Abstract

We analytically predict the existence of both spatial bright and dark counterpropagating solitons in a reflection grating in the presence of the Kerr nonlinearity. The basic trapping mechanism consists of a twofold balance where diffraction is compensated by self-focusing and reflection is altered by the nonlinear-induced interferometric grating. We find that, whenever the spectral soliton profile lies within the grating stop band, bright and dark solitons exist only if the mutual phase of the counterpropagating solitons is π or 0, respectively.

© 2006 Optical Society of America

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  1. W. Chen and D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
    [CrossRef] [PubMed]
  2. S. Trillo and W. Torruellas, Spatial Solitons (Springer, 2001).
  3. D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
    [CrossRef] [PubMed]
  4. R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, Phys. Rev. Lett. 86, 3296 (2001).
    [CrossRef] [PubMed]
  5. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 23902 (2003).
    [CrossRef]
  6. J. W. Fleischer, M. Segev, N. K. Efremidis, D. N. Christodoulides, Nature 422, 147 (2003).
    [CrossRef] [PubMed]
  7. O. Cohen, T. Carmon, M. Segev, and S. Odoulov, Opt. Lett. 15, 2031 (2002).
    [CrossRef]
  8. J. Feng, Opt. Lett. 18, 1302 (1993).
    [CrossRef] [PubMed]
  9. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
    [CrossRef] [PubMed]
  10. O. Cohen, R. Uzdin, T. Carmon, J. W. Flescher, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 1339001 (2002).
  11. Note that, for η=0, symmetric gap solitons in a transmission grating have the same amplitude profiles as in Eqs. .
  12. D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 93904 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]

2005 (1)

2004 (1)

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 93904 (2004).
[CrossRef]

2003 (3)

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 23902 (2003).
[CrossRef]

J. W. Fleischer, M. Segev, N. K. Efremidis, D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

2002 (3)

O. Cohen, R. Uzdin, T. Carmon, J. W. Flescher, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 1339001 (2002).

O. Cohen, T. Carmon, M. Segev, and S. Odoulov, Opt. Lett. 15, 2031 (2002).
[CrossRef]

O. Cohen, S. Lan, T. Carmon, J. A. Giordmaine, and M. Segev, Opt. Lett. 27, 2013 (2002).
[CrossRef]

2001 (1)

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, Phys. Rev. Lett. 86, 3296 (2001).
[CrossRef] [PubMed]

1996 (1)

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

1993 (2)

M. Haelterman, A. P. Sheppard, and A. W. Snyder, Opt. Commun. 103, 145 (1993).
[CrossRef]

J. Feng, Opt. Lett. 18, 1302 (1993).
[CrossRef] [PubMed]

1987 (1)

W. Chen and D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
[CrossRef] [PubMed]

Aitchison, J. S.

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 93904 (2004).
[CrossRef]

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, Phys. Rev. Lett. 86, 3296 (2001).
[CrossRef] [PubMed]

Belic, M.

Carmon, T.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 23902 (2003).
[CrossRef]

O. Cohen, T. Carmon, M. Segev, and S. Odoulov, Opt. Lett. 15, 2031 (2002).
[CrossRef]

O. Cohen, R. Uzdin, T. Carmon, J. W. Flescher, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 1339001 (2002).

O. Cohen, S. Lan, T. Carmon, J. A. Giordmaine, and M. Segev, Opt. Lett. 27, 2013 (2002).
[CrossRef]

Chen, W.

W. Chen and D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
[CrossRef] [PubMed]

Christodoulides, D. N.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 23902 (2003).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef] [PubMed]

Cohen, O.

O. Cohen, R. Uzdin, T. Carmon, J. W. Flescher, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 1339001 (2002).

O. Cohen, S. Lan, T. Carmon, J. A. Giordmaine, and M. Segev, Opt. Lett. 27, 2013 (2002).
[CrossRef]

O. Cohen, T. Carmon, M. Segev, and S. Odoulov, Opt. Lett. 15, 2031 (2002).
[CrossRef]

de Sterke, C. M.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

Denz, C.

Efremidis, N. K.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 23902 (2003).
[CrossRef]

J. W. Fleischer, M. Segev, N. K. Efremidis, D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef] [PubMed]

Eggleton, B. J.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

Eisenberg, H. S.

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, Phys. Rev. Lett. 86, 3296 (2001).
[CrossRef] [PubMed]

Feng, J.

Fleischer, J. W.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 23902 (2003).
[CrossRef]

J. W. Fleischer, M. Segev, N. K. Efremidis, D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef] [PubMed]

Flescher, J. W.

O. Cohen, R. Uzdin, T. Carmon, J. W. Flescher, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 1339001 (2002).

Giordmaine, J. A.

Haelterman, M.

