Abstract

We study a waveguide array with an embedded nonlinear saturable impurity. We solve the impurity problem in closed form and find the nonlinear localized modes. Next, we consider the scattering of a small-amplitude plane wave by a nonlinear impurity mode, and discover regions in parameter space where transmission is fully suppressed. We relate these findings with Fano resonances and propose this setup as a means to control the transport of light across the array.

© 2009 Optical Society of America

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  1. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
    [CrossRef]
  2. S. Flach and A. V. Gorbach, Phys. Rep. 467, 1 (2008).
    [CrossRef]
  3. L. Morales-Molina and R. A. Vicencio, Opt. Lett. 31, 966 (2006).
    [CrossRef] [PubMed]
  4. Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, Phys. Rev. Lett. 99, 133901 (2007).
    [CrossRef] [PubMed]
  5. U. Fano, Phys. Rev. 124, 1866 (1961).
    [CrossRef]
  6. A. E. Miroshnichenko, Y. S. Kivshar, R. A. Vicencio, and M. I. Molina, Opt. Lett. 30, 872 (2005).
    [CrossRef] [PubMed]
  7. R. A. Vicencio, J. Brand, and S. Flach, Phys. Rev. Lett. 98, 184102 (2007).
    [CrossRef] [PubMed]
  8. A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, arXiv:0902.3014.
  9. E. Smirnov, C. E. Ruter, M. Stepic, V. Shandarov, and D. Kip, Opt. Express 14, 11248 (2006).
    [CrossRef] [PubMed]
  10. M. Stepic, E. Smirnov, C. E. Ruter, L. Pronneke, D. Kip, and V. Shandarov, Phys. Rev. E 74, 046614 (2006).
    [CrossRef]
  11. M. I. Molina, R. A. Vicencio, and Y. S. Kivshar, Phys. Lett. A 350, 134 (2006).
    [CrossRef]

2008 (2)

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

S. Flach and A. V. Gorbach, Phys. Rep. 467, 1 (2008).
[CrossRef]

2007 (2)

Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, Phys. Rev. Lett. 99, 133901 (2007).
[CrossRef] [PubMed]

R. A. Vicencio, J. Brand, and S. Flach, Phys. Rev. Lett. 98, 184102 (2007).
[CrossRef] [PubMed]

2006 (4)

M. Stepic, E. Smirnov, C. E. Ruter, L. Pronneke, D. Kip, and V. Shandarov, Phys. Rev. E 74, 046614 (2006).
[CrossRef]

M. I. Molina, R. A. Vicencio, and Y. S. Kivshar, Phys. Lett. A 350, 134 (2006).
[CrossRef]

L. Morales-Molina and R. A. Vicencio, Opt. Lett. 31, 966 (2006).
[CrossRef] [PubMed]

E. Smirnov, C. E. Ruter, M. Stepic, V. Shandarov, and D. Kip, Opt. Express 14, 11248 (2006).
[CrossRef] [PubMed]

2005 (1)

1961 (1)

U. Fano, Phys. Rev. 124, 1866 (1961).
[CrossRef]

Aimez, V.

Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, Phys. Rev. Lett. 99, 133901 (2007).
[CrossRef] [PubMed]

Ares, R.

Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, Phys. Rev. Lett. 99, 133901 (2007).
[CrossRef] [PubMed]

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Bar-Ad, S.

Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, Phys. Rev. Lett. 99, 133901 (2007).
[CrossRef] [PubMed]

Brand, J.

R. A. Vicencio, J. Brand, and S. Flach, Phys. Rev. Lett. 98, 184102 (2007).
[CrossRef] [PubMed]

Christodoulides, D. N.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Fano, U.

U. Fano, Phys. Rev. 124, 1866 (1961).
[CrossRef]

Flach, S.

S. Flach and A. V. Gorbach, Phys. Rep. 467, 1 (2008).
[CrossRef]

R. A. Vicencio, J. Brand, and S. Flach, Phys. Rev. Lett. 98, 184102 (2007).
[CrossRef] [PubMed]

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, arXiv:0902.3014.

Gorbach, A. V.

S. Flach and A. V. Gorbach, Phys. Rep. 467, 1 (2008).
[CrossRef]

Kip, D.

M. Stepic, E. Smirnov, C. E. Ruter, L. Pronneke, D. Kip, and V. Shandarov, Phys. Rev. E 74, 046614 (2006).
[CrossRef]

E. Smirnov, C. E. Ruter, M. Stepic, V. Shandarov, and D. Kip, Opt. Express 14, 11248 (2006).
[CrossRef] [PubMed]

Kivshar, Y. S.

M. I. Molina, R. A. Vicencio, and Y. S. Kivshar, Phys. Lett. A 350, 134 (2006).
[CrossRef]

A. E. Miroshnichenko, Y. S. Kivshar, R. A. Vicencio, and M. I. Molina, Opt. Lett. 30, 872 (2005).
[CrossRef] [PubMed]

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, arXiv:0902.3014.

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Linzon, Y.

Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, Phys. Rev. Lett. 99, 133901 (2007).
[CrossRef] [PubMed]

Miroshnichenko, A. E.

Molina, M. I.

Morales-Molina, L.

Morandotti, R.

Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, Phys. Rev. Lett. 99, 133901 (2007).
[CrossRef] [PubMed]

Pronneke, L.

