Abstract

We investigate the modulational instability of optical pulses propagating in a lossless fiber where both effects of relaxation and saturation of the nonlinearity are taken simultaneously into account. The saturation of the nonlinearity is incorporated in the relaxation dynamics of the Kerr response. We calculate the exact dispersion relation for harmonic perturbations over the stationary solution. In the anomalous dispersive regime, the gain spectrum exhibits two bands in the fast relaxation regime. The low energy band is reduced by the saturation of the nonlinearity but is roughly insensitive to the nonlinearity response time. The high energy band is mainly due to the Raman response. These frequency bands superpose for fast relaxation responses and a new behavior sets up. In the normal dispersive regime, a single instability band sets up associated with the finite response time of the nonlinearity with distinct features for fast and slow nonlinear relaxation.

© 2008 Optical Society of America

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  1. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142-144 (1973).
    [Crossref]
  2. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171-172 (1973).
    [Crossref]
  3. E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, “Soliton stability in plasmas and hydrodynamics,” Phys. Rep. 142, 103-165 (1986).
    [Crossref]
  4. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).
  5. K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 24, 2665-2673 (1989).
    [Crossref]
  6. E. A. Golovchenko and A. N. Pilipetskii, “Unified analysis of 4-photon mixing, modulational instability, and stimulated Raman-scattering under various polarization conditions in fibers,” J. Opt. Soc. Am. B 11, 92-101 (1994).
    [Crossref]
  7. S. B. Cavalcanti and M. L. Lyra, “Modulational instability of ultrashort pulses via a generalized nonlinear Schrödinger equation with deviating argument,” Phys. Lett. A 211, 276-280 (1996).
    [Crossref]
  8. C. Cambournac, H. Maillotte, E. Lantz, J. M. Dudley, and M. Chauvet, “Spatiotemporal behavior of periodic arrays of spatial solitons in a planar waveguide with relaxing Kerr nonlinearity,” J. Opt. Soc. Am. B 19, 574-585 (2002).
    [Crossref]
  9. M. J. Potasek, “Modulational instability in an extendednonlinear Schrödinger equation,” Opt. Lett. 12, 921-923 (1987).
    [Crossref] [PubMed]
  10. X. Liu, J. W. Haus, and S. M. Shahriar, “Modulation instability for a relaxational Kerr medium,” Opt. Commun. 281, 2907-2912 (2008).
    [Crossref]
  11. M. Nurhuda and E. van Groesen, “Effects of delayed Kerr nonlinearity and ionization on the filamentary ultrashort laser pulses in air,” Phys. Rev. E 71, 066502 (2005).
    [Crossref]
  12. S. Pitois and G. Millot, “Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber,” Opt. Commun. 226, 415-422 (2003).
    [Crossref]
  13. P. T. Dinda, C. M. Ngabireng, K. Porsezian, and B. Kalithasan, “Modulational instability in optical fibers with arbitrary higher-order dispersion and delayed Raman response,” Opt. Commun. 266, 142-150 (2006).
    [Crossref]
  14. M. L. Lyra and A. S. Gouveia Neto, “Saturation effects on modulational instability in non-Kerr-like monomode optical fibers,” Opt. Commun. 108, 117-120 (1994).
    [Crossref]
  15. Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stabilization of one-dimensional periodic waves by saturation of the nonlinear response,” Phys. Rev. E 68, 065605(R) (2003).
    [Crossref]
  16. M. Stepic, C. E. Ruter, D. Kip, A. Maluckov, and L. Hadzvievski, “Modulational instability in one-dimensional saturable waveguide arrays: comparison with Kerr nonlinearity,” Opt. Commun. 267, 229-235 (2006).
    [Crossref]
  17. A. Maluckov, Lj. Hadzievski, N. Lazarides, and G. P. Tsironis, “Left-handed metamaterials with saturable nonlinearity,” Phys. Rev. E 77, 046607 (2008).
    [Crossref]
  18. C. P. Jisha, V. C. Kuriakose, and K. Porsezian, “Modulational instability and beam propagation in photorefractive polymer,” J. Opt. Soc. Am. B 25, 674-679 (2008).
    [Crossref]
  19. S. Ambomo, C. M. Ngabireng, P. Tchofo Dinda, A. Labruyère, K. Porsezian, and B. Kalithasan, “Critical behavior with dramatic enhancement of modulational instability gain in fiber systems with periodic variation dispersion,” J. Opt. Soc. Am. B 25, 425-433 (2008).
    [Crossref]
  20. S. C. Wen, W. H. Su, H. Zhang, X. Q. Fu, L. J. Qian, and D. Y. Fan, “Influence of higher-order dispersions and Raman delayed response on modulation instability in microstructured fibres,” Chin. Phys. Lett. 20, 852-854 (2003).
    [Crossref]
  21. S. Gatz and J. Herrmann, “Soliton propagation in materials with saturable nonlinearity,” J. Opt. Soc. Am. B 8, 2296-2302 (1991).
    [Crossref]
  22. G. I. Stegeman, J. Ariyasu, C. T. Seaton, T. P. Chen, and J. V. Moloney, “Nonlinear thin-film guided-waves in non-Kerr media,” Appl. Phys. Lett. 47, 1254-1256 (1985).
    [Crossref]
  23. V. E. Wood, E. D. Evans, and R. P. Kenan, “Soluble saturable refractive-index model,” Opt. Commun. 59, 156-160 (1988).
    [Crossref]

