Abstract

Transverse beam propagation is considered in a layered structure in which Kerr nonlinearity alternates between self-focusing and self-defocusing, which makes it possible to prevent collapse. A structure composed of alternating self-focusing layers with strongly different values of the Kerr coefficient is considered too. By means of both a variational approximation (which is implemented in a completely analytical form, including the stability analysis) and direct simulations, it is demonstrated that stable quasi-stationary (2+1)-dimensional soliton beams exist in these media (direct simulations demonstrate stable propagation over a distance exceeding 100 diffraction lengths of the beam). Quasi-stationary cylindrical solitons with intrinsic vorticity exist too, but they all are unstable, splitting into separating zero-vorticity beams.

© 2002 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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  23. V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, “The theory of spiral laser beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
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  24. W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450–2453 (1997).
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  25. D. V. Petrov and L. Torner, “Second-harmonic generation by intense beams containing phase dislocations: self-breaking into sets of solitons,” Opt. Quantum Electron. 29, 1037–1046 (1997).
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    [CrossRef]
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    [CrossRef]
  30. M. Quiroga-Teixeiro and H. Michinel, “Stable azimuthal stationary state in quintic nonlinear optical media,” J. Opt. Soc. Am. B 14, 2004–2009 (1997).
    [CrossRef]
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    [CrossRef]
  32. L. Torner, D. Mazilu, and D. Mihalache, “Walking solitons in quadratic nonlinear media,” Phys. Rev. Lett. 77, 2455–2458 (1996).
    [CrossRef] [PubMed]
  33. D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, and F. Lederer, “Three-dimensional spinning solitons in the cubic-quintic nonlinear medium,” Phys. Rev. E 61, 7142–7145 (2000).
    [CrossRef]
  34. I. Towers, A. V. Buryak, R. A. Sammut, and B. A. Malomed, “Stable localized vortex solitons,” Phys. Rev. E 63, 055601(R) (2001).
    [CrossRef]

2002 (1)

2001 (3)

Z. H. Musslimani, M. Soljaĉić, M. Segev, and D. N. Christodoulides, “Delayed-action interaction and spin–orbit coupling between solitons,” Phys. Rev. Lett. 86, 799–802 (2001).
[CrossRef] [PubMed]

I. Towers, A. V. Buryak, R. A. Sammut, and B. A. Malomed, “Stable localized vortex solitons,” Phys. Rev. E 63, 055601(R) (2001).
[CrossRef]

B. V. Gisin, A. Kaplan, and B. A. Malomed, “All-optical switching in an antiwaveguiding structure,” Opt. Quantum Electron. 33, 201–208 (2001).
[CrossRef]

2000 (5)

B. Gisin, A. Kaplan, and B. Malomed, “Spontaneous sym-metry breaking and switching in planar nonlinear antiwaveguides,” Phys. Rev. E 62, 2804–2809 (2000).
[CrossRef]

L. Bergé, V. K. Mezentsev, J. Juul Rasmussen, P. L. Christiansen, and Yu. B. Gaididei, “Self-guiding light in layered nonlinear media,” Opt. Lett. 25, 1037–1039 (2000).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, and F. Lederer, “Three-dimensional spinning solitons in the cubic-quintic nonlinear medium,” Phys. Rev. E 61, 7142–7145 (2000).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, and F. Lederer, “Azimuthal instability of spinning spatiotemporal solitons,” Phys. Rev. E 62, R1505–R1508 (2000).
[CrossRef]

L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
[CrossRef]

1999 (3)

1998 (5)

T. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

S. K. Turitsyn and E. G. Shapiro, “Dispersion-managed solitons in optical amplifier transmission systems with zero average dispersion,” Opt. Lett. 23, 682–684 (1998).
[CrossRef]

A. Berntson, N. J. Doran, W. Forysiak, and J. H. B. Nijhof, “Power dependence of dispersion-managed solitons for anomalous, zero, and normal path-average dispersion,” Opt. Lett. 23, 900–902 (1998).
[CrossRef]

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

M. Soljaĉić, S. Sears, and M. Segev, “Self-trapping of ‘necklace’ beams in self-focusing Kerr media,” Phys. Rev. Lett. 81, 4851–4854 (1998).
[CrossRef]

