Abstract

Filamentation, as a universal femtosecond phenomenon that could occur in various nonlinear systems, has aroused extensive interest, owing to its underlying physics, complexity and applicability. It is always anticipated to realize the controllable and designable filamentation. For this aim, the crucial problem is how to actively break the symmetry of light-matter nonlinear interaction. A kind of extensively used approaches is based on the controllable spatial structure of optical fields involving phase, amplitude and polarization. Here we present an idea to control the optical field collapse by introducing optical anisotropy of matter as an additional degree of freedom, associated with polarization structure. Our theoretical prediction and experimental results reveal that the synergy of optical anisotropy and polarization structure is indeed a very effective means for controlling the optical field collapse, which has the robust feature against random noise.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. P. A. Robinson, “Nonlinear wave collapse and strong turbulence,” Rev. Mod. Phys. 69, 507–573 (1997).
    [Crossref]
  2. L. Bergé, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303, 259–370 (1998).
    [Crossref]
  3. G. Fibich and B. Ilan, “Vectorial and random effects in self-focusing and in multiple filamentation,” Physica D 157, 112–146 (2001).
    [Crossref]
  4. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
    [Crossref]
  5. L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
    [Crossref]
  6. S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self- trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).
  7. J. M. Soto-Crespo, E. M. Wright, and N. N. Akhmediev, “Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium,” Phys. Rev. A 45, 3168–3175 (1992).
    [Crossref] [PubMed]
  8. L. Bergé, C. Gouédard, J. Schjoedt-Eriksen, and H. Ward, “Filamentation patterns in Kerr media vs. beam shape robustness, nonlinear saturation and polarization states,” Physica D 176, 181–211 (2003).
    [Crossref]
  9. A. Vincotte and L. Bergé, “Atmospheric propagation of gradient-shaped and spinning femtosecond light pulses,” Physica D 223, 163–173 (2006).
    [Crossref]
  10. A. Dubietis, G. Tamošauskas, G. Fibich, and B. Ilan, “Multiple filamentation induced by input-beam ellipticity,” Opt. Lett. 29, 1126–1128 (2004).
    [Crossref] [PubMed]
  11. T. D. Grow and A. L. Gaeta, “Dependence of multiple filamentation on beam ellipticity,” Opt. Express 13, 4594–4599 (2005).
    [Crossref] [PubMed]
  12. T. Pfeifer, L. Gallmann, M. J. Abel, D. M. Neumark, and S. R. Leone, “Circular phase mask for control and stabilization of single optical filaments,” Opt. Lett. 31, 2326–2328 (2006).
    [Crossref] [PubMed]
  13. D. W. Li, T. T. Xi, L. Z. Zhang, H. Y. Tao, X. Gao, J. Q. Lin, and Z. Q. Hao, “Interference-induced filament array in fused silica,” Opt. Express 25, 23910–23919 (2017).
    [Crossref] [PubMed]
  14. Z. Q. Hao, K. Stelmaszczyk, P. Rohwetter, W. M. Nakaema, and L. Woeste, “Femtosecond laser filament-fringes in fused silica,” Opt. Express 19, 7799–7806 (2011).
    [Crossref] [PubMed]
  15. V. P. Kandidov, N. Akozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80, 267–275 (2005).
    [Crossref]
  16. S. M. Li, Y. N. Li, X. L. Wang, L. J. Kong, K. Lou, C. H. Tu, Y. J. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep.  2, 1007 (2012).
    [Crossref] [PubMed]
  17. S. M. Li, Z. C. Ren, L. J. Kong, S. X. Qian, C. H. Tu, Y. N. Li, and H. T. Wang, “Unveiling stability of multiple filamentation caused by axial symmetry breaking of polarization,” Photon. Res. 4, B29–B34 (2016).
    [Crossref]
  18. J. Kasparian and J. P. Wolf, “Physics and applications of atmospheric nonlinear optics and filamentation,” Opt. Express 16, 466–493 (2008).
    [Crossref] [PubMed]
  19. J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. André, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
    [Crossref] [PubMed]
  20. A. Camino, Z. Hao, X. Liu, and J. Lin, “High spectral power femtosecond supercontinuum source by use of microlens array,” Opt. Lett. 39, 747750 (2014).
    [Crossref]
  21. R. R. Alfano and S. L. Shapiro, “Emission in the region 4000 to 7000 Å via four-photon coupling in glass,” Phys. Rev. Appl. 24, 584–587 (1970).
  22. C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79, 673–677 (2004).
    [Crossref]
  23. H. B. Zhang, Z. J. Yuan, R. Ye, B. He, Y. F. Qi, and J. Zhou, “Filamentation-induced bulk modification in fused silica by excimer laser,” Opt. Mater. Express 7, 3680–3690 (2017).
    [Crossref]
  24. F. Belgiorno, S. L. Cacciatori, M. Clerici, V. Gorini, G. Ortenzi, L. Rizzi, E. Rubino, V. G. Sala, and D. Faccio, “Hawking radiation from ultrashort laser pulse filaments,” Phys. Rev. Lett. 105, 203901 (2010).
    [Crossref]
  25. G. Fibich and B. Ilan, “Deterministic vectorial effects lead to multiple filamentation,” Opt. Lett. 26, 840–842 (2001).
    [Crossref]
  26. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009).
    [Crossref]
  27. X. L. Wang, J. P. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007).
    [Crossref] [PubMed]
  28. X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010).
    [Crossref] [PubMed]
  29. A. A. Ishaaya, L. T. Vuong, T. D. Grow, and A. L. Gaeta, “Self-focusing dynamics of polarization vortices in kerr media,” Opt. Lett. 33, 13–15 (2008).
    [Crossref]
  30. L. T. Vuong, T. D. Grow, A. A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 (2006).
    [Crossref] [PubMed]
  31. M. Born and E. Wolf, Principles of Optics(Cambridge University, 1999).
    [Crossref]
  32. G. Fibich and A. L. Gaeta, “Critical power for self-focusing in bulk media and in hollow waveguides,” Opt. Lett. 25, 335–337 (2000).
    [Crossref]
  33. A. S. Arabanian and R. Massudi, “Modeling of femtosecond pulse propagation inside x-cut and z-cut MgO doped LiNbO3 anisotropic crystals,” Appl. Opt. 52, 4212–4222 (2013).
    [Crossref] [PubMed]
  34. http://www.fabrinet.co.th/custappl/casix/aa/product/prod_cry_linbo3.html
  35. D. Hovhannisyan and K. Stepanyan, “Femtosecond laser pulse propagation in a uniaxial crystal,” J. Mod. Opt. 50, 2201–2211 (2003).
    [Crossref]
  36. G. Agrawal, Nonlinear fiber optics (Academic, 2012).
  37. R. P. Chen, L. X. Zhong, K. H. Chew, T. Y. Zhao, and X. Zhang, “Collapse dynamics of a vector vortex optical field with inhomogeneous states of polarization,” Laser Phys.  25, 075401 (2015).
    [Crossref]

