Dynamics of the optical parametric oscillator near resonance detuning

SIAM Journal on Applied Dynamical Systems

Volumn 4, Issue 4, 2005, Pages 808-831

  Hewitt, Sarah E ^a   Nathan Kutzt, J ^a  

^a UNIVERSITY OF WASHINGTON   (United States)

Author keywords

Bifurcation theory; Parametric oscillators; Stability analysis; Swift Hohenberg

Indexed keywords

BIFURCATION THEORY; STABILITY ANALYSIS; SWIFT-HOHENBERG;

ASYMPTOTIC STABILITY; BIFURCATION (MATHEMATICS); COMPUTATIONAL METHODS; RESONANCE; TUNING;

OPTICAL PARAMETRIC OSCILLATORS;

EID: 25444494711     PISSN: 15360040     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/040610295     Document Type: Article

References (38)

1. T. H. MAIMAN, Stimulated optical radiation in ruby, Nature, 187 (1960), pp. 493-494.
2. P. A. FRANKEN, A. E. HILL, C. W. PETERS, AND G. WEINREICH, Generation of optical harmonics, Phys. Rev. Lett., 7 (1961), pp. 118-119.
3. N. BLOEMBERGER, Nonlinear Optics, Benjamin, New York, 1965.
4. R. L. BYER AND A. PISKARSKAS, EDS., Special Issue on Optical Parametric Oscillation and Amplification, J. Opt. Soc. Amer. B Opt. Phys., 10 (1993), pp. 1655-1791.
5. W. R. BOSENBERG AND R. C. ECKARDT, EDS., Special Issue on Optical Parametric Devices, J. Opt. Soc. Amer. B Opt. Phys., 12 (1995), pp. 2083-2320.
6. M. C. CROSS AND P. C. HOHENBERG, Pattern formation outside of equilibrium, Rev. Modern Phys., 65 (1993), p. 851.
7. G. L. OPPO, M. BRAMBILLA, AND L. A. LUGIATO, Formation and evolution of roll patterns in optical parametric oscillators, Phys. Rev. A (3), 49 (1994), pp. 2028-2032.
8. K. STALIUNAS, Transverse pattern formation in optical parametric oscillators, J. Modern Opt., 42 (1995), pp. 1261-1269.
9. S. LONGHI, Letter hydrodynamic equation model for degenerate optical parametric oscillators, J. Modern Opt., 43 (1996), pp. 1089-1094.
10. S. LONGHI AND A. GERACI, Swift-Hohenberg equation for optical parametric oscillators, Phys. Rev. A (3), 54 (1996), pp. 4581-4584.
11. G. J. DE VALCÁRCEL, K. STALIUNAS, E. ROLDÁN, AND V. J. SÁNCHEZ-MORCILLO, Transverse patterns in degenerate optical parametric oscillation and degenerate four-wave mixing, Phys. Rev. A (3), 54 (1996), pp. 1609-1624.
12. S. TRILLO, M. HAELTERMAN, AND A. SHEPPARD, Stable topological spatial solitons in optical parametric oscillators, Opt. Lett., 22 (1997), pp. 970-972.
13. J. N. KUTZ, T. ERNEUX, S. TRILLO, AND M. HAELTERMAN, Curvature dynamics and stability of topological solitons in the optical parametric oscillator, J. Opt. Soc. Amer. B Opt. Phys., 16 (1999), pp. 1936-1941.
14. M. TIDLI, P. MANDEL, AND R. LEFEVER, Kinetics of localized pattern formation in optical systems, Phys. Rev. Lett., 81 (1998), pp. 979-982.
15. S. TRILLO AND M. HAELTERMAN, Excitation and bistability of self-trapped signal beams in optical parametric oscillators, Opt. Lett., 23 (1998), pp. 1514-1516.
16. K. STALIUNAS AND V. SÁNCHEZ-MORCILLO, Spatial-localized structures in degenerate optical parametric oscillators, Phys. Rev. A (3), 57 (1998), pp. 1454-1457.
17. S. LONGHI, Spatial solitary waves in nondegenerate optical parametric oscillators near an inverted bifurcation, Opt. Comm., 149 (1998), pp. 335-340.
