Abstract

Considering noninstantaneous Kerr nonlinearity, the propagation of a partially coherent optical beam is theoretically investigated by using extensions of the nonlinear Schrödinger equation (NLSE). In order to account for the partial coherence of the beam, a phase-diffusion model is used for the laser beam. To introduce the finite response time of the medium, a time-dependent nonlinear response is incorporated in the system of the NLSE using the Debye relaxation model. We analytically deduce the dispersion relation and numerically compute the gain spectra along with relevant second-order statistical quantities. A detailed study of how these statistical properties are influenced by the group velocity dispersion regime as well as by the delayed nonlinear response of the material is presented. The distinct features for slow and fast delayed nonlinear response are also emphasized.

© 2014 Optical Society of America

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  7. M. Centurion, M. A. Porter, Y. Pu, P. G. Kevrekidis, D. J. Frantzeskakis, and D. Psaltis, “Modulational instability in nonlinearity-managed optical media,” Phys. Rev. A 75, 063804 (2007).
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  9. M. Onorato, A. R. Osborne, and M. Serio, “Modulation instability in crossing sea states: a possible mechanism for the formation of freak waves,” Phys. Rev. Lett. 96, 014503 (2006).
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    [CrossRef]
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  27. B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
    [CrossRef]
  28. D. Kip, M. Soljacić, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
    [CrossRef]
  29. J. Klinger, H. Martin, and Z. Chen, “Experiments on induced modulational instability of an incoherent optical beam,” Opt. Lett. 26, 271–273 (2001).
    [CrossRef]
  30. A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
    [CrossRef]
  31. B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in photonic crystal fiber,” Phys. Rev. E 84, 066605 (2011).
    [CrossRef]
  32. A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: toward a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep., doi:10.1016/2014.03.002 (to be published).
    [CrossRef]
  33. C. Condit, M. A. Schmidt, P. St, J. Russel, and F. Biancalana, “Highly noninstantaneous solitons in liquid-core photonic crystal fibers,” Phys. Rev. Lett. 105, 263902 (2010).
    [CrossRef]

2013 (2)

J. Garnier, G. Xu, S. Trillo, and A. Picozzi, “Incoherent dispersive shocks in the spectral evolution of random waves,” Phys. Rev. Lett. 111, 113902 (2013).
[CrossRef]

K. Nithyanandan and K. Porsezian, “Interplay between relaxation of nonlinear response and coupling coefficient dispersion in the instability spectra of dual core optical fiber,” Opt. Commun. 303, 46–55 (2013).
[CrossRef]

2012 (1)

2011 (1)

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in photonic crystal fiber,” Phys. Rev. E 84, 066605 (2011).
[CrossRef]

2010 (2)

C. Condit, M. A. Schmidt, P. St, J. Russel, and F. Biancalana, “Highly noninstantaneous solitons in liquid-core photonic crystal fibers,” Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

A. A. Canabarro, B. Santos, I. Gleria, M. L. Lyra, and A. S. B. Sombra, “Interplay of XPM and nonlinear response time in the modulational instability of copropagating optical pulses,” J. Opt. Soc. Am. B 27, 1878–1885 (2010).
[CrossRef]

2009 (2)

G. L. da Silva, I. Gleria, M. L. Lyra, and A. S. B. Sombra, “Modulational instability in lossless fibers with saturable delayed nonlinear response,” J. Opt. Soc. Am. B 26, 183–188 (2009).
[CrossRef]

C.-S. Chou and M.-F. Shih, “Slow light achieved by non-instantaneous modulation instability,” J. Opt. A Pure Appl. Opt. 11, 105204 (2009).
[CrossRef]

2008 (4)

M. Stepić, A. Maluckov, M. Stojanović, F. Chen, and D. Kip, “Modulational instability and solitary waves in one-dimensional lattices with intensity-resonant nonlinearity,” Phys. Rev. A 78, 043819 (2008).
[CrossRef]

