Abstract

Using a system of coupled nonlinear Schrödinger equations (CNLSEs), we show that nonlinear light propagation in self-focusing Kerr media can be controlled via a suitable combination of linear and circular birefringences. In particular, magneto-optical effects are taken as a specific physical example, which enables the introduction of both types of birefringences simultaneously via the joint action of the Cotton-Mouton and the Faraday effect. We demonstrate the efficient management of the collapse of (2 + 1)D beams in magneto-optic dielectric media, which may result in either the acceleration or the suppression of the collapse. However, our study also shows that a complete stabilization of the bimodal beams (i.e., the propagation of two-dimensional solitary waves) is not possible under the proposed conditions. The analysis is performed by directly numerically solving the CNLSEs, as well as by using the variational approximation, both showing consistent results. The investigated method allows high-power beam propagation in Kerr media while avoiding collapse, thus offering a viable alternative to the techniques applied in non-instantaneous and/or non-local nonlinear media.

© 2010 OSA

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2010 (2)

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[CrossRef]

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010).
[CrossRef] [PubMed]

2009 (2)

Y. Linzon, K. A. Rutkowska, B. A. Malomed, and R. Morandotti, “Magneto-optical control of light collapse in bulk Kerr media,” Phys. Rev. Lett. 103(5), 053902 (2009).
[CrossRef] [PubMed]

M. C. Sekhar, J.-Y. Hwang, M. Ferrera, Y. Linzon, L. Razzari, C. Harnagea, M. Zaezjev, A. Pignolet, and R. Morandotti, “Strong enhancement of the Faraday rotation in Ce: and Bi: co-modified epitaxial iron garnet thin films,” Appl. Phys. Lett. 94(18), 181916 (2009).
[CrossRef]

2008 (2)

T. Lahaye, J. Metz, B. Fröhlich, T. Koch, M. Meister, A. Griesmaier, T. Pfau, H. Saito, Y. Kawaguchi, and M. Ueda, “d-wave collapse and explosion of a dipolar bose-einstein condensate,” Phys. Rev. Lett. 101(8), 080401 (2008).
[CrossRef] [PubMed]

T. Koch, T. Lahaye, J. Metz, B. Fröhlich, A. Griesmaier, and T. Pfau, “Stabilization of a purely dipolar quantum gas against collapse,” Nat. Phys. 4(3), 218–222 (2008).
[CrossRef]

2007 (2)

2006 (2)

2005 (3)

O. Kamada, T. Nakaya, and S. Higuchi, “Magnetic field optical sensors using Ce:YIG single crystals as a Faraday element,” Sens. Actuators A Phys. 119(2), 345–348 (2005).
[CrossRef]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quant. Semicl. Opt. 7(5), R53–R72 (2005).
[CrossRef]

P. Pedri and L. Santos, “Two-dimensional bright solitons in dipolar Bose-Einstein condensates,” Phys. Rev. Lett. 95(20), 200404 (2005).
[CrossRef] [PubMed]

2004 (2)

R. Kurzynowski and W. A. Wo?niak, “Superposition rule for the magneto-optic effects in isotropic media,” Optik (Stuttg.) 115(10), 473–475 (2004).
[CrossRef]

G. Fibich and B. Ilan, “Optical light bullets in a pure Kerr medium,” Opt. Lett. 29(8), 887–889 (2004).
[CrossRef] [PubMed]

2003 (5)

J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. 28(21), 2094–2096 (2003).
[CrossRef] [PubMed]

D. Cheskis, S. Bar-Ad, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, and D. Ross, “Strong spatiotemporal localization in a silica nonlinear waveguide array,” Phys. Rev. Lett. 91(22), 223901 (2003).
[CrossRef] [PubMed]

F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensation by temporal modulation of the scattering length,” Phys. Rev. A 67(1), 013605 (2003).
[CrossRef]

H. Saito and M. Ueda, “Dynamically stabilized bright solitons in a two-dimensional bose-einstein condensate,” Phys. Rev. Lett. 90(4), 040403 (2003).
[CrossRef] [PubMed]

K. D. Moll, A. L. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the Townes profile,” Phys. Rev. Lett. 90(20), 203902 (2003).
[CrossRef] [PubMed]

2002 (5)

F. Wise and P. di Trapani, “The Hunt for Light Bullets – Spatiotemporal Solitons,” Opt. Photonics News 13(2), 29 (2002).

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4), 046619 (2002).
[CrossRef] [PubMed]

I. Towers and B. A. Malomed, “Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity,” J. Opt. Soc. Am. B 19(3), 537 (2002).
[CrossRef]

M. Peccianti, K. A. Brzdkiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. 27(16), 1460–1462 (2002).
[CrossRef] [PubMed]

B. A. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Progr. Opt. 43, 71 (2002).
[CrossRef]

2000 (5)

G. Fibich and A. L. Gaeta, “Critical power for self-focusing in bulk media and in hollow waveguides,” Opt. Lett. 25(5), 335–337 (2000).
[CrossRef] [PubMed]

L. Bergé, O. Bang, and W. Królikowski, “Influence of four-wave mixing and walk-Off on the self-focusing of coupled waves,” Phys. Rev. Lett. 84(15), 3302–3305 (2000).
[CrossRef] [PubMed]