2008 (4)

2006 (2)

P. T. Dinda, C. M. Ngabireng, K. Porsezian, and B. Kalithasan, “Modulational instability in optical fibers with arbitrary higher-order dispersion and delayed Raman response,” Opt. Commun. 266, 142-150 (2006).
[Crossref]

M. Stepic, C. E. Ruter, D. Kip, A. Maluckov, and L. Hadzvievski, “Modulational instability in one-dimensional saturable waveguide arrays: comparison with Kerr nonlinearity,” Opt. Commun. 267, 229-235 (2006).
[Crossref]

2005 (1)

M. Nurhuda and E. van Groesen, “Effects of delayed Kerr nonlinearity and ionization on the filamentary ultrashort laser pulses in air,” Phys. Rev. E 71, 066502 (2005).
[Crossref]

2003 (3)

S. Pitois and G. Millot, “Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber,” Opt. Commun. 226, 415-422 (2003).
[Crossref]

S. C. Wen, W. H. Su, H. Zhang, X. Q. Fu, L. J. Qian, and D. Y. Fan, “Influence of higher-order dispersions and Raman delayed response on modulation instability in microstructured fibres,” Chin. Phys. Lett. 20, 852-854 (2003).
[Crossref]

Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stabilization of one-dimensional periodic waves by saturation of the nonlinear response,” Phys. Rev. E 68, 065605(R) (2003).
[Crossref]

2002 (1)

1996 (1)

S. B. Cavalcanti and M. L. Lyra, “Modulational instability of ultrashort pulses via a generalized nonlinear Schrödinger equation with deviating argument,” Phys. Lett. A 211, 276-280 (1996).
[Crossref]

1994 (2)

1991 (1)

1989 (1)

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 24, 2665-2673 (1989).
[Crossref]

1988 (1)

V. E. Wood, E. D. Evans, and R. P. Kenan, “Soluble saturable refractive-index model,” Opt. Commun. 59, 156-160 (1988).
[Crossref]

1987 (1)

1986 (1)

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, “Soliton stability in plasmas and hydrodynamics,” Phys. Rep. 142, 103-165 (1986).
[Crossref]

1985 (1)

G. I. Stegeman, J. Ariyasu, C. T. Seaton, T. P. Chen, and J. V. Moloney, “Nonlinear thin-film guided-waves in non-Kerr media,” Appl. Phys. Lett. 47, 1254-1256 (1985).
[Crossref]

1973 (2)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142-144 (1973).
[Crossref]

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171-172 (1973).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

Ambomo, S.

