Abstract

We investigate the phenomenon of polarization attraction in a highly birefringent fiber. This polarization process originates from the nonlinear interaction of two counter-propagating beams. We show that all polarization states of the forward (signal) beam are attracted toward a specific line of polarization states on the surface of the Poincaré sphere, whose characteristics are determined by the polarization state of the injected backward (pump) beam. This phenomenon of polarization attraction takes place without any loss of energy for the signal beam. The stability of different stationary solutions is also discussed through intensive numerical simulations. On the basis of mathematical techniques recently developed for the study of Hamiltonian singularities, we provide a detailed description of this spontaneous polarization process. In several particular cases of interest, the equation of the line of polarization attraction on the Poincaré sphere can be obtained in explicit analytical form.

© 2014 Optical Society of America

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  1. J. E. Heebner, R. S. Bennink, R. W. Boyd, and R. A. Fisher, “Conversion of unpolarized light to polarized light with greater than 50% efficiency by photorefractive two-beam coupling,” Opt. Lett. 25, 257–259 (2000).
    [Crossref]
  2. S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, “Polarization and modal attactors in conservative counterpropagating four-wave interaction,” Europhys. Lett. 70, 88 (2005), doi:10.1209/epl/i2004-10469-9.
    [Crossref]
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    [Crossref]
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    [Crossref]
  5. H. Prakash and D. Singh, “Change in coherence properties and degree of polarization of light propagating in a lossless isotropic nonlinear Kerr medium,” J. Phys. B 41, 045401 (2008).
    [Crossref]
  6. M. Martinelli, M. Cirigliano, M. Ferrario, L. Marazzi, and P. Martelli, “Evidence of Raman-induced polarization pulling,” Opt. Express 17, 947–955 (2009).
    [Crossref]
  7. S. Sergeyev, S. Popov, and A. T. Friberg, “Virtually isotropic transmission media with fiber Raman amplifier,” IEEE J. Quantum Electron. 46, 1492–1497 (2010).
    [Crossref]
  8. L. Ursini, M. Santagiustina, and L. Palmieri, “Raman nonlinear polarization pulling in the pump depleted regime in randomly birefringent fibers,” IEEE Photon. Technol. Lett. 23, 254–256 (2011).
    [Crossref]
  9. S. V. Sergeyev, “Activated polarization pulling and de-correlation of signal and pump states of polarization in a fiber Raman amplifier,” Opt. Express 19, 24268–24279 (2011).
    [Crossref]
  10. V. Kozlov, J. Nuno, J. D. Ania-Castanon, and S. Wabnitz, “Theoretical study of optical fiber Raman polarizers with counterpropagating beams,” J. Lightwave Technol. 29, 341–347 (2011).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  16. M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyanskiy, D. G. Luchinsky, R. Mannella, and P. V. E. McClintock, “Activated escape of periodically driven systems,” Chaos 11, 587–594 (2001).
    [Crossref]
  17. D. Sugny, A. Picozzi, S. Lagrange, and H. R. Jauslin, “Role of singular tori in the dynamics of spatiotemporal nonlinear wave systems,” Phys. Rev. Lett. 103, 034102 (2009).
    [Crossref]
  18. S. Lagrange, D. Sugny, A. Picozzi, and H. R. Jauslin, “Singular tori as attractors of four-wave interaction systems,” Phys. Rev. E 81, 016202 (2010).
    [Crossref]
  19. E. Assémat, S. Lagrange, A. Picozzi, H. R. Jauslin, and D. Sugny, “Complete nonlinear polarization control in an optical fiber system,” Opt. Lett. 35, 2025–2027 (2010).
    [Crossref]
  20. V. V. Kozlov and S. Wabnitz, “Theoretical study of polarization attraction in high-birefringence and spun fibers,” Opt. Lett. 35, 3949–3951 (2010).
    [Crossref]
  21. V. V. Kozlov, J. Nuno, and S. Wabnitz, “Theory of lossless polarization attraction in telecommunication fibers,” J. Opt. Soc. Am. B 28, 100–108 (2011).
    [Crossref]
  22. E. Assémat, D. Dargent, A. Picozzi, H. R. Jauslin, and D. Sugny, “Polarization control in spun and telecommunication optical fibers,” Opt. Lett. 36, 4038–4040 (2011).
    [Crossref]
  23. M. Guasoni, V. V. Kozlov, and S. Wabnitz, “Theory of polarization attraction in parametric amplifiers based on telecommunication fibers,” J. Opt. Soc. Am. B 29, 2710–2720 (2012).
    [Crossref]
  24. M. Guasoni and S. Wabnitz, “Nonlinear polarizers based on four-wave mixing in high-birefringence optical fibers,” J. Opt. Soc. Am. B 29, 1511–1520 (2012).
    [Crossref]
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    [Crossref]
  26. P. Morin, J. Fatome, C. Finot, S. Pitois, R. Claveau, and G. Millot, “All-optical nonlinear processing of both polarization state and intensity profile for 40 Gbit/s regeneration applications,” Opt. Express 19, 17158–17166 (2011).
    [Crossref]
  27. V. V. Kozlov, J. Fatome, P. Morin, S. Pitois, G. Millot, and S. Wabnitz, “Nonlinear repolarization dynamics in optical fibers: transient polarization attraction,” J. Opt. Soc. Am. B 28, 1782–1791 (2011).
    [Crossref]
  28. B. Stiller, P. Morin, D. M. Nguyen, J. Fatome, S. Pitois, E. Lantz, H. Maillotte, C. R. Menyuk, and T. Sylvestre, “Demonstration of polarization pulling using a fiber-optic parametric amplifier,” Opt. Express 20, 27248–27253 (2012).
    [Crossref]
  29. J. Fatome, S. Pitois, P. Morin, D. Sugny, E. Assémat, A. Picozzi, H. R. Jauslin, G. Millot, V. V. Kozlov, and S. Wabnitz, “A universal optical all-fiber omnipolarizer,” Sci. Rep. 2, 938–946 (2012).
    [Crossref]
  30. P.-Y. Bony, M. Guasoni, E. Assémat, S. Pitois, D. Sugny, A. Picozzi, H. R. Jauslin, and J. Fatome, “Optical flip-flop memory and data packet switching operation based on polarization bistability in a telecommunication optical fiber,” J. Opt. Soc. Am. B 30, 2318–2325 (2013).
    [Crossref]
  31. V. V. Kozlov and S. Wabnitz, “Instability of optical solitons in the boundary value problem for a medium of finite extension,” Lett. Math. Phys. 96, 405–413 (2011).
    [Crossref]
  32. V. V. Kozlov, K. Turitsyn, and S. Wabnitz, “Nonlinear repolarization in optical fibers: polarization attraction with copropagating beams,” Opt. Lett. 36, 4050–4052 (2011).
    [Crossref]
  33. V. V. Kozlov, M. Barozzi, A. Vannucci, and S. Wabnitz, “Lossless polarization attraction of co-propagating beams in telecom fibers,” J. Opt. Soc. Am. B 30, 530–540 (2013).
    [Crossref]
  34. R. H. Cushman and L. M. Bates, Global Aspects of Classical Integrable Systems (Birkhauser, 1997).
  35. K. Efstathiou and D. A. Sadovskii, “Normalization and global analysis of perturbations of the hydrogen atom,” Rev. Mod. Phys. 82, 2099–2154 (2010).
    [Crossref]
  36. E. Assémat, A. Picozzi, H. R. Jauslin, and D. Sugny, “Hamiltonian tools for the analysis of optical polarization control,” J. Opt. Soc. Am. B 29, 559–571 (2012).
    [Crossref]
  37. V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, 1989).
  38. K. S. Turitsyn and S. Wabnitz, “Stability analysis of polarization attraction in optical fibers,” Opt. Commun. 307, 62–66 (2013).
    [Crossref]
  39. S. Trillo and S. Wabnitz, “Intermittent spatial chaos in the polarization of counterpropagating beams in a birefringent optical fiber,” Phys. Rev. A 36, 3881–3884 (1987).
    [Crossref]
  40. A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, and P. W. Milonni, “Instabilities and chaos in the polarizations of counterpropagating light fields,” Phys. Rev. Lett. 58, 2432–2435 (1987).
    [Crossref]
  41. D. David, D. D. Holm, and M. V. Tratnik, “Hamiltonian chaos in nonlinear optical polarization dynamics,” Phys. Rep. 187, 281–367 (1990).
    [Crossref]
  42. E. Assémat, A. Picozzi, H. R. Jauslin, and D. Sugny, “Instabilities of optical solitons and Hamiltonian singular solutions in a medium of finite extension,” Phys. Rev. A 84, 013809 (2011).
    [Crossref]

