Abstract

The light ray of a spatial soliton in an optical film whose refractive index is smoothly modulated (wavelength much larger than the typical soliton width) in both spatial directions is shown to possess chaotic regimes for which the propagation is erratic. This is interpreted as a parametric driven pendulum, obtained by what we believe to be a new perturbative approach of the Maxwell’s equation. These findings are then demonstrated to compare well to the eikonal law of light ray propagation (nonlinearity compensates diffraction).

© 2009 Optical Society of America

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2008

D. Song, C. Lou, L. Tang, X. Wang, W. Li, X. Chen, K. J. H. Law, H. Susanto, P. G. Kevrekidis, J. Xu, and Z. Chen, Opt. Express 16, 10110 (2008).
[CrossRef] [PubMed]

Y. Sivan, G. Fibich, N. K. Efremidis, and S. Bar-Ad, Nonlinearity 21, 509 (2008).
[CrossRef]

K. Staliunas, O. Egorov, Y. Kivshar, and F. Lederer, Phys. Rev. Lett. 101, 153903 (2008).
[CrossRef] [PubMed]

2004

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 093904 (2004).
[CrossRef] [PubMed]

2002

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

1998

M. Segev and G. Stegeman, Phys. Today 51, 42 (1998).
[CrossRef]

1997

M Mitchell and M. Segev, Nature 387, 880 (1997).
[CrossRef]

E. M. Gromov, Phys. Lett. A 227, 67 (1997).
[CrossRef]

W. Krolikowski and S. A. Holmstrom, Opt. Lett. 22, 369 (1997).
[CrossRef] [PubMed]

1995

1994

M. Segev, G. C. Valley, B. Crosignani, P. D. Porto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

1993

1991

1974

M. Oikawa and N. Yajima, J. Phys. Soc. Jpn. 37, 486 (1974).
[CrossRef]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Nonlinear Crystals (Academic, 2003).

Aitchison, J. S.

Bar-Ad, S.

Y. Sivan, G. Fibich, N. K. Efremidis, and S. Bar-Ad, Nonlinearity 21, 509 (2008).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 2002).

Carmon, T.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

Carvalho, M. I.

Chen, X.

Chen, Z.

Christodoulides, D. N.

Cohen, O.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

Crosignani, B.

M. Segev, G. C. Valley, B. Crosignani, P. D. Porto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

Efremidis, N. K.

Y. Sivan, G. Fibich, N. K. Efremidis, and S. Bar-Ad, Nonlinearity 21, 509 (2008).
[CrossRef]

Egorov, O.

K. Staliunas, O. Egorov, Y. Kivshar, and F. Lederer, Phys. Rev. Lett. 101, 153903 (2008).
[CrossRef] [PubMed]

Fibich, G.

Y. Sivan, G. Fibich, N. K. Efremidis, and S. Bar-Ad, Nonlinearity 21, 509 (2008).
[CrossRef]

Fleischer, J. W.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

Gromov, E. M.

E. M. Gromov, Phys. Lett. A 227, 67 (1997).
[CrossRef]

Holmstrom, S. A.

Jackel, J. L.

Kevrekidis, P. G.

Kivshar, Y.

K. Staliunas, O. Egorov, Y. Kivshar, and F. Lederer, Phys. Rev. Lett. 101, 153903 (2008).
[CrossRef] [PubMed]

Kivshar, Y. S.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Nonlinear Crystals (Academic, 2003).

Krolikowski, W.

Law, K. J. H.

Leaird, D. E.

Lederer, F.

K. Staliunas, O. Egorov, Y. Kivshar, and F. Lederer, Phys. Rev. Lett. 101, 153903 (2008).
[CrossRef] [PubMed]

Li, W.

Lou, C.

Mandelik, D.

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 093904 (2004).
[CrossRef] [PubMed]

Mitchell, M

M Mitchell and M. Segev, Nature 387, 880 (1997).
[CrossRef]

Morandotti, R.

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 093904 (2004).
[CrossRef] [PubMed]

Odoulov, S.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

Oikawa, M.

M. Oikawa and N. Yajima, J. Phys. Soc. Jpn. 37, 486 (1974).
[CrossRef]

Oliver, M. K.

Porto, P. D.

M. Segev, G. C. Valley, B. Crosignani, P. D. Porto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

Segev, M.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

M. Segev and G. Stegeman, Phys. Today 51, 42 (1998).
[CrossRef]

M Mitchell and M. Segev, Nature 387, 880 (1997).
[CrossRef]

M. Segev, G. C. Valley, B. Crosignani, P. D. Porto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

Sheppard, A. P.

Silberberg, Y.

Sivan, Y.

Y. Sivan, G. Fibich, N. K. Efremidis, and S. Bar-Ad, Nonlinearity 21, 509 (2008).
[CrossRef]

Smith, P. W. E.

Snyder, A. W.

Song, D.

Staliunas, K.

