Abstract

We present a comprehensive analysis of the evolution of a grating envelope, in both transmission and reflection geometry, during the erasure process in various readout configurations in bulk photorefractive media. We derive a single partial differential equation that describes grating evolution. This equation shows that under certain conditions the grating envelope propagates and shares several features with either bright or dark solitons. Details of the grating envelopes directly relate to the dynamics of the observed diffraction efficiency. We derive analytical expressions for diffraction during nondestructive readout. Finally, we discuss the effects of nonzero absorption, finite dark conductivity, fringe bending, fanning, and nonlinear responses on the propagating envelope.

© 1995 Optical Society of America

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  1. N. V. Kukhtarev, "Kinetics of hologram recording and erasure in electrooptic crystals," Pis'ma Zh. Tekh. Fiz. 2, 1114 (1976) [Sov. Tech. Phys. Lett. 2, 438 (1976)].
  2. See, for example, P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications I (Springer- Verlag, Berlin, 1989), Chaps. 1–3.
  3. See, for example, T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, "The photorefractive effect—a review," Prog. Quantum Electron. 10, 77 (1985).
    [CrossRef]
  4. See, for example, F. Vachss, "Non-linear holographic response in photorefractive materials," Ph.D. dissertation (Stanford University, Stanford, Calif., 1988).
  5. See, for example, M. C. Bashaw, M. Jeganathan, and L. Hesselink, "Theory of two-center transport in photorefractive media for low-intensity, continuous-wave illumination in the quasi-steady-state limit," J. Opt. Soc. Am. B 11, 1743 (1994).
    [CrossRef]
  6. See, for example, M. Jeganathan and L. Hesselink, "Diffraction from thermally fixed gratings in a photorefractive medium—steady-state and transient analysis," J. Opt. Soc. Am. B 11, 1791 (1994).
    [CrossRef]
  7. L. B. Au and L. Solymar, "Transients in photorefractive twowave mixing: a numerical study," Appl. Phys. B 49, 339 (1989).
    [CrossRef]
  8. J. M. Heaton and L. Solymar, "Transient energy transfer during hologram formation in photorefractive crystals," Opt. Acta 32, 397 (1985).
    [CrossRef]
  9. D. L. Staebler and J. J. Amodei, "Coupled wave analysis of holographic storage in LiNbO3," J. Appl. Phys. 43, 1042 (1972).
    [CrossRef]
  10. T. K. Gaylord, T. A. Rabson, F. K. Tittel, and C. R. Quick, "Self-enhancement of LiNbO3 holograms," J. Appl. Phys. 44, 896 (1973).
    [CrossRef]
  11. D. W. Vahey, "A nonlinear coupled wave theory of holographic storage in ferroelectric materials," J. Appl. Phys. 46, 3510 (1975).
    [CrossRef]
  12. See, for example, P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications II (Springer-Verlag, Berlin, 1988), Chaps. 4–6.
    [CrossRef]
  13. See, for example, M. P. Petrov, S. I. Stepanov, and A. V. Khomenko, Photorefractive Crystals in Coherent Optical Systems (Springer-Verlag, Berlin, 1991), Chap. 1.
    [CrossRef]
  14. M. Jeganathan, M. C. Bashaw, and L. Hesselink, "Trapping the grating envelope in bulk photorefractive media," Opt. Lett. 19, 1415 (1994).
    [CrossRef] [PubMed]
  15. P. Yeh, "Two-wave mixing in nonlinear media," IEEE J. Quantum Electron. 25, 484 (1989).
    [CrossRef]
  16. J. M. Heaton and L. Solymar, "Transients effects during dynamic hologram formation in BSO crystals: theory and experiment," IEEE J. Quantum Electron. 24, 558 (1988).
    [CrossRef]
  17. J. Otten, A. Bledowski, K. H. Ringhofer, and R. A. Rupp, "Dynamical holographic storage in photorefractive crystals," Comput. Phys. Commun. 69, 187 (1992).
    [CrossRef]
  18. M. Jeganathan, M. C. Bashaw, and L. Hesselink, "Propagation of grating envelope in bulk photorefractive media," presented at the Optical Society of America Annual Meeting, Dallas, Texas, October 1994.
  19. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, "Holographic storage in electrooptic crystals. I. Steady state," Ferroelectrics 22, 949 (1979).
    [CrossRef]
  20. F. Vachss and L. Hesselink, "Nonlinear photorefractive response at high modulation depth," J. Opt. Soc. Am. B 5, 690 (1988).
    [CrossRef]
  21. C. Gu and P. Yeh, "Diffraction properties of fixed gratings in photorefractive media," J. Opt. Soc. Am. B 7, 2339 (1990).
    [CrossRef]
  22. C. Gu, J. Hong, and P. Yeh, "Diffraction properties of momentum mismatched gratings in photorefractive media," J. Opt. Soc. Am. B 9, 1473 (1992).
    [CrossRef]
  23. H. Sasaki, J. Ma, Y. Fainman, S. H. Lee, and Y. Taketomi, "Dynamics of composite grating in photorefractive crystals for memory application," J. Opt. Soc. Am. A 11, 2456 (1994).
    [CrossRef]
  24. Y. M. Osipov and B. I. Sturman, "Transient processes for two-beam interaction in photorefractive crystals," Opt. Commun. 79, 345 (1990).
    [CrossRef]
  25. D. R. Erbschloe and T. Wilson, "A simple analytic solution for transient two-wave mixing in photorefractive materials," Opt. Commun. 72, 135 (1989).
    [CrossRef]
  26. J. Otten, A. Ozols, M. Reinfelde, and K. H. Ringhofer, "Selfenhancement in lithium niobate," Opt. Commun. 72, 175 (1989).
    [CrossRef]
  27. A. Hermanns, C. Benkert, D. M. Lininger, and D. Z. Anderson, "The transfer function and impulse response of photorefractive two-beam coupling," IEEE J. Quantum Electron. 28, 750 (1992).
    [CrossRef]
  28. nmax defined in this paper is half that of the n1 defined in Ref. 15.
  29. H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909 (1969).
  30. J. H. Hong and R. Saxena, "Diffraction efficiency of volume holograms written by coupled beams," Opt. Lett. 16, 180 (1985).
  31. R. De Vrée, M. Jeganathan, J. P. Wilde, and L. Hesselink, "Effect of applied field on the writing and readout of photorefractive gratings," J. Opt. Soc. Am. B 12, 600 (1995).
    [CrossRef]
  32. See, for example, G. P. Agrawal and R. W. Boyd, Contemporary Nonlinear Optics (Academic, San Diego, Calif., 1992), Chap. 2; G. L. Lamb, Jr., Elements of Soliton Theory (Wiley, New York, 1980), Chaps. 5 and 7.
  33. A. A. Zozulya and V. T. Tikhonchuk, "Investigation of stability of four-wave mixing in photorefractive media," Phys. Lett. A 135, 447 (1989).
    [CrossRef]
  34. A. Hermanns, "Theory and applications of the dynamics and signal cross-correlations in photorefractive two-beam coupling," Ph.D. dissertation (University of Colorado, Boulder, Colo., 1993); Semiconductor Research Laboratories, Mitsubishi Electric Corporation, 8-1-1 Tsukaguchi-Honmakchi, Amagasaki, Hyogo 661 Japan (personal communications).
  35. S. Strogatz, Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry, and Engineering (Addison-Wesley, Reading, Mass., 1993).
  36. F. S. Acton, Numerical Methods That Work (Harper and Row, New York, 1970), Chap. 5.
  37. See, for example, P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), Chap. 6.
  38. M. Segev, D. Engin, A. Yariv, and G. C. Valley, "Temporal evolution of photorefractive double phase-conjugate mirrors," Opt. Lett. 18, 1828 (1993).
    [CrossRef] [PubMed]
  39. S. Campbell, P. Yeh, C. Gu, S. H. Lin, C.-J. Chen, and K. Y. Hsu, "Optical self-enhancement of photorefractive holograms," presented at the 1994 IEEE Conference on Nonlinear Optics: Materials, Fundamentals, and Applications, July 25–29, 1994, Waikoloa, Hawaii.
  40. S. Campbell, P. Yeh, C. Gu, S. H. Lin, C. Cheng, and K. Y. Hsu, "Optical restoration of photorefractive holograms through self-enhanced diffraction," Opt. Lett. 20, 330 (1995).
    [CrossRef] [PubMed]
  41. M. Jeganathan, M. C. Bashaw, A. Aharoni, and L. Hesselink, "Effect of self-diffraction on erasure dynamics during readout at different wavelengths and geometries in photorefractive materials," in Proceedings of Conference on Nonlinear Optics (Institute of Electrical and Electronics Engineers, New York, 1994), paper TuP1; A. Aharoni, M. Jeganathan, M. C. Bashaw and L. Hesselink, "Prolonged readout of photorefractive holograms by replay at longer wavelengths," in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CTuJ4.
  42. H. Zhou, F. Zhao, and F. T. S. Yu, "Effects of recordingerasure dynamics of storage capacity of a wavelength-multiplexed reflection-type photorefractive hologram," Appl. Opt. 33, 4339 (1994).
    [CrossRef] [PubMed]
  43. R. Hofmeister, A. Yariv, and S. Yagi, "Spectral response of fixed photorefractive grating interference filters," J. Opt. Soc. Am. A 11, 1342 (1994).
    [CrossRef]
  44. M. Jeganathan, M. C. Bashaw, and L. Hesselink, "Solitonlike propagation of the grating envelope during readout of photorefractive gratings," in Conference on Lasers and Electro-Optics, Vol. 15 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper QTuH5.
  45. M. Segev, Y. Ophir, and B. Fischer, "Nonlinear multi twowave mixing, the fanning process and its bleaching in photorefractive media," Opt. Commun. 77, 265 (1990).
    [CrossRef]
  46. R. A. Vazquez, F. R. Vachss, and R. R. Neurgaonkar, "Large photorefractive coupling coefficient in a thin cerium-doped strontium barium niobate crystal," J. Opt. Soc. Am. B 8, 1932 (1991).
    [CrossRef]
  47. P. Xie, Y-H. Hong, J-H. Dai, Y. Zhu, and H-J. Zhang, "Theoretical and experimental studies of fanning effects in photorefractive crystals," J. Appl. Phys. 74, 813 (1993).
    [CrossRef]
  48. Y-H. Hong, P. Xie, J-H. Dai, Y. Zhu, H-G. Yang, and H-J. Zhang, "Fanning effects in photorefractive crystals," Opt. Lett. 18, 772 (1993).
    [CrossRef] [PubMed]
  49. S. Breugnot, D. Dolfi, H. Rajbenbach, J.-P. Huignard, and M. Defour, "Enhancement of the signal-to-background ratio in photorefractive two wave mixing by mutually incoherent two-beam coupling," Opt. Lett. 19, 1070 (1994).
    [CrossRef] [PubMed]