2013 (3)

2012 (8)

2011 (12)

E. Assémat, A. Picozzi, H. R. Jauslin, and D. Sugny, “Instabilities of optical solitons and Hamiltonian singular solutions in a medium of finite extension,” Phys. Rev. A 84, 013809 (2011).
[Crossref]

Z. Shmilovitch, N. Primerov, A. Zadok, A. Eyal, S. Chin, L. Thevenaz, and M. Tur, “Dual-pump push-pull polarization control using stimulated Brillouin scattering,” Opt. Express 19, 25873–25880 (2011).
[Crossref]

L. Ursini, M. Santagiustina, and L. Palmieri, “Raman nonlinear polarization pulling in the pump depleted regime in randomly birefringent fibers,” IEEE Photon. Technol. Lett. 23, 254–256 (2011).
[Crossref]

S. V. Sergeyev, “Activated polarization pulling and de-correlation of signal and pump states of polarization in a fiber Raman amplifier,” Opt. Express 19, 24268–24279 (2011).
[Crossref]

V. Kozlov, J. Nuno, J. D. Ania-Castanon, and S. Wabnitz, “Theoretical study of optical fiber Raman polarizers with counterpropagating beams,” J. Lightwave Technol. 29, 341–347 (2011).
[Crossref]

N. J. Muga, M. F. S. Ferreira, and A. N. Pinto, “Broadband polarization pulling using Raman amplification,” Opt. Express 19, 18707–18712 (2011).
[Crossref]

P. Morin, J. Fatome, C. Finot, S. Pitois, R. Claveau, and G. Millot, “All-optical nonlinear processing of both polarization state and intensity profile for 40 Gbit/s regeneration applications,” Opt. Express 19, 17158–17166 (2011).
[Crossref]

V. V. Kozlov, J. Fatome, P. Morin, S. Pitois, G. Millot, and S. Wabnitz, “Nonlinear repolarization dynamics in optical fibers: transient polarization attraction,” J. Opt. Soc. Am. B 28, 1782–1791 (2011).
[Crossref]

V. V. Kozlov, J. Nuno, and S. Wabnitz, “Theory of lossless polarization attraction in telecommunication fibers,” J. Opt. Soc. Am. B 28, 100–108 (2011).
[Crossref]

E. Assémat, D. Dargent, A. Picozzi, H. R. Jauslin, and D. Sugny, “Polarization control in spun and telecommunication optical fibers,” Opt. Lett. 36, 4038–4040 (2011).
[Crossref]

V. V. Kozlov and S. Wabnitz, “Instability of optical solitons in the boundary value problem for a medium of finite extension,” Lett. Math. Phys. 96, 405–413 (2011).
[Crossref]

V. V. Kozlov, K. Turitsyn, and S. Wabnitz, “Nonlinear repolarization in optical fibers: polarization attraction with copropagating beams,” Opt. Lett. 36, 4050–4052 (2011).
[Crossref]

2010 (5)

K. Efstathiou and D. A. Sadovskii, “Normalization and global analysis of perturbations of the hydrogen atom,” Rev. Mod. Phys. 82, 2099–2154 (2010).
[Crossref]

S. Lagrange, D. Sugny, A. Picozzi, and H. R. Jauslin, “Singular tori as attractors of four-wave interaction systems,” Phys. Rev. E 81, 016202 (2010).
[Crossref]

E. Assémat, S. Lagrange, A. Picozzi, H. R. Jauslin, and D. Sugny, “Complete nonlinear polarization control in an optical fiber system,” Opt. Lett. 35, 2025–2027 (2010).
[Crossref]

V. V. Kozlov and S. Wabnitz, “Theoretical study of polarization attraction in high-birefringence and spun fibers,” Opt. Lett. 35, 3949–3951 (2010).
[Crossref]

S. Sergeyev, S. Popov, and A. T. Friberg, “Virtually isotropic transmission media with fiber Raman amplifier,” IEEE J. Quantum Electron. 46, 1492–1497 (2010).
[Crossref]

2009 (2)

D. Sugny, A. Picozzi, S. Lagrange, and H. R. Jauslin, “Role of singular tori in the dynamics of spatiotemporal nonlinear wave systems,” Phys. Rev. Lett. 103, 034102 (2009).
[Crossref]

M. Martinelli, M. Cirigliano, M. Ferrario, L. Marazzi, and P. Martelli, “Evidence of Raman-induced polarization pulling,” Opt. Express 17, 947–955 (2009).
[Crossref]

2008 (2)

