Abstract

We discuss the formation of self-trapped localized states near the edge of a semi-infinite array of nonlinear optical waveguides. We study a crossover from nonlinear surface states to discrete solitons by analyzing the families of odd and even modes centered at finite distances from the surface and reveal the physical mechanism of the nonlinearity-induced stabilization of surface modes.

© 2006 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, 102 (1978).
  2. W. J. Tomlinson, Opt. Lett. 5, 323 (1980).
    [CrossRef] [PubMed]
  3. Yu. S. Kivshar, F. Zhang, and S. Takeno, Physica D 113, 248 (1998).
    [CrossRef]
  4. I. E. Tamm, Z. Phys. 76, 849 (1932).
    [CrossRef]
  5. K. G. Markis, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache, Opt. Lett. 30, 2466 (2005).
    [CrossRef]
  6. S. Suntsov, K. G. Makris, D. N. Christodoulides, andG. I. Stegeman, in Nonlinear Guided Waves and Their Applications 2005 Digest (Optical Society of America, 2005), paper ThC4.
  7. D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13, 794 (1988).
    [CrossRef] [PubMed]
  8. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).
  9. Yu. S. Kivshar, F. Zhang, and A. S. Kovalev, Phys. Rev. B 55, 14265 (1997).
    [CrossRef]
  10. S. Darmanyan, A. Kobyakov, F. Lederer, and L. Vázques, Phys. Rev. B 59, 5994 (1999).
    [CrossRef]

2005 (1)

1999 (1)

S. Darmanyan, A. Kobyakov, F. Lederer, and L. Vázques, Phys. Rev. B 59, 5994 (1999).
[CrossRef]

1998 (1)

Yu. S. Kivshar, F. Zhang, and S. Takeno, Physica D 113, 248 (1998).
[CrossRef]

1997 (1)

Yu. S. Kivshar, F. Zhang, and A. S. Kovalev, Phys. Rev. B 55, 14265 (1997).
[CrossRef]

1988 (1)

1980 (1)

1978 (1)

P. Yeh, A. Yariv, and A. Y. Cho, Appl. Phys. Lett. 32, 102 (1978).

1932 (1)

I. E. Tamm, Z. Phys. 76, 849 (1932).
[CrossRef]

Agrawal, G. P.

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

Cho, A. Y.

P. Yeh, A. Yariv, and A. Y. Cho, Appl. Phys. Lett. 32, 102 (1978).

Christodoulides, D. N.

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

D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13, 794 (1988).
[CrossRef] [PubMed]

S. Suntsov, K. G. Makris, D. N. Christodoulides, andG. I. Stegeman, in Nonlinear Guided Waves and Their Applications 2005 Digest (Optical Society of America, 2005), paper ThC4.

Darmanyan, S.

S. Darmanyan, A. Kobyakov, F. Lederer, and L. Vázques, Phys. Rev. B 59, 5994 (1999).
[CrossRef]

Hache, A.

Joseph, R. I.

Kivshar, Yu. S.

Yu. S. Kivshar, F. Zhang, and S. Takeno, Physica D 113, 248 (1998).
[CrossRef]

Yu. S. Kivshar, F. Zhang, and A. S. Kovalev, Phys. Rev. B 55, 14265 (1997).
[CrossRef]

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

Kobyakov, A.

S. Darmanyan, A. Kobyakov, F. Lederer, and L. Vázques, Phys. Rev. B 59, 5994 (1999).
[CrossRef]

Kovalev, A. S.

Yu. S. Kivshar, F. Zhang, and A. S. Kovalev, Phys. Rev. B 55, 14265 (1997).
[CrossRef]

Lederer, F.

S. Darmanyan, A. Kobyakov, F. Lederer, and L. Vázques, Phys. Rev. B 59, 5994 (1999).
[CrossRef]

Makris, K. G.

S. Suntsov, K. G. Makris, D. N. Christodoulides, andG. I. Stegeman, in Nonlinear Guided Waves and Their Applications 2005 Digest (Optical Society of America, 2005), paper ThC4.

