Abstract

We present a closed-form solution for the two-wave coupling process, resulting in a solution with an interesting part, the arbitrary function Λ(t), not being considered in the previous works. In the photorefractive medium, it has been shown that a moving grating will produce a phase shift in the coupling constant. To our knowledge, this has not been derived directly from the grating dynamic equations. We will also show that the temporal dynamics does not have any asymptotically stable equilibrium points of an exponential form. As an example, we have developed resonance conditions resulting in a new expression for the frequency detuning of a resonator with a photorefractive gain medium.

© 2004 Optical Society of America

Full Article  |  PDF Article
Related Articles
Theory of unidirectional photorefractive ring oscillators

Pochi Yeh
J. Opt. Soc. Am. B 2(12) 1924-1928 (1985)

Analysis of transient phase conjugation in photorefractive media

George C. Papen, Bahaa E. A. Saleh, and John A. Tataronis
J. Opt. Soc. Am. B 5(8) 1763-1774 (1988)

References

  • View by:
  • |
  • |
  • |

  1. P. Yeh, Introduction to Photorefractive Nonlinear Optics, (John Wiley and Sons, Inc., New York, 1993).
  2. K.H. Ringhofer, V.P. Kamenov, B.I. Sturman, and A. Chernykh, “Shaping of photorefractive two-wave coupling by fast phase modulation,” Phys. Rev. E 61, 2029 (2000).
    [Crossref]
  3. M. Cronin-Golomb, A.M. Biernacki, C. Lin, and H. Kong, “Photorefractive time differentiation of coherent optical images,” Opt. Lett. 12, 1029 (1987).
    [Crossref] [PubMed]
  4. A. Hermanns, C. Benkert, D. M. Lininger, and D.Z. Anderson, “The transfer function and impulse response of photorefractive two-beam coupling,” IEEE J. Quantum Electron. 28, 750 (1992).
    [Crossref]
  5. D.Z. Anderson and J. Feinberg, “Optical novelty filters,” IEEE J. Quantum Electron. 25, 635 (1989).
    [Crossref]
  6. N.V. Kukhtarev, V.B. Markov, S.G. Odulov, M.S. Soskin, and V.L. Vinetskii, “Holographic storage in electrooptic crystals. I. steady state,” Ferroelectrics 22, 949 (1979).
    [Crossref]
  7. M. Horowitz, D. Kligler, and B. Fischer, “Time-dependent behavior of photorefractive two- and four-wave mixing,” J. Opt. Soc. Am. B 8, 2204 (1991).
    [Crossref]
  8. O. Sandfuchs, F. Kaiser, and M.R. Belic, “Dynamics of transverse waves and zigzag instabilities in photorefractive two-wave mixing with a feedback,” J. Opt. Soc. Am. B 18, 505 (2001).
    [Crossref]
  9. N.V. Kukhtarev, V.B. Markov, and S.G. Odulov, “Transient energy transfer during hologram formation in LiNbO3 in external electric field,” Optics Communications 23, 338 (1977).
    [Crossref]
  10. G. Pauliat, M. Ingold, and P. Günter, “Analysis of the build up of oscillations in self-induced photorefractive light resonators,” IEEE J. Quantum Electron. 25, 201 (1989).
    [Crossref]
  11. A. Bledowski, W. Krolikowski, and A. Kujawski, “Temporal instabilities in single-grating photorefractive four-wave mixing,” J. Opt. Soc. Am. B 6, 1544 (1989).
    [Crossref]
  12. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909 (1969).
  13. B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550 (1989).
    [Crossref]
  14. S. Kwong, M. Cronin-Golomb, and A. Yariv, “Oscillation with photorefractive gain,” IEEE J. Quantum Electron. QE-22, 1508 (1986).
    [Crossref]
  15. M.D. Ewbank and P. Yeh, “Frequency shift and cavity length in photorefractive resonators,” Opt. Lett. 10, 486 (1985).
    [Crossref]

2001 (1)

2000 (1)

K.H. Ringhofer, V.P. Kamenov, B.I. Sturman, and A. Chernykh, “Shaping of photorefractive two-wave coupling by fast phase modulation,” Phys. Rev. E 61, 2029 (2000).
[Crossref]

1992 (1)

