Abstract

We analyze theoretically the polarization properties of a partially coherent optical field that propagates in a nonlinear Kerr medium. We consider the standard model of two resonantly coupled nonlinear Schrödinger equations, which account for a wave-vector mismatch between the orthogonal polarization components. We show that such a phase-mismatch is responsible for the existence of a spontaneous repolarization process of the partially incoherent optical field during its nonlinear propagation. The repolarization process is characterized by an irreversible evolution of the unpolarized beam towards a highly polarized state, without any loss of energy. This unexpected result contrasts with the commonly accepted idea that an optical field undergoes a depolarization process under nonlinear evolution. The repolarization effect can be described in details by simple thermodynamic arguments based on the kinetic wave theory: It is shown to result from the natural tendency of the optical field to approach its thermal equilibrium state. The theory then reveals that it is thermodynamically advantageous for the optical field to evolve towards a highly polarized state, because this permits the optical field to reach the “most disordered state”, i.e., the state of maximum (nonequilibrium) entropy. The theory is in quantitative agreement with the numerical simulations, without adjustable parameters. The physics underlying the reversible property of the repolarization process is briefly discussed in analogy with the celebrated Joule’s experiment of free expansion of a gas. Besides its fundamental interest, the repolarization effect may be exploited to achieve complete polarization of unpolarized incoherent light without loss of energy.

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef]

2008

L. Levi, T. Schwartz, O. Manela, M. Segev, and H. Buljan, "Spontaneous pattern formation upon incoherent waves: From modulation-instability to steady-state," Opt. Express 16, 7818-7831 (2008).
[CrossRef] [PubMed]

C. Rotschild, T. Schwartz, O. Cohen and M. Segev, "Incoherent spatial solitons in effectively instantaneous nonlinear media," Nature Photon. 2, 371 (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] [PubMed]

H. Prakash and D. K. Singh, "Change in coherence properties and degree of polarization of light propagating in a lossless isotropic nonlinear Kerr medium," J. Phys. B: At. Mol. Opt. Phys. 41, 045401 (2008).
[CrossRef]

S. Pitois, J. Fatome, and G. Millot, "Polarization attraction using counter-propagating waves in optical fiber at telecommunication wavelengths," Opt. Express 16, 6646 (2008).
[CrossRef] [PubMed]

2007

2006

M. Wu, P. Krivosik, B. A. Kalinikos, and C. E. Patton, "Random Generation of Coherent Solitary Waves from Incoherent Waves," Phys. Rev. Lett. 96, 227202 (2006).
[CrossRef] [PubMed]

O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, "Incoherent solitons in instantaneous nonlocal nonlinear media," Phys. Rev. E 73, 015601 (2006).
[CrossRef]

A. Picozzi, "Nonequilibrated Oscillations of Coherence in Coupled Nonlinear Wave Systems," Phys. Rev. Lett. 96, 013905 (2006).
[CrossRef] [PubMed]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, "Velocity Locking of Incoherent Nonlinear Wave Packets," Phys. Rev. Lett. 97, 033902 (2006).
[CrossRef] [PubMed]

2005

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, "Condensation of Classical Nonlinear Waves," Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, "Polarization and modal attractors in conservative counterpropagating four-wave interaction," Europhys. Lett. 70, 88 (2005).
[CrossRef]

A. Sauter, S. Pitois, G. Millot, and A. Picozzi, "Incoherent modulation instability in instantaneous nonlinear Kerr media," Opt. Lett. 30, 2143-2145 (2005).
[CrossRef] [PubMed]

V. E. Zakharov and S. V. Nazarenko, "Dynamics of the Bose-Einstein condensation," Physica D 201, 203-211 (2005).
[CrossRef]

A. Picozzi and P. Aschieri, "Influence of dispersion on the resonant interaction between three incoherent waves," Phys. Rev. E 72, 046606 (2005).
[CrossRef]

2004

A. Schauer, I. V. Melnikov, and J. S. Aitchison, "Collisions of orthogonally polarized spatial solitons in AlGaAs slab waveguides," J. Opt. Soc. Am. B 21, 57 (2004).
[CrossRef]

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, "Spectral incoherent solitons: a localized soliton behavior in the frequency domain," Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef] [PubMed]

A. Picozzi and M. Haelterman, "Condensation in Hamiltonian Parametric Wave Interaction," Phys. Rev. Lett. 92, 103901 (2004).
[CrossRef] [PubMed]

