Abstract

We study two-color surface solitons in two-dimensional photonic lattices with quadratic nonlinear response. We demonstrate that such parametrically coupled optical localized modes can exist in the corners or at the edges of a square photonic lattice, and we analyze the impact of the phase mismatch on their properties, stability, and the threshold power for their generation.

© 2009 Optical Society of America

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  1. K. G. Makris, J. Hudock, D. N. Christodoulides, G. Stegeman, O. Manela, and M. Segev, “Surface lattice solitons,” Opt. Lett. 31, 2774-2776 (2006).
    [CrossRef] [PubMed]
  2. R. A. Vicencio, S. Flach, M. I. Molina, and Yu. S. Kivshar, “Discrete surface solitons in two-dimensional anisotropic lattices,” Phys. Lett. A 364, 274-276 (2007).
    [CrossRef]
  3. H. Susanto, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Franzeskakis, “Discrete surface solitons in two dimensions,” Phys. Rev. E 75, 056605 (2007).
    [CrossRef]
  4. 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]
  5. A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and L. Torner, “Observation of two-dimensional surface solitons in asymmetric waveguide arrays,” Phys. Rev. Lett. 98, 173903 (2007).
    [CrossRef]
  6. B. A. Malomed, P. G. Kevrekidis, D. J. Franzeskakis, H. E. Nistazakis, and A. N. Yannacopoulos, “One-and two-dimensional solitons in second-harmonic-generating lattices,” Phys. Rev. E 65, 056606 (2002).
    [CrossRef]
  7. Ya. V. Kartashov, L. Torner, and V. A. Vysloukh, “Multicolor lattice solitons,” Opt. Lett. 29, 1117-1119 (2004).
    [CrossRef] [PubMed]
  8. Z. Xu, Ya. V. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E 71, 016616 (2005).
    [CrossRef]
  9. H. Susano, P. G. Kevrekidis, R. Carretero-Gonzalez, B. A. Malomed, and D. J. Franzeskakis, “Mobility of discrete solitons in quadratically nonlinear media,” Phys. Rev. Lett. 99, 214103 (2007).
    [CrossRef]
  10. G. A. Siviloglou, K. G. Makris, R. Iwanow, R. Schiek, D. N. Christodoulides, G. I. Stegeman, Y. Ming, and W. Sohler, “Observation of discrete quadratic surface solitons,” Indian Chem. Eng. 14, 5508-5516 (2006).
  11. Z. Xu and Yu. S. Kivshar, “Two-color surface lattice solitons,” Opt. Lett. 33, 2551-2553 (2008).
    [CrossRef] [PubMed]
  12. D. Mihalache, D. Mazilu, F. Lederer, and Y. S. Kivshar, “Spatiotemporal surface solitons in two-dimensional photonic lattices,” Opt. Lett. 32, 3173-3175 (2007).
    [CrossRef] [PubMed]
  13. M. Molina, R. Vicencio, and Yu. S. Kivshar, “Discrete solitons and nonlinear surface modes in semi-infinite waveguide arrays,” Opt. Lett. 31, 1693-1695 (2006).
    [CrossRef] [PubMed]

2008

2007

D. Mihalache, D. Mazilu, F. Lederer, and Y. S. Kivshar, “Spatiotemporal surface solitons in two-dimensional photonic lattices,” Opt. Lett. 32, 3173-3175 (2007).
[CrossRef] [PubMed]

R. A. Vicencio, S. Flach, M. I. Molina, and Yu. S. Kivshar, “Discrete surface solitons in two-dimensional anisotropic lattices,” Phys. Lett. A 364, 274-276 (2007).
[CrossRef]

H. Susanto, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Franzeskakis, “Discrete surface solitons in two dimensions,” Phys. Rev. E 75, 056605 (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. Tünnermann, and L. Torner, “Observation of two-dimensional surface solitons in asymmetric waveguide arrays,” Phys. Rev. Lett. 98, 173903 (2007).
[CrossRef]

H. Susano, P. G. Kevrekidis, R. Carretero-Gonzalez, B. A. Malomed, and D. J. Franzeskakis, “Mobility of discrete solitons in quadratically nonlinear media,” Phys. Rev. Lett. 99, 214103 (2007).
[CrossRef]

2006

2005

Z. Xu, Ya. V. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E 71, 016616 (2005).
[CrossRef]

2004

2002

B. A. Malomed, P. G. Kevrekidis, D. J. Franzeskakis, H. E. Nistazakis, and A. N. Yannacopoulos, “One-and two-dimensional solitons in second-harmonic-generating lattices,” Phys. Rev. E 65, 056606 (2002).
[CrossRef]

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]

Carretero-Gonzalez, R.

