Abstract

We investigate temporal stability of stationary solutions for backward degenerate parametric mixing. We show that self-oscillating solutions can be obtained from properly chosen continuous-wave counterpropagating inputs at fundamental and second-harmonic frequencies under general phase-mismatched conditions. The temporal oscillation period near the bifurcation points is predicted by linear stability analysis and verified by numerical simulation of the governing equations.

© 2005 Optical Society of America

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References

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  1. S. E. Harris, "Proposed backward wave oscillation in the infrared," Appl. Phys. Lett. 9, 114-116 (1966).
    [CrossRef]
  2. P. St. J. Russell, "Theoretical study of parametric frequency and wavefront conversion in nonlinear holograms," IEEE J. Quantum Electron. 27, 830-835 (1991).
    [CrossRef]
  3. J. U. Kang, Y. J. Ding, W. K. Burns, and J. S. Melinger, "Backward second-harmonic generation in periodically poled bulk LiNbO3," Opt. Lett. 22, 862-864 (1997).
    [CrossRef] [PubMed]
  4. X. Gu, M. Makarov, Y. J. Ding, J. B. Khurgin, and W. P. Risk, "Backward second-harmonic generation and third-harmonic generation in a periodically poled KTP waveguide," Opt. Lett. 24, 127-129 (1999).
    [CrossRef]
  5. X. Mu, I. B. Zotova, Y. J. Ding, and W. P. Risk, "Backward second-harmonic generation in submicron-period ion-exchanged KTiOPO4 waveguide," Opt. Commun. 181, 153-159 (2000).
    [CrossRef]
  6. C. Conti, S. Trillo, and G. Assanto, "Energy localization in photonic crystals of a purely nonlinear origin," Phys. Rev. Lett. 85, 2502-2505 (2000).
    [CrossRef] [PubMed]
  7. M. Matsumoto and K. Tanaka, "Quasi-phase-matched second-harmonic generation by backward propagating interaction," IEEE J. Quantum Electron. 31, 700-705 (1995).
    [CrossRef]
  8. P. M. Lushnikov, P. Lodhal, and M. Saffman, "Transverse modulational instability of counterpropagating quasi-phase-matching beams in a quadratically nonlinear medium," Opt. Lett. 23, 1650-1652 (1998).
    [CrossRef]
  9. G. D'Alessandro, P. St. J. Russell, and A. A. Wheeler, "Nonlinear dynamics of a backward quasi-phase-matched second-harmonic generator," Phys. Rev. A 55, 3211-3218 (1997).
    [CrossRef]
  10. C. Conti, G. Assanto, and S. Trillo, "Cavityless oscillator through backward quasi-phase-matched second-harmonic generation," Opt. Lett. 24, 1139-1141 (1999).
    [CrossRef]
  11. K. J. McNeil, P. D. Drummond, and D. F. Walls, "Self pulsing in second-harmonic generation," Opt. Commun. 27, 292-294 (1978).
    [CrossRef]
  12. L. A. Lugiato and R. Lefevre, "Spatial dissipative structures in passive optical systems," Phys. Rev. Lett. 58, 2209-2211 (1987).
    [CrossRef] [PubMed]
  13. L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. J. Horowicz, "Bistability, self-pulsing and chaos in optical parametric oscillators," Nuovo Cimento D 10, 959-977 (1988).
    [CrossRef]
  14. S. Trillo and M. Haelterman, "Pulse train generation through modulational instability in intracavity second-harmonic generation," Opt. Lett. 21, 1114-1116 (1996).
    [CrossRef] [PubMed]
  15. H. G. Winful, J. H. Marburger, and E. Garmire, "Theory of bistability in non-linear distributed feedback structures," Appl. Phys. Lett. 35, 379-381 (1979).
    [CrossRef]
  16. H. G. Winful and G. D. Cooperman, "Self-pulsing and chaos in distributed feedback bistable optical devices," Appl. Phys. Lett. 40, 298-300 (1982).
    [CrossRef]
  17. A. Mecozzi, S. Trillo, and S. Wabnitz, "Spatial instabilities, all-optical limiting and thresholding in nonlinear distributed-feedback devices," Opt. Lett. 12, 1008-1010 (1987).
    [CrossRef] [PubMed]
  18. C. M. de Sterke, "Stability analysis of nonlinear periodic media," Phys. Rev. A 45, 8252-8258 (1992).
    [CrossRef] [PubMed]
  19. Y. Silberberg and I. Bar-Joseph, "Instabilities, self-oscillation and chaos in a simple nonlinear optical interaction," Phys. Rev. Lett. 48, 1541-1544 (1982).
    [CrossRef]
  20. C. Richy, K. I. Petsas, E. Giacobino, C. Fabre, and L. Lugiato, "Observation of bistability and delayed bifurcation in a triply resonant optical parametric oscillator," J. Opt. Soc. Am. B 12, 456-461 (1995).
    [CrossRef]
  21. M. Bache, P. Lodhal, A. V. Mamaev, M. Marcus, and M. Saffman, "Observation of self-pulsing in singly resonant optical second-harmonic generation with competing nonlinearities," Phys. Rev. A 65, 033811 (2002).
    [CrossRef]
  22. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, 1983).
    [CrossRef]
  23. G. Cappellini, S. Trillo, S. Wabnitz, and R. Chisari, "Two wave mixing in a quadratic nonlinear medium: bifurcations, spatial instabilities, and chaos," Opt. Lett. 17, 637-639 (1992).
    [CrossRef]
  24. A. V. Buryak, I. Towers, and S. Trillo, "Multistability, homoclinic clamping, and chaos in nonlinear quadratic distributed feedback systems," Phys. Lett. A 267, 319-325 (2000).
    [CrossRef]

