Abstract

Backward beam fanning is numerically investigated with a two-dimensional model on the basis of the solution of the coupled-wave equations between the incident beam and the backward radiation scattered by the inhomogeneities or defects distributed throughout the crystals. The numerical results are consistent with our experimental observations in a BaTiO3:Ce crystal. The physical origin of backward beam fanning is that the coupling constant times the length of the crystal is above the threshold for amplification of the backward radiation but below the threshold for the generation of the phase conjugation by stimulated two-wave mixing.

© 1998 Optical Society of America

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References

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  1. V. V. Voronov, I. R. Dorosh, Yu. S. Kuz’minov, and N. V. Tkachenko, “Photoinduced light scattering in cerium-doped barium strontium niobate crystals,” Sov. J. Quantum Electron. 10, 1346 (1980).
    [CrossRef]
  2. G. Zhang, Q. X. Li, P. P. Ho, S. Liu, Z. Wu, and R. R. Alfano, “Dependence of specklon size on the laser beam size via photoinduced light scattering in  LiNbO 3  :Fe ,” Appl. Opt. 25, 2955 (1986).
    [CrossRef]
  3. G. C. Valley, “Competition between forward- and backward-stimulated photorefractive scattering in  BaTiO 3  ,” J. Opt. Soc. Am. B 4, 14 (1987).
    [CrossRef]
  4. 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 (1990);M. Segev, D. Engin, A. Yariv, and G. C. Valley, “Temporal evolution of fanning in photorefractive materials,” Opt. Lett. 18, 956 (1993).
    [CrossRef] [PubMed]
  5. A. A. Zozulya and D. Z. Anderson, “Spatial structure of light and a nonlinear refractive index generated by fanning in photorefractive media,” Phys. Rev. A 52, 878 (1995).
    [CrossRef] [PubMed]
  6. P. Xie, P. Y. Wang, J. H. Dai, and H. J. Zhang, “Photorefractive image amplification and beam fanning with inclusion of random volume scattering,” Chin. Phys. Lett. 14, 908 (1997).
    [CrossRef]
  7. J. F. Lam, “Origin of phase conjugate waves in self-pumped photorefractive mirrors,” Appl. Phys. Lett. 46, 909 (1985).
    [CrossRef]
  8. P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Self-pumped phase conjugation in photorefractive crystals: reflectivity and spatial fidelity,” Phys. Rev. A 55, 3092 (1997).
    [CrossRef]
  9. P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Temporal behavior and instabilities of the self-pumped phase conjugation in photorefractive crystals,” Phys. Rev. A 56, 936 (1997).
    [CrossRef]
  10. J. Feinberg, “Asymmetric self-defocusing of an optical beam from the photorefractive effect,” J. Opt. Soc. Am. 72, 46 (1982).
    [CrossRef]
  11. M. Snowbell, M. Horowitz, and B. Fischer, “Dynamics of multiple two-wave mixing and fanning in photorefractive materials,” J. Opt. Soc. Am. B 11, 1972 (1994).
    [CrossRef]
  12. P. P. Benerjee and R. M. Misra, “Dependence of photorefractive beam fanning on beam parameters,” Opt. Commun. 100, 166 (1993).
    [CrossRef]
  13. Y. Zhu, P. Bernasconi, M. Zgonik, and P. Gunter, “Low frequency electro-optic coefficient measurements in  Ce:BaTiO 3   crystal,” J. Synthetic Crystals 23, 242 (1994).
  14. H. Risken, The Fokker-Plank Equation: Method of Solution and Applications (Springer-Verlag, Berlin, 1984), pp. 60–62.
  15. Ya. B. Zel’dovich, N. F. Pilipestsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985).

1997 (3)

P. Xie, P. Y. Wang, J. H. Dai, and H. J. Zhang, “Photorefractive image amplification and beam fanning with inclusion of random volume scattering,” Chin. Phys. Lett. 14, 908 (1997).
[CrossRef]

P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Self-pumped phase conjugation in photorefractive crystals: reflectivity and spatial fidelity,” Phys. Rev. A 55, 3092 (1997).
[CrossRef]

P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Temporal behavior and instabilities of the self-pumped phase conjugation in photorefractive crystals,” Phys. Rev. A 56, 936 (1997).
[CrossRef]

1995 (1)

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

1994 (2)

Y. Zhu, P. Bernasconi, M. Zgonik, and P. Gunter, “Low frequency electro-optic coefficient measurements in  Ce:BaTiO 3   crystal,” J. Synthetic Crystals 23, 242 (1994).

