Abstract

We numerically investigate the tunneling of spatial solitons through a focusing Kerr nonlinear optical lattice with longitudinal potential barrier, and find that the position of input beams apparently affects the tunneling behaviors of spatial solitons, which exhibit compression or splitting when passing through the barrier, and that the transverse modulation frequency of lattice and the intensity of input beams strongly affect the ability of tunneling. Based on these properties, we present a scheme for compressing soliton and splitting soliton into stable twin beams. The obtained results may have promising applications in all-optical devices based on spatial solitons.

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]

2008

A. Barak, O. Peleg, C. Stucchio, A. Soffer, and M. Segev, "Observation of Soliton Tunneling Phenomena and Soliton Ejection," Phys. Rev. Lett. 100, 153901 (2008).
[CrossRef] [PubMed]

A. Barak, O. Peleg, A. Soffer, and M. Segev, "Multisoliton ejection from an amplifying potential trap," Opt. Lett. 33, 1798-1800 (2008).
[CrossRef] [PubMed]

2007

J. Zhou, Y. H. Qi, C. H. Xue, S. Y. Lou, and Y. T. Fang, "Different discrete soliton states in periodic optical induced waveguide lattice," Opt. Express 14, 6232-6240 (2007).
[CrossRef]

G. Dekel, V. Fleurov, A. Soffer, and C. Stucchio, "Temporal dynamics of tunneling: Hydrodynamic approach," Phys. Rev. A 75, 043617 (2007).

B. Kibler, P.-A. Lacourt, F. Courvoisier, and J. M. Dudley, "Soliton spectral tunnelling in photonic crystal fibre with sub-wavelength core defect," Electron. Lett. 43, 967-968 (2007).
[CrossRef]

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva "Nonautonomous Solitons in External Potentials," Phys. Rev. Lett. 98, 074102 (2007).
[CrossRef] [PubMed]

H. Zhuo, X. Q. Fu, Y. H. Hu, S. C. Wen, "Compensation of the influence of loss for a spatial soliton in a dissipative modulated Bessel optical lattice," J. Opt. Soc. Am. B 24, 2208-2212 (2007).

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Soliton percolation in random optical lattices," Opt. Express 15, 12409-12417(2007).
[CrossRef] [PubMed]

2006

Y. V. Kartashov, V. A. Vysloukh, and L. Torner. "Soliton Control in Fading Optical Lattices," Opt. Lett. 31, 2181-2183 (2006).
[CrossRef] [PubMed]

Y. J. He, H. Z. Wang, "(1+1)-dimensional dipole solitons supported by optical lattice," Opt. Express 14, 9832-9837 (2006).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, F. Lederer, L.-C. Crasovan, Y. V. Kartashov, L. Torner, B. A. Malomed, "Stable solitons of even and odd parities supported by competing nonlocal nonlinearities," Phys. Rev. E 74, 066614 (2006).

K. Staliunas, R. Herrero, "Nondiffractive Propagation of Light in Photonic Crystals," Phys. Rev. E 73, 016601. (2006).

G. Y. Yang, R. Y. Hao, L. Li, Z. H. Li, and G. S. Zhou, "Cascade compression induced by nonlinear barriers in propagation of optical solitons, " Opt. Commun. 260, 282-287 (2006).
[CrossRef]

2005

2004

2003

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of Distrete Solitons in Optically Induced Real Time Waveguide Arrays," Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

T. Pertsch, U. Peschel, and F. Lederer. "All-optical Switching in Quadratically Nonlinear Waveguide Arrays," Opt. Lett. 28, 102-104 (2003).
[CrossRef] [PubMed]

D. Neshev, E. Ostrovskaya, Y. S. Kivshar, and W. Krolikowski. "Spatial Solitons in Optically Induced Gratings," Opt. Lett. 28, 710-712 (2003).
[CrossRef] [PubMed]

2001

M. J. Ablowitz and Z. H. Musslimani, "Discrete Diffraction Managed Spatial Solitons," Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef] [PubMed]

V. N. Serkin, V. M. Chapela, J. Percino, and T. L. Belyaeva, "Nonlinear tunneling of temporal and spatial optical solitons through organic thin films and polymeric waveguides," Opt. Commun. 192, 237-244 (2001).
[CrossRef]

