Abstract

We report on the existence and stability of solitons in a defect embedded in a square optical lattice based on a photorefractive crystal with focusing saturable nonlinearity. These solitons exist in different bandgaps due to the change of defect intensity. For a positive defect, the solitons only exist in the semi-infinite gap and can be stable in the low power region but not the high power region. For a negative defect, the solitons can exist not only in the semi-infinite gap, but also in the first gap. With increasing the defect depth, these solitons are stable within a moderate power region in the first gap while unstable in the entire semi-infinite gap.

© 2010 OSA

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

2009 (3)

J. Yang, “Newton-conjugate gradient methods for solitary wave computations,” J. Comput. Phys. 228(18), 7007–7024 (2009).
[CrossRef]

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80(4), 043824 (2009).
[CrossRef]

A. Szameit, Y. V. Kartashov, M. Heinrich, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, V. A. Vysloukh, and L. Torner, “Observation of two-dimensional defect surface solitons,” Opt. Lett. 34(6), 797–799 (2009).
[CrossRef] [PubMed]

2008 (2)

J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 227(14), 6862–6876 (2008).
[CrossRef]

J. Wang and J. Yang, “Families of vortex solitons in periodic media,” Phys. Rev. A 77(3), 033834 (2008).
[CrossRef]

2007 (8)

C. Lou, X. Wang, J. Xu, Z. Chen, and J. Yang, “Nonlinear spectrum reshaping and gap-soliton-train trapping in optically induced photonic structures,” Phys. Rev. Lett. 98(21), 213903 (2007).
[CrossRef] [PubMed]

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, “Observation of two-dimensional surface solitons,” Phys. Rev. Lett. 98(12), 123903 (2007).
[CrossRef] [PubMed]

J. Wang, J. Yang, and Z. Chen, “Two-dimensional defect modes in optically induced photonic lattices,” Phys. Rev. A 76(1), 013828 (2007).
[CrossRef]

J. Yang and T. I. Lakoba, “Universally-Convergent Squared-Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations,” Stud. Appl. Math. 118(2), 153–197 (2007).
[CrossRef]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

W. Chen, Y. He, and H. Wang, “Surface defect superlattice solitons,” J. Opt. Soc. Am. B 24(10), 2584–2588 (2007).
[CrossRef]

L. Tang, C. Lou, X. Wang, D. Song, X. Chen, J. Xu, Z. Chen, H. Susanto, K. Law, and P. G. Kevrekidis, “Observation of dipole-like gap solitons in self-defocusing waveguide lattices,” Opt. Lett. 32(20), 3011–3013 (2007).
[CrossRef] [PubMed]

W. H. Chen, Y. J. He, and H. Z. Wang, “Defect superlattice solitons,” Opt. Express 15(22), 14498–14503 (2007).
[CrossRef] [PubMed]

2006 (4)

I. Makasyuk, Z. Chen, and J. Yang, “Band-gap guidance in optically induced photonic lattices with a negative defect,” Phys. Rev. Lett. 96(22), 223903 (2006).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface gap solitons,” Phys. Rev. Lett. 96(7), 073901 (2006).
[CrossRef] [PubMed]

J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(2), 026609 (2006).
[CrossRef] [PubMed]

M. J. Ablowitz, B. Ilan, E. Schonbrun, and R. Piestun, “Solitons in two-dimensional lattices possessing defects, dislocations, and quasicrystal structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 035601 (2006).
[CrossRef] [PubMed]

2004 (1)

2003 (3)

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90(2), 023902 (2003).
[CrossRef] [PubMed]

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

2002 (1)

1993 (1)

Ablowitz, M. J.

M. J. Ablowitz, B. Ilan, E. Schonbrun, and R. Piestun, “Solitons in two-dimensional lattices possessing defects, dislocations, and quasicrystal structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 035601 (2006).
[CrossRef] [PubMed]

Bartal, G.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

Bezryadina, A.

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, “Observation of two-dimensional surface solitons,” Phys. Rev. Lett. 98(12), 123903 (2007).
[CrossRef] [PubMed]

Carmon, T.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90(2), 023902 (2003).
[CrossRef] [PubMed]

Chen, W.

Chen, W. H.

Chen, X.

Chen, Y.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80(4), 043824 (2009).
[CrossRef]

Chen, Z.

