Abstract

We theoretically study gap random-phase lattice solitons (gap-RPLSs) in nonlinear waveguide arrays with self-defocusing nonlinearity. We find that the intensity structure and statistical (coherence) properties of gap-RPLSs conform to the lattice periodicity, while their Floquet-Bloch power spectrum is multi-humped with peaks in the anomalous diffractions regions. It is shown that a gap-RPLS can be generated when a simple incoherent beam with bell-shaped power spectrum and single-hump intensity is launched at a proper angle into the waveguide array. The input incoherent beam evolves in the lattice while shedding off some radiation, and eventually attains the features of gap-RPLS.

© 2005 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  37. A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, �??Incoherent solitons in instantaneous response nonlinear media,�?? Phys. Rev. Lett. 92, 143906 (2004).
    [CrossRef] [PubMed]
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JETP

S. Darmanyan, A. Kobyakov, and F. Lederer, �??Stability of strongly localized excitations in discrete media with cubic nonlinearity�??, JETP 86, 682-686 (1998).
[CrossRef]

Nature

D.N. Christodoulides, F. Lederer, and Y. Silberberg, �??Discretizing light behaviour in linear and nonlinear waveguide lattices,�?? Nature (London) 424 817-823 (2003).
[CrossRef]

J.W. Fleischer, M. Segev, N.K. Efremidis, and D.N. Christodoulides, �??Observation of two-dimensional discrete solitons in optically-induced nonlinear photonic lattices,�?? Nature (London) 422, 147 (2003).
[CrossRef]

O. Cohen, G. Bartal, H. Buljan, J.W. Fleischer, T. Carmon, M. Segev, and D.N. Christodoulides, �??Observation of random-phase lattice solitons,�?? Nature (London) 433, 500 (2005).
[CrossRef]

Opt. Lett.

Optics Express

K. Motzek, A.A. Sukhorukov, F. Kaiser, and Y.S. Kivshar, �??Incoherent multi-gap optical solitons in nonlinear photonic lattices,�?? Optics Express 13 2916 (2005).
[CrossRef] [PubMed]

P. Natl. Acad. Sci. USA

Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides and M. Segev, �??Clustering of solitons in weakly correlated wavefronts,�?? P. Natl. Acad. Sci. USA 99, 5223-5227 (2002).
[CrossRef]

Phys. Rev. E

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, �??Statistical theory for incoherent light propagation in nonlinear media,�?? Phys. Rev. E 65, 035602 (2002).
[CrossRef]

S.A. Ponomarenko, �??Twisted Gaussian Schell-model solitons,�?? Phys. Rev. E 64, 036618 (2001).
[CrossRef]

Phys. Rev. Lett.

M. Mitchell, Z. Chen, M. Shih, and M. Segev, �??Self-trapping of partially spatially incoherent light,�?? Phys. Rev. Lett. 77, 490-493 (1996).
[CrossRef] [PubMed]

D.N. Christodoulides, T.H. Coskun, M. Mitchell, and M. Segev, �??Theory of incoherent self-focusing in biased photorefractive media,�?? Phys. Rev. Lett. 78, 646-649 (1997).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D.N. Christodoulides, �??Theory of self-trapped spatially incoherent light beams,�?? Phys. Rev. Lett. 79, 4990-4993 (1997).
[CrossRef]

W. Chen and D. L. Mills, �??Gap solitons and the nonlinear optical-response of superlattices,�?? Phys. Rev. Lett. 58, 160-163 (1987).
[CrossRef] [PubMed]

G. Bartal, O. Cohen, H. Buljan, J.W. Fleischer, O. Manela, and M. Segev, �??Brillouin zone spectroscopy of nonlinear photonic lattices,�?? Phys. Rev. Lett. 94, 163902 (2005).
[CrossRef] [PubMed]

T. Schwartz, T. Carmon, H. Buljan, and M. Segev, �??Spontaneous pattern formation with incoherent white light,�?? Phys. Rev. Lett. 93, (2004).
[CrossRef] [PubMed]

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, �??Incoherent solitons in instantaneous response nonlinear media,�?? Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef] [PubMed]

V.V. Shkunov and D. Anderson, �??Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media,�?? Phys. Rev. Lett. 81, 2683-2686 (1998).
[CrossRef]

A.W. Snyder and D.J. Mitchell, �??Big incoherent solitons,�?? Phys. Rev. Lett. 80, 1422-1425 (1998).
[CrossRef]

M. Solja¡�?i�?, M. Segev, T.H. Coskun, D.N. Christodoulides, and A. Vishwanath, �??Modulation instability of incoherent beams in noninstantaneous nonlinear media,�?? Phys. Rev. Lett. 84, 467-470 (2000).
[CrossRef] [PubMed]

H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, and J.S. Aitchison, �??Discrete spatial optical solitons in waveguide arrays,�?? Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

H.S. Eisenberg, Y. Silberberg, R. Morandotti, and J.S. Aitchison, �??Diffraction menagement,�?? Phys. Rev. Lett. 85, 1863-1866 (2000)
[CrossRef] [PubMed]