V. N. Serkin and T. L. Belyaeva, "High-energy optical Schrödinger solitons," JETP Lett. 74, 573-577 (2001).
[CrossRef]

2000

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison. "Diffraction Management," Phys. Rev. Lett. 85, 1863-1866 (2000).
[CrossRef] [PubMed]

V. N. Serkin, and A. Hasegawa, "Femtosecond soliton amplification in nonlinear dispersive traps and soliton dispersion management," Proc. SPIE 3927, 302-313 (2000)
[CrossRef]

1999

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

1998

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

1997

G. Kälbermann, "Soliton tunneling," Phys. Rev. E 55, R6360-R6362 (1997).

1994

1993

V. N. Serkin, V. A. Vysloukh, and J. R. Taylor, "Soliton spectral tunnelling effect," Electron. Lett. 29, 12-13 (1993).
[CrossRef]

1988

1978

A. C. Newell, "Nonlinear tunneling," J. Math. Phys. 19, 1126-1133 (1978).
[CrossRef]

Ablowitz, M. J.

M. J. Ablowitz and Z. H. Musslimani, "Discrete Diffraction Managed Spatial Solitons," Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef] [PubMed]

Aitchison, J. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison. "Diffraction Management," Phys. Rev. Lett. 85, 1863-1866 (2000).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Anderson, D.

Barak, A.

A. Barak, O. Peleg, A. Soffer, and M. Segev, "Multisoliton ejection from an amplifying potential trap," Opt. Lett. 33, 1798-1800 (2008).
[CrossRef] [PubMed]

A. Barak, O. Peleg, C. Stucchio, A. Soffer, and M. Segev, "Observation of Soliton Tunneling Phenomena and Soliton Ejection," Phys. Rev. Lett. 100, 153901 (2008).
[CrossRef] [PubMed]

Belyaeva, T. L.

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva "Nonautonomous Solitons in External Potentials," Phys. Rev. Lett. 98, 074102 (2007).
[CrossRef] [PubMed]

V. N. Serkin and T. L. Belyaeva, "High-energy optical Schrödinger solitons," JETP Lett. 74, 573-577 (2001).
[CrossRef]

V. N. Serkin, V. M. Chapela, J. Percino, and T. L. Belyaeva, "Nonlinear tunneling of temporal and spatial optical solitons through organic thin films and polymeric waveguides," Opt. Commun. 192, 237-244 (2001).
[CrossRef]

Boyd, A. R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Carmon, T.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of Distrete Solitons in Optically Induced Real Time Waveguide Arrays," Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

Chapela, V. M.

V. N. Serkin, V. M. Chapela, J. Percino, and T. L. Belyaeva, "Nonlinear tunneling of temporal and spatial optical solitons through organic thin films and polymeric waveguides," Opt. Commun. 192, 237-244 (2001).
[CrossRef]

Chen, Z.

Christodoulides, D. N.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of Distrete Solitons in Optically Induced Real Time Waveguide Arrays," Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides," Opt. Lett. 13, 794-796 (1988).
[CrossRef] [PubMed]

Courvoisier, F.

B. Kibler, P.-A. Lacourt, F. Courvoisier, and J. M. Dudley, "Soliton spectral tunnelling in photonic crystal fibre with sub-wavelength core defect," Electron. Lett. 43, 967-968 (2007).
[CrossRef]

Crasovan, L.-C.

D. Mihalache, D. Mazilu, F. Lederer, L.-C. Crasovan, Y. V. Kartashov, L. Torner, B. A. Malomed, "Stable solitons of even and odd parities supported by competing nonlocal nonlinearities," Phys. Rev. E 74, 066614 (2006).

Dekel, G.

G. Dekel, V. Fleurov, A. Soffer, and C. Stucchio, "Temporal dynamics of tunneling: Hydrodynamic approach," Phys. Rev. A 75, 043617 (2007).

Dudley, J. M.