Y. S. Kivshar and D. E. Pelinovsky, “Self-focusing and transverse instabilities of solitary waves,” Phys. Rep. 331(4), 117–195 (2000).
[CrossRef]

A. L. Gaeta, “Catastrophic collapse of ultrashort pulses,” Phys. Rev. Lett. 84(16), 3582–3585 (2000).
[CrossRef] [PubMed]

D. P. Lathrop, B. W. Zeff, B. Kleber, and J. Fineberg, “Singularity dynamics in curvature collapse and jet eruption on a fluid surface,” Nature 403(6768), 401–404 (2000).
[CrossRef] [PubMed]

1999 (3)

G. I. Stegeman and M. Segev, “Optical Spatial Solitons and Their Interactions: Universality and Diversity,” Science 286(5444), 1518–1523 (1999).
[CrossRef] [PubMed]

O. Bang, L. Bergé, and J. J. Rasmussen, “Fusion, collapse, and stationary bound states of incoherently coupled waves in bulk cubic media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(4), 4600–4613 (1999).
[CrossRef]

O. Bang, D. Edmundson, and W. Królikowski, “Collapse of incoherent light beams in inertial bulk Kerr media,” Phys. Rev. Lett. 83(26), 5479–5482 (1999).
[CrossRef]

1998 (1)

L. Bergé, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303(5-6), 259–370 (1998).
[CrossRef]

1997 (7)

P. A. Robinson, “Nonlinear wave collapse and strong turbulence,” Rev. Mod. Phys. 69(2), 507–574 (1997).
[CrossRef]

L. Bergé, O. Bang, J. J. Rasmussen, and V. K. Mezentsev, “Self-focusing and solitonlike structures in materials with competing quadratic and cubic nonlinearities,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55(3), 3555–3570 (1997).
[CrossRef]

V. Skarka, V. I. Berezhiani, and R. Miklaszewski, “Spatiotemporal soliton propagation in saturating nonlinear optical media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(1), 1080–1087 (1997).
[CrossRef]

S. Gatz and J. Herrmann, “Propagation of optical beams and the properties of two-dimensional spatial solitons in media with a local saturable nonlinear refractive index,” J. Opt. Soc. Am. B 14(7), 1795 (1997).
[CrossRef]

G. Fibich and G. C. Papanicolaou, “Self-focusing in the presence of small time dispersion and nonparaxiality,” Opt. Lett. 22(18), 1379–1381 (1997).
[CrossRef] [PubMed]

Y. Barad and Y. Silberberg, “Polarization Evolution and Polarization Instability of Solitons in a Birefringent Optical Fiber,” Phys. Rev. Lett. 78(17), 3290–3293 (1997).
[CrossRef]

R. J. Ballagh, K. Burnett, and T. F. Scott, “Theory of an Output Coupler for Bose-Einstein Condensated Atoms,” Phys. Rev. Lett. 78(9), 1607–1611 (1997).
[CrossRef]

1995 (1)

E. A. Kuznetsov, J. J. Rasmussen, K. Rypdal, and S. K. Turitsyn, “Shaper criteria for the wave collapse,” Physica D 87(1-4), 273–284 (1995).
[CrossRef]

1994 (1)

O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Subcritical localization in the discrete nonlinear Schrödinger equation with arbitrary power nonlinearity,” Nonlinearity 7(1), 205–218 (1994).
[CrossRef]

1993 (1)

1991 (2)

M. Desaix, D. Anderson, and M. Lisak, “Variational approach to collapse of the optical pulses,” J. Opt. Soc. Am. B 8(10), 2082 (1991).
[CrossRef]

B. A. Malomed, “Polarization dynamics and interactions of solitons in a birefringent optical fiber,” Phys. Rev. A 43(1), 410–423 (1991).
[CrossRef] [PubMed]

1990 (1)

1989 (1)

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Polarized soliton instability and branching in birefringent fibers,” Opt. Commun. 70(2), 166–172 (1989).
[CrossRef]

1987 (1)

1983 (1)

M. Weinstein, “Nonlinear Schrödinger Equations and Sharp Interpolation Estimates,” Commun. Math. Phys. 87(4), 567–576 (1983).
[CrossRef]

1976 (1)

Y. R. Shen, “Recent advances in nonlinear optics,” Rev. Mod. Phys. 48(1), 1–32 (1976).
[CrossRef]

1975 (2)

G. B. Scott, D. E. Lacklison, H. I. Ralph, and J. L. Page, “Magnetic circular dichroism and Faraday rotation spectra of Y3Fe5O12,” Phys. Rev. B 12(7), 2562 (1975).
[CrossRef]

G. A. Smolenskii, R. V. Pisarev, and I. G. Sinii, “Birefringence of light in magnetically ordered crystals,” Usp. Fiziol. Nauk 116, 231 (1975).
[CrossRef]

1970 (1)

J. F. Dillon, J. P. Remeika, and C. R. Staton, “Linear Magnetic Birefringence in the Ferrimagnetic Garnets,” J. Appl. Phys. 41(11), 4613 (1970).
[CrossRef]

1965 (1)

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15(26), 1005–1008 (1965).
[CrossRef]

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-Trapping of Optical Beams,” Phys. Rev. Lett. 13(15), 479–482 (1964).
[CrossRef]

Abdullaev, F. Kh.