1997 (4)

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450–2453 (1997).
[CrossRef]

D. V. Petrov and L. Torner, “Second-harmonic generation by intense beams containing phase dislocations: self-breaking into sets of solitons,” Opt. Quantum Electron. 29, 1037–1046 (1997).
[CrossRef]

M. Quiroga-Teixeiro and H. Michinel, “Stable azimuthal stationary state in quintic nonlinear optical media,” J. Opt. Soc. Am. B 14, 2004–2009 (1997).
[CrossRef]

J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable soliton-like propagation in dispersion managed systems with net anomalous, zero and normal dispersion,” Electron. Lett. 33, 1726–1727 (1997).
[CrossRef]

1996 (1)

L. Torner, D. Mazilu, and D. Mihalache, “Walking solitons in quadratic nonlinear media,” Phys. Rev. Lett. 77, 2455–2458 (1996).
[CrossRef] [PubMed]

1993 (1)

B. A. Malomed, D. F. Parker, and N. F. Smyth, “Resonant shape oscillations and decay of a soliton in periodically inhomogeneous nonlinear optical fiber,” Phys. Rev. E 48, 1418–1425 (1993).
[CrossRef]

1992 (2)

J. Miranda, D. R. Andersen, and S. R. Skinner, “Stability analysis of stationary nonlinear guided waves in self-focusing and self-defocusing Kerr-like media,” Phys. Rev. A 46, 5999–6001 (1992).
[CrossRef] [PubMed]

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, “The theory of spiral laser beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
[CrossRef]

1991 (2)

1990 (1)

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

1985 (1)

A. Barthelemy, S. Maneuf, and C. Froehly, “Propagation soliton et auto-confinement de faisceaux laser par non linearité optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
[CrossRef]

1979 (1)

J. R. Ray and J. L. Reid, “Noether’s theorem, time-dependent invariants and nonlinear equations of motion,” J. Math. Phys. 20, 2054–2057 (1979).
[CrossRef]

1975 (1)

C. J. Elliott and B. R. Suydam, “Self-focusing phenomena in air–glass laser structures,” IEEE J. Quantum Electron. QE-11, 863–866 (1975).
[CrossRef]

1970 (1)

1966 (1)

H. A. Haus, “Higher order trapped light beam solutions,” Appl. Phys. Lett. 8, 128–129 (1966).
[CrossRef]

1964 (1)

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

Andersen, D. R.

J. Miranda, D. R. Andersen, and S. R. Skinner, “Stability analysis of stationary nonlinear guided waves in self-focusing and self-defocusing Kerr-like media,” Phys. Rev. A 46, 5999–6001 (1992).
[CrossRef] [PubMed]

Anderson, D.

Barthelemy, A.

A. Barthelemy, S. Maneuf, and C. Froehly, “Propagation soliton et auto-confinement de faisceaux laser par non linearité optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
[CrossRef]

Bergé, L.

L. Bergé, V. K. Mezentsev, J. Juul Rasmussen, P. L. Christiansen, and Yu. B. Gaididei, “Self-guiding light in layered nonlinear media,” Opt. Lett. 25, 1037–1039 (2000).
[CrossRef]

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

Berntson, A.

Brzozowski, L.

L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
[CrossRef]

Buryak, A. V.

I. Towers, A. V. Buryak, R. A. Sammut, and B. A. Malomed, “Stable localized vortex solitons,” Phys. Rev. E 63, 055601(R) (2001).
[CrossRef]

Chiao, R. Y.

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

Christiansen, P. L.

Christodoulides, D. N.

Z. H. Musslimani, M. Soljaĉić, M. Segev, and D. N. Christodoulides, “Delayed-action interaction and spin–orbit coupling between solitons,” Phys. Rev. Lett. 86, 799–802 (2001).
[CrossRef] [PubMed]

Cornolti, F.

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

Crasovan, L.-C.

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, and F. Lederer, “Azimuthal instability of spinning spatiotemporal solitons,” Phys. Rev. E 62, R1505–R1508 (2000).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, and F. Lederer, “Three-dimensional spinning solitons in the cubic-quintic nonlinear medium,” Phys. Rev. E 61, 7142–7145 (2000).
[CrossRef]

Desaix, M.