S. A. Ponomarenko and G. P. Agrawal, "Asymmetric incoherent vector solitons," Phys. Rev. E 69, 036604 (2004).
[CrossRef]

V. E. Zakharov, F. Dias, and A. Pushkarev, "One-dimensional wave turbulence," Phys. Rep. 398, 1 (2004).
[CrossRef]

A. Picozzi, "Entropy and degree of polarization for nonlinear optical waves," Opt. Lett. 29, 1653 (2004).
[CrossRef] [PubMed]

2002

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]

A. Picozzi, C. Montes, and M. Haelterman, "Coherence properties of the parametric three-wave interaction driven from an incoherent pump," Phys. Rev. E 66, 056605 (2002).
[CrossRef]

2001

B. Rumpf and A. C. Newell, "Coherent Structures and Entropy in Constrained, Modulationally Unstable Nonintegrable Systems," Phys. Rev. Lett. 87, 054102 (2001).
[CrossRef] [PubMed]

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601 (2001).
[CrossRef]

A. Picozzi and M. Haelterman, "Parametric Three-Wave Soliton Generated from Incoherent Light," Phys. Rev. Lett. 86, 2010-2013 (2001).
[CrossRef] [PubMed]

A. C. Newell, S. Nazarenko, and L. Biven, "Wave turbulence and intermittency," Physica D 152, 520 (2001).
[CrossRef]

2000

1999

P. Ohberg and S. Stenholm, "Internal Josephson effect in trapped double condensates," Phys. Rev. A 59, 3890 (1999).
[CrossRef]

1997

D.C. Hutchings, J. S. Aitchison, and J. M. Arnold, "Nonlinear refractive coupling and vector solitons in anisotropic cubic media," J. Opt. Soc. Am. B 14, 869 (1997).
[CrossRef]

M. Mitchell and M. Segev, "Self-trapping of incoherent white light," Nature (London) 387, 880 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, "Theory of Incoherent Self-Focusing in Biased Photorefractive Media," Phys. Rev. Lett. 78, 646 (1997).
[CrossRef]

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, "Theory of Self-Trapped Spatially Incoherent Light Beams," Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

1996

M. Mitchell, Z. Chen, Ming-feng Shih, and M. Segev, "Self-Trapping of Partially Spatially Incoherent Light," Phys. Rev. Lett. 77, 490 (1996).
[CrossRef] [PubMed]

J. U. Kangand, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, "Observation of Manakov Spatial Solitons in AlGaAs Planar Waveguides," Phys. Rev. Lett 76, 3699 (1996).
[CrossRef]

1994

Yu. P. Svirko and N. I. Zheludev, "Propagation of partially polarized light," Phys. Rev. A 50, 709 (1994).
[CrossRef] [PubMed]

1993

1992

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, "Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation," Physica D 57, 96 (1992).
[CrossRef]

1991

C. M. de Sterke and J. E. Sipe, "Polarization instability in a waveguide geometry," Opt. Lett 16, 202 (1991).
[CrossRef] [PubMed]

1986

1968

A. C. Newell, "The closure problem in a system of random gravity waves," Rev. of Geophys. 6, 1-31 (1968).
[CrossRef]

1966

D. J. Benney and P. G. Saffman, "Nonlinear interactions of random waves in a dispersive medium," Proc. R. Soc. London Ser. A 289, 301 (1966).
[CrossRef]

Agrawal, G. P.

Aitchison, J. S.

Akhmediev, N.

J. U. Kangand, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, "Observation of Manakov Spatial Solitons in AlGaAs Planar Waveguides," Phys. Rev. Lett 76, 3699 (1996).
[CrossRef]

Anderson, D.

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]

Arnold, J. M.

Aschieri, P.

A. Picozzi and P. Aschieri, "Influence of dispersion on the resonant interaction between three incoherent waves," Phys. Rev. E 72, 046606 (2005).
[CrossRef]

Barviau, B.

B. Barviau, B. Kibler, S. Coen, and A. Picozzi, "Towards a thermodynamic description of supercontinuum generation," Opt. Lett. (to be published).

Benney, D. J.

D. J. Benney and P. G. Saffman, "Nonlinear interactions of random waves in a dispersive medium," Proc. R. Soc. London Ser. A 289, 301 (1966).
[CrossRef]

Bennink, R.

Biven, L.

A. C. Newell, S. Nazarenko, and L. Biven, "Wave turbulence and intermittency," Physica D 152, 520 (2001).
[CrossRef]

Boyd, R. W.