A. Hermanns, C. Benkert, D. M. Lininger, and D.Z. Anderson, “The transfer function and impulse response of photorefractive two-beam coupling,” IEEE J. Quantum Electron. 28, 750 (1992).
[Crossref]

1991 (1)

1989 (4)

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550 (1989).
[Crossref]

D.Z. Anderson and J. Feinberg, “Optical novelty filters,” IEEE J. Quantum Electron. 25, 635 (1989).
[Crossref]

G. Pauliat, M. Ingold, and P. Günter, “Analysis of the build up of oscillations in self-induced photorefractive light resonators,” IEEE J. Quantum Electron. 25, 201 (1989).
[Crossref]

A. Bledowski, W. Krolikowski, and A. Kujawski, “Temporal instabilities in single-grating photorefractive four-wave mixing,” J. Opt. Soc. Am. B 6, 1544 (1989).
[Crossref]

1987 (1)

1986 (1)

S. Kwong, M. Cronin-Golomb, and A. Yariv, “Oscillation with photorefractive gain,” IEEE J. Quantum Electron. QE-22, 1508 (1986).
[Crossref]

1985 (1)

M.D. Ewbank and P. Yeh, “Frequency shift and cavity length in photorefractive resonators,” Opt. Lett. 10, 486 (1985).
[Crossref]

1979 (1)

N.V. Kukhtarev, V.B. Markov, S.G. Odulov, M.S. Soskin, and V.L. Vinetskii, “Holographic storage in electrooptic crystals. I. steady state,” Ferroelectrics 22, 949 (1979).
[Crossref]

1977 (1)

N.V. Kukhtarev, V.B. Markov, and S.G. Odulov, “Transient energy transfer during hologram formation in LiNbO3 in external electric field,” Optics Communications 23, 338 (1977).
[Crossref]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909 (1969).

Anderson, D.Z.

A. Hermanns, C. Benkert, D. M. Lininger, and D.Z. Anderson, “The transfer function and impulse response of photorefractive two-beam coupling,” IEEE J. Quantum Electron. 28, 750 (1992).
[Crossref]

D.Z. Anderson and J. Feinberg, “Optical novelty filters,” IEEE J. Quantum Electron. 25, 635 (1989).
[Crossref]

Belic, M.R.

Benkert, C.

A. Hermanns, C. Benkert, D. M. Lininger, and D.Z. Anderson, “The transfer function and impulse response of photorefractive two-beam coupling,” IEEE J. Quantum Electron. 28, 750 (1992).
[Crossref]

Biernacki, A.M.

Bledowski, A.

Chernykh, A.

K.H. Ringhofer, V.P. Kamenov, B.I. Sturman, and A. Chernykh, “Shaping of photorefractive two-wave coupling by fast phase modulation,” Phys. Rev. E 61, 2029 (2000).
[Crossref]

Cronin-Golomb, M.

M. Cronin-Golomb, A.M. Biernacki, C. Lin, and H. Kong, “Photorefractive time differentiation of coherent optical images,” Opt. Lett. 12, 1029 (1987).
[Crossref] [PubMed]

S. Kwong, M. Cronin-Golomb, and A. Yariv, “Oscillation with photorefractive gain,” IEEE J. Quantum Electron. QE-22, 1508 (1986).
[Crossref]

Ewbank, M.D.

M.D. Ewbank and P. Yeh, “Frequency shift and cavity length in photorefractive resonators,” Opt. Lett. 10, 486 (1985).
[Crossref]

Feinberg, J.

D.Z. Anderson and J. Feinberg, “Optical novelty filters,” IEEE J. Quantum Electron. 25, 635 (1989).
[Crossref]

Fischer, B.

M. Horowitz, D. Kligler, and B. Fischer, “Time-dependent behavior of photorefractive two- and four-wave mixing,” J. Opt. Soc. Am. B 8, 2204 (1991).
[Crossref]

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550 (1989).
[Crossref]

Günter, P.

G. Pauliat, M. Ingold, and P. Günter, “Analysis of the build up of oscillations in self-induced photorefractive light resonators,” IEEE J. Quantum Electron. 25, 201 (1989).
[Crossref]

Hermanns, A.

A. Hermanns, C. Benkert, D. M. Lininger, and D.Z. Anderson, “The transfer function and impulse response of photorefractive two-beam coupling,” IEEE J. Quantum Electron. 28, 750 (1992).
[Crossref]

Horowitz, M.

