Abstract

A systematic study of degenerate two-wave mixing in the stationary regime in thin nonlinear local-response media is presented. The model is based on the coherent interaction of four waves, the two incident beams, and the two main diffracted waves. Analytical results are obtained in all cases investigated under the assumption of weak-beam amplification. It is demonstrated that the energy transfer arises only from the forward degenerate four-wave mixing terms. In particular, the theoretical results show two new features: (1) the gain factor of the interaction is substantially reduced because of the phase change associated with the two-wave mixing terms and (2) a phase mismatch can compensate for this phase change, leading to an appreciable gain enhancement. Moreover, the general case is solved numerically. Particular attention is given to the influence of separate two-wave and forward degenerate four-wave mixing terms. Self-diffraction effects and the gain dependence versus the ratio of the pump-beam intensity to the signal-beam intensity are investigated.

© 1992 Optical Society of America

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References

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  1. P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
    [CrossRef]
  2. P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications (Springer-Verlag, Berlin, 1988, 1989), Vols. 1 and 2.
    [CrossRef]
  3. I. C. Khoo, R. Normandin, T. H. Liu, and R. R. Michael, “Degenerate multiwave mixing and phase conjugation in silicon,” Phys. Rev. B 40, 7759–7766 (1989).
    [CrossRef]
  4. I. C. Khoo and P. Zhou, “Transient multiwave mixing in a nonlinear medium,” Phys. Rev. A 41, 1544–1555 (1990).
    [CrossRef] [PubMed]
  5. I. C. Khoo, W. Wang, F. Simoni, G. Cipparrone, and D. Duca, “Theory and experiments on stationary degenerate and nearly degenerate optical wave mixing and ring oscillation in a Kerr-like medium,” J. Opt. Soc. Am. B 8, 1433–1441 (1991).
    [CrossRef]
  6. Workshop on Nonlinear Optical Materials, “Research on nonlinear optical materials: an assessment,” Appl. Opt. 26, 211–234 (1987).
    [PubMed]
  7. P. Yeh, “Exact solution of a nonlinear model of two-wave mixing in Kerr media,” J. Opt. Soc. Am. B 3, 747–750 (1986).
    [CrossRef]
  8. P. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiment,” J. Appl. Phys. 58, 45–57 (1985).
    [CrossRef]
  9. J.-P. Huignard, H. Rajbenbach, P. Refregier, and L. Solymar, “Wave mixing in photorefractive BSO crystals and its applications,” Opt. Eng. 24, 586–592 (1985).
  10. F. Sanchez, P. H. Kayoun, and J.-P. Huignard, “Optical amplification at 10.6 μm using thermal nonlinearities in liquid crystals near the phase transition,” in Conference on Lasers and Electro-Optics, Vol. 14 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), pp. 108–109.
  11. F. Sanchez, P. H. Kayoun, and J.-P. Huignard, “Two-wave mixing with gain in liquid crystals at 10.6 μm wavelength,” J. Appl. Phys. 64, 26–31 (1988).
    [CrossRef]
  12. F. Sanchez, P. H. Kayoun, and J.-P. Huignard, “Wave-mixing in liquid crystals and mercury cadmium telluride,” in Advanced Optoelectronic Technology, D. B. Ostrowsky and C. Puech, eds., Proc. Soc. Photo-Opt. Instrum. Eng.864, 104–106 (1987).
    [CrossRef]
  13. H. J. Eichler, M. Glotz, A. Kummrow, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent wave-mixing in silicon,” Phys. Rev. A 35, 4673–4678 (1987).
    [CrossRef] [PubMed]
  14. I. C. Khoo and T. H. Liu, “Probe-beam amplification via two-and four-wave mixings in nematic liquid crystals film,” IEEE J. Quantum Electron. 23, 171–173 (1987).
    [CrossRef]
  15. I. C. Khoo, “Material characteristics and laser-induced responses and wave mixing in nematic liquid crystals,” Opt. Eng. 28, 1108–1113 (1989).
    [CrossRef]
  16. T. H. Liu and I. C. Khoo, “Probe beam amplification via degenerate optical wave mixing in a Kerr medium,” IEEE J. Quantum Electron. 23, 2020–2027 (1987).
    [CrossRef]
  17. L. B. Au and L. Solymar, “Amplification in photorefractive materials via a higher order wave,” Appl. Phys. B 45, 125–128 (1988).
    [CrossRef]
  18. L. B. Au and L. Solymar, “Higher diffraction orders in photorefractive materials,” IEEE J. Quantum Electron. 24, 162–168 (1988).
    [CrossRef]
  19. D. C. Jones and L. Solymar, “Forward four-wave interactions in BSO—theory and experiment,” Appl. Phys. B 50, 355–359 (1990).
    [CrossRef]
  20. A simplified approach to this problem was investigated by R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
    [CrossRef]
  21. V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, and M. S. Soskin, “Dynamic self-diffraction of coherent light beams,” Sov. Phys. Usp. 22, 742–756 (1979).
    [CrossRef]
  22. I. C. Khoo and T. H. Liu, “Theory and experiments on multi-wave-mixing-mediated probe beam amplification,” Phys. Rev. A 39, 4036–4044 (1989).
    [CrossRef] [PubMed]
  23. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, International Series in Pure and Applied Physics (McGraw-Hill, New York, 1953).
  24. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).
  25. A. Maruani, “Propagation analysis of forward degenerate four-wave mixing,” IEEE J. Quantum Electron. 16, 558–566 (1980).
    [CrossRef]

