Abstract

We report on existence and properties of discrete solitons in arrays of alternating waveguides with positive and negative refractive indices. When the nonlinearities of all waveguides are focusing, we found solitons only in the semi-infinite gaps. Finite gap solitons found for waveguides having nonlinearities of different types reveal the symmetry breaking in the Fourier space. It is found that there exist more than one soliton families bifurcating from the gap edges of the linear spectrum. The field distribution in such multichannel couplers reveals nonexponential decay and nonmonotonic dependence of the energy growth in the positive index waveguides on the strength of losses in the negative index waveguides.

© 2012 Optical Society of America

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  1. N. M. Litchinitser and V. M. Shalaev, Laser Phys. Lett. 5, 411 (2008).
    [CrossRef]
  2. H. Chen, B.-I. Wu, and J. A. Kong. J. Electromagn Waves Appl. 20, 2137 (2006).
    [CrossRef]
  3. A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, Phys. Rev. Lett. 91, 037401 (2003).
    [CrossRef]
  4. N. M. Litchinitser, I. R. Gabitov, and A. I. Maimistov, Phys. Rev. Lett. 99, 113902 (2007).
    [CrossRef]
  5. A. I. Maimistov, I. R. Gabitov, and N. M. Litchinitser, Opt. Spectrosc. 104, 253 (2008).
    [CrossRef]
  6. E. Kazantseva, A. I. Maimistov, and S. S. Ozhenko, Phys. Rev. A 80, 043833 (2009).
    [CrossRef]
  7. P. Y. P. Chen and B. A. Malomed, Opt. Commun. 283, 1598 (2010).
    [CrossRef]
  8. Y. Xiang, S. Wen, X. Dai, and D. Fan, Phys. Rev. E 82, 056605 (2010).
    [CrossRef]
  9. A. Maimistov and E. V. Kazantseva, Opt. Spectrosc. 112, 264 (2012).
    [CrossRef]
  10. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
    [CrossRef]
  11. M. I. Stockman, Phys. Rev. Lett. 98, 177404 (2007).
    [CrossRef]
  12. G. Dolling, C. Enkrich, H. Wegener, C. M. Soukoulis, and S. Linden, Opt. Lett. 31, 1800 (2006).
    [CrossRef]

2012 (1)

A. Maimistov and E. V. Kazantseva, Opt. Spectrosc. 112, 264 (2012).
[CrossRef]

2010 (2)

P. Y. P. Chen and B. A. Malomed, Opt. Commun. 283, 1598 (2010).
[CrossRef]

Y. Xiang, S. Wen, X. Dai, and D. Fan, Phys. Rev. E 82, 056605 (2010).
[CrossRef]

2009 (1)

E. Kazantseva, A. I. Maimistov, and S. S. Ozhenko, Phys. Rev. A 80, 043833 (2009).
[CrossRef]

2008 (3)

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

N. M. Litchinitser and V. M. Shalaev, Laser Phys. Lett. 5, 411 (2008).
[CrossRef]

A. I. Maimistov, I. R. Gabitov, and N. M. Litchinitser, Opt. Spectrosc. 104, 253 (2008).
[CrossRef]

2007 (2)

N. M. Litchinitser, I. R. Gabitov, and A. I. Maimistov, Phys. Rev. Lett. 99, 113902 (2007).
[CrossRef]

M. I. Stockman, Phys. Rev. Lett. 98, 177404 (2007).
[CrossRef]

2006 (2)

G. Dolling, C. Enkrich, H. Wegener, C. M. Soukoulis, and S. Linden, Opt. Lett. 31, 1800 (2006).
[CrossRef]

H. Chen, B.-I. Wu, and J. A. Kong. J. Electromagn Waves Appl. 20, 2137 (2006).
[CrossRef]

2003 (1)

A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef]

Assanto, G.

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

Chen, H.

H. Chen, B.-I. Wu, and J. A. Kong. J. Electromagn Waves Appl. 20, 2137 (2006).
[CrossRef]

Chen, P. Y. P.

P. Y. P. Chen and B. A. Malomed, Opt. Commun. 283, 1598 (2010).
[CrossRef]

Christodoulides, D. N.

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

Dai, X.

Y. Xiang, S. Wen, X. Dai, and D. Fan, Phys. Rev. E 82, 056605 (2010).
[CrossRef]

Dolling, G.

Enkrich, C.

Fan, D.

Y. Xiang, S. Wen, X. Dai, and D. Fan, Phys. Rev. E 82, 056605 (2010).
[CrossRef]

Gabitov, I. R.

A. I. Maimistov, I. R. Gabitov, and N. M. Litchinitser, Opt. Spectrosc. 104, 253 (2008).
[CrossRef]

N. M. Litchinitser, I. R. Gabitov, and A. I. Maimistov, Phys. Rev. Lett. 99, 113902 (2007).
[CrossRef]

Kazantseva, E.

E. Kazantseva, A. I. Maimistov, and S. S. Ozhenko, Phys. Rev. A 80, 043833 (2009).
[CrossRef]

Kazantseva, E. V.

A. Maimistov and E. V. Kazantseva, Opt. Spectrosc. 112, 264 (2012).
[CrossRef]

Kivshar, Yu. S.

A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef]

Kong, J. A.

H. Chen, B.-I. Wu, and J. A. Kong. J. Electromagn Waves Appl. 20, 2137 (2006).
[CrossRef]

Lederer, F.

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

Linden, S.

Litchinitser, N. M.

