Abstract

The effect of a position-dependent time constant on nondegenerate photorefractive two-wave mixing is analyzed. For codirectional two-wave mixing the results indicate that the bandwidth of the two-wave-mixing gain decreases when the medium becomes lossy and that the photorefractive phase shift depends on the loss of the medium. In contradirectional two-wave mixing the total local intensity strongly depends on the coupling even if the medium is lossless. Thus the bandwidth is dependent on the energy coupling.

© 1992 Optical Society of America

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References

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  1. See, for example, P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
    [CrossRef]
  2. J. P. Huignard and A. Marrackchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive BSO crystals,” Opt. Commun. 38, 249–254 (1981).
    [CrossRef]
  3. J. P. Huignard and A. Marrackchi, “Two-wave mixing and energy transfer in Bi12SiO20 crystals: amplification and vibration analysis,” Opt. Lett. 6, 622–624 (1981).
    [CrossRef] [PubMed]
  4. P. Yeh, “Contradirectional two-wave mixing in photorefractive media,” Opt. Commun. 45, 323–326 (1983).
    [CrossRef]
  5. P. Yeh, “Photorefractive coupling in ring resonators,” Appl. Opt. 23, 2974–2978 (1984).
    [CrossRef] [PubMed]
  6. J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
    [CrossRef]
  7. M. D. Ewbank and P. Yeh, “Frequency shift and cavity length in photorefractive resonators,” Opt. Lett. 10, 496–498 (1985).
    [CrossRef] [PubMed]
  8. P. Yeh, “Theory of unidirectional photorefractive resonators,” J. Opt. Soc. Am. B 2, 1924–1928 (1985).
    [CrossRef]
  9. D. M. Lininger, D. D. Crouch, P. J. Martin, and D. Z. Anderson, “Theory of bistability and self-pulsing in a ring resonator with saturable photorefractive gain and loss,” Opt. Commun. 76, 89–96 (1990).
    [CrossRef]
  10. D. Erbschloe, L. Solymar, H. Takacs, and T. Wilson, “Two-wave mixing in reflection holograms in photorefractive materials,” IEEE J. Quantum Electron. 24, 830–826 (1988).
    [CrossRef]
  11. D. J. Webb and L. Solymar, “The effect of optical activity and absorption on two-wave mixing in Bi12SiO20,” Opt. Commun. 83, 287–294 (1991).
    [CrossRef]

1991 (1)

D. J. Webb and L. Solymar, “The effect of optical activity and absorption on two-wave mixing in Bi12SiO20,” Opt. Commun. 83, 287–294 (1991).
[CrossRef]

1990 (1)

D. M. Lininger, D. D. Crouch, P. J. Martin, and D. Z. Anderson, “Theory of bistability and self-pulsing in a ring resonator with saturable photorefractive gain and loss,” Opt. Commun. 76, 89–96 (1990).
[CrossRef]

1989 (1)

See, for example, P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

1988 (1)

D. Erbschloe, L. Solymar, H. Takacs, and T. Wilson, “Two-wave mixing in reflection holograms in photorefractive materials,” IEEE J. Quantum Electron. 24, 830–826 (1988).
[CrossRef]

1985 (2)

1984 (1)

1983 (1)

P. Yeh, “Contradirectional two-wave mixing in photorefractive media,” Opt. Commun. 45, 323–326 (1983).
[CrossRef]

1982 (1)

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

1981 (2)

J. P. Huignard and A. Marrackchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive BSO crystals,” Opt. Commun. 38, 249–254 (1981).
[CrossRef]

J. P. Huignard and A. Marrackchi, “Two-wave mixing and energy transfer in Bi12SiO20 crystals: amplification and vibration analysis,” Opt. Lett. 6, 622–624 (1981).
[CrossRef] [PubMed]

Anderson, D. Z.

D. M. Lininger, D. D. Crouch, P. J. Martin, and D. Z. Anderson, “Theory of bistability and self-pulsing in a ring resonator with saturable photorefractive gain and loss,” Opt. Commun. 76, 89–96 (1990).
[CrossRef]

Cronin-Golomb, M.

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

Crouch, D. D.

D. M. Lininger, D. D. Crouch, P. J. Martin, and D. Z. Anderson, “Theory of bistability and self-pulsing in a ring resonator with saturable photorefractive gain and loss,” Opt. Commun. 76, 89–96 (1990).
[CrossRef]

Erbschloe, D.

D. Erbschloe, L. Solymar, H. Takacs, and T. Wilson, “Two-wave mixing in reflection holograms in photorefractive materials,” IEEE J. Quantum Electron. 24, 830–826 (1988).
[CrossRef]

Ewbank, M. D.

Fischer, B.

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

Huignard, J. P.