T. Hansson, D. Anderson, M. Lisak, V. E. Semenov, and U. Österberg, “Quasilinear evolution and saturation of the modulational instability of partially coherent optical waves,” Phys. Rev. A 78, 011807 (2008).
[CrossRef]

A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef]

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

2007 (2)

M. Centurion, M. A. Porter, Y. Pu, P. G. Kevrekidis, D. J. Frantzeskakis, and D. Psaltis, “Modulational instability in nonlinearity-managed optical media,” Phys. Rev. A 75, 063804 (2007).
[CrossRef]

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef]

2006 (2)

M. Onorato, A. R. Osborne, and M. Serio, “Modulation instability in crossing sea states: a possible mechanism for the formation of freak waves,” Phys. Rev. Lett. 96, 014503 (2006).
[CrossRef]

J. Fatome, S. Pitois, and G. Millot, “Measurement of nonlinear and chromatic dispersion parameters of optical fibers using modulation instability,” Opt. Fiber Technol. 12, 243–250 (2006).
[CrossRef]

2004 (1)

L. D. Carr and J. Brand, “Spontaneous soliton formation and modulational instability in Bose–Einstein condensates,” Phys. Rev. Lett. 92, 040401 (2004).
[CrossRef]

2003 (2)

I. Velchev, R. Pattnaik, and J. Toulouse, “Two-beam modulation instability in noninstantaneous nonlinear media,” Phys. Rev. Lett. 91, 093905 (2003).
[CrossRef]

Y. D. Gong, P. Shum, D. Tang, C. Lu, and X. Guo, “660  GHz solitons source based on modulation instability in short cavity,” Opt. Express 11, 2480–2485 (2003).
[CrossRef]

2002 (3)

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]

V. V. Konotop and M. Salerno, “Modulational instability in Bose–Einstein condensates in optical lattices,” Phys. Rev. A 65, 021602 (2002).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
[CrossRef]

2001 (1)

2000 (2)

M. Soljacić, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
[CrossRef]

D. Kip, M. Soljacić, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[CrossRef]

1996 (1)

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef]

1995 (2)

S. B. Cavalcanti, M. Yu, and G. P. Agrawal, “Noise amplification in dispersive nonlinear media,” Phys. Rev. A 51, 4086–4092 (1995).
[CrossRef]

N. Da Dalt, C. De Angelis, G. F. Nalesso, and M. Santagiustina, “Dynamics of induced modulational instability in waveguides with saturable nonlinearity,” Opt. Commun. 121, 69–72 (1995).
[CrossRef]

1989 (1)

1987 (1)

Agrawal, G. P.

S. B. Cavalcanti, M. Yu, and G. P. Agrawal, “Noise amplification in dispersive nonlinear media,” Phys. Rev. A 51, 4086–4092 (1995).
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

Anderson, D.

T. Hansson, D. Anderson, M. Lisak, V. E. Semenov, and U. Österberg, “Quasilinear evolution and saturation of the modulational instability of partially coherent optical waves,” Phys. Rev. A 78, 011807 (2008).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Barviau, B.

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in photonic crystal fiber,” Phys. Rev. E 84, 066605 (2011).
[CrossRef]

Biancalana, F.

C. Condit, M. A. Schmidt, P. St, J. Russel, and F. Biancalana, “Highly noninstantaneous solitons in liquid-core photonic crystal fibers,” Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

Brand, J.

L. D. Carr and J. Brand, “Spontaneous soliton formation and modulational instability in Bose–Einstein condensates,” Phys. Rev. Lett. 92, 040401 (2004).
[CrossRef]

Cambournac, C.

Canabarro, A. A.

Carbou, G.

G. Carbou and B. Hanouzet, “Relaxation approximation of the Kerr model for the impedance initial-boundary value problem,” in Proceedings of the 6th AIMS International Conference (2007), paper 212220.