Buljan, H.

L. Levi, T. Schwartz, O. Manela, M. Segev, and H. Buljan, "Spontaneous pattern formation upon incoherent waves: From modulation-instability to steady-state," Opt. Express 16, 7818-7831 (2008).
[CrossRef] [PubMed]

O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, "Incoherent solitons in instantaneous nonlocal nonlinear media," Phys. Rev. E 73, 015601 (2006).
[CrossRef]

Cada, M.

Chen, Z.

M. Mitchell, Z. Chen, Ming-feng Shih, and M. Segev, "Self-Trapping of Partially Spatially Incoherent Light," Phys. Rev. Lett. 77, 490 (1996).
[CrossRef] [PubMed]

Chernov, V. E.

Christodoulides, D. N.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601 (2001).
[CrossRef]

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, "Theory of Self-Trapped Spatially Incoherent Light Beams," Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, "Theory of Incoherent Self-Focusing in Biased Photorefractive Media," Phys. Rev. Lett. 78, 646 (1997).
[CrossRef]

Coen, S.

B. Barviau, B. Kibler, S. Coen, and A. Picozzi, "Towards a thermodynamic description of supercontinuum generation," Opt. Lett. (to be published).

Cohen, O.

C. Rotschild, T. Schwartz, O. Cohen and M. Segev, "Incoherent spatial solitons in effectively instantaneous nonlinear media," Nature Photon. 2, 371 (2008).
[CrossRef]

O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, "Incoherent solitons in instantaneous nonlocal nonlinear media," Phys. Rev. E 73, 015601 (2006).
[CrossRef]

Connaughton, C.

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, "Condensation of Classical Nonlinear Waves," Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

Coskun, T.

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, "Theory of Self-Trapped Spatially Incoherent Light Beams," Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

Coskun, T. H.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601 (2001).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, "Theory of Incoherent Self-Focusing in Biased Photorefractive Media," Phys. Rev. Lett. 78, 646 (1997).
[CrossRef]

Crosignani, B.

Daino, B.

de Sterke, C. M.

C. M. de Sterke and J. E. Sipe, "Polarization instability in a waveguide geometry," Opt. Lett 16, 202 (1991).
[CrossRef] [PubMed]

Di Porto, P.

Dias, F.

V. E. Zakharov, F. Dias, and A. Pushkarev, "One-dimensional wave turbulence," Phys. Rep. 398, 1 (2004).
[CrossRef]

During, G.

G. During, A. Picozzi, and S. Rica, "Breakdown of wave turbulence and nonlinear wave condensation," Physica D (to be published).

Dyachenko, S.

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, "Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation," Physica D 57, 96 (1992).
[CrossRef]

Eugenieva, E. D.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601 (2001).
[CrossRef]

Fatome, J.

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]

Fisher, R.

Fleischer, J. W.

O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, "Incoherent solitons in instantaneous nonlocal nonlinear media," Phys. Rev. E 73, 015601 (2006).
[CrossRef]

Haelterman, M.

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, "Polarization and modal attractors in conservative counterpropagating four-wave interaction," Europhys. Lett. 70, 88 (2005).
[CrossRef]

A. Picozzi and M. Haelterman, "Condensation in Hamiltonian Parametric Wave Interaction," Phys. Rev. Lett. 92, 103901 (2004).
[CrossRef] [PubMed]

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, "Spectral incoherent solitons: a localized soliton behavior in the frequency domain," Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef] [PubMed]

A. Picozzi, C. Montes, and M. Haelterman, "Coherence properties of the parametric three-wave interaction driven from an incoherent pump," Phys. Rev. E 66, 056605 (2002).
[CrossRef]

A. Picozzi and M. Haelterman, "Parametric Three-Wave Soliton Generated from Incoherent Light," Phys. Rev. Lett. 86, 2010-2013 (2001).
[CrossRef] [PubMed]

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]

Heebner, J. E.

Huang, W.

Hutchings, D.C.

Jauslin, H. R.

S. Lagrange, H. R. Jauslin, and A. Picozzi, "Thermalization of the dispersive three-wave interaction," Europhys. Lett. 79, 64001 (2007).
[CrossRef]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, "Velocity Locking of Incoherent Nonlinear Wave Packets," Phys. Rev. Lett. 97, 033902 (2006).
[CrossRef] [PubMed]

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, "Polarization and modal attractors in conservative counterpropagating four-wave interaction," Europhys. Lett. 70, 88 (2005).
[CrossRef]

Jordan, R.