1995 (2)

1994 (5)

1993 (3)

1992 (3)

A. Hermanns, C. Benkert, D. M. Lininger, and D. Z. Anderson, "The transfer function and impulse response of photorefractive two-beam coupling," IEEE J. Quantum Electron. 28, 750 (1992).
[CrossRef]

J. Otten, A. Bledowski, K. H. Ringhofer, and R. A. Rupp, "Dynamical holographic storage in photorefractive crystals," Comput. Phys. Commun. 69, 187 (1992).
[CrossRef]

C. Gu, J. Hong, and P. Yeh, "Diffraction properties of momentum mismatched gratings in photorefractive media," J. Opt. Soc. Am. B 9, 1473 (1992).
[CrossRef]

1991 (1)

1990 (3)

C. Gu and P. Yeh, "Diffraction properties of fixed gratings in photorefractive media," J. Opt. Soc. Am. B 7, 2339 (1990).
[CrossRef]

Y. M. Osipov and B. I. Sturman, "Transient processes for two-beam interaction in photorefractive crystals," Opt. Commun. 79, 345 (1990).
[CrossRef]

M. Segev, Y. Ophir, and B. Fischer, "Nonlinear multi twowave mixing, the fanning process and its bleaching in photorefractive media," Opt. Commun. 77, 265 (1990).
[CrossRef]

1989 (5)

A. A. Zozulya and V. T. Tikhonchuk, "Investigation of stability of four-wave mixing in photorefractive media," Phys. Lett. A 135, 447 (1989).
[CrossRef]

D. R. Erbschloe and T. Wilson, "A simple analytic solution for transient two-wave mixing in photorefractive materials," Opt. Commun. 72, 135 (1989).
[CrossRef]

J. Otten, A. Ozols, M. Reinfelde, and K. H. Ringhofer, "Selfenhancement in lithium niobate," Opt. Commun. 72, 175 (1989).
[CrossRef]

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

L. B. Au and L. Solymar, "Transients in photorefractive twowave mixing: a numerical study," Appl. Phys. B 49, 339 (1989).
[CrossRef]

1988 (2)

J. M. Heaton and L. Solymar, "Transients effects during dynamic hologram formation in BSO crystals: theory and experiment," IEEE J. Quantum Electron. 24, 558 (1988).
[CrossRef]

F. Vachss and L. Hesselink, "Nonlinear photorefractive response at high modulation depth," J. Opt. Soc. Am. B 5, 690 (1988).
[CrossRef]

1985 (2)

J. H. Hong and R. Saxena, "Diffraction efficiency of volume holograms written by coupled beams," Opt. Lett. 16, 180 (1985).

J. M. Heaton and L. Solymar, "Transient energy transfer during hologram formation in photorefractive crystals," Opt. Acta 32, 397 (1985).
[CrossRef]

1979 (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, "Holographic storage in electrooptic crystals. I. Steady state," Ferroelectrics 22, 949 (1979).
[CrossRef]

1976 (1)

N. V. Kukhtarev, "Kinetics of hologram recording and erasure in electrooptic crystals," Pis'ma Zh. Tekh. Fiz. 2, 1114 (1976) [Sov. Tech. Phys. Lett. 2, 438 (1976)].