H. Susano, P. G. Kevrekidis, R. Carretero-Gonzalez, B. A. Malomed, and D. J. Franzeskakis, “Mobility of discrete solitons in quadratically nonlinear media,” Phys. Rev. Lett. 99, 214103 (2007).
[CrossRef]

Carretero-González, R.

H. Susanto, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Franzeskakis, “Discrete surface solitons in two dimensions,” 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, “Observation of two-dimensional surface solitons,” 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, “Observation of two-dimensional surface solitons,” Phys. Rev. Lett. 98, 123903 (2007).
[CrossRef] [PubMed]

K. G. Makris, J. Hudock, D. N. Christodoulides, G. 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, G. I. Stegeman, Y. Ming, and W. Sohler, “Observation of discrete quadratic surface solitons,” Indian Chem. Eng. 14, 5508-5516 (2006).

Crasovan, L.-C.

Z. Xu, Ya. V. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E 71, 016616 (2005).
[CrossRef]

Dreisow, F.

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

Flach, S.

R. A. Vicencio, S. Flach, M. I. Molina, and Yu. S. Kivshar, “Discrete surface solitons in two-dimensional anisotropic lattices,” Phys. Lett. A 364, 274-276 (2007).
[CrossRef]

Franzeskakis, D. J.

H. Susano, P. G. Kevrekidis, R. Carretero-Gonzalez, B. A. Malomed, and D. J. Franzeskakis, “Mobility of discrete solitons in quadratically nonlinear media,” Phys. Rev. Lett. 99, 214103 (2007).
[CrossRef]

H. Susanto, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Franzeskakis, “Discrete surface solitons in two dimensions,” Phys. Rev. E 75, 056605 (2007).
[CrossRef]

B. A. Malomed, P. G. Kevrekidis, D. J. Franzeskakis, H. E. Nistazakis, and A. N. Yannacopoulos, “One-and two-dimensional solitons in second-harmonic-generating lattices,” Phys. Rev. E 65, 056606 (2002).
[CrossRef]

Hudock, J.

Iwanow, R.

G. A. Siviloglou, K. G. Makris, R. Iwanow, R. Schiek, D. N. Christodoulides, G. I. Stegeman, Y. Ming, and W. Sohler, “Observation of discrete quadratic surface solitons,” Indian Chem. Eng. 14, 5508-5516 (2006).

Kartashov, Y. V.

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

Kartashov, Ya. V.

Z. Xu, Ya. V. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E 71, 016616 (2005).
[CrossRef]

Ya. V. Kartashov, L. Torner, and V. A. Vysloukh, “Multicolor lattice solitons,” Opt. Lett. 29, 1117-1119 (2004).
[CrossRef] [PubMed]

Kevrekidis, P. G.

H. Susano, P. G. Kevrekidis, R. Carretero-Gonzalez, B. A. Malomed, and D. J. Franzeskakis, “Mobility of discrete solitons in quadratically nonlinear media,” Phys. Rev. Lett. 99, 214103 (2007).
[CrossRef]

H. Susanto, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Franzeskakis, “Discrete surface solitons in two dimensions,” Phys. Rev. E 75, 056605 (2007).
[CrossRef]

B. A. Malomed, P. G. Kevrekidis, D. J. Franzeskakis, H. E. Nistazakis, and A. N. Yannacopoulos, “One-and two-dimensional solitons in second-harmonic-generating lattices,” Phys. Rev. E 65, 056606 (2002).
[CrossRef]

Kivshar, Y. S.

Kivshar, Yu. S.

Lederer, F.

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. 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, G. I. Stegeman, Y. Ming, and W. Sohler, “Observation of discrete quadratic surface solitons,” Indian Chem. Eng. 14, 5508-5516 (2006).

Malomed, B. A.

