Abstract

A novel phase-space method is employed for the construction of analytical stationary solitary waves located at the interface between a periodic nonlinear lattice of the Kronig-Penney type and a linear or nonlinear homogeneous medium as well as at the interface between two dissimilar nonlinear lattices. The method provides physical insight and understanding of the shape of the solitary wave profile and results to generic classes of localized solutions having either zero or nonzero semi-infinite backgrounds. For all cases, the method provides conditions involving the values of the propagation constant of the stationary solutions, the linear refractive index and the dimensions of each part in order to assure existence of solutions with specific profile characteristics. The evolution of the analytical solutions under propagation is investigated for cases of realistic configurations and interesting features are presented such as their remarkable robustness which could facilitate their experimental observation.

© 2007 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  33. M. I. Molina and Y. S. Kivshar, “Interface localized modes and hybrid lattice solitons in waveguide arrays,” Phys. Let. A 362, 280–282 (2007).
    [Crossref]
  34. Y. V. Kartashov, V. A. Vysloukh, D. Mihalache, and L. Torner, “Generation of surface soliton arrays,” Opt. Lett. 31, 2329–2331 (2006).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  36. 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, 123903 (2007).
    [Crossref] [PubMed]
  37. A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tunnermann, and L. Torner, “Observation of two-dimensional surface solitons in asymmetric waveguide arrays,” Phys. Rev. Lett. 98, 173903 (2007).
    [Crossref]
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    [Crossref] [PubMed]
  39. I. L. Garanovich, A. A. Sukhorukov, Y. S. Kivshar, and M. Molina, “Surface multi-gap vector solitons,” Opt. Express 14, 4780–4785 (2006).
    [Crossref] [PubMed]
  40. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface lattice kink solitons,” Opt. Express 14, 12365–12372 (2006).
    [Crossref] [PubMed]
  41. Y. V. Kartashov, V. A. Vysloukh, A. A. Egorov, and L. Torner, “Surface vortex solitons,” Opt. Express 14, 4049–4057 (2006).
    [Crossref] [PubMed]
  42. K. Motzek, A. A. Sukhorukov, and Y. S. Kivshar, “Self-trapping of polychromatic light in nonlinear periodic photonic structures,” Opt. Express 14, 9873–9878 (2006).
    [Crossref] [PubMed]
  43. K. Motzek, A. A. Sukhorukov, and Y. S. Kivshar, “Polychromatic interface solitons in nonlinear photonic lattices,” Opt. Lett. 31, 3125–3127 (2006).
    [Crossref] [PubMed]
  44. A. A. Sukhorukov, D. N. Neshev, A. Dreischuh, R. Fischer, S. Ha, W. Krolikowski, J. Bolger, A. Mitchell, B. J. Eggleton, and Y. S. Kivshar, “Polychromatic nonlinear surface modes generated by supercontinuum light,” Opt. Express 14, 11265–11270 (2006).
    [Crossref] [PubMed]
  45. G. A. Siviloglou, K. G. Makris, R. Iwanow, R. Schiek, D. N. Christodoulides, and G. I. Stegeman, “Observation of discrete quadratic surface solitons,” Opt. Express 14, 5508–5516 (2006).
    [Crossref] [PubMed]
  46. Y. V. Kartashov, L. Torner, and V. A. Vysloukh, “Lattice-supported surface solitons in nonlocal nonlinear media,” Opt. Lett. 31, 2595–2597 (2006).
    [Crossref] [PubMed]
  47. Y. Kominis, “Analytical solitary wave solutions of the nonlinear Kronig-Penney model in photonic structures,” Phys. Rev. E 73, 066619 (2006).
    [Crossref]
  48. Y. Kominis and K. Hizanidis, “Lattice solitons in self-defocusing optical media: analytical solutions of the nonlinear KronigPenney model,” Opt. Lett. 31, 2888–2890 (2006).
    [Crossref] [PubMed]
  49. N. Akhmediev, A. Ankiewicz, and R. Grimshaw, “Hamiltonian-versus-energy diagrams in soliton theory,” Phys. Rev. E 59, 6088–6096 (1999).
    [Crossref]
  50. A. Ankiewicz and N. Akhmediev, “Stability analysis for solitons in planar waveguides, fibres and couplers using Hamiltonian concepts,” IEE Proc.-Optoelectron. 150, 519–526 (2003).
    [Crossref]