R. Jordan and C. Josserand, "Self-organization in nonlinear wave turbulence," Phys. Rev. E 61, 1527-1539 (2000).
[CrossRef]

Josserand, C.

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, "Condensation of Classical Nonlinear Waves," Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

R. Jordan and C. Josserand, "Self-organization in nonlinear wave turbulence," Phys. Rev. E 61, 1527-1539 (2000).
[CrossRef]

Kalinikos, B. A.

M. Wu, P. Krivosik, B. A. Kalinikos, and C. E. Patton, "Random Generation of Coherent Solitary Waves from Incoherent Waves," Phys. Rev. Lett. 96, 227202 (2006).
[CrossRef] [PubMed]

Kangand, J. U.

J. U. Kangand, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, "Observation of Manakov Spatial Solitons in AlGaAs Planar Waveguides," Phys. Rev. Lett 76, 3699 (1996).
[CrossRef]

Kibler, B.

B. Barviau, B. Kibler, S. Coen, and A. Picozzi, "Towards a thermodynamic description of supercontinuum generation," Opt. Lett. (to be published).

Krivosik, P.

M. Wu, P. Krivosik, B. A. Kalinikos, and C. E. Patton, "Random Generation of Coherent Solitary Waves from Incoherent Waves," Phys. Rev. Lett. 96, 227202 (2006).
[CrossRef] [PubMed]

Lagrange, S.

S. Lagrange, H. R. Jauslin, and A. Picozzi, "Thermalization of the dispersive three-wave interaction," Europhys. Lett. 79, 64001 (2007).
[CrossRef]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, "Velocity Locking of Incoherent Nonlinear Wave Packets," Phys. Rev. Lett. 97, 033902 (2006).
[CrossRef] [PubMed]

Levi, L.

Lisak, M.

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]

Manela, O.

Melnikov, I. V.

Millot, G.

S. Pitois, J. Fatome, and G. Millot, "Polarization attraction using counter-propagating waves in optical fiber at telecommunication wavelengths," Opt. Express 16, 6646 (2008).
[CrossRef] [PubMed]

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] [PubMed]

A. Sauter, S. Pitois, G. Millot, and A. Picozzi, "Incoherent modulation instability in instantaneous nonlinear Kerr media," Opt. Lett. 30, 2143-2145 (2005).
[CrossRef] [PubMed]

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, "Polarization and modal attractors in conservative counterpropagating four-wave interaction," Europhys. Lett. 70, 88 (2005).
[CrossRef]

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, "Spectral incoherent solitons: a localized soliton behavior in the frequency domain," Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef] [PubMed]

Mitchell, M.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601 (2001).
[CrossRef]

M. Mitchell and M. Segev, "Self-trapping of incoherent white light," Nature (London) 387, 880 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, "Theory of Incoherent Self-Focusing in Biased Photorefractive Media," Phys. Rev. Lett. 78, 646 (1997).
[CrossRef]

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, "Theory of Self-Trapped Spatially Incoherent Light Beams," Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

M. Mitchell, Z. Chen, Ming-feng Shih, and M. Segev, "Self-Trapping of Partially Spatially Incoherent Light," Phys. Rev. Lett. 77, 490 (1996).
[CrossRef] [PubMed]

Montes, C.

A. Picozzi, C. Montes, and M. Haelterman, "Coherence properties of the parametric three-wave interaction driven from an incoherent pump," Phys. Rev. E 66, 056605 (2002).
[CrossRef]

Nazarenko, S.

A. C. Newell, S. Nazarenko, and L. Biven, "Wave turbulence and intermittency," Physica D 152, 520 (2001).
[CrossRef]

Nazarenko, S. V.

V. E. Zakharov and S. V. Nazarenko, "Dynamics of the Bose-Einstein condensation," Physica D 201, 203-211 (2005).
[CrossRef]

Newell, A. C.

A. C. Newell, S. Nazarenko, and L. Biven, "Wave turbulence and intermittency," Physica D 152, 520 (2001).
[CrossRef]

B. Rumpf and A. C. Newell, "Coherent Structures and Entropy in Constrained, Modulationally Unstable Nonintegrable Systems," Phys. Rev. Lett. 87, 054102 (2001).
[CrossRef] [PubMed]

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, "Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation," Physica D 57, 96 (1992).
[CrossRef]

A. C. Newell, "The closure problem in a system of random gravity waves," Rev. of Geophys. 6, 1-31 (1968).
[CrossRef]

Ohberg, P.

