Abstract

The self-structuring of laser light in an artificial optical medium composed of a colloidal suspension of nanoparticles is demonstrated using variational and numerical methods extended to dissipative systems. In such engineered materials, competing nonlinear susceptibilities are enhanced by the light induced migration of nanoparticles. The compensation of diffraction by competing focusing and defocusing nonlinearities, together with a balance between loss and gain, allow for self-organization of light and the formation of stable dissipative breathing vortex solitons. Due to their robustness, the breathers may be used for selective dynamic photonic tweezing of nanoparticles in colloidal nanosuspensions.

© 2017 Optical Society of America

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References

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2015 (3)

D. Pierangeli, F. Di Mei, C. Conti, A. J. Agranat, and E. DelRe, “Spatial rogue waves in photorefractive ferroelectrics,” Phys. Rev. Lett. 115, 093901 (2015).
[Crossref] [PubMed]

B. N. Aleksic, N. B. Aleksic, V. Skarka, and M. Belic, “Stability and nesting of dissipative vortex solitons with high vorticity,” Phys. Rev. A 91, 043832 (2015).
[Crossref]

Albert S. Reyna and Cid B. de Araújo, “An optimization procedure for the design of all-optical switches based on metal-dielectric nanocomposites,” Opt. Express 237659–7666 (2015).
[Crossref] [PubMed]

2014 (6)

V. Shvedov, A. R. Davoyan, C. Hnatovsky, N. Engheta, and W. Krolikowski, “A long-range polarization-controlled optical tractor beam,” Nat. Photon. 8, 846–850 (2014).
[Crossref]

A. S. L. Gomes, M. T. Carvalho, C. T. Dominguez, C. B. de Araújo, and P. N. Prasad, “Direct three-photon excitation of upconversion random laser emission in a weakly scattering organic colloidal system,” Opt. Express 22, 14305–14310 (2014).
[Crossref] [PubMed]

V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90 (2), 023845 (2014).
[Crossref]

S. Fardad, A. Salandrino, M. Heinrich, P. Zhang, Z. Chen, and D. N. Christodoulides, “Plasmonic resonant solitons in metallic nanosuspensions,” Nano Lett.,  14, 2498–2504 (2014).
[Crossref] [PubMed]

Albert S. Reyna and Cid B. de Araújo, “Spatial phase modulation due to quintic and septic nonlinearities in metallic colloids,” Opt. Express 22(19) 22456–22469 (2014).
[Crossref] [PubMed]

S. Roy, A. Marini, and F. Biancalana, “Free-carrier-driven spatiotemporal dynamics in amplifying silicon waveguides,” Phys. Rev. A 89, 053827 (2014).
[Crossref]

2013 (2)

E. L. Falcao-Filho, C. B. de Araújo, G. Boudebs, H. Leblond, and V. Skarka, “Robust two-dimensional spatial soitons in liqid carbon disulfide,” Phys. Rev. Lett. 110, 013901 (2013).
[Crossref]

W. Man, S. Fardad, Z. Zhang, J. Prakash, M. Lau, P. Zhang, M. Heinrich, D. N. Christodoulides, and Z. Chen, “Optical nonlinearities and enhanced light transmission in soft-matter systems with tunable polarizabilities,” Phys. Rev. Lett. 111, 218302 (2013).
[Crossref] [PubMed]

2011 (1)

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nanodisordered ferroelectrics,” Nat. Photon. 5, 39–42 (2011).
[Crossref]

2010 (3)

M. Matuszewski, “Engineering optical soliton bistability in colloidal media,” Phys. Rev. A 81, 013820 (2010).
[Crossref]

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105, 213901 (2010).
[Crossref]

A. M. Brito-Silva, A. Galembeck, A. S. L. Gomes, A. J. Jesus-Silva, and C. B. de Araújo, “Random laser action in dye solutions containing Stöber silica nanoparticles,” J. Appl. Phys. 108, 033508 (2010).
[Crossref]

2009 (4)

M. Matuszewski, W. Krolikowski, and Y. Kivshar, “Soliton interactions and transformations in colloidal media,” Phys. Rev. A 79, 023814 (2009).
[Crossref]

R. El-Ganainy, D. N. Christodoulides, E. M. Wright, W. M. Lee, and K. Dholakia, “Nonlinear optical dynamics in nonideal gases of interacting colloidal nanoparticles,” Phys. Rev. A 80, 053805 (2009).
[Crossref]

A. Armaroli, S. Trillo, and A. Fratalocchi, “Suppression of transverse instabilities of dark solitons and their dispersive shock waves,” Phys. Rev. A 80, 053803 (2009).
[Crossref]

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[Crossref] [PubMed]

2008 (2)

M. Matuszewski, W. Krolikowski, and Y.S. Kivshar, “Spatial solitons and light-induced instabilities in colloidal media,” Opt. Express 16, 1371–1376 (2008).
[Crossref] [PubMed]

V. Skarka, D. V. Timotijević, and N. B. Aleksić, “Extension of the stability criterion for dissipative optical soliton solutions of a two-dimensional Ginzburg–Landau system generated from asymmetric inputs,” J. Opt. A: Pure Appl. Opt. 10, 075102 (2008).
[Crossref]

2007 (2)

2006 (2)

V. Skarka and N.B. Aleksić, “Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimesional complex cubic-quintic Ginzburg-Landau equations,” Phys. Rev. Lett. 96, 013903 (2006).
[Crossref]

J. Chen, H. Mizuno, H. Kawano, A. Miyawaki, and K. Midorikawa, “Two-photon pumping of random lasers by picosecond and nanosecond lasers,” Appl. Phys. B 85, 45–48 (2006).
[Crossref]

