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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  31. 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]
  32. 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]
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    [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]
  35. Y. V. Kartashov, and L. Torner, "Multipole-mode surface solitons," Opt. Lett. 31, 2172-2174 (2006).
    [CrossRef] [PubMed]
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    [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]
  38. Y. V. Kartashov, F. Ye, and L. Torner, "Vector mixed-gap surface solitons," Opt. Express 14, 4808-4814 (2006).
    [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]
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    [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

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]

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]

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]

2006

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

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]

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

Y. V. Kartashov, F. Ye, and L. Torner, "Vector mixed-gap surface solitons," Opt. Express 14, 4808-4814 (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, and L. Torner, "Multipole-mode surface solitons," Opt. Lett. 31, 2172-2174 (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, 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]

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]

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]

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]

K. Motzek, A. A. Sukhorukov, and Y. S. Kivshar, "Polychromatic interface solitons in nonlinear photonic lattices," Opt. Lett. 31, 3125-3127 (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]

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]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Surface lattice kink solitons," Opt. Express 14, 12365-12372 (2006).
[CrossRef] [PubMed]

2005

D. Artigas and L. Torner, "Dyakonov Surface Waves in Photonic Metamaterials," Phys. Rev. Lett. 94, 013901 (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]

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]

2003

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]

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]

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

1999

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

1989

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

1988

1987

1985

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. AkhmedievV. I. Korneev, and Y. V. Kuzmenko, "Excitation of nonlinear surface waves by Gaussian light beams," Sov. Phys. JETP 61, 62-67 (1985).

1983

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

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

1980

1978

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

1966

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

1939

W. Shockley, "On the Surface States Associated with a Periodic Potential," Phys. Rev. 56, 317-323 (1939).
[CrossRef]

1932

I. Tamm, Phys. Z. Sowjetunion 1, 733-746 (1932).

1930

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

Appl. Opt.

Appl. Phys. Lett.

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]

IEEE J. Quantum Electron.

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).
[CrossRef]

J. Opt. Soc. Am.

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

Nature

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

Opt. Commun.

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

Opt. Express

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, F. Ye, and L. Torner, "Vector mixed-gap surface solitons," Opt. Express 14, 4808-4814 (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]

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]

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]

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]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Surface lattice kink solitons," Opt. Express 14, 12365-12372 (2006).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, F. Lederer, and Y. S. Kivshar, "Stable discrete surface light bullets," Opt. Express 15, 589-595 (2007).
[CrossRef] [PubMed]

Opt. Lett.

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]

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]

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]

Y. V. Kartashov, and L. Torner, "Multipole-mode surface solitons," Opt. Lett. 31, 2172-2174 (2006).
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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)

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

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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