Abstract

The fanning behavior of the two interacting laser beams in a double phase-conjugate mirror (DPCM) obviously influences the characteristics of the mirror. We present a two-dimensional experimental study and model of the beam-fanning patterns obtained in nominally undoped and cobalt-doped (20 parts in 106) BaTiO3 samples. We propose a numerical method based on beam coupling theory that allows us to determine, for each crystal, which directions will develop significant fanning. We thus interpret the beam-fanning patterns by focusing on the influence of the incident beam’s width and incidence angle. The time evolution of the intensity in each of these directions is also studied and modeled. This model has been applied to DPCM to enable both incident beams to generate fanning in a common direction, with good results in terms of response time, efficiency, and stability.

© 2001 Optical Society of America

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  1. J. Feinberg, “Asymmetric self-defocusing of an optical beam from the photorefractive effect,” J. Opt. Soc. Am. 72, 46–51 (1982).
    [Crossref]
  2. J. Feinberg, “Self-pumped, continuous-wave conjugator using internal reflection,” Opt. Lett. 7, 486–488 (1982).
    [Crossref] [PubMed]
  3. M. Cronin-Golomb, B. Fisher, J. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
    [Crossref]
  4. S. Weiss, S. Sternklar, and B. Fischer, “Double phase-conjugate mirror: analysis, demonstration, and applications,” Opt. Lett. 12, 114–116 (1987).
    [Crossref] [PubMed]
  5. P. Banerjee and R. Misra, “Dependence of photorefractive beam fanning on beam parameters,” Opt. Commun. 100, 166–172 (1993).
    [Crossref]
  6. M. Segev, D. Engin, A. Yariv, and G. Valley, “Temporal evolution of fanning in photorefractive materials,” Opt. Lett. 18, 956–958 (1993).
    [Crossref] [PubMed]
  7. M. Segev, Y. Ophir, and B. Fischer, “Nonlinear multi two-wave mixing, the fanning process and its bleaching in photorefractive media,” Opt. Commun. 77, 265–274 (1990).
    [Crossref]
  8. 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, 813–818 (1993).
    [Crossref]
  9. Y. H. Hong, P. Xie, J. H. Dai, Y. Zhu, H. G. Yang, and H. J. Zhang, “Fanning effect in photorefractive crystals,” Opt. Lett. 18, 772–774 (1993).
    [Crossref] [PubMed]
  10. P. Xie, J. Dai, P. Wang, and H. Zhang, “A two-dimensional theory and propagation of beam fanning in photorefractive crystals,” J. Appl. Phys. 75, 1891–1895 (1994).
    [Crossref]
  11. A. Zozulya and D. Anderson, “Spatial structure of light and a nonlinear refractive index generated by fanning in photorefractive media,” Phys. Rev. A 52, 878–881 (1995).
    [Crossref] [PubMed]
  12. K. Kamra and K. Singh, “Characterization of beam fanning in BaTiO3 under biasing illumination and its application as log processor,” Opt. Eng. 34, 2266–2273 (1995).
    [Crossref]
  13. G. Li, T. Yang, Y. Y. Teng, F. C. Lin, and M. A. Fiddy, “Scattering and beam fanning in a BaTiO3 crystal,” Waves Random Media 2, 303–315 (1992).
    [Crossref]
  14. G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Gunter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
    [Crossref] [PubMed]
  15. A. Kamshilin, V. Prokoviev, and T. Jaaskelainen, “Beam fanning and double phase conjugation in a fiber-like photorefractive sample,” IEEE J. Quantum Electron. 31, 1642–1647 (1995).
    [Crossref]
  16. A. Kamshilin, H. Tuovinen, V. Prokoviev, and T. Jaaskelainen, “Phase conjugate mirrors on the base of BiTiO20 photorefractive fibre,” Opt. Mater. 4, 399–403 (1995).
    [Crossref]
  17. Q. He, P. Yeh, C. Gu, and R. Neurgaonkar, “Multigrating competition effects in photorefractive mutually pumped phase conjugation,” J. Opt. Soc. Am. B 9, 114–120 (1992).
    [Crossref]
  18. N. Wolffer, P. Gravey, and V. Royer, “Thresholding of two facing double phase conjugate mirror and semilinear phase conjugate mirror with fanning in BTO,” Opt. Commun. 89, 380–384 (1992).
    [Crossref]
  19. P. Gunter and J. Huignard, “Photorefractive properties of BaTiO3,” in Photorefractive Materials and Their Applications, P. Gunter and J. Huignard, eds. (Springer-Verlag, Berlin, 1988), Vol. 1, p. 219.
  20. P. Yeh, “Photorefractive effect,” in Introduction to Photorefractive Nonlinear Optics, J. W. Goodman, ed., Vol. 133 of Wiley Series in Pure and Applied Optics (Wiley-Interscience, New York, 1993), pp. 82–117.
  21. M. Garrett, J. Chang, H. Jenssen, and C. Warde, “High beam-coupling gain and deep- and shallow-trap effects in cobalt-doped barium titanate, BaTiO3:Co,” J. Opt. Soc. Am. B 9, 1407–1415 (1992).
    [Crossref]
  22. Ref. 20, “Wave mixing in photorefractive media,” pp. 118–182.
  23. P. Gunter, E. Voit, and M. Zha, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214 (1985).
    [Crossref]
  24. A. Smout, R. Eason, and M. Gower, “Regular oscillations and self-pulsating in self-pumped BaTiO3,” Opt. Commun. 59, 77–82 (1986).
    [Crossref]
  25. D. Gauthier, P. Narum, and R. Boyd, “Observation of deterministic chaos in a phase-conjugate mirror,” Phys. Rev. Lett. 58, 1640–1643 (1987).
    [Crossref] [PubMed]
  26. A. M. Darwish, M. Dokhanian, P. R. K. Rao, M. Curley, and P. Venkateswarlu, “Oscillations in coherent beam pumped mutual phase conjugate emissions,” in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications, S. Yin and F. T. Yu, eds., Proc. SPIE3470, 137–145 (1998).