H. Prakash and D. Singh, “Change in coherence properties and degree of polarization of light propagating in a lossless isotropic nonlinear Kerr medium,” J. Phys. B 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, 6646–6651 (2008).
[Crossref]

2005 (1)

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, “Polarization and modal attactors in conservative counterpropagating four-wave interaction,” Europhys. Lett. 70, 88 (2005), doi:10.1209/epl/i2004-10469-9.
[Crossref]

2004 (1)

2001 (1)

M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyanskiy, D. G. Luchinsky, R. Mannella, and P. V. E. McClintock, “Activated escape of periodically driven systems,” Chaos 11, 587–594 (2001).
[Crossref]

2000 (1)

1990 (1)

D. David, D. D. Holm, and M. V. Tratnik, “Hamiltonian chaos in nonlinear optical polarization dynamics,” Phys. Rep. 187, 281–367 (1990).
[Crossref]

1987 (2)

S. Trillo and S. Wabnitz, “Intermittent spatial chaos in the polarization of counterpropagating beams in a birefringent optical fiber,” Phys. Rev. A 36, 3881–3884 (1987).
[Crossref]

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, and P. W. Milonni, “Instabilities and chaos in the polarizations of counterpropagating light fields,” Phys. Rev. Lett. 58, 2432–2435 (1987).
[Crossref]

1986 (1)

Ackerhalt, J. R.

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, and P. W. Milonni, “Instabilities and chaos in the polarizations of counterpropagating light fields,” Phys. Rev. Lett. 58, 2432–2435 (1987).
[Crossref]

Ania-Castanon, J. D.

Arnold, V. I.

V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, 1989).

Assémat, E.

Barozzi, M.

Bates, L. M.

R. H. Cushman and L. M. Bates, Global Aspects of Classical Integrable Systems (Birkhauser, 1997).

Bennink, R. S.

Bony, P.-Y.

Boyd, R. W.

J. E. Heebner, R. S. Bennink, R. W. Boyd, and R. A. Fisher, “Conversion of unpolarized light to polarized light with greater than 50% efficiency by photorefractive two-beam coupling,” Opt. Lett. 25, 257–259 (2000).
[Crossref]

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, and P. W. Milonni, “Instabilities and chaos in the polarizations of counterpropagating light fields,” Phys. Rev. Lett. 58, 2432–2435 (1987).
[Crossref]

Chin, S.

Cirigliano, M.

Claveau, R.

Crosignani, B.

Cushman, R. H.

R. H. Cushman and L. M. Bates, Global Aspects of Classical Integrable Systems (Birkhauser, 1997).

Daino, B.

Dargent, D.

David, D.

D. David, D. D. Holm, and M. V. Tratnik, “Hamiltonian chaos in nonlinear optical polarization dynamics,” Phys. Rep. 187, 281–367 (1990).
[Crossref]

Di Porto, P.

Dykman, M. I.

M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyanskiy, D. G. Luchinsky, R. Mannella, and P. V. E. McClintock, “Activated escape of periodically driven systems,” Chaos 11, 587–594 (2001).
[Crossref]

Efstathiou, K.

K. Efstathiou and D. A. Sadovskii, “Normalization and global analysis of perturbations of the hydrogen atom,” Rev. Mod. Phys. 82, 2099–2154 (2010).
[Crossref]

Eyal, A.

Fatome, J.

P.-Y. Bony, M. Guasoni, E. Assémat, S. Pitois, D. Sugny, A. Picozzi, H. R. Jauslin, and J. Fatome, “Optical flip-flop memory and data packet switching operation based on polarization bistability in a telecommunication optical fiber,” J. Opt. Soc. Am. B 30, 2318–2325 (2013).
[Crossref]

B. Stiller, P. Morin, D. M. Nguyen, J. Fatome, S. Pitois, E. Lantz, H. Maillotte, C. R. Menyuk, and T. Sylvestre, “Demonstration of polarization pulling using a fiber-optic parametric amplifier,” Opt. Express 20, 27248–27253 (2012).
[Crossref]

P. Morin, S. Pitois, and J. Fatome, “Simultaneous polarization attraction and Raman amplification of a light beam in optical fibers,” J. Opt. Soc. Am. B 29, 2046–2052 (2012).
[Crossref]

J. Fatome, S. Pitois, P. Morin, D. Sugny, E. Assémat, A. Picozzi, H. R. Jauslin, G. Millot, V. V. Kozlov, and S. Wabnitz, “A universal optical all-fiber omnipolarizer,” Sci. Rep. 2, 938–946 (2012).
[Crossref]

P. Morin, J. Fatome, C. Finot, S. Pitois, R. Claveau, and G. Millot, “All-optical nonlinear processing of both polarization state and intensity profile for 40 Gbit/s regeneration applications,” Opt. Express 19, 17158–17166 (2011).
[Crossref]

V. V. Kozlov, J. Fatome, P. Morin, S. Pitois, G. Millot, and S. Wabnitz, “Nonlinear repolarization dynamics in optical fibers: transient polarization attraction,” J. Opt. Soc. Am. B 28, 1782–1791 (2011).
[Crossref]

S. Pitois, J. Fatome, and G. Millot, “Polarization attraction using counter-propagating waves in optical fiber at telecommunication wavelengths,” Opt. Express 16, 6646–6651 (2008).
[Crossref]

Ferrario, M.

Ferreira, M. F. S.

Finot, C.

Fisher, R. A.

Friberg, A. T.

S. Sergeyev, S. Popov, and A. T. Friberg, “Virtually isotropic transmission media with fiber Raman amplifier,” IEEE J. Quantum Electron. 46, 1492–1497 (2010).
[Crossref]

Gaeta, A. L.

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, and P. W. Milonni, “Instabilities and chaos in the polarizations of counterpropagating light fields,” Phys. Rev. Lett. 58, 2432–2435 (1987).
[Crossref]

Golding, B.

M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyanskiy, D. G. Luchinsky, R. Mannella, and P. V. E. McClintock, “Activated escape of periodically driven systems,” Chaos 11, 587–594 (2001).
[Crossref]

Guasoni, M.

Haelterman, M.

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, “Polarization and modal attactors in conservative counterpropagating four-wave interaction,” Europhys. Lett. 70, 88 (2005), doi:10.1209/epl/i2004-10469-9.
[Crossref]

Heebner, J. E.

Holm, D. D.

D. David, D. D. Holm, and M. V. Tratnik, “Hamiltonian chaos in nonlinear optical polarization dynamics,” Phys. Rep. 187, 281–367 (1990).
[Crossref]

Jauslin, H. R.