M. Haelterman, A. P. Sheppard, and A. W. Snyder, Opt. Commun. 103, 145 (1993).
[CrossRef]

Jander, Ph.

Jovic, D.

Krug, P. A.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

Lan, S.

Lederer, F.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

Mandelik, D.

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 93904 (2004).
[CrossRef]

Mills, D. L.

W. Chen and D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
[CrossRef] [PubMed]

Morandotti, R.

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 93904 (2004).
[CrossRef]

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, Phys. Rev. Lett. 86, 3296 (2001).
[CrossRef] [PubMed]

Odoulov, S.

O. Cohen, R. Uzdin, T. Carmon, J. W. Flescher, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 1339001 (2002).

O. Cohen, T. Carmon, M. Segev, and S. Odoulov, Opt. Lett. 15, 2031 (2002).
[CrossRef]

Petrovic, M.

Schröeder, J.

Segev, M.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 23902 (2003).
[CrossRef]

J. W. Fleischer, M. Segev, N. K. Efremidis, D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef] [PubMed]

O. Cohen, R. Uzdin, T. Carmon, J. W. Flescher, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 1339001 (2002).

O. Cohen, S. Lan, T. Carmon, J. A. Giordmaine, and M. Segev, Opt. Lett. 27, 2013 (2002).
[CrossRef]

O. Cohen, T. Carmon, M. Segev, and S. Odoulov, Opt. Lett. 15, 2031 (2002).
[CrossRef]

Sheppard, A. P.

M. Haelterman, A. P. Sheppard, and A. W. Snyder, Opt. Commun. 103, 145 (1993).
[CrossRef]

Silberberg, Y.

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 93904 (2004).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, Phys. Rev. Lett. 86, 3296 (2001).
[CrossRef] [PubMed]

Sipe, J. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

Slusher, R. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

Snyder, A. W.

M. Haelterman, A. P. Sheppard, and A. W. Snyder, Opt. Commun. 103, 145 (1993).
[CrossRef]

Sorel, M.

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, Phys. Rev. Lett. 86, 3296 (2001).
[CrossRef] [PubMed]

Torruellas, W.

S. Trillo and W. Torruellas, Spatial Solitons (Springer, 2001).

Trillo, S.

S. Trillo and W. Torruellas, Spatial Solitons (Springer, 2001).

Uzdin, R.

O. Cohen, R. Uzdin, T. Carmon, J. W. Flescher, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 1339001 (2002).

Nature (2)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

M. Haelterman, A. P. Sheppard, and A. W. Snyder, Opt. Commun. 103, 145 (1993).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. Lett. (6)

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

O. Cohen, R. Uzdin, T. Carmon, J. W. Flescher, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 1339001 (2002).

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, Phys. Rev. Lett. 86, 3296 (2001).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 23902 (2003).
[CrossRef]

W. Chen and D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
[CrossRef] [PubMed]

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 93904 (2004).
[CrossRef]

Other (2)

S. Trillo and W. Torruellas, Spatial Solitons (Springer, 2001).

Note that, for η=0, symmetric gap solitons in a transmission grating have the same amplitude profiles as in Eqs. .

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Figures (1)

Fig. 1
Fig. 1

Dispersion relation (solid curve) associated with the linearized system of Eqs. (4) in the three possible different regimes (a), (b), and (c). The absolute value of the transverse Fourier spectrum of the bright solitons F [ u b ] (dotted curve) of width L b = 4 η σ g π and of the derivative of the dark solitons F [ d u d d ξ ] (dashed curve) of width L d 2 ( σ g η ) are reported using arbitrary units for their amplitudes.

Equations (10)

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( 2 x 2 + 2 z 2 ) E + k 2 [ 1 + 2 n 0 δ n ( z ) + 2 n 2 n 0 E 2 ] E = 0 ,
E ( x , z ) = n 0 2 n 2 [ exp ( i ζ ) U + ( ξ , ζ ) + exp ( i ζ ) U ( ξ , ζ ) ] ,
( 1 + η 2 ) 2 k = 2 π L N
( i ζ + 1 2 2 ξ 2 ) U + = c N exp ( i η ζ ) U γ 2 [ U + 2 + 2 U 2 ] U + ,
( i ζ 1 2 2 ξ 2 ) U = c N exp ( i η ζ ) U + + γ 2 [ 2 U + 2 + U 2 ] U ,
U ± ( ξ , ζ ) = σ ± exp [ ± i 2 ( η ζ + θ ) ] u ( ξ ) ,
u = ( η σ g ) u 3 γ u 3 ,
δ n eff = 2 n 0 [ γ σ u 2 ( ξ ) + g ] cos ( 2 π N k L ζ + θ ) .
u b ( ξ ) = 2 3 ( η σ g ) sech ( η σ g ξ ) ,
u d ( ξ ) = 1 3 ( σ g η ) tanh ( σ g η 2 ξ ) ,

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