Abstract

We demonstrate the existence of nonlinear Tamm oscillations at the interface between a substrate and a one-dimensional waveguide array with either cubic or saturable, self-focusing or self-defocusing nonlinearity. Light is trapped in the vicinity of the array boundary due to the interplay between the repulsive edge potential and Bragg reflection inside the array. In the special case when this potential is linear these oscillations reduce themselves to surface Bloch oscillations.

© 2007 Optical Society of America

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  1. I. Tamm, Phys. Z. Sowjetunion 1, 733 (1932).
  2. H. Ohno, E. E. Mendez, J. A. Brum, J. M. Hong, F. Agulló-Rueda, L. L. Chang, and L. Esaki, Phys. Rev. Lett. 64, 2555 (1990).
    [CrossRef] [PubMed]
  3. D. Kossel, J. Opt. Soc. Am. 56, 1434 (1966).
  4. K. G. Makris, S. Suntsov, D. N. Christodoulides,G. I. Stegeman, and A. Haché, Opt. Lett. 30, 2466 (2005).
    [CrossRef] [PubMed]
  5. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
    [CrossRef]
  6. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
    [CrossRef] [PubMed]
  7. R. Iwanow, R. Schiek, G. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
    [CrossRef] [PubMed]
  8. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
    [CrossRef]
  9. M. Stepic, C. Wirth, C. E. Rüter, and D. Kip, Opt. Lett. 31, 247 (2006).
    [CrossRef] [PubMed]
  10. M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, Nature 432, 733 (2004).
    [CrossRef] [PubMed]
  11. D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13, 794 (1988).
    [CrossRef] [PubMed]
  12. R. Morandotti, U. Peschel, J. S. Aitchison, H. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
    [CrossRef]
  13. S. Suntsov, K. Makris, D. N. Christodoulides, G. Stegeman, A. Haché, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, Phys. Rev. Lett. 96, 063901 (2006).
    [CrossRef] [PubMed]
  14. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 96, 073901 (2006).
    [CrossRef] [PubMed]
  15. M. I. Molina, R. A. Vicencio, and Yu. S. Kivshar, Opt. Lett. 31, 1693 (2006).
    [CrossRef] [PubMed]
  16. F. Lederer, S. Darmanyan, and A. Kobyakov, in Spatial Solitons, S.Trillo and W.Torruellas, eds. (Springer, 2001), p. 269.
  17. E. Smirnov, M. Stepic, C. E. Rüter, D. Kip, and V. Shandarov, Opt. Lett. 31, 2338 (2006).
    [CrossRef] [PubMed]
  18. G. A. Siviloglou, K. G. Makris, R. Iwanow, R. Schiek, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, Opt. Express 14, 5508 (2006).
    [CrossRef] [PubMed]
  19. Y. V. Kartashov, V. A. Vysloukh, D. Mihalache, and L. Torner, Opt. Lett. 31, 2329 (2006).
    [CrossRef] [PubMed]
  20. U. Peschel, T. Pertsch, and F. Lederer, Opt. Lett. 23, 1701 (1998).
    [CrossRef]

2006 (7)

2005 (1)

2004 (2)

R. Iwanow, R. Schiek, G. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, Nature 432, 733 (2004).
[CrossRef] [PubMed]

2002 (1)

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

2000 (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

1999 (1)

R. Morandotti, U. Peschel, J. S. Aitchison, H. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

1998 (2)

U. Peschel, T. Pertsch, and F. Lederer, Opt. Lett. 23, 1701 (1998).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

1990 (1)

H. Ohno, E. E. Mendez, J. A. Brum, J. M. Hong, F. Agulló-Rueda, L. L. Chang, and L. Esaki, Phys. Rev. Lett. 64, 2555 (1990).
[CrossRef] [PubMed]

1988 (1)

1966 (1)

D. Kossel, J. Opt. Soc. Am. 56, 1434 (1966).

1932 (1)

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

Agulló-Rueda, F.