1991 (1)

1990 (2)

D. C. Jones and L. Solymar, “Forward four-wave interactions in BSO—theory and experiment,” Appl. Phys. B 50, 355–359 (1990).
[CrossRef]

I. C. Khoo and P. Zhou, “Transient multiwave mixing in a nonlinear medium,” Phys. Rev. A 41, 1544–1555 (1990).
[CrossRef] [PubMed]

1989 (4)

I. C. Khoo, “Material characteristics and laser-induced responses and wave mixing in nematic liquid crystals,” Opt. Eng. 28, 1108–1113 (1989).
[CrossRef]

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

I. C. Khoo, R. Normandin, T. H. Liu, and R. R. Michael, “Degenerate multiwave mixing and phase conjugation in silicon,” Phys. Rev. B 40, 7759–7766 (1989).
[CrossRef]

I. C. Khoo and T. H. Liu, “Theory and experiments on multi-wave-mixing-mediated probe beam amplification,” Phys. Rev. A 39, 4036–4044 (1989).
[CrossRef] [PubMed]

1988 (3)

F. Sanchez, P. H. Kayoun, and J.-P. Huignard, “Two-wave mixing with gain in liquid crystals at 10.6 μm wavelength,” J. Appl. Phys. 64, 26–31 (1988).
[CrossRef]

L. B. Au and L. Solymar, “Amplification in photorefractive materials via a higher order wave,” Appl. Phys. B 45, 125–128 (1988).
[CrossRef]

L. B. Au and L. Solymar, “Higher diffraction orders in photorefractive materials,” IEEE J. Quantum Electron. 24, 162–168 (1988).
[CrossRef]

1987 (4)

T. H. Liu and I. C. Khoo, “Probe beam amplification via degenerate optical wave mixing in a Kerr medium,” IEEE J. Quantum Electron. 23, 2020–2027 (1987).
[CrossRef]

H. J. Eichler, M. Glotz, A. Kummrow, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent wave-mixing in silicon,” Phys. Rev. A 35, 4673–4678 (1987).
[CrossRef] [PubMed]

I. C. Khoo and T. H. Liu, “Probe-beam amplification via two-and four-wave mixings in nematic liquid crystals film,” IEEE J. Quantum Electron. 23, 171–173 (1987).
[CrossRef]

Workshop on Nonlinear Optical Materials, “Research on nonlinear optical materials: an assessment,” Appl. Opt. 26, 211–234 (1987).
[PubMed]

1986 (1)

1985 (2)

P. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiment,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

J.-P. Huignard, H. Rajbenbach, P. Refregier, and L. Solymar, “Wave mixing in photorefractive BSO crystals and its applications,” Opt. Eng. 24, 586–592 (1985).

1980 (1)

A. Maruani, “Propagation analysis of forward degenerate four-wave mixing,” IEEE J. Quantum Electron. 16, 558–566 (1980).
[CrossRef]

1979 (1)

V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, and M. S. Soskin, “Dynamic self-diffraction of coherent light beams,” Sov. Phys. Usp. 22, 742–756 (1979).
[CrossRef]

1966 (1)

A simplified approach to this problem was investigated by R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
[CrossRef]

Au, L. B.