P.-Y. Bony, M. Guasoni, E. Assémat, S. Pitois, D. Sugny, A. Picozzi, H. R. Jauslin, and J. Fatome, “Optical flip-flop memory and data packet switching operation based on polarization bistability in a telecommunication optical fiber,” J. Opt. Soc. Am. B 30, 2318–2325 (2013).
[Crossref]

J. Fatome, S. Pitois, P. Morin, D. Sugny, E. Assémat, A. Picozzi, H. R. Jauslin, G. Millot, V. V. Kozlov, and S. Wabnitz, “A universal optical all-fiber omnipolarizer,” Sci. Rep. 2, 938–946 (2012).
[Crossref]

E. Assémat, A. Picozzi, H. R. Jauslin, and D. Sugny, “Hamiltonian tools for the analysis of optical polarization control,” J. Opt. Soc. Am. B 29, 559–571 (2012).
[Crossref]

E. Assémat, D. Dargent, A. Picozzi, H. R. Jauslin, and D. Sugny, “Polarization control in spun and telecommunication optical fibers,” Opt. Lett. 36, 4038–4040 (2011).
[Crossref]

E. Assémat, A. Picozzi, H. R. Jauslin, and D. Sugny, “Instabilities of optical solitons and Hamiltonian singular solutions in a medium of finite extension,” Phys. Rev. A 84, 013809 (2011).
[Crossref]

S. Lagrange, D. Sugny, A. Picozzi, and H. R. Jauslin, “Singular tori as attractors of four-wave interaction systems,” Phys. Rev. E 81, 016202 (2010).
[Crossref]

E. Assémat, S. Lagrange, A. Picozzi, H. R. Jauslin, and D. Sugny, “Complete nonlinear polarization control in an optical fiber system,” Opt. Lett. 35, 2025–2027 (2010).
[Crossref]

D. Sugny, A. Picozzi, S. Lagrange, and H. R. Jauslin, “Role of singular tori in the dynamics of spatiotemporal nonlinear wave systems,” Phys. Rev. Lett. 103, 034102 (2009).
[Crossref]

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, “Polarization and modal attactors in conservative counterpropagating four-wave interaction,” Europhys. Lett. 70, 88 (2005), doi:10.1209/epl/i2004-10469-9.
[Crossref]

Kozlov, V.

Kozlov, V. V.

V. V. Kozlov, M. Barozzi, A. Vannucci, and S. Wabnitz, “Lossless polarization attraction of co-propagating beams in telecom fibers,” J. Opt. Soc. Am. B 30, 530–540 (2013).
[Crossref]

M. Guasoni, V. V. Kozlov, and S. Wabnitz, “Theory of polarization attraction in parametric amplifiers based on telecommunication fibers,” J. Opt. Soc. Am. B 29, 2710–2720 (2012).
[Crossref]

J. Fatome, S. Pitois, P. Morin, D. Sugny, E. Assémat, A. Picozzi, H. R. Jauslin, G. Millot, V. V. Kozlov, and S. Wabnitz, “A universal optical all-fiber omnipolarizer,” Sci. Rep. 2, 938–946 (2012).
[Crossref]

V. V. Kozlov and S. Wabnitz, “Instability of optical solitons in the boundary value problem for a medium of finite extension,” Lett. Math. Phys. 96, 405–413 (2011).
[Crossref]

V. V. Kozlov, K. Turitsyn, and S. Wabnitz, “Nonlinear repolarization in optical fibers: polarization attraction with copropagating beams,” Opt. Lett. 36, 4050–4052 (2011).
[Crossref]

V. V. Kozlov, J. Nuno, and S. Wabnitz, “Theory of lossless polarization attraction in telecommunication fibers,” J. Opt. Soc. Am. B 28, 100–108 (2011).
[Crossref]

V. V. Kozlov, J. Fatome, P. Morin, S. Pitois, G. Millot, and S. Wabnitz, “Nonlinear repolarization dynamics in optical fibers: transient polarization attraction,” J. Opt. Soc. Am. B 28, 1782–1791 (2011).
[Crossref]

V. V. Kozlov and S. Wabnitz, “Theoretical study of polarization attraction in high-birefringence and spun fibers,” Opt. Lett. 35, 3949–3951 (2010).
[Crossref]

Lagrange, S.

E. Assémat, S. Lagrange, A. Picozzi, H. R. Jauslin, and D. Sugny, “Complete nonlinear polarization control in an optical fiber system,” Opt. Lett. 35, 2025–2027 (2010).
[Crossref]

S. Lagrange, D. Sugny, A. Picozzi, and H. R. Jauslin, “Singular tori as attractors of four-wave interaction systems,” Phys. Rev. E 81, 016202 (2010).
[Crossref]

D. Sugny, A. Picozzi, S. Lagrange, and H. R. Jauslin, “Role of singular tori in the dynamics of spatiotemporal nonlinear wave systems,” Phys. Rev. Lett. 103, 034102 (2009).
[Crossref]

Lantz, E.

Luchinsky, D. G.

M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyanskiy, D. G. Luchinsky, R. Mannella, and P. V. E. McClintock, “Activated escape of periodically driven systems,” Chaos 11, 587–594 (2001).
[Crossref]

Maillotte, H.

Mannella, R.

M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyanskiy, D. G. Luchinsky, R. Mannella, and P. V. E. McClintock, “Activated escape of periodically driven systems,” Chaos 11, 587–594 (2001).
[Crossref]

Marazzi, L.

Martelli, P.

Martinelli, M.

McCann, L. I.

M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyanskiy, D. G. Luchinsky, R. Mannella, and P. V. E. McClintock, “Activated escape of periodically driven systems,” Chaos 11, 587–594 (2001).
[Crossref]

McClintock, P. V. E.

M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyanskiy, D. G. Luchinsky, R. Mannella, and P. V. E. McClintock, “Activated escape of periodically driven systems,” Chaos 11, 587–594 (2001).
[Crossref]

Menyuk, C. R.

Millot, G.

Milonni, P. W.

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, and P. W. Milonni, “Instabilities and chaos in the polarizations of counterpropagating light fields,” Phys. Rev. Lett. 58, 2432–2435 (1987).
[Crossref]

Morin, P.

Muga, N. J.

Nguyen, D. M.

Nuno, J.

Palmieri, L.

L. Ursini, M. Santagiustina, and L. Palmieri, “Raman nonlinear polarization pulling in the pump depleted regime in randomly birefringent fibers,” IEEE Photon. Technol. Lett. 23, 254–256 (2011).
[Crossref]

Picozzi, A.