Doran, N. J.

A. Berntson, N. J. Doran, W. Forysiak, and J. H. B. Nijhof, “Power dependence of dispersion-managed solitons for anomalous, zero, and normal path-average dispersion,” Opt. Lett. 23, 900–902 (1998).
[CrossRef]

J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable soliton-like propagation in dispersion managed systems with net anomalous, zero and normal dispersion,” Electron. Lett. 33, 1726–1727 (1997).
[CrossRef]

Elliott, C. J.

C. J. Elliott and B. R. Suydam, “Self-focusing phenomena in air–glass laser structures,” IEEE J. Quantum Electron. QE-11, 863–866 (1975).
[CrossRef]

Firth, W. J.

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450–2453 (1997).
[CrossRef]

Forysiak, W.

A. Berntson, N. J. Doran, W. Forysiak, and J. H. B. Nijhof, “Power dependence of dispersion-managed solitons for anomalous, zero, and normal path-average dispersion,” Opt. Lett. 23, 900–902 (1998).
[CrossRef]

J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable soliton-like propagation in dispersion managed systems with net anomalous, zero and normal dispersion,” Electron. Lett. 33, 1726–1727 (1997).
[CrossRef]

Froehly, C.

A. Barthelemy, S. Maneuf, and C. Froehly, “Propagation soliton et auto-confinement de faisceaux laser par non linearité optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
[CrossRef]

Gaididei, Yu. B.

Garmire, E.

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

Gisin, B.

B. Gisin, A. Kaplan, and B. Malomed, “Spontaneous sym-metry breaking and switching in planar nonlinear antiwaveguides,” Phys. Rev. E 62, 2804–2809 (2000).
[CrossRef]

Gisin, B. V.

A. Kaplan, B. V. Gisin, and B. A. Malomed, “Stable propagation and all-optical switching in planar waveguide–antiwaveguide periodic structures,” J. Opt. Soc. Am. B 19, 522–528 (2002).
[CrossRef]

B. V. Gisin, A. Kaplan, and B. A. Malomed, “All-optical switching in an antiwaveguiding structure,” Opt. Quantum Electron. 33, 201–208 (2001).
[CrossRef]

Hasegawa, A.

Haus, H. A.

H. A. Haus, “Higher order trapped light beam solutions,” Appl. Phys. Lett. 8, 128–129 (1966).
[CrossRef]

Juul Rasmussen, J.

Kaplan, A.

A. Kaplan, B. V. Gisin, and B. A. Malomed, “Stable propagation and all-optical switching in planar waveguide–antiwaveguide periodic structures,” J. Opt. Soc. Am. B 19, 522–528 (2002).
[CrossRef]

B. V. Gisin, A. Kaplan, and B. A. Malomed, “All-optical switching in an antiwaveguiding structure,” Opt. Quantum Electron. 33, 201–208 (2001).
[CrossRef]

B. Gisin, A. Kaplan, and B. Malomed, “Spontaneous sym-metry breaking and switching in planar nonlinear antiwaveguides,” Phys. Rev. E 62, 2804–2809 (2000).
[CrossRef]

Kaup, D. J.

T. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

Knox, F. M.

J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable soliton-like propagation in dispersion managed systems with net anomalous, zero and normal dispersion,” Electron. Lett. 33, 1726–1727 (1997).
[CrossRef]

Kodama, Y.

Kruglov, V. I.

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, “The theory of spiral laser beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
[CrossRef]

Lakoba, T.

T. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

Lederer, F.

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, and F. Lederer, “Azimuthal instability of spinning spatiotemporal solitons,” Phys. Rev. E 62, R1505–R1508 (2000).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, and F. Lederer, “Three-dimensional spinning solitons in the cubic-quintic nonlinear medium,” Phys. Rev. E 61, 7142–7145 (2000).
[CrossRef]

Lisak, M.

Liu, X.

Logvin, Y. A.

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, “The theory of spiral laser beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
[CrossRef]

Lucchesi, M.

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

Malomed, B.