1975 (1)

D. W. Vahey, "A nonlinear coupled wave theory of holographic storage in ferroelectric materials," J. Appl. Phys. 46, 3510 (1975).
[CrossRef]

1973 (1)

T. K. Gaylord, T. A. Rabson, F. K. Tittel, and C. R. Quick, "Self-enhancement of LiNbO3 holograms," J. Appl. Phys. 44, 896 (1973).
[CrossRef]

1972 (1)

D. L. Staebler and J. J. Amodei, "Coupled wave analysis of holographic storage in LiNbO3," J. Appl. Phys. 43, 1042 (1972).
[CrossRef]

1969 (1)

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909 (1969).

Acton, F. S.

F. S. Acton, Numerical Methods That Work (Harper and Row, New York, 1970), Chap. 5.

Agrawal, G. P.

See, for example, G. P. Agrawal and R. W. Boyd, Contemporary Nonlinear Optics (Academic, San Diego, Calif., 1992), Chap. 2; G. L. Lamb, Jr., Elements of Soliton Theory (Wiley, New York, 1980), Chaps. 5 and 7.

Aharoni, A.

M. Jeganathan, M. C. Bashaw, A. Aharoni, and L. Hesselink, "Effect of self-diffraction on erasure dynamics during readout at different wavelengths and geometries in photorefractive materials," in Proceedings of Conference on Nonlinear Optics (Institute of Electrical and Electronics Engineers, New York, 1994), paper TuP1; A. Aharoni, M. Jeganathan, M. C. Bashaw and L. Hesselink, "Prolonged readout of photorefractive holograms by replay at longer wavelengths," in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CTuJ4.

Amodei, J. J.

D. L. Staebler and J. J. Amodei, "Coupled wave analysis of holographic storage in LiNbO3," J. Appl. Phys. 43, 1042 (1972).
[CrossRef]

Anderson, D. Z.

A. Hermanns, C. Benkert, D. M. Lininger, and D. Z. Anderson, "The transfer function and impulse response of photorefractive two-beam coupling," IEEE J. Quantum Electron. 28, 750 (1992).
[CrossRef]

Au, L. B.

L. B. Au and L. Solymar, "Transients in photorefractive twowave mixing: a numerical study," Appl. Phys. B 49, 339 (1989).
[CrossRef]

Bashaw, M. C.

M. Jeganathan, M. C. Bashaw, and L. Hesselink, "Trapping the grating envelope in bulk photorefractive media," Opt. Lett. 19, 1415 (1994).
[CrossRef] [PubMed]

M. Jeganathan, M. C. Bashaw, and L. Hesselink, "Propagation of grating envelope in bulk photorefractive media," presented at the Optical Society of America Annual Meeting, Dallas, Texas, October 1994.

M. Jeganathan, M. C. Bashaw, and L. Hesselink, "Solitonlike propagation of the grating envelope during readout of photorefractive gratings," in Conference on Lasers and Electro-Optics, Vol. 15 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper QTuH5.

M. Jeganathan, M. C. Bashaw, A. Aharoni, and L. Hesselink, "Effect of self-diffraction on erasure dynamics during readout at different wavelengths and geometries in photorefractive materials," in Proceedings of Conference on Nonlinear Optics (Institute of Electrical and Electronics Engineers, New York, 1994), paper TuP1; A. Aharoni, M. Jeganathan, M. C. Bashaw and L. Hesselink, "Prolonged readout of photorefractive holograms by replay at longer wavelengths," in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CTuJ4.

See, for example, M. C. Bashaw, M. Jeganathan, and L. Hesselink, "Theory of two-center transport in photorefractive media for low-intensity, continuous-wave illumination in the quasi-steady-state limit," J. Opt. Soc. Am. B 11, 1743 (1994).
[CrossRef]

Benkert, C.

A. Hermanns, C. Benkert, D. M. Lininger, and D. Z. Anderson, "The transfer function and impulse response of photorefractive two-beam coupling," IEEE J. Quantum Electron. 28, 750 (1992).
[CrossRef]

Bledowski, A.

J. Otten, A. Bledowski, K. H. Ringhofer, and R. A. Rupp, "Dynamical holographic storage in photorefractive crystals," Comput. Phys. Commun. 69, 187 (1992).
[CrossRef]

Boyd, R. W.

See, for example, G. P. Agrawal and R. W. Boyd, Contemporary Nonlinear Optics (Academic, San Diego, Calif., 1992), Chap. 2; G. L. Lamb, Jr., Elements of Soliton Theory (Wiley, New York, 1980), Chaps. 5 and 7.

Breugnot, S.

Campbell, S.

S. Campbell, P. Yeh, C. Gu, S. H. Lin, C. Cheng, and K. Y. Hsu, "Optical restoration of photorefractive holograms through self-enhanced diffraction," Opt. Lett. 20, 330 (1995).
[CrossRef] [PubMed]

S. Campbell, P. Yeh, C. Gu, S. H. Lin, C.-J. Chen, and K. Y. Hsu, "Optical self-enhancement of photorefractive holograms," presented at the 1994 IEEE Conference on Nonlinear Optics: Materials, Fundamentals, and Applications, July 25–29, 1994, Waikoloa, Hawaii.

Chen, C.-J.

S. Campbell, P. Yeh, C. Gu, S. H. Lin, C.-J. Chen, and K. Y. Hsu, "Optical self-enhancement of photorefractive holograms," presented at the 1994 IEEE Conference on Nonlinear Optics: Materials, Fundamentals, and Applications, July 25–29, 1994, Waikoloa, Hawaii.

Cheng, C.

Connors, L. M.

See, for example, T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, "The photorefractive effect—a review," Prog. Quantum Electron. 10, 77 (1985).
[CrossRef]

Dai, J-H.

P. Xie, Y-H. Hong, J-H. Dai, Y. Zhu, and H-J. Zhang, "Theoretical and experimental studies of fanning effects in photorefractive crystals," J. Appl. Phys. 74, 813 (1993).
[CrossRef]

Y-H. Hong, P. Xie, J-H. Dai, Y. Zhu, H-G. Yang, and H-J. Zhang, "Fanning effects in photorefractive crystals," Opt. Lett. 18, 772 (1993).
[CrossRef] [PubMed]

Defour, M.

Dolfi, D.

Engin, D.

Erbschloe, D. R.

D. R. Erbschloe and T. Wilson, "A simple analytic solution for transient two-wave mixing in photorefractive materials," Opt. Commun. 72, 135 (1989).
[CrossRef]

Fainman, Y.

Fischer, B.

M. Segev, Y. Ophir, and B. Fischer, "Nonlinear multi twowave mixing, the fanning process and its bleaching in photorefractive media," Opt. Commun. 77, 265 (1990).
[CrossRef]

Foote, P. D.

See, for example, T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, "The photorefractive effect—a review," Prog. Quantum Electron. 10, 77 (1985).
[CrossRef]

Gaylord, T. K.

T. K. Gaylord, T. A. Rabson, F. K. Tittel, and C. R. Quick, "Self-enhancement of LiNbO3 holograms," J. Appl. Phys. 44, 896 (1973).
[CrossRef]

Gu, C.

Hall, T. J.

See, for example, T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, "The photorefractive effect—a review," Prog. Quantum Electron. 10, 77 (1985).
[CrossRef]

Heaton, J. M.

J. M. Heaton and L. Solymar, "Transients effects during dynamic hologram formation in BSO crystals: theory and experiment," IEEE J. Quantum Electron. 24, 558 (1988).
[CrossRef]

J. M. Heaton and L. Solymar, "Transient energy transfer during hologram formation in photorefractive crystals," Opt. Acta 32, 397 (1985).
[CrossRef]

Hermanns, A.