2003 (1)

D. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

2002 (2)

G. Zacharakis, N. A. Papadogiannis, and T. G. Papazoglou, “Random lasing following two-photon excitation of highly scattering media,” Appl. Phys. Lett. 81, 2511–2513 (2002).
[Crossref]

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99 (2002).
[Crossref]

2001 (1)

V. I. Berezhiani, V. Skarka, and N. B. Aleksić, “Dynamics of localized and nonlocalized optical vortex solitons in cubic-quintic nonlinear media,” Phys. Rev. E 64, 057601 (2001).
[Crossref]

1998 (1)

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A 246, 412–422 (1998).
[Crossref]

1997 (1)

V. Skarka, V. I. Berezhiani, and R. Miklaszewski, “Spatiotemporal soliton propagation in saturating nonlinear optical media,” Phys. Rev. E 56, 1080 (1997).
[Crossref]

1996 (1)

N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190–1201 (1996).
[Crossref]

1994 (1)

N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, “Laser action in strongly scattering media,” Nature 368(6470), 436–438 (1994).
[Crossref]

1986 (1)

Afanasjev, V. V.

N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190–1201 (1996).
[Crossref]

Agranat, A. J.

D. Pierangeli, F. Di Mei, C. Conti, A. J. Agranat, and E. DelRe, “Spatial rogue waves in photorefractive ferroelectrics,” Phys. Rev. Lett. 115, 093901 (2015).
[Crossref] [PubMed]

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nanodisordered ferroelectrics,” Nat. Photon. 5, 39–42 (2011).
[Crossref]

Agrawal, G.P.

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

Akhmediev, N. N.

N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190–1201 (1996).
[Crossref]

N. N. Akhmediev and A. Ankiewicz, Dissipative Solitons: From Optics to Biology and Medicine, Lect. Notes Phys. 751 (Springer, 2008).

Aleksic, B. N.

B. N. Aleksic, N. B. Aleksic, V. Skarka, and M. Belic, “Stability and nesting of dissipative vortex solitons with high vorticity,” Phys. Rev. A 91, 043832 (2015).
[Crossref]

V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90 (2), 023845 (2014).
[Crossref]

Aleksic, N. B.

B. N. Aleksic, N. B. Aleksic, V. Skarka, and M. Belic, “Stability and nesting of dissipative vortex solitons with high vorticity,” Phys. Rev. A 91, 043832 (2015).
[Crossref]

V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90 (2), 023845 (2014).
[Crossref]

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105, 213901 (2010).
[Crossref]

V. Skarka, D. V. Timotijević, and N. B. Aleksić, “Extension of the stability criterion for dissipative optical soliton solutions of a two-dimensional Ginzburg–Landau system generated from asymmetric inputs,” J. Opt. A: Pure Appl. Opt. 10, 075102 (2008).
[Crossref]

V. I. Berezhiani, V. Skarka, and N. B. Aleksić, “Dynamics of localized and nonlocalized optical vortex solitons in cubic-quintic nonlinear media,” Phys. Rev. E 64, 057601 (2001).
[Crossref]

Aleksic, N.B.

V. Skarka and N.B. Aleksić, “Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimesional complex cubic-quintic Ginzburg-Landau equations,” Phys. Rev. Lett. 96, 013903 (2006).
[Crossref]

Ankiewicz, A.

N. N. Akhmediev and A. Ankiewicz, Dissipative Solitons: From Optics to Biology and Medicine, Lect. Notes Phys. 751 (Springer, 2008).

Aranson, I. S.

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99 (2002).
[Crossref]

Armaroli, A.

A. Armaroli, S. Trillo, and A. Fratalocchi, “Suppression of transverse instabilities of dark solitons and their dispersive shock waves,” Phys. Rev. A 80, 053803 (2009).
[Crossref]

Ashkin, A.

Atai, J.

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A 246, 412–422 (1998).
[Crossref]

Balachandran, R. M.

N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, “Laser action in strongly scattering media,” Nature 368(6470), 436–438 (1994).
[Crossref]

Belic, M.

B. N. Aleksic, N. B. Aleksic, V. Skarka, and M. Belic, “Stability and nesting of dissipative vortex solitons with high vorticity,” Phys. Rev. A 91, 043832 (2015).
[Crossref]

Berezhiani, V. I.

V. I. Berezhiani, V. Skarka, and N. B. Aleksić, “Dynamics of localized and nonlocalized optical vortex solitons in cubic-quintic nonlinear media,” Phys. Rev. E 64, 057601 (2001).
[Crossref]

V. Skarka, V. I. Berezhiani, and R. Miklaszewski, “Spatiotemporal soliton propagation in saturating nonlinear optical media,” Phys. Rev. E 56, 1080 (1997).
[Crossref]

Biancalana, F.

S. Roy, A. Marini, and F. Biancalana, “Free-carrier-driven spatiotemporal dynamics in amplifying silicon waveguides,” Phys. Rev. A 89, 053827 (2014).
[Crossref]

Bjorkholm, J. E.

Boudebs, G.

E. L. Falcao-Filho, C. B. de Araújo, G. Boudebs, H. Leblond, and V. Skarka, “Robust two-dimensional spatial soitons in liqid carbon disulfide,” Phys. Rev. Lett. 110, 013901 (2013).
[Crossref]

Brito-Silva, A. M.

A. M. Brito-Silva, A. Galembeck, A. S. L. Gomes, A. J. Jesus-Silva, and C. B. de Araújo, “Random laser action in dye solutions containing Stöber silica nanoparticles,” J. Appl. Phys. 108, 033508 (2010).
[Crossref]

Carvalho, M. T.

Chen, J.