Ingold, M.

G. Pauliat, M. Ingold, and P. Günter, “Analysis of the build up of oscillations in self-induced photorefractive light resonators,” IEEE J. Quantum Electron. 25, 201 (1989).
[Crossref]

Kaiser, F.

Kamenov, V.P.

K.H. Ringhofer, V.P. Kamenov, B.I. Sturman, and A. Chernykh, “Shaping of photorefractive two-wave coupling by fast phase modulation,” Phys. Rev. E 61, 2029 (2000).
[Crossref]

Kligler, D.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909 (1969).

Kong, H.

Krolikowski, W.

Kujawski, A.

Kukhtarev, N.V.

N.V. Kukhtarev, V.B. Markov, S.G. Odulov, M.S. Soskin, and V.L. Vinetskii, “Holographic storage in electrooptic crystals. I. steady state,” Ferroelectrics 22, 949 (1979).
[Crossref]

N.V. Kukhtarev, V.B. Markov, and S.G. Odulov, “Transient energy transfer during hologram formation in LiNbO3 in external electric field,” Optics Communications 23, 338 (1977).
[Crossref]

Kwong, S.

S. Kwong, M. Cronin-Golomb, and A. Yariv, “Oscillation with photorefractive gain,” IEEE J. Quantum Electron. QE-22, 1508 (1986).
[Crossref]

Lin, C.

Lininger, D. M.

A. Hermanns, C. Benkert, D. M. Lininger, and D.Z. Anderson, “The transfer function and impulse response of photorefractive two-beam coupling,” IEEE J. Quantum Electron. 28, 750 (1992).
[Crossref]

Markov, V.B.

N.V. Kukhtarev, V.B. Markov, S.G. Odulov, M.S. Soskin, and V.L. Vinetskii, “Holographic storage in electrooptic crystals. I. steady state,” Ferroelectrics 22, 949 (1979).
[Crossref]

N.V. Kukhtarev, V.B. Markov, and S.G. Odulov, “Transient energy transfer during hologram formation in LiNbO3 in external electric field,” Optics Communications 23, 338 (1977).
[Crossref]

Odulov, S.G.

N.V. Kukhtarev, V.B. Markov, S.G. Odulov, M.S. Soskin, and V.L. Vinetskii, “Holographic storage in electrooptic crystals. I. steady state,” Ferroelectrics 22, 949 (1979).
[Crossref]

N.V. Kukhtarev, V.B. Markov, and S.G. Odulov, “Transient energy transfer during hologram formation in LiNbO3 in external electric field,” Optics Communications 23, 338 (1977).
[Crossref]

Pauliat, G.

G. Pauliat, M. Ingold, and P. Günter, “Analysis of the build up of oscillations in self-induced photorefractive light resonators,” IEEE J. Quantum Electron. 25, 201 (1989).
[Crossref]

Ringhofer, K.H.

K.H. Ringhofer, V.P. Kamenov, B.I. Sturman, and A. Chernykh, “Shaping of photorefractive two-wave coupling by fast phase modulation,” Phys. Rev. E 61, 2029 (2000).
[Crossref]

Sandfuchs, O.

Soskin, M.S.

N.V. Kukhtarev, V.B. Markov, S.G. Odulov, M.S. Soskin, and V.L. Vinetskii, “Holographic storage in electrooptic crystals. I. steady state,” Ferroelectrics 22, 949 (1979).
[Crossref]

Sternklar, S.

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550 (1989).
[Crossref]

Sturman, B.I.

K.H. Ringhofer, V.P. Kamenov, B.I. Sturman, and A. Chernykh, “Shaping of photorefractive two-wave coupling by fast phase modulation,” Phys. Rev. E 61, 2029 (2000).
[Crossref]

Vinetskii, V.L.

N.V. Kukhtarev, V.B. Markov, S.G. Odulov, M.S. Soskin, and V.L. Vinetskii, “Holographic storage in electrooptic crystals. I. steady state,” Ferroelectrics 22, 949 (1979).
[Crossref]

Weiss, S.

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550 (1989).
[Crossref]

Yariv, A.

S. Kwong, M. Cronin-Golomb, and A. Yariv, “Oscillation with photorefractive gain,” IEEE J. Quantum Electron. QE-22, 1508 (1986).
[Crossref]

Yeh, P.