H. Ohno, E. E. Mendez, J. A. Brum, J. M. Hong, F. Agulló-Rueda, L. L. Chang, and L. Esaki, Phys. Rev. Lett. 64, 2555 (1990).
[CrossRef] [PubMed]

Aitchison, J. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

Assanto, G.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, Nature 432, 733 (2004).
[CrossRef] [PubMed]

Boyd, A.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

Brum, J. A.

H. Ohno, E. E. Mendez, J. A. Brum, J. M. Hong, F. Agulló-Rueda, L. L. Chang, and L. Esaki, Phys. Rev. Lett. 64, 2555 (1990).
[CrossRef] [PubMed]

Chang, L. L.

H. Ohno, E. E. Mendez, J. A. Brum, J. M. Hong, F. Agulló-Rueda, L. L. Chang, and L. Esaki, Phys. Rev. Lett. 64, 2555 (1990).
[CrossRef] [PubMed]

Christodoulides, D. N.

Conti, C.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, Nature 432, 733 (2004).
[CrossRef] [PubMed]

Darmanyan, S.

F. Lederer, S. Darmanyan, and A. Kobyakov, in Spatial Solitons, S.Trillo and W.Torruellas, eds. (Springer, 2001), p. 269.

De Luca, A.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, Nature 432, 733 (2004).
[CrossRef] [PubMed]

Efremidis, N. K.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Eisenberg, H.

R. Morandotti, U. Peschel, J. S. Aitchison, H. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

Eisenberg, H. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

Esaki, L.

H. Ohno, E. E. Mendez, J. A. Brum, J. M. Hong, F. Agulló-Rueda, L. L. Chang, and L. Esaki, Phys. Rev. Lett. 64, 2555 (1990).
[CrossRef] [PubMed]

Fleischer, J.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Haché, A.

S. Suntsov, K. Makris, D. N. Christodoulides, G. 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]

Hong, J. M.

H. Ohno, E. E. Mendez, J. A. Brum, J. M. Hong, F. Agulló-Rueda, L. L. Chang, and L. Esaki, Phys. Rev. Lett. 64, 2555 (1990).
[CrossRef] [PubMed]

Iwanow, R.

Joseph, R. I.

Kartashov, Y. V.

Kip, D.

Kivshar, Yu. S.

Kobyakov, A.

F. Lederer, S. Darmanyan, and A. Kobyakov, in Spatial Solitons, S.Trillo and W.Torruellas, eds. (Springer, 2001), p. 269.

Kossel, D.

D. Kossel, J. Opt. Soc. Am. 56, 1434 (1966).

Lederer, F.

R. Iwanow, R. Schiek, G. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

U. Peschel, T. Pertsch, and F. Lederer, Opt. Lett. 23, 1701 (1998).
[CrossRef]

F. Lederer, S. Darmanyan, and A. Kobyakov, in Spatial Solitons, S.Trillo and W.Torruellas, eds. (Springer, 2001), p. 269.

Makris, K.

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

Makris, K. G.

Mendez, E. E.

H. Ohno, E. E. Mendez, J. A. Brum, J. M. Hong, F. Agulló-Rueda, L. L. Chang, and L. Esaki, Phys. Rev. Lett. 64, 2555 (1990).
[CrossRef] [PubMed]

Mihalache, D.

Min, Y.

Molina, M. I.

Morandotti, R.

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

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

Ohno, H.

H. Ohno, E. E. Mendez, J. A. Brum, J. M. Hong, F. Agulló-Rueda, L. L. Chang, and L. Esaki, Phys. Rev. Lett. 64, 2555 (1990).
[CrossRef] [PubMed]

Peccianti, M.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, Nature 432, 733 (2004).
[CrossRef] [PubMed]

Pertsch, T.

R. Iwanow, R. Schiek, G. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

U. Peschel, T. Pertsch, and F. Lederer, Opt. Lett. 23, 1701 (1998).
[CrossRef]

Peschel, U.