B. Kibler, P.-A. Lacourt, F. Courvoisier, and J. M. Dudley, "Soliton spectral tunnelling in photonic crystal fibre with sub-wavelength core defect," Electron. Lett. 43, 967-968 (2007).
[CrossRef]

Efremidis, N. K.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of Distrete Solitons in Optically Induced Real Time Waveguide Arrays," Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

Eisenberg, H. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison. "Diffraction Management," Phys. Rev. Lett. 85, 1863-1866 (2000).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Eugenieva, E. D.

Fang, Y. T.

J. Zhou, Y. H. Qi, C. H. Xue, S. Y. Lou, and Y. T. Fang, "Different discrete soliton states in periodic optical induced waveguide lattice," Opt. Express 14, 6232-6240 (2007).
[CrossRef]

Fleischer, J. W.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of Distrete Solitons in Optically Induced Real Time Waveguide Arrays," Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

Fleurov, V.

G. Dekel, V. Fleurov, A. Soffer, and C. Stucchio, "Temporal dynamics of tunneling: Hydrodynamic approach," Phys. Rev. A 75, 043617 (2007).

Fu, X. Q.

Garanovich, I. L.

Guo, Q.

Q. Guo, "Optical Transmission, Switching, Subsystems," Proc. SPIE 5281, 581-594 (2004).
[CrossRef]

Hao, R. Y.

G. Y. Yang, R. Y. Hao, L. Li, Z. H. Li, and G. S. Zhou, "Cascade compression induced by nonlinear barriers in propagation of optical solitons, " Opt. Commun. 260, 282-287 (2006).
[CrossRef]

Hasegawa, A.

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva "Nonautonomous Solitons in External Potentials," Phys. Rev. Lett. 98, 074102 (2007).
[CrossRef] [PubMed]

V. N. Serkin, and A. Hasegawa, "Femtosecond soliton amplification in nonlinear dispersive traps and soliton dispersion management," Proc. SPIE 3927, 302-313 (2000)
[CrossRef]

He, Y. J.

Herrero, R.

K. Staliunas, R. Herrero, "Nondiffractive Propagation of Light in Photonic Crystals," Phys. Rev. E 73, 016601. (2006).

Hu, Y. H.

Joseph, R. I.

Kälbermann, G.

G. Kälbermann, "Soliton tunneling," Phys. Rev. E 55, R6360-R6362 (1997).

Kartashov, Y. V.

Kibler, B.

B. Kibler, P.-A. Lacourt, F. Courvoisier, and J. M. Dudley, "Soliton spectral tunnelling in photonic crystal fibre with sub-wavelength core defect," Electron. Lett. 43, 967-968 (2007).
[CrossRef]

Kivshar, Y. S.

Krolikowski, W.

Lacourt, P.-A.

B. Kibler, P.-A. Lacourt, F. Courvoisier, and J. M. Dudley, "Soliton spectral tunnelling in photonic crystal fibre with sub-wavelength core defect," Electron. Lett. 43, 967-968 (2007).
[CrossRef]

Lederer, F.

D. Mihalache, D. Mazilu, F. Lederer, L.-C. Crasovan, Y. V. Kartashov, L. Torner, B. A. Malomed, "Stable solitons of even and odd parities supported by competing nonlocal nonlinearities," Phys. Rev. E 74, 066614 (2006).

T. Pertsch, U. Peschel, and F. Lederer. "All-optical Switching in Quadratically Nonlinear Waveguide Arrays," Opt. Lett. 28, 102-104 (2003).
[CrossRef] [PubMed]

Li, L.

G. Y. Yang, R. Y. Hao, L. Li, Z. H. Li, and G. S. Zhou, "Cascade compression induced by nonlinear barriers in propagation of optical solitons, " Opt. Commun. 260, 282-287 (2006).
[CrossRef]

Li, Z. H.

G. Y. Yang, R. Y. Hao, L. Li, Z. H. Li, and G. S. Zhou, "Cascade compression induced by nonlinear barriers in propagation of optical solitons, " Opt. Commun. 260, 282-287 (2006).
[CrossRef]

Lisak, M.

Lou, S. Y.

J. Zhou, Y. H. Qi, C. H. Xue, S. Y. Lou, and Y. T. Fang, "Different discrete soliton states in periodic optical induced waveguide lattice," Opt. Express 14, 6232-6240 (2007).
[CrossRef]

Malomed, B.

