Abstract

Image amplification and beam fanning in a photorefractive medium are described by the multivariable Langevin equations when the random volume scattering is included owing to the inhomogeneities and/or defects distributed throughout the medium. The effects of the random volume scattering on the image amplification and the beam fanning are studied analytically in the undepleted-pump approximation and numerically, and they are compared with the surface scattering.

© 1998 Optical Society of America

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References

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  1. See, for example, Y. Fainman, E. Klancik, and S. H. Lee, “Optical coherent image amplification by two-wave coupling in photorefractive BaTiO3,” Opt. Eng. 25, 228 (1986);P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484 (1989).
    [CrossRef]
  2. J. F. Heanue, M. C. Bashaw, and L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265, 749 (1994).
    [CrossRef] [PubMed]
  3. See, for example, M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Elec- tron. QE-20, 12 (1984);P. Gunter and J. P. Huignard, eds. Photorefractive Materials and Their Applications II, (Springer-Verlag, Berlin, 1989), Chaps. 4–6.
    [CrossRef]
  4. J. H. Hong, A. E. Chiou, and P. Yeh, “Image amplification by two-wave mixing in photorefractive crystals,” Appl. Opt. 29, 3026 (1990).
    [CrossRef] [PubMed]
  5. P. Xie, J. H. Dai, P. Y. Wang, and H. J. Zhang, “Spatial fidelity of image amplification in photorefractive crystals,” Appl. Opt. 35, 7102 (1996).
    [CrossRef] [PubMed]
  6. M. Segev, D. Engin, A. Yariv, and G. C. Valley, “Temporal evolution of fanning in photorefractive materials,” Opt. Lett. 18, 956 (1993).
    [CrossRef] [PubMed]
  7. H. Risken, The Fokker-Plank Equation: Method of Solution and Applications (Springer-Verlag, Berlin, 1984).
  8. K. Meerholz, B. L. Volodin, Sandalphon, B. Kippelen, and N. Peyghambarian, “A photorefractive polymer with high optical gain and diffraction efficiency near 100%,” Nature (London) 371, 497 (1994);A. Grunnet-Jepsen, C. L. Thompson, R. J. Twieg, and W. E. Moerner, “High performance photorefractive polymer with improved stability,” Appl. Phys. Lett. 70, 1515 (1997).
    [CrossRef]
  9. R. L. Honeycutt, “Stochastic Runge–Kutta algorithms. I. White noise,” Phys. Rev. A 45, 600 (1992).
    [CrossRef] [PubMed]
  10. 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]

1997 (1)

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]

1996 (1)

1994 (2)

K. Meerholz, B. L. Volodin, Sandalphon, B. Kippelen, and N. Peyghambarian, “A photorefractive polymer with high optical gain and diffraction efficiency near 100%,” Nature (London) 371, 497 (1994);A. Grunnet-Jepsen, C. L. Thompson, R. J. Twieg, and W. E. Moerner, “High performance photorefractive polymer with improved stability,” Appl. Phys. Lett. 70, 1515 (1997).
[CrossRef]

J. F. Heanue, M. C. Bashaw, and L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265, 749 (1994).
[CrossRef] [PubMed]

1993 (1)

1992 (1)

R. L. Honeycutt, “Stochastic Runge–Kutta algorithms. I. White noise,” Phys. Rev. A 45, 600 (1992).
[CrossRef] [PubMed]

1990 (1)

1986 (1)

See, for example, Y. Fainman, E. Klancik, and S. H. Lee, “Optical coherent image amplification by two-wave coupling in photorefractive BaTiO3,” Opt. Eng. 25, 228 (1986);P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484 (1989).
[CrossRef]

1984 (1)

See, for example, M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Elec- tron. QE-20, 12 (1984);P. Gunter and J. P. Huignard, eds. Photorefractive Materials and Their Applications II, (Springer-Verlag, Berlin, 1989), Chaps. 4–6.
[CrossRef]

Bashaw, M. C.

J. F. Heanue, M. C. Bashaw, and L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265, 749 (1994).
[CrossRef] [PubMed]

Chiou, A. E.

Cronin-Golomb, M.

See, for example, M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Elec- tron. QE-20, 12 (1984);P. Gunter and J. P. Huignard, eds. Photorefractive Materials and Their Applications II, (Springer-Verlag, Berlin, 1989), Chaps. 4–6.
[CrossRef]

Dai, J. H.

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, “Spatial fidelity of image amplification in photorefractive crystals,” Appl. Opt. 35, 7102 (1996).
[CrossRef] [PubMed]

Engin, D.

Fainman, Y.

See, for example, Y. Fainman, E. Klancik, and S. H. Lee, “Optical coherent image amplification by two-wave coupling in photorefractive BaTiO3,” Opt. Eng. 25, 228 (1986);P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484 (1989).
[CrossRef]

Fischer, B.