1995 (5)

G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Gunter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
[Crossref] [PubMed]

A. Kamshilin, V. Prokoviev, and T. Jaaskelainen, “Beam fanning and double phase conjugation in a fiber-like photorefractive sample,” IEEE J. Quantum Electron. 31, 1642–1647 (1995).
[Crossref]

A. Kamshilin, H. Tuovinen, V. Prokoviev, and T. Jaaskelainen, “Phase conjugate mirrors on the base of BiTiO20 photorefractive fibre,” Opt. Mater. 4, 399–403 (1995).
[Crossref]

A. Zozulya and D. Anderson, “Spatial structure of light and a nonlinear refractive index generated by fanning in photorefractive media,” Phys. Rev. A 52, 878–881 (1995).
[Crossref] [PubMed]

K. Kamra and K. Singh, “Characterization of beam fanning in BaTiO3 under biasing illumination and its application as log processor,” Opt. Eng. 34, 2266–2273 (1995).
[Crossref]

1994 (1)

P. Xie, J. Dai, P. Wang, and H. Zhang, “A two-dimensional theory and propagation of beam fanning in photorefractive crystals,” J. Appl. Phys. 75, 1891–1895 (1994).
[Crossref]

1993 (4)

P. Banerjee and R. Misra, “Dependence of photorefractive beam fanning on beam parameters,” Opt. Commun. 100, 166–172 (1993).
[Crossref]

M. Segev, D. Engin, A. Yariv, and G. Valley, “Temporal evolution of fanning in photorefractive materials,” Opt. Lett. 18, 956–958 (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, 813–818 (1993).
[Crossref]

Y. H. Hong, P. Xie, J. H. Dai, Y. Zhu, H. G. Yang, and H. J. Zhang, “Fanning effect in photorefractive crystals,” Opt. Lett. 18, 772–774 (1993).
[Crossref] [PubMed]

1992 (4)

G. Li, T. Yang, Y. Y. Teng, F. C. Lin, and M. A. Fiddy, “Scattering and beam fanning in a BaTiO3 crystal,” Waves Random Media 2, 303–315 (1992).
[Crossref]