P. Ohberg and S. Stenholm, "Internal Josephson effect in trapped double condensates," Phys. Rev. A 59, 3890 (1999).
[CrossRef]

Patton, C. E.

M. Wu, P. Krivosik, B. A. Kalinikos, and C. E. Patton, "Random Generation of Coherent Solitary Waves from Incoherent Waves," Phys. Rev. Lett. 96, 227202 (2006).
[CrossRef] [PubMed]

Picozzi, A.

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] [PubMed]

S. Lagrange, H. R. Jauslin, and A. Picozzi, "Thermalization of the dispersive three-wave interaction," Europhys. Lett. 79, 64001 (2007).
[CrossRef]

A. Picozzi, "Towards a nonequilibrium thermodynamic description of incoherent nonlinear optics," Opt. Express 15, 9063 (2007).
[CrossRef] [PubMed]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, "Velocity Locking of Incoherent Nonlinear Wave Packets," Phys. Rev. Lett. 97, 033902 (2006).
[CrossRef] [PubMed]

A. Picozzi, "Nonequilibrated Oscillations of Coherence in Coupled Nonlinear Wave Systems," Phys. Rev. Lett. 96, 013905 (2006).
[CrossRef] [PubMed]

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, "Condensation of Classical Nonlinear Waves," Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

A. Sauter, S. Pitois, G. Millot, and A. Picozzi, "Incoherent modulation instability in instantaneous nonlinear Kerr media," Opt. Lett. 30, 2143-2145 (2005).
[CrossRef] [PubMed]

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, "Polarization and modal attractors in conservative counterpropagating four-wave interaction," Europhys. Lett. 70, 88 (2005).
[CrossRef]

A. Picozzi and P. Aschieri, "Influence of dispersion on the resonant interaction between three incoherent waves," Phys. Rev. E 72, 046606 (2005).
[CrossRef]

A. Picozzi, "Entropy and degree of polarization for nonlinear optical waves," Opt. Lett. 29, 1653 (2004).
[CrossRef] [PubMed]

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, "Spectral incoherent solitons: a localized soliton behavior in the frequency domain," Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef] [PubMed]

A. Picozzi and M. Haelterman, "Condensation in Hamiltonian Parametric Wave Interaction," Phys. Rev. Lett. 92, 103901 (2004).
[CrossRef] [PubMed]

A. Picozzi, C. Montes, and M. Haelterman, "Coherence properties of the parametric three-wave interaction driven from an incoherent pump," Phys. Rev. E 66, 056605 (2002).
[CrossRef]

A. Picozzi and M. Haelterman, "Parametric Three-Wave Soliton Generated from Incoherent Light," Phys. Rev. Lett. 86, 2010-2013 (2001).
[CrossRef] [PubMed]

G. During, A. Picozzi, and S. Rica, "Breakdown of wave turbulence and nonlinear wave condensation," Physica D (to be published).

B. Barviau, B. Kibler, S. Coen, and A. Picozzi, "Towards a thermodynamic description of supercontinuum generation," Opt. Lett. (to be published).

A. Picozzi and S. Rica, "Coherence absorption induced by thermalization of incoherent fields" Europhys. Lett. (to be published).

Pitois, S.

S. Pitois, J. Fatome, and G. Millot, "Polarization attraction using counter-propagating waves in optical fiber at telecommunication wavelengths," Opt. Express 16, 6646 (2008).
[CrossRef] [PubMed]

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] [PubMed]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, "Velocity Locking of Incoherent Nonlinear Wave Packets," Phys. Rev. Lett. 97, 033902 (2006).
[CrossRef] [PubMed]

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, "Polarization and modal attractors in conservative counterpropagating four-wave interaction," Europhys. Lett. 70, 88 (2005).
[CrossRef]

A. Sauter, S. Pitois, G. Millot, and A. Picozzi, "Incoherent modulation instability in instantaneous nonlinear Kerr media," Opt. Lett. 30, 2143-2145 (2005).
[CrossRef] [PubMed]

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, "Spectral incoherent solitons: a localized soliton behavior in the frequency domain," Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef] [PubMed]

Pomeau, Y.

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, "Condensation of Classical Nonlinear Waves," Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

Ponomarenko, S. A.

Prakash, H.

H. Prakash and D. K. Singh, "Change in coherence properties and degree of polarization of light propagating in a lossless isotropic nonlinear Kerr medium," J. Phys. B: At. Mol. Opt. Phys. 41, 045401 (2008).
[CrossRef]

Pushkarev, A.