H. Susanto, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Franzeskakis, “Discrete surface solitons in two dimensions,” Phys. Rev. E 75, 056605 (2007).
[CrossRef]

H. Susano, P. G. Kevrekidis, R. Carretero-Gonzalez, B. A. Malomed, and D. J. Franzeskakis, “Mobility of discrete solitons in quadratically nonlinear media,” Phys. Rev. Lett. 99, 214103 (2007).
[CrossRef]

B. A. Malomed, P. G. Kevrekidis, D. J. Franzeskakis, H. E. Nistazakis, and A. N. Yannacopoulos, “One-and two-dimensional solitons in second-harmonic-generating lattices,” Phys. Rev. E 65, 056606 (2002).
[CrossRef]

Manela, O.

Mazilu, D.

Mihalache, D.

D. Mihalache, D. Mazilu, F. Lederer, and Y. S. Kivshar, “Spatiotemporal surface solitons in two-dimensional photonic lattices,” Opt. Lett. 32, 3173-3175 (2007).
[CrossRef] [PubMed]

Z. Xu, Ya. V. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E 71, 016616 (2005).
[CrossRef]

Ming, Y.

G. A. Siviloglou, K. G. Makris, R. Iwanow, R. Schiek, D. N. Christodoulides, G. I. Stegeman, Y. Ming, and W. Sohler, “Observation of discrete quadratic surface solitons,” Indian Chem. Eng. 14, 5508-5516 (2006).

Molina, M.

Molina, M. I.

R. A. Vicencio, S. Flach, M. I. Molina, and Yu. S. Kivshar, “Discrete surface solitons in two-dimensional anisotropic lattices,” Phys. Lett. A 364, 274-276 (2007).
[CrossRef]

Nistazakis, H. E.

B. A. Malomed, P. G. Kevrekidis, D. J. Franzeskakis, H. E. Nistazakis, and A. N. Yannacopoulos, “One-and two-dimensional solitons in second-harmonic-generating lattices,” Phys. Rev. E 65, 056606 (2002).
[CrossRef]

Nolte, S.

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

Pertsch, T.

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

Schiek, R.

G. A. Siviloglou, K. G. Makris, R. Iwanow, R. Schiek, D. N. Christodoulides, G. I. Stegeman, Y. Ming, and W. Sohler, “Observation of discrete quadratic surface solitons,” Indian Chem. Eng. 14, 5508-5516 (2006).

Segev, M.

Siviloglou, G. A.

G. A. Siviloglou, K. G. Makris, R. Iwanow, R. Schiek, D. N. Christodoulides, G. I. Stegeman, Y. Ming, and W. Sohler, “Observation of discrete quadratic surface solitons,” Indian Chem. Eng. 14, 5508-5516 (2006).

Sohler, W.

G. A. Siviloglou, K. G. Makris, R. Iwanow, R. Schiek, D. N. Christodoulides, G. I. Stegeman, Y. Ming, and W. Sohler, “Observation of discrete quadratic surface solitons,” Indian Chem. Eng. 14, 5508-5516 (2006).

Stegeman, G.

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

G. A. Siviloglou, K. G. Makris, R. Iwanow, R. Schiek, D. N. Christodoulides, G. I. Stegeman, Y. Ming, and W. Sohler, “Observation of discrete quadratic surface solitons,” Indian Chem. Eng. 14, 5508-5516 (2006).

Susano, H.

H. Susano, P. G. Kevrekidis, R. Carretero-Gonzalez, B. A. Malomed, and D. J. Franzeskakis, “Mobility of discrete solitons in quadratically nonlinear media,” Phys. Rev. Lett. 99, 214103 (2007).
[CrossRef]

Susanto, H.

H. Susanto, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Franzeskakis, “Discrete surface solitons in two dimensions,” Phys. Rev. E 75, 056605 (2007).
[CrossRef]

Szameit, A.

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

Torner, L.

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

Z. Xu, Ya. V. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E 71, 016616 (2005).
[CrossRef]

Ya. V. Kartashov, L. Torner, and V. A. Vysloukh, “Multicolor lattice solitons,” Opt. Lett. 29, 1117-1119 (2004).
[CrossRef] [PubMed]

Tünnermann, A.

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

Vicencio, R.

Vicencio, R. A.

R. A. Vicencio, S. Flach, M. I. Molina, and Yu. S. Kivshar, “Discrete surface solitons in two-dimensional anisotropic lattices,” Phys. Lett. A 364, 274-276 (2007).
[CrossRef]

Vysloukh, V. A.

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).
[CrossRef] [PubMed]

Xu, Z.