F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensation by temporal modulation of the scattering length,” Phys. Rev. A 67(1), 013605 (2003).
[CrossRef]

Aitchison, J. S.

D. Cheskis, S. Bar-Ad, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, and D. Ross, “Strong spatiotemporal localization in a silica nonlinear waveguide array,” Phys. Rev. Lett. 91(22), 223901 (2003).
[CrossRef] [PubMed]

Akhmediev, N.

Anderson, D.

Ankiewicz, A.

Assanto, G.

Ballagh, R. J.

R. J. Ballagh, K. Burnett, and T. F. Scott, “Theory of an Output Coupler for Bose-Einstein Condensated Atoms,” Phys. Rev. Lett. 78(9), 1607–1611 (1997).
[CrossRef]

Bang, O.

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4), 046619 (2002).
[CrossRef] [PubMed]

L. Bergé, O. Bang, and W. Królikowski, “Influence of four-wave mixing and walk-Off on the self-focusing of coupled waves,” Phys. Rev. Lett. 84(15), 3302–3305 (2000).
[CrossRef] [PubMed]

O. Bang, L. Bergé, and J. J. Rasmussen, “Fusion, collapse, and stationary bound states of incoherently coupled waves in bulk cubic media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(4), 4600–4613 (1999).
[CrossRef]

O. Bang, D. Edmundson, and W. Królikowski, “Collapse of incoherent light beams in inertial bulk Kerr media,” Phys. Rev. Lett. 83(26), 5479–5482 (1999).
[CrossRef]

L. Bergé, O. Bang, J. J. Rasmussen, and V. K. Mezentsev, “Self-focusing and solitonlike structures in materials with competing quadratic and cubic nonlinearities,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55(3), 3555–3570 (1997).
[CrossRef]

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J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
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T. Koch, T. Lahaye, J. Metz, B. Fröhlich, A. Griesmaier, and T. Pfau, “Stabilization of a purely dipolar quantum gas against collapse,” Nat. Phys. 4(3), 218–222 (2008).
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G. B. Scott, D. E. Lacklison, H. I. Ralph, and J. L. Page, “Magnetic circular dichroism and Faraday rotation spectra of Y3Fe5O12,” Phys. Rev. B 12(7), 2562 (1975).
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T. Lahaye, J. Metz, B. Fröhlich, T. Koch, M. Meister, A. Griesmaier, T. Pfau, H. Saito, Y. Kawaguchi, and M. Ueda, “d-wave collapse and explosion of a dipolar bose-einstein condensate,” Phys. Rev. Lett. 101(8), 080401 (2008).
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Metz, J.

T. Lahaye, J. Metz, B. Fröhlich, T. Koch, M. Meister, A. Griesmaier, T. Pfau, H. Saito, Y. Kawaguchi, and M. Ueda, “d-wave collapse and explosion of a dipolar bose-einstein condensate,” Phys. Rev. Lett. 101(8), 080401 (2008).
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T. Koch, T. Lahaye, J. Metz, B. Fröhlich, A. Griesmaier, and T. Pfau, “Stabilization of a purely dipolar quantum gas against collapse,” Nat. Phys. 4(3), 218–222 (2008).
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L. Bergé, O. Bang, J. J. Rasmussen, and V. K. Mezentsev, “Self-focusing and solitonlike structures in materials with competing quadratic and cubic nonlinearities,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55(3), 3555–3570 (1997).
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V. Skarka, V. I. Berezhiani, and R. Miklaszewski, “Spatiotemporal soliton propagation in saturating nonlinear optical media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(1), 1080–1087 (1997).
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K. D. Moll, A. L. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the Townes profile,” Phys. Rev. Lett. 90(20), 203902 (2003).
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Y. Linzon, K. A. Rutkowska, B. A. Malomed, and R. Morandotti, “Magneto-optical control of light collapse in bulk Kerr media,” Phys. Rev. Lett. 103(5), 053902 (2009).
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Nakaya, T.

O. Kamada, T. Nakaya, and S. Higuchi, “Magnetic field optical sensors using Ce:YIG single crystals as a Faraday element,” Sens. Actuators A Phys. 119(2), 345–348 (2005).
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G. B. Scott, D. E. Lacklison, H. I. Ralph, and J. L. Page, “Magnetic circular dichroism and Faraday rotation spectra of Y3Fe5O12,” Phys. Rev. B 12(7), 2562 (1975).
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Pfau, T.