2002 (1)

M. Bache, P. Lodhal, A. V. Mamaev, M. Marcus, and M. Saffman, "Observation of self-pulsing in singly resonant optical second-harmonic generation with competing nonlinearities," Phys. Rev. A 65, 033811 (2002).
[CrossRef]

2000 (3)

A. V. Buryak, I. Towers, and S. Trillo, "Multistability, homoclinic clamping, and chaos in nonlinear quadratic distributed feedback systems," Phys. Lett. A 267, 319-325 (2000).
[CrossRef]

X. Mu, I. B. Zotova, Y. J. Ding, and W. P. Risk, "Backward second-harmonic generation in submicron-period ion-exchanged KTiOPO4 waveguide," Opt. Commun. 181, 153-159 (2000).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, "Energy localization in photonic crystals of a purely nonlinear origin," Phys. Rev. Lett. 85, 2502-2505 (2000).
[CrossRef] [PubMed]

1999 (2)

1998 (1)

1997 (2)

J. U. Kang, Y. J. Ding, W. K. Burns, and J. S. Melinger, "Backward second-harmonic generation in periodically poled bulk LiNbO3," Opt. Lett. 22, 862-864 (1997).
[CrossRef] [PubMed]

G. D'Alessandro, P. St. J. Russell, and A. A. Wheeler, "Nonlinear dynamics of a backward quasi-phase-matched second-harmonic generator," Phys. Rev. A 55, 3211-3218 (1997).
[CrossRef]

1996 (1)

1995 (2)

C. Richy, K. I. Petsas, E. Giacobino, C. Fabre, and L. Lugiato, "Observation of bistability and delayed bifurcation in a triply resonant optical parametric oscillator," J. Opt. Soc. Am. B 12, 456-461 (1995).
[CrossRef]

M. Matsumoto and K. Tanaka, "Quasi-phase-matched second-harmonic generation by backward propagating interaction," IEEE J. Quantum Electron. 31, 700-705 (1995).
[CrossRef]

1992 (2)

1991 (1)

P. St. J. Russell, "Theoretical study of parametric frequency and wavefront conversion in nonlinear holograms," IEEE J. Quantum Electron. 27, 830-835 (1991).
[CrossRef]

1988 (1)

L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. J. Horowicz, "Bistability, self-pulsing and chaos in optical parametric oscillators," Nuovo Cimento D 10, 959-977 (1988).
[CrossRef]

1987 (2)

1982 (2)

H. G. Winful and G. D. Cooperman, "Self-pulsing and chaos in distributed feedback bistable optical devices," Appl. Phys. Lett. 40, 298-300 (1982).
[CrossRef]