L. B. Au and L. Solymar, “Amplification in photorefractive materials via a higher order wave,” Appl. Phys. B 45, 125–128 (1988).
[CrossRef]

L. B. Au and L. Solymar, “Higher diffraction orders in photorefractive materials,” IEEE J. Quantum Electron. 24, 162–168 (1988).
[CrossRef]

Chiao, R. Y.

A simplified approach to this problem was investigated by R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
[CrossRef]

Cipparrone, G.

Duca, D.

Eichler, H. J.

H. J. Eichler, M. Glotz, A. Kummrow, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent wave-mixing in silicon,” Phys. Rev. A 35, 4673–4678 (1987).
[CrossRef] [PubMed]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, International Series in Pure and Applied Physics (McGraw-Hill, New York, 1953).

Garmire, E.

A simplified approach to this problem was investigated by R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
[CrossRef]

Glotz, M.

H. J. Eichler, M. Glotz, A. Kummrow, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent wave-mixing in silicon,” Phys. Rev. A 35, 4673–4678 (1987).
[CrossRef] [PubMed]

Huignard, J.-P.

F. Sanchez, P. H. Kayoun, and J.-P. Huignard, “Two-wave mixing with gain in liquid crystals at 10.6 μm wavelength,” J. Appl. Phys. 64, 26–31 (1988).
[CrossRef]

P. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiment,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

J.-P. Huignard, H. Rajbenbach, P. Refregier, and L. Solymar, “Wave mixing in photorefractive BSO crystals and its applications,” Opt. Eng. 24, 586–592 (1985).

F. Sanchez, P. H. Kayoun, and J.-P. Huignard, “Wave-mixing in liquid crystals and mercury cadmium telluride,” in Advanced Optoelectronic Technology, D. B. Ostrowsky and C. Puech, eds., Proc. Soc. Photo-Opt. Instrum. Eng.864, 104–106 (1987).
[CrossRef]

F. Sanchez, P. H. Kayoun, and J.-P. Huignard, “Optical amplification at 10.6 μm using thermal nonlinearities in liquid crystals near the phase transition,” in Conference on Lasers and Electro-Optics, Vol. 14 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), pp. 108–109.

Jones, D. C.

D. C. Jones and L. Solymar, “Forward four-wave interactions in BSO—theory and experiment,” Appl. Phys. B 50, 355–359 (1990).
[CrossRef]

Kayoun, P. H.

F. Sanchez, P. H. Kayoun, and J.-P. Huignard, “Two-wave mixing with gain in liquid crystals at 10.6 μm wavelength,” J. Appl. Phys. 64, 26–31 (1988).
[CrossRef]

F. Sanchez, P. H. Kayoun, and J.-P. Huignard, “Optical amplification at 10.6 μm using thermal nonlinearities in liquid crystals near the phase transition,” in Conference on Lasers and Electro-Optics, Vol. 14 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), pp. 108–109.

F. Sanchez, P. H. Kayoun, and J.-P. Huignard, “Wave-mixing in liquid crystals and mercury cadmium telluride,” in Advanced Optoelectronic Technology, D. B. Ostrowsky and C. Puech, eds., Proc. Soc. Photo-Opt. Instrum. Eng.864, 104–106 (1987).
[CrossRef]

Kelley, P. L.

A simplified approach to this problem was investigated by R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
[CrossRef]

Khoo, I. C.

I. C. Khoo, W. Wang, F. Simoni, G. Cipparrone, and D. Duca, “Theory and experiments on stationary degenerate and nearly degenerate optical wave mixing and ring oscillation in a Kerr-like medium,” J. Opt. Soc. Am. B 8, 1433–1441 (1991).
[CrossRef]

I. C. Khoo and P. Zhou, “Transient multiwave mixing in a nonlinear medium,” Phys. Rev. A 41, 1544–1555 (1990).
[CrossRef] [PubMed]

I. C. Khoo, R. Normandin, T. H. Liu, and R. R. Michael, “Degenerate multiwave mixing and phase conjugation in silicon,” Phys. Rev. B 40, 7759–7766 (1989).
[CrossRef]