M. Snowbell, M. Horowitz, and B. Fischer, “Dynamics of multiple two-wave mixing and fanning in photorefractive materials,” J. Opt. Soc. Am. B 11, 1972 (1994).
[CrossRef]

1993 (1)

P. P. Benerjee and R. M. Misra, “Dependence of photorefractive beam fanning on beam parameters,” Opt. Commun. 100, 166 (1993).
[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 (1990);M. Segev, D. Engin, A. Yariv, and G. C. Valley, “Temporal evolution of fanning in photorefractive materials,” Opt. Lett. 18, 956 (1993).
[CrossRef] [PubMed]

1987 (1)

1986 (1)

1985 (1)

J. F. Lam, “Origin of phase conjugate waves in self-pumped photorefractive mirrors,” Appl. Phys. Lett. 46, 909 (1985).
[CrossRef]

1982 (1)

1980 (1)

V. V. Voronov, I. R. Dorosh, Yu. S. Kuz’minov, and N. V. Tkachenko, “Photoinduced light scattering in cerium-doped barium strontium niobate crystals,” Sov. J. Quantum Electron. 10, 1346 (1980).
[CrossRef]

Alfano, R. R.

Anderson, D. Z.

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

Benerjee, P. P.

P. P. Benerjee and R. M. Misra, “Dependence of photorefractive beam fanning on beam parameters,” Opt. Commun. 100, 166 (1993).
[CrossRef]

Bernasconi, P.

Y. Zhu, P. Bernasconi, M. Zgonik, and P. Gunter, “Low frequency electro-optic coefficient measurements in  Ce:BaTiO 3   crystal,” J. Synthetic Crystals 23, 242 (1994).

Dai, J. H.

P. Xie, P. Y. Wang, J. H. Dai, and H. J. Zhang, “Photorefractive image amplification and beam fanning with inclusion of random volume scattering,” Chin. Phys. Lett. 14, 908 (1997).
[CrossRef]

P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Self-pumped phase conjugation in photorefractive crystals: reflectivity and spatial fidelity,” Phys. Rev. A 55, 3092 (1997).
[CrossRef]

P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Temporal behavior and instabilities of the self-pumped phase conjugation in photorefractive crystals,” Phys. Rev. A 56, 936 (1997).
[CrossRef]

Dorosh, I. R.

V. V. Voronov, I. R. Dorosh, Yu. S. Kuz’minov, and N. V. Tkachenko, “Photoinduced light scattering in cerium-doped barium strontium niobate crystals,” Sov. J. Quantum Electron. 10, 1346 (1980).
[CrossRef]

Feinberg, J.

Fischer, B.

M. Snowbell, M. Horowitz, and B. Fischer, “Dynamics of multiple two-wave mixing and fanning in photorefractive materials,” J. Opt. Soc. Am. B 11, 1972 (1994).
[CrossRef]

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 (1990);M. Segev, D. Engin, A. Yariv, and G. C. Valley, “Temporal evolution of fanning in photorefractive materials,” Opt. Lett. 18, 956 (1993).
[CrossRef] [PubMed]

Gunter, P.

Y. Zhu, P. Bernasconi, M. Zgonik, and P. Gunter, “Low frequency electro-optic coefficient measurements in  Ce:BaTiO 3   crystal,” J. Synthetic Crystals 23, 242 (1994).

Ho, P. P.

Horowitz, M.

Kuz’minov, Yu. S.

V. V. Voronov, I. R. Dorosh, Yu. S. Kuz’minov, and N. V. Tkachenko, “Photoinduced light scattering in cerium-doped barium strontium niobate crystals,” Sov. J. Quantum Electron. 10, 1346 (1980).
[CrossRef]

Lam, J. F.

J. F. Lam, “Origin of phase conjugate waves in self-pumped photorefractive mirrors,” Appl. Phys. Lett. 46, 909 (1985).
[CrossRef]

Li, Q. X.

Liu, S.

Misra, R. M.

P. P. Benerjee and R. M. Misra, “Dependence of photorefractive beam fanning on beam parameters,” Opt. Commun. 100, 166 (1993).
[CrossRef]

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 (1990);M. Segev, D. Engin, A. Yariv, and G. C. Valley, “Temporal evolution of fanning in photorefractive materials,” Opt. Lett. 18, 956 (1993).
[CrossRef] [PubMed]

Pilipestsky, N. F.

Ya. B. Zel’dovich, N. F. Pilipestsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985).

Risken, H.

H. Risken, The Fokker-Plank Equation: Method of Solution and Applications (Springer-Verlag, Berlin, 1984), pp. 60–62.

Segev, M.

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 (1990);M. Segev, D. Engin, A. Yariv, and G. C. Valley, “Temporal evolution of fanning in photorefractive materials,” Opt. Lett. 18, 956 (1993).
[CrossRef] [PubMed]

Shkunov, V. V.

Ya. B. Zel’dovich, N. F. Pilipestsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985).