M.D. Ewbank and P. Yeh, “Frequency shift and cavity length in photorefractive resonators,” Opt. Lett. 10, 486 (1985).
[Crossref]

P. Yeh, Introduction to Photorefractive Nonlinear Optics, (John Wiley and Sons, Inc., New York, 1993).

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909 (1969).

Ferroelectrics (1)

N.V. Kukhtarev, V.B. Markov, S.G. Odulov, M.S. Soskin, and V.L. Vinetskii, “Holographic storage in electrooptic crystals. I. steady state,” Ferroelectrics 22, 949 (1979).
[Crossref]

IEEE J. Quantum Electron. (5)

A. Hermanns, C. Benkert, D. M. Lininger, and D.Z. Anderson, “The transfer function and impulse response of photorefractive two-beam coupling,” IEEE J. Quantum Electron. 28, 750 (1992).
[Crossref]

D.Z. Anderson and J. Feinberg, “Optical novelty filters,” IEEE J. Quantum Electron. 25, 635 (1989).
[Crossref]

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. 25, 550 (1989).
[Crossref]

S. Kwong, M. Cronin-Golomb, and A. Yariv, “Oscillation with photorefractive gain,” IEEE J. Quantum Electron. QE-22, 1508 (1986).
[Crossref]

G. Pauliat, M. Ingold, and P. Günter, “Analysis of the build up of oscillations in self-induced photorefractive light resonators,” IEEE J. Quantum Electron. 25, 201 (1989).
[Crossref]

J. Opt. Soc. Am. B (3)

Opt. Lett. (2)

M. Cronin-Golomb, A.M. Biernacki, C. Lin, and H. Kong, “Photorefractive time differentiation of coherent optical images,” Opt. Lett. 12, 1029 (1987).
[Crossref] [PubMed]

M.D. Ewbank and P. Yeh, “Frequency shift and cavity length in photorefractive resonators,” Opt. Lett. 10, 486 (1985).
[Crossref]

Optics Communications (1)

N.V. Kukhtarev, V.B. Markov, and S.G. Odulov, “Transient energy transfer during hologram formation in LiNbO3 in external electric field,” Optics Communications 23, 338 (1977).
[Crossref]

Phys. Rev. E (1)

K.H. Ringhofer, V.P. Kamenov, B.I. Sturman, and A. Chernykh, “Shaping of photorefractive two-wave coupling by fast phase modulation,” Phys. Rev. E 61, 2029 (2000).
[Crossref]

Other (1)

P. Yeh, Introduction to Photorefractive Nonlinear Optics, (John Wiley and Sons, Inc., New York, 1993).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1.
Fig. 1.

Two-wave coupling.

Fig. 2.
Fig. 2.

Unidirectional photorefractive resonator. The two-wave coupling configuration inside the photorefractive material is similar to that of Fig. 1.

Fig. 3.
Fig. 3.

Resonator intensity (I 10) vs. detuning frequency (Ω) when β=0, the pure diffusion case.

Fig. 4.
Fig. 4.

Resonator intensity (I 10) vs. detuning frequency (Ω) when γ=0.

Fig. 5.
Fig. 5.

Detuning frequency (Ω) vs. resonator loss (1-ρ) when β=0, the pure diffusion case.

Fig. 6.
Fig. 6.

Detuning frequency (Ω) vs. resonator loss (1-ρ) when γ=0.

Fig. 7.
Fig. 7.

Detuning frequency (Ω) vs. cavity length D when β=0, the pure diffusion case.

Fig. 8.
Fig. 8.

Detuning frequency (Ω) vs. cavity length D when γ=0.

Equations (34)