2007 (5)

D. Mihalache, D. Mazilu, F. Lederer, and Y. S. Kivshar, “Stable discrete surface light bullets,” Opt. Express 15, 589–595 (2007).
[Crossref] [PubMed]

M. Stepic, E. Smirnov, C. E. Ruter, D. Kip, A. Maluckov, and L. Hadzievski, “Tamm oscillations in semi-infinite nonlinear waveguide arrays,” Opt. Lett. 32, 823–825 (2007).
[Crossref]

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, 123903 (2007).
[Crossref] [PubMed]

A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tunnermann, and L. Torner, “Observation of two-dimensional surface solitons in asymmetric waveguide arrays,” Phys. Rev. Lett. 98, 173903 (2007).
[Crossref]

M. I. Molina and Y. S. Kivshar, “Interface localized modes and hybrid lattice solitons in waveguide arrays,” Phys. Let. A 362, 280–282 (2007).
[Crossref]

2006 (19)

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

Y. V. Kartashov and L. Torner, “Multipole-mode surface solitons,” Opt. Lett. 31, 2172–2174 (2006).
[Crossref] [PubMed]

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

I. L. Garanovich, A. A. Sukhorukov, Y. S. Kivshar, and M. Molina, “Surface multi-gap vector solitons,” Opt. Express 14, 4780–4785 (2006).
[Crossref] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface lattice kink solitons,” Opt. Express 14, 12365–12372 (2006).
[Crossref] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, A. A. Egorov, and L. Torner, “Surface vortex solitons,” Opt. Express 14, 4049–4057 (2006).
[Crossref] [PubMed]

K. Motzek, A. A. Sukhorukov, and Y. S. Kivshar, “Self-trapping of polychromatic light in nonlinear periodic photonic structures,” Opt. Express 14, 9873–9878 (2006).
[Crossref] [PubMed]

K. Motzek, A. A. Sukhorukov, and Y. S. Kivshar, “Polychromatic interface solitons in nonlinear photonic lattices,” Opt. Lett. 31, 3125–3127 (2006).
[Crossref] [PubMed]

A. A. Sukhorukov, D. N. Neshev, A. Dreischuh, R. Fischer, S. Ha, W. Krolikowski, J. Bolger, A. Mitchell, B. J. Eggleton, and Y. S. Kivshar, “Polychromatic nonlinear surface modes generated by supercontinuum light,” Opt. Express 14, 11265–11270 (2006).
[Crossref] [PubMed]

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

Y. V. Kartashov, L. Torner, and V. A. Vysloukh, “Lattice-supported surface solitons in nonlocal nonlinear media,” Opt. Lett. 31, 2595–2597 (2006).
[Crossref] [PubMed]

Y. Kominis, “Analytical solitary wave solutions of the nonlinear Kronig-Penney model in photonic structures,” Phys. Rev. E 73, 066619 (2006).
[Crossref]

Y. Kominis and K. Hizanidis, “Lattice solitons in self-defocusing optical media: analytical solutions of the nonlinear KronigPenney model,” Opt. Lett. 31, 2888–2890 (2006).
[Crossref] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface gap solitons,” Phys. Rev. Lett. 96, 073901 (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. Ruter, D. Kip, and V. Shandarov, “Observation of staggered surface solitary waves in one-dimensional waveguide arrays,” Opt. Lett. 31, 2338–2340 (2006).
[Crossref] [PubMed]

K. G. Makris, J. Hudock, D. N. Christodoulides, G. I. Stegeman, O. Manela, and M. Segev, “Surface lattice solitons,” Opt. Lett. 31, 2774–2776 (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, 2332–2334 (2006).
[Crossref] [PubMed]

M. I. Molina, R. A. Vicencio, and Y. S. Kivshar, “Discrete solitons and nonlinear surface modes in semi-infinite waveguide arrays,” Opt. Lett. 31, 1693–1695 (2006).
[Crossref] [PubMed]

2005 (4)

J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13, 1780–1796 (2005).
[Crossref] [PubMed]

K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache, “Discrete surface solitons,” Opt. Lett. 30, 2466–2468 (2005).
[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 (2005).
[Crossref]

D. Artigas and L. Torner, “Dyakonov Surface Waves in Photonic Metamaterials,” Phys. Rev. Lett. 94, 013901 (2005).
[Crossref] [PubMed]

2003 (3)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon sub-wavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003).
[Crossref]

A. Ankiewicz and N. Akhmediev, “Stability analysis for solitons in planar waveguides, fibres and couplers using Hamiltonian concepts,” IEE Proc.-Optoelectron. 150, 519–526 (2003).
[Crossref]

1999 (1)

N. Akhmediev, A. Ankiewicz, and R. Grimshaw, “Hamiltonian-versus-energy diagrams in soliton theory,” Phys. Rev. E 59, 6088–6096 (1999).
[Crossref]

1989 (1)

D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” Prog. Opt. 27, 229–313 (1989).

