Abstract

The study of the polarization dynamics of two counterpropagating beams in optical fibers has recently been the subject of a growing renewed interest, from both the theoretical and experimental points of view. This system exhibits a phenomenon of polarization attraction, which can be used to achieve a complete polarization of an initially unpolarized signal beam, almost without any loss of energy. Along the same way, an arbitrary polarization state of the signal beam can be controlled and converted into any other desired state of polarization, by adjusting the polarization state of the counterpropagating pump beam. These properties have been demonstrated in various different types of optical fibers, i.e., isotropic fibers, spun fibers, and telecommunication optical fibers. This article is aimed at providing a rather complete understanding of this phenomenon of polarization attraction on the basis of new mathematical techniques recently developed for the study of Hamiltonian singularities. In particular, we show the essential role that play the peculiar topological properties of singular tori in the process of polarization attraction. We provide here a pedagogical introduction to this geometric approach of Hamiltonian singularities and give a unified description of the polarization attraction phenomenon in various types of optical fiber systems.

© 2012 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]
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  3. 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, 88–94 (2005).
    [CrossRef]
  4. 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]
  5. 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]
  6. E. Assemat, 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]
  7. V. V. Kozlov and S. Wabnitz, “Theoretical study of polarization attraction in high birefringence and spun fibers,” Opt. Lett. 35, 3949–3951 (2010).
    [CrossRef]
  8. 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]
  9. 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]
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    [CrossRef]
  11. E. Assémat, D. Dargent, A. Picozzi, H. R. Jauslin, and D. Sugny, “Polarization control in spun and telecommunication optical fibers,” Opt. Lett. 36, 4038–4041 (2011).
    [CrossRef]
  12. E. Assémat, C. Michel, A. Picozzi, H. R. Jauslin, and D. Sugny, “Manifestation of Hamiltonian monodromy in nonlinear wave systems,” Phys. Rev. Lett. 106, 014101 (2011).
    [CrossRef]
  13. S. Pitois, J. Fatome, and G. Millot, “Polarization attraction using counterpropagating waves in optical fiber at telecommunication wavelengths,” Opt. Express 16, 6646–6651 (2008).
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    [CrossRef]
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    [CrossRef]
  16. 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]
  17. D. J. Gauthier, M. S. Malcuit, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams,” Phys. Rev. Lett. 64, 1721–1724 (1990).
    [CrossRef]
  18. G. Gregori and S. Wabnitz, “New exact solutions and bifurcations in the spatial distribution of polarization in third-order nonlinear optical interactions,” Phys. Rev. Lett. 56, 600–603 (1986).
    [CrossRef]
  19. 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]
  20. M. V. Tratnik and J. E. Sipe, “Nonlinear polarization dynamics. III. Spatial polarization chaos in counterpropagating beams,” Phys. Rev. A 36, 4817–4822 (1987).
    [CrossRef]
  21. 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]
  22. M. V. Tratnik and J. E. Sipe, “Nonlinear polarization dynamics. II. Counterpropagating-beam equations: New simple solutions and the possibilities for chaos,” Phys. Rev. A 35, 2976–2988 (1987).
    [CrossRef]
  23. A. L. Gaeta and R. W. Boyd, “Transverse instabilities in the polarizations and intensities of counterpropagating light waves,” Phys. Rev. A 48, 1610–1624 (1993).
    [CrossRef]
  24. S. Pitois, G. Millot, and S. Wabnitz, “Polarization domain wall solitons with counterpropagating laser beams,” Phys. Rev. Lett. 81, 1409–1412 (1998).
    [CrossRef]
  25. S. Pitois, G. Millot, and S. Wabnitz, “Nonlinear polarization dynamics of counterpropagating waves in an isotropic optical fiber: theory and experiments,” J. Opt. Soc. Am. B 18, 432–443 (2001).
    [CrossRef]
  26. S. Wabnitz, “Chiral polarization solitons in elliptically birefringent spun optical fibers,” Opt. Lett. 34, 908–910 (2009).
    [CrossRef]
  27. D. David, D. D. Holm, and M. V. Tratnik, “Hamiltonian chaos in nonlinear optical polarization dynamics,” Phys. Rep. 187, 281–367 (1990).
    [CrossRef]
  28. R. H. Cushman and L. M. Bates, Global Aspects of Classical Integrable Systems (Birkhauser, 1997).
  29. K. Efstathiou and D. A. Sadovskii, “Normalization and global analysis of perturbations of the hydrogen atom,” Rev. Mod. Phys. 82, 2099–2154 (2010).
    [CrossRef]
  30. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, 2003).
  31. E. Assémat, K. Efstathiou, M. Joyeux, and D. Sugny, “Fractional bidromy in the vibrational spectrum of HOCl,” Phys. Rev. Lett. 104, 113002 (2010).
    [CrossRef]
  32. R. H. Cushman, H. R. Dullin, A. Giacobbe, D. D. Holm, M. Joyeux, P. Lynch, D. A. Sadovskii, and B. I. Zhilinskii, “CO2 molecule as a quantum realization of the 1∶1∶2 resonant swing-spring with monodromy,” Phys. Rev. Lett. 93, 024302 (2004).
    [CrossRef]
  33. D. Sugny, P. Mardesic, M. Pelletier, J. Jebrane, and H. R. Jauslin, “Fractional Hamiltonian monodromy from a Gauss-Manin monodromy,” J. Math. Phys. 49, 042701 (2008).
    [CrossRef]
  34. V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, 1989).
  35. M. Grenier, H.-R. Jauslin, C. Klein, and V. B. Matveev, “Wave attraction in resonant counter-propagating wave systems,” J. Math. Phys. 52, 082704 (2011).
    [CrossRef]
  36. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, 1965).
  37. M. Joyeux, “Classical dynamics of the 1∶1, 1∶2 and 1∶3 resonance Hamiltonians,” Chem. Phys. 203, 281–307 (1996).
    [CrossRef]

