Abstract

We study the role of polarization in modulational instabilities in a synchronously pumped ring resonator that is filled with an isotropic nonlinear dispersive medium. To describe nonlinear propagation of the polarized field through the ring, we introduce two coupled driven and damped nonlinear Schrödinger equations. These equations, which result from averaging propagation and boundary conditions over each circulation through the ring, permit a simple stability analysis. This analysis predicts polarization multistability in the steady state as well as the emergence of stable pulse trains whose polarization state is either parallel or orthogonal to a linearly polarized synchronous pump beam. The analytical predictions are confirmed and extended by numerical simulations of polarized wave propagation in the cavity.

© 1994 Optical Society of America

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  1. H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, London, 1985).
  2. R. Vallée, Opt. Commun. 81, 419 (1991).
    [CrossRef]
  3. M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992).
    [CrossRef]
  4. M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Lett. 17, 745 (1992).
    [CrossRef] [PubMed]
  5. M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 93, 343 (1992); Phys. Rev. A 47, 2344 (1993).
    [CrossRef]
  6. D. W. McLaughlin, J. V. Moloney, and A. C. Newell, in Chaos in Nonlinear Dynamical Systems, J. Chandra, ed. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1984); J. V. Moloney, in Nonlinear Phenomena and Chaos, S. Sarkar, ed. (Hilger, Bristol, UK, 1986), pp. 214–245.
  7. G. S. McDonald and W. J. Firth, J. Opt. Soc. Am. B 7, 1328 (1990).
    [CrossRef]
  8. B. K. Nayar, K. J. Blow, and N. Doran, Opt. Comput. Process. 1, 81 (1991).
  9. F. Ouellette and M. Piché, J. Opt. Soc. Am. B 5, 1228 (1988).
    [CrossRef]
  10. M. Haelterman, S. Trillo, and S. Wabnitz, Electron. Lett. 29, 119 (1993).
    [CrossRef]
  11. E. P. Ippen, H. A. Haus, and L. Y. Liu, J. Opt. Soc. Am. B 6, 1736 (1989).
    [CrossRef]
  12. M. Nakazawa, K. Suzuki, and H. A. Haus, IEEE J. Quantum Electron. 25, 2036 (1989); M. Nakazawa, K. Suzuki, H. Kubota, and H. A. Haus, IEEE J. Quantum Electron. 25, 2045 (1989).
    [CrossRef]
  13. G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
    [CrossRef] [PubMed]
  14. A. L. Berkhoer and V. E. Zakharov, Sov. Phys. JETP 31, 486 (1970).
  15. S. Trillo and S. Wabnitz, J. Opt. Soc. Am. B 6, 238 (1989).
    [CrossRef]
  16. S. Wabnitz, Phys. Rev. A 38, 2018 (1988).
    [CrossRef] [PubMed]
  17. P. D. Maker, B. W. Terhune, and C. M. Savage, Phys. Rev. Lett. 12, 507 (1964).
    [CrossRef]
  18. I. P. Areshev, T. A. Murina, N. N. Rosanov, and V. K. Subashiev, Opt. Commun. 47, 414 (1983).
    [CrossRef]
  19. M. Haelterman and P. Mandel, Opt. Lett. 15, 1412 (1990).
    [CrossRef] [PubMed]
  20. B. Daino, G. Gregori, and S. Wabinitz, Opt. Lett. 11, 42 (1986); H. G. Winful, Opt. Lett. 11, 33 (1986).
    [CrossRef]
  21. S. Trillo and S. Wabnitz, Opt. Lett. 16, 986, 1566 (1991).
    [CrossRef] [PubMed]
  22. M. J. Ablowitz and J. F. Ladik, Stud. Appl. Math. 55, 213 (1976).
  23. M. Haelterman, Opt. Lett. 17, 792 (1992).
    [CrossRef]

1993 (1)

M. Haelterman, S. Trillo, and S. Wabnitz, Electron. Lett. 29, 119 (1993).
[CrossRef]

1992 (4)

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Lett. 17, 745 (1992).
[CrossRef] [PubMed]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 93, 343 (1992); Phys. Rev. A 47, 2344 (1993).
[CrossRef]