P.-Y. Bony, M. Guasoni, E. Assémat, S. Pitois, D. Sugny, A. Picozzi, H. R. Jauslin, and J. Fatome, “Optical flip-flop memory and data packet switching operation based on polarization bistability in a telecommunication optical fiber,” J. Opt. Soc. Am. B 30, 2318–2325 (2013).
[Crossref]

J. Fatome, S. Pitois, P. Morin, D. Sugny, E. Assémat, A. Picozzi, H. R. Jauslin, G. Millot, V. V. Kozlov, and S. Wabnitz, “A universal optical all-fiber omnipolarizer,” Sci. Rep. 2, 938–946 (2012).
[Crossref]

E. Assémat, A. Picozzi, H. R. Jauslin, and D. Sugny, “Hamiltonian tools for the analysis of optical polarization control,” J. Opt. Soc. Am. B 29, 559–571 (2012).
[Crossref]

E. Assémat, D. Dargent, A. Picozzi, H. R. Jauslin, and D. Sugny, “Polarization control in spun and telecommunication optical fibers,” Opt. Lett. 36, 4038–4040 (2011).
[Crossref]

E. Assémat, A. Picozzi, H. R. Jauslin, and D. Sugny, “Instabilities of optical solitons and Hamiltonian singular solutions in a medium of finite extension,” Phys. Rev. A 84, 013809 (2011).
[Crossref]

S. Lagrange, D. Sugny, A. Picozzi, and H. R. Jauslin, “Singular tori as attractors of four-wave interaction systems,” Phys. Rev. E 81, 016202 (2010).
[Crossref]

E. Assémat, S. Lagrange, A. Picozzi, H. R. Jauslin, and D. Sugny, “Complete nonlinear polarization control in an optical fiber system,” Opt. Lett. 35, 2025–2027 (2010).
[Crossref]

D. Sugny, A. Picozzi, S. Lagrange, and H. R. Jauslin, “Role of singular tori in the dynamics of spatiotemporal nonlinear wave systems,” Phys. Rev. Lett. 103, 034102 (2009).
[Crossref]

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, “Polarization and modal attactors in conservative counterpropagating four-wave interaction,” Europhys. Lett. 70, 88 (2005), doi:10.1209/epl/i2004-10469-9.
[Crossref]

A. Picozzi, “Entropy and degree of polarization for nonlinear optical waves,” Opt. Lett. 29, 1653–1655 (2004).
[Crossref]

Pinto, A. N.

Pitois, S.

P.-Y. Bony, M. Guasoni, E. Assémat, S. Pitois, D. Sugny, A. Picozzi, H. R. Jauslin, and J. Fatome, “Optical flip-flop memory and data packet switching operation based on polarization bistability in a telecommunication optical fiber,” J. Opt. Soc. Am. B 30, 2318–2325 (2013).
[Crossref]

B. Stiller, P. Morin, D. M. Nguyen, J. Fatome, S. Pitois, E. Lantz, H. Maillotte, C. R. Menyuk, and T. Sylvestre, “Demonstration of polarization pulling using a fiber-optic parametric amplifier,” Opt. Express 20, 27248–27253 (2012).
[Crossref]

P. Morin, S. Pitois, and J. Fatome, “Simultaneous polarization attraction and Raman amplification of a light beam in optical fibers,” J. Opt. Soc. Am. B 29, 2046–2052 (2012).
[Crossref]

J. Fatome, S. Pitois, P. Morin, D. Sugny, E. Assémat, A. Picozzi, H. R. Jauslin, G. Millot, V. V. Kozlov, and S. Wabnitz, “A universal optical all-fiber omnipolarizer,” Sci. Rep. 2, 938–946 (2012).
[Crossref]

P. Morin, J. Fatome, C. Finot, S. Pitois, R. Claveau, and G. Millot, “All-optical nonlinear processing of both polarization state and intensity profile for 40 Gbit/s regeneration applications,” Opt. Express 19, 17158–17166 (2011).
[Crossref]

V. V. Kozlov, J. Fatome, P. Morin, S. Pitois, G. Millot, and S. Wabnitz, “Nonlinear repolarization dynamics in optical fibers: transient polarization attraction,” J. Opt. Soc. Am. B 28, 1782–1791 (2011).
[Crossref]

S. Pitois, J. Fatome, and G. Millot, “Polarization attraction using counter-propagating waves in optical fiber at telecommunication wavelengths,” Opt. Express 16, 6646–6651 (2008).
[Crossref]

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, “Polarization and modal attactors in conservative counterpropagating four-wave interaction,” Europhys. Lett. 70, 88 (2005), doi:10.1209/epl/i2004-10469-9.
[Crossref]

Popov, S.

S. Sergeyev and S. Popov, “Two-section fiber optic Raman polarizer,” IEEE J. Quantum Electron. 48, 56–60 (2012).
[Crossref]

S. Sergeyev, S. Popov, and A. T. Friberg, “Virtually isotropic transmission media with fiber Raman amplifier,” IEEE J. Quantum Electron. 46, 1492–1497 (2010).
[Crossref]

Prakash, H.

H. Prakash and D. Singh, “Change in coherence properties and degree of polarization of light propagating in a lossless isotropic nonlinear Kerr medium,” J. Phys. B 41, 045401 (2008).
[Crossref]

Primerov, N.

Sadovskii, D. A.

K. Efstathiou and D. A. Sadovskii, “Normalization and global analysis of perturbations of the hydrogen atom,” Rev. Mod. Phys. 82, 2099–2154 (2010).
[Crossref]

Santagiustina, M.

L. Ursini, M. Santagiustina, and L. Palmieri, “Raman nonlinear polarization pulling in the pump depleted regime in randomly birefringent fibers,” IEEE Photon. Technol. Lett. 23, 254–256 (2011).
[Crossref]

Sergeyev, S.

S. Sergeyev and S. Popov, “Two-section fiber optic Raman polarizer,” IEEE J. Quantum Electron. 48, 56–60 (2012).
[Crossref]

S. Sergeyev, “Fiber Raman amplification in a two-scale spun fiber,” Opt. Mater. Express 2, 1683–1689 (2012).
[Crossref]

S. Sergeyev, S. Popov, and A. T. Friberg, “Virtually isotropic transmission media with fiber Raman amplifier,” IEEE J. Quantum Electron. 46, 1492–1497 (2010).
[Crossref]

Sergeyev, S. V.

Shmilovitch, Z.

Singh, D.

H. Prakash and D. Singh, “Change in coherence properties and degree of polarization of light propagating in a lossless isotropic nonlinear Kerr medium,” J. Phys. B 41, 045401 (2008).
[Crossref]

Smelyanskiy, V. N.

M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyanskiy, D. G. Luchinsky, R. Mannella, and P. V. E. McClintock, “Activated escape of periodically driven systems,” Chaos 11, 587–594 (2001).
[Crossref]

Stiller, B.

Sugny, D.