J. Chen, H. Mizuno, H. Kawano, A. Miyawaki, and K. Midorikawa, “Two-photon pumping of random lasers by picosecond and nanosecond lasers,” Appl. Phys. B 85, 45–48 (2006).
[Crossref]

Chen, Z.

S. Fardad, A. Salandrino, M. Heinrich, P. Zhang, Z. Chen, and D. N. Christodoulides, “Plasmonic resonant solitons in metallic nanosuspensions,” Nano Lett.,  14, 2498–2504 (2014).
[Crossref] [PubMed]

W. Man, S. Fardad, Z. Zhang, J. Prakash, M. Lau, P. Zhang, M. Heinrich, D. N. Christodoulides, and Z. Chen, “Optical nonlinearities and enhanced light transmission in soft-matter systems with tunable polarizabilities,” Phys. Rev. Lett. 111, 218302 (2013).
[Crossref] [PubMed]

Christodoulides, D. N.

S. Fardad, A. Salandrino, M. Heinrich, P. Zhang, Z. Chen, and D. N. Christodoulides, “Plasmonic resonant solitons in metallic nanosuspensions,” Nano Lett.,  14, 2498–2504 (2014).
[Crossref] [PubMed]

W. Man, S. Fardad, Z. Zhang, J. Prakash, M. Lau, P. Zhang, M. Heinrich, D. N. Christodoulides, and Z. Chen, “Optical nonlinearities and enhanced light transmission in soft-matter systems with tunable polarizabilities,” Phys. Rev. Lett. 111, 218302 (2013).
[Crossref] [PubMed]

R. El-Ganainy, D. N. Christodoulides, E. M. Wright, W. M. Lee, and K. Dholakia, “Nonlinear optical dynamics in nonideal gases of interacting colloidal nanoparticles,” Phys. Rev. A 80, 053805 (2009).
[Crossref]

R. El-Ganainy, D. N. Christodoulides, C. Rotschild, and M. Segev, “Soliton dynamics and self-induced transparency in nonlinear nanosuspensions,” Opt. Express 15, 10207–10218 (2007).
[Crossref] [PubMed]

Chu, S.

Conti, C.

D. Pierangeli, F. Di Mei, C. Conti, A. J. Agranat, and E. DelRe, “Spatial rogue waves in photorefractive ferroelectrics,” Phys. Rev. Lett. 115, 093901 (2015).
[Crossref] [PubMed]

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nanodisordered ferroelectrics,” Nat. Photon. 5, 39–42 (2011).
[Crossref]

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[Crossref] [PubMed]

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, “Shocks in nonlocal media,” Phys. Rev. Lett. 99, 043903 (2007).
[Crossref] [PubMed]

Davoyan, A. R.

V. Shvedov, A. R. Davoyan, C. Hnatovsky, N. Engheta, and W. Krolikowski, “A long-range polarization-controlled optical tractor beam,” Nat. Photon. 8, 846–850 (2014).
[Crossref]

de Araújo, C. B.

A. S. L. Gomes, M. T. Carvalho, C. T. Dominguez, C. B. de Araújo, and P. N. Prasad, “Direct three-photon excitation of upconversion random laser emission in a weakly scattering organic colloidal system,” Opt. Express 22, 14305–14310 (2014).
[Crossref] [PubMed]

E. L. Falcao-Filho, C. B. de Araújo, G. Boudebs, H. Leblond, and V. Skarka, “Robust two-dimensional spatial soitons in liqid carbon disulfide,” Phys. Rev. Lett. 110, 013901 (2013).
[Crossref]

A. M. Brito-Silva, A. Galembeck, A. S. L. Gomes, A. J. Jesus-Silva, and C. B. de Araújo, “Random laser action in dye solutions containing Stöber silica nanoparticles,” J. Appl. Phys. 108, 033508 (2010).
[Crossref]

de Araújo, Cid B.

DelRe, E.

D. Pierangeli, F. Di Mei, C. Conti, A. J. Agranat, and E. DelRe, “Spatial rogue waves in photorefractive ferroelectrics,” Phys. Rev. Lett. 115, 093901 (2015).
[Crossref] [PubMed]

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nanodisordered ferroelectrics,” Nat. Photon. 5, 39–42 (2011).
[Crossref]

Dholakia, K.

R. El-Ganainy, D. N. Christodoulides, E. M. Wright, W. M. Lee, and K. Dholakia, “Nonlinear optical dynamics in nonideal gases of interacting colloidal nanoparticles,” Phys. Rev. A 80, 053805 (2009).
[Crossref]

Dominguez, C. T.

Dziedzic, J. M.

El-Ganainy, R.

R. El-Ganainy, D. N. Christodoulides, E. M. Wright, W. M. Lee, and K. Dholakia, “Nonlinear optical dynamics in nonideal gases of interacting colloidal nanoparticles,” Phys. Rev. A 80, 053805 (2009).
[Crossref]

R. El-Ganainy, D. N. Christodoulides, C. Rotschild, and M. Segev, “Soliton dynamics and self-induced transparency in nonlinear nanosuspensions,” Opt. Express 15, 10207–10218 (2007).
[Crossref] [PubMed]

Engheta, N.

V. Shvedov, A. R. Davoyan, C. Hnatovsky, N. Engheta, and W. Krolikowski, “A long-range polarization-controlled optical tractor beam,” Nat. Photon. 8, 846–850 (2014).
[Crossref]

Falcao-Filho, E. L.

E. L. Falcao-Filho, C. B. de Araújo, G. Boudebs, H. Leblond, and V. Skarka, “Robust two-dimensional spatial soitons in liqid carbon disulfide,” Phys. Rev. Lett. 110, 013901 (2013).
[Crossref]

Fardad, S.