I. C. Khoo and T. H. Liu, “Theory and experiments on multi-wave-mixing-mediated probe beam amplification,” Phys. Rev. A 39, 4036–4044 (1989).
[CrossRef] [PubMed]

I. C. Khoo, “Material characteristics and laser-induced responses and wave mixing in nematic liquid crystals,” Opt. Eng. 28, 1108–1113 (1989).
[CrossRef]

I. C. Khoo and T. H. Liu, “Probe-beam amplification via two-and four-wave mixings in nematic liquid crystals film,” IEEE J. Quantum Electron. 23, 171–173 (1987).
[CrossRef]

T. H. Liu and I. C. Khoo, “Probe beam amplification via degenerate optical wave mixing in a Kerr medium,” IEEE J. Quantum Electron. 23, 2020–2027 (1987).
[CrossRef]

Kukhtarev, N. V.

V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, and M. S. Soskin, “Dynamic self-diffraction of coherent light beams,” Sov. Phys. Usp. 22, 742–756 (1979).
[CrossRef]

Kummrow, A.

H. J. Eichler, M. Glotz, A. Kummrow, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent wave-mixing in silicon,” Phys. Rev. A 35, 4673–4678 (1987).
[CrossRef] [PubMed]

Liu, T. H.

I. C. Khoo and T. H. Liu, “Theory and experiments on multi-wave-mixing-mediated probe beam amplification,” Phys. Rev. A 39, 4036–4044 (1989).
[CrossRef] [PubMed]

I. C. Khoo, R. Normandin, T. H. Liu, and R. R. Michael, “Degenerate multiwave mixing and phase conjugation in silicon,” Phys. Rev. B 40, 7759–7766 (1989).
[CrossRef]

I. C. Khoo and T. H. Liu, “Probe-beam amplification via two-and four-wave mixings in nematic liquid crystals film,” IEEE J. Quantum Electron. 23, 171–173 (1987).
[CrossRef]

T. H. Liu and I. C. Khoo, “Probe beam amplification via degenerate optical wave mixing in a Kerr medium,” IEEE J. Quantum Electron. 23, 2020–2027 (1987).
[CrossRef]

Maruani, A.

A. Maruani, “Propagation analysis of forward degenerate four-wave mixing,” IEEE J. Quantum Electron. 16, 558–566 (1980).
[CrossRef]

Michael, R. R.

I. C. Khoo, R. Normandin, T. H. Liu, and R. R. Michael, “Degenerate multiwave mixing and phase conjugation in silicon,” Phys. Rev. B 40, 7759–7766 (1989).
[CrossRef]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, International Series in Pure and Applied Physics (McGraw-Hill, New York, 1953).

Normandin, R.

I. C. Khoo, R. Normandin, T. H. Liu, and R. R. Michael, “Degenerate multiwave mixing and phase conjugation in silicon,” Phys. Rev. B 40, 7759–7766 (1989).
[CrossRef]

Odulov, S. G.

V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, and M. S. Soskin, “Dynamic self-diffraction of coherent light beams,” Sov. Phys. Usp. 22, 742–756 (1979).
[CrossRef]

Rajbenbach, H.

J.-P. Huignard, H. Rajbenbach, P. Refregier, and L. Solymar, “Wave mixing in photorefractive BSO crystals and its applications,” Opt. Eng. 24, 586–592 (1985).

P. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiment,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Refregier, P.

P. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiment,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

J.-P. Huignard, H. Rajbenbach, P. Refregier, and L. Solymar, “Wave mixing in photorefractive BSO crystals and its applications,” Opt. Eng. 24, 586–592 (1985).

Richter, K.

H. J. Eichler, M. Glotz, A. Kummrow, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent wave-mixing in silicon,” Phys. Rev. A 35, 4673–4678 (1987).
[CrossRef] [PubMed]

Sanchez, F.

F. Sanchez, P. H. Kayoun, and J.-P. Huignard, “Two-wave mixing with gain in liquid crystals at 10.6 μm wavelength,” J. Appl. Phys. 64, 26–31 (1988).
[CrossRef]

F. Sanchez, P. H. Kayoun, and J.-P. Huignard, “Wave-mixing in liquid crystals and mercury cadmium telluride,” in Advanced Optoelectronic Technology, D. B. Ostrowsky and C. Puech, eds., Proc. Soc. Photo-Opt. Instrum. Eng.864, 104–106 (1987).
[CrossRef]

F. Sanchez, P. H. Kayoun, and J.-P. Huignard, “Optical amplification at 10.6 μm using thermal nonlinearities in liquid crystals near the phase transition,” in Conference on Lasers and Electro-Optics, Vol. 14 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), pp. 108–109.

