Abstract

The nonlinear problem of speckle-beam propagation in a photorefractive crystal with a strong diffusion-grating response has been solved. The approximate self-sustaining solution has been found that conserves the Gaussian shape of an angular spectrum of the beam but gradually tilts its propagation direction. The rate of angular tilting is proportional to the square of the beam’s angular divergence and depends on the crystal-axis orientation.

© 1994 Optical Society of America

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References

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  1. P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications, Vols. 61 and 62 of Springer Topics in Applied Physics (Springer-Verlag, Heidelberg, 1988, 1989).
    [CrossRef]
  2. J. Feinberg, “Self-pumped continuous-wave phase conjugator using internal reflection,” Opt. Lett. 7, 486–488 (1982).
    [CrossRef] [PubMed]
  3. A. M. C. Smout and R. W. Eason, “Bistability and noncom-mutative behavior of multiple-beam self-pulsing and self-pumping in BaTiO3,” Opt. Lett. 12, 51–53 (1987).
    [CrossRef]
  4. M. D. Ewbank, “Mechanism for photorefractive phase conjugation using incoherent beams,” Opt. Lett. 13, 47–49 (1988).
    [CrossRef] [PubMed]
  5. J. Feinberg, “Asymmetric self-defocusing of an optical beam from the photorefractive effect,” J. Opt. Soc. Am. 72, 46–51 (1982).
    [CrossRef]
  6. A. A. Esayan, A. A. Zozulya, and V. T. Tikhonchuk, “Self-bending of a light beam in a photorefractive crystal,” Lebedev Phys. Inst. Short Commun. Phys. 5, 45–47 (1990).
  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. J. F. Lam, “Origin of phase conjugate waves in self-pumped photorefractive mirrors,” Appl. Phys. Lett. 46, 909 (1985).
    [CrossRef]
  9. T. R. Moore, Ph.D. dissertation (Naval Post-graduate School, Monterey, Calif., 1987).
  10. B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Heidelberg, 1985).
    [CrossRef]
  11. S. Weiss, S. Sternklar, and B. Fischer, “Double-phase-conjugate mirror: analysis, demonstration, and application,” Opt. Lett. 12, 114–116 (1987).
    [CrossRef] [PubMed]
  12. N. I. Beldyugina, A. V. Mamaev, and V. V. Shkunov, “Dynamics of generation self-starting for a semilinear phase-conjugate mirror with a multi-transverse-mode nonlinear cavity,” Kvantovaya Elektron. 19, 691–697 (1992).
  13. B. Ya. Zel’dovich and V. V. Shkunov, “Specklon,” Izv. Akad. Nauk Fiz. 48, 1545 (1984).
  14. V. G. Sidorovich, “To the theory of a ‘Brillouin mirror,’” Sov. Phys. Tech. Phys. 21, 1270 (1976).
  15. M. Segev, D. Engin, A. Yariv, and G. S. Valley, “Temporal evolution of fanning in photorefractive crystals,” Opt. Lett. 18, 956 (1993).
    [CrossRef] [PubMed]
  16. M. Segev, D. Engin, A. Yariv, and G. S. Valley, “Temporal evolution and spatial cross talk in photorefractive double-phase-conjugate mirrors,” Opt. Lett. 18, 1828 (1993).
    [CrossRef] [PubMed]

1993 (2)

1992 (1)

N. I. Beldyugina, A. V. Mamaev, and V. V. Shkunov, “Dynamics of generation self-starting for a semilinear phase-conjugate mirror with a multi-transverse-mode nonlinear cavity,” Kvantovaya Elektron. 19, 691–697 (1992).

1990 (2)

A. A. Esayan, A. A. Zozulya, and V. T. Tikhonchuk, “Self-bending of a light beam in a photorefractive crystal,” Lebedev Phys. Inst. Short Commun. Phys. 5, 45–47 (1990).