S. Fardad, A. Salandrino, M. Heinrich, P. Zhang, Z. Chen, and D. N. Christodoulides, “Plasmonic resonant solitons in metallic nanosuspensions,” Nano Lett.,  14, 2498–2504 (2014).
[Crossref] [PubMed]

W. Man, S. Fardad, Z. Zhang, J. Prakash, M. Lau, P. Zhang, M. Heinrich, D. N. Christodoulides, and Z. Chen, “Optical nonlinearities and enhanced light transmission in soft-matter systems with tunable polarizabilities,” Phys. Rev. Lett. 111, 218302 (2013).
[Crossref] [PubMed]

Fratalocchi, A.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[Crossref] [PubMed]

A. Armaroli, S. Trillo, and A. Fratalocchi, “Suppression of transverse instabilities of dark solitons and their dispersive shock waves,” Phys. Rev. A 80, 053803 (2009).
[Crossref]

Galembeck, A.

A. M. Brito-Silva, A. Galembeck, A. S. L. Gomes, A. J. Jesus-Silva, and C. B. de Araújo, “Random laser action in dye solutions containing Stöber silica nanoparticles,” J. Appl. Phys. 108, 033508 (2010).
[Crossref]

Ghofraniha, N.

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, “Shocks in nonlocal media,” Phys. Rev. Lett. 99, 043903 (2007).
[Crossref] [PubMed]

Gomes, A. S. L.

A. S. L. Gomes, M. T. Carvalho, C. T. Dominguez, C. B. de Araújo, and P. N. Prasad, “Direct three-photon excitation of upconversion random laser emission in a weakly scattering organic colloidal system,” Opt. Express 22, 14305–14310 (2014).
[Crossref] [PubMed]

A. M. Brito-Silva, A. Galembeck, A. S. L. Gomes, A. J. Jesus-Silva, and C. B. de Araújo, “Random laser action in dye solutions containing Stöber silica nanoparticles,” J. Appl. Phys. 108, 033508 (2010).
[Crossref]

N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, “Laser action in strongly scattering media,” Nature 368(6470), 436–438 (1994).
[Crossref]

Grier, D.

D. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

Heinrich, M.

S. Fardad, A. Salandrino, M. Heinrich, P. Zhang, Z. Chen, and D. N. Christodoulides, “Plasmonic resonant solitons in metallic nanosuspensions,” Nano Lett.,  14, 2498–2504 (2014).
[Crossref] [PubMed]

W. Man, S. Fardad, Z. Zhang, J. Prakash, M. Lau, P. Zhang, M. Heinrich, D. N. Christodoulides, and Z. Chen, “Optical nonlinearities and enhanced light transmission in soft-matter systems with tunable polarizabilities,” Phys. Rev. Lett. 111, 218302 (2013).
[Crossref] [PubMed]

Hnatovsky, C.

V. Shvedov, A. R. Davoyan, C. Hnatovsky, N. Engheta, and W. Krolikowski, “A long-range polarization-controlled optical tractor beam,” Nat. Photon. 8, 846–850 (2014).
[Crossref]

Jesus-Silva, A. J.

A. M. Brito-Silva, A. Galembeck, A. S. L. Gomes, A. J. Jesus-Silva, and C. B. de Araújo, “Random laser action in dye solutions containing Stöber silica nanoparticles,” J. Appl. Phys. 108, 033508 (2010).
[Crossref]

Kawano, H.

J. Chen, H. Mizuno, H. Kawano, A. Miyawaki, and K. Midorikawa, “Two-photon pumping of random lasers by picosecond and nanosecond lasers,” Appl. Phys. B 85, 45–48 (2006).
[Crossref]

Kivshar, Y.

M. Matuszewski, W. Krolikowski, and Y. Kivshar, “Soliton interactions and transformations in colloidal media,” Phys. Rev. A 79, 023814 (2009).
[Crossref]

Kivshar, Y. S.

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

Kivshar, Y.S.

Kramer, L.

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99 (2002).
[Crossref]

Krolikowski, W.

V. Shvedov, A. R. Davoyan, C. Hnatovsky, N. Engheta, and W. Krolikowski, “A long-range polarization-controlled optical tractor beam,” Nat. Photon. 8, 846–850 (2014).
[Crossref]

M. Matuszewski, W. Krolikowski, and Y. Kivshar, “Soliton interactions and transformations in colloidal media,” Phys. Rev. A 79, 023814 (2009).
[Crossref]

M. Matuszewski, W. Krolikowski, and Y.S. Kivshar, “Spatial solitons and light-induced instabilities in colloidal media,” Opt. Express 16, 1371–1376 (2008).
[Crossref] [PubMed]

Lau, M.

W. Man, S. Fardad, Z. Zhang, J. Prakash, M. Lau, P. Zhang, M. Heinrich, D. N. Christodoulides, and Z. Chen, “Optical nonlinearities and enhanced light transmission in soft-matter systems with tunable polarizabilities,” Phys. Rev. Lett. 111, 218302 (2013).
[Crossref] [PubMed]

Lawandy, N. M.

N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, “Laser action in strongly scattering media,” Nature 368(6470), 436–438 (1994).
[Crossref]

Leblond, H.

V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90 (2), 023845 (2014).
[Crossref]

E. L. Falcao-Filho, C. B. de Araújo, G. Boudebs, H. Leblond, and V. Skarka, “Robust two-dimensional spatial soitons in liqid carbon disulfide,” Phys. Rev. Lett. 110, 013901 (2013).
[Crossref]

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105, 213901 (2010).
[Crossref]

Lee, W. M.