2017 (2)

2016 (1)

2015 (1)

R. P. Chen, L. X. Zhong, K. H. Chew, T. Y. Zhao, and X. Zhang, “Collapse dynamics of a vector vortex optical field with inhomogeneous states of polarization,” Laser Phys.  25, 075401 (2015).
[Crossref]

2014 (1)

A. Camino, Z. Hao, X. Liu, and J. Lin, “High spectral power femtosecond supercontinuum source by use of microlens array,” Opt. Lett. 39, 747750 (2014).
[Crossref]

2013 (1)

2012 (1)

S. M. Li, Y. N. Li, X. L. Wang, L. J. Kong, K. Lou, C. H. Tu, Y. J. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep.  2, 1007 (2012).
[Crossref] [PubMed]

2011 (1)

2010 (2)

F. Belgiorno, S. L. Cacciatori, M. Clerici, V. Gorini, G. Ortenzi, L. Rizzi, E. Rubino, V. G. Sala, and D. Faccio, “Hawking radiation from ultrashort laser pulse filaments,” Phys. Rev. Lett. 105, 203901 (2010).
[Crossref]

X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (2)

2007 (3)

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
[Crossref]

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[Crossref]

X. L. Wang, J. P. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007).
[Crossref] [PubMed]

2006 (3)

L. T. Vuong, T. D. Grow, A. A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 (2006).
[Crossref] [PubMed]

A. Vincotte and L. Bergé, “Atmospheric propagation of gradient-shaped and spinning femtosecond light pulses,” Physica D 223, 163–173 (2006).
[Crossref]

T. Pfeifer, L. Gallmann, M. J. Abel, D. M. Neumark, and S. R. Leone, “Circular phase mask for control and stabilization of single optical filaments,” Opt. Lett. 31, 2326–2328 (2006).
[Crossref] [PubMed]

2005 (2)

V. P. Kandidov, N. Akozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80, 267–275 (2005).
[Crossref]

T. D. Grow and A. L. Gaeta, “Dependence of multiple filamentation on beam ellipticity,” Opt. Express 13, 4594–4599 (2005).
[Crossref] [PubMed]

2004 (2)

A. Dubietis, G. Tamošauskas, G. Fibich, and B. Ilan, “Multiple filamentation induced by input-beam ellipticity,” Opt. Lett. 29, 1126–1128 (2004).
[Crossref] [PubMed]

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79, 673–677 (2004).
[Crossref]

2003 (3)

D. Hovhannisyan and K. Stepanyan, “Femtosecond laser pulse propagation in a uniaxial crystal,” J. Mod. Opt. 50, 2201–2211 (2003).
[Crossref]

L. Bergé, C. Gouédard, J. Schjoedt-Eriksen, and H. Ward, “Filamentation patterns in Kerr media vs. beam shape robustness, nonlinear saturation and polarization states,” Physica D 176, 181–211 (2003).
[Crossref]

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. André, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[Crossref] [PubMed]

2001 (2)

G. Fibich and B. Ilan, “Vectorial and random effects in self-focusing and in multiple filamentation,” Physica D 157, 112–146 (2001).
[Crossref]

G. Fibich and B. Ilan, “Deterministic vectorial effects lead to multiple filamentation,” Opt. Lett. 26, 840–842 (2001).
[Crossref]

2000 (1)

1998 (1)

L. Bergé, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303, 259–370 (1998).
[Crossref]

1997 (1)

P. A. Robinson, “Nonlinear wave collapse and strong turbulence,” Rev. Mod. Phys. 69, 507–573 (1997).
[Crossref]

1992 (1)

J. M. Soto-Crespo, E. M. Wright, and N. N. Akhmediev, “Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium,” Phys. Rev. A 45, 3168–3175 (1992).
[Crossref] [PubMed]

1970 (1)

R. R. Alfano and S. L. Shapiro, “Emission in the region 4000 to 7000 Å via four-photon coupling in glass,” Phys. Rev. Appl. 24, 584–587 (1970).