Ariyasu, J.

G. I. Stegeman, J. Ariyasu, C. T. Seaton, T. P. Chen, and J. V. Moloney, “Nonlinear thin-film guided-waves in non-Kerr media,” Appl. Phys. Lett. 47, 1254-1256 (1985).
[Crossref]

Blow, K. J.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 24, 2665-2673 (1989).
[Crossref]

Cambournac, C.

Cavalcanti, S. B.

S. B. Cavalcanti and M. L. Lyra, “Modulational instability of ultrashort pulses via a generalized nonlinear Schrödinger equation with deviating argument,” Phys. Lett. A 211, 276-280 (1996).
[Crossref]

Chauvet, M.

Chen, T. P.

G. I. Stegeman, J. Ariyasu, C. T. Seaton, T. P. Chen, and J. V. Moloney, “Nonlinear thin-film guided-waves in non-Kerr media,” Appl. Phys. Lett. 47, 1254-1256 (1985).
[Crossref]

Dinda, P. T.

P. T. Dinda, C. M. Ngabireng, K. Porsezian, and B. Kalithasan, “Modulational instability in optical fibers with arbitrary higher-order dispersion and delayed Raman response,” Opt. Commun. 266, 142-150 (2006).
[Crossref]

Dudley, J. M.

Egorov, A. A.

Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stabilization of one-dimensional periodic waves by saturation of the nonlinear response,” Phys. Rev. E 68, 065605(R) (2003).
[Crossref]

Evans, E. D.

V. E. Wood, E. D. Evans, and R. P. Kenan, “Soluble saturable refractive-index model,” Opt. Commun. 59, 156-160 (1988).
[Crossref]

Fan, D. Y.

S. C. Wen, W. H. Su, H. Zhang, X. Q. Fu, L. J. Qian, and D. Y. Fan, “Influence of higher-order dispersions and Raman delayed response on modulation instability in microstructured fibres,” Chin. Phys. Lett. 20, 852-854 (2003).
[Crossref]

Fu, X. Q.

S. C. Wen, W. H. Su, H. Zhang, X. Q. Fu, L. J. Qian, and D. Y. Fan, “Influence of higher-order dispersions and Raman delayed response on modulation instability in microstructured fibres,” Chin. Phys. Lett. 20, 852-854 (2003).
[Crossref]

Gatz, S.

Golovchenko, E. A.

Gouveia Neto, A. S.

M. L. Lyra and A. S. Gouveia Neto, “Saturation effects on modulational instability in non-Kerr-like monomode optical fibers,” Opt. Commun. 108, 117-120 (1994).
[Crossref]

Hadzievski, Lj.

A. Maluckov, Lj. Hadzievski, N. Lazarides, and G. P. Tsironis, “Left-handed metamaterials with saturable nonlinearity,” Phys. Rev. E 77, 046607 (2008).
[Crossref]

Hadzvievski, L.

M. Stepic, C. E. Ruter, D. Kip, A. Maluckov, and L. Hadzvievski, “Modulational instability in one-dimensional saturable waveguide arrays: comparison with Kerr nonlinearity,” Opt. Commun. 267, 229-235 (2006).
[Crossref]

Hasegawa, A.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142-144 (1973).
[Crossref]

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171-172 (1973).
[Crossref]

Haus, J. W.

X. Liu, J. W. Haus, and S. M. Shahriar, “Modulation instability for a relaxational Kerr medium,” Opt. Commun. 281, 2907-2912 (2008).
[Crossref]

Herrmann, J.

Jisha, C. P.

Kalithasan, B.