T. Koch, T. Lahaye, J. Metz, B. Fröhlich, A. Griesmaier, and T. Pfau, “Stabilization of a purely dipolar quantum gas against collapse,” Nat. Phys. 4(3), 218–222 (2008).
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O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4), 046619 (2002).
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M. C. Sekhar, J.-Y. Hwang, M. Ferrera, Y. Linzon, L. Razzari, C. Harnagea, M. Zaezjev, A. Pignolet, and R. Morandotti, “Strong enhancement of the Faraday rotation in Ce: and Bi: co-modified epitaxial iron garnet thin films,” Appl. Phys. Lett. 94(18), 181916 (2009).
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A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
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Y. Linzon, K. A. Rutkowska, B. A. Malomed, and R. Morandotti, “Magneto-optical control of light collapse in bulk Kerr media,” Phys. Rev. Lett. 103(5), 053902 (2009).
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E. A. Kuznetsov, J. J. Rasmussen, K. Rypdal, and S. K. Turitsyn, “Shaper criteria for the wave collapse,” Physica D 87(1-4), 273–284 (1995).
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T. Lahaye, J. Metz, B. Fröhlich, T. Koch, M. Meister, A. Griesmaier, T. Pfau, H. Saito, Y. Kawaguchi, and M. Ueda, “d-wave collapse and explosion of a dipolar bose-einstein condensate,” Phys. Rev. Lett. 101(8), 080401 (2008).
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[CrossRef] [PubMed]

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G. B. Scott, D. E. Lacklison, H. I. Ralph, and J. L. Page, “Magnetic circular dichroism and Faraday rotation spectra of Y3Fe5O12,” Phys. Rev. B 12(7), 2562 (1975).
[CrossRef]

Scott, T. F.

R. J. Ballagh, K. Burnett, and T. F. Scott, “Theory of an Output Coupler for Bose-Einstein Condensated Atoms,” Phys. Rev. Lett. 78(9), 1607–1611 (1997).
[CrossRef]

Segev, M.

G. I. Stegeman and M. Segev, “Optical Spatial Solitons and Their Interactions: Universality and Diversity,” Science 286(5444), 1518–1523 (1999).
[CrossRef] [PubMed]

Sekhar, M. C.

M. C. Sekhar, J.-Y. Hwang, M. Ferrera, Y. Linzon, L. Razzari, C. Harnagea, M. Zaezjev, A. Pignolet, and R. Morandotti, “Strong enhancement of the Faraday rotation in Ce: and Bi: co-modified epitaxial iron garnet thin films,” Appl. Phys. Lett. 94(18), 181916 (2009).
[CrossRef]

Shen, Y. R.

Y. R. Shen, “Recent advances in nonlinear optics,” Rev. Mod. Phys. 48(1), 1–32 (1976).
[CrossRef]

Silberberg, Y.

D. Cheskis, S. Bar-Ad, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, and D. Ross, “Strong spatiotemporal localization in a silica nonlinear waveguide array,” Phys. Rev. Lett. 91(22), 223901 (2003).
[CrossRef] [PubMed]

Y. Barad and Y. Silberberg, “Polarization Evolution and Polarization Instability of Solitons in a Birefringent Optical Fiber,” Phys. Rev. Lett. 78(17), 3290–3293 (1997).
[CrossRef]

Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. 15(22), 1282–1284 (1990).
[CrossRef] [PubMed]

Sinii, I. G.

G. A. Smolenskii, R. V. Pisarev, and I. G. Sinii, “Birefringence of light in magnetically ordered crystals,” Usp. Fiziol. Nauk 116, 231 (1975).
[CrossRef]

Sivan, Y.

Skarka, V.

V. Skarka, V. I. Berezhiani, and R. Miklaszewski, “Spatiotemporal soliton propagation in saturating nonlinear optical media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(1), 1080–1087 (1997).
[CrossRef]

Smolenskii, G. A.

G. A. Smolenskii, R. V. Pisarev, and I. G. Sinii, “Birefringence of light in magnetically ordered crystals,” Usp. Fiziol. Nauk 116, 231 (1975).
[CrossRef]

Soto-Crespo, J. M.

Staton, C. R.

J. F. Dillon, J. P. Remeika, and C. R. Staton, “Linear Magnetic Birefringence in the Ferrimagnetic Garnets,” J. Appl. Phys. 41(11), 4613 (1970).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman and M. Segev, “Optical Spatial Solitons and Their Interactions: Universality and Diversity,” Science 286(5444), 1518–1523 (1999).
[CrossRef] [PubMed]

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Polarized soliton instability and branching in birefringent fibers,” Opt. Commun. 70(2), 166–172 (1989).
[CrossRef]

Szameit, A.

Torner, L.

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quant. Semicl. Opt. 7(5), R53–R72 (2005).
[CrossRef]

Towers, I.

I. Towers and B. A. Malomed, “Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity,” J. Opt. Soc. Am. B 19(3), 537 (2002).
[CrossRef]

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-Trapping of Optical Beams,” Phys. Rev. Lett. 13(15), 479–482 (1964).
[CrossRef]

Trillo, S.

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Polarized soliton instability and branching in birefringent fibers,” Opt. Commun. 70(2), 166–172 (1989).
[CrossRef]

Tünnermann, A.

Turitsyn, S. K.

E. A. Kuznetsov, J. J. Rasmussen, K. Rypdal, and S. K. Turitsyn, “Shaper criteria for the wave collapse,” Physica D 87(1-4), 273–284 (1995).
[CrossRef]

Ueda, M.