M. Haelterman, Opt. Lett. 17, 792 (1992).
[CrossRef]

1991 (3)

S. Trillo and S. Wabnitz, Opt. Lett. 16, 986, 1566 (1991).
[CrossRef] [PubMed]

B. K. Nayar, K. J. Blow, and N. Doran, Opt. Comput. Process. 1, 81 (1991).

R. Vallée, Opt. Commun. 81, 419 (1991).
[CrossRef]

1990 (2)

1989 (3)

E. P. Ippen, H. A. Haus, and L. Y. Liu, J. Opt. Soc. Am. B 6, 1736 (1989).
[CrossRef]

M. Nakazawa, K. Suzuki, and H. A. Haus, IEEE J. Quantum Electron. 25, 2036 (1989); M. Nakazawa, K. Suzuki, H. Kubota, and H. A. Haus, IEEE J. Quantum Electron. 25, 2045 (1989).
[CrossRef]

S. Trillo and S. Wabnitz, J. Opt. Soc. Am. B 6, 238 (1989).
[CrossRef]

1988 (2)

1987 (1)

G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[CrossRef] [PubMed]

1986 (1)

1983 (1)

I. P. Areshev, T. A. Murina, N. N. Rosanov, and V. K. Subashiev, Opt. Commun. 47, 414 (1983).
[CrossRef]

1976 (1)

M. J. Ablowitz and J. F. Ladik, Stud. Appl. Math. 55, 213 (1976).

1970 (1)

A. L. Berkhoer and V. E. Zakharov, Sov. Phys. JETP 31, 486 (1970).

1964 (1)

P. D. Maker, B. W. Terhune, and C. M. Savage, Phys. Rev. Lett. 12, 507 (1964).
[CrossRef]

Ablowitz, M. J.

M. J. Ablowitz and J. F. Ladik, Stud. Appl. Math. 55, 213 (1976).

Agrawal, G. P.

G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[CrossRef] [PubMed]

Areshev, I. P.

I. P. Areshev, T. A. Murina, N. N. Rosanov, and V. K. Subashiev, Opt. Commun. 47, 414 (1983).
[CrossRef]

Berkhoer, A. L.

A. L. Berkhoer and V. E. Zakharov, Sov. Phys. JETP 31, 486 (1970).

Blow, K. J.

B. K. Nayar, K. J. Blow, and N. Doran, Opt. Comput. Process. 1, 81 (1991).

Daino, B.

Doran, N.

B. K. Nayar, K. J. Blow, and N. Doran, Opt. Comput. Process. 1, 81 (1991).

Firth, W. J.

Gibbs, H. M.

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, London, 1985).

Gregori, G.

Haelterman, M.

M. Haelterman, S. Trillo, and S. Wabnitz, Electron. Lett. 29, 119 (1993).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 93, 343 (1992); Phys. Rev. A 47, 2344 (1993).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Lett. 17, 745 (1992).
[CrossRef] [PubMed]

M. Haelterman, Opt. Lett. 17, 792 (1992).
[CrossRef]

M. Haelterman and P. Mandel, Opt. Lett. 15, 1412 (1990).
[CrossRef] [PubMed]

Haus, H. A.

M. Nakazawa, K. Suzuki, and H. A. Haus, IEEE J. Quantum Electron. 25, 2036 (1989); M. Nakazawa, K. Suzuki, H. Kubota, and H. A. Haus, IEEE J. Quantum Electron. 25, 2045 (1989).
[CrossRef]

E. P. Ippen, H. A. Haus, and L. Y. Liu, J. Opt. Soc. Am. B 6, 1736 (1989).
[CrossRef]

Ippen, E. P.

Ladik, J. F.

M. J. Ablowitz and J. F. Ladik, Stud. Appl. Math. 55, 213 (1976).

Liu, L. Y.

Maker, P. D.

P. D. Maker, B. W. Terhune, and C. M. Savage, Phys. Rev. Lett. 12, 507 (1964).
[CrossRef]

Mandel, P.

McDonald, G. S.

McLaughlin, D. W.

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, in Chaos in Nonlinear Dynamical Systems, J. Chandra, ed. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1984); J. V. Moloney, in Nonlinear Phenomena and Chaos, S. Sarkar, ed. (Hilger, Bristol, UK, 1986), pp. 214–245.

