Abstract

We analyze localization of light at the interfaces separating square and hexagonal photonic lattices, as recently realized experimentally for two-dimensional laser-written waveguides in silica glass with self-focusing nonlinearity [Opt. Lett. 33, 663 (2008) ]. We find the conditions for the existence of linear and nonlinear surface states substantially influenced by the lattice topology, and study the effect of different symmetries and couplings on the stability of two-dimensional interface solitons.

© 2008 Optical Society of America

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  1. K. G. Makris, J. Hudock, D. N. Christodoulides, G. Stegeman, O. Manela, and M. Segev, Opt. Lett. 31, 2774 (2006).
    [CrossRef] [PubMed]
  2. R. A. Vicencio, S. Flach, M. I. Molina, and Yu. S. Kivshar, Phys. Lett. A 364, 274 (2007).
    [CrossRef]
  3. H. Susanto, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Franzeskakis, Phys. Rev. E 75, 056605 (2007).
    [CrossRef]
  4. X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, Phys. Rev. Lett. 98, 123903 (2007).
    [CrossRef] [PubMed]
  5. A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and L. Torner, Phys. Rev. Lett. 98, 173903 (2007).
    [CrossRef]
  6. A. Szameit, Y. V. Kartashov, V. A. Vysloukh, M. Heinrich, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, and L. Torner, Opt. Lett. 33, 1542 (2008).
    [CrossRef] [PubMed]
  7. K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Haché, Opt. Lett. 30, 2466 (2005).
    [CrossRef] [PubMed]
  8. S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Haché, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, Phys. Rev. Lett. 96, 063901 (2006).
    [CrossRef] [PubMed]
  9. M. Molina, R. Vicencio, and Yu. S. Kivshar, Opt. Lett. 31, 1693 (2006).
    [CrossRef] [PubMed]
  10. A. Szameit, Y. V. Kartashov, F. Dreisow, M. Heinrich, V. A. Vysloukh, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, and L. Torner, Opt. Lett. 33, 663 (2008).
    [CrossRef] [PubMed]
  11. See, for instance, M. I. Molina, Y. V. Kartashov, L. Torner and Y. S. Kivshar, Phys. Rev. A 77, 053813 (2008).
    [CrossRef]
  12. A detailed analysis of twisted modes at the interface of two dissimilar square saturable nonlinear lattices, can be found in Y. V. Kartashov and L. Torner, Opt. Lett. 31, 2172 (2005).
    [CrossRef]

2008

2007

R. A. Vicencio, S. Flach, M. I. Molina, and Yu. S. Kivshar, Phys. Lett. A 364, 274 (2007).
[CrossRef]

H. Susanto, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Franzeskakis, Phys. Rev. E 75, 056605 (2007).
[CrossRef]

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, Phys. Rev. Lett. 98, 123903 (2007).
[CrossRef] [PubMed]

A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and L. Torner, Phys. Rev. Lett. 98, 173903 (2007).
[CrossRef]

2006

2005

Bezryadina, A.

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, Phys. Rev. Lett. 98, 123903 (2007).
[CrossRef] [PubMed]

Carretero-González, R.

H. Susanto, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Franzeskakis, Phys. Rev. E 75, 056605 (2007).
[CrossRef]

Chen, Z.

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, Phys. Rev. Lett. 98, 123903 (2007).
[CrossRef] [PubMed]

Christodoulides, D. N.

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, Phys. Rev. Lett. 98, 123903 (2007).
[CrossRef] [PubMed]

K. G. Makris, J. Hudock, D. N. Christodoulides, G. Stegeman, O. Manela, and M. Segev, Opt. Lett. 31, 2774 (2006).
[CrossRef] [PubMed]

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

K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Haché, Opt. Lett. 30, 2466 (2005).
[CrossRef] [PubMed]

Dreisow, F.

Flach, S.

R. A. Vicencio, S. Flach, M. I. Molina, and Yu. S. Kivshar, Phys. Lett. A 364, 274 (2007).
[CrossRef]

Franzeskakis, D. J.

H. Susanto, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Franzeskakis, Phys. Rev. E 75, 056605 (2007).
[CrossRef]

Haché, A.

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

K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Haché, Opt. Lett. 30, 2466 (2005).
[CrossRef] [PubMed]

Heinrich, M.

Hudock, J.

Kartashov, Y. V.

Kevrekidis, P. G.

H. Susanto, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Franzeskakis, Phys. Rev. E 75, 056605 (2007).
[CrossRef]

Kivshar, S.

Kivshar, Y. S.

See, for instance, M. I. Molina, Y. V. Kartashov, L. Torner and Y. S. Kivshar, Phys. Rev. A 77, 053813 (2008).
[CrossRef]

Kivshar, Yu. S.

R. A. Vicencio, S. Flach, M. I. Molina, and Yu. S. Kivshar, Phys. Lett. A 364, 274 (2007).
[CrossRef]

Lederer, F.

Makris, K. G.