T. Lahaye, J. Metz, B. Fröhlich, T. Koch, M. Meister, A. Griesmaier, T. Pfau, H. Saito, Y. Kawaguchi, and M. Ueda, “d-wave collapse and explosion of a dipolar bose-einstein condensate,” Phys. Rev. Lett. 101(8), 080401 (2008).
[CrossRef] [PubMed]

H. Saito and M. Ueda, “Dynamically stabilized bright solitons in a two-dimensional bose-einstein condensate,” Phys. Rev. Lett. 90(4), 040403 (2003).
[CrossRef] [PubMed]

Vuong, L. T.

T. D. Grow, A. A. Ishaaya, L. T. Vuong, and A. L. Gaeta, “Collapse and stability of necklace beams in Kerr media,” Phys. Rev. Lett. 99(13), 133902 (2007).
[CrossRef] [PubMed]

Wabnitz, S.

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Polarized soliton instability and branching in birefringent fibers,” Opt. Commun. 70(2), 166–172 (1989).
[CrossRef]

Weinstein, M.

M. Weinstein, “Nonlinear Schrödinger Equations and Sharp Interpolation Estimates,” Commun. Math. Phys. 87(4), 567–576 (1983).
[CrossRef]

Wise, F.

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quant. Semicl. Opt. 7(5), R53–R72 (2005).
[CrossRef]

F. Wise and P. di Trapani, “The Hunt for Light Bullets – Spatiotemporal Solitons,” Opt. Photonics News 13(2), 29 (2002).

Wise, F. W.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[CrossRef]

Wozniak, W. A.

R. Kurzynowski and W. A. Wo?niak, “Superposition rule for the magneto-optic effects in isotropic media,” Optik (Stuttg.) 115(10), 473–475 (2004).
[CrossRef]

Wright, E. M.

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Polarized soliton instability and branching in birefringent fibers,” Opt. Commun. 70(2), 166–172 (1989).
[CrossRef]

Wyller, J.

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4), 046619 (2002).
[CrossRef] [PubMed]

Yang, J.

Zaezjev, M.

M. C. Sekhar, J.-Y. Hwang, M. Ferrera, Y. Linzon, L. Razzari, C. Harnagea, M. Zaezjev, A. Pignolet, and R. Morandotti, “Strong enhancement of the Faraday rotation in Ce: and Bi: co-modified epitaxial iron garnet thin films,” Appl. Phys. Lett. 94(18), 181916 (2009).
[CrossRef]

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D. P. Lathrop, B. W. Zeff, B. Kleber, and J. Fineberg, “Singularity dynamics in curvature collapse and jet eruption on a fluid surface,” Nature 403(6768), 401–404 (2000).
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Appl. Phys. Lett. (1)

M. C. Sekhar, J.-Y. Hwang, M. Ferrera, Y. Linzon, L. Razzari, C. Harnagea, M. Zaezjev, A. Pignolet, and R. Morandotti, “Strong enhancement of the Faraday rotation in Ce: and Bi: co-modified epitaxial iron garnet thin films,” Appl. Phys. Lett. 94(18), 181916 (2009).
[CrossRef]

Commun. Math. Phys. (1)

M. Weinstein, “Nonlinear Schrödinger Equations and Sharp Interpolation Estimates,” Commun. Math. Phys. 87(4), 567–576 (1983).
[CrossRef]

J. Appl. Phys. (1)

J. F. Dillon, J. P. Remeika, and C. R. Staton, “Linear Magnetic Birefringence in the Ferrimagnetic Garnets,” J. Appl. Phys. 41(11), 4613 (1970).
[CrossRef]

J. Opt. Soc. Am. B (1)

I. Towers and B. A. Malomed, “Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity,” J. Opt. Soc. Am. B 19(3), 537 (2002).
[CrossRef]

J. Opt. B: Quant. Semicl. Opt. (1)

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quant. Semicl. Opt. 7(5), R53–R72 (2005).
[CrossRef]

J. Opt. Soc. Am. B (2)

Nat. Photonics (1)

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[CrossRef]

Nat. Phys. (1)

T. Koch, T. Lahaye, J. Metz, B. Fröhlich, A. Griesmaier, and T. Pfau, “Stabilization of a purely dipolar quantum gas against collapse,” Nat. Phys. 4(3), 218–222 (2008).
[CrossRef]

Nature (1)

D. P. Lathrop, B. W. Zeff, B. Kleber, and J. Fineberg, “Singularity dynamics in curvature collapse and jet eruption on a fluid surface,” Nature 403(6768), 401–404 (2000).
[CrossRef] [PubMed]

Nonlinearity (1)

O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Subcritical localization in the discrete nonlinear Schrödinger equation with arbitrary power nonlinearity,” Nonlinearity 7(1), 205–218 (1994).
[CrossRef]

Opt. Commun. (1)

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Polarized soliton instability and branching in birefringent fibers,” Opt. Commun. 70(2), 166–172 (1989).
[CrossRef]

Opt. Express (2)

Opt. Lett. (8)

Opt. Photonics News (1)

F. Wise and P. di Trapani, “The Hunt for Light Bullets – Spatiotemporal Solitons,” Opt. Photonics News 13(2), 29 (2002).

Optik (Stuttg.) (1)