A. Hermanns, C. Benkert, D. M. Lininger, and D. Z. Anderson, "The transfer function and impulse response of photorefractive two-beam coupling," IEEE J. Quantum Electron. 28, 750 (1992).
[CrossRef]

A. Hermanns, "Theory and applications of the dynamics and signal cross-correlations in photorefractive two-beam coupling," Ph.D. dissertation (University of Colorado, Boulder, Colo., 1993); Semiconductor Research Laboratories, Mitsubishi Electric Corporation, 8-1-1 Tsukaguchi-Honmakchi, Amagasaki, Hyogo 661 Japan (personal communications).

Hesselink, L.

R. De Vrée, M. Jeganathan, J. P. Wilde, and L. Hesselink, "Effect of applied field on the writing and readout of photorefractive gratings," J. Opt. Soc. Am. B 12, 600 (1995).
[CrossRef]

M. Jeganathan, M. C. Bashaw, and L. Hesselink, "Trapping the grating envelope in bulk photorefractive media," Opt. Lett. 19, 1415 (1994).
[CrossRef] [PubMed]

F. Vachss and L. Hesselink, "Nonlinear photorefractive response at high modulation depth," J. Opt. Soc. Am. B 5, 690 (1988).
[CrossRef]

M. Jeganathan, M. C. Bashaw, A. Aharoni, and L. Hesselink, "Effect of self-diffraction on erasure dynamics during readout at different wavelengths and geometries in photorefractive materials," in Proceedings of Conference on Nonlinear Optics (Institute of Electrical and Electronics Engineers, New York, 1994), paper TuP1; A. Aharoni, M. Jeganathan, M. C. Bashaw and L. Hesselink, "Prolonged readout of photorefractive holograms by replay at longer wavelengths," in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CTuJ4.

See, for example, M. C. Bashaw, M. Jeganathan, and L. Hesselink, "Theory of two-center transport in photorefractive media for low-intensity, continuous-wave illumination in the quasi-steady-state limit," J. Opt. Soc. Am. B 11, 1743 (1994).
[CrossRef]

See, for example, M. Jeganathan and L. Hesselink, "Diffraction from thermally fixed gratings in a photorefractive medium—steady-state and transient analysis," J. Opt. Soc. Am. B 11, 1791 (1994).
[CrossRef]

M. Jeganathan, M. C. Bashaw, and L. Hesselink, "Solitonlike propagation of the grating envelope during readout of photorefractive gratings," in Conference on Lasers and Electro-Optics, Vol. 15 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper QTuH5.

M. Jeganathan, M. C. Bashaw, and L. Hesselink, "Propagation of grating envelope in bulk photorefractive media," presented at the Optical Society of America Annual Meeting, Dallas, Texas, October 1994.

Hofmeister, R.

Hong, J.

Hong, J. H.

Hong, Y-H.

Y-H. Hong, P. Xie, J-H. Dai, Y. Zhu, H-G. Yang, and H-J. Zhang, "Fanning effects in photorefractive crystals," Opt. Lett. 18, 772 (1993).
[CrossRef] [PubMed]

P. Xie, Y-H. Hong, J-H. Dai, Y. Zhu, and H-J. Zhang, "Theoretical and experimental studies of fanning effects in photorefractive crystals," J. Appl. Phys. 74, 813 (1993).
[CrossRef]

Hsu, K. Y.

S. Campbell, P. Yeh, C. Gu, S. H. Lin, C. Cheng, and K. Y. Hsu, "Optical restoration of photorefractive holograms through self-enhanced diffraction," Opt. Lett. 20, 330 (1995).
[CrossRef] [PubMed]

S. Campbell, P. Yeh, C. Gu, S. H. Lin, C.-J. Chen, and K. Y. Hsu, "Optical self-enhancement of photorefractive holograms," presented at the 1994 IEEE Conference on Nonlinear Optics: Materials, Fundamentals, and Applications, July 25–29, 1994, Waikoloa, Hawaii.

Huignard, J.-P.

Jaura, R.

See, for example, T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, "The photorefractive effect—a review," Prog. Quantum Electron. 10, 77 (1985).
[CrossRef]

Jeganathan, M.

R. De Vrée, M. Jeganathan, J. P. Wilde, and L. Hesselink, "Effect of applied field on the writing and readout of photorefractive gratings," J. Opt. Soc. Am. B 12, 600 (1995).
[CrossRef]

M. Jeganathan, M. C. Bashaw, and L. Hesselink, "Trapping the grating envelope in bulk photorefractive media," Opt. Lett. 19, 1415 (1994).
[CrossRef] [PubMed]

M. Jeganathan, M. C. Bashaw, and L. Hesselink, "Propagation of grating envelope in bulk photorefractive media," presented at the Optical Society of America Annual Meeting, Dallas, Texas, October 1994.

See, for example, M. Jeganathan and L. Hesselink, "Diffraction from thermally fixed gratings in a photorefractive medium—steady-state and transient analysis," J. Opt. Soc. Am. B 11, 1791 (1994).
[CrossRef]

M. Jeganathan, M. C. Bashaw, and L. Hesselink, "Solitonlike propagation of the grating envelope during readout of photorefractive gratings," in Conference on Lasers and Electro-Optics, Vol. 15 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper QTuH5.

See, for example, M. C. Bashaw, M. Jeganathan, and L. Hesselink, "Theory of two-center transport in photorefractive media for low-intensity, continuous-wave illumination in the quasi-steady-state limit," J. Opt. Soc. Am. B 11, 1743 (1994).
[CrossRef]

M. Jeganathan, M. C. Bashaw, A. Aharoni, and L. Hesselink, "Effect of self-diffraction on erasure dynamics during readout at different wavelengths and geometries in photorefractive materials," in Proceedings of Conference on Nonlinear Optics (Institute of Electrical and Electronics Engineers, New York, 1994), paper TuP1; A. Aharoni, M. Jeganathan, M. C. Bashaw and L. Hesselink, "Prolonged readout of photorefractive holograms by replay at longer wavelengths," in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CTuJ4.

Khomenko, A. V.

See, for example, M. P. Petrov, S. I. Stepanov, and A. V. Khomenko, Photorefractive Crystals in Coherent Optical Systems (Springer-Verlag, Berlin, 1991), Chap. 1.
[CrossRef]

Kogelnik, H.

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909 (1969).

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, "Holographic storage in electrooptic crystals. I. Steady state," Ferroelectrics 22, 949 (1979).
[CrossRef]

N. V. Kukhtarev, "Kinetics of hologram recording and erasure in electrooptic crystals," Pis'ma Zh. Tekh. Fiz. 2, 1114 (1976) [Sov. Tech. Phys. Lett. 2, 438 (1976)].

Lee, S. H.

Lin, S. H.

S. Campbell, P. Yeh, C. Gu, S. H. Lin, C. Cheng, and K. Y. Hsu, "Optical restoration of photorefractive holograms through self-enhanced diffraction," Opt. Lett. 20, 330 (1995).
[CrossRef] [PubMed]

S. Campbell, P. Yeh, C. Gu, S. H. Lin, C.-J. Chen, and K. Y. Hsu, "Optical self-enhancement of photorefractive holograms," presented at the 1994 IEEE Conference on Nonlinear Optics: Materials, Fundamentals, and Applications, July 25–29, 1994, Waikoloa, Hawaii.