V. E. Zakharov, F. Dias, and A. Pushkarev, "One-dimensional wave turbulence," Phys. Rep. 398, 1 (2004).
[CrossRef]

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, "Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation," Physica D 57, 96 (1992).
[CrossRef]

Rica, S.

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, "Condensation of Classical Nonlinear Waves," Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

G. During, A. Picozzi, and S. Rica, "Breakdown of wave turbulence and nonlinear wave condensation," Physica D (to be published).

A. Picozzi and S. Rica, "Coherence absorption induced by thermalization of incoherent fields" Europhys. Lett. (to be published).

Rotschild, C.

C. Rotschild, T. Schwartz, O. Cohen and M. Segev, "Incoherent spatial solitons in effectively instantaneous nonlinear media," Nature Photon. 2, 371 (2008).
[CrossRef]

Rumpf, B.

B. Rumpf and A. C. Newell, "Coherent Structures and Entropy in Constrained, Modulationally Unstable Nonintegrable Systems," Phys. Rev. Lett. 87, 054102 (2001).
[CrossRef] [PubMed]

Saffman, P. G.

D. J. Benney and P. G. Saffman, "Nonlinear interactions of random waves in a dispersive medium," Proc. R. Soc. London Ser. A 289, 301 (1966).
[CrossRef]

Sauter, A.

Schauer, A.

Schwartz, T.

C. Rotschild, T. Schwartz, O. Cohen and M. Segev, "Incoherent spatial solitons in effectively instantaneous nonlinear media," Nature Photon. 2, 371 (2008).
[CrossRef]

L. Levi, T. Schwartz, O. Manela, M. Segev, and H. Buljan, "Spontaneous pattern formation upon incoherent waves: From modulation-instability to steady-state," Opt. Express 16, 7818-7831 (2008).
[CrossRef] [PubMed]

O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, "Incoherent solitons in instantaneous nonlocal nonlinear media," Phys. Rev. E 73, 015601 (2006).
[CrossRef]

Segev, M.

C. Rotschild, T. Schwartz, O. Cohen and M. Segev, "Incoherent spatial solitons in effectively instantaneous nonlinear media," Nature Photon. 2, 371 (2008).
[CrossRef]

L. Levi, T. Schwartz, O. Manela, M. Segev, and H. Buljan, "Spontaneous pattern formation upon incoherent waves: From modulation-instability to steady-state," Opt. Express 16, 7818-7831 (2008).
[CrossRef] [PubMed]

O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, "Incoherent solitons in instantaneous nonlocal nonlinear media," Phys. Rev. E 73, 015601 (2006).
[CrossRef]

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601 (2001).
[CrossRef]

M. Mitchell and M. Segev, "Self-trapping of incoherent white light," Nature (London) 387, 880 (1997).
[CrossRef]

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, "Theory of Self-Trapped Spatially Incoherent Light Beams," Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, "Theory of Incoherent Self-Focusing in Biased Photorefractive Media," Phys. Rev. Lett. 78, 646 (1997).
[CrossRef]

Semenov, V. E.

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]

Singh, D. K.

H. Prakash and D. K. Singh, "Change in coherence properties and degree of polarization of light propagating in a lossless isotropic nonlinear Kerr medium," J. Phys. B: At. Mol. Opt. Phys. 41, 045401 (2008).
[CrossRef]

Sipe, J. E.

C. M. de Sterke and J. E. Sipe, "Polarization instability in a waveguide geometry," Opt. Lett 16, 202 (1991).
[CrossRef] [PubMed]

Stegeman, G. I.

J. U. Kangand, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, "Observation of Manakov Spatial Solitons in AlGaAs Planar Waveguides," Phys. Rev. Lett 76, 3699 (1996).
[CrossRef]

Stenholm, S.

P. Ohberg and S. Stenholm, "Internal Josephson effect in trapped double condensates," Phys. Rev. A 59, 3890 (1999).
[CrossRef]

Svirko, Yu. P.

Yu. P. Svirko and N. I. Zheludev, "Propagation of partially polarized light," Phys. Rev. A 50, 709 (1994).
[CrossRef] [PubMed]

Wolf, E.

Wu, M.

M. Wu, P. Krivosik, B. A. Kalinikos, and C. E. Patton, "Random Generation of Coherent Solitary Waves from Incoherent Waves," Phys. Rev. Lett. 96, 227202 (2006).
[CrossRef] [PubMed]

Zakharov, V. E.

