Abstract

We consider the theory of photorefractive ring oscillators, using our unified solution method. Both unidirectional and bidirectional ring resonators are analyzed, based on the two-wave mixing process with crossed polarization and the four-wave mixing process with parallel polarization in photorefractive crystals. We highlight symmetries between the transmission and the reflection geometries of these processes and use them to write analytical expressions for oscillation conditions in all the cases. Symmetry breaking is noted in the four-wave mixing between the transmission and the reflection grating cases. An optical transistor based on photorefractive rings is proposed.

© 1995 Optical Society of America

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References

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  1. See, for example, C. M. Bowden, H. M. Gibbs, and S. L. McCall, eds., Optical Bistability 1 (Plenum, New York, 1982);Optical Bistability 2 (Plenum, New York, 1984);special issue, IEEE J. Quantum Electron. QE-21(9) (1985);feature on Transverse effects in nonlinear-optical systems, J. Opt. Soc. Am. B 7, 948–1373 (1990);feature on Photorefractive materials, effects, and devices, J. Opt. Soc. Am. B9, 140–175 (1992).
  2. P. Yeh, Photorefractive Nonlinear Optics (Wiley, New York, 1993).
  3. P. Yeh, J. Opt. Soc. Am. B 2, 1924 (1985);C. Gu and P. Yeh, J. Opt. Soc. Am. B 8, 1428 (1991).
    [Crossref]
  4. M. Belić and M. Petrović, J. Opt. Soc. Am. B 11, 481 (1994).
    [Crossref]
  5. M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984);A. Bledowski and W. Krolikowski, Opt. Lett. 13, 146 (1988).
    [Crossref]
  6. M. R. MacDonald and J. Fienberg, Phys. Rev. Lett. 55, 821 (1985);M. Cronin-Golomb and A. Yariv, Opt. Lett. 11, 242 (1986);B. Fischer, S. Sternklar, and S. Weiss, IEEE J. Quantum Electron. 25, 550 (1989);G. Pauliat, M. Ingold, and P. Gunter, IEEE J. Quantum Electron. 25, 201 (1989).
    [Crossref] [PubMed]
  7. S. L. Wang and H. F. Pan, Opt. Commun. 103, 116 (1993);S. L. Wang and H. F. Pan, Zhejiung University, Hangzhou 310027, China (personal communication).
    [Crossref]
  8. M. Belić, M. Petrović, and F. Kaiser, “Photorefractive circuitry and optical transistors,” Opt. Commun. (to be published).

1994 (1)

1993 (1)

S. L. Wang and H. F. Pan, Opt. Commun. 103, 116 (1993);S. L. Wang and H. F. Pan, Zhejiung University, Hangzhou 310027, China (personal communication).
[Crossref]

1985 (2)

P. Yeh, J. Opt. Soc. Am. B 2, 1924 (1985);C. Gu and P. Yeh, J. Opt. Soc. Am. B 8, 1428 (1991).
[Crossref]

M. R. MacDonald and J. Fienberg, Phys. Rev. Lett. 55, 821 (1985);M. Cronin-Golomb and A. Yariv, Opt. Lett. 11, 242 (1986);B. Fischer, S. Sternklar, and S. Weiss, IEEE J. Quantum Electron. 25, 550 (1989);G. Pauliat, M. Ingold, and P. Gunter, IEEE J. Quantum Electron. 25, 201 (1989).
[Crossref] [PubMed]

1984 (1)

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984);A. Bledowski and W. Krolikowski, Opt. Lett. 13, 146 (1988).
[Crossref]

Belic, M.

M. Belić and M. Petrović, J. Opt. Soc. Am. B 11, 481 (1994).
[Crossref]

M. Belić, M. Petrović, and F. Kaiser, “Photorefractive circuitry and optical transistors,” Opt. Commun. (to be published).

Cronin-Golomb, M.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984);A. Bledowski and W. Krolikowski, Opt. Lett. 13, 146 (1988).
[Crossref]

Fienberg, J.

M. R. MacDonald and J. Fienberg, Phys. Rev. Lett. 55, 821 (1985);M. Cronin-Golomb and A. Yariv, Opt. Lett. 11, 242 (1986);B. Fischer, S. Sternklar, and S. Weiss, IEEE J. Quantum Electron. 25, 550 (1989);G. Pauliat, M. Ingold, and P. Gunter, IEEE J. Quantum Electron. 25, 201 (1989).
[Crossref] [PubMed]

Fischer, B.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984);A. Bledowski and W. Krolikowski, Opt. Lett. 13, 146 (1988).
[Crossref]

Kaiser, F.