1966 (1)

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self- trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).

Abel, M. J.

Agrawal, G.

G. Agrawal, Nonlinear fiber optics (Academic, 2012).

Akhmanov, S. A.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self- trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).

Akhmediev, N. N.

J. M. Soto-Crespo, E. M. Wright, and N. N. Akhmediev, “Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium,” Phys. Rev. A 45, 3168–3175 (1992).
[Crossref] [PubMed]

Akozbek, N.

V. P. Kandidov, N. Akozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80, 267–275 (2005).
[Crossref]

Alfano, R. R.

R. R. Alfano and S. L. Shapiro, “Emission in the region 4000 to 7000 Å via four-photon coupling in glass,” Phys. Rev. Appl. 24, 584–587 (1970).

André, Y. B.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. André, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[Crossref] [PubMed]

Arabanian, A. S.

Belgiorno, F.

F. Belgiorno, S. L. Cacciatori, M. Clerici, V. Gorini, G. Ortenzi, L. Rizzi, E. Rubino, V. G. Sala, and D. Faccio, “Hawking radiation from ultrashort laser pulse filaments,” Phys. Rev. Lett. 105, 203901 (2010).
[Crossref]

Bergé, L.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[Crossref]

A. Vincotte and L. Bergé, “Atmospheric propagation of gradient-shaped and spinning femtosecond light pulses,” Physica D 223, 163–173 (2006).
[Crossref]

L. Bergé, C. Gouédard, J. Schjoedt-Eriksen, and H. Ward, “Filamentation patterns in Kerr media vs. beam shape robustness, nonlinear saturation and polarization states,” Physica D 176, 181–211 (2003).
[Crossref]

L. Bergé, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303, 259–370 (1998).
[Crossref]

Biegert, J.

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79, 673–677 (2004).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics(Cambridge University, 1999).
[Crossref]

Bourayou, R.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. André, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[Crossref] [PubMed]

Cacciatori, S. L.

F. Belgiorno, S. L. Cacciatori, M. Clerici, V. Gorini, G. Ortenzi, L. Rizzi, E. Rubino, V. G. Sala, and D. Faccio, “Hawking radiation from ultrashort laser pulse filaments,” Phys. Rev. Lett. 105, 203901 (2010).
[Crossref]

Camino, A.

A. Camino, Z. Hao, X. Liu, and J. Lin, “High spectral power femtosecond supercontinuum source by use of microlens array,” Opt. Lett. 39, 747750 (2014).
[Crossref]

Chen, J.

Chen, R. P.

R. P. Chen, L. X. Zhong, K. H. Chew, T. Y. Zhao, and X. Zhang, “Collapse dynamics of a vector vortex optical field with inhomogeneous states of polarization,” Laser Phys.  25, 075401 (2015).
[Crossref]

Chew, K. H.

R. P. Chen, L. X. Zhong, K. H. Chew, T. Y. Zhao, and X. Zhang, “Collapse dynamics of a vector vortex optical field with inhomogeneous states of polarization,” Laser Phys.  25, 075401 (2015).
[Crossref]

Chin, S. L.

V. P. Kandidov, N. Akozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80, 267–275 (2005).
[Crossref]

Clerici, M.

F. Belgiorno, S. L. Cacciatori, M. Clerici, V. Gorini, G. Ortenzi, L. Rizzi, E. Rubino, V. G. Sala, and D. Faccio, “Hawking radiation from ultrashort laser pulse filaments,” Phys. Rev. Lett. 105, 203901 (2010).
[Crossref]

Couairon, A.

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
[Crossref]

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79, 673–677 (2004).
[Crossref]

Ding, J. P.

Dubietis, A.

Eliel, E. R.

L. T. Vuong, T. D. Grow, A. A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 (2006).
[Crossref] [PubMed]

Faccio, D.

F. Belgiorno, S. L. Cacciatori, M. Clerici, V. Gorini, G. Ortenzi, L. Rizzi, E. Rubino, V. G. Sala, and D. Faccio, “Hawking radiation from ultrashort laser pulse filaments,” Phys. Rev. Lett. 105, 203901 (2010).
[Crossref]

Fibich, G.

Frey, S.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. André, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[Crossref] [PubMed]

Gaeta, A. L.

Gallmann, L.

Gao, X.

Gorini, V.