Y. Silberberg and I. Bar-Joseph, "Instabilities, self-oscillation and chaos in a simple nonlinear optical interaction," Phys. Rev. Lett. 48, 1541-1544 (1982).
[CrossRef]

1979 (1)

H. G. Winful, J. H. Marburger, and E. Garmire, "Theory of bistability in non-linear distributed feedback structures," Appl. Phys. Lett. 35, 379-381 (1979).
[CrossRef]

1978 (1)

K. J. McNeil, P. D. Drummond, and D. F. Walls, "Self pulsing in second-harmonic generation," Opt. Commun. 27, 292-294 (1978).
[CrossRef]

1966 (1)

S. E. Harris, "Proposed backward wave oscillation in the infrared," Appl. Phys. Lett. 9, 114-116 (1966).
[CrossRef]

Assanto, G.

C. Conti, S. Trillo, and G. Assanto, "Energy localization in photonic crystals of a purely nonlinear origin," Phys. Rev. Lett. 85, 2502-2505 (2000).
[CrossRef] [PubMed]

C. Conti, G. Assanto, and S. Trillo, "Cavityless oscillator through backward quasi-phase-matched second-harmonic generation," Opt. Lett. 24, 1139-1141 (1999).
[CrossRef]

Bache, M.

M. Bache, P. Lodhal, A. V. Mamaev, M. Marcus, and M. Saffman, "Observation of self-pulsing in singly resonant optical second-harmonic generation with competing nonlinearities," Phys. Rev. A 65, 033811 (2002).
[CrossRef]

Bar-Joseph, I.

Y. Silberberg and I. Bar-Joseph, "Instabilities, self-oscillation and chaos in a simple nonlinear optical interaction," Phys. Rev. Lett. 48, 1541-1544 (1982).
[CrossRef]

Burns, W. K.

Buryak, A. V.

A. V. Buryak, I. Towers, and S. Trillo, "Multistability, homoclinic clamping, and chaos in nonlinear quadratic distributed feedback systems," Phys. Lett. A 267, 319-325 (2000).
[CrossRef]

Cappellini, G.

Chisari, R.

Conti, C.

C. Conti, S. Trillo, and G. Assanto, "Energy localization in photonic crystals of a purely nonlinear origin," Phys. Rev. Lett. 85, 2502-2505 (2000).
[CrossRef] [PubMed]

C. Conti, G. Assanto, and S. Trillo, "Cavityless oscillator through backward quasi-phase-matched second-harmonic generation," Opt. Lett. 24, 1139-1141 (1999).
[CrossRef]

Cooperman, G. D.

H. G. Winful and G. D. Cooperman, "Self-pulsing and chaos in distributed feedback bistable optical devices," Appl. Phys. Lett. 40, 298-300 (1982).
[CrossRef]

D'Alessandro, G.

G. D'Alessandro, P. St. J. Russell, and A. A. Wheeler, "Nonlinear dynamics of a backward quasi-phase-matched second-harmonic generator," Phys. Rev. A 55, 3211-3218 (1997).
[CrossRef]

de Sterke, C. M.

C. M. de Sterke, "Stability analysis of nonlinear periodic media," Phys. Rev. A 45, 8252-8258 (1992).
[CrossRef] [PubMed]

Ding, Y. J.

Drummond, P. D.

K. J. McNeil, P. D. Drummond, and D. F. Walls, "Self pulsing in second-harmonic generation," Opt. Commun. 27, 292-294 (1978).
[CrossRef]

Fabre, C.

C. Richy, K. I. Petsas, E. Giacobino, C. Fabre, and L. Lugiato, "Observation of bistability and delayed bifurcation in a triply resonant optical parametric oscillator," J. Opt. Soc. Am. B 12, 456-461 (1995).
[CrossRef]

L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. J. Horowicz, "Bistability, self-pulsing and chaos in optical parametric oscillators," Nuovo Cimento D 10, 959-977 (1988).
[CrossRef]

Garmire, E.

H. G. Winful, J. H. Marburger, and E. Garmire, "Theory of bistability in non-linear distributed feedback structures," Appl. Phys. Lett. 35, 379-381 (1979).
[CrossRef]

Giacobino, E.