R. El-Ganainy, D. N. Christodoulides, E. M. Wright, W. M. Lee, and K. Dholakia, “Nonlinear optical dynamics in nonideal gases of interacting colloidal nanoparticles,” Phys. Rev. A 80, 053805 (2009).
[Crossref]

Lekic, M.

V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90 (2), 023845 (2014).
[Crossref]

Malomed, B. A.

V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90 (2), 023845 (2014).
[Crossref]

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105, 213901 (2010).
[Crossref]

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A 246, 412–422 (1998).
[Crossref]

Man, W.

W. Man, S. Fardad, Z. Zhang, J. Prakash, M. Lau, P. Zhang, M. Heinrich, D. N. Christodoulides, and Z. Chen, “Optical nonlinearities and enhanced light transmission in soft-matter systems with tunable polarizabilities,” Phys. Rev. Lett. 111, 218302 (2013).
[Crossref] [PubMed]

Marini, A.

S. Roy, A. Marini, and F. Biancalana, “Free-carrier-driven spatiotemporal dynamics in amplifying silicon waveguides,” Phys. Rev. A 89, 053827 (2014).
[Crossref]

Matuszewski, M.

M. Matuszewski, “Engineering optical soliton bistability in colloidal media,” Phys. Rev. A 81, 013820 (2010).
[Crossref]

M. Matuszewski, W. Krolikowski, and Y. Kivshar, “Soliton interactions and transformations in colloidal media,” Phys. Rev. A 79, 023814 (2009).
[Crossref]

M. Matuszewski, W. Krolikowski, and Y.S. Kivshar, “Spatial solitons and light-induced instabilities in colloidal media,” Opt. Express 16, 1371–1376 (2008).
[Crossref] [PubMed]

Mei, F. Di

D. Pierangeli, F. Di Mei, C. Conti, A. J. Agranat, and E. DelRe, “Spatial rogue waves in photorefractive ferroelectrics,” Phys. Rev. Lett. 115, 093901 (2015).
[Crossref] [PubMed]

Midorikawa, K.

J. Chen, H. Mizuno, H. Kawano, A. Miyawaki, and K. Midorikawa, “Two-photon pumping of random lasers by picosecond and nanosecond lasers,” Appl. Phys. B 85, 45–48 (2006).
[Crossref]

Mihalache, D.

V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90 (2), 023845 (2014).
[Crossref]

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105, 213901 (2010).
[Crossref]

Miklaszewski, R.

V. Skarka, V. I. Berezhiani, and R. Miklaszewski, “Spatiotemporal soliton propagation in saturating nonlinear optical media,” Phys. Rev. E 56, 1080 (1997).
[Crossref]

Miyawaki, A.

J. Chen, H. Mizuno, H. Kawano, A. Miyawaki, and K. Midorikawa, “Two-photon pumping of random lasers by picosecond and nanosecond lasers,” Appl. Phys. B 85, 45–48 (2006).
[Crossref]

Mizuno, H.

J. Chen, H. Mizuno, H. Kawano, A. Miyawaki, and K. Midorikawa, “Two-photon pumping of random lasers by picosecond and nanosecond lasers,” Appl. Phys. B 85, 45–48 (2006).
[Crossref]

Nicolis, G.

G. Nicolis and I. Prigogine, Self-organization in Nonequilibrium Systems (Wiley, 1997).

Papadogiannis, N. A.

G. Zacharakis, N. A. Papadogiannis, and T. G. Papazoglou, “Random lasing following two-photon excitation of highly scattering media,” Appl. Phys. Lett. 81, 2511–2513 (2002).
[Crossref]

Papazoglou, T. G.

G. Zacharakis, N. A. Papadogiannis, and T. G. Papazoglou, “Random lasing following two-photon excitation of highly scattering media,” Appl. Phys. Lett. 81, 2511–2513 (2002).
[Crossref]

Peccianti, M.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[Crossref] [PubMed]

Pierangeli, D.

D. Pierangeli, F. Di Mei, C. Conti, A. J. Agranat, and E. DelRe, “Spatial rogue waves in photorefractive ferroelectrics,” Phys. Rev. Lett. 115, 093901 (2015).
[Crossref] [PubMed]

Prakash, J.

W. Man, S. Fardad, Z. Zhang, J. Prakash, M. Lau, P. Zhang, M. Heinrich, D. N. Christodoulides, and Z. Chen, “Optical nonlinearities and enhanced light transmission in soft-matter systems with tunable polarizabilities,” Phys. Rev. Lett. 111, 218302 (2013).
[Crossref] [PubMed]

Prasad, P. N.

Prigogine, I.

G. Nicolis and I. Prigogine, Self-organization in Nonequilibrium Systems (Wiley, 1997).

Reyna, Albert S.

Rotschild, C.

Roy, S.

S. Roy, A. Marini, and F. Biancalana, “Free-carrier-driven spatiotemporal dynamics in amplifying silicon waveguides,” Phys. Rev. A 89, 053827 (2014).
[Crossref]

Ruocco, G.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[Crossref] [PubMed]

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, “Shocks in nonlocal media,” Phys. Rev. Lett. 99, 043903 (2007).
[Crossref] [PubMed]

Salandrino, A.

S. Fardad, A. Salandrino, M. Heinrich, P. Zhang, Z. Chen, and D. N. Christodoulides, “Plasmonic resonant solitons in metallic nanosuspensions,” Nano Lett.,  14, 2498–2504 (2014).
[Crossref] [PubMed]

Sauvain, E.

N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, “Laser action in strongly scattering media,” Nature 368(6470), 436–438 (1994).
[Crossref]

Segev, M.

Shvedov, V.