V. E. Zakharov and S. V. Nazarenko, "Dynamics of the Bose-Einstein condensation," Physica D 201, 203-211 (2005).
[CrossRef]

V. E. Zakharov, F. Dias, and A. Pushkarev, "One-dimensional wave turbulence," Phys. Rep. 398, 1 (2004).
[CrossRef]

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, "Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation," Physica D 57, 96 (1992).
[CrossRef]

Zheludev, N. I.

Yu. P. Svirko and N. I. Zheludev, "Propagation of partially polarized light," Phys. Rev. A 50, 709 (1994).
[CrossRef] [PubMed]

Zon, B. A.

Europhys. Lett.

S. Lagrange, H. R. Jauslin, and A. Picozzi, "Thermalization of the dispersive three-wave interaction," Europhys. Lett. 79, 64001 (2007).
[CrossRef]

A. Picozzi and S. Rica, "Coherence absorption induced by thermalization of incoherent fields" Europhys. Lett. (to be published).

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, "Polarization and modal attractors in conservative counterpropagating four-wave interaction," Europhys. Lett. 70, 88 (2005).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. B: At. Mol. Opt. Phys.

H. Prakash and D. K. Singh, "Change in coherence properties and degree of polarization of light propagating in a lossless isotropic nonlinear Kerr medium," J. Phys. B: At. Mol. Opt. Phys. 41, 045401 (2008).
[CrossRef]

Nature (London)

M. Mitchell and M. Segev, "Self-trapping of incoherent white light," Nature (London) 387, 880 (1997).
[CrossRef]

Nature Photon.

C. Rotschild, T. Schwartz, O. Cohen and M. Segev, "Incoherent spatial solitons in effectively instantaneous nonlinear media," Nature Photon. 2, 371 (2008).
[CrossRef]

Opt. Express

Opt. Lett

C. M. de Sterke and J. E. Sipe, "Polarization instability in a waveguide geometry," Opt. Lett 16, 202 (1991).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rep.

V. E. Zakharov, F. Dias, and A. Pushkarev, "One-dimensional wave turbulence," Phys. Rep. 398, 1 (2004).
[CrossRef]

Phys. Rev. A

Yu. P. Svirko and N. I. Zheludev, "Propagation of partially polarized light," Phys. Rev. A 50, 709 (1994).
[CrossRef] [PubMed]

P. Ohberg and S. Stenholm, "Internal Josephson effect in trapped double condensates," Phys. Rev. A 59, 3890 (1999).
[CrossRef]

Phys. Rev. E

A. Picozzi, C. Montes, and M. Haelterman, "Coherence properties of the parametric three-wave interaction driven from an incoherent pump," Phys. Rev. E 66, 056605 (2002).
[CrossRef]

A. Picozzi and P. Aschieri, "Influence of dispersion on the resonant interaction between three incoherent waves," Phys. Rev. E 72, 046606 (2005).
[CrossRef]

R. Jordan and C. Josserand, "Self-organization in nonlinear wave turbulence," Phys. Rev. E 61, 1527-1539 (2000).
[CrossRef]

O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, "Incoherent solitons in instantaneous nonlocal nonlinear media," Phys. Rev. E 73, 015601 (2006).
[CrossRef]

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601 (2001).
[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]

S. A. Ponomarenko and G. P. Agrawal, "Asymmetric incoherent vector solitons," Phys. Rev. E 69, 036604 (2004).
[CrossRef]

Phys. Rev. Lett

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] [PubMed]

J. U. Kangand, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, "Observation of Manakov Spatial Solitons in AlGaAs Planar Waveguides," Phys. Rev. Lett 76, 3699 (1996).
[CrossRef]

Phys. Rev. Lett.

B. Rumpf and A. C. Newell, "Coherent Structures and Entropy in Constrained, Modulationally Unstable Nonintegrable Systems," Phys. Rev. Lett. 87, 054102 (2001).
[CrossRef] [PubMed]

A. Picozzi and M. Haelterman, "Condensation in Hamiltonian Parametric Wave Interaction," Phys. Rev. Lett. 92, 103901 (2004).
[CrossRef] [PubMed]

A. Picozzi, "Nonequilibrated Oscillations of Coherence in Coupled Nonlinear Wave Systems," Phys. Rev. Lett. 96, 013905 (2006).
[CrossRef] [PubMed]

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, "Condensation of Classical Nonlinear Waves," Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

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The elements Ji j (z) do not depend on the spatial position r because of the assumption of homogeneous statistics [5, 10, 11]. Note also that the degree of polarization do not depend on the basis of polarization in which it is calculated [9].