Moloney, J. V.

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, in Chaos in Nonlinear Dynamical Systems, J. Chandra, ed. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1984); J. V. Moloney, in Nonlinear Phenomena and Chaos, S. Sarkar, ed. (Hilger, Bristol, UK, 1986), pp. 214–245.

Murina, T. A.

I. P. Areshev, T. A. Murina, N. N. Rosanov, and V. K. Subashiev, Opt. Commun. 47, 414 (1983).
[CrossRef]

Nakazawa, M.

M. Nakazawa, K. Suzuki, and H. A. Haus, IEEE J. Quantum Electron. 25, 2036 (1989); M. Nakazawa, K. Suzuki, H. Kubota, and H. A. Haus, IEEE J. Quantum Electron. 25, 2045 (1989).
[CrossRef]

Nayar, B. K.

B. K. Nayar, K. J. Blow, and N. Doran, Opt. Comput. Process. 1, 81 (1991).

Newell, A. C.

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, in Chaos in Nonlinear Dynamical Systems, J. Chandra, ed. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1984); J. V. Moloney, in Nonlinear Phenomena and Chaos, S. Sarkar, ed. (Hilger, Bristol, UK, 1986), pp. 214–245.

Ouellette, F.

Piché, M.

Rosanov, N. N.

I. P. Areshev, T. A. Murina, N. N. Rosanov, and V. K. Subashiev, Opt. Commun. 47, 414 (1983).
[CrossRef]

Savage, C. M.

P. D. Maker, B. W. Terhune, and C. M. Savage, Phys. Rev. Lett. 12, 507 (1964).
[CrossRef]

Subashiev, V. K.

I. P. Areshev, T. A. Murina, N. N. Rosanov, and V. K. Subashiev, Opt. Commun. 47, 414 (1983).
[CrossRef]

Suzuki, K.

M. Nakazawa, K. Suzuki, and H. A. Haus, IEEE J. Quantum Electron. 25, 2036 (1989); M. Nakazawa, K. Suzuki, H. Kubota, and H. A. Haus, IEEE J. Quantum Electron. 25, 2045 (1989).
[CrossRef]

Terhune, B. W.

P. D. Maker, B. W. Terhune, and C. M. Savage, Phys. Rev. Lett. 12, 507 (1964).
[CrossRef]

Trillo, S.

M. Haelterman, S. Trillo, and S. Wabnitz, Electron. Lett. 29, 119 (1993).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 93, 343 (1992); Phys. Rev. A 47, 2344 (1993).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Lett. 17, 745 (1992).
[CrossRef] [PubMed]

S. Trillo and S. Wabnitz, Opt. Lett. 16, 986, 1566 (1991).
[CrossRef] [PubMed]

S. Trillo and S. Wabnitz, J. Opt. Soc. Am. B 6, 238 (1989).
[CrossRef]

Vallée, R.

R. Vallée, Opt. Commun. 81, 419 (1991).
[CrossRef]

Wabinitz, S.

Wabnitz, S.

M. Haelterman, S. Trillo, and S. Wabnitz, Electron. Lett. 29, 119 (1993).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 93, 343 (1992); Phys. Rev. A 47, 2344 (1993).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Lett. 17, 745 (1992).
[CrossRef] [PubMed]

S. Trillo and S. Wabnitz, Opt. Lett. 16, 986, 1566 (1991).
[CrossRef] [PubMed]

S. Trillo and S. Wabnitz, J. Opt. Soc. Am. B 6, 238 (1989).
[CrossRef]

S. Wabnitz, Phys. Rev. A 38, 2018 (1988).
[CrossRef] [PubMed]

Zakharov, V. E.