Carr, L. D.

L. D. Carr and J. Brand, “Spontaneous soliton formation and modulational instability in Bose–Einstein condensates,” Phys. Rev. Lett. 92, 040401 (2004).
[CrossRef]

Cavalcanti, S. B.

S. B. Cavalcanti, M. Yu, and G. P. Agrawal, “Noise amplification in dispersive nonlinear media,” Phys. Rev. A 51, 4086–4092 (1995).
[CrossRef]

Centurion, M.

M. Centurion, M. A. Porter, Y. Pu, P. G. Kevrekidis, D. J. Frantzeskakis, and D. Psaltis, “Modulational instability in nonlinearity-managed optical media,” Phys. Rev. A 75, 063804 (2007).
[CrossRef]

Chauvet, M.

Chen, F.

M. Stepić, A. Maluckov, M. Stojanović, F. Chen, and D. Kip, “Modulational instability and solitary waves in one-dimensional lattices with intensity-resonant nonlinearity,” Phys. Rev. A 78, 043819 (2008).
[CrossRef]

Chen, Z.

J. Klinger, H. Martin, and Z. Chen, “Experiments on induced modulational instability of an incoherent optical beam,” Opt. Lett. 26, 271–273 (2001).
[CrossRef]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef]

Chou, C.-S.

C.-S. Chou and M.-F. Shih, “Slow light achieved by non-instantaneous modulation instability,” J. Opt. A Pure Appl. Opt. 11, 105204 (2009).
[CrossRef]

Christodoulides, D. N.

D. Kip, M. Soljacić, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[CrossRef]

M. Soljacić, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
[CrossRef]

A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: toward a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep., doi:10.1016/2014.03.002 (to be published).
[CrossRef]

Condit, C.

C. Condit, M. A. Schmidt, P. St, J. Russel, and F. Biancalana, “Highly noninstantaneous solitons in liquid-core photonic crystal fibers,” Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

Coskun, T.

M. Soljacić, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
[CrossRef]

Da Dalt, N.

N. Da Dalt, C. De Angelis, G. F. Nalesso, and M. Santagiustina, “Dynamics of induced modulational instability in waveguides with saturable nonlinearity,” Opt. Commun. 121, 69–72 (1995).
[CrossRef]

da Silva, G. L.

De Angelis, C.

N. Da Dalt, C. De Angelis, G. F. Nalesso, and M. Santagiustina, “Dynamics of induced modulational instability in waveguides with saturable nonlinearity,” Opt. Commun. 121, 69–72 (1995).
[CrossRef]

Dudley, J. M.

Eugenieva, E.

D. Kip, M. Soljacić, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[CrossRef]

Fatome, J.

J. Fatome, S. Pitois, and G. Millot, “Measurement of nonlinear and chromatic dispersion parameters of optical fibers using modulation instability,” Opt. Fiber Technol. 12, 243–250 (2006).
[CrossRef]

Fedele, R.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Frantzeskakis, D. J.

M. Centurion, M. A. Porter, Y. Pu, P. G. Kevrekidis, D. J. Frantzeskakis, and D. Psaltis, “Modulational instability in nonlinearity-managed optical media,” Phys. Rev. A 75, 063804 (2007).
[CrossRef]

Garnier, J.

J. Garnier, G. Xu, S. Trillo, and A. Picozzi, “Incoherent dispersive shocks in the spectral evolution of random waves,” Phys. Rev. Lett. 111, 113902 (2013).
[CrossRef]

B. Kliber, C. Michel, J. Garnier, and A. Picozzi, “Temporal dynamics of incoherent waves in noninstantaneous response nonlinear Kerr media,” Opt. Lett. 37, 2472–2474 (2012).
[CrossRef]

A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: toward a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep., doi:10.1016/2014.03.002 (to be published).
[CrossRef]

Gleria, I.

Gong, Y. D.

Guo, X.