B. Gisin, A. Kaplan, and B. Malomed, “Spontaneous sym-metry breaking and switching in planar nonlinear antiwaveguides,” Phys. Rev. E 62, 2804–2809 (2000).
[CrossRef]

Malomed, B. A.

A. Kaplan, B. V. Gisin, and B. A. Malomed, “Stable propagation and all-optical switching in planar waveguide–antiwaveguide periodic structures,” J. Opt. Soc. Am. B 19, 522–528 (2002).
[CrossRef]

B. V. Gisin, A. Kaplan, and B. A. Malomed, “All-optical switching in an antiwaveguiding structure,” Opt. Quantum Electron. 33, 201–208 (2001).
[CrossRef]

I. Towers, A. V. Buryak, R. A. Sammut, and B. A. Malomed, “Stable localized vortex solitons,” Phys. Rev. E 63, 055601(R) (2001).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, and F. Lederer, “Three-dimensional spinning solitons in the cubic-quintic nonlinear medium,” Phys. Rev. E 61, 7142–7145 (2000).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, and F. Lederer, “Azimuthal instability of spinning spatiotemporal solitons,” Phys. Rev. E 62, R1505–R1508 (2000).
[CrossRef]

T. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

B. A. Malomed, D. F. Parker, and N. F. Smyth, “Resonant shape oscillations and decay of a soliton in periodically inhomogeneous nonlinear optical fiber,” Phys. Rev. E 48, 1418–1425 (1993).
[CrossRef]

Maneuf, S.

A. Barthelemy, S. Maneuf, and C. Froehly, “Propagation soliton et auto-confinement de faisceaux laser par non linearité optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
[CrossRef]

Mazilu, D.

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, and F. Lederer, “Azimuthal instability of spinning spatiotemporal solitons,” Phys. Rev. E 62, R1505–R1508 (2000).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, and F. Lederer, “Three-dimensional spinning solitons in the cubic-quintic nonlinear medium,” Phys. Rev. E 61, 7142–7145 (2000).
[CrossRef]

L. Torner, D. Mazilu, and D. Mihalache, “Walking solitons in quadratic nonlinear media,” Phys. Rev. Lett. 77, 2455–2458 (1996).
[CrossRef] [PubMed]

Mezentsev, V. K.

Michinel, H.

Mihalache, D.

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, and F. Lederer, “Three-dimensional spinning solitons in the cubic-quintic nonlinear medium,” Phys. Rev. E 61, 7142–7145 (2000).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, and F. Lederer, “Azimuthal instability of spinning spatiotemporal solitons,” Phys. Rev. E 62, R1505–R1508 (2000).
[CrossRef]

L. Torner, D. Mazilu, and D. Mihalache, “Walking solitons in quadratic nonlinear media,” Phys. Rev. Lett. 77, 2455–2458 (1996).
[CrossRef] [PubMed]

Miranda, J.

J. Miranda, D. R. Andersen, and S. R. Skinner, “Stability analysis of stationary nonlinear guided waves in self-focusing and self-defocusing Kerr-like media,” Phys. Rev. A 46, 5999–6001 (1992).
[CrossRef] [PubMed]

Musslimani, Z. H.

Z. H. Musslimani, M. Soljaĉić, M. Segev, and D. N. Christodoulides, “Delayed-action interaction and spin–orbit coupling between solitons,” Phys. Rev. Lett. 86, 799–802 (2001).
[CrossRef] [PubMed]

Nijhof, J. H. B.

A. Berntson, N. J. Doran, W. Forysiak, and J. H. B. Nijhof, “Power dependence of dispersion-managed solitons for anomalous, zero, and normal path-average dispersion,” Opt. Lett. 23, 900–902 (1998).
[CrossRef]

J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable soliton-like propagation in dispersion managed systems with net anomalous, zero and normal dispersion,” Electron. Lett. 33, 1726–1727 (1997).
[CrossRef]

Parker, D. F.

B. A. Malomed, D. F. Parker, and N. F. Smyth, “Resonant shape oscillations and decay of a soliton in periodically inhomogeneous nonlinear optical fiber,” Phys. Rev. E 48, 1418–1425 (1993).
[CrossRef]

Petrishchev, V. A.