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]

1988 (1)

1987 (2)

1985 (1)

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

1984 (1)

B. Ya. Zel’dovich and V. V. Shkunov, “Specklon,” Izv. Akad. Nauk Fiz. 48, 1545 (1984).

1982 (2)

1976 (1)

V. G. Sidorovich, “To the theory of a ‘Brillouin mirror,’” Sov. Phys. Tech. Phys. 21, 1270 (1976).

Beldyugina, N. I.

N. I. Beldyugina, A. V. Mamaev, and V. V. Shkunov, “Dynamics of generation self-starting for a semilinear phase-conjugate mirror with a multi-transverse-mode nonlinear cavity,” Kvantovaya Elektron. 19, 691–697 (1992).

Eason, R. W.

Engin, D.

Esayan, A. A.

A. A. Esayan, A. A. Zozulya, and V. T. Tikhonchuk, “Self-bending of a light beam in a photorefractive crystal,” Lebedev Phys. Inst. Short Commun. Phys. 5, 45–47 (1990).

Ewbank, M. D.

Feinberg, J.

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 application,” Opt. Lett. 12, 114–116 (1987).
[CrossRef] [PubMed]

Lam, J. F.

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

Mamaev, A. V.

N. I. Beldyugina, A. V. Mamaev, and V. V. Shkunov, “Dynamics of generation self-starting for a semilinear phase-conjugate mirror with a multi-transverse-mode nonlinear cavity,” Kvantovaya Elektron. 19, 691–697 (1992).

Moore, T. R.

T. R. Moore, Ph.D. dissertation (Naval Post-graduate School, Monterey, Calif., 1987).

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]

Pilipetsky, N. F.

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Heidelberg, 1985).
[CrossRef]

Segev, M.

Shkunov, V. V.

N. I. Beldyugina, A. V. Mamaev, and V. V. Shkunov, “Dynamics of generation self-starting for a semilinear phase-conjugate mirror with a multi-transverse-mode nonlinear cavity,” Kvantovaya Elektron. 19, 691–697 (1992).

B. Ya. Zel’dovich and V. V. Shkunov, “Specklon,” Izv. Akad. Nauk Fiz. 48, 1545 (1984).

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Heidelberg, 1985).
[CrossRef]

Sidorovich, V. G.

V. G. Sidorovich, “To the theory of a ‘Brillouin mirror,’” Sov. Phys. Tech. Phys. 21, 1270 (1976).

Smout, A. M. C.

Sternklar, S.

Tikhonchuk, V. T.

A. A. Esayan, A. A. Zozulya, and V. T. Tikhonchuk, “Self-bending of a light beam in a photorefractive crystal,” Lebedev Phys. Inst. Short Commun. Phys. 5, 45–47 (1990).

Valley, G. S.

Weiss, S.

Yariv, A.

Zel’dovich, B. Ya.

B. Ya. Zel’dovich and V. V. Shkunov, “Specklon,” Izv. Akad. Nauk Fiz. 48, 1545 (1984).

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Heidelberg, 1985).
[CrossRef]

Zozulya, A. A.

A. A. Esayan, A. A. Zozulya, and V. T. Tikhonchuk, “Self-bending of a light beam in a photorefractive crystal,” Lebedev Phys. Inst. Short Commun. Phys. 5, 45–47 (1990).

Appl. Phys. Lett. (1)

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

Izv. Akad. Nauk Fiz. (1)

B. Ya. Zel’dovich and V. V. Shkunov, “Specklon,” Izv. Akad. Nauk Fiz. 48, 1545 (1984).

J. Opt. Soc. Am. (1)

Kvantovaya Elektron. (1)

N. I. Beldyugina, A. V. Mamaev, and V. V. Shkunov, “Dynamics of generation self-starting for a semilinear phase-conjugate mirror with a multi-transverse-mode nonlinear cavity,” Kvantovaya Elektron. 19, 691–697 (1992).

Lebedev Phys. Inst. Short Commun. Phys. (1)

A. A. Esayan, A. A. Zozulya, and V. T. Tikhonchuk, “Self-bending of a light beam in a photorefractive crystal,” Lebedev Phys. Inst. Short Commun. Phys. 5, 45–47 (1990).

