x = 0, y = 0 in the direction y = x of the grating wave vector. (c) The profile at t = 180τ, close to the maximum of the cycle. The parameters are as in Fig. 3, except that Γ = 3, = 10−5, and the nonzero ϕ = 0.01.

Fig. 5
Fig. 5

(a) Total transmissivities in the steady-convection state when numerical absorption is introduced at the edge of the crystal. The parameters are as in Fig. 4, except for the higher value of ϕ (ϕ = 0.03). (b) Transverse profile of I3. (c) Transverse profile of I3 represented as a contour plot, displaying the yx symmetry built into the model. This symmetry fixes the direction of convection.

Fig. 6
Fig. 6

Reflectivities R = R0 = Rd as functions of the coupling strength Γ for β = 0. The amplitudes of the Gaussian pump beams are chosen equal, C2 = C4 = 1, whereas the input of PC beams |C1|2 = |C3|2 = is varied. (a) PW results for different beam seeds : solid curve, = 10−1; dashed curve, = 10−2; dotted–dashed curve, = 10−5; dotted curve, = 10−9. The reflectivities are calculated by both the analytical formulas and the numerical method (with ϕ = 10−20). The threshold condition Γth = 2 for oscillation is clearly visible. (b) Reflectivities in the transverse case for different ϕ and the fixed seed ( = 109): ϕ = 10−20 (solid curve), ϕ = 10−2 (dashed curve), ϕ = 101 (dotted–dashed curve), ϕ = 1 (dotted curve). The inset shows an enlargement of the take-off region for ϕ = 0.1 and ϕ = 1. The threshold is now smeared over an interval, and for higher ϕ it might even not exist.

Fig. 7
Fig. 7

Reflectivities versus coupling strength on the logarithmic scale for different seeds . The existence of an oscillation threshold as goes to zero is evident. Below the threshold DPCM is an amplifier; above the threshold the saturation of reflectivities is noted. The shift of the threshold and the reduction of derivatives are shown for three values of ϕ = f: squares, = 10−1; circles, = 10−2; triangles, = 10−5; crosses, = 10−9. The curves are polynomial fits through the points, drawn to guide the eye.

Fig. 8
Fig. 8

Dynamics of the oscillation switch on, for two values of the transverse displacement β and for different values of the coupling strength Γ. Total reflectivities are presented as functions of time for different values of the seed (solid curves, = 10−1; dashed curves, = 10−2; dotted–dashed curves, = 10−5; dotted curves, = 10−9). For (a)–(c) β = 0.1, and the device acts as an amplifier. For (d)–(f) β = 0.01, and the device acts as an oscillator. In (a) (when the whole curve for = 10−9 is multiplied by 50) and (d) Γ = 3, in (b) and (e) Γ = 4, and in (c) and (f) Γ = 5. Here ϕ = 0.

Fig. 9
Fig. 9

Transverse profiles of the beam I30 for ϕ = 0 and = 10−5 and for different values of the coupling strength Γ and of the transverse displacement β: (a) Γ = 3, (b) Γ = 5. The value of β is given in each figure. The dashed curves are profiles of one of the pumps (for β = 0.1). (c) Spatial transverse distribution of beams I2 and I3 in the crystal for β = 0.2 and Γ = 5. A small amount of seed ( = 105) and of diffraction (ϕ = 2.72 × 10−4) is included. (d) Same as (c) for the total intensity.

Fig. 10
Fig. 10

(a) Dynamics of the total transmissivities T = T0 = Td for ϕ = 0.05 and Γ = 3. (b) Contour plot of the transverse profile of I3 close to the cycle minimum at t = 215τ. (c) Contour plot at the cycle maximum, t = 240τ.

Fig. 11
Fig. 11

Transverse patterns of the PC field I3 at four locations during one cycle in Fig. 10. (a) t = 200τ, (b) t = 205τ, (c) t = 215τ, (d) t = 240τ. A periodic rise of the convective pulse is observed. The pulse is absorbed at the edge of the crystal.

Fig. 12
Fig. 12

Comparison with the experimental results of Ref. 2: (a) Reflectivities R0 and Rd at both sides of the crystal as functions of the pump ratio r. Filled circles are the experimental values of R0, and crossed circles are the values of Rd. Dashed curves are polynomial fits through the experimental points. Solid curves are numerical curves with Γ = 4 and ϕ = 0.121. (b) Corresponding transmissivities T0 and Td. One of the solid curves is a fourth-order polynomial fit through the experimental points for T0, and the other is the corresponding numerical curve. Dashed curves are the same for Td.

Fig. 13
Fig. 13

Temporal signal for an ordered PW state with the external electric field applied transversely across the crystal (along the x direction). (a) Transmissivities, (b) phase portrait of the lower, stable state, showing that indeed it is a limit cycle. The parameters are |C1|2 = |C3|2 = 10−5, |C2|2 = |C4|2 = 1, Γ0 = 4, ϕ = 10−20, EM = 100, Eq = 5, ED = 1, and E0 = 4 (arbitrary units).

Fig. 14
Fig. 14

Temporal behavior with the inclusion of transverse effects. Left column, transmissivity T = T0 = Td; right column, phase portrait of A3 (at z = 0) in the center of the beam after 1000τ steps. The same parameters as in Fig. 13, except for (a) ϕ = 0.006, (b) ϕ = 0.003, (c) ϕ = 0.002, (d) ϕ = 0.001.

Fig. 15
Fig. 15

Temporal transverse signal of the field I3(x) after the transients have died away. The parameters are as in Fig. 14.

Fig. 16
Fig. 16

Filamented transverse profiles of I3 (solid curves) and I2 (dashed curves) at z = 0 at different instants: (a) t = 1000τ, (b) t = 1300τ, (c) t = 1500τ. The simulation corresponds to that of Figs. 14(d) and 15(d). Note the partial anticorrelation between the I3 and the I2 peaks.

