Abstract

We study surface modes at the edge of a semi-infinite chirped photonic lattice in the framework of an effective discrete nonlinear model. We demonstrate that the lattice chirp can change dramatically the conditions for the mode localization near the surface, and we find numerically the families of discrete surface solitons in this case. Such solitons do not require any minimum power to exist provided the chirp parameter exceeds some critical value. We also analyze how the chirp modifies the interaction of a soliton with the lattice edge.

© 2007 Optical Society of America

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References

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  1. P. Yeh, A. Yariv, and A. Y. Cho, Appl. Phys. Lett. 32, 104 (1978).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).
  5. M. Molina, R. Vicencio, and Yu. S. Kivshar, Opt. Lett. 31, 1693 (2006).
    [CrossRef] [PubMed]
  6. Ya. V. Kartashov, V. V. Vysloukh, and L. Torner, Phys. Rev. Lett. 96, 073901 (2006).
    [CrossRef] [PubMed]
  7. C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Yu. S. Kivshar, Phys. Rev. Lett. 97, 083901 (2006).
    [CrossRef] [PubMed]
  8. E. Smirnov, M. Stepic, C. E. Ruter, D. Kip, and V. Shandarov, Opt. Lett. 31, 2338 (2006).
    [CrossRef] [PubMed]
  9. K. G. Makris, J. Hudock, D. N. Christodoulides, G. I. Stegeman, O. Manela, and M. Segev, Opt. Lett. 31, 2774 (2006).
    [CrossRef] [PubMed]
  10. K. Motzek, A. A. Sukhorukov, and Yu. S. Kivshar, Opt. Express 14, 9873 (2006).
    [CrossRef] [PubMed]
  11. M. I. Molina and Yu. S. Kivshar, Phys. Lett. A 362, 280 (2007).
    [CrossRef]
  12. Ya. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 76, 013831 (2007).
    [CrossRef]

2007 (2)

M. I. Molina and Yu. S. Kivshar, Phys. Lett. A 362, 280 (2007).
[CrossRef]

Ya. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 76, 013831 (2007).
[CrossRef]

2006 (7)

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]

Ya. V. Kartashov, V. V. Vysloukh, and L. Torner, 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 Yu. S. Kivshar, Phys. Rev. Lett. 97, 083901 (2006).
[CrossRef] [PubMed]

M. Molina, R. Vicencio, and Yu. S. Kivshar, Opt. Lett. 31, 1693 (2006).
[CrossRef] [PubMed]

E. Smirnov, M. Stepic, C. E. Ruter, D. Kip, and V. Shandarov, Opt. Lett. 31, 2338 (2006).
[CrossRef] [PubMed]

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

K. Motzek, A. A. Sukhorukov, and Yu. S. Kivshar, Opt. Express 14, 9873 (2006).
[CrossRef] [PubMed]

2005 (1)

1978 (1)

P. Yeh, A. Yariv, and A. Y. Cho, Appl. Phys. Lett. 32, 104 (1978).
[CrossRef]

Appl. Phys. Lett. (1)

P. Yeh, A. Yariv, and A. Y. Cho, Appl. Phys. Lett. 32, 104 (1978).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Phys. Lett. A (1)

M. I. Molina and Yu. S. Kivshar, Phys. Lett. A 362, 280 (2007).
[CrossRef]

Phys. Rev. A (1)

Ya. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 76, 013831 (2007).
[CrossRef]

Phys. Rev. Lett. (3)

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]

Ya. V. Kartashov, V. V. Vysloukh, and L. Torner, 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 Yu. S. Kivshar, Phys. Rev. Lett. 97, 083901 (2006).
[CrossRef] [PubMed]

Other (1)

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

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Figures (5)

Fig. 1
Fig. 1

Transverse profile of refraction index for a chirped array of nonlinear optical waveguides. n 0 is the linear index of the substrate.

Fig. 2
Fig. 2

(a),(b) Linear surface modes for (a) σ = 0.04 and (b) σ = 0.5 . (c),(d) Nonlinear surface modes for (c) P = 2.69 , β = 2.64 , σ = 0.1 ; and (d) P = 2.54 , β = 2.64 , σ = 0.5 [points C and D in Fig. 4a]. ( N = 51 ) .

Fig. 3
Fig. 3

(a) Position of the mode maximum in the waveguide array as a function of the chirp parameter. (b) Propagation constant of localized mode as a function of the chirp parameter ( N = 51 ) .

Fig. 4
Fig. 4

(a) Power versus propagation constant curves of surface mode for two values of the chirp parameter. Solid (dashed) curve denotes stable (unstable) mode. (b) Minimum power to create a surface mode in terms of the chirp parameter ( N = 51 ) .

Fig. 5
Fig. 5

Propagation dynamics of discrete solitons launched toward the surface for various values of the chirp parameter: (a) A = 1.1 , k = 0.7 , σ = 0 ; (b) A = 1.1 , k = 0.7 , σ = 0.03 ; (c) A = 1.1 , k = 0.7 , σ = 0.5 ( n c = 10 and N = 101 ).

Equations (4)

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i d E 0 d ξ + λ 0 V 0 E 0 + E 1 + γ E 0 2 E 0 = 0 , at n = 0 ,
i d E n d ξ + ( λ 0 V 0 f ( n ) + γ E n 2 ) E n + f ( n ) E n + 1 + f ( n 1 ) E n 1 = 0 ,
β u 0 + u 1 + γ u 0 2 u 0 = 0 ,
Λ n u n + f ( n ) u n + 1 + f ( n 1 ) u n 1 + γ u n 2 u n = 0 ,

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