S. Ambomo, C. M. Ngabireng, P. Tchofo Dinda, A. Labruyère, K. Porsezian, and B. Kalithasan, “Critical behavior with dramatic enhancement of modulational instability gain in fiber systems with periodic variation dispersion,” J. Opt. Soc. Am. B 25, 425-433 (2008).
[Crossref]

P. T. Dinda, C. M. Ngabireng, K. Porsezian, and B. Kalithasan, “Modulational instability in optical fibers with arbitrary higher-order dispersion and delayed Raman response,” Opt. Commun. 266, 142-150 (2006).
[Crossref]

Kartashov, Y. V.

Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stabilization of one-dimensional periodic waves by saturation of the nonlinear response,” Phys. Rev. E 68, 065605(R) (2003).
[Crossref]

Kenan, R. P.

V. E. Wood, E. D. Evans, and R. P. Kenan, “Soluble saturable refractive-index model,” Opt. Commun. 59, 156-160 (1988).
[Crossref]

Kip, D.

M. Stepic, C. E. Ruter, D. Kip, A. Maluckov, and L. Hadzvievski, “Modulational instability in one-dimensional saturable waveguide arrays: comparison with Kerr nonlinearity,” Opt. Commun. 267, 229-235 (2006).
[Crossref]

Kuriakose, V. C.

Kuznetsov, E. A.

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, “Soliton stability in plasmas and hydrodynamics,” Phys. Rep. 142, 103-165 (1986).
[Crossref]

Labruyère, A.

Lantz, E.

Lazarides, N.

A. Maluckov, Lj. Hadzievski, N. Lazarides, and G. P. Tsironis, “Left-handed metamaterials with saturable nonlinearity,” Phys. Rev. E 77, 046607 (2008).
[Crossref]

Liu, X.

X. Liu, J. W. Haus, and S. M. Shahriar, “Modulation instability for a relaxational Kerr medium,” Opt. Commun. 281, 2907-2912 (2008).
[Crossref]

Lyra, M. L.

S. B. Cavalcanti and M. L. Lyra, “Modulational instability of ultrashort pulses via a generalized nonlinear Schrödinger equation with deviating argument,” Phys. Lett. A 211, 276-280 (1996).
[Crossref]

M. L. Lyra and A. S. Gouveia Neto, “Saturation effects on modulational instability in non-Kerr-like monomode optical fibers,” Opt. Commun. 108, 117-120 (1994).
[Crossref]

Maillotte, H.

Maluckov, A.

A. Maluckov, Lj. Hadzievski, N. Lazarides, and G. P. Tsironis, “Left-handed metamaterials with saturable nonlinearity,” Phys. Rev. E 77, 046607 (2008).
[Crossref]

M. Stepic, C. E. Ruter, D. Kip, A. Maluckov, and L. Hadzvievski, “Modulational instability in one-dimensional saturable waveguide arrays: comparison with Kerr nonlinearity,” Opt. Commun. 267, 229-235 (2006).
[Crossref]

Millot, G.

S. Pitois and G. Millot, “Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber,” Opt. Commun. 226, 415-422 (2003).
[Crossref]

Moloney, J. V.

G. I. Stegeman, J. Ariyasu, C. T. Seaton, T. P. Chen, and J. V. Moloney, “Nonlinear thin-film guided-waves in non-Kerr media,” Appl. Phys. Lett. 47, 1254-1256 (1985).
[Crossref]

Ngabireng, C. M.

S. Ambomo, C. M. Ngabireng, P. Tchofo Dinda, A. Labruyère, K. Porsezian, and B. Kalithasan, “Critical behavior with dramatic enhancement of modulational instability gain in fiber systems with periodic variation dispersion,” J. Opt. Soc. Am. B 25, 425-433 (2008).
[Crossref]

P. T. Dinda, C. M. Ngabireng, K. Porsezian, and B. Kalithasan, “Modulational instability in optical fibers with arbitrary higher-order dispersion and delayed Raman response,” Opt. Commun. 266, 142-150 (2006).
[Crossref]

Nurhuda, M.

M. Nurhuda and E. van Groesen, “Effects of delayed Kerr nonlinearity and ionization on the filamentary ultrashort laser pulses in air,” Phys. Rev. E 71, 066502 (2005).
[Crossref]

Pilipetskii, A. N.