R. Morandotti, U. Peschel, J. S. Aitchison, H. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

U. Peschel, T. Pertsch, and F. Lederer, Opt. Lett. 23, 1701 (1998).
[CrossRef]

Rüter, C. E.

Salamo, G.

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

Schiek, R.

Sears, S.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Segev, M.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Shandarov, V.

Silberberg, Y.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

Siviloglou, G. A.

Smirnov, E.

Sohler, W.

Sorel, M.

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

Stegeman, G.

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

R. Iwanow, R. Schiek, G. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

Stegeman, G. I.

Stepic, M.

Suntsov, S.

S. Suntsov, K. Makris, D. N. Christodoulides, G. 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]

Tamm, I.

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

Torner, L.

Umeton, C.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, Nature 432, 733 (2004).
[CrossRef] [PubMed]

Vicencio, R. A.

Vysloukh, V. A.

Wirth, C.

Yang, H.

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

J. Opt. Soc. Am. (1)

D. Kossel, J. Opt. Soc. Am. 56, 1434 (1966).

Nature (1)

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, Nature 432, 733 (2004).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (7)

Phys. Rev. E (1)

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Phys. Rev. Lett. (7)

H. Ohno, E. E. Mendez, J. A. Brum, J. M. Hong, F. Agulló-Rueda, L. L. Chang, and L. Esaki, Phys. Rev. Lett. 64, 2555 (1990).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

R. Iwanow, R. Schiek, G. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

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

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

Phys. Z. Sowjetunion (1)

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

Other (1)

F. Lederer, S. Darmanyan, and A. Kobyakov, in Spatial Solitons, S.Trillo and W.Torruellas, eds. (Springer, 2001), p. 269.

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Figures (4)

Fig. 1
Fig. 1

Tamm oscillations of (a) mode A with C = 0.5 , β = 3.34 , and soliton frequency ν = 2.3 launched into the first channel, and (b) mode B with C = 0.5 , β = 3.34 , and soliton frequency ν = 4.83 launched into the second and third channels.

Fig. 2
Fig. 2

Dependence of the effective repulsive potential on the distance from the boundary. (a) Self-focusing case: β = 3.1 , F 0 2 = 0.24 , and different values of C. (b) Self-defocusing case: C = 0.5 , F 0 2 = 0.172 , and different values of β.

Fig. 3
Fig. 3

Tamm oscillations in a WA with period 8.4 μ m and Δ n 0 = 3.7 × 10 4 . Input patterns with amplitude ratios 1 : 0.5 : 0.1 are launched into the first channel. (a) Self-defocusing saturable case Δ n n l = 6.2 × 10 4 , r = I I d = 6 ; (b) self-defocusing cubic case Δ n n l = 4.42 × 10 4 ; (c) self-focusing saturable case Δ n n l = 3.34 × 10 4 , r = 6 ; and (d) self-focusing cubic case Δ n n l = 2.62 × 10 4 .

Fig. 4
Fig. 4

Saturable defocusing WA with input profile and lattice parameters from Fig. 3, but Δ n n l = 6.21 × 10 4 and r = 6 . Input beam is launched into (a) first and (b) third channels.

Equations (4)

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

i d U n d ξ + C ( U n + 1 + U n 1 2 U n ) g ( U n 2 ) U n = 0 ,
V r e p s ( p ) = β i = 0 p 1 [ ln ( 1 + α 2 ( n a i ) F 2 ) F 2 α 2 ( n a i ) ] + F 2 C [ α 2 ( n a p ) ( 1 + α ) 2 i = 1 p α 2 ( n a i ) ] ,
V r e p c ( p ) = β 2 F 4 i = 0 p 1 α 4 ( n a i ) + C F 2 α 2 ( n p a ) C F 2 ( 1 + α ) 2 i = 1 p α 2 ( n i a )
i E y + 1 2 k 2 E z 2 + k n ( z ) + Δ n n l n s E = 0 .

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