Snowbell, M.

Tkachenko, N. V.

V. V. Voronov, I. R. Dorosh, Yu. S. Kuz’minov, and N. V. Tkachenko, “Photoinduced light scattering in cerium-doped barium strontium niobate crystals,” Sov. J. Quantum Electron. 10, 1346 (1980).
[CrossRef]

Valley, G. C.

Voronov, V. V.

V. V. Voronov, I. R. Dorosh, Yu. S. Kuz’minov, and N. V. Tkachenko, “Photoinduced light scattering in cerium-doped barium strontium niobate crystals,” Sov. J. Quantum Electron. 10, 1346 (1980).
[CrossRef]

Wang, P. Y.

P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Self-pumped phase conjugation in photorefractive crystals: reflectivity and spatial fidelity,” Phys. Rev. A 55, 3092 (1997).
[CrossRef]

P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Temporal behavior and instabilities of the self-pumped phase conjugation in photorefractive crystals,” Phys. Rev. A 56, 936 (1997).
[CrossRef]

P. Xie, P. Y. Wang, J. H. Dai, and H. J. Zhang, “Photorefractive image amplification and beam fanning with inclusion of random volume scattering,” Chin. Phys. Lett. 14, 908 (1997).
[CrossRef]

Wu, Z.

Xie, P.

P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Temporal behavior and instabilities of the self-pumped phase conjugation in photorefractive crystals,” Phys. Rev. A 56, 936 (1997).
[CrossRef]

P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Self-pumped phase conjugation in photorefractive crystals: reflectivity and spatial fidelity,” Phys. Rev. A 55, 3092 (1997).
[CrossRef]

P. Xie, P. Y. Wang, J. H. Dai, and H. J. Zhang, “Photorefractive image amplification and beam fanning with inclusion of random volume scattering,” Chin. Phys. Lett. 14, 908 (1997).
[CrossRef]

Zel’dovich, Ya. B.

Ya. B. Zel’dovich, N. F. Pilipestsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985).

Zgonik, M.

Y. Zhu, P. Bernasconi, M. Zgonik, and P. Gunter, “Low frequency electro-optic coefficient measurements in  Ce:BaTiO 3   crystal,” J. Synthetic Crystals 23, 242 (1994).

Zhang, G.

Zhang, H. J.

P. Xie, P. Y. Wang, J. H. Dai, and H. J. Zhang, “Photorefractive image amplification and beam fanning with inclusion of random volume scattering,” Chin. Phys. Lett. 14, 908 (1997).
[CrossRef]

P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Self-pumped phase conjugation in photorefractive crystals: reflectivity and spatial fidelity,” Phys. Rev. A 55, 3092 (1997).
[CrossRef]

P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Temporal behavior and instabilities of the self-pumped phase conjugation in photorefractive crystals,” Phys. Rev. A 56, 936 (1997).
[CrossRef]

Zhu, Y.

Y. Zhu, P. Bernasconi, M. Zgonik, and P. Gunter, “Low frequency electro-optic coefficient measurements in  Ce:BaTiO 3   crystal,” J. Synthetic Crystals 23, 242 (1994).

Zozulya, A. A.

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

Appl. Opt. (1)

Appl. Phys. Lett. (1)

J. F. Lam, “Origin of phase conjugate waves in self-pumped photorefractive mirrors,” Appl. Phys. Lett. 46, 909 (1985).
[CrossRef]

Chin. Phys. Lett. (1)

P. Xie, P. Y. Wang, J. H. Dai, and H. J. Zhang, “Photorefractive image amplification and beam fanning with inclusion of random volume scattering,” Chin. Phys. Lett. 14, 908 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Synthetic Crystals (1)

Y. Zhu, P. Bernasconi, M. Zgonik, and P. Gunter, “Low frequency electro-optic coefficient measurements in  Ce:BaTiO 3   crystal,” J. Synthetic Crystals 23, 242 (1994).

Opt. Commun. (2)

P. P. Benerjee and R. M. Misra, “Dependence of photorefractive beam fanning on beam parameters,” Opt. Commun. 100, 166 (1993).
[CrossRef]

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 (1990);M. Segev, D. Engin, A. Yariv, and G. C. Valley, “Temporal evolution of fanning in photorefractive materials,” Opt. Lett. 18, 956 (1993).
[CrossRef] [PubMed]

Phys. Rev. A (3)

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

P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Self-pumped phase conjugation in photorefractive crystals: reflectivity and spatial fidelity,” Phys. Rev. A 55, 3092 (1997).
[CrossRef]

P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Temporal behavior and instabilities of the self-pumped phase conjugation in photorefractive crystals,” Phys. Rev. A 56, 936 (1997).
[CrossRef]

Sov. J. Quantum Electron. (1)

V. V. Voronov, I. R. Dorosh, Yu. S. Kuz’minov, and N. V. Tkachenko, “Photoinduced light scattering in cerium-doped barium strontium niobate crystals,” Sov. J. Quantum Electron. 10, 1346 (1980).
[CrossRef]

Other (2)

H. Risken, The Fokker-Plank Equation: Method of Solution and Applications (Springer-Verlag, Berlin, 1984), pp. 60–62.

Ya. B. Zel’dovich, N. F. Pilipestsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985).