Lininger, D. M.

A. Hermanns, C. Benkert, D. M. Lininger, and D. Z. Anderson, "The transfer function and impulse response of photorefractive two-beam coupling," IEEE J. Quantum Electron. 28, 750 (1992).
[CrossRef]

Ma, J.

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, "Holographic storage in electrooptic crystals. I. Steady state," Ferroelectrics 22, 949 (1979).
[CrossRef]

Neurgaonkar, R. R.

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, "Holographic storage in electrooptic crystals. I. Steady state," Ferroelectrics 22, 949 (1979).
[CrossRef]

Ophir, Y.

M. Segev, Y. Ophir, and B. Fischer, "Nonlinear multi twowave mixing, the fanning process and its bleaching in photorefractive media," Opt. Commun. 77, 265 (1990).
[CrossRef]

Osipov, Y. M.

Y. M. Osipov and B. I. Sturman, "Transient processes for two-beam interaction in photorefractive crystals," Opt. Commun. 79, 345 (1990).
[CrossRef]

Otten, J.

J. Otten, A. Bledowski, K. H. Ringhofer, and R. A. Rupp, "Dynamical holographic storage in photorefractive crystals," Comput. Phys. Commun. 69, 187 (1992).
[CrossRef]

J. Otten, A. Ozols, M. Reinfelde, and K. H. Ringhofer, "Selfenhancement in lithium niobate," Opt. Commun. 72, 175 (1989).
[CrossRef]

Ozols, A.

J. Otten, A. Ozols, M. Reinfelde, and K. H. Ringhofer, "Selfenhancement in lithium niobate," Opt. Commun. 72, 175 (1989).
[CrossRef]

Petrov, M. P.

See, for example, M. P. Petrov, S. I. Stepanov, and A. V. Khomenko, Photorefractive Crystals in Coherent Optical Systems (Springer-Verlag, Berlin, 1991), Chap. 1.
[CrossRef]

Quick, C. R.

T. K. Gaylord, T. A. Rabson, F. K. Tittel, and C. R. Quick, "Self-enhancement of LiNbO3 holograms," J. Appl. Phys. 44, 896 (1973).
[CrossRef]

Rabson, T. A.

T. K. Gaylord, T. A. Rabson, F. K. Tittel, and C. R. Quick, "Self-enhancement of LiNbO3 holograms," J. Appl. Phys. 44, 896 (1973).
[CrossRef]

Rajbenbach, H.

Reinfelde, M.

J. Otten, A. Ozols, M. Reinfelde, and K. H. Ringhofer, "Selfenhancement in lithium niobate," Opt. Commun. 72, 175 (1989).
[CrossRef]

Ringhofer, K. H.

J. Otten, A. Bledowski, K. H. Ringhofer, and R. A. Rupp, "Dynamical holographic storage in photorefractive crystals," Comput. Phys. Commun. 69, 187 (1992).
[CrossRef]

J. Otten, A. Ozols, M. Reinfelde, and K. H. Ringhofer, "Selfenhancement in lithium niobate," Opt. Commun. 72, 175 (1989).
[CrossRef]

Rupp, R. A.

J. Otten, A. Bledowski, K. H. Ringhofer, and R. A. Rupp, "Dynamical holographic storage in photorefractive crystals," Comput. Phys. Commun. 69, 187 (1992).
[CrossRef]

Sasaki, H.

Saxena, R.

Segev, M.

M. Segev, D. Engin, A. Yariv, and G. C. Valley, "Temporal evolution of photorefractive double phase-conjugate mirrors," Opt. Lett. 18, 1828 (1993).
[CrossRef] [PubMed]

M. Segev, Y. Ophir, and B. Fischer, "Nonlinear multi twowave mixing, the fanning process and its bleaching in photorefractive media," Opt. Commun. 77, 265 (1990).
[CrossRef]

Solymar, L.

L. B. Au and L. Solymar, "Transients in photorefractive twowave mixing: a numerical study," Appl. Phys. B 49, 339 (1989).
[CrossRef]

J. M. Heaton and L. Solymar, "Transients effects during dynamic hologram formation in BSO crystals: theory and experiment," IEEE J. Quantum Electron. 24, 558 (1988).
[CrossRef]

J. M. Heaton and L. Solymar, "Transient energy transfer during hologram formation in photorefractive crystals," Opt. Acta 32, 397 (1985).
[CrossRef]

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, "Holographic storage in electrooptic crystals. I. Steady state," Ferroelectrics 22, 949 (1979).
[CrossRef]

Staebler, D. L.

D. L. Staebler and J. J. Amodei, "Coupled wave analysis of holographic storage in LiNbO3," J. Appl. Phys. 43, 1042 (1972).
[CrossRef]

Stepanov, S. I.

See, for example, M. P. Petrov, S. I. Stepanov, and A. V. Khomenko, Photorefractive Crystals in Coherent Optical Systems (Springer-Verlag, Berlin, 1991), Chap. 1.
[CrossRef]

Strogatz, S.

S. Strogatz, Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry, and Engineering (Addison-Wesley, Reading, Mass., 1993).

Sturman, B. I.

Y. M. Osipov and B. I. Sturman, "Transient processes for two-beam interaction in photorefractive crystals," Opt. Commun. 79, 345 (1990).
[CrossRef]

Taketomi, Y.

Tikhonchuk, V. T.

A. A. Zozulya and V. T. Tikhonchuk, "Investigation of stability of four-wave mixing in photorefractive media," Phys. Lett. A 135, 447 (1989).
[CrossRef]

Tittel, F. K.

T. K. Gaylord, T. A. Rabson, F. K. Tittel, and C. R. Quick, "Self-enhancement of LiNbO3 holograms," J. Appl. Phys. 44, 896 (1973).
[CrossRef]

Vachss, F.

F. Vachss and L. Hesselink, "Nonlinear photorefractive response at high modulation depth," J. Opt. Soc. Am. B 5, 690 (1988).
[CrossRef]

See, for example, F. Vachss, "Non-linear holographic response in photorefractive materials," Ph.D. dissertation (Stanford University, Stanford, Calif., 1988).

Vachss, F. R.

Vahey, D. W.

D. W. Vahey, "A nonlinear coupled wave theory of holographic storage in ferroelectric materials," J. Appl. Phys. 46, 3510 (1975).
[CrossRef]

Valley, G. C.

Vazquez, R. A.

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, "Holographic storage in electrooptic crystals. I. Steady state," Ferroelectrics 22, 949 (1979).
[CrossRef]

Vrée, R. De

Wilde, J. P.

Wilson, T.

D. R. Erbschloe and T. Wilson, "A simple analytic solution for transient two-wave mixing in photorefractive materials," Opt. Commun. 72, 135 (1989).
[CrossRef]

Xie, P.