Simoni, F.

Solymar, L.

D. C. Jones and L. Solymar, “Forward four-wave interactions in BSO—theory and experiment,” Appl. Phys. B 50, 355–359 (1990).
[CrossRef]

L. B. Au and L. Solymar, “Higher diffraction orders in photorefractive materials,” IEEE J. Quantum Electron. 24, 162–168 (1988).
[CrossRef]

L. B. Au and L. Solymar, “Amplification in photorefractive materials via a higher order wave,” Appl. Phys. B 45, 125–128 (1988).
[CrossRef]

P. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiment,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

J.-P. Huignard, H. Rajbenbach, P. Refregier, and L. Solymar, “Wave mixing in photorefractive BSO crystals and its applications,” Opt. Eng. 24, 586–592 (1985).

Soskin, M. S.

V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, and M. S. Soskin, “Dynamic self-diffraction of coherent light beams,” Sov. Phys. Usp. 22, 742–756 (1979).
[CrossRef]

Vinetskii, V. L.

V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, and M. S. Soskin, “Dynamic self-diffraction of coherent light beams,” Sov. Phys. Usp. 22, 742–756 (1979).
[CrossRef]

Wang, W.

Yang, X.

H. J. Eichler, M. Glotz, A. Kummrow, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent wave-mixing in silicon,” Phys. Rev. A 35, 4673–4678 (1987).
[CrossRef] [PubMed]

Yeh, P.

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

P. Yeh, “Exact solution of a nonlinear model of two-wave mixing in Kerr media,” J. Opt. Soc. Am. B 3, 747–750 (1986).
[CrossRef]

Zhou, P.

I. C. Khoo and P. Zhou, “Transient multiwave mixing in a nonlinear medium,” Phys. Rev. A 41, 1544–1555 (1990).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. B (2)

L. B. Au and L. Solymar, “Amplification in photorefractive materials via a higher order wave,” Appl. Phys. B 45, 125–128 (1988).
[CrossRef]

D. C. Jones and L. Solymar, “Forward four-wave interactions in BSO—theory and experiment,” Appl. Phys. B 50, 355–359 (1990).
[CrossRef]

IEEE J. Quantum Electron. (5)

T. H. Liu and I. C. Khoo, “Probe beam amplification via degenerate optical wave mixing in a Kerr medium,” IEEE J. Quantum Electron. 23, 2020–2027 (1987).
[CrossRef]

A. Maruani, “Propagation analysis of forward degenerate four-wave mixing,” IEEE J. Quantum Electron. 16, 558–566 (1980).
[CrossRef]

L. B. Au and L. Solymar, “Higher diffraction orders in photorefractive materials,” IEEE J. Quantum Electron. 24, 162–168 (1988).
[CrossRef]

I. C. Khoo and T. H. Liu, “Probe-beam amplification via two-and four-wave mixings in nematic liquid crystals film,” IEEE J. Quantum Electron. 23, 171–173 (1987).
[CrossRef]

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

J. Appl. Phys. (2)

P. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiment,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

F. Sanchez, P. H. Kayoun, and J.-P. Huignard, “Two-wave mixing with gain in liquid crystals at 10.6 μm wavelength,” J. Appl. Phys. 64, 26–31 (1988).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Eng. (2)

J.-P. Huignard, H. Rajbenbach, P. Refregier, and L. Solymar, “Wave mixing in photorefractive BSO crystals and its applications,” Opt. Eng. 24, 586–592 (1985).