1988 (1)

1987 (1)

1985 (2)

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 774–783 (1985).
[Crossref]

N. N. Akhmediev, V. I. Korneev, and Y. V. Kuzmenko, “Excitation of nonlinear surface waves by Gaussian light beams,” Sov. Phys. JETP 61, 62–67 (1985).

1983 (1)

U. Langbein, F. Lederer, and H. E. Ponath, “A new type of non-linear slab-guided waves,” Opt. Commun. 46, 167–169 (1983).
[Crossref]

1982 (1)

V. K. Fedyanin and D. Mihalache, “P-polarized nonlinear surface polaritons in layered structures,” Z. Phys. B 47, 167–173 (1982).
[Crossref]

1980 (1)

1978 (2)

P. Yeh, A. Yariv, and A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104–105 (1978).
[Crossref]

W. Ng, P. Yeh, P. C. Chen, and A. Yariv, “Optical surface waves in periodic layered medium grown by liquid phase epitaxy,” Appl. Phys. Lett. 32, 370–371 (1978).
[Crossref]

1974 (1)

1966 (1)

D. Kossel, “Analogies between thin-film optics and electron. band theory of solids,” J. Opt. Soc. Am. 56, 1434 (1966).

1939 (1)

W. Shockley, “On the Surface States Associated with a Periodic Potential,” Phys. Rev. 56, 317–323 (1939).
[Crossref]

1930 (1)

R. de, L. Kronig, and W. G. Penney, “Quantum Mechanics of Electrons in Crystal Lattices,” Proc. R. Soc. London 130, 499–513 (1930).

Akhmediev, N.

A. Ankiewicz and N. Akhmediev, “Stability analysis for solitons in planar waveguides, fibres and couplers using Hamiltonian concepts,” IEE Proc.-Optoelectron. 150, 519–526 (2003).
[Crossref]

N. Akhmediev, A. Ankiewicz, and R. Grimshaw, “Hamiltonian-versus-energy diagrams in soliton theory,” Phys. Rev. E 59, 6088–6096 (1999).
[Crossref]

Akhmediev, N. N.

N. N. Akhmediev, V. I. Korneev, and Y. V. Kuzmenko, “Excitation of nonlinear surface waves by Gaussian light beams,” Sov. Phys. JETP 61, 62–67 (1985).

Ankiewicz, A.

A. Ankiewicz and N. Akhmediev, “Stability analysis for solitons in planar waveguides, fibres and couplers using Hamiltonian concepts,” IEE Proc.-Optoelectron. 150, 519–526 (2003).
[Crossref]

N. Akhmediev, A. Ankiewicz, and R. Grimshaw, “Hamiltonian-versus-energy diagrams in soliton theory,” Phys. Rev. E 59, 6088–6096 (1999).
[Crossref]

Arnaud, J. A.

Artigas, D.

D. Artigas and L. Torner, “Dyakonov Surface Waves in Photonic Metamaterials,” Phys. Rev. Lett. 94, 013901 (2005).
[Crossref] [PubMed]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon sub-wavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

Bartal, G.

Bertolotti, M.

D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” Prog. Opt. 27, 229–313 (1989).

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, 123903 (2007).
[Crossref] [PubMed]

Boardman, A. D.

D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, “Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface,” Opt. Lett. 12, 187–189 (1987).
[Crossref] [PubMed]

A. D. Boardman, P. Egan, F. Lederer, U. Langbein, and D. MihalacheH. E. Ponath and G. I. Stegeman, Nonlinear Surface Electromagnetic Phenomena, edited byNorth-Holland, Amsterdam, 29, 73 (1991).

Bolger, J.

Buljan, H.

Chen, P. C.

W. Ng, P. Yeh, P. C. Chen, and A. Yariv, “Optical surface waves in periodic layered medium grown by liquid phase epitaxy,” Appl. Phys. Lett. 32, 370–371 (1978).
[Crossref]

Chen, Z.