Hall, B.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Hanouzet, B.

G. Carbou and B. Hanouzet, “Relaxation approximation of the Kerr model for the impedance initial-boundary value problem,” in Proceedings of the 6th AIMS International Conference (2007), paper 212220.

Hansson, T.

T. Hansson, D. Anderson, M. Lisak, V. E. Semenov, and U. Österberg, “Quasilinear evolution and saturation of the modulational instability of partially coherent optical waves,” Phys. Rev. A 78, 011807 (2008).
[CrossRef]

A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: toward a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep., doi:10.1016/2014.03.002 (to be published).
[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]

Jalali, B.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef]

Kevrekidis, P. G.

M. Centurion, M. A. Porter, Y. Pu, P. G. Kevrekidis, D. J. Frantzeskakis, and D. Psaltis, “Modulational instability in nonlinearity-managed optical media,” Phys. Rev. A 75, 063804 (2007).
[CrossRef]

Kibler, B.

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in photonic crystal fiber,” Phys. Rev. E 84, 066605 (2011).
[CrossRef]

Kip, D.

M. Stepić, A. Maluckov, M. Stojanović, F. Chen, and D. Kip, “Modulational instability and solitary waves in one-dimensional lattices with intensity-resonant nonlinearity,” Phys. Rev. A 78, 043819 (2008).
[CrossRef]

D. Kip, M. Soljacić, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[CrossRef]

Kliber, B.

Klinger, J.

Konotop, V. V.

V. V. Konotop and M. Salerno, “Modulational instability in Bose–Einstein condensates in optical lattices,” Phys. Rev. A 65, 021602 (2002).
[CrossRef]

Koonath, P.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef]

Kudlinski, A.

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in photonic crystal fiber,” Phys. Rev. E 84, 066605 (2011).
[CrossRef]

Lantz, E.

Lisak, M.

T. Hansson, D. Anderson, M. Lisak, V. E. Semenov, and U. Österberg, “Quasilinear evolution and saturation of the modulational instability of partially coherent optical waves,” Phys. Rev. A 78, 011807 (2008).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
[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]

Lu, C.

Lyra, M. L.

Maillotte, H.

Maluckov, A.

M. Stepić, A. Maluckov, M. Stojanović, F. Chen, and D. Kip, “Modulational instability and solitary waves in one-dimensional lattices with intensity-resonant nonlinearity,” Phys. Rev. A 78, 043819 (2008).
[CrossRef]

Martin, H.

Michel, C.

B. Kliber, C. Michel, J. Garnier, and A. Picozzi, “Temporal dynamics of incoherent waves in noninstantaneous response nonlinear Kerr media,” Opt. Lett. 37, 2472–2474 (2012).
[CrossRef]

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in photonic crystal fiber,” Phys. Rev. E 84, 066605 (2011).
[CrossRef]

Millot, G.

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in photonic crystal fiber,” Phys. Rev. E 84, 066605 (2011).
[CrossRef]

A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef]

J. Fatome, S. Pitois, and G. Millot, “Measurement of nonlinear and chromatic dispersion parameters of optical fibers using modulation instability,” Opt. Fiber Technol. 12, 243–250 (2006).
[CrossRef]

A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: toward a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep., doi:10.1016/2014.03.002 (to be published).
[CrossRef]

Mitchell, M.

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef]

Nalesso, G. F.

N. Da Dalt, C. De Angelis, G. F. Nalesso, and M. Santagiustina, “Dynamics of induced modulational instability in waveguides with saturable nonlinearity,” Opt. Commun. 121, 69–72 (1995).
[CrossRef]

Nithyanandan, K.

K. Nithyanandan and K. Porsezian, “Interplay between relaxation of nonlinear response and coupling coefficient dispersion in the instability spectra of dual core optical fiber,” Opt. Commun. 303, 46–55 (2013).
[CrossRef]

Onorato, M.