P.-Y. Bony, M. Guasoni, E. Assémat, S. Pitois, D. Sugny, A. Picozzi, H. R. Jauslin, and J. Fatome, “Optical flip-flop memory and data packet switching operation based on polarization bistability in a telecommunication optical fiber,” J. Opt. Soc. Am. B 30, 2318–2325 (2013).
[Crossref]

E. Assémat, A. Picozzi, H. R. Jauslin, and D. Sugny, “Hamiltonian tools for the analysis of optical polarization control,” J. Opt. Soc. Am. B 29, 559–571 (2012).
[Crossref]

J. Fatome, S. Pitois, P. Morin, D. Sugny, E. Assémat, A. Picozzi, H. R. Jauslin, G. Millot, V. V. Kozlov, and S. Wabnitz, “A universal optical all-fiber omnipolarizer,” Sci. Rep. 2, 938–946 (2012).
[Crossref]

E. Assémat, A. Picozzi, H. R. Jauslin, and D. Sugny, “Instabilities of optical solitons and Hamiltonian singular solutions in a medium of finite extension,” Phys. Rev. A 84, 013809 (2011).
[Crossref]

E. Assémat, D. Dargent, A. Picozzi, H. R. Jauslin, and D. Sugny, “Polarization control in spun and telecommunication optical fibers,” Opt. Lett. 36, 4038–4040 (2011).
[Crossref]

E. Assémat, S. Lagrange, A. Picozzi, H. R. Jauslin, and D. Sugny, “Complete nonlinear polarization control in an optical fiber system,” Opt. Lett. 35, 2025–2027 (2010).
[Crossref]

S. Lagrange, D. Sugny, A. Picozzi, and H. R. Jauslin, “Singular tori as attractors of four-wave interaction systems,” Phys. Rev. E 81, 016202 (2010).
[Crossref]

D. Sugny, A. Picozzi, S. Lagrange, and H. R. Jauslin, “Role of singular tori in the dynamics of spatiotemporal nonlinear wave systems,” Phys. Rev. Lett. 103, 034102 (2009).
[Crossref]

Sylvestre, T.

Thevenaz, L.

Tratnik, M. V.

D. David, D. D. Holm, and M. V. Tratnik, “Hamiltonian chaos in nonlinear optical polarization dynamics,” Phys. Rep. 187, 281–367 (1990).
[Crossref]

Trillo, S.

S. Trillo and S. Wabnitz, “Intermittent spatial chaos in the polarization of counterpropagating beams in a birefringent optical fiber,” Phys. Rev. A 36, 3881–3884 (1987).
[Crossref]

Tur, M.

Turitsyn, K.

Turitsyn, K. S.

K. S. Turitsyn and S. Wabnitz, “Stability analysis of polarization attraction in optical fibers,” Opt. Commun. 307, 62–66 (2013).
[Crossref]

Ursini, L.

L. Ursini, M. Santagiustina, and L. Palmieri, “Raman nonlinear polarization pulling in the pump depleted regime in randomly birefringent fibers,” IEEE Photon. Technol. Lett. 23, 254–256 (2011).
[Crossref]

Vannucci, A.

Wabnitz, S.

V. V. Kozlov, M. Barozzi, A. Vannucci, and S. Wabnitz, “Lossless polarization attraction of co-propagating beams in telecom fibers,” J. Opt. Soc. Am. B 30, 530–540 (2013).
[Crossref]

K. S. Turitsyn and S. Wabnitz, “Stability analysis of polarization attraction in optical fibers,” Opt. Commun. 307, 62–66 (2013).
[Crossref]

J. Fatome, S. Pitois, P. Morin, D. Sugny, E. Assémat, A. Picozzi, H. R. Jauslin, G. Millot, V. V. Kozlov, and S. Wabnitz, “A universal optical all-fiber omnipolarizer,” Sci. Rep. 2, 938–946 (2012).
[Crossref]

M. Guasoni and S. Wabnitz, “Nonlinear polarizers based on four-wave mixing in high-birefringence optical fibers,” J. Opt. Soc. Am. B 29, 1511–1520 (2012).
[Crossref]

M. Guasoni, V. V. Kozlov, and S. Wabnitz, “Theory of polarization attraction in parametric amplifiers based on telecommunication fibers,” J. Opt. Soc. Am. B 29, 2710–2720 (2012).
[Crossref]

V. V. Kozlov, K. Turitsyn, and S. Wabnitz, “Nonlinear repolarization in optical fibers: polarization attraction with copropagating beams,” Opt. Lett. 36, 4050–4052 (2011).
[Crossref]

V. V. Kozlov, J. Nuno, and S. Wabnitz, “Theory of lossless polarization attraction in telecommunication fibers,” J. Opt. Soc. Am. B 28, 100–108 (2011).
[Crossref]

V. V. Kozlov, J. Fatome, P. Morin, S. Pitois, G. Millot, and S. Wabnitz, “Nonlinear repolarization dynamics in optical fibers: transient polarization attraction,” J. Opt. Soc. Am. B 28, 1782–1791 (2011).
[Crossref]

V. Kozlov, J. Nuno, J. D. Ania-Castanon, and S. Wabnitz, “Theoretical study of optical fiber Raman polarizers with counterpropagating beams,” J. Lightwave Technol. 29, 341–347 (2011).
[Crossref]

V. V. Kozlov and S. Wabnitz, “Instability of optical solitons in the boundary value problem for a medium of finite extension,” Lett. Math. Phys. 96, 405–413 (2011).
[Crossref]

V. V. Kozlov and S. Wabnitz, “Theoretical study of polarization attraction in high-birefringence and spun fibers,” Opt. Lett. 35, 3949–3951 (2010).
[Crossref]

S. Trillo and S. Wabnitz, “Intermittent spatial chaos in the polarization of counterpropagating beams in a birefringent optical fiber,” Phys. Rev. A 36, 3881–3884 (1987).
[Crossref]

Zadok, A.

