Abstract

We report that surface defect superlattice solitons can be supported by a defect on the interface between the optical superlattices and the homogeneous media with saturable nonlinearity. The surface defect of superlattices offers such types of soliton some unique properties. For the positive defect, solitons exist in the semi-infinite bandgap, whereas for the negative defect, solitons exist not only in the semi-infinite bandgap but also in the first finite bandgap. When defect intensity approximates to the intensity of optical superlattices, solitons exist in the semi-infinite bandgap. The stable regions of solitons with various defects are given.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2007 (1)

2006 (16)

W. H. Chen, Y. J. He, and H. Z. Wang, "Surface defect gap solitons," Opt. Express 14, 11271-11276 (2006).
[CrossRef] [PubMed]

M. A. Porter, P. G. Kevrekidis, R. Carretero-González, D. J. Frantzeskakis, "Dynamics and manipulation of matter-wave solitons in optical superlattices," Phys. Lett. A 352, 210-215 (2006).
[CrossRef]

K. Yagasakai, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, "Gap solitons in Bragg gratings with a harmonic superlattice," Europhys. Lett. 74, 1006-1012 (2006).
[CrossRef]

H. Sakaguchi and B. A. Malomed, "Gap solitons in quasiperiodic optical lattices," Phys. Rev. E 74, 026601 (2006).
[CrossRef]

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

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, "Surface vortex solitons," Opt. Express 14, 4049-4057 (2006).
[CrossRef] [PubMed]

I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, "Surface multi-gap vector solitons," Opt. Express 14, 4780-4785 (2006).
[CrossRef] [PubMed]

Y. V. Kartashov, F. Ye, and L. Torner, "Vector mixed-gap surface solitons," Opt. Express 14, 4808-4814 (2006).
[CrossRef] [PubMed]

M. I. Molina, I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, "Discrete surface solitons in semi-infinite binary waveguide arrays," Opt. Lett. 31, 2332-3334 (2006).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, D. Mihalache, and L. Torner, "Generation of surface soliton arrays," Opt. Lett. 31, 2329-2331 (2006).
[CrossRef] [PubMed]

M. I. Molina, "Nonlinear surface impurity in a semi-infinite two-dimensional square lattice: Green function approach," Phys. Rev. B 74, 045412 (2006).
[CrossRef]

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, "Observation of discrete surface solitons," Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef] [PubMed]

C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, "Observation of surface gap solitons in semi-infinite waveguide arrays," Phys. Rev. Lett. 97, 083901 (2006).
[CrossRef] [PubMed]

E. Smirnov, M. Stepic, C. E. Rüter, D. Kip, and V. Shandarov, "Observation of staggered surface solitary waves in one-dimensional waveguide arrays," Opt. Lett. 31, 2338-2340 (2006).
[CrossRef] [PubMed]

G. A. Siviloglou, K. G. Makris, R. Iwanow, R. Schiek, D. N. Christodoulides, and G. I. Stegeman, "Observation of discrete quadratic solitons," Opt. Express 14, 5508-5516 (2006).
[CrossRef] [PubMed]

J. Yang and Z. Chen, "Defect solitons in photonic lattices," Phys. Rev. E 73, 026609 (2006).
[CrossRef]

2005 (6)

F. Fedele, J. Yang, and Z. Chen, "Defect modes in one-dimensional photonic lattices," Opt. Lett. 30, 1506-1508 (2005).
[CrossRef] [PubMed]

F. Fedele, J. Yang, and Z. Chen, "Properties of defect modes in one-dimensional optically induced photonic lattices," Stud. Appl. Math. 115, 277-299 (2005).
[CrossRef]

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, "Dispersion control for matter waves and gap solitons in optical superlattices," Phys. Rev. A 71, 023612 (2005).
[CrossRef]

K. G. Makris, S. Suntsov, D. N. Christodoulides, and G. I. Stegeman, "Discrete surface solitons," Opt. Lett. 30, 2466-2468 (2005).
[CrossRef] [PubMed]

J. Hudock, S. Suntsov, D. N. Christodoulides, and G. I. Stegeman, "Vector discrete nonlinear surface waves," Opt. Express 13, 7720-7725 (2005).
[CrossRef] [PubMed]

B. A. Malomed, T. Mayteevarunyoo, E. A. Ostrovskaya, and Y. S. Kivshar, "Coupled-mode theory for spatial gap solitons in optically induced lattices," Phys. Rev. E 71, 056616 (2005).
[CrossRef]