M. Onorato, A. R. Osborne, and M. Serio, “Modulation instability in crossing sea states: a possible mechanism for the formation of freak waves,” Phys. Rev. Lett. 96, 014503 (2006).
[CrossRef]

Osborne, A. R.

M. Onorato, A. R. Osborne, and M. Serio, “Modulation instability in crossing sea states: a possible mechanism for the formation of freak waves,” Phys. Rev. Lett. 96, 014503 (2006).
[CrossRef]

Österberg, U.

T. Hansson, D. Anderson, M. Lisak, V. E. Semenov, and U. Österberg, “Quasilinear evolution and saturation of the modulational instability of partially coherent optical waves,” Phys. Rev. A 78, 011807 (2008).
[CrossRef]

Pattnaik, R.

I. Velchev, R. Pattnaik, and J. Toulouse, “Two-beam modulation instability in noninstantaneous nonlinear media,” Phys. Rev. Lett. 91, 093905 (2003).
[CrossRef]

Picozzi, A.

J. Garnier, G. Xu, S. Trillo, and A. Picozzi, “Incoherent dispersive shocks in the spectral evolution of random waves,” Phys. Rev. Lett. 111, 113902 (2013).
[CrossRef]

B. Kliber, C. Michel, J. Garnier, and A. Picozzi, “Temporal dynamics of incoherent waves in noninstantaneous response nonlinear Kerr media,” Opt. Lett. 37, 2472–2474 (2012).
[CrossRef]

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in photonic crystal fiber,” Phys. Rev. E 84, 066605 (2011).
[CrossRef]

A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef]

A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: toward a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep., doi:10.1016/2014.03.002 (to be published).
[CrossRef]

Pitois, S.

A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef]

J. Fatome, S. Pitois, and G. Millot, “Measurement of nonlinear and chromatic dispersion parameters of optical fibers using modulation instability,” Opt. Fiber Technol. 12, 243–250 (2006).
[CrossRef]

Porsezian, K.

K. Nithyanandan and K. Porsezian, “Interplay between relaxation of nonlinear response and coupling coefficient dispersion in the instability spectra of dual core optical fiber,” Opt. Commun. 303, 46–55 (2013).
[CrossRef]

Porter, M. A.

M. Centurion, M. A. Porter, Y. Pu, P. G. Kevrekidis, D. J. Frantzeskakis, and D. Psaltis, “Modulational instability in nonlinearity-managed optical media,” Phys. Rev. A 75, 063804 (2007).
[CrossRef]

Potasek, M. J.

Psaltis, D.

M. Centurion, M. A. Porter, Y. Pu, P. G. Kevrekidis, D. J. Frantzeskakis, and D. Psaltis, “Modulational instability in nonlinearity-managed optical media,” Phys. Rev. A 75, 063804 (2007).
[CrossRef]

Pu, Y.

M. Centurion, M. A. Porter, Y. Pu, P. G. Kevrekidis, D. J. Frantzeskakis, and D. Psaltis, “Modulational instability in nonlinearity-managed optical media,” Phys. Rev. A 75, 063804 (2007).
[CrossRef]

Randoux, S.

A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: toward a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep., doi:10.1016/2014.03.002 (to be published).
[CrossRef]

Ropers, C.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef]

Russel, J.

C. Condit, M. A. Schmidt, P. St, J. Russel, and F. Biancalana, “Highly noninstantaneous solitons in liquid-core photonic crystal fibers,” Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

Salerno, M.

V. V. Konotop and M. Salerno, “Modulational instability in Bose–Einstein condensates in optical lattices,” Phys. Rev. A 65, 021602 (2002).
[CrossRef]

Santagiustina, M.

N. Da Dalt, C. De Angelis, G. F. Nalesso, and M. Santagiustina, “Dynamics of induced modulational instability in waveguides with saturable nonlinearity,” Opt. Commun. 121, 69–72 (1995).
[CrossRef]

Santos, B.