Y-H. Hong, P. Xie, J-H. Dai, Y. Zhu, H-G. Yang, and H-J. Zhang, "Fanning effects in photorefractive crystals," Opt. Lett. 18, 772 (1993).
[CrossRef] [PubMed]

P. Xie, Y-H. Hong, J-H. Dai, Y. Zhu, and H-J. Zhang, "Theoretical and experimental studies of fanning effects in photorefractive crystals," J. Appl. Phys. 74, 813 (1993).
[CrossRef]

Yagi, S.

Yang, H-G.

Yariv, A.

Yeh, P.

S. Campbell, P. Yeh, C. Gu, S. H. Lin, C. Cheng, and K. Y. Hsu, "Optical restoration of photorefractive holograms through self-enhanced diffraction," Opt. Lett. 20, 330 (1995).
[CrossRef] [PubMed]

C. Gu, J. Hong, and P. Yeh, "Diffraction properties of momentum mismatched gratings in photorefractive media," J. Opt. Soc. Am. B 9, 1473 (1992).
[CrossRef]

C. Gu and P. Yeh, "Diffraction properties of fixed gratings in photorefractive media," J. Opt. Soc. Am. B 7, 2339 (1990).
[CrossRef]

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

S. Campbell, P. Yeh, C. Gu, S. H. Lin, C.-J. Chen, and K. Y. Hsu, "Optical self-enhancement of photorefractive holograms," presented at the 1994 IEEE Conference on Nonlinear Optics: Materials, Fundamentals, and Applications, July 25–29, 1994, Waikoloa, Hawaii.

See, for example, P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), Chap. 6.

Yu, F. T. S.

Zhang, H-J.

Y-H. Hong, P. Xie, J-H. Dai, Y. Zhu, H-G. Yang, and H-J. Zhang, "Fanning effects in photorefractive crystals," Opt. Lett. 18, 772 (1993).
[CrossRef] [PubMed]

P. Xie, Y-H. Hong, J-H. Dai, Y. Zhu, and H-J. Zhang, "Theoretical and experimental studies of fanning effects in photorefractive crystals," J. Appl. Phys. 74, 813 (1993).
[CrossRef]

Zhao, F.

Zhou, H.

Zhu, Y.

Y-H. Hong, P. Xie, J-H. Dai, Y. Zhu, H-G. Yang, and H-J. Zhang, "Fanning effects in photorefractive crystals," Opt. Lett. 18, 772 (1993).
[CrossRef] [PubMed]

P. Xie, Y-H. Hong, J-H. Dai, Y. Zhu, and H-J. Zhang, "Theoretical and experimental studies of fanning effects in photorefractive crystals," J. Appl. Phys. 74, 813 (1993).
[CrossRef]

Zozulya, A. A.

A. A. Zozulya and V. T. Tikhonchuk, "Investigation of stability of four-wave mixing in photorefractive media," Phys. Lett. A 135, 447 (1989).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

L. B. Au and L. Solymar, "Transients in photorefractive twowave mixing: a numerical study," Appl. Phys. B 49, 339 (1989).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909 (1969).

Comput. Phys. Commun. (1)

J. Otten, A. Bledowski, K. H. Ringhofer, and R. A. Rupp, "Dynamical holographic storage in photorefractive crystals," Comput. Phys. Commun. 69, 187 (1992).
[CrossRef]

Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, "Holographic storage in electrooptic crystals. I. Steady state," Ferroelectrics 22, 949 (1979).
[CrossRef]

IEEE J. Quantum Electron. (3)

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

J. M. Heaton and L. Solymar, "Transients effects during dynamic hologram formation in BSO crystals: theory and experiment," IEEE J. Quantum Electron. 24, 558 (1988).
[CrossRef]

A. Hermanns, C. Benkert, D. M. Lininger, and D. Z. Anderson, "The transfer function and impulse response of photorefractive two-beam coupling," IEEE J. Quantum Electron. 28, 750 (1992).
[CrossRef]

J. Appl. Phys. (4)

D. L. Staebler and J. J. Amodei, "Coupled wave analysis of holographic storage in LiNbO3," J. Appl. Phys. 43, 1042 (1972).
[CrossRef]

T. K. Gaylord, T. A. Rabson, F. K. Tittel, and C. R. Quick, "Self-enhancement of LiNbO3 holograms," J. Appl. Phys. 44, 896 (1973).
[CrossRef]

D. W. Vahey, "A nonlinear coupled wave theory of holographic storage in ferroelectric materials," J. Appl. Phys. 46, 3510 (1975).
[CrossRef]

P. Xie, Y-H. Hong, J-H. Dai, Y. Zhu, and H-J. Zhang, "Theoretical and experimental studies of fanning effects in photorefractive crystals," J. Appl. Phys. 74, 813 (1993).
[CrossRef]

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

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

Opt. Acta (1)

J. M. Heaton and L. Solymar, "Transient energy transfer during hologram formation in photorefractive crystals," Opt. Acta 32, 397 (1985).
[CrossRef]

Opt. Commun. (4)

Y. M. Osipov and B. I. Sturman, "Transient processes for two-beam interaction in photorefractive crystals," Opt. Commun. 79, 345 (1990).
[CrossRef]

D. R. Erbschloe and T. Wilson, "A simple analytic solution for transient two-wave mixing in photorefractive materials," Opt. Commun. 72, 135 (1989).
[CrossRef]

J. Otten, A. Ozols, M. Reinfelde, and K. H. Ringhofer, "Selfenhancement in lithium niobate," Opt. Commun. 72, 175 (1989).
[CrossRef]

M. Segev, Y. Ophir, and B. Fischer, "Nonlinear multi twowave mixing, the fanning process and its bleaching in photorefractive media," Opt. Commun. 77, 265 (1990).
[CrossRef]

Opt. Lett. (6)

Phys. Lett. A (1)

A. A. Zozulya and V. T. Tikhonchuk, "Investigation of stability of four-wave mixing in photorefractive media," Phys. Lett. A 135, 447 (1989).
[CrossRef]

Pis'ma Zh. Tekh. Fiz. (1)

N. V. Kukhtarev, "Kinetics of hologram recording and erasure in electrooptic crystals," Pis'ma Zh. Tekh. Fiz. 2, 1114 (1976) [Sov. Tech. Phys. Lett. 2, 438 (1976)].

Other (17)

See, for example, P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications I (Springer- Verlag, Berlin, 1989), Chaps. 1–3.

See, for example, T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, "The photorefractive effect—a review," Prog. Quantum Electron. 10, 77 (1985).
[CrossRef]

See, for example, F. Vachss, "Non-linear holographic response in photorefractive materials," Ph.D. dissertation (Stanford University, Stanford, Calif., 1988).

See, for example, M. C. Bashaw, M. Jeganathan, and L. Hesselink, "Theory of two-center transport in photorefractive media for low-intensity, continuous-wave illumination in the quasi-steady-state limit," J. Opt. Soc. Am. B 11, 1743 (1994).
[CrossRef]

See, for example, M. Jeganathan and L. Hesselink, "Diffraction from thermally fixed gratings in a photorefractive medium—steady-state and transient analysis," J. Opt. Soc. Am. B 11, 1791 (1994).
[CrossRef]

See, for example, P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications II (Springer-Verlag, Berlin, 1988), Chaps. 4–6.
[CrossRef]

See, for example, M. P. Petrov, S. I. Stepanov, and A. V. Khomenko, Photorefractive Crystals in Coherent Optical Systems (Springer-Verlag, Berlin, 1991), Chap. 1.
[CrossRef]

A. Hermanns, "Theory and applications of the dynamics and signal cross-correlations in photorefractive two-beam coupling," Ph.D. dissertation (University of Colorado, Boulder, Colo., 1993); Semiconductor Research Laboratories, Mitsubishi Electric Corporation, 8-1-1 Tsukaguchi-Honmakchi, Amagasaki, Hyogo 661 Japan (personal communications).