L. Tang, C. Lou, X. Wang, D. Song, X. Chen, J. Xu, Z. Chen, H. Susanto, K. Law, and P. G. Kevrekidis, “Observation of dipole-like gap solitons in self-defocusing waveguide lattices,” Opt. Lett. 32(20), 3011–3013 (2007).
[CrossRef] [PubMed]

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, “Observation of two-dimensional surface solitons,” Phys. Rev. Lett. 98(12), 123903 (2007).
[CrossRef] [PubMed]

J. Wang, J. Yang, and Z. Chen, “Two-dimensional defect modes in optically induced photonic lattices,” Phys. Rev. A 76(1), 013828 (2007).
[CrossRef]

C. Lou, X. Wang, J. Xu, Z. Chen, and J. Yang, “Nonlinear spectrum reshaping and gap-soliton-train trapping in optically induced photonic structures,” Phys. Rev. Lett. 98(21), 213903 (2007).
[CrossRef] [PubMed]

J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(2), 026609 (2006).
[CrossRef] [PubMed]

I. Makasyuk, Z. Chen, and J. Yang, “Band-gap guidance in optically induced photonic lattices with a negative defect,” Phys. Rev. Lett. 96(22), 223903 (2006).
[CrossRef] [PubMed]

Z. Chen and K. McCarthy, “Spatial soliton pixels from partially incoherent light,” Opt. Lett. 27(22), 2019–2021 (2002).
[CrossRef]

Christodoulides, D. N.

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, “Observation of two-dimensional surface solitons,” Phys. Rev. Lett. 98(12), 123903 (2007).
[CrossRef] [PubMed]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90(2), 023902 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[CrossRef] [PubMed]

Dreisow, F.

Efremidis, N. K.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90(2), 023902 (2003).
[CrossRef] [PubMed]

Fleischer, J. W.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90(2), 023902 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[CrossRef] [PubMed]

Freedman, B.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

He, Y.

He, Y. J.

Heinrich, M.

Ilan, B.

M. J. Ablowitz, B. Ilan, E. Schonbrun, and R. Piestun, “Solitons in two-dimensional lattices possessing defects, dislocations, and quasicrystal structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 035601 (2006).
[CrossRef] [PubMed]

Kartashov, Y. V.

Kevrekidis, P. G.

Kivshar, Y. S.

Lakoba, T. I.

J. Yang and T. I. Lakoba, “Universally-Convergent Squared-Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations,” Stud. Appl. Math. 118(2), 153–197 (2007).
[CrossRef]

Law, K.

Lederer, F.

Li, Y.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80(4), 043824 (2009).
[CrossRef]

Lou, C.

L. Tang, C. Lou, X. Wang, D. Song, X. Chen, J. Xu, Z. Chen, H. Susanto, K. Law, and P. G. Kevrekidis, “Observation of dipole-like gap solitons in self-defocusing waveguide lattices,” Opt. Lett. 32(20), 3011–3013 (2007).
[CrossRef] [PubMed]

C. Lou, X. Wang, J. Xu, Z. Chen, and J. Yang, “Nonlinear spectrum reshaping and gap-soliton-train trapping in optically induced photonic structures,” Phys. Rev. Lett. 98(21), 213903 (2007).
[CrossRef] [PubMed]

Makasyuk, I.

I. Makasyuk, Z. Chen, and J. Yang, “Band-gap guidance in optically induced photonic lattices with a negative defect,” Phys. Rev. Lett. 96(22), 223903 (2006).
[CrossRef] [PubMed]

Makris, K. G.

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, “Observation of two-dimensional surface solitons,” Phys. Rev. Lett. 98(12), 123903 (2007).
[CrossRef] [PubMed]

Manela, O.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

McCarthy, K.

Musslimani, Z. H.

Nolte, S.

Pang, W.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80(4), 043824 (2009).
[CrossRef]

Peleg, O.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

Pertsch, T.

Piestun, R.

M. J. Ablowitz, B. Ilan, E. Schonbrun, and R. Piestun, “Solitons in two-dimensional lattices possessing defects, dislocations, and quasicrystal structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 035601 (2006).
[CrossRef] [PubMed]

Schonbrun, E.

M. J. Ablowitz, B. Ilan, E. Schonbrun, and R. Piestun, “Solitons in two-dimensional lattices possessing defects, dislocations, and quasicrystal structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 035601 (2006).
[CrossRef] [PubMed]

Segev, M.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90(2), 023902 (2003).
[CrossRef] [PubMed]

Song, D.