C. Richy, K. I. Petsas, E. Giacobino, C. Fabre, and L. Lugiato, "Observation of bistability and delayed bifurcation in a triply resonant optical parametric oscillator," J. Opt. Soc. Am. B 12, 456-461 (1995).
[CrossRef]

L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. J. Horowicz, "Bistability, self-pulsing and chaos in optical parametric oscillators," Nuovo Cimento D 10, 959-977 (1988).
[CrossRef]

Gu, X.

Guckenheimer, J.

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, 1983).
[CrossRef]

Haelterman, M.

Harris, S. E.

S. E. Harris, "Proposed backward wave oscillation in the infrared," Appl. Phys. Lett. 9, 114-116 (1966).
[CrossRef]

Holmes, P.

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, 1983).
[CrossRef]

Horowicz, R. J.

L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. J. Horowicz, "Bistability, self-pulsing and chaos in optical parametric oscillators," Nuovo Cimento D 10, 959-977 (1988).
[CrossRef]

Kang, J. U.

Khurgin, J. B.

Lefevre, R.

L. A. Lugiato and R. Lefevre, "Spatial dissipative structures in passive optical systems," Phys. Rev. Lett. 58, 2209-2211 (1987).
[CrossRef] [PubMed]

Lodhal, P.

M. Bache, P. Lodhal, A. V. Mamaev, M. Marcus, and M. Saffman, "Observation of self-pulsing in singly resonant optical second-harmonic generation with competing nonlinearities," Phys. Rev. A 65, 033811 (2002).
[CrossRef]

P. M. Lushnikov, P. Lodhal, and M. Saffman, "Transverse modulational instability of counterpropagating quasi-phase-matching beams in a quadratically nonlinear medium," Opt. Lett. 23, 1650-1652 (1998).
[CrossRef]

Lugiato, L.

Lugiato, L. A.

L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. J. Horowicz, "Bistability, self-pulsing and chaos in optical parametric oscillators," Nuovo Cimento D 10, 959-977 (1988).
[CrossRef]

L. A. Lugiato and R. Lefevre, "Spatial dissipative structures in passive optical systems," Phys. Rev. Lett. 58, 2209-2211 (1987).
[CrossRef] [PubMed]

Lushnikov, P. M.

Makarov, M.

Mamaev, A. V.

M. Bache, P. Lodhal, A. V. Mamaev, M. Marcus, and M. Saffman, "Observation of self-pulsing in singly resonant optical second-harmonic generation with competing nonlinearities," Phys. Rev. A 65, 033811 (2002).
[CrossRef]

Marburger, J. H.

H. G. Winful, J. H. Marburger, and E. Garmire, "Theory of bistability in non-linear distributed feedback structures," Appl. Phys. Lett. 35, 379-381 (1979).
[CrossRef]

Marcus, M.

M. Bache, P. Lodhal, A. V. Mamaev, M. Marcus, and M. Saffman, "Observation of self-pulsing in singly resonant optical second-harmonic generation with competing nonlinearities," Phys. Rev. A 65, 033811 (2002).
[CrossRef]

Matsumoto, M.

M. Matsumoto and K. Tanaka, "Quasi-phase-matched second-harmonic generation by backward propagating interaction," IEEE J. Quantum Electron. 31, 700-705 (1995).
[CrossRef]

McNeil, K. J.

K. J. McNeil, P. D. Drummond, and D. F. Walls, "Self pulsing in second-harmonic generation," Opt. Commun. 27, 292-294 (1978).
[CrossRef]

Mecozzi, A.

Melinger, J. S.

Mu, X.

X. Mu, I. B. Zotova, Y. J. Ding, and W. P. Risk, "Backward second-harmonic generation in submicron-period ion-exchanged KTiOPO4 waveguide," Opt. Commun. 181, 153-159 (2000).
[CrossRef]

Oldano, C.

L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. J. Horowicz, "Bistability, self-pulsing and chaos in optical parametric oscillators," Nuovo Cimento D 10, 959-977 (1988).
[CrossRef]

Petsas, K. I.

Richy, C.

Risk, W. P.