Q. He, P. Yeh, C. Gu, and R. Neurgaonkar, “Multigrating competition effects in photorefractive mutually pumped phase conjugation,” J. Opt. Soc. Am. B 9, 114–120 (1992).
[Crossref]

N. Wolffer, P. Gravey, and V. Royer, “Thresholding of two facing double phase conjugate mirror and semilinear phase conjugate mirror with fanning in BTO,” Opt. Commun. 89, 380–384 (1992).
[Crossref]

M. Garrett, J. Chang, H. Jenssen, and C. Warde, “High beam-coupling gain and deep- and shallow-trap effects in cobalt-doped barium titanate, BaTiO3:Co,” J. Opt. Soc. Am. B 9, 1407–1415 (1992).
[Crossref]

1990 (1)

M. Segev, Y. Ophir, and B. Fischer, “Nonlinear multi two-wave mixing, the fanning process and its bleaching in photorefractive media,” Opt. Commun. 77, 265–274 (1990).
[Crossref]

1987 (2)

S. Weiss, S. Sternklar, and B. Fischer, “Double phase-conjugate mirror: analysis, demonstration, and applications,” Opt. Lett. 12, 114–116 (1987).
[Crossref] [PubMed]

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

1986 (1)

A. Smout, R. Eason, and M. Gower, “Regular oscillations and self-pulsating in self-pumped BaTiO3,” Opt. Commun. 59, 77–82 (1986).
[Crossref]

1985 (1)

P. Gunter, E. Voit, and M. Zha, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214 (1985).
[Crossref]

1984 (1)

M. Cronin-Golomb, B. Fisher, J. 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 (2)

Anderson, D.

A. Zozulya and D. Anderson, “Spatial structure of light and a nonlinear refractive index generated by fanning in photorefractive media,” Phys. Rev. A 52, 878–881 (1995).
[Crossref] [PubMed]

Anderson, D. Z.

G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Gunter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
[Crossref] [PubMed]

Banerjee, P.

P. Banerjee and R. Misra, “Dependence of photorefractive beam fanning on beam parameters,” Opt. Commun. 100, 166–172 (1993).
[Crossref]

Boyd, R.

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

Chang, J.

Cronin-Golomb, M.

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

Curley, M.

A. M. Darwish, M. Dokhanian, P. R. K. Rao, M. Curley, and P. Venkateswarlu, “Oscillations in coherent beam pumped mutual phase conjugate emissions,” in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications, S. Yin and F. T. Yu, eds., Proc. SPIE3470, 137–145 (1998).

Czaia, L.

G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Gunter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
[Crossref] [PubMed]

Dai, J.

P. Xie, J. Dai, P. Wang, and H. Zhang, “A two-dimensional theory and propagation of beam fanning in photorefractive crystals,” J. Appl. Phys. 75, 1891–1895 (1994).
[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, 813–818 (1993).
[Crossref]

Y. H. Hong, P. Xie, J. H. Dai, Y. Zhu, H. G. Yang, and H. J. Zhang, “Fanning effect in photorefractive crystals,” Opt. Lett. 18, 772–774 (1993).
[Crossref] [PubMed]

Darwish, A. M.

A. M. Darwish, M. Dokhanian, P. R. K. Rao, M. Curley, and P. Venkateswarlu, “Oscillations in coherent beam pumped mutual phase conjugate emissions,” in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications, S. Yin and F. T. Yu, eds., Proc. SPIE3470, 137–145 (1998).

Dokhanian, M.

A. M. Darwish, M. Dokhanian, P. R. K. Rao, M. Curley, and P. Venkateswarlu, “Oscillations in coherent beam pumped mutual phase conjugate emissions,” in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications, S. Yin and F. T. Yu, eds., Proc. SPIE3470, 137–145 (1998).

Eason, R.

A. Smout, R. Eason, and M. Gower, “Regular oscillations and self-pulsating in self-pumped BaTiO3,” Opt. Commun. 59, 77–82 (1986).
[Crossref]

Engin, D.

Feinberg, J.

Fiddy, M. A.