I. C. Khoo, “Material characteristics and laser-induced responses and wave mixing in nematic liquid crystals,” Opt. Eng. 28, 1108–1113 (1989).
[CrossRef]

Phys. Rev. A (3)

H. J. Eichler, M. Glotz, A. Kummrow, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent wave-mixing in silicon,” Phys. Rev. A 35, 4673–4678 (1987).
[CrossRef] [PubMed]

I. C. Khoo and P. Zhou, “Transient multiwave mixing in a nonlinear medium,” Phys. Rev. A 41, 1544–1555 (1990).
[CrossRef] [PubMed]

I. C. Khoo and T. H. Liu, “Theory and experiments on multi-wave-mixing-mediated probe beam amplification,” Phys. Rev. A 39, 4036–4044 (1989).
[CrossRef] [PubMed]

Phys. Rev. B (1)

I. C. Khoo, R. Normandin, T. H. Liu, and R. R. Michael, “Degenerate multiwave mixing and phase conjugation in silicon,” Phys. Rev. B 40, 7759–7766 (1989).
[CrossRef]

Phys. Rev. Lett. (1)

A simplified approach to this problem was investigated by R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
[CrossRef]

Sov. Phys. Usp. (1)

V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, and M. S. Soskin, “Dynamic self-diffraction of coherent light beams,” Sov. Phys. Usp. 22, 742–756 (1979).
[CrossRef]

Other (5)

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, International Series in Pure and Applied Physics (McGraw-Hill, New York, 1953).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).

P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications (Springer-Verlag, Berlin, 1988, 1989), Vols. 1 and 2.
[CrossRef]

F. Sanchez, P. H. Kayoun, and J.-P. Huignard, “Optical amplification at 10.6 μm using thermal nonlinearities in liquid crystals near the phase transition,” in Conference on Lasers and Electro-Optics, Vol. 14 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), pp. 108–109.

F. Sanchez, P. H. Kayoun, and J.-P. Huignard, “Wave-mixing in liquid crystals and mercury cadmium telluride,” in Advanced Optoelectronic Technology, D. B. Ostrowsky and C. Puech, eds., Proc. Soc. Photo-Opt. Instrum. Eng.864, 104–106 (1987).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

(a) Schematic of the two-wave mixing experiment. Beams 1 and 2 are, respectively, the signal and the pump; beams 3 and 4 are the two diffraction orders involved in this model, (b) Wave-vector diagram, showing the phase mismatch present for large incident angles.

Fig. 2
Fig. 2

Evolution of G4 versus l. α = 50 cm −1, Δk = 0.

Fig. 3
Fig. 3

Evolution of G5 versus l. α = 50 cm−1, Δk = 0.

Fig. 4
Fig. 4

Evolution of G6 versus l. α = 50 cm −1, Δk = 0.

Fig. 5
Fig. 5

Evolution of G8 versus l for increasing values of Δk. g = 200 cm−1, α = 50 cm−1.

Fig. 6
Fig. 6

Evolution of G9 versus l for increasing values of Δk. g = 200 cm−1, α = 50 cm−1.

Fig. 7
Fig. 7

(a), (b) Evolution of the gain G10 versus l for several values of Δk. (c) Evolution of G10 versus Δk for different interaction lengths, g = 200 cm−1, α = 50 cm−1.

Fig. 8
Fig. 8

Evolution of S = sin ( Φ 1 + Φ 3 + Δ k l ) versus l for increasing values of Δk: (a) with the TWM terms, (b) without the TWM terms. g = 200 cm−1, α = 50 cm−1.

Fig. 9
Fig. 9

Evolution along the propagation axis of the normalized intensities of the four waves: (a) without the TWM terms, (b) with the TWM terms, g = 300 cm−1, α = 50 cm−1, Δk = 0, β = 1. The normalization is made with respect to I1(0).

Fig. 10
Fig. 10

Evolution along the propagation of the normalized intensities of the four waves: (a) without the TWM terms, (b) with the TWM terms. g = 200 cm−1, α = 50 cm−1, Δk = 0, β = 100. I1, I3, and I4 are normalized versus I1(0), whereas I2 is normalized versus I2(0).

Fig. 11
Fig. 11

Gain evolution versus β. g = 200 cm−1,α = 50 cm−1. (a) Without the TWM terms, Δk = 0. (b), (c) With the TWM terms: (b) Δk = 0, (c) Δk = 200 cm−1

Fig. 12
Fig. 12

Evolution along the propagation of the normalized intensity of the signal beam E1(z) for increasing values of β. The solid and the dotted curves correspond, respectively, to solutions of systems (5) and (A2) when absorption is taken into account. g = 200 cm−1, α = 50 cm−1, Δk = 200 cm−1.