X. Mu, I. B. Zotova, Y. J. Ding, and W. P. Risk, "Backward second-harmonic generation in submicron-period ion-exchanged KTiOPO4 waveguide," Opt. Commun. 181, 153-159 (2000).
[CrossRef]

X. Gu, M. Makarov, Y. J. Ding, J. B. Khurgin, and W. P. Risk, "Backward second-harmonic generation and third-harmonic generation in a periodically poled KTP waveguide," Opt. Lett. 24, 127-129 (1999).
[CrossRef]

Russell, P. St. J.

G. D'Alessandro, P. St. J. Russell, and A. A. Wheeler, "Nonlinear dynamics of a backward quasi-phase-matched second-harmonic generator," Phys. Rev. A 55, 3211-3218 (1997).
[CrossRef]

P. St. J. Russell, "Theoretical study of parametric frequency and wavefront conversion in nonlinear holograms," IEEE J. Quantum Electron. 27, 830-835 (1991).
[CrossRef]

Saffman, M.

M. Bache, P. Lodhal, A. V. Mamaev, M. Marcus, and M. Saffman, "Observation of self-pulsing in singly resonant optical second-harmonic generation with competing nonlinearities," Phys. Rev. A 65, 033811 (2002).
[CrossRef]

P. M. Lushnikov, P. Lodhal, and M. Saffman, "Transverse modulational instability of counterpropagating quasi-phase-matching beams in a quadratically nonlinear medium," Opt. Lett. 23, 1650-1652 (1998).
[CrossRef]

Silberberg, Y.

Y. Silberberg and I. Bar-Joseph, "Instabilities, self-oscillation and chaos in a simple nonlinear optical interaction," Phys. Rev. Lett. 48, 1541-1544 (1982).
[CrossRef]

Tanaka, K.

M. Matsumoto and K. Tanaka, "Quasi-phase-matched second-harmonic generation by backward propagating interaction," IEEE J. Quantum Electron. 31, 700-705 (1995).
[CrossRef]

Towers, I.

A. V. Buryak, I. Towers, and S. Trillo, "Multistability, homoclinic clamping, and chaos in nonlinear quadratic distributed feedback systems," Phys. Lett. A 267, 319-325 (2000).
[CrossRef]

Trillo, S.

Wabnitz, S.

Walls, D. F.

K. J. McNeil, P. D. Drummond, and D. F. Walls, "Self pulsing in second-harmonic generation," Opt. Commun. 27, 292-294 (1978).
[CrossRef]

Wheeler, A. A.

G. D'Alessandro, P. St. J. Russell, and A. A. Wheeler, "Nonlinear dynamics of a backward quasi-phase-matched second-harmonic generator," Phys. Rev. A 55, 3211-3218 (1997).
[CrossRef]

Winful, H. G.

H. G. Winful and G. D. Cooperman, "Self-pulsing and chaos in distributed feedback bistable optical devices," Appl. Phys. Lett. 40, 298-300 (1982).
[CrossRef]

H. G. Winful, J. H. Marburger, and E. Garmire, "Theory of bistability in non-linear distributed feedback structures," Appl. Phys. Lett. 35, 379-381 (1979).
[CrossRef]

Zotova, I. B.

X. Mu, I. B. Zotova, Y. J. Ding, and W. P. Risk, "Backward second-harmonic generation in submicron-period ion-exchanged KTiOPO4 waveguide," Opt. Commun. 181, 153-159 (2000).
[CrossRef]

Appl. Phys. Lett. (3)

S. E. Harris, "Proposed backward wave oscillation in the infrared," Appl. Phys. Lett. 9, 114-116 (1966).
[CrossRef]

H. G. Winful, J. H. Marburger, and E. Garmire, "Theory of bistability in non-linear distributed feedback structures," Appl. Phys. Lett. 35, 379-381 (1979).
[CrossRef]

H. G. Winful and G. D. Cooperman, "Self-pulsing and chaos in distributed feedback bistable optical devices," Appl. Phys. Lett. 40, 298-300 (1982).
[CrossRef]

IEEE J. Quantum Electron. (2)

P. St. J. Russell, "Theoretical study of parametric frequency and wavefront conversion in nonlinear holograms," IEEE J. Quantum Electron. 27, 830-835 (1991).
[CrossRef]