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, 123903 (2007).
[Crossref] [PubMed]

Chilwell, J. T.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 774–783 (1985).
[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.

Cohen, O.

de, R.

R. de, L. Kronig, and W. G. Penney, “Quantum Mechanics of Electrons in Crystal Lattices,” Proc. R. Soc. London 130, 499–513 (1930).

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon sub-wavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

Dreischuh, A.

Dreisow, F.

A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tunnermann, and L. Torner, “Observation of two-dimensional surface solitons in asymmetric waveguide arrays,” Phys. Rev. Lett. 98, 173903 (2007).
[Crossref]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon sub-wavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

Efremidis, N. K.

Egan, P.

A. D. Boardman, P. Egan, F. Lederer, U. Langbein, and D. MihalacheH. E. Ponath and G. I. Stegeman, Nonlinear Surface Electromagnetic Phenomena, edited byNorth-Holland, Amsterdam, 29, 73 (1991).

Eggleton, B. J.

Egorov, A. A.

Eisenberg, H. S.

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003).
[Crossref]

Fedyanin, V. K.

V. K. Fedyanin and D. Mihalache, “P-polarized nonlinear surface polaritons in layered structures,” Z. Phys. B 47, 167–173 (1982).
[Crossref]

Fischer, R.

Fleischer, J. W.

Freedman, B.

Garanovich, I. L.

Grimshaw, R.

N. Akhmediev, A. Ankiewicz, and R. Grimshaw, “Hamiltonian-versus-energy diagrams in soliton theory,” Phys. Rev. E 59, 6088–6096 (1999).
[Crossref]

Ha, S.

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 (2005).
[Crossref]

K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache, “Discrete surface solitons,” Opt. Lett. 30, 2466–2468 (2005).
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Hizanidis, K.

Hudock, J.

Iwanow, R.

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Kartashov, Y. V.

Kip, D.

Kivshar, Y. S.

M. I. Molina and Y. S. Kivshar, “Interface localized modes and hybrid lattice solitons in waveguide arrays,” Phys. Let. A 362, 280–282 (2007).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, and Y. S. Kivshar, “Stable discrete surface light bullets,” Opt. Express 15, 589–595 (2007).
[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]

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, 2332–2334 (2006).
[Crossref] [PubMed]

M. I. Molina, R. A. Vicencio, and Y. S. Kivshar, “Discrete solitons and nonlinear surface modes in semi-infinite waveguide arrays,” Opt. Lett. 31, 1693–1695 (2006).
[Crossref] [PubMed]

I. L. Garanovich, A. A. Sukhorukov, Y. S. Kivshar, and M. Molina, “Surface multi-gap vector solitons,” Opt. Express 14, 4780–4785 (2006).
[Crossref] [PubMed]

K. Motzek, A. A. Sukhorukov, and Y. S. Kivshar, “Self-trapping of polychromatic light in nonlinear periodic photonic structures,” Opt. Express 14, 9873–9878 (2006).
[Crossref] [PubMed]

K. Motzek, A. A. Sukhorukov, and Y. S. Kivshar, “Polychromatic interface solitons in nonlinear photonic lattices,” Opt. Lett. 31, 3125–3127 (2006).
[Crossref] [PubMed]

A. A. Sukhorukov, D. N. Neshev, A. Dreischuh, R. Fischer, S. Ha, W. Krolikowski, J. Bolger, A. Mitchell, B. J. Eggleton, and Y. S. Kivshar, “Polychromatic nonlinear surface modes generated by supercontinuum light,” Opt. Express 14, 11265–11270 (2006).
[Crossref] [PubMed]

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003).
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N. N. Akhmediev, V. I. Korneev, and Y. V. Kuzmenko, “Excitation of nonlinear surface waves by Gaussian light beams,” Sov. Phys. JETP 61, 62–67 (1985).

Kossel, D.

D. Kossel, “Analogies between thin-film optics and electron. band theory of solids,” J. Opt. Soc. Am. 56, 1434 (1966).

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).
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A. A. Sukhorukov, D. N. Neshev, A. Dreischuh, R. Fischer, S. Ha, W. Krolikowski, J. Bolger, A. Mitchell, B. J. Eggleton, and Y. S. Kivshar, “Polychromatic nonlinear surface modes generated by supercontinuum light,” Opt. Express 14, 11265–11270 (2006).
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R. de, L. Kronig, and W. G. Penney, “Quantum Mechanics of Electrons in Crystal Lattices,” Proc. R. Soc. London 130, 499–513 (1930).