2011 (8)

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

E. Assemat, 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, 4038–4041 (2011).
[CrossRef]

E. Assémat, C. Michel, A. Picozzi, H. R. Jauslin, and D. Sugny, “Manifestation of Hamiltonian monodromy in nonlinear wave systems,” Phys. Rev. Lett. 106, 014101 (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. 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]

M. Grenier, H.-R. Jauslin, C. Klein, and V. B. Matveev, “Wave attraction in resonant counter-propagating wave systems,” J. Math. Phys. 52, 082704 (2011).
[CrossRef]

2010 (6)

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

E. Assémat, K. Efstathiou, M. Joyeux, and D. Sugny, “Fractional bidromy in the vibrational spectrum of HOCl,” Phys. Rev. Lett. 104, 113002 (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. Assemat, 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]

J. Fatome, S. Pitois, P. Morin, and G. Millot, “Observation of light-by-light polarization control and stabilization in optical fibre for telecommunication applications,” Opt. Express 18, 15311–15317 (2010).
[CrossRef]

2009 (2)

S. Wabnitz, “Chiral polarization solitons in elliptically birefringent spun optical fibers,” Opt. Lett. 34, 908–910 (2009).
[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]

2008 (3)

2005 (1)

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, 88–94 (2005).
[CrossRef]

2004 (1)

R. H. Cushman, H. R. Dullin, A. Giacobbe, D. D. Holm, M. Joyeux, P. Lynch, D. A. Sadovskii, and B. I. Zhilinskii, “CO2 molecule as a quantum realization of the 1∶1∶2 resonant swing-spring with monodromy,” Phys. Rev. Lett. 93, 024302 (2004).
[CrossRef]

2001 (1)

2000 (1)

1998 (1)

S. Pitois, G. Millot, and S. Wabnitz, “Polarization domain wall solitons with counterpropagating laser beams,” Phys. Rev. Lett. 81, 1409–1412 (1998).
[CrossRef]

1996 (1)

M. Joyeux, “Classical dynamics of the 1∶1, 1∶2 and 1∶3 resonance Hamiltonians,” Chem. Phys. 203, 281–307 (1996).
[CrossRef]

1993 (1)

A. L. Gaeta and R. W. Boyd, “Transverse instabilities in the polarizations and intensities of counterpropagating light waves,” Phys. Rev. A 48, 1610–1624 (1993).
[CrossRef]

1990 (2)

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

D. J. Gauthier, M. S. Malcuit, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams,” Phys. Rev. Lett. 64, 1721–1724 (1990).
[CrossRef]

1987 (4)

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]

M. V. Tratnik and J. E. Sipe, “Nonlinear polarization dynamics. III. Spatial polarization chaos in counterpropagating beams,” Phys. Rev. A 36, 4817–4822 (1987).
[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]

M. V. Tratnik and J. E. Sipe, “Nonlinear polarization dynamics. II. Counterpropagating-beam equations: New simple solutions and the possibilities for chaos,” Phys. Rev. A 35, 2976–2988 (1987).
[CrossRef]

1986 (1)

G. Gregori and S. Wabnitz, “New exact solutions and bifurcations in the spatial distribution of polarization in third-order nonlinear optical interactions,” Phys. Rev. Lett. 56, 600–603 (1986).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, 1965).