See, for example, M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Elec- tron. QE-20, 12 (1984);P. Gunter and J. P. Huignard, eds. Photorefractive Materials and Their Applications II, (Springer-Verlag, Berlin, 1989), Chaps. 4–6.
[CrossRef]

Heanue, J. F.

J. F. Heanue, M. C. Bashaw, and L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265, 749 (1994).
[CrossRef] [PubMed]

Hesselink, L.

J. F. Heanue, M. C. Bashaw, and L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265, 749 (1994).
[CrossRef] [PubMed]

Honeycutt, R. L.

R. L. Honeycutt, “Stochastic Runge–Kutta algorithms. I. White noise,” Phys. Rev. A 45, 600 (1992).
[CrossRef] [PubMed]

Hong, J. H.

Kippelen, B.

K. Meerholz, B. L. Volodin, Sandalphon, B. Kippelen, and N. Peyghambarian, “A photorefractive polymer with high optical gain and diffraction efficiency near 100%,” Nature (London) 371, 497 (1994);A. Grunnet-Jepsen, C. L. Thompson, R. J. Twieg, and W. E. Moerner, “High performance photorefractive polymer with improved stability,” Appl. Phys. Lett. 70, 1515 (1997).
[CrossRef]

Klancik, E.

See, for example, Y. Fainman, E. Klancik, and S. H. Lee, “Optical coherent image amplification by two-wave coupling in photorefractive BaTiO3,” Opt. Eng. 25, 228 (1986);P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484 (1989).
[CrossRef]

Lee, S. H.

See, for example, Y. Fainman, E. Klancik, and S. H. Lee, “Optical coherent image amplification by two-wave coupling in photorefractive BaTiO3,” Opt. Eng. 25, 228 (1986);P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484 (1989).
[CrossRef]

Meerholz, K.

K. Meerholz, B. L. Volodin, Sandalphon, B. Kippelen, and N. Peyghambarian, “A photorefractive polymer with high optical gain and diffraction efficiency near 100%,” Nature (London) 371, 497 (1994);A. Grunnet-Jepsen, C. L. Thompson, R. J. Twieg, and W. E. Moerner, “High performance photorefractive polymer with improved stability,” Appl. Phys. Lett. 70, 1515 (1997).
[CrossRef]

Peyghambarian, N.

K. Meerholz, B. L. Volodin, Sandalphon, B. Kippelen, and N. Peyghambarian, “A photorefractive polymer with high optical gain and diffraction efficiency near 100%,” Nature (London) 371, 497 (1994);A. Grunnet-Jepsen, C. L. Thompson, R. J. Twieg, and W. E. Moerner, “High performance photorefractive polymer with improved stability,” Appl. Phys. Lett. 70, 1515 (1997).
[CrossRef]

Risken, H.

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

Sandalphon,

K. Meerholz, B. L. Volodin, Sandalphon, B. Kippelen, and N. Peyghambarian, “A photorefractive polymer with high optical gain and diffraction efficiency near 100%,” Nature (London) 371, 497 (1994);A. Grunnet-Jepsen, C. L. Thompson, R. J. Twieg, and W. E. Moerner, “High performance photorefractive polymer with improved stability,” Appl. Phys. Lett. 70, 1515 (1997).
[CrossRef]

Segev, M.

Valley, G. C.

Volodin, B. L.

K. Meerholz, B. L. Volodin, Sandalphon, B. Kippelen, and N. Peyghambarian, “A photorefractive polymer with high optical gain and diffraction efficiency near 100%,” Nature (London) 371, 497 (1994);A. Grunnet-Jepsen, C. L. Thompson, R. J. Twieg, and W. E. Moerner, “High performance photorefractive polymer with improved stability,” Appl. Phys. Lett. 70, 1515 (1997).
[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, “Spatial fidelity of image amplification in photorefractive crystals,” Appl. Opt. 35, 7102 (1996).
[CrossRef] [PubMed]

White, J. O.

See, for example, M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Elec- tron. QE-20, 12 (1984);P. Gunter and J. P. Huignard, eds. Photorefractive Materials and Their Applications II, (Springer-Verlag, Berlin, 1989), Chaps. 4–6.
[CrossRef]

Xie, P.

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, “Spatial fidelity of image amplification in photorefractive crystals,” Appl. Opt. 35, 7102 (1996).
[CrossRef] [PubMed]

Yariv, A.

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

See, for example, M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Elec- tron. QE-20, 12 (1984);P. Gunter and J. P. Huignard, eds. Photorefractive Materials and Their Applications II, (Springer-Verlag, Berlin, 1989), Chaps. 4–6.
[CrossRef]

Yeh, P.

Zhang, H. J.