Chaos (1)

M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyanskiy, D. G. Luchinsky, R. Mannella, and P. V. E. McClintock, “Activated escape of periodically driven systems,” Chaos 11, 587–594 (2001).
[Crossref]

Europhys. Lett. (1)

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, “Polarization and modal attactors in conservative counterpropagating four-wave interaction,” Europhys. Lett. 70, 88 (2005), doi:10.1209/epl/i2004-10469-9.
[Crossref]

IEEE J. Quantum Electron. (2)

S. Sergeyev, S. Popov, and A. T. Friberg, “Virtually isotropic transmission media with fiber Raman amplifier,” IEEE J. Quantum Electron. 46, 1492–1497 (2010).
[Crossref]

S. Sergeyev and S. Popov, “Two-section fiber optic Raman polarizer,” IEEE J. Quantum Electron. 48, 56–60 (2012).
[Crossref]

IEEE Photon. Technol. Lett. (1)

L. Ursini, M. Santagiustina, and L. Palmieri, “Raman nonlinear polarization pulling in the pump depleted regime in randomly birefringent fibers,” IEEE Photon. Technol. Lett. 23, 254–256 (2011).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (9)

B. Crosignani, B. Daino, and P. Di Porto, “Depolarization of light due to the optical Kerr effect in low-birefringence single-mode fibers,” J. Opt. Soc. Am. B 3, 1120–1123 (1986).
[Crossref]

P. Morin, S. Pitois, and J. Fatome, “Simultaneous polarization attraction and Raman amplification of a light beam in optical fibers,” J. Opt. Soc. Am. B 29, 2046–2052 (2012).
[Crossref]

M. Guasoni, V. V. Kozlov, and S. Wabnitz, “Theory of polarization attraction in parametric amplifiers based on telecommunication fibers,” J. Opt. Soc. Am. B 29, 2710–2720 (2012).
[Crossref]

M. Guasoni and S. Wabnitz, “Nonlinear polarizers based on four-wave mixing in high-birefringence optical fibers,” J. Opt. Soc. Am. B 29, 1511–1520 (2012).
[Crossref]

V. V. Kozlov, J. Nuno, and S. Wabnitz, “Theory of lossless polarization attraction in telecommunication fibers,” J. Opt. Soc. Am. B 28, 100–108 (2011).
[Crossref]

V. V. Kozlov, J. Fatome, P. Morin, S. Pitois, G. Millot, and S. Wabnitz, “Nonlinear repolarization dynamics in optical fibers: transient polarization attraction,” J. Opt. Soc. Am. B 28, 1782–1791 (2011).
[Crossref]

P.-Y. Bony, M. Guasoni, E. Assémat, S. Pitois, D. Sugny, A. Picozzi, H. R. Jauslin, and J. Fatome, “Optical flip-flop memory and data packet switching operation based on polarization bistability in a telecommunication optical fiber,” J. Opt. Soc. Am. B 30, 2318–2325 (2013).
[Crossref]

V. V. Kozlov, M. Barozzi, A. Vannucci, and S. Wabnitz, “Lossless polarization attraction of co-propagating beams in telecom fibers,” J. Opt. Soc. Am. B 30, 530–540 (2013).
[Crossref]

E. Assémat, A. Picozzi, H. R. Jauslin, and D. Sugny, “Hamiltonian tools for the analysis of optical polarization control,” J. Opt. Soc. Am. B 29, 559–571 (2012).
[Crossref]

J. Phys. B (1)

H. Prakash and D. Singh, “Change in coherence properties and degree of polarization of light propagating in a lossless isotropic nonlinear Kerr medium,” J. Phys. B 41, 045401 (2008).
[Crossref]

Lett. Math. Phys. (1)

V. V. Kozlov and S. Wabnitz, “Instability of optical solitons in the boundary value problem for a medium of finite extension,” Lett. Math. Phys. 96, 405–413 (2011).
[Crossref]

Opt. Commun. (1)

K. S. Turitsyn and S. Wabnitz, “Stability analysis of polarization attraction in optical fibers,” Opt. Commun. 307, 62–66 (2013).
[Crossref]

Opt. Express (7)

Opt. Lett. (6)

Opt. Mater. Express (1)

Phys. Rep. (1)

D. David, D. D. Holm, and M. V. Tratnik, “Hamiltonian chaos in nonlinear optical polarization dynamics,” Phys. Rep. 187, 281–367 (1990).
[Crossref]

Phys. Rev. A (2)

E. Assémat, A. Picozzi, H. R. Jauslin, and D. Sugny, “Instabilities of optical solitons and Hamiltonian singular solutions in a medium of finite extension,” Phys. Rev. A 84, 013809 (2011).
[Crossref]

S. Trillo and S. Wabnitz, “Intermittent spatial chaos in the polarization of counterpropagating beams in a birefringent optical fiber,” Phys. Rev. A 36, 3881–3884 (1987).
[Crossref]

Phys. Rev. E (1)

S. Lagrange, D. Sugny, A. Picozzi, and H. R. Jauslin, “Singular tori as attractors of four-wave interaction systems,” Phys. Rev. E 81, 016202 (2010).
[Crossref]

Phys. Rev. Lett. (2)

D. Sugny, A. Picozzi, S. Lagrange, and H. R. Jauslin, “Role of singular tori in the dynamics of spatiotemporal nonlinear wave systems,” Phys. Rev. Lett. 103, 034102 (2009).
[Crossref]

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, and P. W. Milonni, “Instabilities and chaos in the polarizations of counterpropagating light fields,” Phys. Rev. Lett. 58, 2432–2435 (1987).
[Crossref]

Rev. Mod. Phys. (1)

K. Efstathiou and D. A. Sadovskii, “Normalization and global analysis of perturbations of the hydrogen atom,” Rev. Mod. Phys. 82, 2099–2154 (2010).
[Crossref]

Sci. Rep. (1)

J. Fatome, S. Pitois, P. Morin, D. Sugny, E. Assémat, A. Picozzi, H. R. Jauslin, G. Millot, V. V. Kozlov, and S. Wabnitz, “A universal optical all-fiber omnipolarizer,” Sci. Rep. 2, 938–946 (2012).
[Crossref]

Other (2)

V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, 1989).

R. H. Cushman and L. M. Bates, Global Aspects of Classical Integrable Systems (Birkhauser, 1997).

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Figures (8)

Fig. 1.
Fig. 1.

Energy momentum diagram, H versus K, of the stationary system associated with Eq. (3) for identical powers of the counter-propagating beams, S0=J0=1. The singular states are plotted with red and blue lines. The inside red line corresponds to a line of bitori, the points of the blue line correspond to circles, and the red point on the top of the diagram refers to a point in the original phase space (see Ref. [36] for details concerning the definitions of singular tori and singular states).

Fig. 2.
Fig. 2.

Numerical simulations of the spatio-temporal Eq. (3): input (a) and output (b)–(d) signal SOPs S⃗(L) over the Poincaré sphere for three different values of L: (b) L=5, (c) L=8, (d) L=20. The blue line denotes the singular line of polarization attraction. The input pump J⃗(L)=(1/3,1/3,1/3) is fixed (green dot). Note that the efficiency of the attraction process toward the singular line increases as the normalized fiber length L increases. Also note that the spreading of the signal SOPs on the line of polarization attraction decreases as L increases.

Fig. 3.
Fig. 3.