F. Belgiorno, S. L. Cacciatori, M. Clerici, V. Gorini, G. Ortenzi, L. Rizzi, E. Rubino, V. G. Sala, and D. Faccio, “Hawking radiation from ultrashort laser pulse filaments,” Phys. Rev. Lett. 105, 203901 (2010).
[Crossref]

Gouédard, C.

L. Bergé, C. Gouédard, J. Schjoedt-Eriksen, and H. Ward, “Filamentation patterns in Kerr media vs. beam shape robustness, nonlinear saturation and polarization states,” Physica D 176, 181–211 (2003).
[Crossref]

Grow, T. D.

Guo, C. S.

Hao, Z.

A. Camino, Z. Hao, X. Liu, and J. Lin, “High spectral power femtosecond supercontinuum source by use of microlens array,” Opt. Lett. 39, 747750 (2014).
[Crossref]

Hao, Z. Q.

Hauri, C. P.

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79, 673–677 (2004).
[Crossref]

He, B.

Heinrich, A.

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79, 673–677 (2004).
[Crossref]

Helbing, F. W.

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79, 673–677 (2004).
[Crossref]

Hooft, G. W. ’t

L. T. Vuong, T. D. Grow, A. A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 (2006).
[Crossref] [PubMed]

Hosseini, S. A.

V. P. Kandidov, N. Akozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80, 267–275 (2005).
[Crossref]

Hovhannisyan, D.

D. Hovhannisyan and K. Stepanyan, “Femtosecond laser pulse propagation in a uniaxial crystal,” J. Mod. Opt. 50, 2201–2211 (2003).
[Crossref]

Ilan, B.

Ishaaya, A. A.

A. A. Ishaaya, L. T. Vuong, T. D. Grow, and A. L. Gaeta, “Self-focusing dynamics of polarization vortices in kerr media,” Opt. Lett. 33, 13–15 (2008).
[Crossref]

L. T. Vuong, T. D. Grow, A. A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 (2006).
[Crossref] [PubMed]

Kandidov, V. P.

V. P. Kandidov, N. Akozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80, 267–275 (2005).
[Crossref]

Kasparian, J.

J. Kasparian and J. P. Wolf, “Physics and applications of atmospheric nonlinear optics and filamentation,” Opt. Express 16, 466–493 (2008).
[Crossref] [PubMed]

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[Crossref]

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. André, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[Crossref] [PubMed]

Keller, U.

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79, 673–677 (2004).
[Crossref]

Khokhlov, R. V.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self- trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).

Kong, L. J.

Kornelis, W.

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79, 673–677 (2004).
[Crossref]

Kosareva, O. G.

V. P. Kandidov, N. Akozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80, 267–275 (2005).
[Crossref]

Leone, S. R.

Li, D. W.

Li, S. M.

Li, Y. N.

Lin, J.

A. Camino, Z. Hao, X. Liu, and J. Lin, “High spectral power femtosecond supercontinuum source by use of microlens array,” Opt. Lett. 39, 747750 (2014).
[Crossref]

Lin, J. Q.

Liu, X.

A. Camino, Z. Hao, X. Liu, and J. Lin, “High spectral power femtosecond supercontinuum source by use of microlens array,” Opt. Lett. 39, 747750 (2014).
[Crossref]

Lou, K.

S. M. Li, Y. N. Li, X. L. Wang, L. J. Kong, K. Lou, C. H. Tu, Y. J. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep.  2, 1007 (2012).
[Crossref] [PubMed]

Luo, Q.

V. P. Kandidov, N. Akozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80, 267–275 (2005).
[Crossref]

Massudi, R.

Méjean, G.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. André, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[Crossref] [PubMed]

Mysyrowicz, A.

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
[Crossref]

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79, 673–677 (2004).
[Crossref]

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. André, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[Crossref] [PubMed]

Nakaema, W. M.

Neumark, D. M.

Ni, W. J.

Nuter, R.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[Crossref]

Nyakk, A. V.

V. P. Kandidov, N. Akozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80, 267–275 (2005).
[Crossref]

Ortenzi, G.

F. Belgiorno, S. L. Cacciatori, M. Clerici, V. Gorini, G. Ortenzi, L. Rizzi, E. Rubino, V. G. Sala, and D. Faccio, “Hawking radiation from ultrashort laser pulse filaments,” Phys. Rev. Lett. 105, 203901 (2010).
[Crossref]

Pfeifer, T.

Qi, Y. F.

Qian, S. X.

Ren, Z. C.

Rizzi, L.

F. Belgiorno, S. L. Cacciatori, M. Clerici, V. Gorini, G. Ortenzi, L. Rizzi, E. Rubino, V. G. Sala, and D. Faccio, “Hawking radiation from ultrashort laser pulse filaments,” Phys. Rev. Lett. 105, 203901 (2010).
[Crossref]

Robinson, P. A.

P. A. Robinson, “Nonlinear wave collapse and strong turbulence,” Rev. Mod. Phys. 69, 507–573 (1997).
[Crossref]

Rodriguez, M.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. André, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[Crossref] [PubMed]

Rohwetter, P.

Rubino, E.

F. Belgiorno, S. L. Cacciatori, M. Clerici, V. Gorini, G. Ortenzi, L. Rizzi, E. Rubino, V. G. Sala, and D. Faccio, “Hawking radiation from ultrashort laser pulse filaments,” Phys. Rev. Lett. 105, 203901 (2010).
[Crossref]

Sala, V. G.