Pitois, S.

S. Pitois and G. Millot, “Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber,” Opt. Commun. 226, 415-422 (2003).
[Crossref]

Porsezian, K.

Potasek, M. J.

Qian, L. J.

S. C. Wen, W. H. Su, H. Zhang, X. Q. Fu, L. J. Qian, and D. Y. Fan, “Influence of higher-order dispersions and Raman delayed response on modulation instability in microstructured fibres,” Chin. Phys. Lett. 20, 852-854 (2003).
[Crossref]

Rubenchik, A. M.

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, “Soliton stability in plasmas and hydrodynamics,” Phys. Rep. 142, 103-165 (1986).
[Crossref]

Ruter, C. E.

M. Stepic, C. E. Ruter, D. Kip, A. Maluckov, and L. Hadzvievski, “Modulational instability in one-dimensional saturable waveguide arrays: comparison with Kerr nonlinearity,” Opt. Commun. 267, 229-235 (2006).
[Crossref]

Seaton, C. T.

G. I. Stegeman, J. Ariyasu, C. T. Seaton, T. P. Chen, and J. V. Moloney, “Nonlinear thin-film guided-waves in non-Kerr media,” Appl. Phys. Lett. 47, 1254-1256 (1985).
[Crossref]

Shahriar, S. M.

X. Liu, J. W. Haus, and S. M. Shahriar, “Modulation instability for a relaxational Kerr medium,” Opt. Commun. 281, 2907-2912 (2008).
[Crossref]

Stegeman, G. I.

G. I. Stegeman, J. Ariyasu, C. T. Seaton, T. P. Chen, and J. V. Moloney, “Nonlinear thin-film guided-waves in non-Kerr media,” Appl. Phys. Lett. 47, 1254-1256 (1985).
[Crossref]

Stepic, M.

M. Stepic, C. E. Ruter, D. Kip, A. Maluckov, and L. Hadzvievski, “Modulational instability in one-dimensional saturable waveguide arrays: comparison with Kerr nonlinearity,” Opt. Commun. 267, 229-235 (2006).
[Crossref]

Su, W. H.

S. C. Wen, W. H. Su, H. Zhang, X. Q. Fu, L. J. Qian, and D. Y. Fan, “Influence of higher-order dispersions and Raman delayed response on modulation instability in microstructured fibres,” Chin. Phys. Lett. 20, 852-854 (2003).
[Crossref]

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171-172 (1973).
[Crossref]

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142-144 (1973).
[Crossref]

Tchofo Dinda, P.

Torner, L.

Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stabilization of one-dimensional periodic waves by saturation of the nonlinear response,” Phys. Rev. E 68, 065605(R) (2003).
[Crossref]

Tsironis, G. P.

A. Maluckov, Lj. Hadzievski, N. Lazarides, and G. P. Tsironis, “Left-handed metamaterials with saturable nonlinearity,” Phys. Rev. E 77, 046607 (2008).
[Crossref]

van Groesen, E.

M. Nurhuda and E. van Groesen, “Effects of delayed Kerr nonlinearity and ionization on the filamentary ultrashort laser pulses in air,” Phys. Rev. E 71, 066502 (2005).
[Crossref]

Vysloukh, V. A.

Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stabilization of one-dimensional periodic waves by saturation of the nonlinear response,” Phys. Rev. E 68, 065605(R) (2003).
[Crossref]

Wen, S. C.

S. C. Wen, W. H. Su, H. Zhang, X. Q. Fu, L. J. Qian, and D. Y. Fan, “Influence of higher-order dispersions and Raman delayed response on modulation instability in microstructured fibres,” Chin. Phys. Lett. 20, 852-854 (2003).
[Crossref]

Wood, D.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 24, 2665-2673 (1989).
[Crossref]

Wood, V. E.