N. M. Litchinitser and V. M. Shalaev, Laser Phys. Lett. 5, 411 (2008).
[CrossRef]

A. I. Maimistov, I. R. Gabitov, and N. M. Litchinitser, Opt. Spectrosc. 104, 253 (2008).
[CrossRef]

N. M. Litchinitser, I. R. Gabitov, and A. I. Maimistov, Phys. Rev. Lett. 99, 113902 (2007).
[CrossRef]

Maimistov, A.

A. Maimistov and E. V. Kazantseva, Opt. Spectrosc. 112, 264 (2012).
[CrossRef]

Maimistov, A. I.

E. Kazantseva, A. I. Maimistov, and S. S. Ozhenko, Phys. Rev. A 80, 043833 (2009).
[CrossRef]

A. I. Maimistov, I. R. Gabitov, and N. M. Litchinitser, Opt. Spectrosc. 104, 253 (2008).
[CrossRef]

N. M. Litchinitser, I. R. Gabitov, and A. I. Maimistov, Phys. Rev. Lett. 99, 113902 (2007).
[CrossRef]

Malomed, B. A.

P. Y. P. Chen and B. A. Malomed, Opt. Commun. 283, 1598 (2010).
[CrossRef]

Ozhenko, S. S.

E. Kazantseva, A. I. Maimistov, and S. S. Ozhenko, Phys. Rev. A 80, 043833 (2009).
[CrossRef]

Segev, M.

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

Shadrivov, I. V.

A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef]

Shalaev, V. M.

N. M. Litchinitser and V. M. Shalaev, Laser Phys. Lett. 5, 411 (2008).
[CrossRef]

Silberberg, Y.

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

Soukoulis, C. M.

Stegeman, G. I.

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

Stockman, M. I.

M. I. Stockman, Phys. Rev. Lett. 98, 177404 (2007).
[CrossRef]

Wegener, H.

Wen, S.

Y. Xiang, S. Wen, X. Dai, and D. Fan, Phys. Rev. E 82, 056605 (2010).
[CrossRef]

Wu, B.-I.

H. Chen, B.-I. Wu, and J. A. Kong. J. Electromagn Waves Appl. 20, 2137 (2006).
[CrossRef]

Xiang, Y.

Y. Xiang, S. Wen, X. Dai, and D. Fan, Phys. Rev. E 82, 056605 (2010).
[CrossRef]

Zharov, A. A.

A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef]

J. Electromagn Waves Appl. (1)

H. Chen, B.-I. Wu, and J. A. Kong. J. Electromagn Waves Appl. 20, 2137 (2006).
[CrossRef]

Laser Phys. Lett. (1)

N. M. Litchinitser and V. M. Shalaev, Laser Phys. Lett. 5, 411 (2008).
[CrossRef]

Opt. Commun. (1)

P. Y. P. Chen and B. A. Malomed, Opt. Commun. 283, 1598 (2010).
[CrossRef]

Opt. Lett. (1)

Opt. Spectrosc. (2)

A. Maimistov and E. V. Kazantseva, Opt. Spectrosc. 112, 264 (2012).
[CrossRef]

A. I. Maimistov, I. R. Gabitov, and N. M. Litchinitser, Opt. Spectrosc. 104, 253 (2008).
[CrossRef]

Phys. Rep. (1)

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

Phys. Rev. A (1)

E. Kazantseva, A. I. Maimistov, and S. S. Ozhenko, Phys. Rev. A 80, 043833 (2009).
[CrossRef]

Phys. Rev. E (1)

Y. Xiang, S. Wen, X. Dai, and D. Fan, Phys. Rev. E 82, 056605 (2010).
[CrossRef]

Phys. Rev. Lett. (3)

M. I. Stockman, Phys. Rev. Lett. 98, 177404 (2007).
[CrossRef]

A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef]

N. M. Litchinitser, I. R. Gabitov, and A. I. Maimistov, Phys. Rev. Lett. 99, 113902 (2007).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Array of alternating PIM and NIM waveguides. Arrows show oppositely directed Poynting vectors.

Fig. 2.
Fig. 2.

(a) Families of solitons in the array with: ν=2, κ=0.5, and χ1,2=1. The finite gap is |b|<3; the infinite gaps are b<2 and b>2. Stable (unstable) solitons correspond to solid (dashed) fragments of the curves. (b) and (c) Profiles of the stable soliton belonging to the lower families at b=2.1 and b=2.1, respectively. (d) Profile of the unstable soliton belonging to the upper family at b=2.1. Filled (empty) circles indicate PIM (NIM) waveguides.

Fig. 3.
Fig. 3.

(a) Families of solitons for the array with χ1=χ2=1 (other parameters as in Fig. 2). Profiles of stable (b) and unstable (c) solitons in a finite gap at b=0.5. (d) Profile of a stable soliton in the semi-infinite gap at b=2.1.

Fig. 4.
Fig. 4.

(b) and (c) Propagation with the input of PIM waveguides |un|2 and the output of the NIM waveguides |vn|2 (both given at z=0) coinciding with solitons shown, respectively, in Figs. 2(b) and 2(c), in the presence of dissipation γ=0.1. (b’) and (c’) Coefficients K computed for the propagations with different γ and the same input (output) for PIM (NIM) waveguides as in panels (b) and (c) and for the fixed length of the array L=10. Diamonds, triangles, and circles correspond to Ku, Kv, and logK.

Equations (2)

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iu˙n+νun+κ(vn1+vn)+χ1|un|2un=0,iv˙nνvnκ(un+un+1)χ2|vn|2vn=iγvn.
βκ2d2U0dx2b2[1β2]U0+[χ1+β4χ2]|U0|2U0=0.

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