F. Belgiorno, S. L. Cacciatori, M. Clerici, V. Gorini, G. Ortenzi, L. Rizzi, E. Rubino, V. G. Sala, and D. Faccio, “Hawking radiation from ultrashort laser pulse filaments,” Phys. Rev. Lett. 105, 203901 (2010).
[Crossref]

Salmon, E.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. André, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[Crossref] [PubMed]

Sauerbrey, R.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. André, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[Crossref] [PubMed]

Scalora, M.

V. P. Kandidov, N. Akozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80, 267–275 (2005).
[Crossref]

Schjoedt-Eriksen, J.

L. Bergé, C. Gouédard, J. Schjoedt-Eriksen, and H. Ward, “Filamentation patterns in Kerr media vs. beam shape robustness, nonlinear saturation and polarization states,” Physica D 176, 181–211 (2003).
[Crossref]

Shapiro, S. L.

R. R. Alfano and S. L. Shapiro, “Emission in the region 4000 to 7000 Å via four-photon coupling in glass,” Phys. Rev. Appl. 24, 584–587 (1970).

Skupin, S.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[Crossref]

Soto-Crespo, J. M.

J. M. Soto-Crespo, E. M. Wright, and N. N. Akhmediev, “Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium,” Phys. Rev. A 45, 3168–3175 (1992).
[Crossref] [PubMed]

Stelmaszczyk, K.

Stepanyan, K.

D. Hovhannisyan and K. Stepanyan, “Femtosecond laser pulse propagation in a uniaxial crystal,” J. Mod. Opt. 50, 2201–2211 (2003).
[Crossref]

Sukhorukov, A. P.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self- trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).

Tamošauskas, G.

Tao, H. Y.

Tian, Y. J.

S. M. Li, Y. N. Li, X. L. Wang, L. J. Kong, K. Lou, C. H. Tu, Y. J. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep.  2, 1007 (2012).
[Crossref] [PubMed]

Tu, C. H.

Vincotte, A.

A. Vincotte and L. Bergé, “Atmospheric propagation of gradient-shaped and spinning femtosecond light pulses,” Physica D 223, 163–173 (2006).
[Crossref]

Vuong, L. T.

A. A. Ishaaya, L. T. Vuong, T. D. Grow, and A. L. Gaeta, “Self-focusing dynamics of polarization vortices in kerr media,” Opt. Lett. 33, 13–15 (2008).
[Crossref]

L. T. Vuong, T. D. Grow, A. A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 (2006).
[Crossref] [PubMed]

Wang, H. T.

Wang, X. L.

Ward, H.

L. Bergé, C. Gouédard, J. Schjoedt-Eriksen, and H. Ward, “Filamentation patterns in Kerr media vs. beam shape robustness, nonlinear saturation and polarization states,” Physica D 176, 181–211 (2003).
[Crossref]

Wille, H.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. André, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[Crossref] [PubMed]

Woeste, L.

Wolf, E.

M. Born and E. Wolf, Principles of Optics(Cambridge University, 1999).
[Crossref]

Wolf, J. P.

J. Kasparian and J. P. Wolf, “Physics and applications of atmospheric nonlinear optics and filamentation,” Opt. Express 16, 466–493 (2008).
[Crossref] [PubMed]

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[Crossref]

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. André, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[Crossref] [PubMed]

Wöste, L.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. André, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[Crossref] [PubMed]

Wright, E. M.

J. M. Soto-Crespo, E. M. Wright, and N. N. Akhmediev, “Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium,” Phys. Rev. A 45, 3168–3175 (1992).
[Crossref] [PubMed]

Xi, T. T.

Ye, R.

Yu, J.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. André, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[Crossref] [PubMed]

Yuan, Z. J.

Zhan, Q.

Zhang, H. B.

Zhang, L. Z.

Zhang, X.

R. P. Chen, L. X. Zhong, K. H. Chew, T. Y. Zhao, and X. Zhang, “Collapse dynamics of a vector vortex optical field with inhomogeneous states of polarization,” Laser Phys.  25, 075401 (2015).
[Crossref]

Zhao, T. Y.

R. P. Chen, L. X. Zhong, K. H. Chew, T. Y. Zhao, and X. Zhang, “Collapse dynamics of a vector vortex optical field with inhomogeneous states of polarization,” Laser Phys.  25, 075401 (2015).
[Crossref]

Zhong, L. X.

R. P. Chen, L. X. Zhong, K. H. Chew, T. Y. Zhao, and X. Zhang, “Collapse dynamics of a vector vortex optical field with inhomogeneous states of polarization,” Laser Phys.  25, 075401 (2015).
[Crossref]

Zhou, J.