V. E. Wood, E. D. Evans, and R. P. Kenan, “Soluble saturable refractive-index model,” Opt. Commun. 59, 156-160 (1988).
[Crossref]

Zakharov, V. E.

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, “Soliton stability in plasmas and hydrodynamics,” Phys. Rep. 142, 103-165 (1986).
[Crossref]

Zelenina, A. S.

Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stabilization of one-dimensional periodic waves by saturation of the nonlinear response,” Phys. Rev. E 68, 065605(R) (2003).
[Crossref]

Zhang, H.

S. C. Wen, W. H. Su, H. Zhang, X. Q. Fu, L. J. Qian, and D. Y. Fan, “Influence of higher-order dispersions and Raman delayed response on modulation instability in microstructured fibres,” Chin. Phys. Lett. 20, 852-854 (2003).
[Crossref]

Appl. Phys. Lett. (3)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142-144 (1973).
[Crossref]

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171-172 (1973).
[Crossref]

G. I. Stegeman, J. Ariyasu, C. T. Seaton, T. P. Chen, and J. V. Moloney, “Nonlinear thin-film guided-waves in non-Kerr media,” Appl. Phys. Lett. 47, 1254-1256 (1985).
[Crossref]

Chin. Phys. Lett. (1)

S. C. Wen, W. H. Su, H. Zhang, X. Q. Fu, L. J. Qian, and D. Y. Fan, “Influence of higher-order dispersions and Raman delayed response on modulation instability in microstructured fibres,” Chin. Phys. Lett. 20, 852-854 (2003).
[Crossref]

IEEE J. Quantum Electron. (1)

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 24, 2665-2673 (1989).
[Crossref]

J. Opt. Soc. Am. B (5)

Opt. Commun. (6)

V. E. Wood, E. D. Evans, and R. P. Kenan, “Soluble saturable refractive-index model,” Opt. Commun. 59, 156-160 (1988).
[Crossref]

S. Pitois and G. Millot, “Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber,” Opt. Commun. 226, 415-422 (2003).
[Crossref]

P. T. Dinda, C. M. Ngabireng, K. Porsezian, and B. Kalithasan, “Modulational instability in optical fibers with arbitrary higher-order dispersion and delayed Raman response,” Opt. Commun. 266, 142-150 (2006).
[Crossref]

M. L. Lyra and A. S. Gouveia Neto, “Saturation effects on modulational instability in non-Kerr-like monomode optical fibers,” Opt. Commun. 108, 117-120 (1994).
[Crossref]

X. Liu, J. W. Haus, and S. M. Shahriar, “Modulation instability for a relaxational Kerr medium,” Opt. Commun. 281, 2907-2912 (2008).
[Crossref]

M. Stepic, C. E. Ruter, D. Kip, A. Maluckov, and L. Hadzvievski, “Modulational instability in one-dimensional saturable waveguide arrays: comparison with Kerr nonlinearity,” Opt. Commun. 267, 229-235 (2006).
[Crossref]

Opt. Lett. (1)

Phys. Lett. A (1)

S. B. Cavalcanti and M. L. Lyra, “Modulational instability of ultrashort pulses via a generalized nonlinear Schrödinger equation with deviating argument,” Phys. Lett. A 211, 276-280 (1996).
[Crossref]

Phys. Rep. (1)

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, “Soliton stability in plasmas and hydrodynamics,” Phys. Rep. 142, 103-165 (1986).
[Crossref]

Phys. Rev. E (3)

A. Maluckov, Lj. Hadzievski, N. Lazarides, and G. P. Tsironis, “Left-handed metamaterials with saturable nonlinearity,” Phys. Rev. E 77, 046607 (2008).
[Crossref]

M. Nurhuda and E. van Groesen, “Effects of delayed Kerr nonlinearity and ionization on the filamentary ultrashort laser pulses in air,” Phys. Rev. E 71, 066502 (2005).
[Crossref]

Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stabilization of one-dimensional periodic waves by saturation of the nonlinear response,” Phys. Rev. E 68, 065605(R) (2003).
[Crossref]

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

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Figures (5)

Fig. 1
Fig. 1

Gain spectrum of modulational instability in waveguides with saturable delayed nonlinear response. The presence of a finite response time induces the emergence of a second instability band at large frequencies. The saturation of the nonlinearity reduces both the range of unstable frequencies and the instability gain.