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

E sc ( x , t ) = E sc [ cos ( Kx + ϕ ) + exp ( t τ g ) cos ( Kx + ω g t + ϕ ) ] ,
τ G ( z , t ) t = G ( z , t ) + Γ 2 I 0 E 1 ( z , t ) E 2 * ( z , t ) ,
E 1 ( z , t ) z = α 2 E 1 ( z , t ) + G ( z , t ) E 2 ( z , t ) ,
E 2 ( z , t ) z = α 2 E 2 ( z , t ) G * ( z , t ) E 1 ( z , t ) ,
G ˜ ( z , ς ) = g ( z ) g ¯ ( ς ) ,
E ˜ 1 ( z , ς ) = e 1 ( z ) e ¯ 1 ( ς ) ,
E ˜ 2 ( z , ς ) = e 2 ( z ) e ¯ 2 ( ς ) ,
1 + Γ 2 [ e ¯ 1 e ¯ 2 * I ˜ 0 g ¯ ] e 1 e 2 * g = ip 0 = g ¯ ˙ g ¯ ,
e 1 ge 2 = p 1 = g ¯ e ¯ 2 e ¯ 1 ,
e 2 g * e 1 = p 2 = g ¯ * e ¯ 1 e ¯ 2 ,
g = Γ 2 p 1 I 12 ( 1 + ip 0 ) e 1 e 2 * ,
e 1 = Γ e ¯ 2 2 2 I ˜ 0 ( 1 + ip 0 ) e 1 e 2 2 ,
e 2 = Γ * e ¯ 1 2 2 I ˜ 0 ( 1 + ip 0 ) e 2 e 1 2 .
E ˜ 1 ( z , ς ) = Γ ˜ 2 I ˜ 0 E ˜ 1 ( z , ς ) E ˜ 2 ( z , ς ) 2 ,
E ˜ 2 ( z , ς ) = Γ ˜ * 2 I ˜ 0 E ˜ 2 ( z , ς ) E ˜ 1 ( z , ς ) 2 ,
E ˜ 1 ( z , ς ) = E ˜ 1 ( 0 , ς ) 1 + r 1 + re γ ˜ z e i φ 1 ( z ) ,
E ˜ 2 ( z , ς ) = E ˜ 2 ( 0 , ς ) 1 + r 1 + e γ ˜ z e i φ 2 ( z ) ,
φ 1 ( z ) = β ˜ γ ˜ ln ( 1 + r 1 + re γ ˜ z ) ,
φ 2 ( z ) = β ˜ γ ˜ ln ( 1 + r r + e γ ˜ z ) .
g ¯ ( ς ) = g ¯ ( 0 ) e i p 0 ς ,
E ˜ 2 ( 0 , ς ) = E ˜ 2 ( 0 , 0 ) Λ ( τ ς ) ,
E ˜ 1 ( 0 , ς ) = E ˜ 1 ( 0 , 0 ) Λ ( τ ς ) e i p 0 ς ,
G ( z , t ) = G ( 0 , 0 ) ( 1 + r ) e i [ φ 1 ( z ) φ 2 ( z ) ] e i Ω t ( 1 + r e γ ˜ z ) ( r + e γ ˜ z ) ,
E 1 ( z , t ) = E 1 ( 0 , 0 ) 1 + r r + r e γ ˜ z e α z 2 e i φ 1 ( z ) e i Ω t Λ ( t ) ,
E 2 ( z , t ) = E 2 ( 0 , 0 ) 1 + r r + e γ ˜ z e α z 2 e i φ 2 ( z ) Λ ( t ) ,
E 1 ( 0 , t ) = ρ e i Φ E 1 ( d , t ) ,
ρ 1 + r 1 + r e γ ˜ d e α d 2 = 1 , Φ + φ 1 ( d ) = 2 π m ,
ρ e d 2 ( α γ ˜ ) = 1 , Φ + β ˜ d = 2 π m .
Ω = 1 τ ( 2 ln ρ α d ) [ β d ± β 2 d 2 ( d γ α d + 2 ln ρ ) ( 2 ln ρ α d ) ] ,
I 0 ρ 2 e α d = I 10 + I 20 e γ ˜ d
D ω c + β ˜ γ ˜ ln ( I 0 I 10 + I 20 e γ ˜ d ) = 2 π m ,
Ω τ = 2 ( α d 2 ln ρ ) ( ω p D c 2 π m ) I 10 = ρ 2 e α d e γd 1 + ( Ω τ ) 2 1 ρ 2 e α d I 20 } when β = 0
Ω τ = [ 2 ( α d 2 ln ρ ) ( ω p c 2 π m ) ] 1 I 10 = ρ 2 e α d e 2 β Ω τ d 1 + ( Ω τ ) 2 1 ρ 2 e α d I 20 } when γ = 0
D ( ω p + Ω ) c + β ˜ γ ˜ ( α d 2 ln ρ ) = 2 π m .

Metrics