R. Kurzynowski and W. A. Wo?niak, “Superposition rule for the magneto-optic effects in isotropic media,” Optik (Stuttg.) 115(10), 473–475 (2004).
[CrossRef]

Phys. Rev. A (1)

F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensation by temporal modulation of the scattering length,” Phys. Rev. A 67(1), 013605 (2003).
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Phys. Rep. (2)

Y. S. Kivshar and D. E. Pelinovsky, “Self-focusing and transverse instabilities of solitary waves,” Phys. Rep. 331(4), 117–195 (2000).
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L. Bergé, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303(5-6), 259–370 (1998).
[CrossRef]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

O. Bang, L. Bergé, and J. J. Rasmussen, “Fusion, collapse, and stationary bound states of incoherently coupled waves in bulk cubic media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(4), 4600–4613 (1999).
[CrossRef]

Phys. Rev. Lett. (1)

Y. Barad and Y. Silberberg, “Polarization Evolution and Polarization Instability of Solitons in a Birefringent Optical Fiber,” Phys. Rev. Lett. 78(17), 3290–3293 (1997).
[CrossRef]

Phys. Rev. A (1)

B. A. Malomed, “Polarization dynamics and interactions of solitons in a birefringent optical fiber,” Phys. Rev. A 43(1), 410–423 (1991).
[CrossRef] [PubMed]

Phys. Rev. B (1)

G. B. Scott, D. E. Lacklison, H. I. Ralph, and J. L. Page, “Magnetic circular dichroism and Faraday rotation spectra of Y3Fe5O12,” Phys. Rev. B 12(7), 2562 (1975).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4), 046619 (2002).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

V. Skarka, V. I. Berezhiani, and R. Miklaszewski, “Spatiotemporal soliton propagation in saturating nonlinear optical media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(1), 1080–1087 (1997).
[CrossRef]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

L. Bergé, O. Bang, J. J. Rasmussen, and V. K. Mezentsev, “Self-focusing and solitonlike structures in materials with competing quadratic and cubic nonlinearities,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55(3), 3555–3570 (1997).
[CrossRef]

Phys. Rev. Lett. (14)

O. Bang, D. Edmundson, and W. Królikowski, “Collapse of incoherent light beams in inertial bulk Kerr media,” Phys. Rev. Lett. 83(26), 5479–5482 (1999).
[CrossRef]

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010).
[CrossRef] [PubMed]

A. L. Gaeta, “Catastrophic collapse of ultrashort pulses,” Phys. Rev. Lett. 84(16), 3582–3585 (2000).
[CrossRef] [PubMed]

D. Cheskis, S. Bar-Ad, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, and D. Ross, “Strong spatiotemporal localization in a silica nonlinear waveguide array,” Phys. Rev. Lett. 91(22), 223901 (2003).
[CrossRef] [PubMed]

P. Pedri and L. Santos, “Two-dimensional bright solitons in dipolar Bose-Einstein condensates,” Phys. Rev. Lett. 95(20), 200404 (2005).
[CrossRef] [PubMed]

H. Saito and M. Ueda, “Dynamically stabilized bright solitons in a two-dimensional bose-einstein condensate,” Phys. Rev. Lett. 90(4), 040403 (2003).
[CrossRef] [PubMed]

L. Bergé, O. Bang, and W. Królikowski, “Influence of four-wave mixing and walk-Off on the self-focusing of coupled waves,” Phys. Rev. Lett. 84(15), 3302–3305 (2000).
[CrossRef] [PubMed]

T. D. Grow, A. A. Ishaaya, L. T. Vuong, and A. L. Gaeta, “Collapse and stability of necklace beams in Kerr media,” Phys. Rev. Lett. 99(13), 133902 (2007).
[CrossRef] [PubMed]

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15(26), 1005–1008 (1965).
[CrossRef]

T. Lahaye, J. Metz, B. Fröhlich, T. Koch, M. Meister, A. Griesmaier, T. Pfau, H. Saito, Y. Kawaguchi, and M. Ueda, “d-wave collapse and explosion of a dipolar bose-einstein condensate,” Phys. Rev. Lett. 101(8), 080401 (2008).
[CrossRef] [PubMed]

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-Trapping of Optical Beams,” Phys. Rev. Lett. 13(15), 479–482 (1964).
[CrossRef]

K. D. Moll, A. L. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the Townes profile,” Phys. Rev. Lett. 90(20), 203902 (2003).
[CrossRef] [PubMed]

Y. Linzon, K. A. Rutkowska, B. A. Malomed, and R. Morandotti, “Magneto-optical control of light collapse in bulk Kerr media,” Phys. Rev. Lett. 103(5), 053902 (2009).
[CrossRef] [PubMed]

R. J. Ballagh, K. Burnett, and T. F. Scott, “Theory of an Output Coupler for Bose-Einstein Condensated Atoms,” Phys. Rev. Lett. 78(9), 1607–1611 (1997).
[CrossRef]

Physica D (1)