V. Shvedov, A. R. Davoyan, C. Hnatovsky, N. Engheta, and W. Krolikowski, “A long-range polarization-controlled optical tractor beam,” Nat. Photon. 8, 846–850 (2014).
[Crossref]

Skarka, V.

B. N. Aleksic, N. B. Aleksic, V. Skarka, and M. Belic, “Stability and nesting of dissipative vortex solitons with high vorticity,” Phys. Rev. A 91, 043832 (2015).
[Crossref]

V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90 (2), 023845 (2014).
[Crossref]

E. L. Falcao-Filho, C. B. de Araújo, G. Boudebs, H. Leblond, and V. Skarka, “Robust two-dimensional spatial soitons in liqid carbon disulfide,” Phys. Rev. Lett. 110, 013901 (2013).
[Crossref]

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105, 213901 (2010).
[Crossref]

V. Skarka, D. V. Timotijević, and N. B. Aleksić, “Extension of the stability criterion for dissipative optical soliton solutions of a two-dimensional Ginzburg–Landau system generated from asymmetric inputs,” J. Opt. A: Pure Appl. Opt. 10, 075102 (2008).
[Crossref]

V. Skarka and N.B. Aleksić, “Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimesional complex cubic-quintic Ginzburg-Landau equations,” Phys. Rev. Lett. 96, 013903 (2006).
[Crossref]

V. I. Berezhiani, V. Skarka, and N. B. Aleksić, “Dynamics of localized and nonlocalized optical vortex solitons in cubic-quintic nonlinear media,” Phys. Rev. E 64, 057601 (2001).
[Crossref]

V. Skarka, V. I. Berezhiani, and R. Miklaszewski, “Spatiotemporal soliton propagation in saturating nonlinear optical media,” Phys. Rev. E 56, 1080 (1997).
[Crossref]

Soto-Crespo, J. M.

N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190–1201 (1996).
[Crossref]

Spinozzi, E.

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nanodisordered ferroelectrics,” Nat. Photon. 5, 39–42 (2011).
[Crossref]

Timotijevic, D. V.

V. Skarka, D. V. Timotijević, and N. B. Aleksić, “Extension of the stability criterion for dissipative optical soliton solutions of a two-dimensional Ginzburg–Landau system generated from asymmetric inputs,” J. Opt. A: Pure Appl. Opt. 10, 075102 (2008).
[Crossref]

Trillo, S.

A. Armaroli, S. Trillo, and A. Fratalocchi, “Suppression of transverse instabilities of dark solitons and their dispersive shock waves,” Phys. Rev. A 80, 053803 (2009).
[Crossref]

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[Crossref] [PubMed]

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, “Shocks in nonlocal media,” Phys. Rev. Lett. 99, 043903 (2007).
[Crossref] [PubMed]

Wright, E. M.

R. El-Ganainy, D. N. Christodoulides, E. M. Wright, W. M. Lee, and K. Dholakia, “Nonlinear optical dynamics in nonideal gases of interacting colloidal nanoparticles,” Phys. Rev. A 80, 053805 (2009).
[Crossref]

Zacharakis, G.

G. Zacharakis, N. A. Papadogiannis, and T. G. Papazoglou, “Random lasing following two-photon excitation of highly scattering media,” Appl. Phys. Lett. 81, 2511–2513 (2002).
[Crossref]

Zhang, P.

S. Fardad, A. Salandrino, M. Heinrich, P. Zhang, Z. Chen, and D. N. Christodoulides, “Plasmonic resonant solitons in metallic nanosuspensions,” Nano Lett.,  14, 2498–2504 (2014).
[Crossref] [PubMed]

W. Man, S. Fardad, Z. Zhang, J. Prakash, M. Lau, P. Zhang, M. Heinrich, D. N. Christodoulides, and Z. Chen, “Optical nonlinearities and enhanced light transmission in soft-matter systems with tunable polarizabilities,” Phys. Rev. Lett. 111, 218302 (2013).
[Crossref] [PubMed]

Zhang, Z.

W. Man, S. Fardad, Z. Zhang, J. Prakash, M. Lau, P. Zhang, M. Heinrich, D. N. Christodoulides, and Z. Chen, “Optical nonlinearities and enhanced light transmission in soft-matter systems with tunable polarizabilities,” Phys. Rev. Lett. 111, 218302 (2013).
[Crossref] [PubMed]

Appl. Phys. B (1)

J. Chen, H. Mizuno, H. Kawano, A. Miyawaki, and K. Midorikawa, “Two-photon pumping of random lasers by picosecond and nanosecond lasers,” Appl. Phys. B 85, 45–48 (2006).
[Crossref]

Appl. Phys. Lett. (1)

G. Zacharakis, N. A. Papadogiannis, and T. G. Papazoglou, “Random lasing following two-photon excitation of highly scattering media,” Appl. Phys. Lett. 81, 2511–2513 (2002).
[Crossref]

J. Appl. Phys. (1)

A. M. Brito-Silva, A. Galembeck, A. S. L. Gomes, A. J. Jesus-Silva, and C. B. de Araújo, “Random laser action in dye solutions containing Stöber silica nanoparticles,” J. Appl. Phys. 108, 033508 (2010).
[Crossref]

J. Opt. A: Pure Appl. Opt. (1)

V. Skarka, D. V. Timotijević, and N. B. Aleksić, “Extension of the stability criterion for dissipative optical soliton solutions of a two-dimensional Ginzburg–Landau system generated from asymmetric inputs,” J. Opt. A: Pure Appl. Opt. 10, 075102 (2008).
[Crossref]

Nano Lett. (1)