Energy–momentum diagrams, H versus K, and signal SOPs corresponding to the configurations of Fig. 2. The red points represent the positions of the final SOPs in the (H,K)-diagram, the panels (a)–(d) are related to the same panels of Fig. 2. Panel (a) displays the initial condition and thus the distribution of point spreads all over the diagram. In panels (b)–(d), we observe an increasing attraction toward the parabolic line of equation H=4(7/2)K2. A zoom has been used in the panels (b)–(d).

Fig. 4.
Fig. 4.

Evolution of the distance d, defined in Eq. (12) as a function of the fiber length L for the input pump SOPs J⃗(L)=(1/3,1/3,1/3) (solid line) and J⃗(L)=(1,0,0) (dashed line). The distance is normalized with respect to its value for L=0 to improve the visualization of the convergence toward the singular line of attraction.

Fig. 5.
Fig. 5.

(a) Three particular stationary states on the line of polarization attraction corresponding to the simulations of Figs. 1 and 2, and corresponding spatial evolutions for (b) L=5, (c) L=8, and (d) L=20 For the sake of clarity, only the S2-component has been reported. In panel (a): S⃗a=(0.80,0.58,0.41) [gray dot in (a), gray lines in (b)–(d)], S⃗b=(0.25,0.79,0.63) [red dot in (a), red lines in (b)–(d)], and S⃗c=(0.25,0.90,0.31) [black dot in (a), black lines in (b)–(d)].

Fig. 6.
Fig. 6.

Numerical simulations of the spatio-temporal Eq. (3) analogous to those reported in Figs. 2 and 3, except that the pump SOP is now J⃗(L)=(1,0,0) [green dot in panels (a), (c)]. The initial condition of the signal SOPs is the same as in Fig. 2(a). (a), (b) and (c), (d) report the output signal SOPs S⃗(L) and the corresponding positions on the energy–momentum diagram for two different normalized fiber lengths [L=5 (a), (b) and L=20 (c), (d)]. The blue line in panels (a), (c) represents the line of singular polarization states on the Poincaré sphere. Note that, for L=20, we observe a stronger attraction toward the singular attraction line.

Fig. 7.
Fig. 7.

Numerical simulation of the spatio-temporal Eq. (3) analogous to that reported in Figs. 2 and 3, except that the pump SOPs is now J⃗(L)=(0,1,0) [green dot in (a)]. The initial condition is the same as in Fig. 2(a). The fiber length is fixed to L=30. In panel (a), the singular line is a blue circle of the equation S2=0.6. The black and red dots on the Poincaré sphere and the corresponding red dots in the energy–momentum diagram refer to the signal SOPs at the fiber output S⃗(L). In the energy–momentum diagram, the value of K is close to 0.4. Note that some signal SOPs are attracted toward the state S2=1 [red dots in Fig. 7(a)]. The SOP S2=1 is associated to the singular line, which delimits the boundary of the energy–momentum diagram, represented by a cyan line in panel (b). This singular line then also plays a role of attractor for the system [see red open circles in panel (b)]. In the energy–momentum diagram, such points are characterized by K0.

Fig. 8.
Fig. 8.

Numerical simulation of the spatio-temporal Eq. (3) analogous to that reported in Figs. 2 and 3, except that now the pump SOPs J⃗(L)=(1,0,0) and the powers are S0=1.5, J0=1.0. The initial condition is the same as in Fig. 2(a). Two different fiber lengths have been considered: L=5 (a), (b) and L=20 (c), (d). The pink lines in (a) and (c) denote the singular line of polarization attraction. The blue and the pink lines in the energy–momentum diagrams (b), (d) represent the lines of singular states for the cases S0=J0=1 and S0=1.5,J0=1.0, respectively. The black points in panels (a), (c) depict signal SOPs associated with a stationary dynamics while gray points are associated to nonstationary dynamics (plotted at time t=900). In the energy–momentum diagrams (b) and (d), only the stationary states have been plotted.

Equations (16)

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

(t+vgz)Ex=iγ(|Ex|2+23|Ey|2+2|E¯x|2+23|E¯y|2)Ex+2iγ3EyEyEx¯*(t+vgz)Ey=iγ(|Ey|2+23|Ex|2+2|E¯y|2+23|E¯x|2)Ey+2iγ3ExExEy¯*(tvgz)E¯x=iγ(|E¯x|2+23|E¯y|2+2|Ex|2+23|Ey|2)E¯x+2iγ3E¯yEyEx*(tvgz)E¯y=iγ(|E¯y|2+23|E¯x|2+2|Ey|2+23|Ex|2)E¯y+2iγ3E¯xExEy*,
u=Ex+iEy12,v=ExiEy12,u¯=E¯x+iE¯y12,v¯=E¯xiE¯y12,
(t+vgz)u=iγ(5|u|2+6|v|2+6|u¯|2+10|v¯|2)u+iγ(6v¯*u¯+2u¯*v¯+vu*)v(t+vgz)v=iγ(5|v|2+6|u|2+6|v¯|2+10|u¯|2)v+iγ(6u¯*v¯+2v¯*u¯+uv*)u(tvgz)u¯=iγ(5|u¯|2+6|v¯|2+6|u|2+10|v|2)u¯+iγ(6v*u+2u*v+v¯u¯*)v¯(tvgz)v¯=iγ(5|v¯|2+6|u¯|2+6|v|2+10|u|2)v¯+iγ(6u*v+2v*u+u¯v¯*)u¯.
{S1=i(u*vuv*)S2=u*v+uv*S3=|u|2|v|2,{J1=i(u¯*v¯u¯v¯*)J2=u¯*v¯+u¯v¯*J3=|u¯|2|v¯|2.
{S⃗t+S⃗z=S⃗×(IsS⃗)+S⃗×(IiJ⃗)J⃗tJ⃗z=J⃗×(IsJ⃗)+J⃗×(IiS⃗),
H=4S1J18S2J2+4S3J3S22J22,
H=472K2,
H=6K2±25K,
H=432K2.
S3(L)=±1S1(L)2S2(L)2S2(L)=KJ2(L),
H=4S1(L)J1(L)8(KJ2(L))J2(L)±4J3(L)1S1(L)2(KJ2(L))2(KJ2(L))2J2(L)2.
472K2=4S1(L)J1(L)8(KJ2(L))J2(L)±4J3(L)1S1(L)2(KJ2(L))2(KJ2(L))2J2(L)2.
π1=S2J2π2=J3S3J1S1π3=S1J3+S3J1,
π32+π22(S0214(K+π1)2)(J0214(Kπ1)2)=0.
H=4π2+32π152K2.
d(L)=i(HiHs)2,

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