E. A. Kuznetsov, J. J. Rasmussen, K. Rypdal, and S. K. Turitsyn, “Shaper criteria for the wave collapse,” Physica D 87(1-4), 273–284 (1995).
[CrossRef]

Progr. Opt. (1)

B. A. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Progr. Opt. 43, 71 (2002).
[CrossRef]

Rev. Mod. Phys. (3)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
[CrossRef]

P. A. Robinson, “Nonlinear wave collapse and strong turbulence,” Rev. Mod. Phys. 69(2), 507–574 (1997).
[CrossRef]

Y. R. Shen, “Recent advances in nonlinear optics,” Rev. Mod. Phys. 48(1), 1–32 (1976).
[CrossRef]

Science (1)

G. I. Stegeman and M. Segev, “Optical Spatial Solitons and Their Interactions: Universality and Diversity,” Science 286(5444), 1518–1523 (1999).
[CrossRef] [PubMed]

Sens. Actuators A Phys. (1)

O. Kamada, T. Nakaya, and S. Higuchi, “Magnetic field optical sensors using Ce:YIG single crystals as a Faraday element,” Sens. Actuators A Phys. 119(2), 345–348 (2005).
[CrossRef]

Usp. Fiziol. Nauk (1)

G. A. Smolenskii, R. V. Pisarev, and I. G. Sinii, “Birefringence of light in magnetically ordered crystals,” Usp. Fiziol. Nauk 116, 231 (1975).
[CrossRef]

Other (10)

A. K. Zvezdin, and V. A. Kotov, Modern Magnetooptics and Magnetooptical Materials (Taylor & Francis Group, New York 1997).

J.-M. Liu, Photonic Devices, Cambridge Univ. Press, New York 2005.

U. M. Ascher, Numerical methods for evolutionary differential equations (SIAM, Philadelphia 2008).

J. C. Butcher, Numerical Methods for Ordinary Differential Equations (Wiley & Sons, West Sussex 2008).

C. Sulem and P. L. Sulem, The nonlinear Schrödinger equation: self-focusing and wave collapse (Springer, New York 1999).

E. Infeld and G. Rowlands, Nonlinear Waves, Solitons and Chaos (Cambridge Univ. Press, Cambridge 2000).

R. W. Boyd, Nonlinear Optics (Academic Press, San Diego 2008).

B. A. Malomed, “Nonlinear Schrödinger equations,” in Encyclopedia of Nonlinear Science, A. Scott, ed., (Routledge, New York 2005).

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

S. Trillo and W. Torruellas, Spatial Solitons (Springer-Verlag, Berlin 2001).

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Figures (5)

Fig. 2
Fig. 2

Results of direct simulations (a)-(c), and simulations performed in the framework of the variational approximation (d). Panel (a) shows the normalized width of the output beam, for an input amplitude A0 = 1.135 and a propagation distance z = 100LD, as a function of the magnetically-induced birefringences. Solutions that collapse before z = 100LD are represented by the grey color in panel (a), and the corresponding collapse distance is shown in panel (b). Panel (c) shows the beam propagation for three particular combinations of the birefringence coefficients, with c = 0.055 and b = bA,B,C. Panel (d) displays the results of the numerical solution of the variational equations [Eqs. (7)], for the cases corresponding to those presented in panel (a). For further details, see section 3.2.

Fig. 3
Fig. 3

The normalized stable propagation length L/LD (defined by the collapse point, or by the largest propagation length for which the normalized beam waist radius is equal to 1) as a function of the circular birefringence coefficient b for a fixed value of the input amplitude, A0 = 1.135, and for two different values of the linear birefringence coefficient, c = 0.055 (a1) and c = 0.07 (a2). The values of the critical circular birefringence coefficients bi are indicated in both cases. On the other hand, the corresponding evolution of the normalized beam waist radius for specific values of b, in proximity of the critical values, is presented in (b1) and (b2).

Fig. 1
Fig. 1

Results of numerical simulations for the beam propagation at low (a) and high (b)-(e) input powers with zero (a)-(b) and nonzero (c)-(e) magnetically-induced birefringences. Numerical parameters are indicated in each panel, where A0 is the amplitude of the input Gaussian beam, whereas b and c are the coefficients of the circular and linear birefringences, respectively. All cases are summarized in panel (f), where the normalized beam waist radius is plotted versus the scaled propagation distance z. Note that the results of the simulations obtained after the collapse point are not relevant and should not be taken into consideration.

Fig. 4
Fig. 4

The quasi-stabilization line, as predicted by the direct numerical simulations of Eqs. (1) (square symbols), indicating pairs of coefficients (bB, c) for which the magnetically-induced birefringences allow for a prolongation of the stable beam propagation (a). The blue squares denote cases for which only a reduction of the beam spatial divergence is achieved (rather than the extension of the quasi-stable propagation that is observed for the black squares) as shown in (b1) and (b2). The quasi-stabilization line predicted by the variational approximation for N ˜ 0 =0.68 (red) is compared to that (black) produced by the direct simulations in panel (a).