2003 (1)

2002 (1)

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, "Discrete solitons in photorefractive optically induced photonic lattices," Phys. Rev. E 66, 046602 (2002).
[CrossRef]

2001 (1)

A. A. Sukhorukov and Y. S. Kivshar, "Nonlinear localized waves in a periodic medium," Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

1999 (2)

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Nonlinearly induced escape from a defect state in waveguide arrays," Appl. Phys. Lett. 75, 1348-1350 (1999).
[CrossRef]

J. Schöllmann, R. Scheibenzuber, A. S. Kovalev, A. P. Mayer, and A. A. Maradudin, "Stability of stationary gap solitary waves at periodically modulated surfaces," Phys. Rev. E 59, 4618-4629 (1999).
[CrossRef]

1996 (1)

1995 (2)

1978 (1)

P. Yeh, A. Yariv, and A. Y. Cho, "Optical surface waves in periodic layered media," Appl. Phys. Lett. 32, 104-105 (1978).
[CrossRef]

Aitchison, J. S.

R. Morandotti, H. S. Eisenberg, D. Dandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, and J. S. Aitchison, "Interactions of discrete solitons with structural defects," Opt. Lett. 28, 834-836 (2003).
[CrossRef] [PubMed]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Nonlinearly induced escape from a defect state in waveguide arrays," Appl. Phys. Lett. 75, 1348-1350 (1999).
[CrossRef]

Carretero-González, R.

M. A. Porter, P. G. Kevrekidis, R. Carretero-González, D. J. Frantzeskakis, "Dynamics and manipulation of matter-wave solitons in optical superlattices," Phys. Lett. A 352, 210-215 (2006).
[CrossRef]

Carvalho, M. I.

Champneys, A. R.

K. Yagasakai, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, "Gap solitons in Bragg gratings with a harmonic superlattice," Europhys. Lett. 74, 1006-1012 (2006).
[CrossRef]

Chen, W. H.

Chen, Z.

J. Yang and Z. Chen, "Defect solitons in photonic lattices," Phys. Rev. E 73, 026609 (2006).
[CrossRef]

F. Fedele, J. Yang, and Z. Chen, "Defect modes in one-dimensional photonic lattices," Opt. Lett. 30, 1506-1508 (2005).
[CrossRef] [PubMed]

F. Fedele, J. Yang, and Z. Chen, "Properties of defect modes in one-dimensional optically induced photonic lattices," Stud. Appl. Math. 115, 277-299 (2005).
[CrossRef]

Cho, A. Y.

P. Yeh, A. Yariv, and A. Y. Cho, "Optical surface waves in periodic layered media," Appl. Phys. Lett. 32, 104-105 (1978).
[CrossRef]

Christodoulides, D. N.

Cronin-Golomb, M.

Dandelik, D.

Efremidis, N. K.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, "Discrete solitons in photorefractive optically induced photonic lattices," Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Egorov, A. A.

Eisenberg, H. S.

R. Morandotti, H. S. Eisenberg, D. Dandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, and J. S. Aitchison, "Interactions of discrete solitons with structural defects," Opt. Lett. 28, 834-836 (2003).
[CrossRef] [PubMed]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Nonlinearly induced escape from a defect state in waveguide arrays," Appl. Phys. Lett. 75, 1348-1350 (1999).
[CrossRef]

Fedele, F.

F. Fedele, J. Yang, and Z. Chen, "Properties of defect modes in one-dimensional optically induced photonic lattices," Stud. Appl. Math. 115, 277-299 (2005).
[CrossRef]

F. Fedele, J. Yang, and Z. Chen, "Defect modes in one-dimensional photonic lattices," Opt. Lett. 30, 1506-1508 (2005).
[CrossRef] [PubMed]

Fleischer, J. W.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, "Discrete solitons in photorefractive optically induced photonic lattices," Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Frantzeskakis, D. J.

M. A. Porter, P. G. Kevrekidis, R. Carretero-González, D. J. Frantzeskakis, "Dynamics and manipulation of matter-wave solitons in optical superlattices," Phys. Lett. A 352, 210-215 (2006).
[CrossRef]

Garanovich, I. L.

Hache, A.

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, "Observation of discrete surface solitons," Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef] [PubMed]

He, Y. J.

Hudock, J.

Iwanow, R.

Kartashov, Y. V.

Kevrekidis, P. G.