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]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, 2003).

Arnold, V. I.

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

Assemat, E.

E. Assemat, 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. Assemat, 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]

Assémat, E.

E. Assémat, D. Dargent, A. Picozzi, H. R. Jauslin, and D. Sugny, “Polarization control in spun and telecommunication optical fibers,” Opt. Lett. 36, 4038–4041 (2011).
[CrossRef]

E. Assémat, C. Michel, A. Picozzi, H. R. Jauslin, and D. Sugny, “Manifestation of Hamiltonian monodromy in nonlinear wave systems,” Phys. Rev. Lett. 106, 014101 (2011).
[CrossRef]

E. Assémat, K. Efstathiou, M. Joyeux, and D. Sugny, “Fractional bidromy in the vibrational spectrum of HOCl,” Phys. Rev. Lett. 104, 113002 (2010).
[CrossRef]

Bates, L. M.

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

Bennink, R. S.

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 and R. W. Boyd, “Transverse instabilities in the polarizations and intensities of counterpropagating light waves,” Phys. Rev. A 48, 1610–1624 (1993).
[CrossRef]

D. J. Gauthier, M. S. Malcuit, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams,” Phys. Rev. Lett. 64, 1721–1724 (1990).
[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]

Claveau, R.

Cushman, R. H.

R. H. Cushman, H. R. Dullin, A. Giacobbe, D. D. Holm, M. Joyeux, P. Lynch, D. A. Sadovskii, and B. I. Zhilinskii, “CO2 molecule as a quantum realization of the 1∶1∶2 resonant swing-spring with monodromy,” Phys. Rev. Lett. 93, 024302 (2004).
[CrossRef]

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

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]

Dullin, H. R.

R. H. Cushman, H. R. Dullin, A. Giacobbe, D. D. Holm, M. Joyeux, P. Lynch, D. A. Sadovskii, and B. I. Zhilinskii, “CO2 molecule as a quantum realization of the 1∶1∶2 resonant swing-spring with monodromy,” Phys. Rev. Lett. 93, 024302 (2004).
[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]

E. Assémat, K. Efstathiou, M. Joyeux, and D. Sugny, “Fractional bidromy in the vibrational spectrum of HOCl,” Phys. Rev. Lett. 104, 113002 (2010).
[CrossRef]

Fatome, J.

Finot, C.

Fisher, R. A.

Gaeta, A. L.

A. L. Gaeta and R. W. Boyd, “Transverse instabilities in the polarizations and intensities of counterpropagating light waves,” Phys. Rev. A 48, 1610–1624 (1993).
[CrossRef]

D. J. Gauthier, M. S. Malcuit, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams,” Phys. Rev. Lett. 64, 1721–1724 (1990).
[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]

Gauthier, D. J.

D. J. Gauthier, M. S. Malcuit, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams,” Phys. Rev. Lett. 64, 1721–1724 (1990).
[CrossRef]

Giacobbe, A.

R. H. Cushman, H. R. Dullin, A. Giacobbe, D. D. Holm, M. Joyeux, P. Lynch, D. A. Sadovskii, and B. I. Zhilinskii, “CO2 molecule as a quantum realization of the 1∶1∶2 resonant swing-spring with monodromy,” Phys. Rev. Lett. 93, 024302 (2004).
[CrossRef]

Gregori, G.

G. Gregori and S. Wabnitz, “New exact solutions and bifurcations in the spatial distribution of polarization in third-order nonlinear optical interactions,” Phys. Rev. Lett. 56, 600–603 (1986).
[CrossRef]

Grenier, M.

M. Grenier, H.-R. Jauslin, C. Klein, and V. B. Matveev, “Wave attraction in resonant counter-propagating wave systems,” J. Math. Phys. 52, 082704 (2011).
[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, 88–94 (2005).
[CrossRef]

Heebner, J. E.