Fig. 17
Fig. 17

Transverse distribution of the field I3 in the crystal at different times. (a) t = 1000τ, (b) t = 1300τ, (c) t = 1500τ. The simulation corresponds to that of Figs. 14(d) and 15(d).

Equations (42)

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z A 1 + β K ^ · T A 1 + i ϕ T 2 A 1 = Q A 4 ,
z A 2 + β K ^ · T A 2 - i ϕ T 2 A 2 = Q ¯ A 3 ,
z A 3 - β K ^ · T A 3 - i ϕ T 2 A 3 = - Q A 2 ,
z A 4 - β K ^ · T A 4 + i ϕ T 2 A 4 = - Q ¯ A 1 ,
A 4 , 1 ( x , y , 0 ) = C 4 , 1 G ( - ζ , ρ ) , A 2 , 3 ( x , y , d ) = C 2 , 3 G ( ζ , ρ ) ,
G ( ζ , ρ ) = exp { i [ tan - 1 ( ζ ) - ρ 2 ζ + ζ - 1 ] - ρ 2 1 + ζ 2 } 1 + ζ 2 .
τ t Q + η Q = Γ I ( A 1 A ¯ 4 + A ¯ 2 A 3 ) ,
A 30 = C 2 sin ( u ) ,             A 1 d = C 4 sin ( u ) ,
tan ( Θ 0 ) = exp ( - a Γ 2 ) ( a - q * a + q * ) 1 / 2 , tan ( Θ d ) = exp ( a Γ 2 ) ( a - q * a + q * ) 1 / 2 ,
a = tanh ( a Γ / 2 ) .
Γ = Γ 0 E q + E D E D E D + i E 0 E M + E D + i E 0 ,
η = E D + E q + i E 0 E M + E D + i E 0 ,
z A 1 + i 2 k Δ T A 1 = Q A 4 ,
z A 2 - i 2 k Δ T A 2 = Q ¯ A 3 ,
z A 3 - i 2 k Δ T A 3 = - Q A 2 ,
z A 4 + i 2 k Δ T A 4 = - Q ¯ A 1 ,
z A 1 + θ x A 1 + i 2 k ( x 2 - 2 θ x z ) A 1 = Q A 4 ,
z A 2 + θ x A 2 - i 2 k ( x 2 - 2 θ x z ) A 2 = Q ¯ A 3 ,
z A 3 - θ x A 3 - i 2 k ( x 2 + 2 θ x z ) A 3 = - Q A 2 ,
z A 4 - θ x A 4 + i 2 k ( x 2 + 2 θ x z ) A 4 = - Q ¯ A 1 ,
z A 1 + θ δ x A 1 + i ϕ ( x 2 - 2 θ δ x z ) A 1 = Q A 4 ,
z A 2 + θ δ x A 2 - i ϕ ( x 2 - 2 θ δ x z ) A 2 = Q ¯ A 3 ,
z A 3 - θ δ x A 3 - i ϕ ( x 2 + 2 θ δ x z ) A 3 = - Q A 2 ,
z A 4 - θ δ x A 4 + i ϕ ( x 2 + 2 θ δ x z ) A 4 = - Q ¯ A 1 ,
Θ = Γ Q I ,
A 1 = C 1 cos ( Θ - Θ 0 ) + C 4 sin ( Θ - Θ 0 ) ,
A 4 = C 4 cos ( Θ - Θ 0 ) - C 1 sin ( Θ - Θ 0 ) ,
A 3 = C 3 cos ( Θ d - Θ ) + C 2 sin ( Θ d - Θ ) ,
A 2 = C 2 cos ( Θ d - Θ ) - C 3 sin ( Θ d - Θ ) ,
2 Q = a I sin ( 2 Θ ) ,
tan ( Θ ) = tan ( Θ 0 ) exp ( a Γ z ) .
tan ( u ) = q b - v ,
tan ( s ) = q b w b - c ,
b sin ( u ) = a I sin ( s ) ,
b 2 q 2 + ( w b - c ) 2 = a 2 I 2 [ q 2 + ( b - v ) 2 ] .
a = 1 I [ 4 I 2 q 2 + ( 2 I w - c Γ ) 2 ( q Γ ) 2 + ( 2 I - v Γ ) 2 ] 1 / 2 .
( z + i β k - i ϕ k 2 ) A ˜ 1 ( z ) = Q A ˜ 4 ( z ) ,
A ˜ 1 ( z ) = exp [ i ( ϕ k 2 - β k ) z ] × { A ˜ 1 ( 0 ) + 0 z Q A ˜ 4 ( z ) exp [ - i ( ϕ k 2 - β k ) z ] d z }
A ˜ 1 ( z + δ z ) = exp [ i ( ϕ k 2 - β k ) δ z ] A ˜ 1 ( z ) + exp [ i ( ϕ k 2 - β k ) ( z + δ z ) ] z z + δ z Q A ˜ 4 ( z ) × exp [ - i ( ϕ k 2 - β k ) z ] d z .
A ˜ 1 ( z + δ z ) = exp [ i ( ϕ k 2 - β k ) δ z ] A ˜ 1 ( z ) + i Q A ˜ 4 ( z ) 1 - exp [ i ( ϕ k 2 - β k ) δ z ] ϕ k 2 - β k ,
Q ( t ) = exp ( - η t τ ) [ Q ( 0 ) + Γ 0 t q ( t ) I ( t ) exp ( η t τ ) d t τ ] .
Q ( t + δ t ) = exp ( - η δ t τ ) Q ( t ) + Γ q ( t ) η I ( t ) [ 1 - exp ( - η δ t τ ) ] .

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