M. A. Porter, P. G. Kevrekidis, R. Carretero-González, D. J. Frantzeskakis, "Dynamics and manipulation of matter-wave solitons in optical superlattices," Phys. Lett. A 352, 210-215 (2006).
[CrossRef]

Kip, D.

Kivshar, Y. S.

C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, "Observation of surface gap solitons in semi-infinite waveguide arrays," Phys. Rev. Lett. 97, 083901 (2006).
[CrossRef] [PubMed]

M. I. Molina, I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, "Discrete surface solitons in semi-infinite binary waveguide arrays," Opt. Lett. 31, 2332-3334 (2006).
[CrossRef] [PubMed]

I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, "Surface multi-gap vector solitons," Opt. Express 14, 4780-4785 (2006).
[CrossRef] [PubMed]

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, "Dispersion control for matter waves and gap solitons in optical superlattices," Phys. Rev. A 71, 023612 (2005).
[CrossRef]

B. A. Malomed, T. Mayteevarunyoo, E. A. Ostrovskaya, and Y. S. Kivshar, "Coupled-mode theory for spatial gap solitons in optically induced lattices," Phys. Rev. E 71, 056616 (2005).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, "Nonlinear localized waves in a periodic medium," Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

Kovalev, A. S.

J. Schöllmann, R. Scheibenzuber, A. S. Kovalev, A. P. Mayer, and A. A. Maradudin, "Stability of stationary gap solitary waves at periodically modulated surfaces," Phys. Rev. E 59, 4618-4629 (1999).
[CrossRef]

Krolikowski, W.

C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, "Observation of surface gap solitons in semi-infinite waveguide arrays," Phys. Rev. Lett. 97, 083901 (2006).
[CrossRef] [PubMed]

Louis, P. J. Y.

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, "Dispersion control for matter waves and gap solitons in optical superlattices," Phys. Rev. A 71, 023612 (2005).
[CrossRef]

Makris, K. G.

Malomed, B. A.

Y. J. He, W. H. Chen, H. Z. Wang, and B. A. Malomed, "Surface superlattices gap solitons," Opt. Lett. 32, 1390-1892 (2007).
[CrossRef] [PubMed]

H. Sakaguchi and B. A. Malomed, "Gap solitons in quasiperiodic optical lattices," Phys. Rev. E 74, 026601 (2006).
[CrossRef]

K. Yagasakai, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, "Gap solitons in Bragg gratings with a harmonic superlattice," Europhys. Lett. 74, 1006-1012 (2006).
[CrossRef]

B. A. Malomed, T. Mayteevarunyoo, E. A. Ostrovskaya, and Y. S. Kivshar, "Coupled-mode theory for spatial gap solitons in optically induced lattices," Phys. Rev. E 71, 056616 (2005).
[CrossRef]

Maradudin, A. A.

J. Schöllmann, R. Scheibenzuber, A. S. Kovalev, A. P. Mayer, and A. A. Maradudin, "Stability of stationary gap solitary waves at periodically modulated surfaces," Phys. Rev. E 59, 4618-4629 (1999).
[CrossRef]

Mayer, A. P.

J. Schöllmann, R. Scheibenzuber, A. S. Kovalev, A. P. Mayer, and A. A. Maradudin, "Stability of stationary gap solitary waves at periodically modulated surfaces," Phys. Rev. E 59, 4618-4629 (1999).
[CrossRef]

Mayteevarunyoo, T.

B. A. Malomed, T. Mayteevarunyoo, E. A. Ostrovskaya, and Y. S. Kivshar, "Coupled-mode theory for spatial gap solitons in optically induced lattices," Phys. Rev. E 71, 056616 (2005).
[CrossRef]

Merhasin, I. M.

K. Yagasakai, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, "Gap solitons in Bragg gratings with a harmonic superlattice," Europhys. Lett. 74, 1006-1012 (2006).
[CrossRef]

Mihalache, D.

Mitchell, A.

C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, "Observation of surface gap solitons in semi-infinite waveguide arrays," Phys. Rev. Lett. 97, 083901 (2006).
[CrossRef] [PubMed]

Modotto, D.

Molina, M. I.