Fig. 2
Fig. 2

Gain spectrum for a fixed saturation parameter and increasing response times. The second band moves to smaller frequencies when the response time is increased. The maximum gain becomes strongly reduced in slowly responding media for which the instantaneous and Raman bands becomes superposed.

Fig. 3
Fig. 3

Frequencies at which the gain spectrum reaches a maximum as a function of the response time τ for a fixed saturation parameter. (a) Γ = 0.1 , (b) Γ = 1 . For small values of τ the optimal frequency at the Raman band decreases as 1 τ , while it stays roughly constant at the instantaneous band. In the case of larger saturation, the optimal frequencies in both bands are smaller. After the coalescence of the bands, the optimal frequency continues to decrease with increasing response times.

Fig. 4
Fig. 4

Maximum gain at each instability band as a function of the response time τ and fixed saturation parameter. (a) Γ = 0.1 , (b) Γ = 1 . The maximum gains remain roughly constant for small values of τ, being twice larger at the instantaneous band in comparison with the Raman band. After the coalescence of the instability bands, the maximum gain develops a strong decay with increasing relaxation times. The instability gain is substantially reduced by saturation.

Fig. 5
Fig. 5

(a) Gain spectrum in the regime of normal dispersion for distinct response times and a fixed saturation parameter. In this case, there is a single instability band irrespective to the response regime. (b) Maximum gain (dashed curve) and respective frequency (solid curve) as a function of the response time. These quantities depict quite distinct dependences on the response time in the regimes of fast and slow relaxation.

Equations (17)

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i E ( z , t ) z = β 2 2 2 E ( z , t ) t 2 γ E ( z , t ) 2 E ( z , t ) ,
f ( Γ E 2 ) = Γ E 2 1 + Γ E 2 ,
i E z = ± 1 2 2 E t 2 + N E ,
N t = 1 τ ( N + f ̃ ( Γ , E 2 ) ) ,
E CW = E 0 e i f ̃ ( Γ E 0 2 ) z ,
N CW = f ̃ ( Γ , E 0 2 ) .
E = ( E 0 + e ( z , t ) ) e i f ̃ ( Γ E 0 2 ) z ,
N = f ̃ ( Γ , E 0 2 ) + n ( z , t ) ,
i e z = ± 1 2 2 e t 2 + n E 0 ,
n t = 1 τ [ n f ̃ ( Γ , E 0 2 ) + f ̃ ( Γ , E 0 + e 2 ) ] .
i e z = ± 1 2 2 e t 2 + n E 0 ,
n t = 1 τ [ n + Γ f ̃ ( Γ , E 0 2 ) E 0 ( e + e * ) ] ,
( k ± Ω 2 2 ) e ( Ω , k ) n ( Ω , k ) E 0 = 0 ,
( i Ω τ + 1 ) n ( Ω , k ) = f ̃ ( Γ , E 0 2 ) Γ E 0 ( e ( Ω , k ) + e * ( Ω , k ) ) ,
e ( Ω , k ) ( k ± Ω 2 2 ) = f ̃ ( Γ , E 0 2 ) Γ E 0 2 1 + i Ω τ ( e ( Ω , k ) + e * ( Ω , k ) ) .
e * ( Ω , k ) ( k ± Ω 2 2 ) = f ̃ ( Γ , E 0 2 ) Γ E 0 2 1 + i Ω τ ( e ( Ω , k ) + e * ( Ω , k ) ) .
k 2 = Ω 4 4 Ω 2 E 0 2 f ( Γ , E 0 2 ) 1 + Ω 2 τ 2 ( 1 i Ω τ ) ,

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