G. Li, T. Yang, Y. Y. Teng, F. C. Lin, and M. A. Fiddy, “Scattering and beam fanning in a BaTiO3 crystal,” Waves Random Media 2, 303–315 (1992).
[Crossref]

Fischer, B.

M. Segev, Y. Ophir, and B. Fischer, “Nonlinear multi two-wave mixing, the fanning process and its bleaching in photorefractive media,” Opt. Commun. 77, 265–274 (1990).
[Crossref]

S. Weiss, S. Sternklar, and B. Fischer, “Double phase-conjugate mirror: analysis, demonstration, and applications,” Opt. Lett. 12, 114–116 (1987).
[Crossref] [PubMed]

Fisher, B.

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

Garrett, M.

Gauthier, D.

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

Gower, M.

A. Smout, R. Eason, and M. Gower, “Regular oscillations and self-pulsating in self-pumped BaTiO3,” Opt. Commun. 59, 77–82 (1986).
[Crossref]

Gravey, P.

N. Wolffer, P. Gravey, and V. Royer, “Thresholding of two facing double phase conjugate mirror and semilinear phase conjugate mirror with fanning in BTO,” Opt. Commun. 89, 380–384 (1992).
[Crossref]

Gu, C.

Gunter, P.

G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Gunter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
[Crossref] [PubMed]

P. Gunter, E. Voit, and M. Zha, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214 (1985).
[Crossref]

P. Gunter and J. Huignard, “Photorefractive properties of BaTiO3,” in Photorefractive Materials and Their Applications, P. Gunter and J. Huignard, eds. (Springer-Verlag, Berlin, 1988), Vol. 1, p. 219.

He, Q.

Hong, Y. H.

Y. H. Hong, P. Xie, J. H. Dai, Y. Zhu, H. G. Yang, and H. J. Zhang, “Fanning effect in photorefractive crystals,” Opt. Lett. 18, 772–774 (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, 813–818 (1993).
[Crossref]

Huignard, J.

P. Gunter and J. Huignard, “Photorefractive properties of BaTiO3,” in Photorefractive Materials and Their Applications, P. Gunter and J. Huignard, eds. (Springer-Verlag, Berlin, 1988), Vol. 1, p. 219.

Jaaskelainen, T.

A. Kamshilin, H. Tuovinen, V. Prokoviev, and T. Jaaskelainen, “Phase conjugate mirrors on the base of BiTiO20 photorefractive fibre,” Opt. Mater. 4, 399–403 (1995).
[Crossref]

A. Kamshilin, V. Prokoviev, and T. Jaaskelainen, “Beam fanning and double phase conjugation in a fiber-like photorefractive sample,” IEEE J. Quantum Electron. 31, 1642–1647 (1995).
[Crossref]

Jenssen, H.

Kamra, K.

K. Kamra and K. Singh, “Characterization of beam fanning in BaTiO3 under biasing illumination and its application as log processor,” Opt. Eng. 34, 2266–2273 (1995).
[Crossref]

Kamshilin, A.

A. Kamshilin, H. Tuovinen, V. Prokoviev, and T. Jaaskelainen, “Phase conjugate mirrors on the base of BiTiO20 photorefractive fibre,” Opt. Mater. 4, 399–403 (1995).
[Crossref]

A. Kamshilin, V. Prokoviev, and T. Jaaskelainen, “Beam fanning and double phase conjugation in a fiber-like photorefractive sample,” IEEE J. Quantum Electron. 31, 1642–1647 (1995).
[Crossref]

Li, G.

G. Li, T. Yang, Y. Y. Teng, F. C. Lin, and M. A. Fiddy, “Scattering and beam fanning in a BaTiO3 crystal,” Waves Random Media 2, 303–315 (1992).
[Crossref]

Lin, F. C.

G. Li, T. Yang, Y. Y. Teng, F. C. Lin, and M. A. Fiddy, “Scattering and beam fanning in a BaTiO3 crystal,” Waves Random Media 2, 303–315 (1992).
[Crossref]

Misra, R.

P. Banerjee and R. Misra, “Dependence of photorefractive beam fanning on beam parameters,” Opt. Commun. 100, 166–172 (1993).
[Crossref]

Montemezzani, G.