Stegeman, G. I.

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, “Observation of two-dimensional surface solitons,” Phys. Rev. Lett. 98(12), 123903 (2007).
[CrossRef] [PubMed]

Susanto, H.

Szameit, A.

Tang, L.

Torner, L.

Tünnermann, A.

Vysloukh, V.

Vysloukh, V. A.

Wang, H.

Wang, H. Z.

Wang, J.

J. Wang and J. Yang, “Families of vortex solitons in periodic media,” Phys. Rev. A 77(3), 033834 (2008).
[CrossRef]

J. Wang, J. Yang, and Z. Chen, “Two-dimensional defect modes in optically induced photonic lattices,” Phys. Rev. A 76(1), 013828 (2007).
[CrossRef]

Wang, X.

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, “Observation of two-dimensional surface solitons,” Phys. Rev. Lett. 98(12), 123903 (2007).
[CrossRef] [PubMed]

L. Tang, C. Lou, X. Wang, D. Song, X. Chen, J. Xu, Z. Chen, H. Susanto, K. Law, and P. G. Kevrekidis, “Observation of dipole-like gap solitons in self-defocusing waveguide lattices,” Opt. Lett. 32(20), 3011–3013 (2007).
[CrossRef] [PubMed]

C. Lou, X. Wang, J. Xu, Z. Chen, and J. Yang, “Nonlinear spectrum reshaping and gap-soliton-train trapping in optically induced photonic structures,” Phys. Rev. Lett. 98(21), 213903 (2007).
[CrossRef] [PubMed]

Xu, J.

C. Lou, X. Wang, J. Xu, Z. Chen, and J. Yang, “Nonlinear spectrum reshaping and gap-soliton-train trapping in optically induced photonic structures,” Phys. Rev. Lett. 98(21), 213903 (2007).
[CrossRef] [PubMed]

L. Tang, C. Lou, X. Wang, D. Song, X. Chen, J. Xu, Z. Chen, H. Susanto, K. Law, and P. G. Kevrekidis, “Observation of dipole-like gap solitons in self-defocusing waveguide lattices,” Opt. Lett. 32(20), 3011–3013 (2007).
[CrossRef] [PubMed]

Yang, J.

J. Yang, “Newton-conjugate gradient methods for solitary wave computations,” J. Comput. Phys. 228(18), 7007–7024 (2009).
[CrossRef]

J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 227(14), 6862–6876 (2008).
[CrossRef]

J. Wang and J. Yang, “Families of vortex solitons in periodic media,” Phys. Rev. A 77(3), 033834 (2008).
[CrossRef]

C. Lou, X. Wang, J. Xu, Z. Chen, and J. Yang, “Nonlinear spectrum reshaping and gap-soliton-train trapping in optically induced photonic structures,” Phys. Rev. Lett. 98(21), 213903 (2007).
[CrossRef] [PubMed]

J. Wang, J. Yang, and Z. Chen, “Two-dimensional defect modes in optically induced photonic lattices,” Phys. Rev. A 76(1), 013828 (2007).
[CrossRef]

J. Yang and T. I. Lakoba, “Universally-Convergent Squared-Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations,” Stud. Appl. Math. 118(2), 153–197 (2007).
[CrossRef]

I. Makasyuk, Z. Chen, and J. Yang, “Band-gap guidance in optically induced photonic lattices with a negative defect,” Phys. Rev. Lett. 96(22), 223903 (2006).
[CrossRef] [PubMed]

J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(2), 026609 (2006).
[CrossRef] [PubMed]

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

Yu, Z.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80(4), 043824 (2009).
[CrossRef]

Zhang, H.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80(4), 043824 (2009).
[CrossRef]

Zhou, J.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80(4), 043824 (2009).
[CrossRef]

J. Comput. Phys. (2)

J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 227(14), 6862–6876 (2008).
[CrossRef]

J. Yang, “Newton-conjugate gradient methods for solitary wave computations,” J. Comput. Phys. 228(18), 7007–7024 (2009).
[CrossRef]

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

Nature (1)

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (5)

Phys. Rev. A (3)

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80(4), 043824 (2009).
[CrossRef]