Equations (37)

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E = j = 1 4 E j exp ( i k j r ) .
Δ E = ( n 2 ω 2 / c 2 ) E .
n ( r ) = n + Δ n ( r ) ,
Δ n ( r ) n 2 [ | j = 1 4 E j exp ( i k j r ) | 2 j = 1 4 | E j | 2 ] ,
E 1 z = α 2 E 1 + i q | E 2 | 2 E 1 + i q [ E 2 2 E 3 * exp ( i Δ k z ) + 2 E 2 E 4 E 1 * exp ( i Δ k z ) ] , E 2 z = α 2 E 2 + i q | E 1 | 2 E 2 + i q [ E 1 2 E 4 * exp ( i Δ k z ) + 2 E 1 E 3 E 2 * exp ( i Δ k z ) ] , E 3 z = α 2 E 3 + i q ( | E 2 | 2 + | E 1 | 2 ) E 3 + i q E 2 2 E 1 * exp ( i Δ k z ) , E 4 z = α 2 E 4 + i q ( | E 2 | 2 + | E 1 | 2 ) E 4 + i q E 1 2 E 2 * exp ( i Δ k z ) ,
E 1 z = α 2 E 1 + i q [ | E 2 | 2 E 1 + E 2 2 E 3 * exp ( i Δ k z ) ] , E 2 z = α 2 E 2 , E 3 z = α 2 E 3 + i q [ | E 2 | 2 E 3 + E 2 2 E 1 * exp ( i Δ k z ) ] .
E 1 z = i g E 1 + i g E 3 * , E 3 z = i g E 3 + i g E 1 * ,
E 1 ( z ) = E 1 ( 0 ) ch( g z ) , E 3 ( z ) = i E 1 * ( 0 ) sh( g z ) .
G 1 ( l ) = | E 1 ( l ) E 1 ( 0 ) | 2 = ch 2 ( g l ) .
G 2 ( l ) = 1 + g 2 l 2 .
E 1 z = i g E 1 + i g E 3 * α 2 E 1 , E 3 z = i g E 3 + i g E 1 * α 2 E 3 ,
G 3 ( l ) = ch 2 ( g l ) exp ( α l ) ,
G 4 ( l ) = ( 1 + g 2 l 2 ) exp ( α l ) .
E 1 z = i g ( E 1 + E 3 * ) exp ( α z ) α 2 E 1 , E 3 z = i g ( E 3 + E 1 * ) exp ( α z ) α 2 E 3 .
E 1 ( z ) = E 1 ( 0 ) ch { g α [ 1 exp ( α z ) ] } exp ( α z / 2 ) , E 3 ( z ) = i E 1 * ( 0 ) sh { g α [ 1 exp ( α z ) ] } exp ( α z / 2 ) .
G 5 ( l ) = ch 2 { g α [ 1 exp ( α l ) ] } exp ( α l ) .
G 6 ( l ) = { 1 + g 2 α 2 [ 1 exp ( α l ) ] 2 } exp ( α l ) .
E 1 z = i g [ E 1 + E 3 * exp ( i Δ k z ) ] α 2 E 1 , E 3 z = i g [ E 3 + E 1 * exp ( i Δ k z ) ] α 2 E 3 .
E 1 ( z ) = E 1 ( 0 ) ch λ z exp ( i Δ k z 2 ) exp ( α 2 z ) , E 3 ( z ) = i g λ E 1 * ( 0 ) sh λ z exp ( i Δ k z 2 ) exp ( α 2 z ) ,
E 1 ( z ) = E 1 ( 0 ) [ cos β z + i g ( Δ k / 2 ) β sin β z ] × exp ( i Δ k z 2 ) exp ( α 2 z ) , E 3 ( z ) = i g β E 1 * ( 0 ) sin β z exp ( i Δ k z 2 ) exp ( α 2 z ) ,
G 7 ( l ) = ch 2 λ l exp ( α l ) ,
G 8 ( l ) = { cos 2 β l + [ g ( Δ k / 2 ) ] 2 β 2 sin 2 β l } exp ( α l ) .
E 1 z = i g [ E 1 + E 3 * exp ( i Δ k z ) ] exp ( α z ) α 2 E 1 , E 3 z = i g [ E 3 + E 1 * exp ( i Δ k z ) ] exp ( α z ) α 2 E 3 .