Schmidt, M. A.

C. Condit, M. A. Schmidt, P. St, J. Russel, and F. Biancalana, “Highly noninstantaneous solitons in liquid-core photonic crystal fibers,” Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

Segev, M.

M. Soljacić, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
[CrossRef]

D. Kip, M. Soljacić, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[CrossRef]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef]

Semenov, V. E.

T. Hansson, D. Anderson, M. Lisak, V. E. Semenov, and U. Österberg, “Quasilinear evolution and saturation of the modulational instability of partially coherent optical waves,” Phys. Rev. A 78, 011807 (2008).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Serio, M.

M. Onorato, A. R. Osborne, and M. Serio, “Modulation instability in crossing sea states: a possible mechanism for the formation of freak waves,” Phys. Rev. Lett. 96, 014503 (2006).
[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]

Shih, M.

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef]

Shih, M.-F.

C.-S. Chou and M.-F. Shih, “Slow light achieved by non-instantaneous modulation instability,” J. Opt. A Pure Appl. Opt. 11, 105204 (2009).
[CrossRef]

Shum, P.

Soljacic, M.

D. Kip, M. Soljacić, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[CrossRef]

M. Soljacić, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
[CrossRef]

Solli, D. R.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef]

Sombra, A. S. B.

St, P.

C. Condit, M. A. Schmidt, P. St, J. Russel, and F. Biancalana, “Highly noninstantaneous solitons in liquid-core photonic crystal fibers,” Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

Stegeman, G. I.

Stepic, M.

M. Stepić, A. Maluckov, M. Stojanović, F. Chen, and D. Kip, “Modulational instability and solitary waves in one-dimensional lattices with intensity-resonant nonlinearity,” Phys. Rev. A 78, 043819 (2008).
[CrossRef]

Stojanovic, M.

M. Stepić, A. Maluckov, M. Stojanović, F. Chen, and D. Kip, “Modulational instability and solitary waves in one-dimensional lattices with intensity-resonant nonlinearity,” Phys. Rev. A 78, 043819 (2008).
[CrossRef]

Suret, P.

A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: toward a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep., doi:10.1016/2014.03.002 (to be published).
[CrossRef]

Tang, D.

Toulouse, J.

I. Velchev, R. Pattnaik, and J. Toulouse, “Two-beam modulation instability in noninstantaneous nonlinear media,” Phys. Rev. Lett. 91, 093905 (2003).
[CrossRef]

Trillo, S.

Velchev, I.

I. Velchev, R. Pattnaik, and J. Toulouse, “Two-beam modulation instability in noninstantaneous nonlinear media,” Phys. Rev. Lett. 91, 093905 (2003).
[CrossRef]

Vishwanath, A.

M. Soljacić, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
[CrossRef]

Wabnitz, S.

Wright, E. M.

Xu, G.

J. Garnier, G. Xu, S. Trillo, and A. Picozzi, “Incoherent dispersive shocks in the spectral evolution of random waves,” Phys. Rev. Lett. 111, 113902 (2013).
[CrossRef]

Yu, M.

S. B. Cavalcanti, M. Yu, and G. P. Agrawal, “Noise amplification in dispersive nonlinear media,” Phys. Rev. A 51, 4086–4092 (1995).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

C.-S. Chou and M.-F. Shih, “Slow light achieved by non-instantaneous modulation instability,” J. Opt. A Pure Appl. Opt. 11, 105204 (2009).
[CrossRef]

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

Nature (1)

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef]

Opt. Commun. (3)

N. Da Dalt, C. De Angelis, G. F. Nalesso, and M. Santagiustina, “Dynamics of induced modulational instability in waveguides with saturable nonlinearity,” Opt. Commun. 121, 69–72 (1995).
[CrossRef]