C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, "Observation of surface gap solitons in semi-infinite waveguide arrays," Phys. Rev. Lett. 97, 083901 (2006).
[CrossRef] [PubMed]

M. I. Molina, "Nonlinear surface impurity in a semi-infinite two-dimensional square lattice: Green function approach," Phys. Rev. B 74, 045412 (2006).
[CrossRef]

M. I. Molina, I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, "Discrete surface solitons in semi-infinite binary waveguide arrays," Opt. Lett. 31, 2332-3334 (2006).
[CrossRef] [PubMed]

Morandotti, R.

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, "Observation of discrete surface solitons," Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef] [PubMed]

R. Morandotti, H. S. Eisenberg, D. Dandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, and J. S. Aitchison, "Interactions of discrete solitons with structural defects," Opt. Lett. 28, 834-836 (2003).
[CrossRef] [PubMed]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Nonlinearly induced escape from a defect state in waveguide arrays," Appl. Phys. Lett. 75, 1348-1350 (1999).
[CrossRef]

Neshev, D. N.

C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, "Observation of surface gap solitons in semi-infinite waveguide arrays," Phys. Rev. Lett. 97, 083901 (2006).
[CrossRef] [PubMed]

Ostrovskaya, E. A.

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, "Dispersion control for matter waves and gap solitons in optical superlattices," Phys. Rev. A 71, 023612 (2005).
[CrossRef]

B. A. Malomed, T. Mayteevarunyoo, E. A. Ostrovskaya, and Y. S. Kivshar, "Coupled-mode theory for spatial gap solitons in optically induced lattices," Phys. Rev. E 71, 056616 (2005).
[CrossRef]

Peschel, U.

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Nonlinearly induced escape from a defect state in waveguide arrays," Appl. Phys. Lett. 75, 1348-1350 (1999).
[CrossRef]

Porter, M. A.

M. A. Porter, P. G. Kevrekidis, R. Carretero-González, D. J. Frantzeskakis, "Dynamics and manipulation of matter-wave solitons in optical superlattices," Phys. Lett. A 352, 210-215 (2006).
[CrossRef]

Rosberg, C. R.

C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, "Observation of surface gap solitons in semi-infinite waveguide arrays," Phys. Rev. Lett. 97, 083901 (2006).
[CrossRef] [PubMed]

Rüter, C. E.

Sakaguchi, H.

H. Sakaguchi and B. A. Malomed, "Gap solitons in quasiperiodic optical lattices," Phys. Rev. E 74, 026601 (2006).
[CrossRef]

Salamo, G.

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, "Observation of discrete surface solitons," Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef] [PubMed]

Scheibenzuber, R.

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[CrossRef]

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N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, "Discrete solitons in photorefractive optically induced photonic lattices," Phys. Rev. E 66, 046602 (2002).
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Appl. Phys. Lett. (2)

P. Yeh, A. Yariv, and A. Y. Cho, "Optical surface waves in periodic layered media," Appl. Phys. Lett. 32, 104-105 (1978).
[CrossRef]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Nonlinearly induced escape from a defect state in waveguide arrays," Appl. Phys. Lett. 75, 1348-1350 (1999).
[CrossRef]

Europhys. Lett. (1)

K. Yagasakai, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, "Gap solitons in Bragg gratings with a harmonic superlattice," Europhys. Lett. 74, 1006-1012 (2006).
[CrossRef]

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

Opt. Express (6)

Opt. Lett. (8)

Phys. Lett. A (1)

M. A. Porter, P. G. Kevrekidis, R. Carretero-González, D. J. Frantzeskakis, "Dynamics and manipulation of matter-wave solitons in optical superlattices," Phys. Lett. A 352, 210-215 (2006).
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Phys. Rev. A (1)

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, "Dispersion control for matter waves and gap solitons in optical superlattices," Phys. Rev. A 71, 023612 (2005).
[CrossRef]

Phys. Rev. B (1)

M. I. Molina, "Nonlinear surface impurity in a semi-infinite two-dimensional square lattice: Green function approach," Phys. Rev. B 74, 045412 (2006).
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Phys. Rev. E (5)

J. Schöllmann, R. Scheibenzuber, A. S. Kovalev, A. P. Mayer, and A. A. Maradudin, "Stability of stationary gap solitary waves at periodically modulated surfaces," Phys. Rev. E 59, 4618-4629 (1999).
[CrossRef]

J. Yang and Z. Chen, "Defect solitons in photonic lattices," Phys. Rev. E 73, 026609 (2006).
[CrossRef]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, "Discrete solitons in photorefractive optically induced photonic lattices," Phys. Rev. E 66, 046602 (2002).
[CrossRef]