Holm, D. D.

R. H. Cushman, H. R. Dullin, A. Giacobbe, D. D. Holm, M. Joyeux, P. Lynch, D. A. Sadovskii, and B. I. Zhilinskii, “CO2 molecule as a quantum realization of the 1∶1∶2 resonant swing-spring with monodromy,” Phys. Rev. Lett. 93, 024302 (2004).
[CrossRef]

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.

E. Assemat, 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, C. Michel, A. Picozzi, H. R. Jauslin, and D. Sugny, “Manifestation of Hamiltonian monodromy in nonlinear wave systems,” Phys. Rev. Lett. 106, 014101 (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, 4038–4041 (2011).
[CrossRef]

E. Assemat, 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]

D. Sugny, P. Mardesic, M. Pelletier, J. Jebrane, and H. R. Jauslin, “Fractional Hamiltonian monodromy from a Gauss-Manin monodromy,” J. Math. Phys. 49, 042701 (2008).
[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, 88–94 (2005).
[CrossRef]

Jauslin, H.-R.

M. Grenier, H.-R. Jauslin, C. Klein, and V. B. Matveev, “Wave attraction in resonant counter-propagating wave systems,” J. Math. Phys. 52, 082704 (2011).
[CrossRef]

Jebrane, J.

D. Sugny, P. Mardesic, M. Pelletier, J. Jebrane, and H. R. Jauslin, “Fractional Hamiltonian monodromy from a Gauss-Manin monodromy,” J. Math. Phys. 49, 042701 (2008).
[CrossRef]

Joyeux, M.

E. Assémat, K. Efstathiou, M. Joyeux, and D. Sugny, “Fractional bidromy in the vibrational spectrum of HOCl,” Phys. Rev. Lett. 104, 113002 (2010).
[CrossRef]

R. H. Cushman, H. R. Dullin, A. Giacobbe, D. D. Holm, M. Joyeux, P. Lynch, D. A. Sadovskii, and B. I. Zhilinskii, “CO2 molecule as a quantum realization of the 1∶1∶2 resonant swing-spring with monodromy,” Phys. Rev. Lett. 93, 024302 (2004).
[CrossRef]

M. Joyeux, “Classical dynamics of the 1∶1, 1∶2 and 1∶3 resonance Hamiltonians,” Chem. Phys. 203, 281–307 (1996).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, 2003).

Klein, C.

M. Grenier, H.-R. Jauslin, C. Klein, and V. B. Matveev, “Wave attraction in resonant counter-propagating wave systems,” J. Math. Phys. 52, 082704 (2011).
[CrossRef]

Kozlov, V. V.

Lagrange, S.

E. Assemat, 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]

Lynch, P.

R. H. Cushman, H. R. Dullin, A. Giacobbe, D. D. Holm, M. Joyeux, P. Lynch, D. A. Sadovskii, and B. I. Zhilinskii, “CO2 molecule as a quantum realization of the 1∶1∶2 resonant swing-spring with monodromy,” Phys. Rev. Lett. 93, 024302 (2004).
[CrossRef]

Malcuit, M. S.

D. J. Gauthier, M. S. Malcuit, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams,” Phys. Rev. Lett. 64, 1721–1724 (1990).
[CrossRef]

Mardesic, P.

D. Sugny, P. Mardesic, M. Pelletier, J. Jebrane, and H. R. Jauslin, “Fractional Hamiltonian monodromy from a Gauss-Manin monodromy,” J. Math. Phys. 49, 042701 (2008).
[CrossRef]

Matveev, V. B.

M. Grenier, H.-R. Jauslin, C. Klein, and V. B. Matveev, “Wave attraction in resonant counter-propagating wave systems,” J. Math. Phys. 52, 082704 (2011).
[CrossRef]

Michel, C.

E. Assémat, C. Michel, A. Picozzi, H. R. Jauslin, and D. Sugny, “Manifestation of Hamiltonian monodromy in nonlinear wave systems,” Phys. Rev. Lett. 106, 014101 (2011).
[CrossRef]

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.

Nuno, J.

Pelletier, M.

D. Sugny, P. Mardesic, M. Pelletier, J. Jebrane, and H. R. Jauslin, “Fractional Hamiltonian monodromy from a Gauss-Manin monodromy,” J. Math. Phys. 49, 042701 (2008).
[CrossRef]

Picozzi, A.