Petrov, D. V.

D. V. Petrov and L. Torner, “Second-harmonic generation by intense beams containing phase dislocations: self-breaking into sets of solitons,” Opt. Quantum Electron. 29, 1037–1046 (1997).
[CrossRef]

Qian, L.

Qian, L. J.

Quiroga-Teixeiro, M.

Ray, J. R.

J. R. Ray and J. L. Reid, “Noether’s theorem, time-dependent invariants and nonlinear equations of motion,” J. Math. Phys. 20, 2054–2057 (1979).
[CrossRef]

Reid, J. L.

J. R. Ray and J. L. Reid, “Noether’s theorem, time-dependent invariants and nonlinear equations of motion,” J. Math. Phys. 20, 2054–2057 (1979).
[CrossRef]

Sammut, R. A.

I. Towers, A. V. Buryak, R. A. Sammut, and B. A. Malomed, “Stable localized vortex solitons,” Phys. Rev. E 63, 055601(R) (2001).
[CrossRef]

Sargent, E. H.

L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
[CrossRef]

Sears, S.

M. Soljaĉić, S. Sears, and M. Segev, “Self-trapping of ‘necklace’ beams in self-focusing Kerr media,” Phys. Rev. Lett. 81, 4851–4854 (1998).
[CrossRef]

Segev, M.

Z. H. Musslimani, M. Soljaĉić, M. Segev, and D. N. Christodoulides, “Delayed-action interaction and spin–orbit coupling between solitons,” Phys. Rev. Lett. 86, 799–802 (2001).
[CrossRef] [PubMed]

M. Soljaĉić, S. Sears, and M. Segev, “Self-trapping of ‘necklace’ beams in self-focusing Kerr media,” Phys. Rev. Lett. 81, 4851–4854 (1998).
[CrossRef]

Shapiro, E. G.

Skinner, S. R.

J. Miranda, D. R. Andersen, and S. R. Skinner, “Stability analysis of stationary nonlinear guided waves in self-focusing and self-defocusing Kerr-like media,” Phys. Rev. A 46, 5999–6001 (1992).
[CrossRef] [PubMed]

Skryabin, D. V.

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450–2453 (1997).
[CrossRef]

Smyth, N. F.

B. A. Malomed, D. F. Parker, and N. F. Smyth, “Resonant shape oscillations and decay of a soliton in periodically inhomogeneous nonlinear optical fiber,” Phys. Rev. E 48, 1418–1425 (1993).
[CrossRef]

Soljacic, M.

Z. H. Musslimani, M. Soljaĉić, M. Segev, and D. N. Christodoulides, “Delayed-action interaction and spin–orbit coupling between solitons,” Phys. Rev. Lett. 86, 799–802 (2001).
[CrossRef] [PubMed]

M. Soljaĉić, S. Sears, and M. Segev, “Self-trapping of ‘necklace’ beams in self-focusing Kerr media,” Phys. Rev. Lett. 81, 4851–4854 (1998).
[CrossRef]

Suydam, B. R.

C. J. Elliott and B. R. Suydam, “Self-focusing phenomena in air–glass laser structures,” IEEE J. Quantum Electron. QE-11, 863–866 (1975).
[CrossRef]

Talanov, V. I.

Torner, L.

L. Torner, “Walk-off compensated dispersion-mapped quadratic solitons,” IEEE Photon. Technol. Lett. 11, 1268–1270 (1999).
[CrossRef]

D. V. Petrov and L. Torner, “Second-harmonic generation by intense beams containing phase dislocations: self-breaking into sets of solitons,” Opt. Quantum Electron. 29, 1037–1046 (1997).
[CrossRef]

L. Torner, D. Mazilu, and D. Mihalache, “Walking solitons in quadratic nonlinear media,” Phys. Rev. Lett. 77, 2455–2458 (1996).
[CrossRef] [PubMed]

Towers, I.

I. Towers, A. V. Buryak, R. A. Sammut, and B. A. Malomed, “Stable localized vortex solitons,” Phys. Rev. E 63, 055601(R) (2001).
[CrossRef]

Townes, C. H.

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

Turitsyn, S. K.