M. Belić, M. Petrović, and F. Kaiser, “Photorefractive circuitry and optical transistors,” Opt. Commun. (to be published).

MacDonald, M. R.

M. R. MacDonald and J. Fienberg, Phys. Rev. Lett. 55, 821 (1985);M. Cronin-Golomb and A. Yariv, Opt. Lett. 11, 242 (1986);B. Fischer, S. Sternklar, and S. Weiss, IEEE J. Quantum Electron. 25, 550 (1989);G. Pauliat, M. Ingold, and P. Gunter, IEEE J. Quantum Electron. 25, 201 (1989).
[Crossref] [PubMed]

Pan, H. F.

S. L. Wang and H. F. Pan, Opt. Commun. 103, 116 (1993);S. L. Wang and H. F. Pan, Zhejiung University, Hangzhou 310027, China (personal communication).
[Crossref]

Petrovic, M.

M. Belić and M. Petrović, J. Opt. Soc. Am. B 11, 481 (1994).
[Crossref]

M. Belić, M. Petrović, and F. Kaiser, “Photorefractive circuitry and optical transistors,” Opt. Commun. (to be published).

Wang, S. L.

S. L. Wang and H. F. Pan, Opt. Commun. 103, 116 (1993);S. L. Wang and H. F. Pan, Zhejiung University, Hangzhou 310027, China (personal communication).
[Crossref]

White, J. O.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984);A. Bledowski and W. Krolikowski, Opt. Lett. 13, 146 (1988).
[Crossref]

Yariv, A.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984);A. Bledowski and W. Krolikowski, Opt. Lett. 13, 146 (1988).
[Crossref]

Yeh, P.

IEEE J. Quantum Electron. (1)

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984);A. Bledowski and W. Krolikowski, Opt. Lett. 13, 146 (1988).
[Crossref]

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

Opt. Commun. (1)

S. L. Wang and H. F. Pan, Opt. Commun. 103, 116 (1993);S. L. Wang and H. F. Pan, Zhejiung University, Hangzhou 310027, China (personal communication).
[Crossref]

Phys. Rev. Lett. (1)

M. R. MacDonald and J. Fienberg, Phys. Rev. Lett. 55, 821 (1985);M. Cronin-Golomb and A. Yariv, Opt. Lett. 11, 242 (1986);B. Fischer, S. Sternklar, and S. Weiss, IEEE J. Quantum Electron. 25, 550 (1989);G. Pauliat, M. Ingold, and P. Gunter, IEEE J. Quantum Electron. 25, 201 (1989).
[Crossref] [PubMed]

Other (3)

M. Belić, M. Petrović, and F. Kaiser, “Photorefractive circuitry and optical transistors,” Opt. Commun. (to be published).

See, for example, C. M. Bowden, H. M. Gibbs, and S. L. McCall, eds., Optical Bistability 1 (Plenum, New York, 1982);Optical Bistability 2 (Plenum, New York, 1984);special issue, IEEE J. Quantum Electron. QE-21(9) (1985);feature on Transverse effects in nonlinear-optical systems, J. Opt. Soc. Am. B 7, 948–1373 (1990);feature on Photorefractive materials, effects, and devices, J. Opt. Soc. Am. B9, 140–175 (1992).

P. Yeh, Photorefractive Nonlinear Optics (Wiley, New York, 1993).

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Figures (8)

Fig. 1
Fig. 1

Photorefractive ring resonators considered: (a) 2WM unidirectional ring with cross coupling in TG, (b) 2WM unidirectional ring in RG, (c) 4WM bidirectional ring with parallel coupling in either TG or RG.

Fig. 2
Fig. 2

Threshold conditions for 2WM rings from Table 1 presented graphically: NP, not possible. For these values of overall reflectivity of the cavity |r| and coupling strength aΓd the ring will not oscillate.

Fig. 3
Fig. 3

Oscillation intensities |Asd|2 and |Apd|2 (TG side) and |Bsd|2 and |Bpd|2 (RG side) in the 2WM ring as functions of coupling Γd for two values of the pump intensity ratio. Perfect symmetry between TG and RG is visible. Here |r| = 0.5 and α = 0.

Fig. 4
Fig. 4

Integration constant δ / I 0 = T / 2 I 0 in the 4WM TG case as a function of the coupling coefficient Γd for three values of the pump ratio I10/I2d. If the symmetry between TG and RG were to hold, this constant would be the same for any value of Γd (depicted by the corresponding horizontal lines). As it can be seen, the symmetry holds only approximately, and better so for higher values of the pump ratio. Other parameters are |r| = 0.5 and I2d = 1.