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N. N. Akhmediev, V. I. Korneev, and Y. V. Kuzmenko, “Excitation of nonlinear surface waves by Gaussian light beams,” Sov. Phys. JETP 61, 62–67 (1985).

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U. Langbein, F. Lederer, and H. E. Ponath, “A new type of non-linear slab-guided waves,” Opt. Commun. 46, 167–169 (1983).
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Lederer, F.

D. Mihalache, D. Mazilu, F. Lederer, and Y. S. Kivshar, “Stable discrete surface light bullets,” Opt. Express 15, 589–595 (2007).
[Crossref] [PubMed]

U. Langbein, F. Lederer, and H. E. Ponath, “A new type of non-linear slab-guided waves,” Opt. Commun. 46, 167–169 (1983).
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A. D. Boardman, P. Egan, F. Lederer, U. Langbein, and D. MihalacheH. E. Ponath and G. I. Stegeman, Nonlinear Surface Electromagnetic Phenomena, edited byNorth-Holland, Amsterdam, 29, 73 (1991).

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, 123903 (2007).
[Crossref] [PubMed]

K. G. Makris, J. Hudock, D. N. Christodoulides, G. I. Stegeman, O. Manela, and M. Segev, “Surface lattice solitons,” Opt. Lett. 31, 2774–2776 (2006).
[Crossref] [PubMed]

G. A. Siviloglou, K. G. Makris, R. Iwanow, R. Schiek, D. N. Christodoulides, and G. I. Stegeman, “Observation of discrete quadratic surface solitons,” Opt. Express 14, 5508–5516 (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 (2005).
[Crossref]

K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache, “Discrete surface solitons,” Opt. Lett. 30, 2466–2468 (2005).
[Crossref] [PubMed]

Maluckov, A.

Manela, O.

Mazilu, D.

Mihalache, D.

D. Mihalache, D. Mazilu, F. Lederer, and Y. S. Kivshar, “Stable discrete surface light bullets,” Opt. Express 15, 589–595 (2007).
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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]

A. A. Sukhorukov, D. N. Neshev, A. Dreischuh, R. Fischer, S. Ha, W. Krolikowski, J. Bolger, A. Mitchell, B. J. Eggleton, and Y. S. Kivshar, “Polychromatic nonlinear surface modes generated by supercontinuum light,” Opt. Express 14, 11265–11270 (2006).
[Crossref] [PubMed]

Molina, M.

Molina, M. I.

M. I. Molina and Y. S. Kivshar, “Interface localized modes and hybrid lattice solitons in waveguide arrays,” Phys. Let. A 362, 280–282 (2007).
[Crossref]

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, 2332–2334 (2006).
[Crossref] [PubMed]

M. I. Molina, R. A. Vicencio, and Y. S. Kivshar, “Discrete solitons and nonlinear surface modes in semi-infinite waveguide arrays,” Opt. Lett. 31, 1693–1695 (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 (2005).
[Crossref]

Motzek, K.

Neshev, D. N.

A. A. Sukhorukov, D. N. Neshev, A. Dreischuh, R. Fischer, S. Ha, W. Krolikowski, J. Bolger, A. Mitchell, B. J. Eggleton, and Y. S. Kivshar, “Polychromatic nonlinear surface modes generated by supercontinuum light,” Opt. Express 14, 11265–11270 (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).
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W. Ng, P. Yeh, P. C. Chen, and A. Yariv, “Optical surface waves in periodic layered medium grown by liquid phase epitaxy,” Appl. Phys. Lett. 32, 370–371 (1978).
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A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tunnermann, and L. Torner, “Observation of two-dimensional surface solitons in asymmetric waveguide arrays,” Phys. Rev. Lett. 98, 173903 (2007).
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A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tunnermann, and L. Torner, “Observation of two-dimensional surface solitons in asymmetric waveguide arrays,” Phys. Rev. Lett. 98, 173903 (2007).
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U. Langbein, F. Lederer, and H. E. Ponath, “A new type of non-linear slab-guided waves,” Opt. Commun. 46, 167–169 (1983).
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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).
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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 (2005).
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Schiek, R.