B. A. Malomed, T. Mayteevarunyoo, E. A. Ostrovskaya, and Y. S. Kivshar, "Coupled-mode theory for spatial gap solitons in optically induced lattices," Phys. Rev. E 71, 056616 (2005).
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H. Sakaguchi and B. A. Malomed, "Gap solitons in quasiperiodic optical lattices," Phys. Rev. E 74, 026601 (2006).
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Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Surface gap solitons," Phys. Rev. Lett. 96, 073901 (2006).
[CrossRef] [PubMed]

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, "Observation of discrete surface solitons," Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef] [PubMed]

C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, "Observation of surface gap solitons in semi-infinite waveguide arrays," Phys. Rev. Lett. 97, 083901 (2006).
[CrossRef] [PubMed]

Stud. Appl. Math. (1)

F. Fedele, J. Yang, and Z. Chen, "Properties of defect modes in one-dimensional optically induced photonic lattices," Stud. Appl. Math. 115, 277-299 (2005).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Lattice intensity profile with I 0 = 3 and (a) ϵ 1 = 0.3 : (b) ϵ 2 = 0.4 , (c) ϵ 2 = 0.4 and (d) ϵ 2 = 0 . Applied dc field parameter E 0 versus the propagation constant μ; the gray regions are Bloch bands.

Fig. 2
Fig. 2

(a) Power versus the propagation constant (black regions are Bloch bands) in the semi-infinite bandgap for ϵ 2 = 0.4 ; the solid curve is stable and the dashed curve is unstable. (b) Real part of perturbation growth rate versus the propagation constant. (c) Stable SDSSs at point A ( μ = 1.8 ) (d) unstable SDSSs at point B ( μ = 1.6 ) . (e) and (f) SDSSs propagation corresponding to (c) and (d).

Fig. 3
Fig. 3

(a) The power versus the propagation constant (black regions are Bloch bands) in the semi-infinite bandgap for ϵ 2 = 0 ; the solid curve (except near point B) is stable and the dashed curve is unstable. (b) Real part of perturbation growth rate versus the propagation constant. (c) Stable SDSSs at point C ( μ = 2.0 ) , (e) at point A ( μ = 2.5 ) ; (d) unstable SDSSs at point B ( μ = 2.45 ) , and (f) SDSSs propagation for (d).

Fig. 4
Fig. 4

(a) The power versus the propagation constant (black regions are Bloch bands) in the semi-infinite and first finite bandgap (E; μ = 3.31 ; D; μ = 3.75 ) for ϵ 2 = 0.4 . In the semi-infinite bandgap, the solid curve (except near point B) is stable and the dashed curve is unstable. (b) Real part of perturbation growth rate versus the propagation constant. (c) Stable SDSSs at point C ( μ = 2.1 ) , (e) at point A ( μ = 2.5 ) ; (d) unstable SDSSs at point B ( μ = 2.35 ) , and (f) SDSSs propagation for (d).

Fig. 5
Fig. 5

In the first finite bandgap, stable SDSSs at (a) μ = 3.31 and (b) μ = 3.75 , corresponding to the points E and D in Fig. 4, respectively. (c) and (d) correspond to (a) and (b).

Fig. 6
Fig. 6

Stable (gray) and unstable (white) domains ( ϵ 2 , μ ) of SDSSs in the semi-infinite bandgap (a) and (b), and the first finite bandgap (c).

Equations (7)

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i q z + 2 q x 2 E 0 1 + I L ( x ) + q 2 q = 0 ,
I L ( x ) = { I 0 { ϵ 1 sin 2 [ Ω 1 ( x + π 2 ) ] + ( 1 ϵ 1 ) sin 2 [ Ω 2 ( x + π 2 ) ] } , x π 2 I 0 ( 0.85 + ϵ 2 ) sin 2 [ Ω 1 ( x + π 2 ) ] , π 2 x π 2 0 , x π 2 } .
μ f = d 2 f d x 2 2 i k d f d x + k 2 f + E 0 1 + I L ( x ) f ,
2 f x 2 E 0 1 + I L ( x ) + f 2 f + μ f = 0 .
P = + f 2 ( x ) d x .
δ h = 2 e x 2 μ e + E 0 e ( 1 + I L + f 2 ) ,
δ e = 2 h x 2 + μ h E 0 h [ 1 ( 1 + I L + f 2 ) 2 f 2 ( 1 + I L + f 2 ) 2 ] .

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