J. P. Huignard and A. Marrackchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive BSO crystals,” Opt. Commun. 38, 249–254 (1981).
[CrossRef]

J. P. Huignard and A. Marrackchi, “Two-wave mixing and energy transfer in Bi12SiO20 crystals: amplification and vibration analysis,” Opt. Lett. 6, 622–624 (1981).
[CrossRef] [PubMed]

Lininger, D. M.

D. M. Lininger, D. D. Crouch, P. J. Martin, and D. Z. Anderson, “Theory of bistability and self-pulsing in a ring resonator with saturable photorefractive gain and loss,” Opt. Commun. 76, 89–96 (1990).
[CrossRef]

Marrackchi, A.

J. P. Huignard and A. Marrackchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive BSO crystals,” Opt. Commun. 38, 249–254 (1981).
[CrossRef]

J. P. Huignard and A. Marrackchi, “Two-wave mixing and energy transfer in Bi12SiO20 crystals: amplification and vibration analysis,” Opt. Lett. 6, 622–624 (1981).
[CrossRef] [PubMed]

Martin, P. J.

D. M. Lininger, D. D. Crouch, P. J. Martin, and D. Z. Anderson, “Theory of bistability and self-pulsing in a ring resonator with saturable photorefractive gain and loss,” Opt. Commun. 76, 89–96 (1990).
[CrossRef]

Solymar, L.

D. J. Webb and L. Solymar, “The effect of optical activity and absorption on two-wave mixing in Bi12SiO20,” Opt. Commun. 83, 287–294 (1991).
[CrossRef]

D. Erbschloe, L. Solymar, H. Takacs, and T. Wilson, “Two-wave mixing in reflection holograms in photorefractive materials,” IEEE J. Quantum Electron. 24, 830–826 (1988).
[CrossRef]

Takacs, H.

D. Erbschloe, L. Solymar, H. Takacs, and T. Wilson, “Two-wave mixing in reflection holograms in photorefractive materials,” IEEE J. Quantum Electron. 24, 830–826 (1988).
[CrossRef]

Webb, D. J.

D. J. Webb and L. Solymar, “The effect of optical activity and absorption on two-wave mixing in Bi12SiO20,” Opt. Commun. 83, 287–294 (1991).
[CrossRef]

White, J. O.

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

Wilson, T.

D. Erbschloe, L. Solymar, H. Takacs, and T. Wilson, “Two-wave mixing in reflection holograms in photorefractive materials,” IEEE J. Quantum Electron. 24, 830–826 (1988).
[CrossRef]

Yariv, A.

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

Yeh, P.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

IEEE J. Quantum Electron. (2)

See, for example, P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

D. Erbschloe, L. Solymar, H. Takacs, and T. Wilson, “Two-wave mixing in reflection holograms in photorefractive materials,” IEEE J. Quantum Electron. 24, 830–826 (1988).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (4)

D. M. Lininger, D. D. Crouch, P. J. Martin, and D. Z. Anderson, “Theory of bistability and self-pulsing in a ring resonator with saturable photorefractive gain and loss,” Opt. Commun. 76, 89–96 (1990).
[CrossRef]

D. J. Webb and L. Solymar, “The effect of optical activity and absorption on two-wave mixing in Bi12SiO20,” Opt. Commun. 83, 287–294 (1991).
[CrossRef]

J. P. Huignard and A. Marrackchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive BSO crystals,” Opt. Commun. 38, 249–254 (1981).
[CrossRef]

P. Yeh, “Contradirectional two-wave mixing in photorefractive media,” Opt. Commun. 45, 323–326 (1983).
[CrossRef]

Opt. Lett. (2)

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Figures (4)

Fig. 1
Fig. 1

(a) Schematic drawing of codirectional two-wave mixing. (b) Schematic drawing of contradirectional two-wave mixing.

Fig. 2
Fig. 2

Codirectional two-wave mixing gain versus L for various values of m. Solid curves, position-dependent time constant; dashed curves, position-independent time constant.

Fig. 3
Fig. 3

Codirectional two-wave-mixing gain and photorefractive phase shift as functions of Ωτ0 for various values of m. Solid curves, position-dependent time constant; dashed curves, position-independent time constant.

Fig. 4
Fig. 4

Contradirectional two-wave-mixing gain and photorefractive phase shift as functions of Ωτ0 (or Ωτ) for various values of m. Solid curves, position-dependent time constant (as a function of Ωτ0); dashed curves, position-independent time constant (as function of Ωτ).