M. Stepic, E. Smirnov, C. E. Ruter, L. Pronneke, D. Kip, and V. Shandarov, Phys. Rev. E 74, 046614 (2006).
[CrossRef]

Ruter, C. E.

M. Stepic, E. Smirnov, C. E. Ruter, L. Pronneke, D. Kip, and V. Shandarov, Phys. Rev. E 74, 046614 (2006).
[CrossRef]

E. Smirnov, C. E. Ruter, M. Stepic, V. Shandarov, and D. Kip, Opt. Express 14, 11248 (2006).
[CrossRef] [PubMed]

Segev, M.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Shandarov, V.

M. Stepic, E. Smirnov, C. E. Ruter, L. Pronneke, D. Kip, and V. Shandarov, Phys. Rev. E 74, 046614 (2006).
[CrossRef]

E. Smirnov, C. E. Ruter, M. Stepic, V. Shandarov, and D. Kip, Opt. Express 14, 11248 (2006).
[CrossRef] [PubMed]

Silberberg, Y.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Smirnov, E.

M. Stepic, E. Smirnov, C. E. Ruter, L. Pronneke, D. Kip, and V. Shandarov, Phys. Rev. E 74, 046614 (2006).
[CrossRef]

E. Smirnov, C. E. Ruter, M. Stepic, V. Shandarov, and D. Kip, Opt. Express 14, 11248 (2006).
[CrossRef] [PubMed]

Stegeman, G. I.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Stepic, M.

M. Stepic, E. Smirnov, C. E. Ruter, L. Pronneke, D. Kip, and V. Shandarov, Phys. Rev. E 74, 046614 (2006).
[CrossRef]

E. Smirnov, C. E. Ruter, M. Stepic, V. Shandarov, and D. Kip, Opt. Express 14, 11248 (2006).
[CrossRef] [PubMed]

Vicencio, R. A.

R. A. Vicencio, J. Brand, and S. Flach, Phys. Rev. Lett. 98, 184102 (2007).
[CrossRef] [PubMed]

L. Morales-Molina and R. A. Vicencio, Opt. Lett. 31, 966 (2006).
[CrossRef] [PubMed]

M. I. Molina, R. A. Vicencio, and Y. S. Kivshar, Phys. Lett. A 350, 134 (2006).
[CrossRef]

A. E. Miroshnichenko, Y. S. Kivshar, R. A. Vicencio, and M. I. Molina, Opt. Lett. 30, 872 (2005).
[CrossRef] [PubMed]

Volatier, M.

Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, Phys. Rev. Lett. 99, 133901 (2007).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (2)

Phys. Lett. A (1)

M. I. Molina, R. A. Vicencio, and Y. S. Kivshar, Phys. Lett. A 350, 134 (2006).
[CrossRef]

Phys. Rep. (2)

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

S. Flach and A. V. Gorbach, Phys. Rep. 467, 1 (2008).
[CrossRef]

Phys. Rev. (1)

U. Fano, Phys. Rev. 124, 1866 (1961).
[CrossRef]

Phys. Rev. E (1)

M. Stepic, E. Smirnov, C. E. Ruter, L. Pronneke, D. Kip, and V. Shandarov, Phys. Rev. E 74, 046614 (2006).
[CrossRef]

Phys. Rev. Lett. (2)

R. A. Vicencio, J. Brand, and S. Flach, Phys. Rev. Lett. 98, 184102 (2007).
[CrossRef] [PubMed]

Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, Phys. Rev. Lett. 99, 133901 (2007).
[CrossRef] [PubMed]

Other (1)

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, arXiv:0902.3014.

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Figures (4)

Fig. 1
Fig. 1

(a) Impurity setup. (b) P versus λ diagram for NLM. (c) λ versus U 0 for NLM (black curves) and ω c c versus U 0 for LLM (gray curves).

Fig. 2
Fig. 2

Transmission coefficient T as a function of beam angle k and amplitude U 0 . Bright (dark) regions denote high (low) T. The dashed curve marks places where ω k = ω c c .

Fig. 3
Fig. 3

T versus k for U 0 = 0.5 (squares), 1.36 (diamonds), and 2 (circles). Curves correspond to theoretical T and symbols to numerical simulations.

Fig. 4
Fig. 4

Linear output profile for different k. The dashed line marks the position of the impurity site. U 0 = 1.36 .

Equations (7)

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

i d u n d z = ( u n + 1 + u n 1 ) + ( ϵ β 1 + | u n | 2 ) u n δ n , n c .
i d ϕ n d z = ( ϕ n + 1 + ϕ n 1 ) + [ ( ϵ g 2 β ) ϕ n + g 2 β U 0 2 e 2 i λ z ϕ n ] δ n , n c .
ω a n = ( a n + 1 + a n 1 ) + [ ( ϵ g 2 / β ) a n + ( g 2 U 0 2 / β ) b n ] δ n , n c ,
( 2 λ ω ) b n = ( b n + 1 + b n 1 ) + [ ( ϵ g 2 / β ) b n + ( g 2 U 0 2 / β ) a n ] δ n , n c .
a n = { B e i k ( n n c ) + D e i k ( n n c ) n < n c F e i k ( n n c ) n n c } ,
b n = b 0 y | n n c | .
Ω ( k ) = ϵ g 2 β + g 4 U 0 4 / β ( g 2 / β ) ϵ ± 2 ( λ cos   k ) 2 1 ,

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