G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Gunter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
[Crossref] [PubMed]

Narum, P.

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

Neurgaonkar, R.

Ophir, Y.

M. Segev, Y. Ophir, and B. Fischer, “Nonlinear multi two-wave mixing, the fanning process and its bleaching in photorefractive media,” Opt. Commun. 77, 265–274 (1990).
[Crossref]

Prokoviev, V.

A. Kamshilin, V. Prokoviev, and T. Jaaskelainen, “Beam fanning and double phase conjugation in a fiber-like photorefractive sample,” IEEE J. Quantum Electron. 31, 1642–1647 (1995).
[Crossref]

A. Kamshilin, H. Tuovinen, V. Prokoviev, and T. Jaaskelainen, “Phase conjugate mirrors on the base of BiTiO20 photorefractive fibre,” Opt. Mater. 4, 399–403 (1995).
[Crossref]

Rao, P. R. K.

A. M. Darwish, M. Dokhanian, P. R. K. Rao, M. Curley, and P. Venkateswarlu, “Oscillations in coherent beam pumped mutual phase conjugate emissions,” in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications, S. Yin and F. T. Yu, eds., Proc. SPIE3470, 137–145 (1998).

Royer, V.

N. Wolffer, P. Gravey, and V. Royer, “Thresholding of two facing double phase conjugate mirror and semilinear phase conjugate mirror with fanning in BTO,” Opt. Commun. 89, 380–384 (1992).
[Crossref]

Segev, M.

M. Segev, D. Engin, A. Yariv, and G. Valley, “Temporal evolution of fanning in photorefractive materials,” Opt. Lett. 18, 956–958 (1993).
[Crossref] [PubMed]

M. Segev, Y. Ophir, and B. Fischer, “Nonlinear multi two-wave mixing, the fanning process and its bleaching in photorefractive media,” Opt. Commun. 77, 265–274 (1990).
[Crossref]

Singh, K.

K. Kamra and K. Singh, “Characterization of beam fanning in BaTiO3 under biasing illumination and its application as log processor,” Opt. Eng. 34, 2266–2273 (1995).
[Crossref]

Smout, A.

A. Smout, R. Eason, and M. Gower, “Regular oscillations and self-pulsating in self-pumped BaTiO3,” Opt. Commun. 59, 77–82 (1986).
[Crossref]

Sternklar, S.

Teng, Y. Y.

G. Li, T. Yang, Y. Y. Teng, F. C. Lin, and M. A. Fiddy, “Scattering and beam fanning in a BaTiO3 crystal,” Waves Random Media 2, 303–315 (1992).
[Crossref]

Tuovinen, H.

A. Kamshilin, H. Tuovinen, V. Prokoviev, and T. Jaaskelainen, “Phase conjugate mirrors on the base of BiTiO20 photorefractive fibre,” Opt. Mater. 4, 399–403 (1995).
[Crossref]

Valley, G.

Venkateswarlu, P.

A. M. Darwish, M. Dokhanian, P. R. K. Rao, M. Curley, and P. Venkateswarlu, “Oscillations in coherent beam pumped mutual phase conjugate emissions,” in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications, S. Yin and F. T. Yu, eds., Proc. SPIE3470, 137–145 (1998).

Voit, E.

P. Gunter, E. Voit, and M. Zha, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214 (1985).
[Crossref]

Wang, P.

P. Xie, J. Dai, P. Wang, and H. Zhang, “A two-dimensional theory and propagation of beam fanning in photorefractive crystals,” J. Appl. Phys. 75, 1891–1895 (1994).
[Crossref]

Warde, C.

Weiss, S.

White, J.

M. Cronin-Golomb, B. Fisher, J. 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, and V. Royer, “Thresholding of two facing double phase conjugate mirror and semilinear phase conjugate mirror with fanning in BTO,” Opt. Commun. 89, 380–384 (1992).
[Crossref]

Xie, P.