Adv. Opt. Photon. (1)

Appl. Opt. (1)

Appl. Phys. B (2)

V. P. Kandidov, N. Akozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80, 267–275 (2005).
[Crossref]

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79, 673–677 (2004).
[Crossref]

J. Mod. Opt. (1)

D. Hovhannisyan and K. Stepanyan, “Femtosecond laser pulse propagation in a uniaxial crystal,” J. Mod. Opt. 50, 2201–2211 (2003).
[Crossref]

Laser Phys (1)

R. P. Chen, L. X. Zhong, K. H. Chew, T. Y. Zhao, and X. Zhang, “Collapse dynamics of a vector vortex optical field with inhomogeneous states of polarization,” Laser Phys.  25, 075401 (2015).
[Crossref]

Opt. Express (5)

Opt. Lett. (7)

Opt. Mater. Express (1)

Photon. Res. (1)

Phys. Rep. (2)

L. Bergé, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303, 259–370 (1998).
[Crossref]

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
[Crossref]

Phys. Rev. A (1)

J. M. Soto-Crespo, E. M. Wright, and N. N. Akhmediev, “Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium,” Phys. Rev. A 45, 3168–3175 (1992).
[Crossref] [PubMed]

Phys. Rev. Appl. (1)

R. R. Alfano and S. L. Shapiro, “Emission in the region 4000 to 7000 Å via four-photon coupling in glass,” Phys. Rev. Appl. 24, 584–587 (1970).

Phys. Rev. Lett. (2)

F. Belgiorno, S. L. Cacciatori, M. Clerici, V. Gorini, G. Ortenzi, L. Rizzi, E. Rubino, V. G. Sala, and D. Faccio, “Hawking radiation from ultrashort laser pulse filaments,” Phys. Rev. Lett. 105, 203901 (2010).
[Crossref]

L. T. Vuong, T. D. Grow, A. A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 (2006).
[Crossref] [PubMed]

Physica D (3)

L. Bergé, C. Gouédard, J. Schjoedt-Eriksen, and H. Ward, “Filamentation patterns in Kerr media vs. beam shape robustness, nonlinear saturation and polarization states,” Physica D 176, 181–211 (2003).
[Crossref]

A. Vincotte and L. Bergé, “Atmospheric propagation of gradient-shaped and spinning femtosecond light pulses,” Physica D 223, 163–173 (2006).
[Crossref]

G. Fibich and B. Ilan, “Vectorial and random effects in self-focusing and in multiple filamentation,” Physica D 157, 112–146 (2001).
[Crossref]

Rep. Prog. Phys. (1)

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[Crossref]

Rev. Mod. Phys. (1)

P. A. Robinson, “Nonlinear wave collapse and strong turbulence,” Rev. Mod. Phys. 69, 507–573 (1997).
[Crossref]

Sci. Rep (1)

S. M. Li, Y. N. Li, X. L. Wang, L. J. Kong, K. Lou, C. H. Tu, Y. J. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep.  2, 1007 (2012).
[Crossref] [PubMed]

Science (1)

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. André, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[Crossref] [PubMed]

Sov. Phys. JETP (1)

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self- trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).

Other (3)

M. Born and E. Wolf, Principles of Optics(Cambridge University, 1999).
[Crossref]

http://www.fabrinet.co.th/custappl/casix/aa/product/prod_cry_linbo3.html

G. Agrawal, Nonlinear fiber optics (Academic, 2012).

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Figures (9)

Fig. 1
Fig. 1 Schematic of configuration for investigating the collapse of VOFs in an anisotropic MgO:LiNbO3 crystal.
Fig. 2
Fig. 2 The collapsing behaviors of AV-LP-VOFs (m = 1 and 2) for three different initial phases (δ0 = 0, π/4 and π/2). Measured patterns of total intensity and o -polarized component of the created vector fields in the x -cut MgO:LiNbO3 crystal are shown in the first and second rows. The third and fourth rows correspond to the measured and simulated collapsing patterns, respectively.
Fig. 3
Fig. 3 The collapsing behaviors of AV-LP-VOFs (m = 3, 4, 5, 6) for two different initial phases (δ0 = 0, π/2). The first three columns correspond to four AV-LP-VOFs (m = 3, 4, 5, 6) with δ0 = 0, while the last three columns correspond to four AV-LP-VOFs (m = 3, 4, 5, 6) with δ0 = π/2. Measured patterns of o -polarized component of the created AV-LP-VOFs in the uniaxial crystal are shown in the first and fourth columns. The second (fifth) and third (sixth) columns correspond to the measured and simulated collapsing patterns, respectively.
Fig. 4
Fig. 4 The relationship between the collapsing patterns of AV-LP-VFs and the orientation of the crystal. The collapsing patterns of RP-VOF (m = 1, δ0 = 0) and AV-LP-VOF (m = 2, δ0 = 0) rotated with the optic axis of the MgO:LiNbO3 crystal are shown in Figs. 4(a) and 4(b), respectively.
Fig. 5
Fig. 5 Measured and simulated collapsing behaviors of AV-LP-VFs with m = 1 but with different δ0 = 0, π/4, π/2 and 3π/2 in BaF2 with the linear isotropy but nonlinear anisotropy (β > 1/3).
Fig. 6
Fig. 6 The simulated collapsing behaviors of RP-VOF (AV-LP-VOF with m = 1 and δ0 = 0) in the nonlinear media with the linear isotropy but the nonlinear anisotropy. For comparison, the first and second columns show the collapsing behavior of RP-VOF in the traditional isotropic Kerr medium.
Fig. 7
Fig. 7 The simulated evolution S3 of the azimuthal-variant polarization states for RP-VOF (AV-LP-VF with m = 1 and δ0 = 0) in the media with the linear isotropy but the nonlinear anisotropy. (a) β = 0.292 > 1/3 and (b) β = 0.374 > 1/3.
Fig. 8
Fig. 8 The simulated collapsing behaviors of RP-VOF (AV-LP-VOF with m = 1 and δ0 = 0) in the media with both the linear and nonlinear anisotropies. For comparison, the first and second columns show the collapsing behaviors of RP-VOF in the media with the linear anisotropy but the nonlinear isotropy β = 1/3. In particular, the last column shows the collapsing behavior of RP-VOF with the larger linear anisotropy (N = 1.1).
Fig. 9
Fig. 9 The normalized AMI growth rate of RP-VOF in different nonlinear media with (a) N ≥ 1, β = 1/3; (b) N ≥ 1, β< 1/3; (c) N ≥ 1, β> 1/3