Fig. 5
Fig. 5

Oscillation intensities I3d for the RG case and I40 for the TG case of the 4WM bidirectional ring. This figure should be compared with Fig. 3. The dashed curves on the TG side display the symmetric approximate solution obtained by analogy with the 2WM ring. A further breakup of symmetry is noted for small values of the pump ratio I10/I2d: whereas in the RG there is no solution in the TG one obtains multiple solutions. The parameters are as in Fig. 4.

Fig. 6
Fig. 6

Grating action u as a function of the coupling constant Γd for different values of the pump ratio in the 4WM case. The dashed curves depict the approximate symmetric solution. The parameters are as in Fig. 4. Again, multistability is noted.

Fig. 7
Fig. 7

Threshold value of Γthd as a function of the pump ratio for different values of |r|. RG is assumed. (Results for TG are identical to those of Gu and Yeh.)

Fig. 8
Fig. 8

Transistor action of a 2WM TG ring: (a) The amplification effect. Here β = 100.2 for |r| = 0.995 and Γd = –5. The parameter α is the phase difference between the pump components. (b) The saturation effect. (c) Volt-ampere characteristics of an optical transistor. For απ/2 one notes the region of negative resistance.

Tables (2)

Tables Icon

Table 1 Threshold Conditions

Tables Icon

Table 2 Summary of Results

Equations (75)