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D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, “Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface,” Opt. Lett. 12, 187–189 (1987).
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C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 774–783 (1985).
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D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” Prog. Opt. 27, 229–313 (1989).

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A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003).
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Smirnov, E.

Smith, S. D.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 774–783 (1985).
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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 (2005).
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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, 123903 (2007).
[Crossref] [PubMed]

K. G. Makris, J. Hudock, D. N. Christodoulides, G. I. Stegeman, O. Manela, and M. Segev, “Surface lattice solitons,” Opt. Lett. 31, 2774–2776 (2006).
[Crossref] [PubMed]

G. A. Siviloglou, K. G. Makris, R. Iwanow, R. Schiek, D. N. Christodoulides, and G. I. Stegeman, “Observation of discrete quadratic surface solitons,” Opt. Express 14, 5508–5516 (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 (2005).
[Crossref]

K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache, “Discrete surface solitons,” Opt. Lett. 30, 2466–2468 (2005).
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D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, “Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface,” Opt. Lett. 12, 187–189 (1987).
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Sukhorukov, A. A.

Suntsov, S.

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 (2005).
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K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache, “Discrete surface solitons,” Opt. Lett. 30, 2466–2468 (2005).
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A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tunnermann, and L. Torner, “Observation of two-dimensional surface solitons in asymmetric waveguide arrays,” Phys. Rev. Lett. 98, 173903 (2007).
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Valera, J. D.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 774–783 (1985).
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Vicencio, R. 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).
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M. I. Molina, R. A. Vicencio, and Y. S. Kivshar, “Discrete solitons and nonlinear surface modes in semi-infinite waveguide arrays,” Opt. Lett. 31, 1693–1695 (2006).
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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, 123903 (2007).
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Yang, H.

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 (2005).
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Yeh, P.

W. Ng, P. Yeh, P. C. Chen, and A. Yariv, “Optical surface waves in periodic layered medium grown by liquid phase epitaxy,” Appl. Phys. Lett. 32, 370–371 (1978).
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Appl. Opt. (1)

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).
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W. Ng, P. Yeh, P. C. Chen, and A. Yariv, “Optical surface waves in periodic layered medium grown by liquid phase epitaxy,” Appl. Phys. Lett. 32, 370–371 (1978).
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A. Ankiewicz and N. Akhmediev, “Stability analysis for solitons in planar waveguides, fibres and couplers using Hamiltonian concepts,” IEE Proc.-Optoelectron. 150, 519–526 (2003).
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IEEE J. Quantum Electron. (2)

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 774–783 (1985).
[Crossref]

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003).
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J. Opt. Soc. Am. (1)

D. Kossel, “Analogies between thin-film optics and electron. band theory of solids,” J. Opt. Soc. Am. 56, 1434 (1966).

Nature (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon sub-wavelength optics,” Nature 424, 824–830 (2003).
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Opt. Commun. (1)

U. Langbein, F. Lederer, and H. E. Ponath, “A new type of non-linear slab-guided waves,” Opt. Commun. 46, 167–169 (1983).
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Opt. Express (9)

J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13, 1780–1796 (2005).
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D. Mihalache, D. Mazilu, F. Lederer, and Y. S. Kivshar, “Stable discrete surface light bullets,” Opt. Express 15, 589–595 (2007).
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Y. V. Kartashov, F. Ye, and L. Torner, “Vector mixed-gap surface solitons,” Opt. Express 14, 4808–4814 (2006).
[Crossref] [PubMed]

I. L. Garanovich, A. A. Sukhorukov, Y. S. Kivshar, and M. Molina, “Surface multi-gap vector solitons,” Opt. Express 14, 4780–4785 (2006).
[Crossref] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface lattice kink solitons,” Opt. Express 14, 12365–12372 (2006).
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Y. V. Kartashov, V. A. Vysloukh, A. A. Egorov, and L. Torner, “Surface vortex solitons,” Opt. Express 14, 4049–4057 (2006).
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K. Motzek, A. A. Sukhorukov, and Y. S. Kivshar, “Self-trapping of polychromatic light in nonlinear periodic photonic structures,” Opt. Express 14, 9873–9878 (2006).
[Crossref] [PubMed]