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, “Spatial fidelity of image amplification in photorefractive crystals,” Appl. Opt. 35, 7102 (1996).
[CrossRef] [PubMed]

Appl. Opt. (2)

IEEE J. Quantum Elec- tron. (1)

See, for example, M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Elec- tron. QE-20, 12 (1984);P. Gunter and J. P. Huignard, eds. Photorefractive Materials and Their Applications II, (Springer-Verlag, Berlin, 1989), Chaps. 4–6.
[CrossRef]

Nature (London) (1)

K. Meerholz, B. L. Volodin, Sandalphon, B. Kippelen, and N. Peyghambarian, “A photorefractive polymer with high optical gain and diffraction efficiency near 100%,” Nature (London) 371, 497 (1994);A. Grunnet-Jepsen, C. L. Thompson, R. J. Twieg, and W. E. Moerner, “High performance photorefractive polymer with improved stability,” Appl. Phys. Lett. 70, 1515 (1997).
[CrossRef]

Opt. Eng. (1)

See, for example, Y. Fainman, E. Klancik, and S. H. Lee, “Optical coherent image amplification by two-wave coupling in photorefractive BaTiO3,” Opt. Eng. 25, 228 (1986);P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484 (1989).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (2)

R. L. Honeycutt, “Stochastic Runge–Kutta algorithms. I. White noise,” Phys. Rev. A 45, 600 (1992).
[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]

Science (1)

J. F. Heanue, M. C. Bashaw, and L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265, 749 (1994).
[CrossRef] [PubMed]

Other (1)

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

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Figures (6)

Fig. 1
Fig. 1

Spatial-intensity distributions of the amplified image beam for γL=4: (a) γ=1 and L=4, (b) γ=10 and L=0.4; q=10-9/mm.

Fig. 2
Fig. 2

Spatial-intensity distributions of the amplified image beam in a BaTiO3 crystal of L=5 mm with (a) q=0, (b) q=10-9, and (c) q=10-8, in units of mm-1.

Fig. 3
Fig. 3

(a) Spatial fidelity of the image amplification versus the volume-scattering strength q; (b) [A(x, L)-G×A(x, 0)]2dx versus q. The solid line is the fit curve.

Fig. 4
Fig. 4

Solid curves represent {exp[2γ(-α, θ)L/cos θ]}/2γ(-α, θ)/cos θ versus θ, and dashed curves represent exp[2γ(-α, θ)L/cos θ] versus θ: (a) L=0.5 mm, (b) L=1.2 mm.

Fig. 5
Fig. 5

Angular-intensity distributions of the fanning beam in a BaTiO3:Ce crystal of L=0.8 mm for (a) volume scattering and (b) surface scattering. The input angle of the input wave is α=15°, and the c axis makes an angle of 45° with respect to the direction of the pump wave.

Fig. 6
Fig. 6

Numerical simulation of the beam path in a BaTiO3 crystal with the angle between the crystal c axis and the propagating direction of the input beam being 35°. q=4×10-9/mm.

Equations (20)

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

Ep(x, z, t)Ap(z)exp[i(k sin αx+k cos αz-ωt)]+c.c.,
Es(x, z, t)=As(x, z) exp[i(kz-ωt)]+c.c.,
As(x, z)=θfs(θ, z)×exp[ik sin θx+i(k cos θ-k)z],
dAp(z)dz=-1cos θθγ(-α, θ)I0(z) fs(θ, z)f0*(θ, z)Ap(z)-αL2 Ap(z),
dfs(θ, z)dz=1cos θ γ(-α, θ)I0(z) Ap(z)Ap*(z)fs(θ, z)-θγ(θ, θ)I0(z) fs(θ, z)fs*(θ, z)fs(θ, z)-αL2 fs(θ, z)+σ(θ, z)I0(z).
σP(θ, z)=0,
σP(θ, z)σP(θ, z)=2qδPPδθθδ(z-z),
dXidz=hi({X}, z)+j[gij({X}, z)Γj(z)],
dXdz=γX+Γ(z),
P(X, z)z=-γ X (XP)+D 2X2 P.
P(X, z|X, z)=-γ2πD{1-exp[2γ(z-z)]}×expγ(X-exp[γ(z-z)]X)22D{1-exp[2γ(z-z)]}.
W(X, L)=γ2πD exp(2γL)× exp-γ2D exp(2γL) [X-C exp(γL)]2,
X=C exp(γL),
(X-X)2=Dγ exp(2γL)=qI0(0)2γ exp(2γL)qr2γ [C exp(γL)]2,
(X-X)2=[(1±εr)C exp(γL)-C exp(γL)]2εr[C exp(γL)]2.
X2=qIP(0)2γ exp(2γL),
X2=εIP(0) exp(2γL).
FI=As(x, 0)As(x, L)dx|As(x, 0)|2dx|As(x, L)|2dx,
exp[2γ(-α, θ)L/cos θ]2γ(-α, θ)/cos θ.
exp[2γ(-α,θ)L/cos θ]2γ(-α,θ)/cos θ,

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