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Figures (8)

Fig. 1
Fig. 1

Illustration defining external angle αe and internal angle α between the normal of the crystal surface and the incident beam. The counterclockwise angles shown in the figure are defined as positive and the clockwise angles as negative. The +z direction is taken along the propagation direction of the incident beam inside the crystal, and the z axis is symmetrical to the z axis with respect to the normal of the surface ad.

Fig. 2
Fig. 2

Photographs of the crystal taken at the time of t=3 s, i.e., after 3 s of the beam is incident upon the crystal. The upper arrow indicates the direction of the incident beam outside the crystal.

Fig. 3
Fig. 3

Angular intensity distributions of the (a) input beam and (b) backward-fanning beam at z=0 inside the crystal.

Fig. 4
Fig. 4

Spatial-intensity distribution of the backward fanning beam inside the crystal after the Fresnel reflection by the input face ad. The sizes of the calculation region are 10.24 mm and 5.5 mm along coordinates x and z, respectively.

Fig. 5
Fig. 5

Reflectivity of the backward fanning beam versus the incident angle α.

Fig. 6
Fig. 6

Output intensity distribution (left) and output phase (right) of the backward beam in spatial-frequency space for crystal length of (a) 7 mm (1=5×10-5), (b) 8 mm (1=1.7×10-5), (c) 10 mm (1=2×10-6), and (d) 14 mm (1=1×10-8). Dashed curves, input.

Fig. 7
Fig. 7

Spatial fidelity as a function of coupling constant times length γL for the boundary conditions (a) of diffuse backward scattering and (b) of retroreflection with a random phase.

Fig. 8
Fig. 8

(a) Output intensity distribution and (b) output phase of the backward beam at input surface for crystal length L=16 mm with volume-scattering strength q=2×10-19 mm-1. Dashed curves, input.

Tables (1)

Tables Icon

Table 1 Parameters of BaTiO3:Ce at Wavelength λ=632.8 nm and Temperature T=300 K 9,13

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

Ej(x, z, t)=Aj(x, z, t)exp(-iωt)+c.c.=θfj(θ, z, t)exp[±ik(sin θ)x+(cos θ)z]×exp(-iωt)+c.c.(j=F, B),
fF(θ, z, t)z=1cos θ θθ[QT(θ, θ, z, t)fF(θ, z, t)+QR(θ, θ, z, t)fB(θ, z, t)]+QR(θ, θ, z, t)fB(θ, z, t)+σF(θ, z)(1+cos2 θ)I0(z, t),
fB*(θ, z, t)z=1cos θ θθ[QT(θ, θ, z, t)fB*(θ, z, t)+QR(θ, θ, z, t)fF*(θ, z, t)]+QR(θ, θ, z, t)fF*(θ, z, t)+σB(θ, z)(1+cos2 θ)I0(z, t).
σjP(θ, z)=0,
σjP(θ, z)σjP(θ, z)=2qδjjδPPδθθδ(z-z),
τT(θ, θ) QT(θ, θ, z, t)t+QT(θ, θ, z, t)
=γT(θ, θ)I0(z, t) [fF(θ, z, t)fF*(θ, z, t)+fB*(θ, z, t)fB(θ, z, t)],
τR(θ, θ) QR(θ, θ, z, t)t+QR(θ, θ, z, t)
=γR(θ, θ)I0(z, t) [fF(θ, z, t)fB*(θ, z, t)+fB*(θ, z, t)fF(θ, z, t)],whenθθ,
τR(θ, θ) QR(θ, θ, z, t)t+QR(θ, θ, z, t)
=γR(θ, θ)I0(z) fF(θ, z, t)fB*(θ, z, t),
τl(θ, θ)=τ 1+Edl(θ, θ)/EMl(θ, θ)1+Edl(θ, θ)/Eql(θ, θ),
QT(θ, θ, z)=0,
QR(θ, θ, z)=γR(θ, θ)I0(z) [fF(θ, z)fB*(θ, z)+fB*(θ, z)fF(θ, z)],whenθθ,
QR(θ, θ, z)=γR(θ, θ)I0(z) fF(θ, z)fB*(θ, z),
R=θ|fB(θ, z=0)|2θ|fF(θ, z=0)|2

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