J. Wang, J. Yang, and Z. Chen, “Two-dimensional defect modes in optically induced photonic lattices,” Phys. Rev. A 76(1), 013828 (2007).
[CrossRef]

J. Wang and J. Yang, “Families of vortex solitons in periodic media,” Phys. Rev. A 77(3), 033834 (2008).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(2), 026609 (2006).
[CrossRef] [PubMed]

M. J. Ablowitz, B. Ilan, E. Schonbrun, and R. Piestun, “Solitons in two-dimensional lattices possessing defects, dislocations, and quasicrystal structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 035601 (2006).
[CrossRef] [PubMed]

Phys. Rev. Lett. (6)

I. Makasyuk, Z. Chen, and J. Yang, “Band-gap guidance in optically induced photonic lattices with a negative defect,” Phys. Rev. Lett. 96(22), 223903 (2006).
[CrossRef] [PubMed]

C. Lou, X. Wang, J. Xu, Z. Chen, and J. Yang, “Nonlinear spectrum reshaping and gap-soliton-train trapping in optically induced photonic structures,” Phys. Rev. Lett. 98(21), 213903 (2007).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface gap solitons,” Phys. Rev. Lett. 96(7), 073901 (2006).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90(2), 023902 (2003).
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X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, “Observation of two-dimensional surface solitons,” Phys. Rev. Lett. 98(12), 123903 (2007).
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Stud. Appl. Math. (1)

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[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

(Color online) (a) Band structure of square optical lattices. (b) The square optical lattices with a negative defect ε = −0.5. (c) The square optical lattices with a positive defect ε = 0.5.

Fig. 2
Fig. 2

(Color online) ε = −0.5. (a) Power versus propagation constant (blue regions are Bloch bands). (b) Perturbation growth rate Re(δ) for the unstable DSs. (c) Profile of DS for μ = 2.8 (point A), and its profile at (d) z = 100, and (e) z = 200. (f) Profile of DS for μ = 3.4 (point B), and its profile at (g) z = 100, and (h) z = 200.

Fig. 3
Fig. 3

(Color online)ε = −0.5. (a) Profile of DS for μ = 2.4, and its profile at (b) z = 100, and (c) z = 200. (d) Profile of DS for μ = 4.5, and its profile at (e) z = 100, and (f) z = 200. (g) Profile of DS for μ = 4.7, and its profile at (h) z = 100, and (i) z = 200.

Fig. 4
Fig. 4

(Color online) ε = −1. (a) Power versus propagation constant. (b) Perturbation growth rate Re(δ) for the unstable DSs. (c) Profile of DS for μ = 5.0, and its profile at (d) z = 100, and (e) z = 200. (f) Profile of DS for μ = 4.68, and its profile at (g) z = 100, and (h) z = 200.

Fig. 5
Fig. 5

(Color online)ε = −1. (a) Profile of DS for μ = 5.08, and its profile at (b) z = 100, and (c) z = 200. (d) Profile of DS for μ = 3.0, and its profile at (e) z = 100, and (f) z = 200.

Fig. 6
Fig. 6

(Color online)ε = 0.5. (a) Power versus propagation constant. (The solid and dashed lines present the stable and unstable regions, respectively). (b) The perturbation growth rate Re(δ) for the unstable DSs. (c) Profile of DS for μ = 3.0, and its profile at (d) z = 100, and (e) z = 200. (f) Profile of DS for μ = 1.4, and its profiles at (g) z = 100, and (h) z = 200.

Equations (5)

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i U z + 2 U x 2 + 2 U y 2 E 0 1 + I L + | U | 2 U = 0 ,
I L = I 0 cos 2 ( x ) cos 2 ( y ) { 1 + ε exp [ ( x 2 + y 2 ) 4 / 128 ] } .
2 u x 2 + 2 u y 2 + 2 i k x u x + 2 i k y u y ( k x 2 + k y 2 ) u E 0 1 + I L ( x , y ) u = μ u ,
2 u x 2 + 2 u y 2 E 0 1 + I L + | u | 2 u = μ u .
{ δ v = i [ ( 2 w / x 2 + 2 w / y 2 ) + μ w E 0 w / ( 1 + I L + u 2 ) ] ,           δ w = i [ ( 2 v / x 2 + 2 v / y 2 ) + μ v E 0 v ( 1 + I L u 2 ) / ( 1 + I L + u 2 ) 2 ] .

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