E. Assémat, C. Michel, A. Picozzi, H. R. Jauslin, and D. Sugny, “Manifestation of Hamiltonian monodromy in nonlinear wave systems,” Phys. Rev. Lett. 106, 014101 (2011).
[CrossRef]

E. Assemat, 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, 4038–4041 (2011).
[CrossRef]

E. Assemat, 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]

A. Picozzi, “Spontaneous polarization induced by natural thermalization of incoherent light,” Opt. Express 16, 17171–17185 (2008).
[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, 88–94 (2005).
[CrossRef]

Pitois, S.

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]

R. H. Cushman, H. R. Dullin, A. Giacobbe, D. D. Holm, M. Joyeux, P. Lynch, D. A. Sadovskii, and B. I. Zhilinskii, “CO2 molecule as a quantum realization of the 1∶1∶2 resonant swing-spring with monodromy,” Phys. Rev. Lett. 93, 024302 (2004).
[CrossRef]

Sipe, J. E.

M. V. Tratnik and J. E. Sipe, “Nonlinear polarization dynamics. II. Counterpropagating-beam equations: New simple solutions and the possibilities for chaos,” Phys. Rev. A 35, 2976–2988 (1987).
[CrossRef]

M. V. Tratnik and J. E. Sipe, “Nonlinear polarization dynamics. III. Spatial polarization chaos in counterpropagating beams,” Phys. Rev. A 36, 4817–4822 (1987).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, 1965).

Sugny, D.

E. Assémat, D. Dargent, A. Picozzi, H. R. Jauslin, and D. Sugny, “Polarization control in spun and telecommunication optical fibers,” Opt. Lett. 36, 4038–4041 (2011).
[CrossRef]

E. Assémat, C. Michel, A. Picozzi, H. R. Jauslin, and D. Sugny, “Manifestation of Hamiltonian monodromy in nonlinear wave systems,” Phys. Rev. Lett. 106, 014101 (2011).
[CrossRef]

E. Assemat, 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, K. Efstathiou, M. Joyeux, and D. Sugny, “Fractional bidromy in the vibrational spectrum of HOCl,” Phys. Rev. Lett. 104, 113002 (2010).
[CrossRef]

E. Assemat, 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]

D. Sugny, P. Mardesic, M. Pelletier, J. Jebrane, and H. R. Jauslin, “Fractional Hamiltonian monodromy from a Gauss-Manin monodromy,” J. Math. Phys. 49, 042701 (2008).
[CrossRef]

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]

M. V. Tratnik and J. E. Sipe, “Nonlinear polarization dynamics. II. Counterpropagating-beam equations: New simple solutions and the possibilities for chaos,” Phys. Rev. A 35, 2976–2988 (1987).
[CrossRef]

M. V. Tratnik and J. E. Sipe, “Nonlinear polarization dynamics. III. Spatial polarization chaos in counterpropagating beams,” Phys. Rev. A 36, 4817–4822 (1987).
[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]

Wabnitz, S.

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, 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]

S. Wabnitz, “Chiral polarization solitons in elliptically birefringent spun optical fibers,” Opt. Lett. 34, 908–910 (2009).
[CrossRef]

S. Pitois, G. Millot, and S. Wabnitz, “Nonlinear polarization dynamics of counterpropagating waves in an isotropic optical fiber: theory and experiments,” J. Opt. Soc. Am. B 18, 432–443 (2001).
[CrossRef]

S. Pitois, G. Millot, and S. Wabnitz, “Polarization domain wall solitons with counterpropagating laser beams,” Phys. Rev. Lett. 81, 1409–1412 (1998).
[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]

G. Gregori and S. Wabnitz, “New exact solutions and bifurcations in the spatial distribution of polarization in third-order nonlinear optical interactions,” Phys. Rev. Lett. 56, 600–603 (1986).
[CrossRef]

Zhilinskii, B. I.