Equations (46)

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

E ( r , ϕ ) = A ( r ) ( cos δ e ^ y + sin δ e ^ z ) .
ψ o ζ = j 2 ψ o + j 4 α P P o C [ | ψ o | 2 ψ o + χ 16 χ 11 ( 2 | ψ e | 2 ψ o + ψ e 2 ψ o * ) ] ,
ψ e ζ = j n o 0 n e 0 2 ψ e + j 4 α P P e C [ | ψ e | 2 ψ e + χ 16 χ 33 ( 2 | ψ o | 2 ψ e + ψ o 2 ψ e * ) ] ,
ψ o ζ = j 2 ψ o + j 8 P P C [ | ψ o | 2 ψ o + β ( 2 | ψ e | 2 ψ o + ψ e 2 ψ o * ) ] ,
ψ e ζ = j N 2 ψ e + j N 8 P P C [ | ψ e | 2 ψ e + β ( 2 | ψ o | 2 ψ e + ψ o 2 ψ e * ) ] .
Δ ψ o = j [ m 2 B ( cos 2 δ + 3 β sin 2 δ ) ] Z cos δ ,
Δ ψ e = j N [ m 2 B ( sin 2 δ + 3 β cos 2 δ ) ] Z sin δ ,
Δ I ϕ = 1 6 m Z 2 W sin ( 2 δ ) ,
W = U V cos ( 2 δ ) ( N 2 1 ) ( U + V cos 2 δ ) V sin 2 δ ,
U = 4 m 2 3 B ( 1 + β ) = 4 m 2 [ 1 36 P ( 1 + β ) / 5 P m C ] ,
V = 3 B ( 1 3 β ) .
2 Δ I ϕ 2 = 1 6 m Z 2 [ W ϕ sin ( 2 δ ) + 2 m W cos ( 2 δ ) ] .
Δ I ϕ = 1 12 m Z 2 U V sin ( 4 δ ) ,
2 Δ I ϕ 2 = 1 3 m 2 Z 2 U V cos ( 4 δ ) .
γ m 2 = m 2 ρ ¯ 2 { B [ ( N 2 cos 2 δ + sin 2 δ ) + ( N 2 cos 2 δ sin 2 δ ) 2 + N 2 ( 3 β ) 2 sin 2 ( 2 δ ) ] m 2 ρ ¯ 2 } ,
γ m 2 = m 2 ρ ¯ 2 { B [ 1 + 1 + ( 3 β + 1 ) ( 3 β 1 ) sin 2 ( 2 δ ) ] m 2 ρ ¯ 2 } .
2 Δ I ϕ 2 | sin ( 2 δ ) = 0 = 1 3 m 2 Z 2 { U V , when 2 δ = 2 n π N 2 U V , when 2 δ = 2 n π + π
W = 0 U V cos ( 2 δ ) ( N 2 1 ) ( U + V cos 2 δ ) V sin 2 δ = 0 U cos ( 2 δ ) = ( N 2 1 ) ( U + V cos 2 δ ) sin 2 δ U 2 U sin 2 δ = ( N 2 1 ) U sin 2 δ + ( N 2 1 ) V sin 2 δ cos 2 δ U 2 U sin 2 δ ( N 2 1 ) U sin 2 δ = ( N 2 1 ) V sin 2 δ cos 2 δ U U sin 2 δ N 2 U sin 2 δ = ( N 2 1 ) V sin 2 δ cos 2 δ U cos 2 δ N 2 U + N 2 U cos 2 δ = ( N 2 1 ) V sin 2 δ cos 2 δ U cos 2 δ + N 2 U cos 2 δ ( N 2 1 ) V sin 2 δ cos 2 δ = N 2 U [ ( N 2 + 1 ) U ( N 2 1 ) V sin 2 δ ] cos 2 δ = N 2 U ( N 2 1 ) V = U N 2 sin 2 δ cos 2 δ sin 2 δ cos 2 δ .
W ϕ = ϕ [ U V cos ( 2 δ ) ( N 2 1 ) ( U + V cos 2 δ ) V sin 2 δ ] = 2 m U V sin ( 2 δ ) + ( N 2 1 ) m V 2 sin ( 2 δ ) sin 2 δ ( N 2 1 ) m ( U + V cos 2 δ ) V sin ( 2 δ ) = m [ 2 U + ( N 2 1 ) V sin 2 δ ( N 2 1 ) ( U + V cos 2 δ ) ] V sin ( 2 δ ) = m [ 2 U + ( N 2 1 ) V sin 2 δ ( N 2 1 ) U ( N 2 1 ) V cos 2 δ ] V sin ( 2 δ ) = m [ ( N 2 + 1 ) U ( N 2 1 ) V cos ( 2 δ ) ] V sin ( 2 δ ) .