M. Matsumoto and K. Tanaka, "Quasi-phase-matched second-harmonic generation by backward propagating interaction," IEEE J. Quantum Electron. 31, 700-705 (1995).
[CrossRef]

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

Nuovo Cimento D (1)

L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. J. Horowicz, "Bistability, self-pulsing and chaos in optical parametric oscillators," Nuovo Cimento D 10, 959-977 (1988).
[CrossRef]

Opt. Commun. (2)

K. J. McNeil, P. D. Drummond, and D. F. Walls, "Self pulsing in second-harmonic generation," Opt. Commun. 27, 292-294 (1978).
[CrossRef]

X. Mu, I. B. Zotova, Y. J. Ding, and W. P. Risk, "Backward second-harmonic generation in submicron-period ion-exchanged KTiOPO4 waveguide," Opt. Commun. 181, 153-159 (2000).
[CrossRef]

Opt. Lett. (7)

Phys. Lett. A (1)

A. V. Buryak, I. Towers, and S. Trillo, "Multistability, homoclinic clamping, and chaos in nonlinear quadratic distributed feedback systems," Phys. Lett. A 267, 319-325 (2000).
[CrossRef]

Phys. Rev. A (3)

M. Bache, P. Lodhal, A. V. Mamaev, M. Marcus, and M. Saffman, "Observation of self-pulsing in singly resonant optical second-harmonic generation with competing nonlinearities," Phys. Rev. A 65, 033811 (2002).
[CrossRef]

C. M. de Sterke, "Stability analysis of nonlinear periodic media," Phys. Rev. A 45, 8252-8258 (1992).
[CrossRef] [PubMed]

G. D'Alessandro, P. St. J. Russell, and A. A. Wheeler, "Nonlinear dynamics of a backward quasi-phase-matched second-harmonic generator," Phys. Rev. A 55, 3211-3218 (1997).
[CrossRef]

Phys. Rev. Lett. (3)

L. A. Lugiato and R. Lefevre, "Spatial dissipative structures in passive optical systems," Phys. Rev. Lett. 58, 2209-2211 (1987).
[CrossRef] [PubMed]

C. Conti, S. Trillo, and G. Assanto, "Energy localization in photonic crystals of a purely nonlinear origin," Phys. Rev. Lett. 85, 2502-2505 (2000).
[CrossRef] [PubMed]

Y. Silberberg and I. Bar-Joseph, "Instabilities, self-oscillation and chaos in a simple nonlinear optical interaction," Phys. Rev. Lett. 48, 1541-1544 (1982).
[CrossRef]

Other (1)

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, 1983).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Spatiotemporal evolution of field moduli at FF [ a 1 ( ξ , τ ) 2 ] and SH [ a 2 ( ξ , τ ) ] for δ k = 1 , γ = 1 , and D 1 , 2 = 0 . Boundary values, a 10 2 = a 20 = 2.4 and β 0 = π 2 .

Fig. 2
Fig. 2

Stationary ( τ = 0 ) solutions that correspond to the evolutions shown in Fig. 1. Solid (dashed) curves, FF (SH) field.

Fig. 3
Fig. 3

Left, temporal evolution of an unstable stationary solution. Right, phase portrait; the thick curve represents the limit cycle. a 10 = 2.4 2 , a 20 = 2.4 , β 0 = π 2 , and δ k = 1 .

Fig. 4
Fig. 4

Marginal stability surfaces in space ( β 0 , a 20 , a 10 ) , and corresponding periods of self-oscillating solutions for three values of mismatch δ k ( = 0 , 1 , 3 ) .

Fig. 5
Fig. 5

Period of the oscillation at bifurcation points for δ k = 1 and a 20 = 4 : Comparison between the values calculated from the linear analysis (solid curve) and those obtained from direct integration of the governing equations (circles). Values of a 10 2 are displayed as a dashed curve.

Fig. 6
Fig. 6

Input and output intensities for a self-oscillating solution. Estimated period, T = 2.03 ; real period, T = 2 . Dashed curve, FF input, thicker solid curve, FF output; thinner solid curve, SH output. The values of the parameters are the same as in Fig. 3.