Malomed, B. A.

D. Mihalache, D. Mazilu, F. Lederer, L.-C. Crasovan, Y. V. Kartashov, L. Torner, B. A. Malomed, "Stable solitons of even and odd parities supported by competing nonlocal nonlinearities," Phys. Rev. E 74, 066614 (2006).

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, "Spatiotemporal optical solitons," J. Opt. B: Quantum Semiclass. Opt. 7, R53-R72 (2005).
[CrossRef]

Martin, H.

Mazilu, D.

D. Mihalache, D. Mazilu, F. Lederer, L.-C. Crasovan, Y. V. Kartashov, L. Torner, B. A. Malomed, "Stable solitons of even and odd parities supported by competing nonlocal nonlinearities," Phys. Rev. E 74, 066614 (2006).

Mihalache, D.

D. Mihalache, D. Mazilu, F. Lederer, L.-C. Crasovan, Y. V. Kartashov, L. Torner, B. A. Malomed, "Stable solitons of even and odd parities supported by competing nonlocal nonlinearities," Phys. Rev. E 74, 066614 (2006).

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, "Spatiotemporal optical solitons," J. Opt. B: Quantum Semiclass. Opt. 7, R53-R72 (2005).
[CrossRef]

Morandotti, R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison. "Diffraction Management," Phys. Rev. Lett. 85, 1863-1866 (2000).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Musslimani, Z. H.

M. J. Ablowitz and Z. H. Musslimani, "Discrete Diffraction Managed Spatial Solitons," Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef] [PubMed]

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[CrossRef]

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Peleg, O.

A. Barak, O. Peleg, C. Stucchio, A. Soffer, and M. Segev, "Observation of Soliton Tunneling Phenomena and Soliton Ejection," Phys. Rev. Lett. 100, 153901 (2008).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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V. N. Serkin, V. M. Chapela, J. Percino, and T. L. Belyaeva, "Nonlinear tunneling of temporal and spatial optical solitons through organic thin films and polymeric waveguides," Opt. Commun. 192, 237-244 (2001).
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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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V. N. Serkin, A. Hasegawa, and T. L. Belyaeva "Nonautonomous Solitons in External Potentials," Phys. Rev. Lett. 98, 074102 (2007).
[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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A. Barak, O. Peleg, C. Stucchio, A. Soffer, and M. Segev, "Observation of Soliton Tunneling Phenomena and Soliton Ejection," Phys. Rev. Lett. 100, 153901 (2008).
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A. Barak, O. Peleg, C. Stucchio, A. Soffer, and M. Segev, "Observation of Soliton Tunneling Phenomena and Soliton Ejection," Phys. Rev. Lett. 100, 153901 (2008).
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J. Zhou, Y. H. Qi, C. H. Xue, S. Y. Lou, and Y. T. Fang, "Different discrete soliton states in periodic optical induced waveguide lattice," Opt. Express 14, 6232-6240 (2007).
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G. Y. Yang, R. Y. Hao, L. Li, Z. H. Li, and G. S. Zhou, "Cascade compression induced by nonlinear barriers in propagation of optical solitons, " Opt. Commun. 260, 282-287 (2006).
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Zelenina, A. S.

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G. Y. Yang, R. Y. Hao, L. Li, Z. H. Li, and G. S. Zhou, "Cascade compression induced by nonlinear barriers in propagation of optical solitons, " Opt. Commun. 260, 282-287 (2006).
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J. Zhou, Y. H. Qi, C. H. Xue, S. Y. Lou, and Y. T. Fang, "Different discrete soliton states in periodic optical induced waveguide lattice," Opt. Express 14, 6232-6240 (2007).
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[CrossRef]

V. N. Serkin, V. M. Chapela, J. Percino, and T. L. Belyaeva, "Nonlinear tunneling of temporal and spatial optical solitons through organic thin films and polymeric waveguides," Opt. Commun. 192, 237-244 (2001).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

G. Dekel, V. Fleurov, A. Soffer, and C. Stucchio, "Temporal dynamics of tunneling: Hydrodynamic approach," Phys. Rev. A 75, 043617 (2007).