R. H. Cushman, H. R. Dullin, A. Giacobbe, D. D. Holm, M. Joyeux, P. Lynch, D. A. Sadovskii, and B. I. Zhilinskii, “CO2 molecule as a quantum realization of the 1∶1∶2 resonant swing-spring with monodromy,” Phys. Rev. Lett. 93, 024302 (2004).
[CrossRef]

Chem. Phys. (1)

M. Joyeux, “Classical dynamics of the 1∶1, 1∶2 and 1∶3 resonance Hamiltonians,” Chem. Phys. 203, 281–307 (1996).
[CrossRef]

Europhys. Lett. (1)

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, 88–94 (2005).
[CrossRef]

J. Math. Phys. (2)

D. Sugny, P. Mardesic, M. Pelletier, J. Jebrane, and H. R. Jauslin, “Fractional Hamiltonian monodromy from a Gauss-Manin monodromy,” J. Math. Phys. 49, 042701 (2008).
[CrossRef]

M. Grenier, H.-R. Jauslin, C. Klein, and V. B. Matveev, “Wave attraction in resonant counter-propagating wave systems,” J. Math. Phys. 52, 082704 (2011).
[CrossRef]

J. Opt. Soc. Am. B (3)

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. Express (4)

Opt. Lett. (5)

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 (5)

M. V. Tratnik and J. E. Sipe, “Nonlinear polarization dynamics. III. Spatial polarization chaos in counterpropagating beams,” Phys. Rev. A 36, 4817–4822 (1987).
[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]

M. V. Tratnik and J. E. Sipe, “Nonlinear polarization dynamics. II. Counterpropagating-beam equations: New simple solutions and the possibilities for chaos,” Phys. Rev. A 35, 2976–2988 (1987).
[CrossRef]

A. L. Gaeta and R. W. Boyd, “Transverse instabilities in the polarizations and intensities of counterpropagating light waves,” Phys. Rev. A 48, 1610–1624 (1993).
[CrossRef]

E. Assemat, 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]

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. (8)

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]

E. Assémat, C. Michel, A. Picozzi, H. R. Jauslin, and D. Sugny, “Manifestation of Hamiltonian monodromy in nonlinear wave systems,” Phys. Rev. Lett. 106, 014101 (2011).
[CrossRef]

D. J. Gauthier, M. S. Malcuit, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams,” Phys. Rev. Lett. 64, 1721–1724 (1990).
[CrossRef]

G. Gregori and S. Wabnitz, “New exact solutions and bifurcations in the spatial distribution of polarization in third-order nonlinear optical interactions,” Phys. Rev. Lett. 56, 600–603 (1986).
[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]

S. Pitois, G. Millot, and S. Wabnitz, “Polarization domain wall solitons with counterpropagating laser beams,” Phys. Rev. Lett. 81, 1409–1412 (1998).
[CrossRef]

E. Assémat, K. Efstathiou, M. Joyeux, and D. Sugny, “Fractional bidromy in the vibrational spectrum of HOCl,” Phys. Rev. Lett. 104, 113002 (2010).
[CrossRef]

R. H. Cushman, H. R. Dullin, A. Giacobbe, D. D. Holm, M. Joyeux, P. Lynch, D. A. Sadovskii, and B. I. Zhilinskii, “CO2 molecule as a quantum realization of the 1∶1∶2 resonant swing-spring with monodromy,” Phys. Rev. Lett. 93, 024302 (2004).
[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]

Other (4)

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, 2003).

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).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, 1965).

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Figures (15)

Fig. 1.
Fig. 1.

Polarization attraction toward a continuous line of polarization states in HSBFs. The attraction effect is demonstrated by integrating numerically the spatiotemporal Eq. (1) on the Poincaré sphere for φ=π/4. The green and red dots denote respectively the initial (S(0)) and final (S(L)) SOPs of the signal. The yellow dot represents the SOP of the pump, which is kept fixed at the fiber output: J(L)=(0,1,0) for a fiber length L=5. The blue line is calculated analytically in Sec. 4. Note that the dashed part corresponds to unstable solutions that cannot be reached by the spatiotemporal dynamics.

Fig. 2.
Fig. 2.

Examples of tori of an integrable Hamiltonian phase space of dimension four. A system can freely oscillate around a regular (standard) torus (a), but its evolution can also be blocked by the presence of a pinched point in a singly (b) or doubly (c) pinched torus. The singular pinched tori can be viewed as a two-dimensional generalization of the concept of separatrix, well-known for systems with 1 degree of freedom. A bitorus and a curled torus are represented in (d) and (e). They can be constructed by gluing two regular tori along a circle, with an additional twist in the case of a curled torus.