2 Δ I ϕ 2 | W = 0 = 1 6 m Z 2 [ W ϕ sin ( 2 δ ) + 2 m W cos ( 2 δ ) ] | W = 0 = 1 6 m Z 2 [ W ϕ sin ( 2 δ ) ] | W = 0 = 1 6 m 2 Z 2 [ ( N 2 + 1 ) U ( N 2 1 ) V cos ( 2 δ ) ] V sin 2 ( 2 δ ) = 1 6 m 2 Z 2 [ ( N 2 + 1 ) U + U N 2 sin 2 δ cos 2 δ sin 2 δ cos 2 δ cos ( 2 δ ) ] V sin 2 ( 2 δ ) = 1 6 m 2 Z 2 [ ( N 2 + 1 ) U + U N 2 sin 2 δ cos 2 δ sin 2 δ cos 2 δ ( cos 2 δ sin 2 δ ) ] V sin 2 ( 2 δ ) = 1 6 m 2 Z 2 [ cos 2 δ sin 2 δ + N 2 sin 2 δ cos 2 δ ] U V sin 2 ( 2 δ ) .
W = 0 ( N 2 1 ) V = U N 2 sin 2 δ cos 2 δ sin 2 δ cos 2 δ N 2 cos 2 δ 1 sin 2 δ = ( N 2 1 ) V U .
1 cos 2 δ 1 sin 2 δ 0.
N cos 2 δ 1 sin 2 δ < 0 tan 2 δ < 1 N .
ψ o = E o exp ( j λ o ζ ) ,
ψ e = E e exp ( j λ e ζ ) ,
λ o E o + 2 E o + B [ | E o | 2 + β ( 2 | E e | 2 + E e 2 E o * E o ) ] E o = 0 ,
λ e E e + N 2 E e + N B [ | E e | 2 + β ( 2 | E o | 2 + E o 2 E e * E e ) ] E e = 0.
ψ o = [ E o + ϵ ( υ o + j ω o ) ] exp ( j λ o ζ ) ,
ψ e = [ E e + ϵ ( υ e + j ω e ) ] exp ( j λ e ζ ) ,
υ o ζ = L o 0 ω o ,
ω o ζ = L o 1 υ o + [ 2 B β E e E o ( 2 + E o * E o ) ] υ e ,
υ e ζ = L e 0 ω e ,
ω e ζ = L e 1 υ e + [ 2 N B β E o E e ( 2 + E e * E e ) ] υ o .
L o 0 = λ o 2 B | E o | 2 2 B β | E e | 2 B β E e 2 E o * E o ,
L e 0 = λ e N 2 N B | E e | 2 2 N B β | E o | 2 N B β E o 2 E e * E e ,
L o 1 = λ o 2 3 B | E o | 2 2 B β | E e | 2 B β E e 2 E o * E o ,
L o 1 = λ e N 2 3 N B | E e | 2 2 N B β | E o | 2 N B β E o 2 E e * E e .
ψ o = E o exp { j B [ | E o | 2 + β ( 2 | E e | 2 + E e 2 E o * E o ) ] ζ } ,
ψ e = E e exp { j N B [ | E e | 2 + β ( 2 | E o | 2 + E e 2 E e * E e ) ] ζ } .
E o = N cos δ ,
E e = N sin δ ,
γ m , o 2 υ o = m 2 ρ ¯ 2 ( 2 B | E o | 2 m 2 ρ ¯ 2 ) υ o + m 2 ρ ¯ 2 [ 2 B β E e E o ( 2 + E o * E o ) ] υ e ,
1 N 2 γ m , e 2 υ e = m 2 ρ ¯ 2 ( 2 B | E e | 2 m 2 ρ ¯ 2 ) υ e + m 2 ρ ¯ 2 [ 2 B β E o E e ( 2 + E e * E e ) ] υ o .
γ m 2 = m 2 ρ ¯ 2 ( 2 Δ ± m 2 ρ ¯ 2 )
2 Δ ± = B [ ( | E o | 2 + | E e | 2 ) ± ( | E o | 2 | E e | 2 ) 2 + ( 2 β E o E e ) 2 ( 2 + E o * E o ) ( 2 + E e * E e ) ] .
γ m 2 = m 2 ρ ¯ 2 { B [ ( N 2 cos 2 δ + sin 2 δ ) + ( N 2 cos 2 δ sin 2 δ ) 2 + N 2 ( 3 β ) 2 sin 2 ( 2 δ ) ] m 2 ρ ¯ 2 } .

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