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Phys. Rev. Lett.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of Distrete Solitons in Optically Induced Real Time Waveguide Arrays," Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison. "Diffraction Management," Phys. Rev. Lett. 85, 1863-1866 (2000).
[CrossRef] [PubMed]

M. J. Ablowitz and Z. H. Musslimani, "Discrete Diffraction Managed Spatial Solitons," Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef] [PubMed]

A. Barak, O. Peleg, C. Stucchio, A. Soffer, and M. Segev, "Observation of Soliton Tunneling Phenomena and Soliton Ejection," Phys. Rev. Lett. 100, 153901 (2008).
[CrossRef] [PubMed]

Z. Xu, Y. V. Kartashov, and L.Torner, "Soliton Mobility in Nonlocal Optical Lattices," Phys. Rev. Lett. 95, 113901 (2005).
[CrossRef] [PubMed]

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva "Nonautonomous Solitons in External Potentials," Phys. Rev. Lett. 98, 074102 (2007).
[CrossRef] [PubMed]

Proc. SPIE

Q. Guo, "Optical Transmission, Switching, Subsystems," Proc. SPIE 5281, 581-594 (2004).
[CrossRef]

V. N. Serkin, and A. Hasegawa, "Femtosecond soliton amplification in nonlinear dispersive traps and soliton dispersion management," Proc. SPIE 3927, 302-313 (2000)
[CrossRef]

Science

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

Other

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

N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams, (Chapman and Hall, London, 1997).

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

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Figures (7)

Fig. 1.
Fig. 1.

Profile of a steep lattice potential barrier described by Eq. (2) with Ω η =4,δ=1 and ξB=5.

Fig. 2.
Fig. 2.

Propagation of spatial solitons with A=1and η 0=0 across the barrier given by Eq. (2) with Ω η =4, (a) p=6; (b) p=36; (c) and (d) corresponding to the cases of (a) and (b), respectively.

Fig. 3.
Fig. 3.

Propagation of spatial solitons with A=1 and η 0=0 across the barrier given by Eq. (2) with Ω η =8, (a) p=36; (b) p=150; (c) and (d) corresponding to the cases of (a) and (b), respectively.

Fig. 4.
Fig. 4.

Tunneling of spatial solitons with A=2 and η 0=0 through the barrier given by Eq.(2) with (a) Ω η =4, p=36; (b) Ω η =8, p=150; (c) and (d) corresponding to the cases of (a) and (b), respectively.

Fig. 5.
Fig. 5.

Propagation of spatial solitons with η 0 = π 2 Ω η across the barrier given by Eq. (2) with Ω η =4, (a) A=1, p=10; (b) A=1, p=50; (c) A=2, p=50; (d), (e) and (f) corresponding to the cases of (a), (b) and (c), respectively

Fig. 6.
Fig. 6.

Compression of spatial solitons with A=2 and η 0=0 through the lattice system given by Eq. (3) with Ω η =4 (a) δ=1, p=30; (b) δ=0 (i.e. harmonic lattice), p=30. (c) and (d) corresponding to the beam shapes of (a) and (b) at different distances and different potentials, respectively.

Fig. 7.
Fig. 7.

Splitting of spatial solitons with A=2 and η 0 = π 2 Ω η through the lattice system given by Eq. (3) with Ω η =4 (a) δ=1, p=50 ; (b) δ=0 (i.e. harmonic lattice), p=50 ; (c) beam shapes at different distances corresponding to the case of (a);(d) the parameters are the same as in (b) except p=100.

Equations (3)

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

i q ξ = 1 2 2 q η 2 q q 2 p R ( η , ξ ) q .
R ( η , ξ ) = { cos 2 ( Ω η η ) exp [ δ ( ξ ξ B ) ] 0 ξ < ξ B cos 2 ( Ω η η ) exp [ δ ( ξ ξ B ) ] ξ B ξ < 2 ξ B ,
R ( η , ξ ) = { cos 2 ( Ω η η ) exp [ δ ( ξ ξ B ) ] 0 ξ < ξ B cos 2 ( Ω η η ) ξ B ξ < 2 ξ B

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