| E 1 | 2 z = α | E 1 | 2 + 2 g exp ( α z ) Re [ i E 1 * E 3 * exp ( i Δ k z ) ] ,
sin ( Φ 1 + Φ 3 Δ k z ) > 0 ,
E 1 z = i g ( 2 E 1 + E 3 * exp { 2 i g α [ 1 exp ( α z ) ] } × exp ( i Δ k z ) ) exp ( α z ) α 2 E 1 , E 3 z = i g ( 2 E 3 + E 1 * exp { 2 i g α [ 1 exp ( α z ) ] } × exp ( i Δ k z ) ) exp ( α z ) α 2 E 3 .
Δ E = n 2 ω 2 c 2 ( 1 + 2 n 2 n | E 2 | ) E .
E 1 z = i q ( | E 1 | 2 + 2 | E 2 | 2 + 2 | E 3 | 2 + 2 | E 4 | 2 ) E 1 + i q [ E 2 2 E 3 * exp ( i Δ k z ) + 2 E 4 E 1 * E 2 exp ( i Δ k z ) ] , E 2 z = i q ( | E 2 | 2 + 2 | E 1 | 2 + 2 | E 3 | 2 + 2 | E 4 | 2 ) E 2 + i q [ E 1 2 E 4 * exp ( i Δ k z ) + 2 E 1 E 2 * E 3 exp ( i Δ k z ) ] , E 3 z = i q ( | E 3 | 2 + 2 | E 1 | 2 + 2 | E 2 | 2 + 2 | E 4 | 2 ) E 3 + i q E 2 2 E 1 * exp ( i Δ k z ) , E 4 z = i q ( | E 4 | 2 + 2 | E 1 | 2 + 2 | E 2 | 2 + 2 | E 3 | 2 ) E 4 + i q E 1 2 E 2 * exp ( i Δ k z ) .
| E 1 | 2 + | E 2 | 2 + | E 3 | 2 + | E 4 | 2 = const . = | E 0 | 2 .
A i = E i exp ( i q | E 0 | 2 z ) .
A 1 z = i q ( | A 2 | 2 + | A 3 | 2 + | A 4 | 2 ) A 1 + i q [ A 2 2 A 3 * exp ( i Δ k z ) + 2 A 4 A 1 * A 2 exp ( i Δ k z ) ] , A 2 z = i q ( | A 1 | 2 + | A 3 | 2 + | A 4 | 2 ) A 2 + i q [ A 1 2 A 4 * exp ( i Δ k z ) + 2 A 1 A 2 * A 3 exp ( i Δ k z ) ] , A 3 z = i q ( | A 1 | 2 + | A 2 | 2 + | A 4 | 2 ) A 3 + i q A 2 2 A 1 * exp ( i Δ k z ) , A 4 z = i q ( | A 1 | 2 + | A 2 | 2 + | A 3 | 2 ) A 4 + i q A 1 2 A 2 * exp ( i Δ k z ) .
A 1 z = i q | A 2 | 2 A 1 + i q [ A 2 2 A 3 * exp ( i Δ k z ) + 2 A 4 A 1 * A 2 exp ( i Δ k z ) ] , A 2 z = i q | A 1 | 2 A 2 + i q [ A 1 2 A 4 * exp ( i Δ k z ) + 2 A 1 A 2 * A 3 exp ( i Δ k z ) ] , A 3 z = i q ( | A 1 | 2 | A 2 | 2 ) A 3 + i q A 2 2 A 1 * exp ( i Δ k z ) , A 4 z = i q ( | A 1 | 2 | A 2 | 2 ) A 4 + i q A 1 2 A 2 * exp ( i Δ k z ) .
A 1 z = i g exp ( α z ) exp ( i Δ k z ) A 3 * , A 3 * z = i g exp ( α z ) exp ( i Δ k z ) A 1 .
2 A 1 z 2 + ( α i Δ k ) A 1 z g 2 exp ( α z ) A 1 = 0 .
2 A 1 x 2 = i Δ k α 1 x A 1 x g 2 α 2 A 1 = 0 ,
A 1 ( x ) = A ( a , b , a + b + 1 , x ) + x a b ( b , a , 1 a b , x ) ,
( a , b ) = 1 2 { i Δ k α 1 ± [ ( i Δ k α 1 ) 2 4 g 2 α a ] 1 / 2 } .

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