P. Xie, J. Dai, P. Wang, and H. Zhang, “A two-dimensional theory and propagation of beam fanning in photorefractive crystals,” J. Appl. Phys. 75, 1891–1895 (1994).
[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, 813–818 (1993).
[Crossref]

Y. H. Hong, P. Xie, J. H. Dai, Y. Zhu, H. G. Yang, and H. J. Zhang, “Fanning effect in photorefractive crystals,” Opt. Lett. 18, 772–774 (1993).
[Crossref] [PubMed]

Yang, H. G.

Yang, T.

G. Li, T. Yang, Y. Y. Teng, F. C. Lin, and M. A. Fiddy, “Scattering and beam fanning in a BaTiO3 crystal,” Waves Random Media 2, 303–315 (1992).
[Crossref]

Yariv, A.

M. Segev, D. Engin, A. Yariv, and G. Valley, “Temporal evolution of fanning in photorefractive materials,” Opt. Lett. 18, 956–958 (1993).
[Crossref] [PubMed]

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

Yeh, P.

Q. He, P. Yeh, C. Gu, and R. Neurgaonkar, “Multigrating competition effects in photorefractive mutually pumped phase conjugation,” J. Opt. Soc. Am. B 9, 114–120 (1992).
[Crossref]

P. Yeh, “Photorefractive effect,” in Introduction to Photorefractive Nonlinear Optics, J. W. Goodman, ed., Vol. 133 of Wiley Series in Pure and Applied Optics (Wiley-Interscience, New York, 1993), pp. 82–117.

Zgonik, M.

G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson, M. Zgonik, and P. Gunter, “Origin of the lobe structure in photorefractive beam fanning,” Phys. Rev. A 52, 1791–1794 (1995).
[Crossref] [PubMed]

Zha, M.

P. Gunter, E. Voit, and M. Zha, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214 (1985).
[Crossref]

Zhang, H.

P. Xie, J. Dai, P. Wang, and H. Zhang, “A two-dimensional theory and propagation of beam fanning in photorefractive crystals,” J. Appl. Phys. 75, 1891–1895 (1994).
[Crossref]

Zhang, H. J.

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, 813–818 (1993).
[Crossref]

Y. H. Hong, P. Xie, J. H. Dai, Y. Zhu, H. G. Yang, and H. J. Zhang, “Fanning effect in photorefractive crystals,” Opt. Lett. 18, 772–774 (1993).
[Crossref] [PubMed]

Zhu, Y.

Y. H. Hong, P. Xie, J. H. Dai, Y. Zhu, H. G. Yang, and H. J. Zhang, “Fanning effect in photorefractive crystals,” Opt. Lett. 18, 772–774 (1993).
[Crossref] [PubMed]

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[Crossref]

Zozulya, A.

A. Zozulya and D. Anderson, “Spatial structure of light and a nonlinear refractive index generated by fanning in photorefractive media,” Phys. Rev. A 52, 878–881 (1995).
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Zozulya, A. A.

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref] [PubMed]

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Ref. 20, “Wave mixing in photorefractive media,” pp. 118–182.

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Figures (12)

Fig. 1
Fig. 1

Experimental setup used for study of beam fanning. (a) Description of the experimental apparatus used with either (b) shutter 1 or (c) shutter 2 opened, for propagation of light in the direction of the c axis or in the opposite direction.

Fig. 2
Fig. 2

Evolution of the width of the beam inside the crystal as a function of incident angle.

Fig. 3
Fig. 3

Fanning distribution in an undoped BaTiO3 crystal for an incident angle θi=23° on entrance face +c. Fanning for 200- and 100-µW intensities are shown. The curve referred to as “2*100 µW” represents the pattern obtained with 100 µW magnified by a factor of 2. The angles are taken inside the crystal.

Fig. 4
Fig. 4

Fanning patterns obtained in a doped BaTiO3 crystal with θi=18.6° and 100-µW incident intensity, depending on the beam width.

Fig. 5
Fig. 5

Dependence of fanning angle on incident angle for several beam waists.

Fig. 6
Fig. 6

Comparison of measured (filled circles) and theoretically determined (curves) fanning angles as a function of incident angle for Co-doped and undoped BaTiO3 samples. Upper and lower dashed curves, theoretical maxima of gain for NA=0.4×1016 cm-3; solid curves, NA=0.15×1016 cm-3. Dark filled circles, experimental points of the doped crystal; pale filled circles, experimental points of the undoped crystal. Curves and filled circles located over the first bisecting line correspond to -c orientation; others, to +c orientation.