Fig. 3.
Fig. 3.

Schematic illustration of the reduction process, which maps the main phase space, i.e., the two Poincaré spheres, S2×S2, toward the reduced phase space, which has the form of a deformed sphere with two conical singularities. The reduced phase space is defined by Eq. (10) for the IF and Eq. (13) for the RBF and HBSF.

Fig. 4.
Fig. 4.

IF model: Intersection of the reduced phase space (red) and of the Hamiltonian surface (blue) with S0=J0=1, K=0, and H=1. We observe that the reduced phase space for K=0 has two points with noncontinuous derivative. The intersection with the Hamiltonian surface H=cst contains both points, which as a consequence is a singular torus (see the text).

Fig. 5.
Fig. 5.

IF model: Illustration of the relation between the points of the reduced phase space and the points of the singular torus. The dashed line depicts the intersection with the Hamiltonian surface.

Fig. 6.
Fig. 6.

IF model: Energy-momentum diagram for S0=J0=1. The gray region corresponds to regular tori in the phase space and the red dot to a doubly pinched torus.

Fig. 7.
Fig. 7.

RBF model: Energy-momentum diagram for S0=J0=1. The gray region corresponds to regular tori in the phase space and the red dot to a sphere.

Fig. 8.
Fig. 8.

RBF model: The Hamiltonian surface (blue surface) and the reduced phase space (red surface) intersect along a segment (red line) that draws a sphere in the main phase space.

Fig. 9.
Fig. 9.

HBSF model: Intersection of the reduced phase space defined by Eq. (13) and the Hamiltonian of Eq. (15) with φ=π/4, K=0, and H=0.5.

Fig. 10.
Fig. 10.

HBSF model: Energy-momentum diagram with φ=π/4. Each point of the gray region corresponds to a regular torus in the main phase space and each point of the red line corresponds to a bitorus.

Fig. 11.
Fig. 11.

RBF model: The distance of the pump SOP to the predicted point decreases as L increases. The distance plotted here is the average of the distances of the 64 different signal inputs uniformly distributed on the Poincaré sphere (green dots in Fig. 1) with Jx(L)=Jy(L)=0.7 and Jz(L)=2/10.

Fig. 12.
Fig. 12.

RBF model: Numerical simulations of the spatiotemporal system on the Poincaré sphere with Δ=0.05. The green and red dots denote respectively the initial (S(0)) and final (S(L)) SOPs of the signal. The yellow dot displays the fixed pump SOP: J(L)=(1,0,0) for L=15.

Fig. 13.
Fig. 13.

HBSF model: The gray domain depicts the possible values of K as a function of Jz(L) when Jx(L)=0.

Fig. 14.
Fig. 14.

HBSF model: Monotonic (up) and nonmonotonic (bottom) behaviors of the stationary solutions corresponding respectively to the stable and unstable parts of the eight-shaped line. Only the monotonic stationary solutions are stable, whereas the stationary solutions that exhibit an oscillatory behavior are unstable. This explains why only half of the eight-shaped closed curve plays the role of an attractor in the spatiotemporal dynamics.

Fig. 15.
Fig. 15.

HBSF model: Two series of numerical simulations with two different values of J(L) that show the possibility of producing an elliptic polarization with two linear polarizations. The plain and dashed black lines depict respectively the stable and unstable parts of the figure eight. The two large yellow dots are associated to the two values of the pump J(L). The small dots on the equator depict the injected polarization signal S(0) and the small dots outside the equator (on eight-shaped curves) the outgoing polarization of the signal after the interaction.

Equations (53)