S. Fardad, A. Salandrino, M. Heinrich, P. Zhang, Z. Chen, and D. N. Christodoulides, “Plasmonic resonant solitons in metallic nanosuspensions,” Nano Lett.,  14, 2498–2504 (2014).
[Crossref] [PubMed]

Nat. Photon. (2)

V. Shvedov, A. R. Davoyan, C. Hnatovsky, N. Engheta, and W. Krolikowski, “A long-range polarization-controlled optical tractor beam,” Nat. Photon. 8, 846–850 (2014).
[Crossref]

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nanodisordered ferroelectrics,” Nat. Photon. 5, 39–42 (2011).
[Crossref]

Nature (2)

D. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, “Laser action in strongly scattering media,” Nature 368(6470), 436–438 (1994).
[Crossref]

Opt. Express (5)

Opt. Lett. (1)

Phys. Lett. A (1)

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A 246, 412–422 (1998).
[Crossref]

Phys. Rev. A (7)

V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90 (2), 023845 (2014).
[Crossref]

A. Armaroli, S. Trillo, and A. Fratalocchi, “Suppression of transverse instabilities of dark solitons and their dispersive shock waves,” Phys. Rev. A 80, 053803 (2009).
[Crossref]

S. Roy, A. Marini, and F. Biancalana, “Free-carrier-driven spatiotemporal dynamics in amplifying silicon waveguides,” Phys. Rev. A 89, 053827 (2014).
[Crossref]

B. N. Aleksic, N. B. Aleksic, V. Skarka, and M. Belic, “Stability and nesting of dissipative vortex solitons with high vorticity,” Phys. Rev. A 91, 043832 (2015).
[Crossref]

M. Matuszewski, W. Krolikowski, and Y. Kivshar, “Soliton interactions and transformations in colloidal media,” Phys. Rev. A 79, 023814 (2009).
[Crossref]

M. Matuszewski, “Engineering optical soliton bistability in colloidal media,” Phys. Rev. A 81, 013820 (2010).
[Crossref]

R. El-Ganainy, D. N. Christodoulides, E. M. Wright, W. M. Lee, and K. Dholakia, “Nonlinear optical dynamics in nonideal gases of interacting colloidal nanoparticles,” Phys. Rev. A 80, 053805 (2009).
[Crossref]

Phys. Rev. E (3)

V. I. Berezhiani, V. Skarka, and N. B. Aleksić, “Dynamics of localized and nonlocalized optical vortex solitons in cubic-quintic nonlinear media,” Phys. Rev. E 64, 057601 (2001).
[Crossref]

V. Skarka, V. I. Berezhiani, and R. Miklaszewski, “Spatiotemporal soliton propagation in saturating nonlinear optical media,” Phys. Rev. E 56, 1080 (1997).
[Crossref]

N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190–1201 (1996).
[Crossref]

Phys. Rev. Lett. (7)

V. Skarka and N.B. Aleksić, “Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimesional complex cubic-quintic Ginzburg-Landau equations,” Phys. Rev. Lett. 96, 013903 (2006).
[Crossref]

E. L. Falcao-Filho, C. B. de Araújo, G. Boudebs, H. Leblond, and V. Skarka, “Robust two-dimensional spatial soitons in liqid carbon disulfide,” Phys. Rev. Lett. 110, 013901 (2013).
[Crossref]

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, “Shocks in nonlocal media,” Phys. Rev. Lett. 99, 043903 (2007).
[Crossref] [PubMed]

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[Crossref] [PubMed]

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105, 213901 (2010).
[Crossref]

D. Pierangeli, F. Di Mei, C. Conti, A. J. Agranat, and E. DelRe, “Spatial rogue waves in photorefractive ferroelectrics,” Phys. Rev. Lett. 115, 093901 (2015).
[Crossref] [PubMed]

W. Man, S. Fardad, Z. Zhang, J. Prakash, M. Lau, P. Zhang, M. Heinrich, D. N. Christodoulides, and Z. Chen, “Optical nonlinearities and enhanced light transmission in soft-matter systems with tunable polarizabilities,” Phys. Rev. Lett. 111, 218302 (2013).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

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[Crossref]

Other (3)

N. N. Akhmediev and A. Ankiewicz, Dissipative Solitons: From Optics to Biology and Medicine, Lect. Notes Phys. 751 (Springer, 2008).

G. Nicolis and I. Prigogine, Self-organization in Nonequilibrium Systems (Wiley, 1997).

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

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Figures (9)

Fig. 1
Fig. 1 Illustrating the dependence of packing fraction sgn (θ) (ηη0) on light intensity |E|2, for few values of the background packing fraction η0. Solid squares represent the fitting formula, Eq. (7), for the case η0 = 0.01
Fig. 2
Fig. 2 Illustrating the stability of the beam amplitude A as a function of the nonlinear gain ε. The upper (solid line) branch corresponds to the stable A+ solution and the lower (dashed line) branch corresponds to the unstable A solution.
Fig. 3
Fig. 3 Stability domain (in yellow) produced by the VA-generated fixed points, in the plane of the nonlinear-gain strength ε and the nonlinear-loss strength μ (both dimensionless). The stability of vortex solitons is confirmed by direct simulations of Eq. (8) for parameters inside the region delimited by dashed lines.
Fig. 4
Fig. 4 Amplitude A, power P, and angular momentum M monitored after each propagation step, from z = 0 to z = 21500 (for dissipative parameters ε = 0.38 and μ = 2). The ratio M/P corresponds to the constant value of S = 1.
Fig. 5
Fig. 5 Development of a stable breathing vortex with S = 1. Initial vortex (at z = 0) corresponds to the input Gaussian with parameters β = 0.05, δ = 0.01, μ = 2, and ε = 0.38 from the stability domain. Breathing vortex soliton is self-generated after z = 7000 steps. It remains stable after z = 21000 steps.
Fig. 6
Fig. 6 Regular breathing behavior of the beam characteristics. It corresponds to the vortex soliton self-organization. Different scales are used for amplitude A, power P, angular momentum M, and width L.
Fig. 7
Fig. 7 Robust cyclic behavior with period T = 240 from a “millstone” at z = 21160 through various domes, to the same “millstone” at z = 21400, confirming the stability of a breathing vortex soliton.
Fig. 8
Fig. 8 Similar to Fig. 4 but for larger gain and loss parameters, ε = 1.8 and μ = 3. The amplitude, power, and angular momentum oscillate with a short period TA = 4 and a superimposed beat period, TB = 500, as a double breather. The breathing with shorter period TA = 4 starts already after 500 steps.
Fig. 9
Fig. 9 Perfect correspondence between domes and “millstones” appearing successively after TA + 1 = 5, in two cycles distant for 20 steps.