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

K. Nithyanandan and K. Porsezian, “Interplay between relaxation of nonlinear response and coupling coefficient dispersion in the instability spectra of dual core optical fiber,” Opt. Commun. 303, 46–55 (2013).
[CrossRef]

Opt. Express (1)

Opt. Fiber Technol. (1)

J. Fatome, S. Pitois, and G. Millot, “Measurement of nonlinear and chromatic dispersion parameters of optical fibers using modulation instability,” Opt. Fiber Technol. 12, 243–250 (2006).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (5)

T. Hansson, D. Anderson, M. Lisak, V. E. Semenov, and U. Österberg, “Quasilinear evolution and saturation of the modulational instability of partially coherent optical waves,” Phys. Rev. A 78, 011807 (2008).
[CrossRef]

S. B. Cavalcanti, M. Yu, and G. P. Agrawal, “Noise amplification in dispersive nonlinear media,” Phys. Rev. A 51, 4086–4092 (1995).
[CrossRef]

M. Centurion, M. A. Porter, Y. Pu, P. G. Kevrekidis, D. J. Frantzeskakis, and D. Psaltis, “Modulational instability in nonlinearity-managed optical media,” Phys. Rev. A 75, 063804 (2007).
[CrossRef]

M. Stepić, A. Maluckov, M. Stojanović, F. Chen, and D. Kip, “Modulational instability and solitary waves in one-dimensional lattices with intensity-resonant nonlinearity,” Phys. Rev. A 78, 043819 (2008).
[CrossRef]

V. V. Konotop and M. Salerno, “Modulational instability in Bose–Einstein condensates in optical lattices,” Phys. Rev. A 65, 021602 (2002).
[CrossRef]

Phys. Rev. E (2)

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
[CrossRef]

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in photonic crystal fiber,” Phys. Rev. E 84, 066605 (2011).
[CrossRef]

Phys. Rev. Lett. (8)

A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef]

C. Condit, M. A. Schmidt, P. St, J. Russel, and F. Biancalana, “Highly noninstantaneous solitons in liquid-core photonic crystal fibers,” Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

L. D. Carr and J. Brand, “Spontaneous soliton formation and modulational instability in Bose–Einstein condensates,” Phys. Rev. Lett. 92, 040401 (2004).
[CrossRef]

M. Onorato, A. R. Osborne, and M. Serio, “Modulation instability in crossing sea states: a possible mechanism for the formation of freak waves,” Phys. Rev. Lett. 96, 014503 (2006).
[CrossRef]

I. Velchev, R. Pattnaik, and J. Toulouse, “Two-beam modulation instability in noninstantaneous nonlinear media,” Phys. Rev. Lett. 91, 093905 (2003).
[CrossRef]

J. Garnier, G. Xu, S. Trillo, and A. Picozzi, “Incoherent dispersive shocks in the spectral evolution of random waves,” Phys. Rev. Lett. 111, 113902 (2013).
[CrossRef]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef]

M. Soljacić, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
[CrossRef]

Science (1)

D. Kip, M. Soljacić, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[CrossRef]

Other (3)

A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: toward a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep., doi:10.1016/2014.03.002 (to be published).
[CrossRef]

G. Carbou and B. Hanouzet, “Relaxation approximation of the Kerr model for the impedance initial-boundary value problem,” in Proceedings of the 6th AIMS International Conference (2007), paper 212220.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

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Figures (12)

Fig. 1.
Fig. 1.

Gain spectra for several values of the finite response time τ for β2=0.06ps2/m (normal GVD regime) and fast responding medium.

Fig. 2.
Fig. 2.

Gain spectra for several values of the finite response time τ for β2=0.06ps2/m (normal GVD regime) and slow responding medium.

Fig. 3.
Fig. 3.

Maximum gain as a function of frequency Ω and nonlinear response time τ for normal GVD regime.

Fig. 4.
Fig. 4.