A. L. Berkhoer and V. E. Zakharov, Sov. Phys. JETP 31, 486 (1970).

Electron. Lett. (1)

M. Haelterman, S. Trillo, and S. Wabnitz, Electron. Lett. 29, 119 (1993).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Nakazawa, K. Suzuki, and H. A. Haus, IEEE J. Quantum Electron. 25, 2036 (1989); M. Nakazawa, K. Suzuki, H. Kubota, and H. A. Haus, IEEE J. Quantum Electron. 25, 2045 (1989).
[CrossRef]

J. Opt. Soc. Am. B (4)

Opt. Commun. (4)

R. Vallée, Opt. Commun. 81, 419 (1991).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 93, 343 (1992); Phys. Rev. A 47, 2344 (1993).
[CrossRef]

I. P. Areshev, T. A. Murina, N. N. Rosanov, and V. K. Subashiev, Opt. Commun. 47, 414 (1983).
[CrossRef]

Opt. Comput. Process. (1)

B. K. Nayar, K. J. Blow, and N. Doran, Opt. Comput. Process. 1, 81 (1991).

Opt. Lett. (5)

Phys. Rev. A (1)

S. Wabnitz, Phys. Rev. A 38, 2018 (1988).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

P. D. Maker, B. W. Terhune, and C. M. Savage, Phys. Rev. Lett. 12, 507 (1964).
[CrossRef]

G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

A. L. Berkhoer and V. E. Zakharov, Sov. Phys. JETP 31, 486 (1970).

Stud. Appl. Math. (1)

M. J. Ablowitz and J. F. Ladik, Stud. Appl. Math. 55, 213 (1976).

Other (2)

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, in Chaos in Nonlinear Dynamical Systems, J. Chandra, ed. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1984); J. V. Moloney, in Nonlinear Phenomena and Chaos, S. Sarkar, ed. (Hilger, Bristol, UK, 1986), pp. 214–245.

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, London, 1985).

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Figures (10)

Fig. 1
Fig. 1

Schematic of the nonlinear dispersive fiber ring cavity; θ and ρ are the transmission and reflection coefficients for the amplitudes, respectively. The polarization beam splitter (PBS) is used to discriminate the two x- and y-polarization components in the output, and the pump is linearly polarized along the x axis.

Fig. 2
Fig. 2

Steady-state bistable response of the cavity: cavity intensity components Y = |u+|2 and Z = |u|2 versus the input intensity component X = S2 calculated for Δ = 3. The dashed (solid) curves represent unstable (stable) steady-state solutions.

Fig. 3
Fig. 3

Regions of MI in the plane of parameters Y − Δ for both normal (η = 1) and anomalous (η = −1) dispersion regimes.

Fig. 4
Fig. 4

Dependence of the PMI gain on the sideband detuning Ω and the steady-state intensity components Y for a caviity detuning Δ = 4 and a fiber operating in (a) the anomalous dispersion regime (η = −1) and (b) the normal dispersion regime (η = 1).

Fig. 5
Fig. 5

Scalar pulse train generation from a modulationally unstable x-polarized cw in the anomalous dispersion regime (η = −1). The initial perturbation at the optimum frequency for SMI, Ω = Ω1max = (2)1/2, is directed along the y axis. Here we chose Y = Z = 1.5 and Δ = 2. The time t′ is measured in units of the modulation period tp = 2π/Ω.

Fig. 6
Fig. 6

Evolution of the linearly polarized components from a modulationally unstable x-polarized cw in the anomalous dispersion regime (η = −1). Here we chose Y = Z = 3.2, Δ = 1, and the optimum frequency for PMI, Ω = Ω2max = 1.5.

Fig. 7
Fig. 7

Onset of linearly polarized chaos in the cavity for the parameter value of Fig. 6 and Ω = 0.5.

Fig. 8
Fig. 8

Stable emission of a pulse train from an unstable cw directed along the x axis in the normal dispersion regime (η = 1). Here Y = Z = 5, Δ = 4, and Ω = Ω2max = 1.155.

Fig. 9
Fig. 9

Evolution of the circularly polarized components for the case of modulationally induced downswitching: Y = Z = 4, Δ = 4, Ω = 1.155, and a normal dispersion regime.

Fig. 10
Fig. 10

As in Fig. 9 for modulationally induced symmetry breaking. Here Y = Z = 7, Δ = 4, and Ω = 1.