K. Staliunas, O. Egorov, Y. Kivshar, and F. Lederer, Phys. Rev. Lett. 101, 153903 (2008).
[CrossRef] [PubMed]

Starrett, J.

J. Starrett and R. Tagg, Phys. Rev. Lett. 74, 1974 (1995).
[CrossRef] [PubMed]

Stegeman, G.

M. Segev and G. Stegeman, Phys. Today 51, 42 (1998).
[CrossRef]

Susanto, H.

Tagg, R.

J. Starrett and R. Tagg, Phys. Rev. Lett. 74, 1974 (1995).
[CrossRef] [PubMed]

Tang, L.

Uzdin, R.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

Valley, G. C.

M. Segev, G. C. Valley, B. Crosignani, P. D. Porto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

Wang, X.

Weiner, A. M.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 2002).

Xu, J.

Yajima, N.

M. Oikawa and N. Yajima, J. Phys. Soc. Jpn. 37, 486 (1974).
[CrossRef]

Yariv, A.

M. Segev, G. C. Valley, B. Crosignani, P. D. Porto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B

J. Phys. Soc. Jpn.

M. Oikawa and N. Yajima, J. Phys. Soc. Jpn. 37, 486 (1974).
[CrossRef]

Nature

M Mitchell and M. Segev, Nature 387, 880 (1997).
[CrossRef]

Nonlinearity

Y. Sivan, G. Fibich, N. K. Efremidis, and S. Bar-Ad, Nonlinearity 21, 509 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Lett. A

E. M. Gromov, Phys. Lett. A 227, 67 (1997).
[CrossRef]

Phys. Rev. Lett.

J. Starrett and R. Tagg, Phys. Rev. Lett. 74, 1974 (1995).
[CrossRef] [PubMed]

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 093904 (2004).
[CrossRef] [PubMed]

K. Staliunas, O. Egorov, Y. Kivshar, and F. Lederer, Phys. Rev. Lett. 101, 153903 (2008).
[CrossRef] [PubMed]

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. D. Porto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

Phys. Today

M. Segev and G. Stegeman, Phys. Today 51, 42 (1998).
[CrossRef]

Other

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Nonlinear Crystals (Academic, 2003).

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 2002).

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Figures (1)

Fig. 1
Fig. 1

Graph (a), contour plots of the electric field envelope E ( x , z ) solution of the NLS equation (3) in the potential [Eq. (4)] with initial conditions [Eq. (5)] for two slightly different values of the injection point x 0 = 460 for the red (r) ray and x 0 = 461 for the blue (b) one. The refractive index landscape [Eq. (4)] is plotted for reference. Graph (b) is the driven pendulum approximation [Eq. (13)] and graph (c) is the eikonal model [Eq. (15)], both simulated with the same initial conditions ( A = 2 in all of those three cases). The lower graphs (d)–(f) describe the corresponding simulations with the same initial conditions and parameters except for the longitudinal modulation Ω replaced with 2 Ω (then here A = 0.5 ).

Equations (19)

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( 2 x 2 + 2 z 2 ) E 1 c 2 2 t 2 { n 2 E + χ E 3 } = 0.
E = 2 / 3 E ( ε x , ε 2 z ) e i [ ω t k z ] + c .c . + O ( ε 2 ) .
i E z + 1 2 2 E x 2 + χ | E | 2 E n 0 n ( x , z ) n 0 E = 0 ,
n ( x , z ) = n 0 [ 1 χ V 0 sin 2 ( K x ) sin 2 ( Ω z ) ] ,
E ( x , 0 ) = a   sech [ a χ ( x x 0 ) ] ,
a = 0.9 ,     χ = 0.01 ,     V 0 = 0.2 ,
K = χ π / 25 ,     Ω = K χ V 0 ,
E = ε 2 3 χ Ψ ( ξ 1 , ζ ) e i θ ( ξ 2 , ζ ) e i [ ω t k z ] + c.c. + ... ,
( n ( x , z ) n 0 ) / n 0 = ε 2 V ( ξ 2 , ζ ) ,
ξ 1 = ε [ x ϕ ( ξ 2 , ζ ) ] ,     ξ 2 = ε 2 x ,     ζ = ε 2 z .
i Ψ ζ + 1 2 2 Ψ ξ 1 2 + | Ψ | 2 Ψ = 0 ,     θ ζ = V .
ϕ ζ = θ ξ 2 .
2 ϕ z 2 = 1 n 0 n ( x , z ) x .
Ψ s = a e ( i / 2 ) a 2 χ z   sech [ a χ ( x ϕ ) ] ,
d 2 x s d z 2 2 ϕ ( x s , z ) z 2 = 1 n 0 n ( x s , z ) x s ,
d 2 q d t 2 = A sin 2 ( t ) sin ( q ) ,     A = 2 χ V 0 K 2 Ω 2 ,
d d s ( n d r d s ) = n ,
x = [ 1 + ( x ) 2 ] 1 n ( n x x n z ) ,
x 1 n 0 n x ,

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