Equations (24)

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α = 3 V p ε 0 n b 2 θ ,
α ( 4 k B T ) 1 | U ( η ) | 2 = sgn ( θ ) [ g ( η ) g ( η 0 ) ] ,
2 U + k 0 2 n eff 2 U = 0 ,
U ( r ) ( 3 V p ε 0 n b 2 | θ | / 4 k B T ) 1 / 2 = E ( r ) exp [ i k 0 n b ( 1 + 3 θ η 0 ) 1 / 2 Z ] ,
| E | 2 = sgn ( θ ) [ g ( η ) g ( η 0 ) ] .
i E z + γ Δ E + sgn ( θ ) ( η η 0 ) E + i Γ η E = 0 ,
sgn ( θ ) ( η η 0 ) = ( κ + σ | E | 2 ν | E | 4 )
i E z + γ Δ E + ( κ + σ | E | 2 ν | E | 4 ) E = i δ E + i ε | E | 2 E i μ | E | 4 E + i β Δ E .
L = i 2 ( E * z E E z E * ) + γ | E | 2 κ | E | 2 σ 2 | E | 4 + ν 3 | E | 6 .
E = B A ( r G R ) S exp [ r 2 G 2 R 2 + i C G 2 r 2 + i S φ + i Ψ ] ,
G 2 4 γ ( R 2 d A d z + A R d R d z ) + A [ R 2 ( δ 0 2 ε σ A 2 + 3 μ γ 2 σ 2 A 4 ) + β γ ( 1 + 4 C 2 R 4 ) ] = 0 .
G 2 2 γ d ψ d z + R 2 ( 4 C 2 + G 2 γ d C d z ) + 1 R 2 κ 0 2 A 2 + 3 2 ν γ σ 2 A 4 = 0 .
G 2 2 γ d ψ d z + 2 R 2 ( 4 C 2 + G 2 γ d C d z ) κ 0 A 2 + ν γ 2 σ 2 A 4 + 4 β γ C = 0 .
G 2 γ ( R 2 d A d z + 2 A R d R d z ) + A [ R 2 ( 4 C + 2 δ 0 3 ε σ A 2 + 2 μ γ σ 2 A 4 ) + β γ ( 1 + 12 C 2 R 4 ) ] = 0 ,
d A d z = γ A G 2 [ 5 ε σ A 2 4 ( C + μ γ σ 2 A 4 + β γ R 2 C 2 ) 3 β γ R 2 2 δ 0 ] ,
d R d z = γ R G 2 ( 4 C ε σ A 2 + μ γ σ 2 A 4 + β γ R 2 4 β γ C 2 R 2 ) ,
d C d z = γ G 2 ( 1 R 4 4 C 2 A 2 R 2 + ν γ σ 2 A 4 R 2 4 β γ C R 2 ) ,
d ψ d z = γ G 2 ( κ 0 2 R 2 + 3 A 2 5 2 ν γ σ 2 A 4 + 4 β γ C ) .
A ± = ( 2 ε / σ β / γ ) ± ( 2 ε / σ β / γ ) 2 2 δ 0 ( 3 μ γ / σ 2 2 β ν / σ 2 ) ( 3 μ γ / σ 2 2 β ν / σ 2 ) .
λ 3 + α 1 λ 2 + α 2 λ + α 3 = 0
α 1 = γ G 2 [ 6 β γ R 2 + 16 μ γ σ 2 A 4 10 ε σ A 2 + 8 C ( 1 + β γ CR 2 ) ] > 0 ,
α 2 = γ 2 G 4 8 R 4 [ 1 + β 2 γ 2 A 2 R 2 ( 1 + 6 β ε γ σ 9 β μ σ 2 A 2 2 ν γ σ 2 A 2 ) 2 CA 2 R 4 ( β γ + 5 ε σ 8 μ γ σ 2 A 2 2 β ν σ 2 A 2 ) + 4 C 2 R 4 ( 1 + β 2 γ 2 3 β ε γ σ A 2 R 2 + 5 β μ σ 2 A 4 R 2 ) ] > 0 ,
α 3 = γ 3 G 6 32 A 2 R 4 [ β γ 2 ε σ + β 2 γ 2 ε σ + A 2 ( 3 μ γ σ 2 + β 2 μ γ σ 2 2 β ν σ 2 ) 4 CR 2 β γ ( ε σ β γ 2 μ γ σ 2 A 2 + 2 β ν σ 2 A 2 ) 4 C 2 R 4 ( 2 ε σ + β γ + 3 β 2 γ 2 ε σ 3 μ γ σ 2 A 2 5 β 2 μ γ σ 2 A 2 2 β ν σ 2 A 2 ) ] > 0 ,
α 123 = α 1 α 2 α 3 > 0 .

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