Equations (24)

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d d z I 1 ( z ) = γ I 1 ( z ) I 2 ( z ) I 1 ( z ) + I 2 ( z ) α I 1 ( z ) ,
d d z I 2 ( z ) = σ γ I 1 ( z ) I 2 ( z ) I 1 ( z ) + I 2 ( z ) σ α I 2 ( z ) ,
d d z ψ 1 ( z ) = β I 2 ( z ) I 1 ( z ) + I 2 ( z ) ,
d d z ψ 2 ( z ) = σ β I 1 ( z ) I 1 ( z ) + I 2 ( z ) ,
γ = γ 0 1 + { Ω τ 0 [ I 1 ( z ) + I 2 ( z ) ] } 2 , β = γ 2 Ω τ 0 I 1 ( z ) + I 2 ( z ) ,
I 1 ( z ) + I 2 ( z ) = [ I 1 ( z = 0 ) + I 2 ( z = 0 ) ] exp ( α z ) ,
d d z [ I 1 ( z ) I 2 ( z ) ] = γ 0 1 + { Ω τ 0 / [ I 1 ( z ) + I 2 ( z ) ] } 2 I 1 ( z ) I 2 ( z ) .
I 1 ( z ) I 2 ( z ) = m [ 1 + { Ω τ 0 / [ I 1 ( 0 ) + I 2 ( 0 ) ] } 2 exp ( 2 α z ) ( 1 + { Ω τ 0 / [ I 1 ( 0 ) + I 2 ( 0 ) ] } 2 ) exp ( 2 α z ) ] γ 0 / 2 α ,
I 1 ( z ) I 1 ( 0 ) = ( 1 + m 1 ) exp ( α z ) 1 + m 1 [ 1 + { Ω τ 0 / [ I 1 ( 0 ) + I 2 ( 0 ) ] } 2 exp ( 2 α z ) ( 1 + { Ω τ 0 / [ I 1 ( 0 ) + I 2 ( 0 ) ] } 2 ) exp ( 2 α z ) ] γ 0 / 2 α ,
I 2 ( z ) I 2 ( 0 ) = ( 1 + m ) exp ( α z ) 1 + m [ 1 + { Ω τ 0 / [ I 1 ( 0 ) + I 2 ( 0 ) ] } 2 exp ( 2 α z ) ( 1 + { Ω τ 0 / [ I 1 ( 0 ) + I 2 ( 0 ) ] } 2 ) exp ( 2 α z ) ] γ 0 / 2 α .
gain I 2 ( L ) I 2 ( 0 ) = ( 1 + m ) exp ( α L ) 1 + m [ 1 + { Ω τ 0 / [ I 1 ( 0 ) + I 2 ( 0 ) ] } 2 exp ( 2 α L ) ( 1 + { Ω τ 0 / [ I 1 ( 0 ) + I 2 ( 0 ) ] } 2 ) exp ( 2 α L ) ] γ 0 / 2 α .
gain = ( 1 + m ) exp ( α L ) 1 + m exp ( γ L ) ,
Δ ψ = β γ log ( 1 + m ) 1 + m exp ( γ L ) ,
γ = γ 0 1 + { Ω τ 0 / [ I 1 ( 0 ) + I 2 ( 0 ) ] } 2 , β = γ 2 Ω τ 0 I 1 ( 0 ) + I 2 ( 0 ) .
I 1 ( z ) I 2 ( z ) = 2 C = I 1 ( 0 ) I 2 ( 0 ) = I 1 ( L ) I 2 ( L ) ,
d d z X ( z ) = γ 0 1 + { Ω τ 0 [ I 1 ( z ) + I 2 ( z ) ] } 2 X ( z ) ,
U ( z ) [ U ( z ) X ( 0 ) + C 2 X ( z ) + C 2 ] ( Ω τ 0 ) 2 / 4 C 2 = exp ( γ 0 z ) ,
[ I 1 ( z ) + I 2 ( z ) ] 2 4 X ( z ) = 4 C 2 = [ I 1 ( 0 ) + I 2 ( 0 ) ] 2 4 X ( 0 ) ,
X ( 0 ) + C 2 X ( z ) + C 2 = [ I 1 ( 0 ) C ] 2 C 2 + U ( z ) [ I 1 2 ( 0 ) 2 C I 1 ( 0 ) ] ,
2 C = I 1 2 ( 0 ) U ( L ) I 2 2 ( L ) I 2 ( L ) + I 1 ( 0 ) U ( L ) ,
U ( L ) [ U ( L ) [ I 1 ( 0 ) C ] 2 C 2 + U ( L ) [ I 1 2 ( 0 ) 2 C I 1 ( 0 ) ] ] ( Ω τ 0 ) 2 / 4 C 2 = exp ( γ 0 L ) .
gain = g I 2 ( 0 ) I 2 ( L ) = 1 + m 1 + m U ( L ) ,
I 1 ( z ) = C + [ C 2 + g I 1 ( 0 ) I 2 ( L ) U ( z ) ] 1 / 2 ,
I 2 ( z ) = C + [ C 2 + g I 1 ( 0 ) I 2 ( L ) U ( z ) ] 1 / 2 ,

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