Phase variance as a function of time for some values of the delayed nonlinear response time for β2=0.06ps2/m: (a) Δν=0.008THz and (b) Δν=0.08THz.

Fig. 5.
Fig. 5.

Coherence function as a function of time for some values of the delayed nonlinear response time for β2=0.06ps2/m: (a) Δν=0.008THz and (b) Δν=0.08THz.

Fig. 6.
Fig. 6.

Spectral line shapes corresponding to the coherence functions displayed in Fig. 5: (a) Δν=0.008THz and (b) Δν=0.08THz.

Fig. 7.
Fig. 7.

Gain spectra for several values of the finite response time τ for β2=0.06ps2/m (anomalous GVD regime). Fast responding medium.

Fig. 8.
Fig. 8.

Gain spectra for several values of the finite response time τ for β2=0.06ps2/m (anomalous GVD regime). Slow responding medium.

Fig. 9.
Fig. 9.

Maximum gain as a function of frequency Ω and nonlinear response time τ for anomalous GVD regime.

Fig. 10.
Fig. 10.

Phase variance as a function of time for some values of the delayed nonlinear response time for anomalous GVD regime.

Fig. 11.
Fig. 11.

Coherence function as a function of time for some values of the delayed nonlinear response time for anomalous GVD regime.

Fig. 12.
Fig. 12.

Spectral line shapes corresponding to the coherence functions displayed in Fig. 11: (a) Δν=0.008THz and (b) Δν=0.08THz for β2=0.06ps2/m.

Equations (23)

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

iEz=12β22Et2γ|E|2E,
P|E|2E,
PχE,
χt=1τχ+1τ|E|2.
iEz=12β22Et2NE,Nt=1τ(N+γ|E|2),
E(z,t)=[E0+δE(z,t)]ei(Φ0+δΦ(z,t)),N(z,t)=N0+δN(z,t),
Γ(z,t)=E*(z,0)E(z,t),
S(z,Ω)=Γ(z,t)eiΩtdt.
Γ(z,t)|E0|2eΔΦ22,
ΔΦ=δΦ(z,t)δΦ(z,0)
β222δEt2+E0δΦz+δNE0=0,β22E02δΦt2δEz=0,δNt=1τ(2γE0δEδN),
δE(z,t)=12π+δe(z,Ω)eiΩtdΩ,δΦ(z,t)=12π+δϕ(z,Ω)eiΩtdΩ,δN(z,t)=12π+δn(z,Ω)eiΩtdΩ,
2δez2(2γ|E0|2Ωτ1+Ω2τ2)δez+β2Ω22(β2Ω22+2γ|E0|21+Ω2τ2)δe=0,2δϕz2(2γ|E0|2Ωτ1+Ω2τ2)δϕz+β2Ω22(β2Ω22+2γ|E0|21+Ω2τ2)δϕ=0,
2ez2α1ez+α2e=0,2ϕz2α1ϕz+α2ϕ=0,
k2+ikα1α2=0,
[ki(α1+α124α22)]×[ki(α1α124α22)]=0,
α1=2γ|E0|2Ωτ1+Ω2τ2,α2=β2Ω22(β2Ω22+2γ|E0|21+Ω2τ2),
g(Ω)=α1±α124α2.
δe(z,Ω)=δe(0,Ω)eg(Ω)z/2,δϕ(z,Ω)=δϕ(0,Ω)eg(Ω)z/2,
ΔΦ=12π0δϕ(0,Ω)eikz[e(iΩt)1]dΩ+c.c.,
δϕ*(0,Ω)δϕ(0,Ω)=SΦ(0,Ω)δ(ΩΩ),
(ΔΦ)2(z,t)=2π0SΦ(0,Ω)[1cos(Ωt)]×[eg(Ω)z1]dΩ.
(ΔΦ)2(L,t)=2πΔν{t+2π0Ωc[1cos(Ωt)][eg(Ω)L1]dΩΩ2},

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