Note that the property |E+ |/|E+>(z) = |E- |2(z) has not been observed for an intensity of the field grater than the critical intensity of polarization instability.

Conversely, in the case of incoherently coupled NLS equations, one has to introduce two distinct Lagrange�??s multipliers μx and μy for each conserved quantity Ix and Iy, as discussed in details in Ref.[31].

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Figures (3)

Fig. 1.
Fig. 1.

Spontaneous polarization of an unpolarized field during its propagation in a loss-less Kerr medium in the presence of a phase-mismatch between the orthogonal polarization components (δ=3.2×103m-1). Numerical simulations of Eqs.(1-2) showing the evolution during the propagation of the degree of polarization �� (a), the kinetic and resonant energies (b), and the nonequilibrium entropy S(c). The process of entropy production is saturated once the equilibrium state is reached, as described by the H-theorem of entropy growth (z is in units of the nonlinear length Lnl =1/Γ��=0.24mm).

Fig. 2.
Fig. 2.

(a) Interpretation of the repolarization process as a thermalization of intensity fluctuations from the x-polarized (‘excited’) state towards the y-polarized (‘ground’) state. (b) Degree of polarization at equilibrium �� eq vs the energy ��, for a fixed value of the field intensity ��. The continuous line refers to the theory [Eqs.(13,14)], the points (◇) to the numerical simulations. The parameters are the same as in Fig. 1 (the energy �� is varied by varying the coherence of the field: the coherence is degraded as the energy �� increases).

Fig. 3.
Fig. 3.

Schematic illustration of the repolarization process on the Poincaré sphere representation: (a) An unpolarized field evolves towards a highly polarized state. Because of the reversibility of wave propagation, the phase conjugation (p.c.) of the repolarized field will regain its original unpolarized state at the entry of the medium [dashed arrow in (a)]. Such a depolarization process only refers to a transient evolution: The field would again reach a polarized state in a medium twice as long (b), in complete analogy with the celebrated Joule’s experiment of free expansion of a gas (c).

Equations (22)

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i z E + = α 2 E + + δ E ( E + 2 + 2 E 2 ) E + ,
i z E = α 2 E + δ E + ( E 2 + 2 E + 2 ) E ,
α 2 α y y + 1 2 β t t
P ( z ) = [ 1 4 Det ( J ) ( Tr J ) 2 ] 1 2 ,
E kin = α A A E 2 d r ,
E res = δ A A ( E + E * + E + * E ) d r .
i z E x = α 2 E x + δ E x + γ ( E x 2 + 2 3 E y 2 ) E x + γ 3 E x * E y 2 ,
i z E y = α 2 E y δ E y + γ ( E y 2 + 2 3 E x 2 ) E y + γ 3 E y * E x 2 .
K x ( k ) = α k 2 + δ ,
K y ( k ) = α k 2 + δ ,
n x e q ( k ) = T α k 2 + δ μ ,
n y e q ( k ) = T α k 2 δ μ ,
P e q ( μ ) = 1 4 Σ k U k · Σ k U k + ( Σ k U k + + U k ) 2 ,
E ( μ ) = I Σ k V k + + V k Σ k U k + + U k ,
U k ± = 1 α k 2 ± δ μ ,
V k ± = ( α k 2 ± δ ) U k ± .
z n x ( k 1 , z ) = C o l l S [ n x ] + C o l l X [ n x , n y ] + C o l l R [ n x , n y ] ,
z n y ( k 1 , z ) = C o l l S [ n y ] + C o l l X [ n y , n x ] + C o l l R [ n y , n x ] ,
C o l l X [ n x , n y ] = d k 2 d k 3 d k 4 N X W X x , y ,
N X = n x ( k 1 ) n y ( k 2 ) n y ( k 3 ) n x ( k 4 ) [ n x 1 ( k 1 ) + n y 1 ( k 2 ) n y 1 ( k 3 ) n x 1 ( k 4 ) ] ,
C o l l R [ n x , n y ] = d k 2 d k 3 d k 4 N R W R x , y ,
N R = n x ( k 1 ) n x ( k 2 ) n y ( k 3 ) n y ( k 4 ) [ n x 1 ( k 1 ) + n x 1 ( k 2 ) n y 1 ( k 3 ) n y 1 ( k 4 ) ] ,

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