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{St+Sξ=S×(IsS)+S×(IiJ)JtJξ=J×(IsJ)+J×(IiS),
{Sx=S02If2cosφfSy=S02If2sinφfSz=Ifand{Jx=J02Ib2cosφbJy=J02Ib2sinφbJz=Ib
{g1,g2}=i=f,bg1φig2Iig1Iig2φi.
ξIb,f={Ib,f,H}=Hφb,fξφb,f={φb,f,H}=HIb,f.
{Sx,Sy}=SxφfSyIfSxIfSyφf=Sz{Jx,Jy}=JxφbJyIbJxIbJyφb=Jz,
dSidξ={Si,H}anddJidξ={Ji,H}.
H=S.IiJ12(S.IsS+J.IsJ)
{ξSx=SzSy+2SzJyξSy=SzSx2SzJxξSz=2JxSy2SxJy
{ξJx=JzJy2JzSyξJy=JzJx+2JzSxξJz=2JxSy2SxJy.
H=2(SyJy+SxJx)12(Sz2+Jz2),
{ξSx=SyJzSzJyξSy=JxSz+SxJzξSz=JySx+SyJx
{ξJx=SyJz+JySzξJy=JzSxSzJxξJz=JySxJxSy.
H=SxJxSyJy+SzJz,
{ξSx=α(SzJy2SyJz)+βSySzξSy=α(SzJx+2SxJz)βSxSzξSz=α(JySx+SyJx)
{ξJx=α(2SzJySyJz)βJyJzξJy=α(2SzJx+SxJz)+βJxJzξJz=α(JySx+JxSy),
H=α(SyJySxJx+2SzJz)β2(Sz2+Jz2)
dSidz={Si,K}anddJidz={Ji,K},
{S˙y=SxS˙x=SyS˙z=0;{J˙y=JxJ˙x=JyJ˙z=0.
{x0=K=SzJzx1=Sz+Jzx2=S.Jx3=SxJySyJx.
x32+(x2+14(x02x12)2)2(S0214(x0+x1)2)(J0214(x0x1)2)=0,
H=2x2+14x0234x12.
x1=±2S04S02K2K2x22x22S02x2.
H=2S0234K2andH=2S02+14K2.
{S˙x=SyS˙y=SxS˙z=0;{J˙x=JyJ˙y=JxJ˙z=0,
{x0=K3=Sz+Jzx1=SzJzx2=SyJySxJxx3=SxJy+SyJx.
x32+x22+(S0214(x0+x1)2)(J0214(x0x1)2)=0.
H=14(x02x12)x2.
H=K322S02andH=S02.
H=α(x2+x02x122)β4(x02+x12).
x1=±K2+4S02±4S02K2+x22.
{H=εα2(K24+S02)2S02K2)+αK22βK24H=2αS02S02ββK22+|K|4αβS02+S02β2+3S02α2,
H=cos2φ(K2S02)sin2φ2K2.
{Sx=S02If2cosφfSy=S02If2sinφfSz=If;{Jx=J02Ib2cosφbJy=J02Ib2sinφbJz=Ib
H=2(S02If2)(S02Ib2)cos(φfφb)If2+Ib22.
(S02e2)(2cos(φfφb)+1)=0.
φf=φb±2π3.
Sy=K2S02.
Sx=±S02K2(S02K2)2.
{Sx=±S02K2(S02K2)2Sy=K2S02Sz=K.
Klim=32Jz±43Jz2±41Jz2.
ϕs=FIs=χϕp=FIp=χ+ψK=Fχ=Is+IpJ=Fψ=Ip.
H=2(J02J2)(S02(KJ)2)cosψ12(J2+(KJ)2),
dJdξ=Hψ=2(S02J2)(J02(KJ)2)sinψ,
(dJdξ)2=e(Jα)(Jβ)(Jγ)(Jδ),
J(ξ)=β+γβ1ηsn2(λξ|μ),
J(ξ)=δ(βγ)β(δγ)e2λξε1/2βγ(δγ)e2λξε1/2+o(ε).
K2±16(15K2+12H±12(K4+16S02K2+4HK2+4S04+8S02H+4H2)1/2+48S02)1/2.
K=ρcosθH=ρsinθS02,
J(ξ)=S0tanh(λξ)+O(ρ).
L=24k2So(2ln2+lnS02ln2cos2θ2+2S02k2cos2θ8S02cos2θ(k2)2(k+2)2lnρ)+O(ρ),
H=2(S02J2)cosψJ2,
Sx=S0cos(χ(0))sech(3S0ξ)Sy=S0tanh(2S0ξ)Sz=S0sin(χ(0))sech(3S0ξ)Jx=S0cos(χ(0)+s2π3)sech(3S0ξ)Jy=S0tanh(3S0ξ)Jz=S0sin(χ(0)+s2π3)sech(3S0ξ),
Sx=S032sech(3S0ξ)Sy=S0tanh(3S0ξ)Sz=S012sech(3S0ξ)Jz=S032sech(3S0ξ)Jy=S0tanh(3S0ξ)Jz=S012sech(3S0ξ).

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