Markis, K. G.

Stegeman, G. I.

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

S. Suntsov, K. G. Makris, D. N. Christodoulides, andG. I. Stegeman, in Nonlinear Guided Waves and Their Applications 2005 Digest (Optical Society of America, 2005), paper ThC4.

Suntsov, S.

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

S. Suntsov, K. G. Makris, D. N. Christodoulides, andG. I. Stegeman, in Nonlinear Guided Waves and Their Applications 2005 Digest (Optical Society of America, 2005), paper ThC4.

Takeno, S.

Yu. S. Kivshar, F. Zhang, and S. Takeno, Physica D 113, 248 (1998).
[CrossRef]

Tamm, I. E.

I. E. Tamm, Z. Phys. 76, 849 (1932).
[CrossRef]

Tomlinson, W. J.

Vázques, L.

S. Darmanyan, A. Kobyakov, F. Lederer, and L. Vázques, Phys. Rev. B 59, 5994 (1999).
[CrossRef]

Yariv, A.

P. Yeh, A. Yariv, and A. Y. Cho, Appl. Phys. Lett. 32, 102 (1978).

Yeh, P.

P. Yeh, A. Yariv, and A. Y. Cho, Appl. Phys. Lett. 32, 102 (1978).

Zhang, F.

Yu. S. Kivshar, F. Zhang, and S. Takeno, Physica D 113, 248 (1998).
[CrossRef]

Yu. S. Kivshar, F. Zhang, and A. S. Kovalev, Phys. Rev. B 55, 14265 (1997).
[CrossRef]

Appl. Phys. Lett. (1)

P. Yeh, A. Yariv, and A. Y. Cho, Appl. Phys. Lett. 32, 102 (1978).

Opt. Lett. (3)

Phys. Rev. B (2)

Yu. S. Kivshar, F. Zhang, and A. S. Kovalev, Phys. Rev. B 55, 14265 (1997).
[CrossRef]

S. Darmanyan, A. Kobyakov, F. Lederer, and L. Vázques, Phys. Rev. B 59, 5994 (1999).
[CrossRef]

Physica D (1)

Yu. S. Kivshar, F. Zhang, and S. Takeno, Physica D 113, 248 (1998).
[CrossRef]

Z. Phys. (1)

I. E. Tamm, Z. Phys. 76, 849 (1932).
[CrossRef]

Other (2)

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

S. Suntsov, K. G. Makris, D. N. Christodoulides, andG. I. Stegeman, in Nonlinear Guided Waves and Their Applications 2005 Digest (Optical Society of America, 2005), paper ThC4.

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Figures (5)

Fig. 1
Fig. 1

Examples of surface localized modes at β = 3 in an array of focusing waveguides ( γ = + 1 ) centered at distances d of ( a ) 0, (b) 1, (c) 2, (d) 3 from the array edge.

Fig. 2
Fig. 2

Examples of localized surface modes at β = 3 in an array of defocusing waveguides ( γ = 1 ) located at distances d of ( a ) 0, (b) 1, (c) 2, (d) 3 from the array edge.

Fig. 3
Fig. 3

Normalized power versus propagation constant β for the surface modes shown in Fig. 1 located at distances d = 0 , 1 , 2 , 3 from the surface. The darkest curve corresponds to the discrete soliton in an infinite array.

Fig. 4
Fig. 4

Effective energy of surface modes versus coordinate X near the edge of the array: (a) below ( P = 2.85 ) and (b) above ( P = 4.05 ) threshold. Filled circles, stationary solutions found without constraint.

Fig. 5
Fig. 5

Examples of stable flat-topped localized surface modes at β = 4 in the array of defocusing waveguides ( γ = 1 ) centered between various sites near the edge.

Equations (2)

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i d E 1 d z + α E 1 + E 2 + γ E 1 2 E 1 = 0 ,
i d E n d z + α E n + ( E n + 1 + E n 1 ) + γ E n 2 E n = 0 ,

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