A. A. Sukhorukov, D. N. Neshev, A. Dreischuh, R. Fischer, S. Ha, W. Krolikowski, J. Bolger, A. Mitchell, B. J. Eggleton, and Y. S. Kivshar, “Polychromatic nonlinear surface modes generated by supercontinuum light,” Opt. Express 14, 11265–11270 (2006).
[Crossref] [PubMed]

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

Opt. Lett. (14)

Y. V. Kartashov, L. Torner, and V. A. Vysloukh, “Lattice-supported surface solitons in nonlocal nonlinear media,” Opt. Lett. 31, 2595–2597 (2006).
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K. Motzek, A. A. Sukhorukov, and Y. S. Kivshar, “Polychromatic interface solitons in nonlinear photonic lattices,” Opt. Lett. 31, 3125–3127 (2006).
[Crossref] [PubMed]

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

K. G. Makris, J. Hudock, D. N. Christodoulides, G. I. Stegeman, O. Manela, and M. Segev, “Surface lattice solitons,” Opt. Lett. 31, 2774–2776 (2006).
[Crossref] [PubMed]

Y. Kominis and K. Hizanidis, “Lattice solitons in self-defocusing optical media: analytical solutions of the nonlinear KronigPenney model,” Opt. Lett. 31, 2888–2890 (2006).
[Crossref] [PubMed]

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Figures (10)

Fig. 1.
Fig. 1.

Transverse profile of the photonic structure consisting of a nonlinear lattice (parts D 1 and D 2) and a homogeneous linear (a) or nonlinear (b) homogeneous medium (part D 3) as well as two dissimilar nonlinear lattices having different widths of the corresponding nonlinear parts (c). Shaded areas denote nonlinear medium while nonshaded areas denote linear medium.

Fig. 2.
Fig. 2.

Phase space for each part of the structure: (a) nonlinear part (D 1) for β>ε 1, (b) linear part (D 2 or D 3) for β<εi (i=2 or 3) and, (c) linear part (D 2 or D 3) for β>εi (i=2 or 3).

Fig. 3.
Fig. 3.

Phase space representation of the constructed solutions for n even. (a) Nonlinear homogeneous part, (b) Linear homogeneous part having β<ε 3, and (c) Linear homogeneous part having β>ε 3. Dotted line denotes the solution in the lattice part and solid line denotes the solution in the homogeneous part.

Fig. 4.
Fig. 4.

Band structure of the linearized system (propagation constant β vs Bloch wave number q) for the two lattices having ε 1=0, ε 2=0.3, L=4π and N 1=2π (blue line), N 2=π (red line). Ti =L+Ni, i=1,2 is the period of each lattice. Circles depict the location of the analytically obtained localized modes for n=1, 2.

Fig. 5.
Fig. 5.

Profiles of stationary surface lattice solutions for the case of a nonlinear homogeneous medium having the same material characteristics with the nonlinear part of the lattice, for n=1,2 (top to bottom) and x 0=0,π,2π (left to right).

Fig. 6.
Fig. 6.

Propagation of the stationary solutions shown in Fig. 5.

Fig. 7.
Fig. 7.

Profiles of stationary surface lattice solutions for the case of a linear homogeneous medium having ε 3=0.1, for n=1,2 (top to bottom) and x 0=-π,0,π (left to right).

Fig. 8.
Fig. 8.

Propagation of the stationary solutions shown in Fig. 7.

Fig. 9.
Fig. 9.

Profiles of stationary surface lattice solutions for the case of the interface between two dissimilar nonlinear waveguide arrays, for n=1,2 (top to bottom), and x 0=0,-π,-3π/2 (left to right).

Fig. 10.
Fig. 10.

Propagation of the stationary solutions shown in Fig. 9.

Equations (6)

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i ψ z + 2 ψ x 2 + ε ( x ) ψ + g ( x ) ψ 2 ψ = 0
d 2 u d x 2 + [ ε ( x ) β ] u + g ( x ) u 3 = 0
( ε ( x ) , g ( x ) ) = { ( ε 1 , 2 ) , x D 1 ( ε 2 , 0 ) , x D 2 ( ε 3 , 0 ) , x D 3
β n = ε 2 ( n π L ) 2 , n = 1 , 2 ,
u ( x ; β n , x 0 ) = { v ( x ; β n , x 0 ) x D 1 a k sin ( ε 2 , 3 β n x + ϕ k ) x D 2 , D 3
x 0 = 1 β ε 1 sech 1 ( ε 3 ε 1 β ε 1 ) N 1 2

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