Opt. Commun. (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]

Opt. Lett. (6)

Sov. Phys. Tech. Phys. (1)

V. G. Sidorovich, “To the theory of a ‘Brillouin mirror,’” Sov. Phys. Tech. Phys. 21, 1270 (1976).

Other (3)

P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications, Vols. 61 and 62 of Springer Topics in Applied Physics (Springer-Verlag, Heidelberg, 1988, 1989).
[CrossRef]

T. R. Moore, Ph.D. dissertation (Naval Post-graduate School, Monterey, Calif., 1987).

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Heidelberg, 1985).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Experimental trajectories of curving wide-angle part of the beam (solid boundaries) and untilted central component (dashed boundaries), from photographs presented in (a) Ref. 2 and (b) Ref. 3.

Fig. 2
Fig. 2

Orientation of the local frame (ξ, ζ), which is tightly fixed at the beam trajectory with respect to the optical axis C.

Fig. 3
Fig. 3

Experimental (dashed boundaries) trajectories of the self-curving beams from the photograph presented in Ref. 4 for a BaTiO3 crystal and the theoretical curves (solid curves) calculated with the proposed model at Δθ ≈ 0.05.

Equations (19)

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E ζ - i 2 k Δ E = - i γ E 2 / ξ E 2 E ( r , ζ ) .
E ( r , ζ ) = q C ( q , ζ ) exp ( i qr - i q 2 ζ / 2 k ) ,
P ( R ) = P E * E 2 E ( R ) + [ P ( R ) - P E * E 2 E ( R ) ] .
- i γ ( E ξ + E E * E * ξ ) .
( E E * E * ξ ) coh = E E * / ξ E 2 E ( R ) = - i q ¯ x E ( R ) ,
q ¯ x ( ζ ) = 1 E 2 q q x C ( q , ζ ) 2 .
C ( q , ζ ) ζ = γ ( q x - q ¯ x ) C ( q , ζ ) .
C ( q , ζ ) ζ = γ E 2 q 1 q 2 q 3 ( q 1 - q 2 ) x × C ( q 1 ) C * ( q 2 ) C ( q 3 ) × exp [ i ( q 1 - q 2 + q 3 - q ) r - i ( q 1 2 - q 2 2 + q 3 2 - q 2 ) ζ / 2 k ] .
C ( q 1 ) C * ( q 2 ) - C ( q 1 ) 2 δ ( 2 ) ( q 1 - q 2 ) .
C ( q , ζ ) ζ = γ E 2 [ p ( q - p ) x 1 + ( q - p ) 2 r D 2 C ( p ) 2 ] C ( q , ζ ) ,
C ( q , ζ ) = C ( q , ζ 0 ) m ( ζ ) exp [ q x ζ 0 ζ γ ( ζ ) d ζ ] ,
m ( ζ ) = { q C ( q , ζ 0 ) 2 exp [ 2 q x ζ ζ γ ( ζ ) d ζ ] } - 1 / 2
C ( q , ζ ) 2 = j ( q , ζ ) = j 0 exp [ - ( q x - k v ) 2 b x 2 - q y 2 b y 2 ] ,
j ( q , ζ ) = I 1 ( ζ ) π b x b y exp [ - ( q x - k v ) 2 b x 2 - q y 2 b y 2 ] + I 2 ( ζ ) δ ( 2 ) ( q - q 0 ) ,
I 1 ( ζ ) = I 1 0 r + 1 s r + 1
I 2 ( ζ ) = I 2 0 s ( r + 1 ) s r + 1
z ( x ) = 0 x tan [ α ( x ) ] d x = - 1 ρ 2 × ln | cos [ ρ 2 ρ 1 x + arctan ( ρ 2 / ρ 1 tan α 0 ) ] cos [ arctan ( ρ 2 ρ 1 tan α 0 ) ] | ,
E ( r , ζ ) = [ E + ( r , ζ ) exp ( i k R ) + E - ( r , ζ ) exp ( - i k R ) ] ,
δ ɛ ( R ) ( E + 2 + E - 2 ) / ξ ( E + 2 + E - 2 ) .

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