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I A s = σ Γ Q B p , I A p = σ Γ Q B s ,
1 B s = Γ Q ¯ A p , I B p = Γ Q ¯ A s ,
I A 1 = σ Γ Q A 3 , I A 3 = Γ Q ¯ A 1 ,
I A 2 = σ Γ Q ¯ A 4 , I A 4 = Γ Q A 2 ,
θ = Γ | Q | / I ,
A s = A s 0 c ( Θ ) + σ B p 0 exp ( i ϕ ) s ( Θ ) ,
B p = B p 0 c ( Θ ) + A s 0 exp ( i ϕ ) s ( Θ ) ,
A p = A p 0 c ( Θ ) + σ B s 0 exp ( i ϕ ) s ( Θ ) ,
B s = B s 0 c ( Θ ) + A p 0 exp ( i ϕ ) s ( Θ ) ,
A s 0 = A s d sech ( u ) B p 0 tanh ( u ) exp ( i ϕ ) ,
A p 0 = A p d sech ( u ) B s 0 tanh ( u ) exp ( i ϕ ) ,
TG : tan ( Θ ) = | Q 0 | δ coth ( δ Γ z I 0 ) P 0 ,
RG : Γ z = I 0 J 0 | Q 0 | P 0 Θ + J 0 | Q 0 | I 0 P 0 2 ( | Q 0 | 2 P 0 2 ) × ln | ( | Q 0 | + P 0 ) exp ( 4 Θ ) + | Q 0 | P 0 2 | Q 0 | | ,
A p 0 = r exp ( i ψ ) A p d , A s 0 = r exp ( i ψ ) A s d ,
ψ + ϕ r = 2 m π ,
κ = A s 0 B p 0 exp ( i ϕ ) = A p 0 B s 0 exp ( i ϕ ) = sin ( u ) cos ( u ) 1 | r | .
tan ( u ) = κ ( b c ) b + κ 2 c ,
cos ( u ) = b + c | r | 2 | r | ( b + c ) ,
Γ th d = ln ( | r | ) / a ,
| A p d | 2 = | B s 0 | 2 1 | r | 2 ( 1 τ 2 | r | 2 ) ,
| A s d | 2 = | B p 0 | 2 1 | r | 2 ( 1 τ 2 | r | 2 ) .
B p 0 = r exp ( i ψ ) B p d , B s 0 = r exp ( i ψ ) B s d .
κ = B p 0 A s d exp ( i ϕ ) = B s 0 A p d exp ( i ϕ ) ,
κ = sinh ( u ) 1 cosh ( u ) | r | .
Γ d = I 0 J 0 | Q 0 | P 0 u + J 0 | Q 0 | I 0 P 0 2 ( | Q 0 | 2 P 0 2 ) × ln | ( | Q 0 | + P 0 ) exp ( 4 u ) + | Q 0 | P 0 2 | Q 0 | | ,
| Q 0 | = a κ I A sech ( u ) [ 1 + κ sinh ( u ) ] ,
2 P 0 = a I A sech 2 ( u ) [ 1 + 2 κ sinh ( u ) + κ 2 cosh ( 2 u ) ] ,
I 0 = I A sech 2 ( u ) [ 1 + 2 κ sinh ( u ) + κ 2 cosh ( 2 u ) ] ,
J 0 = 2 κ I A sech ( u ) [ 1 + κ sinh ( u ) ] ,
I 0 | Q 0 | = J 0 P 0 ,
a Γ d = ln | | Q 0 | cosh ( 2 u ) + P 0 sinh ( 2 u ) | Q 0 | | .
sinh ( u ) = κ ( b c ) b + κ 2 c .
cosh ( u ) = | r | ( b + c ) b + c | r | 2 .
A 1 = A 10 cosh ( Θ ) + A 30 exp ( i ϕ ) sinh ( Θ ) ,
A 3 = A 30 cosh ( Θ ) + A 10 exp ( i ϕ ) sinh ( Θ ) ,
A 2 = A 20 cosh ( Θ ) + A 40 exp ( i ϕ ) sinh ( Θ ) ,
A 4 = A 40 cosh ( Θ ) + A 20 exp ( i ϕ ) sinh ( Θ ) .
A 30 = A 3 d sech ( u ) A 10 exp ( i ϕ ) tanh ( u ) ,
A 20 = A 2 d sech ( u ) A 40 exp ( i ϕ ) tanh ( u ) ,
A 40 = r exp ( i ψ ) A 4 d , A 3 d = r exp ( i ψ ) A 30 ,
κ = A 3 d A 10 exp ( i ϕ ) = A 40 A 2 d exp ( i ϕ )
R = I 2 4 | Q | 2
sinh ( 2 θ ) = sinh ( 2 θ 0 ) exp ( Γ z ) .
sinh ( u ) = [ exp ( Γ d ) 1 ] | A 10 3 d + 2 d A 40 | exp ( Γ d ) ( I 10 + I 40 ) + I 2 d + I 3 d ,
A 1 = A 10 cos ( Θ ) A 40 exp ( i ϕ ) sin ( Θ ) ,
A 4 = A 40 cos ( Θ ) A 10 exp ( i ϕ ) sin ( Θ ) ,
A 2 = A 20 cos ( Θ ) A 30 exp ( i ϕ ) sin ( Θ ) ,
A 3 = A 30 cos ( Θ ) + A 20 exp ( i ϕ ) sin ( Θ ) ,
A 20 = A 2 d cos ( u ) + A 3 d exp ( i ϕ ) sin ( u ) ,
A 30 = A 3 d cos ( u ) A 2 d exp ( i ϕ ) sin ( u ) .
A 40 = r exp ( i ψ ) A 4 d , A 3 d = r exp ( i ψ ) A 30 ,
κ = A 40 A 10 exp ( i ϕ ) = A 3 d A 2 d exp ( i ϕ ) = sin ( u ) cos ( u ) 1 | r | .
T = F 2 + 4 | Q | 2 ,
tan ( θ ) = tan ( θ 0 ) exp ( T Γ z I 0 ) .
tan ( u ) = q T coth ( T Γ d 2 I 0 ) υ ,
tanh ( u ) = 2 κ ( B s 0 B ¯ p 0 + c . c . ) 2 δ coth ( δ Γ d I 0 ) ( κ 2 1 ) ( B s 0 B ¯ P 0 + c . c . ) ,
tanh ( u ) = 2 κ ( I 2 d I 10 ) T coth ( T Γ d 2 I 0 ) ( κ 2 1 ) ( I 2 d I 10 ) ,
exp ( Γ th d ) = p | r | p | r | 1 ,
p < 1 | r | for | r | > 1 , p > 1 | r | for | r | < 1 ,
p > | r | for | r | > 1 , p < | r | for | r | < 1 .
β = κ 2 | r | 2 = 1 τ 2 | r | 2 1 | r | 2 .
τ = I 2 d + I 10 exp ( Γ d ) I 10 + I 2 d exp ( Γ d )
τ I 10 + I 2 d exp ( Γ d ) I 2 d + I 10 exp ( Γ d )
cosh ( u ) = | r | ( τ + 1 ) τ + | r | 2
sec ( u ) = | r | ( τ + 1 ) τ + | r | 2
cosh ( u ) = | r | ( τ + 1 ) τ + | r | 2
sec ( u ) | r | ( τ + 1 ) τ + | r | 2
κ = | r | sinh ( u ) | r | cosh ( u )
κ = | r | sin ( u ) | r | cosh ( u ) 1
κ = | r | sinh ( u ) | r | cosh ( u )
κ = | r | sin ( u ) | r | cos ( u ) 1
sinh ( u ) = κ ( τ 1 ) τ + κ 2
tan ( u ) = κ ( τ 1 ) τ + κ 2
sinh ( u ) = κ ( τ 1 ) τ + κ 2
tan ( u ) κ ( τ 1 ) τ + κ 2

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