S. Strogatz, Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry, and Engineering (Addison-Wesley, Reading, Mass., 1993).

F. S. Acton, Numerical Methods That Work (Harper and Row, New York, 1970), Chap. 5.

See, for example, P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), Chap. 6.

S. Campbell, P. Yeh, C. Gu, S. H. Lin, C.-J. Chen, and K. Y. Hsu, "Optical self-enhancement of photorefractive holograms," presented at the 1994 IEEE Conference on Nonlinear Optics: Materials, Fundamentals, and Applications, July 25–29, 1994, Waikoloa, Hawaii.

M. Jeganathan, M. C. Bashaw, A. Aharoni, and L. Hesselink, "Effect of self-diffraction on erasure dynamics during readout at different wavelengths and geometries in photorefractive materials," in Proceedings of Conference on Nonlinear Optics (Institute of Electrical and Electronics Engineers, New York, 1994), paper TuP1; A. Aharoni, M. Jeganathan, M. C. Bashaw and L. Hesselink, "Prolonged readout of photorefractive holograms by replay at longer wavelengths," in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CTuJ4.

M. Jeganathan, M. C. Bashaw, and L. Hesselink, "Solitonlike propagation of the grating envelope during readout of photorefractive gratings," in Conference on Lasers and Electro-Optics, Vol. 15 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper QTuH5.

See, for example, G. P. Agrawal and R. W. Boyd, Contemporary Nonlinear Optics (Academic, San Diego, Calif., 1992), Chap. 2; G. L. Lamb, Jr., Elements of Soliton Theory (Wiley, New York, 1980), Chaps. 5 and 7.

nmax defined in this paper is half that of the n1 defined in Ref. 15.

M. Jeganathan, M. C. Bashaw, and L. Hesselink, "Propagation of grating envelope in bulk photorefractive media," presented at the Optical Society of America Annual Meeting, Dallas, Texas, October 1994.

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Figures (13)

Fig. 1
Fig. 1

Recording schematic for (a) symmetric transmission geometry and (b) symmetric reflection geometry, showing the coordinate system used. Readout schematic for (c) symmetric transmission geometry and (d) symmetric reflection geometry. The direction of the c axis has been left out in (c) and (d) so that the beams correspond to a general representation of reference and signal.

Fig. 2
Fig. 2

Shape of the grating envelope at steady-state recording in the transmission geometry. The various boxes represent crystal boundaries, and the part of the sech function inside a box represents the grating envelope. For box A r0 = 1000 and γd = 5; for box B r0 = 100 and γd = 10; for box C r0 = 1 and γd = 10; for box D r0 = 0.01 and γd = 5.

Fig. 3
Fig. 3

(A) Transients during erasure with I ref ρ. Diffraction efficiency is normalized such that η(t = 0) = 1 and time is normalized to the photorefractive time constant τ (this convention will be followed for subsequent figures unless otherwise stated). γ = 1 mm−1 and r0 = 1 during recording. (a) d = 5 mm, (b) d = 10 mm, (c) d = 15 mm, (d) d = 20 mm. (B) Evolution of the grating envelope during readout with I ref ρ for case (d) in (A). The shape of the grating envelope is shown at approximately equal time intervals of Δt′ = 1 unit from t′ = 0 to t′ = 30. The darker curve shows the initial grating envelope shape (r0 = 1, γ = 1 mm−1, d = 20 mm).

Fig. 4
Fig. 4

(a) A, Normalized index amplitude n(u), which is the shape of the propagating grating envelope (AHSF) during erasure in the transmission geometry. B, ΘT(u), which is proportional to the integral of n(u). C, Plot of sech(u/2) scaled and shifted to match the maximum of n(u). We computed the functions n(u) and ΘT(u) by using mathematica with initial conditions n(u = 33.7) = 10−5 and ΘT(u = 33.7) = 1. (b) Similar to (a) but with the vertical axis on a log scale. A, The normalized index amplitude n(u) in the moving coordinate system. B, Plot of sech(u/2). Dashed curves C and D show the asymptotes of the AHSF with slopes +1 and 1 - 2, respectively.

Fig. 5
Fig. 5

Dispersion: dependence of propagation velocity on grating period for various screening and diffusion lengths: A, rD = 0.1 μm and ls = 1 μm; B, rD = 1 μm and ls = 1 μm; C, rD = 10 μm and ls = 1 μm; D, rD = 0.1 μm and ls = 0.1 μm.

Fig. 6
Fig. 6

Evolution of the grating envelope with time for nonzero absorption in the normal transmission geometry (readout with I ref ρ). The shape of the grating envelope is plotted at approximately equal time intervals of Δt′ = 1 unit from t′ = 0 to t′ = 40, clearly showing the slowing down of envelope propagation [compare with Fig. 3(B)]. r0 = 1, γ = 1 mm−1, d = 20 mm, α/γ cos θ = 0.2.

Fig. 7
Fig. 7

(A) Transients during erasure with I ref σ. r0 = 1, γ = 1 mm−1, (a) d = 5 mm, (b) d = 10 mm, (c) d = 15 mm, (d) d = 20 mm. Compare time scales with Fig. 3(A). (B) Evolution of the grating envelope during readout with I ref σ for case (d) in (A). The 3D plot shows the ripples described in the text. r0 = 1, γ = 1 mm−1, d = 20 mm.

Fig. 8
Fig. 8

Normalized diffraction efficiency versus normalized time during readout of a grating written to steady state (with r0 = 1, γ = 1 mm−1, and d = 5 mm) with (a) I ref ρ, (b) I ref σ, (c) I ref - σ, (d) I ref - ρ.

Fig. 9
Fig. 9

Shape of the grating envelope at steady-state recording in the reflection geometry. The various boxes represent crystal boundaries, and the part of the function inside a box represents the grating envelope. For box A, B/C2 = 1000 and γd = 5; for box B, B/C2 = 100 and γd = 10; for box C, B/C2 = 1 and γd = 10; for box D, B/C2 = 0.01 and γd = 5.

Fig. 10
Fig. 10

Plot of the parameter B/C2 as a function of the writing beam ratio r0. A, γd = 0.1; B, γd = 1; C, γd = 10; D, γd = −10. Note that, at ln r0 = −γd/2, B/C2 has a singularity at which the recorded grating envelope is uniform.

Fig. 11
Fig. 11

(A) Transients during coherent erasure with I ref ρ in the reflection geometry. To give a uniform initial grating ln(B/C2) is taken to be 25. γ = 1 mm−1. (a) d = 20 mm, (b) d = 15 mm, (c) d = 10 mm, (d) d = 5 mm. (B) Evolution of the grating envelope during readout with I ref ρ for case (a) in (A). The shape of the grating envelope is plotted at approximately equal time intervals of Δt′ = 1 unit from t′ = 0 to t′ = 30. The initial grating has a constant amplitude of 1. ln(B/C2) = 25, γ = 1 mm−1, d = 20 mm. Note that the envelope propagates to the left.