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, Phys. Rev. Lett. 98, 123903 (2007).
[CrossRef] [PubMed]

K. G. Makris, J. Hudock, D. N. Christodoulides, G. Stegeman, O. Manela, and M. Segev, Opt. Lett. 31, 2774 (2006).
[CrossRef] [PubMed]

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

K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Haché, Opt. Lett. 30, 2466 (2005).
[CrossRef] [PubMed]

Malomed, B. A.

H. Susanto, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Franzeskakis, Phys. Rev. E 75, 056605 (2007).
[CrossRef]

Manela, O.

Molina, M.

Molina, M. I.

See, for instance, M. I. Molina, Y. V. Kartashov, L. Torner and Y. S. Kivshar, Phys. Rev. A 77, 053813 (2008).
[CrossRef]

R. A. Vicencio, S. Flach, M. I. Molina, and Yu. S. Kivshar, Phys. Lett. A 364, 274 (2007).
[CrossRef]

Morandotti, R.

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

Nolte, S.

Pertsch, T.

Salamo, G.

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

Segev, M.

Sorel, M.

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

Stegeman, G.

Stegeman, G. I.

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, Phys. Rev. Lett. 98, 123903 (2007).
[CrossRef] [PubMed]

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

K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Haché, Opt. Lett. 30, 2466 (2005).
[CrossRef] [PubMed]

Suntsov, S.

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

K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Haché, Opt. Lett. 30, 2466 (2005).
[CrossRef] [PubMed]

Susanto, H.

H. Susanto, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Franzeskakis, Phys. Rev. E 75, 056605 (2007).
[CrossRef]

Szameit, A.

Torner, L.

Tünnermann, A.

Vicencio, R.

Vicencio, R. A.

R. A. Vicencio, S. Flach, M. I. Molina, and Yu. S. Kivshar, Phys. Lett. A 364, 274 (2007).
[CrossRef]

Vysloukh, V. A.

Wang, X.

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, Phys. Rev. Lett. 98, 123903 (2007).
[CrossRef] [PubMed]

Yang, H.

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

Opt. Lett.

Phys. Lett. A

R. A. Vicencio, S. Flach, M. I. Molina, and Yu. S. Kivshar, Phys. Lett. A 364, 274 (2007).
[CrossRef]

Phys. Rev. A

See, for instance, M. I. Molina, Y. V. Kartashov, L. Torner and Y. S. Kivshar, Phys. Rev. A 77, 053813 (2008).
[CrossRef]

Phys. Rev. E

H. Susanto, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Franzeskakis, Phys. Rev. E 75, 056605 (2007).
[CrossRef]

Phys. Rev. Lett.

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, Phys. Rev. Lett. 98, 123903 (2007).
[CrossRef] [PubMed]

A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and L. Torner, Phys. Rev. Lett. 98, 173903 (2007).
[CrossRef]

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

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Figures (4)

Fig. 1
Fig. 1

Top: transverse view of a square–hexagonal photonic lattice. Bottom: examples of a low-order nonlinear interface mode centered (left) on the site belonging to the boundary of the square lattice (which ends at x = 1 ) and (right) on the site belonging to the boundary of the hexagonal lattice (which starts at x = 0 ).

Fig. 2
Fig. 2

Linear interface localized modes. (a) Existence diagram with the shaded area for the modes centered at the interface. Right: example of a linear localized mode for V b V s = 2 , V h V s = 0.5 . The square lattice ends at x = 1 while the triangular lattice starts at x = 0 .

Fig. 3
Fig. 3

Nonlinear case ( γ > 0 ) . Top left, minimum power to create a localized mode at the boundary of the square lattice, as a function of the couplings. Top right, example of an interface localized mode centered on a site belonging to the boundary of the square lattice ( V b V s = 1 = V h V s , β = 5.9 ) . Bottom left, power versus propagation constant for interface modes localized at the boundary of the square lattice; V b V s = 2 , V h V s = 0.5 (curve with Power min = 0 ) and V b V s = 1 = V h / V s (curve with Power min 5.8 ). The dashed portion of the curve denotes an unstable regime. Bottom right, example of a higher-order interface mode.

Fig. 4
Fig. 4

Top, average power fraction remaining at the square–triangular interface waveguide, after some propagation distance z max = 20 V s , as a function of the input power (solid curve). The dotted (dashed) curves refer to the cases of fully square (hexagonal) lattices. Left, V b V s = 1 = V h V s . Right, V b V s = 2 , V h V s = 0.5 . Bottom, dynamical excitation of an interface localized mode. Left, localized mode at the boundary of the square lattice. Right, localized mode at the boundary of the hexagonal lattice ( V b V s = 1 , V h V s = 2 3 , input power: 6.76). The hexagonal lattice starts at x = 0 .

Equations (2)

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i d E n d z + V n 1 , n 2 E m + γ E n 2 E n = 0 ,
β E n + V n 1 , n 2 E m + γ E n 2 E n = 0 .

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