18. S. LONGHI, Localized structures in optical parametric oscillation, Phys. Scripta, 56 (1997), pp. 611-618.
19. M. TIDLI, P. MANDEL, AND M. HAELTERMAN, Spatiotemporal patterns and localized structures in nonlinear optics, Phys. Rev. E (3), 56 (1997), pp. 6524-6530.
20. G. A. BUKAUSKAS, V. I. KABELKA, A. S. PISKARSKAS, AND A. Yu. STABINIS, Features of three-photon parametric interaction of ultrashort light packets in the nonlinear amplification regime, Sov. J. Quant. Elec., 4 (1974), pp. 290-292.
21. R. DANELYUS, G. DIKCHYUS, V. KABELKA, A. PISKARSKAS, AND A. STABINIS, Parametric excitation of light in the picosecond range, Sov. J. Quant. Elec., 7 (1977), pp. 1360-1368.
22. S. H. HEWITT, K. INTRACHAT, AND J. N. KUTZ, Dynamics and stability of a new class of periodic solutions of the optical parametric oscillator, Opt. Comm., 240 (2004), pp. 423-436.
23. A. C. NEWELL AND J. A. WHITEHEAD, Finite bandwidth, finite amplitude convection, J. Fluid Mech., 38 (1969), pp. 279-303.
24. L. A. SEGEL, Distant side-walls cause slow amplitude modulation of cellular convection, J. Fluid Mech., 38 (1969), pp. 203-224.
25. B. FRIEDMAN, Principles and Techniques of Applied Mathematics, John Wiley & Sons, New York, 1956.
26. E. ZAUDERER, Partial Differential Equations and Applications, John Wiley & Sons, New York, 1989.
27. T. KAPITULA, Bifurcating bright and dark solitary waves for the perturbed cubic-quintic nonlinear Schrödinger equation, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), pp. 585-629.
28. J. N. KUTZ AND W. L. KATH, Stability of pulses in nonlinear optical fibers using phase-sensitive amplifiers, SIAM J. Appl. Math., 56 (1996), pp. 611-626.
29. J. N. KUTZ, W. L. KATH, R.-D. LI, AND P. KUMAR, Long-distance pulse propagation in nonlinear optical fibers by using periodically spaced parametric amplifiers, Opt. Lett., 18 (1993), pp. 802-804.
30. M. WEINSTEIN, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), pp. 472-491.
31. J. N. KUTZ, C. V. HILE, W. L. KATH, R.-D. LI, AND P. KUMAR, Pulse propagation in nonlinear optical fiber-lines that employ phase-sensitive parametric amplifiers, J. Opt. Soc. Amer. B Opt. Phys., 11 (1994), pp. 2112-2123.
32. J. C. ALEXANDER, M. G. GRILLAKIS, C. K. R. T. JONES, AND B. SANDSTEDE, Stability of pulses on optical fibers with phase-sensitive amplifiers, Z. Angew. Math. Phys., 48 (1997), pp. 175-192.
33. K. PROMISLOW AND J. N. KUTZ, Bifurcation and asymptotic stability in the large detuning limit of the optical parametric oscillator, Nonlinearity, 13 (2000), pp. 675-698.
34. J. GUCKENHEIMER AND P. HOLMES, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer-Verlag, New York, 1980.
35. B. SANDSTEDE, C. K. R. T. JONES, AND J. C. ALEXANDER, Existence and stability of N-pulses on optical fibers with phase-sensitive amplifiers, Phys. D, 106 (1997), pp. 167-206.
36. M. ABRAMOWITZ AND I. STEGUN, Handbook of Mathematical Functions, Dover, New York, 1972.
37. H. T. DAVIS, Introduction to Nonlinear Differential and Integral Equations, Dover, New York, 1962.
38. E. L. INCE, Ordinary Differential Equations, Dover, New York, 1944.