Vlasov, S. N.

Volkov, V. M.

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, “The theory of spiral laser beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
[CrossRef]

Wise, F.

Wise, F. W.

Yang, J.

T. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

Zambon, B.

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

H. A. Haus, “Higher order trapped light beam solutions,” Appl. Phys. Lett. 8, 128–129 (1966).
[CrossRef]

Electron. Lett. (1)

J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable soliton-like propagation in dispersion managed systems with net anomalous, zero and normal dispersion,” Electron. Lett. 33, 1726–1727 (1997).
[CrossRef]

IEEE J. Quantum Electron. (2)

C. J. Elliott and B. R. Suydam, “Self-focusing phenomena in air–glass laser structures,” IEEE J. Quantum Electron. QE-11, 863–866 (1975).
[CrossRef]

L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

L. Torner, “Walk-off compensated dispersion-mapped quadratic solitons,” IEEE Photon. Technol. Lett. 11, 1268–1270 (1999).
[CrossRef]

J. Math. Phys. (1)

J. R. Ray and J. L. Reid, “Noether’s theorem, time-dependent invariants and nonlinear equations of motion,” J. Math. Phys. 20, 2054–2057 (1979).
[CrossRef]

J. Mod. Opt. (1)

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, “The theory of spiral laser beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
[CrossRef]

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

Opt. Commun. (3)

T. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

A. Barthelemy, S. Maneuf, and C. Froehly, “Propagation soliton et auto-confinement de faisceaux laser par non linearité optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
[CrossRef]

Opt. Lett. (6)

Opt. Quantum Electron. (2)

B. V. Gisin, A. Kaplan, and B. A. Malomed, “All-optical switching in an antiwaveguiding structure,” Opt. Quantum Electron. 33, 201–208 (2001).
[CrossRef]

D. V. Petrov and L. Torner, “Second-harmonic generation by intense beams containing phase dislocations: self-breaking into sets of solitons,” Opt. Quantum Electron. 29, 1037–1046 (1997).
[CrossRef]

Phys. Rep. (1)

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

Phys. Rev. A (1)

J. Miranda, D. R. Andersen, and S. R. Skinner, “Stability analysis of stationary nonlinear guided waves in self-focusing and self-defocusing Kerr-like media,” Phys. Rev. A 46, 5999–6001 (1992).
[CrossRef] [PubMed]

Phys. Rev. E (5)

B. A. Malomed, D. F. Parker, and N. F. Smyth, “Resonant shape oscillations and decay of a soliton in periodically inhomogeneous nonlinear optical fiber,” Phys. Rev. E 48, 1418–1425 (1993).
[CrossRef]

B. Gisin, A. Kaplan, and B. Malomed, “Spontaneous sym-metry breaking and switching in planar nonlinear antiwaveguides,” Phys. Rev. E 62, 2804–2809 (2000).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, and F. Lederer, “Azimuthal instability of spinning spatiotemporal solitons,” Phys. Rev. E 62, R1505–R1508 (2000).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, and F. Lederer, “Three-dimensional spinning solitons in the cubic-quintic nonlinear medium,” Phys. Rev. E 61, 7142–7145 (2000).
[CrossRef]

I. Towers, A. V. Buryak, R. A. Sammut, and B. A. Malomed, “Stable localized vortex solitons,” Phys. Rev. E 63, 055601(R) (2001).
[CrossRef]

Phys. Rev. Lett. (5)

L. Torner, D. Mazilu, and D. Mihalache, “Walking solitons in quadratic nonlinear media,” Phys. Rev. Lett. 77, 2455–2458 (1996).
[CrossRef] [PubMed]

Z. H. Musslimani, M. Soljaĉić, M. Segev, and D. N. Christodoulides, “Delayed-action interaction and spin–orbit coupling between solitons,” Phys. Rev. Lett. 86, 799–802 (2001).
[CrossRef] [PubMed]

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450–2453 (1997).
[CrossRef]

M. Soljaĉić, S. Sears, and M. Segev, “Self-trapping of ‘necklace’ beams in self-focusing Kerr media,” Phys. Rev. Lett. 81, 4851–4854 (1998).
[CrossRef]

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

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Figures (6)

Fig. 1
Fig. 1

Existence and stability regions of the fundamental fixed points in the parameter plane (Γ, L) of the variational model. The fixed point is stable in the speckled area.