Fig. 12
Fig. 12

A, Shape of the propagating envelope (solid curve) in the reflection geometry. B, The function ΘR. For comparison, C shows the similarity of the grating envelope to the tanh function, which is shifted and scaled to match the propagating envelope {i.e., 0.5 + 0.5 tanh[0.5(u − 15.63)]}. We computed the functions n(u) and ΘR(u) by using mathematica with initial conditions n(u = 0) = 1 and ΘR(u = 0) = 1.6.

Fig. 13
Fig. 13

(A) Transients during erasure with I ref σ. Here ln(B/C2) = 15, γ = 1 mm−1. (a) d = 20 mm, (b) d = 15 mm, (c) d = 10 mm, (d) d = 5 mm. Unlike for the transmission geometry, the oscillations here are too small to be observed on the scale of the figure. Compare with the time scale of Fig. 11(A). (B) Evolution of the grating envelope during readout with I ref σ for case (a) in (A). The shape of the grating envelope is plotted at approximately equal time intervals of Δt′ = 0.1 unit from t′ = 0 to t′ = 2. The thicker curves show the initial envelope shape. ln(B/C2) = 15, γ = 1 mm−1, d = 20 mm.

Equations (47)

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E SC = - i m E max = - i m E D 1 + E D / E q = - i m k B t e k g 1 + l s 2 k g 2 ,
d n 1 d t = - n 1 τ + n ss τ ,
1 τ = 1 + l s 2 k g 2 1 + r D 2 k g 2 Γ die ,
Δ n ( x , z ) = n p ( z ) cos ( k g · r ) .
A 1 ( z , t ) z = - i π λ cos θ n p ( z , t ) A 2 ( z , t ) - α 2 cos θ A 1 ( z , t ) ,
p A 2 ( z , t ) z = - i π λ cos θ n p ( z , t ) A 1 ( z , t ) - α 2 cos θ A 2 ( z , t ) ,
t n p ( z , t ) = - n p ( z , t ) τ + i n max m ( z , t ) τ ,
m ( z , t ) = 2 A 1 * ( z , t ) A 2 ( z , t ) A 1 * ( z , t ) 2 + A 2 * ( z , t ) 2 .
E ˜ ρ ( r ) = A ρ ( z ) exp ( i k ρ · r ) , E ˜ σ ( r ) = A σ ( z ) exp ( i k σ · r )
A ρ d z = - i 2 π n max λ cos θ A ρ A σ * I A σ - α 2 cos θ A ρ , p d A σ d z = - i 2 π n max λ cos θ A ρ * A σ I A ρ - α 2 cos θ A σ .
γ = 4 π n max λ cos θ sin ϕ ,
r 0 = I p ( z = 0 ) I σ ( z = 0 ) ,
m ( z ) = 2 [ I ρ ( z ) I σ ( z ) ] 1 / 2 I ρ ( z ) + I σ ( z ) .
d A ref ( z ) d z = - i π λ cos θ n p ( z ) A sig ( z ) - α 2 cos θ A ref ( z ) , p d A sig ( z ) d z = - i π λ cos θ n p ( z ) A ref ( z ) - α 2 cos θ A sig ( z ) ,
d r s d z = 1 R r d S i d z - S i R r 2 d R r d z = - p π λ cos θ n p ( z ) ( 1 + p r s 2 ) .
d r s 1 + p r s 2 = - p π λ cos θ n p ( z ) d z ,
I ρ ( z ) = I ρ ( 0 ) 1 + r 0 r 0 + exp ( γ z ) exp ( - α z / cos θ ) , I σ ( z ) = I σ ( 0 ) 1 + r 0 1 + r 0 exp ( - γ z ) exp ( - α z / cos θ ) ,
n p ( z ) = n max m ( z ) = n max sech ( γ z - ln r 0 2 ) .
η = sin 2 [ π λ cos θ 0 d n p ( z ) d z ] ,
η = sin 2 [ π n max λ cos θ 0 d m ( z ) d z ] .
n p ( z ) = n max 2 exp ( - γ z - ln r 0 2 ) ,
η = sin 2 { r 0 4 [ 1 - exp ( - γ d / 2 ) ] } .
η = sin 2 [ 2 arctan ( tanh γ d - ln r 0 4 ) + 2 arctan ( tanh ln r 0 4 ) ] .
Θ T ( z , t ) = 2 λ cos θ 0 z n p ( z , t ) d z .
n p ( z , t ) = λ cos θ 2 d Θ T ( z , t ) d z .
I sig ( z , t ) = I 0 sin 2 [ ( π / 2 ) Θ T ( z , t ) ] ,
m ( z , t ) = 2 [ I sig ( z , t ) I ref ( z , t ) ] 1 / 2 I sig ( z , t ) + I ref ( z , t ) = sin π Θ T ( z , t ) .
t n p ( z , t ) = - n p ( z , t ) τ + n max τ sin [ 2 π λ cos θ 0 z n p ( z , t ) d z ] ,
2 π 2 Θ T t z + 2 π Θ T z - sin π Θ T = 0.
n = 2 π Θ T z ,
η ( t ) = sin 2 [ π 2 Θ T ( γ d , t / τ ) ] .
2 χ ξ ζ = sin χ ,
2 π c d 2 Θ T d u 2 - 2 π d Θ T d u + sin π Θ T = 0.
2 π exp ( - α γ cos θ z ) 2 Θ T t z + 2 π Θ T z - sin π Θ T = 0.
2 π 2 Θ T t z + 2 π Θ T z + sin π Θ T = 0 ,
I ρ = - C + [ C 2 + B exp ( - γ z ) ] 1 / 2 , I σ = C + [ C 2 + B exp ( - γ z ) ] 1 / 2 ,
n p ( z ) = n max m ( z ) = n max { 1 + exp [ γ z - ln ( B / C 2 ) ] } - 1 / 2 .
I ref ( z ) I sig ( z ) = tanh 2 [ π λ z d n p ( z ) d z ]
η = tanh 2 { B 2 C [ 1 - exp ( - γ d / 2 ) ] } .
η = tanh 2 ( ln { [ s ( d ) - 1 ] [ s ( 0 ) + 1 ] [ s ( d ) + 1 ] [ s ( 0 ) - 1 ] } ) ,
Θ R ( z , t ) = 2 λ z d n p ( z , t ) d z .
I sig ( z , t ) = I 0 ( 1 - η 0 ) sinh 2 π 2 Θ R ( z , t ) ,
m ( z , t ) = 2 [ I sig ( z , t ) I ref ( z , t ) ] 1 / 2 I sig ( z , t ) + I ref ( z , t ) = tanh π Θ R ( z , t ) .
t n p ( z , t ) = - n p ( z , t ) τ + n max τ tanh [ 2 π λ z d n p ( z , t ) d z ] .
2 π 2 Θ R t z + 2 π Θ R z + tanh π Θ R = 0.
c d 2 Θ R d u 2 - d Θ R d u - 1 2 π tanh π Θ R = 0.
2 π 2 Θ R t z + 2 π Θ R z - tanh π Θ R = 0.

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