Fig. 7
Fig. 7

Real part of the leading eigenvalue of M as a function of transmitted intensity P and effective mismatch δ k (the domain labeled NOT ACCESSIBLE corresponds to P > P max ( δ k ) where solutions diverge; see Appendix A). Note that the narrow purely real branch tends to form an island of instability (where it emerges above zero) for relatively large values of P and δ k .

Fig. 8
Fig. 8

(a) Real (solid curve) and imaginary (dashed line) part of the leading eigenvalue for δ k = 12 : Note that the unstable eigenvalue is purely real. (b) Input–output transmission curve δ k = 12 (enlarged portion of Fig. 12). Note that values of P located between the abscissas of points labeled 1 and 2 in (a) correspond to the negative slope branch in (b).

Fig. 9
Fig. 9

Left, temporal evolution of an unstable steady solution, corresponding to vanishing backward input with real positive eigenvalue of M. Right, phase portrait: The system evolves from an unstable point to a stable point. δ k = 15.7 and P = 15 .

Fig. 10
Fig. 10

Bifurcation diagrams η e versus δ k for fixed P: P = 1 and P = 1 . Dashed (solid) curves, saddle points (centers). Equilibrium points η e < P [shaded region in (a)] are not physically accessible.

Fig. 11
Fig. 11

Phase plane associated with Eqs. (A1) for δ k slightly above the bifurcation points in Fig. 10: (a) P = 1 , δ k = 2.5 ; (b) P = 1 , δ k = 4 . Thicker curves represent separatrices emanating from the saddle points, shaded region η < P = 1 in (a) is not physically accessible. Note also that, for δ k δ k , one obtains identical phase-space portraits, except for ϕ ϕ + π .

Fig. 12
Fig. 12

Stationary field intensity at FF (solid curves) and SH (dashed curves) for δ k = 12 and several values of P near the critical value P c = δ k 2 16 = 9 : (a) P = 8.8 < P c , (b) P = P c , (c) P = 9.2 > P c . (d) Corresponding transmission curve P = a 1 ( ξ = 1 ) 2 2 versus P in = a 1 ( ξ = 0 ) 2 2 compared with the transmission obtained in phase matching ( δ k = 0 ) . Dashed curves correspond to critical output intensity P = P c .

Equations (15)

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

i ξ a 1 + i τ a 1 D 1 τ τ 2 a 1 + a 2 a 1 * = 0 ,
i ξ a 2 + i γ τ a 2 D 2 τ τ 2 a 2 + δ k a 2 + ( a 1 2 2 ) = 0 ,
a 1 ( 0 , τ ) = a 10 , a 2 ( 1 , τ ) = a 20 exp ( i β 0 ) ,
i ξ a ¯ 1 = H a ¯ 1 * = a ¯ 2 a ¯ 1 * ,
i ξ a ¯ 2 = H a ¯ 2 * = δ k a ¯ 2 + a ¯ 1 2 2 ,
τ p 1 + ξ p 1 = i a ¯ 2 p 1 * + i a ¯ 1 * p 2 ,
τ p 2 ξ p 2 = i a ¯ 1 p 1 + i δ k p 2 .
p ̇ ( τ ) = M ( a 10 , a 20 , β 0 , δ k ) p ( τ ) ,
p = [ R { p 1 } , I { p 1 } , R { p 2 } , I { p 2 } ] T ,
p 1 , 2 = [ p 1 , 2 ( ξ 1 , τ ) , , p 1 , 2 ( ξ N , τ ) ] T ,
M = [ A B I a 2 R a 2 I a 1 R a 1 R a 2 A B + I a 2 R a 1 I a 1 I a 1 R a 1 A F δ k I N R a 1 I a 1 δ k I N A F ] ,
η ̇ = H r ϕ , ϕ ̇ = H r η ,
H r = δ k η + 2 η ( η + P ) cos ϕ ,
1 ξ d ξ = ( ξ 1 ) = η ( 1 ) = 0 η ( ξ ) d η [ 4 η ( η η + ) ( η η ) ] 1 2 ,
η ± = K { [ η ± + Re ( η ± ) 2 η ± ] 1 2 } ,

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