Equations (26)

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i E + n Z n - - β 2 2 E + n T 2 + γ ( 1 - B 2 E + n 2 + 1 + B 2 E - n 2 ) E + n = 0 ,
i E - n Z n - β 2 2 E - n T 2 + γ ( 1 - B 2 E - n 2 + 1 + B 2 E + n 2 ) E - n = 0 ,
E ± n + 1 ( Z n = 0 , T ) = θ E i ( T ) + ρ exp ( - i ϕ ) E ± n ( Z n = L , T ) ,
q ± n ( ξ = ɛ ) = q ± n ( ξ = 0 ) - i η ɛ 2 2 q ± n τ 2 + i ɛ ( 1 - B 2 q ± n 2 + 1 + B 2 q n 2 ) q ± n + O ( ɛ 2 ) ,
q ± n + 1 ( 0 , τ ) - q ± n ( 0 , τ ) = θ q i - θ 2 2 q ± n ( 0 , τ ) - i ϕ q ± n ( 0 , τ ) - i η ɛ 2 2 q ± n ( 0 , τ ) τ 2 + i ɛ [ 1 - B 2 q ± n ( 0 , τ ) 2 + 1 + B 2 q n ( 0 , τ ) 2 ] × q ± n ( 0 , τ ) + O ( ɛ 2 ) ,
q ± ( ξ , τ ) ξ | ξ = n ɛ = q ± [ ξ = ( n + 1 ) ɛ , τ ] - q ± ( ξ = n ɛ , τ ) ɛ .
i u + z - η 2 2 u + t 2 + ( 1 - B 2 u + 2 + 1 + B 2 u - 2 ) u + + i ( 1 + i Δ ) u + - i S = 0 ,
i u - z - η 2 2 u - t 2 + ( 1 - B 2 u - 2 + 1 + B 2 u + 2 ) u - + i ( 1 + i Δ ) u - - i S = 0 ,
[ 1 + ( Δ - 1 - B 2 Y - 1 + B 2 Z ) 2 ] Y = X ,
[ 1 + ( Δ - 1 - B 2 Z - 1 + B 2 Y ) 2 ] Z = X ,
( Z - Y ) [ ( Y 2 + Y Z + Z 2 ) / 9 - 2 Δ ( Y + Z ) / 3 + ( 1 + Δ 2 ) ] = 0.
Y 3 - 2 Δ Y 2 + ( 1 + Δ 2 ) Y = X .
Y ± = Z ± = 2 Δ ± ( Δ 2 - 3 ) 1 / 2 .
u ± ( z , t ) = U s + a ± ( z ) cos Ω t .
i a + z + ( η Ω 2 2 + i - Δ + 3 - B 2 U s 2 ) a + + 1 + B 2 U s 2 a - + U s 2 ( 1 - B 2 a + * + 1 + B 2 a - * ) = i S ,
i a - z + ( η Ω 2 2 + i - Δ + 3 - B 2 U s 2 ) a - + 1 + B 2 U s 2 a + + U s 2 ( 1 - B 2 a - * + 1 + B 2 a + * ) = i S .
λ 1 = - 1 + [ 4 ( Δ - η Ω 2 / 2 ) Y - ( Δ - η Ω 2 / 2 ) 2 - 3 Y 2 ] 1 / 2 ,
λ 2 = - 1 + [ 4 3 ( Δ - η Ω 2 / 2 ) Y - ( Δ - η Ω 2 / 2 ) 2 - Y 2 3 ] 1 / 2 ,
λ 2 max = Y / 3 - 1.
Ω 2 max = [ 2 η ( Δ - 2 Y / 3 ) ] 1 / 2 ,
f 2 max = [ η ( 2 ϕ / L - 2 γ P x / 3 ) / β ] 1 / 2 / 2 π .
Y > 3 for Δ 2 , Y > 3 Δ / 2 for Δ > 2.
3 < γ < 3 Δ / 2 ,             with Δ > 2.
- i A y , - Ω z = B 2 U 0 x 2 A y , + Ω * exp ( - i Δ k z ) ,
- i A y , + Ω z = B 2 ( U 0 x * ) 2 A y , - Ω exp ( + i Δ k z ) ,
i u ± j z - η 2 u ± j + 1 + u ± j - 1 - 2 u ± j Δ t 2 + ( 1 - B 2 u ± j 2 + 1 + B 2 u j 2 ) u ± j + 1 + u ± j - 1 2 + i ( 1 + i Δ ) u ± j - i S = 0 ,

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