Z. Xu and Yu. S. Kivshar, “Two-color surface lattice solitons,” Opt. Lett. 33, 2551-2553 (2008).
[CrossRef] [PubMed]

Z. Xu, Ya. V. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E 71, 016616 (2005).
[CrossRef]

Yannacopoulos, A. N.

B. A. Malomed, P. G. Kevrekidis, D. J. Franzeskakis, H. E. Nistazakis, and A. N. Yannacopoulos, “One-and two-dimensional solitons in second-harmonic-generating lattices,” Phys. Rev. E 65, 056606 (2002).
[CrossRef]

Indian Chem. Eng.

G. A. Siviloglou, K. G. Makris, R. Iwanow, R. Schiek, D. N. Christodoulides, G. I. Stegeman, Y. Ming, and W. Sohler, “Observation of discrete quadratic surface solitons,” Indian Chem. Eng. 14, 5508-5516 (2006).

Opt. Lett.

Phys. Lett. A

R. A. Vicencio, S. Flach, M. I. Molina, and Yu. S. Kivshar, “Discrete surface solitons in two-dimensional anisotropic lattices,” Phys. Lett. A 364, 274-276 (2007).
[CrossRef]

Phys. Rev. E

H. Susanto, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Franzeskakis, “Discrete surface solitons in two dimensions,” Phys. Rev. E 75, 056605 (2007).
[CrossRef]

B. A. Malomed, P. G. Kevrekidis, D. J. Franzeskakis, H. E. Nistazakis, and A. N. Yannacopoulos, “One-and two-dimensional solitons in second-harmonic-generating lattices,” Phys. Rev. E 65, 056606 (2002).
[CrossRef]

Z. Xu, Ya. V. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E 71, 016616 (2005).
[CrossRef]

Phys. Rev. Lett.

H. Susano, P. G. Kevrekidis, R. Carretero-Gonzalez, B. A. Malomed, and D. J. Franzeskakis, “Mobility of discrete solitons in quadratically nonlinear media,” Phys. Rev. Lett. 99, 214103 (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. Tünnermann, and L. Torner, “Observation of two-dimensional surface solitons in asymmetric waveguide arrays,” Phys. Rev. Lett. 98, 173903 (2007).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Examples of the corner (left) and edge (right) localized modes in two-dimensional quadratic lattice. Shown are the amplitudes of the FF components for C u = 1 , C v = 0.5 , β = 0 , and λ = 5 .

Fig. 2
Fig. 2

Total power versus propagation constant of the two-dimensional modes localized in the corner (left) and at the edge (right) of the lattice for C u = 1 , C v = 0.5 , and for three different values of the mismatch parameter β.

Fig. 3
Fig. 3

Minimum power versus mismatch for the corner (circle) and edge (square) modes for (a) C u = 1 , C v = 0.5 and (b) C u = 1 , C v = 0 . For most values of the mismatch parameter, the corner mode requires less power than the edge mode.

Fig. 4
Fig. 4

Example of a two-dimensional twisted mode at the lattice corner. Shown are the amplitudes of the FF (left) and SH (right) fields for C u = 1 , C v = 0.5 , λ = 5 , and β = 0 .

Fig. 5
Fig. 5

Excitation of two-color surface modes at the lattice corner. Shown are the dynamics of the intensity at the corner site for FF (solid) and SH (dashed) components for (a, b) C u = 1 , C v = 0.5 ; β = 0 with (a) U 0 = V 0 = 5 and (b) U 0 = 20 , V 0 = 0 ; and (c, d) C u = 1 , C v = 0 , with (c) U 0 = 5 , V 0 = 0 , β = 0 , and U 0 = 20 , V 0 = 0 , β = 5 .

Equations (7)

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

i d u n , m d z + C u Δ 2 u n , m + 2 γ u n , m * v n , m exp ( + i β z ) = 0 ,
i d v n , m d z + C v Δ 2 v n , m + γ u n , m 2 exp ( i β z ) = 0 ,
Δ 2 u n , m = i = ± 1 ( u n + i , m + u n , m + i ) .
P = P u + P v = n , m ( u n , m 2 + 2 v n , m 2 ) .
λ U n , m + C u Δ 2 U n , m + 2 γ U n , m V n , m = 0 ,
2 λ V n , m + C v Δ 2 V n , m + β V n , m + γ U n , m 2 = 0 ,
i ( d d z ) u n , m + C u Δ 2 u n , m + 2 i γ 2 u n , m * 0 z u n , m 2 ( s ) d s = 0 .

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