Fig. 7
Fig. 7

Theoretical determination of the successive fanning angles in the undoped crystal for θi=23°. Long-dashed line, the first bisecting line; solid curves, the couples (θi, θf) that correspond to γ/θf=0 for +c and -c propagation. Short-dashed lines, successive fanning angles that can be determined for a+c propagation. Inset, corresponding BF pattern experimentally observed.

Fig. 8
Fig. 8

Definition of the effective interaction length leff of two beams inside the crystal, showing the difference in the influence on leff of narrow and broad beams.

Fig. 9
Fig. 9

Temporal evolution of the energy distribution in the fanning pattern for various incident intensities recorded in an undoped BaTiO3 sample. The beam width is the same, and the incident angle is 23°.

Fig. 10
Fig. 10

(a) Theoretical temporal evolution of the energy distribution in the fanning pattern with the first model and (b) results of the calculation performed for the improved model with depletion of the pumps taken into account. Both models are for an undoped BaTiO3 sample. The parameters were chosen to fit as closely as possible the steady-state experimental ratio of the intensities in the various fanning directions.

Fig. 11
Fig. 11

Influence of η on the theoretical temporal evolution of the intensity in four expected fanning directions with parameters of a nominally undoped crystal. The parameters used here are those used in the experiment represented in Fig. 9.

Fig. 12
Fig. 12

Temporal evolution of the two efficiency coefficients of a DPCM in the undoped crystal. Darker curve, reflectivity of the conjugate beam coming from the +c face; lighter curve, for the -c face. Inset, enlargement representing the first 300 s of the same curves.

Equations (19)

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l(θi)=2w0 cos(θi)cos{arcsin[n sin(θi)]}.
Is(z)Is(0)=Is(0)+Ip(0)Ip(0)+Is(0)exp(γ z) exp[(γ-α)z].
Is(z)=Is(0)exp[(γ-α)z],
γ(θi, θf)=(2π)2n3kBTcos(θi-θf/2)eλ nenor42 cosθi+θf2×sin(θi+θf) ΛΛ2+ΛD2,
Λ=λ2×sin(θi-θf/2),
ΛD=2π[εaε0 cos2(θi+θf/2)+εcε0 sin2(θi+θf/2)]1/2eNAkBT,
ΛD2πεaε0 cos(θi+θf/2)e(NA/kBT)1/2.
γ(θi, θf)=(2π)2n3kBTe nenor42 cos(θi+θf/2)sin(θi+θf)sin(θi-θf)λ2 sin(θi+θf/2)2+2πkBT(εaε0/NA)1/2 cos(θi+θf/2)q2.
γ(θi, θf)θf=0.
leffNB=lcos2(θi+θf/2)(tan θi-tan θf).
leffBB=l0cos(θi+θf/2).
leffBB=leffNB,
lcos2(θi+θf/2)(tan θi-tan θf)=l0cos(θi+θf/2),
If [θf(n)]=I0[θf(n)]i=1n-1 exp {γ[θf(i), θf(n)]leff [θf(i), θf(n)]}×exp {γ[θi, θf(n)]leff [θi, θf(n)]}.
If[θf(n)]
=I0[θf(n)]i=1n-1×expγ(θf(i), θf(n))1-exp-tτleff [θf(i), θf(n)]×expγ[θi, θf(n)]-exp-tτleff [θi, θf(n)]-αl0cos θf(n)-j=n+1 Io[θf(j)]expγ[θf(n), θf(j)]×1-exp-tτleff [θf(n), θf(j)].
kD2=kOD2+kOT2,
0<η(Is)=1kD2 kOD2+kOT21+β/sT Is(0)1,
γ=(2π)2n3kBTe nenor42 cos(θi+θf/2)sin(θi+θf)sin(θi-θf)λ2 sin(θi+θf/2)2+2πkBT(εaε0/NA)1/2 cos(θi+θf/2)q2η(I).

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