Fig. 5
Fig. 5

Results for the simulations of the variational Eqs. (7). The normalized distance of the stable propagation is shown as a function of the circular-birefringence coefficient b in panel (a), where the red symbols correspond to the collapse of the beam, while the black ones indicate its diffraction. Solid dots represent the cases in which the width of the oscillating beam is much larger than the initial value (we note that in these cases the VA - which postulates that the profile of the solution is represented by the trial function along propagation – fails to be reliable). The evolution of the beam width along the propagation distance for some specific cases is shown in panel (b). The parameters used in the simulations are indicated in the panels.

Equations (24)

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i u R z ˜ + 1 2 ( 2 r ˜ 2 + 1 r ˜ r ˜ ) u R +b u R +c u L +( | u R | 2 +2 | u L | 2 ) u R =0,
i u L z ˜ + 1 2 ( 2 r ˜ 2 + 1 r ˜ r ˜ ) u L b u L +c u R +( | u L | 2 +2 | u R | 2 ) u L =0.
Λ= i 2 [ u R * u R + u R ( u R * ) + u L * u L + u L ( u L * ) ] 1 2 ( | u R | 2 + | u L | 2 ) + 1 2 ( | u R | 4 + | u L | 4 )+2 | u R | 2 | u L | 2 +b( | u R | 2 | u L | 2 )+c( u R * u L + u R u L * ),
{ u R (r,z) u L (r,z) }=A  exp( iφ )  { exp( iψ/2 )cosθ exp( iψ/2 )sinθ }{ exp( r 2 2 w R 2 + i 2 ξ R r 2 ) exp( r 2 2 w L 2 + i 2 ξ L r 2 ) },
{ u R (r,z) u L (r,z) }=A  exp( iφ )  { exp( iψ/2 )cosθ exp( iψ/2 )sinθ }exp( r 2 2 w 2 + i 2 ξ r 2 ).
N2π N ˜ =2π ( | u R ( r ) | 2 + | u L ( r ) | 2 )dr =π A 2 w 2 .
L eff =π N ˜ ( 2 ϕ + ψ cos( 2θ )+ w 2 ξ )π N ˜ ( 1 w 2 + ξ 2 w 2 )+π N ˜ 2 w 2 ( 1+ 1 2 sin 2 ( 2θ ) )+ +2π N ˜ ( bcos( 2θ )+csin( 2θ )cosψ ).
d 2 w d z 2 = 1 w 3 [ 1 N ˜ ( 1+ 1 2 sin 2 ( 2θ ) ) ].
dθ dz =csinψ,
dψ dz =2b N ˜ w 2 cos( 2θ )2ccot( 2θ )cosψ.
Η ^ = π N ˜ w 2 +π N ˜ ( w ) 2 π N ˜ 2 w 2 ( 1+ 1 2 sin 2 ( 2θ ) )2π N ˜ ( bcos( 2θ )+csin( 2θ )cosψ ).
d 2 w d z 2 = 1 w 3 [ 1 N ˜ ( 1+ 1 4 [ ( 1+ K 2 )+( 1 K 2 )cos( 4cz ) ] ) ],
d 2 w d z 2 = 1 w 3 [ 1 N ˜ ( 1+ 1 2 c 2 b 2 cos 2 ( 2bz ) ) ].
ε= n 0 2 [ 1 iQcosα 0 iQcosα 1 iQsinα 0 iQsinα 1 ]+[ B 1 sin 2 α 0 B 2 sinαcosα 0 0 0 B 2 sinαcosα 0 B 1 cos 2 α ],
( ε 3 cosα ε 1 2 ε 3 2 ) 2 =( ε 3 ε 1 2 ε 3 2 1 n 2 )( ε 3 cos 2 α ε 1 2 ε 3 2 + sin 2 α ε 2 1 n 2 )
n 1,2 2 = sec 2 α 2 ε 2   { ε 1 ε 2 ( 1+ cos 2 α )+( ε 1 2 ε 3 2 ) sin 2 α ± [ ( ε 1 2 ε 3 2 ) sin 2 α+ ε 1 ε 2 ( 1+ cos 2 α ) ] 2 4( ε 1 2 ε 3 2 ) ε 2 2 cos 2 α }.
n CM,1 2 = ε 1 2 ε 3 2 ε 1 , n CM,1 2 = ε 2  for  ε 3 0,
n F,1,2 2 = ε 1 ± ε 3  for  ε 1 = ε 2 n 0 2  and  ε 3 n 0 2 .
( Δn ) 2 = ( Δ n l ) 2 + ( Δ n c ) 2 ,
( Δn ) 2 = ( Δ n l ) 2 + ( Δ n c ) 2 = ( Δ n CM ) 2 sin 4 α+ ( Δ n F ) 2 cos 2 α,
u R ( r,z=0 )= u L ( r,z=0 )= A 0 2 2 exp( r ˜ 2 2 ),
P= 3π n 0 2η ( | u R ( r ) | 2 + | u L ( r ) | 2 )rdr = 3π A 0 2 2 n 0 n 2 k 0 2 .
R( z )= [ 0 ( | u R | 2 + | u L | 2 )   r ˜ 2 d r ˜ ] [ 0 ( | u R | 2 + | u L | 2 )d r ˜ ] 1 .
L~ ( b A b ) α A  and  L~ ( b b C ) α C .

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