Fig. 2
Fig. 2

Amplitudes A, as predicted by variational ansatz (5) for L=1.0 and Γ=-1.3, versus the value of nonlinearity γ-. For a given γ-, two distinct solutions may exist, with different powers.

Fig. 3
Fig. 3

Evolution of the solitary beam produced by direct simulations of Eq. (1). The initial configuration is taken in the form of Eq. (5) with the width predicted by the variational approximation from Eq. (20) for the values of the parameters Γ=-1.3, L=1.0, γ-=-1.0, γ+=4.4, and L-=L+=0.001. At the top, the evolution of the beam’s peak amplitude as it propagates in z is shown. The amplitude quickly levels out after some relaxation. At the bottom, a cross-sectional view of the propagating beam is displayed.

Fig. 4
Fig. 4

Evolution of the solitary beam’s amplitude, as found from direct simulations of Eq. (1) performed for several values of the initial amplitude, starting from the ansatz predicted by the variational approximation (as in Fig. 3). In all the cases shown, the widths of the alternating layers are L-=0.01 and L+=0.02. Nonlinearity coefficient γ- is -1.0 in all the cases except for data represented by squares and triangles, where γ-=+1.0 (which corresponds to a case when both layers are self-focusing). The data shown by squares, the dashed curve, and triangles all pertain to Γ=-1.0; the dashed–dotted curve pertains to Γ=-0.8; the dotted curve, to Γ=-0.6; and the solid curve, to Γ=-0.55.

Fig. 5
Fig. 5

Evolution of the beam’s cross section as it propagates in z. As can be seen, the initial configuration taken according to the variational ansatz rapidly relaxes to a stable soliton beam. The parameters are the same as in the case represented by the solid curve in Fig. 4, i.e., L=2.0, Γ=-0.55. The other parameters are L-=0.01, L+=0.02, γ-=-1.0, and γ+=2.64.

Fig. 6
Fig. 6

Evolution of the peak amplitude of one of two identical secondary beams with zero vorticity that are generated by the breakup of a primary beam with the intrinsic vorticity S=1. The primary spinning beam was taken as predicted by the variational approximation, i.e., according to Eqs. (5) and (20). In this case, L=1 and Γ=-1.3. Notice that, although the initial shape of the secondary beam is quite different from the genuine stable stationary form, the relaxation to it is very fast.

Equations (27)

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

iuz+½2u+γ(z)|u|2u=0,
P2π0|u(r)|2rdr
u(z, r, θ)=exp(iSθ)U(z, r),
iUz+12Urr+1rUr-S2r2U+γ(z)|U|2U=0,
U=A(z)rS exp[ib(z)r2+iϕ(z)]sechrW(z),
A2W2(S+1)=constP/(2πI1),
b(z)=(2W)-1dW/dz,
d2Wdz2=2I1W3[I2-I4γ(z)],
I1=0x2S+3 sech2 xdx,
I2=120SxS-1 sech x-xSsinh xcosh2 x2+S2x2S-2 sech2 xxdx,
I4=120 x4S+1 sech4 xdx.
I1=98ζ(3)1.3523,I2=2 ln 2+1120.1987,
I4=4 ln 2-1120.1477,
dVdz2+Γ=HV,
VW2,
Γ8[I2/I1-(I4/I1)γ],
Vj={V0+(L+/2V0)[(V0)2+Γ+]}2+Γ+(V0)2+Γ+V0,
Vj=V0+L+2V0[(V0)2+Γ+],
VrescVΓ-L-,Vresc=VΓ-,
L-1, L+LL+/L-,
Γ-1,Γ+ΓΓ+/Γ-,
(V0)FP=-L(T-1)4L+1-1-LT,
(V0)FP=-1-LTL+1.
Γ<-1/L.
γ¯L+γ++L-γ-L++L-=8I2(L+1)-I1Γ-(LΓ+1)8I4(L+1)
r=r-qz,
u(r, z)=u(r, z)exp(½iq2z+iq·r),

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