R. Morandotti, H.S. Eisenberg, Y. Silberberg, M. Sorel, and J.S. Aitchison, �??Self-Focusing and defocusing in waveguide arrays,�?? Phys. Rev. Lett. 86 3296 (2001).
[CrossRef] [PubMed]

J.W. Fleischer, T. Carmon, M. Segev, N.K. Efremidis, and D.N. Christodoulides, �??Observation of discrete solitons in optically-induced real time waveguide arrays�?? Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, �??Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,�?? Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, �??Gap solitons in waveguide arrays,�?? Phys. Rev. Lett. 92, 093904 (2004).
[CrossRef] [PubMed]

D. Neshev, A. A. Sukhorukov, B. Hanna,W. Krolikowski, and Yu. S. Kivshar, �??Controlled generation and steering of spatial gap solitons,�?? Phys. Rev. Lett. 93, 0839054 (2004).
[CrossRef]

H. Buljan, O. Cohen, J.W. Fleischer, T. Schwartz, M. Segev, Z.H. Musslimani, N.K. Efremidis, and D.N. Christodoulides, �??Random-phase solitons in nonlinear periodic lattices,�?? Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef] [PubMed]

O. Cohen, T. Schwartz, J.W. Fleischer, M. Segev, and D.N. Christodoulides, �??Multiband vector lattice solitons�?? Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef] [PubMed]

A.A. Sukhorukov and Y.S. Kivshar, �??Multigap discrete vector solitons,�?? Phys. Rev. Lett. 91, 113902 (2003).
[CrossRef] [PubMed]

Science

D. Kip, M. Solja¡�?i�?, M. Segev, E. Eugenieva, and D. N. Christodoulides, �??Modulation instability and pattern formation in spatially incoherent light beams,�?? Science 290, 495-498 (2000).
[CrossRef] [PubMed]

Z.G. Chen, M. Mitchell, M. Segev, T.H. Coskun, and D.N. Christodoulides, �??Self-trapping of dark incoherent light beams,�?? Science 280, 889-892 (1998).
[CrossRef] [PubMed]

Stud. Appl. Math

H. Buljan, G. Bartal, O. Cohen, T. Schwartz, O. Manela, T. Carmon, M. Segev, J.W. Fleischer, and D.N. Christodoulides, �??Partially coherent waves in nonlinear periodic lattices,�?? Stud. Appl. Math., in print (2005).
[CrossRef]

Zh. Tekh. Fiz.

Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin, �??Stationary waves in nonlinear, periodically modulated media with higher group retardation,�?? Zh. Tekh. Fiz. 51, 902-907 (1981).

Other

M. Segev and D.N. Christodoulides, Incoherent Solitons in Spatial Solitons, S. Trillo and W. Torruellas eds. (Springer, Berlin, 2001) pp. 87-125.

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Figures (6)

Fig. 1.
Fig. 1.

The band-gap structure of the lattice, and the power spectra of the first gap-RPLS example. (a) Band-gap structure of the lattice (solid lines), and propagation constants of the RPLS modes (closed circles). (b) Fourier power spectrum of a gap-RPLS in the self-defocusing nonlinear waveguide array (solid line); the humps in the spectrum are located mainly in the anomalous diffraction regions. The Fourier power spectrum of an input incoherent beam launched at an angle is indicated by the dot-dashed line, and spectrum of the output beam is given by the dashed line (the dot-dashed line and the dashed line almost fully coincide). (c) Same as in (b) but with the Floquet-Bloch power spectrum of the beams. See text for details.

Fig. 2.
Fig. 2.

The intensity structure and coherence properties of the first example of a gap-RPLS. (a) The complex coherence factor µ(x,x )∊[-1,1] of the gap-RPLS; black (white) corresponds to a value of -1 (1). (b) The intensity profile I(x)/IS (solid line) of a gap-RPLS. The intensity structure of an incoherent beam launched at angle θ=0.7π/Dk, and propagated for 50 mm (dashed line). Dotted line illustrates the diffraction-broadened beam after z=30 mm propagation in a linear lattice, for an input beam with a wavefunction identical to that of a gap-RPLS. The vertical lines represent the lattice sites.

Fig. 3.
Fig. 3.

The stable evolution of the (a) intensity structure, (b) Fourier power spectrum, and (c) Floquet-Bloch power spectrum of the first gap-RPLS example. The evolution of an incoherent beam launched at an angle θ=0.7π/Dk into the nonlinear waveguide array; (d) intensity structure, (e) Fourier power spectrum, and (f) Floquet-Bloch power spectrum.

Fig. 4.
Fig. 4.

Same as in Figure 1 for the second example of a gap-RPLS. See text for details.

Fig. 5.
Fig. 5.

Same as Figure 2 for the second example of a gap-RPLS. See text.

Fig. 6.
Fig. 6.

Same as Figure 3 for the second example of a gap-RPLS. See text for details.

Equations (2)

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i ψ m z + 1 2 k 2 ψ m x 2 + V ( x , z ) k n 0 ψ m ( x , z ) = 0 .
1 2 k d 2 u n , l d x 2 + V ( x ) k n 0 u n , 1 ( x , z ) = κ n , l u n , l ,

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