Discrete vector solitons in two-dimensional nonlinear waveguide arrays: solutions, stability, and dynamics.

We identify and investigate bimodal (vector) solitons in models of square-lattice arrays of nonlinear optical waveguides. These vector self-localized states are, in fact, self-induced channels in a nonlinear photonic-crystal matrix. Such two-dimensional discrete vector solitons are possible in waveguide arrays in which each element carries two light beams that are either orthogonally polarized or have different carrier wavelengths. Estimates of the physical parameters necessary to support such soliton solutions in waveguide arrays are given. Using Newton relaxation methods, we obtain stationary vector-soliton solutions, and examine their stability through the computation of linearized eigenvalues for small perturbations. Our results may also be applicable to other systems such as two-component Bose-Einstein condensates trapped in a two-dimensional optical lattice.

[1]  A R Bishop,et al.  Twisted localized modes. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Y. A. Li,et al.  Structural stability of non-ground state traveling waves of coupled nonlinear Schro , 1998 .

[3]  Andrey Kobyakov,et al.  Dark spatial solitons in discrete cubic media with self- and cross-phase modulation , 1998 .

[4]  A. Davydov,et al.  The theory of contraction of proteins under their excitation. , 1973, Journal of theoretical biology.

[5]  Segev,et al.  Composite multihump vector solitons carrying topological charge , 2000, Physical review letters.

[6]  Pouget,et al.  Energy self-localization and gap local pulses in a two-dimensional nonlinear lattice. , 1993, Physical review. B, Condensed matter.

[7]  Joseph,et al.  Slow Bragg solitons in nonlinear periodic structures. , 1989, Physical review letters.

[8]  Resolved-Sideband Raman Cooling to the Ground State of an Optical Lattice , 1998, quant-ph/9801025.

[9]  Kim Ø. Rasmussen,et al.  Dynamics of breathers in discrete nonlinear Schro¨dinger models , 1998 .

[10]  A. Scott,et al.  Binding energy versus nonlinearity for a small stationary soliton , 1983 .

[11]  R. Morandotti,et al.  Optical discrete solitons in waveguide arrays. I. Soliton formation , 2002 .

[12]  J. S. Aitchison,et al.  Discrete Spatial Optical Solitons in Waveguide Arrays , 1998 .

[13]  Turitsyn,et al.  Stability of discrete solitons and quasicollapse to intrinsically localized modes. , 1994, Physical review letters.

[14]  Willis,et al.  Hamiltonian equations for multiple-collective-variable theories of nonlinear Klein-Gordon equations: A projection-operator approach. , 1988, Physical review. B, Condensed matter.

[15]  B. Malomed,et al.  Discrete vortex solitons. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Todd Kapitula,et al.  Exponentially small splitting of heteroclinc orbits: from the rapidly forced pendulum to discrete solitons , 2000 .

[17]  Alejandro B. Aceves,et al.  Self-induced transparency solitons in nonlinear refractive periodic media , 1989, Annual Meeting Optical Society of America.

[18]  A. Sievers,et al.  Intrinsic localized modes in anharmonic crystals. , 1988, Physical review letters.

[19]  R. Morandotti,et al.  Dynamics of discrete solitons in optical waveguide arrays , 1999 .

[20]  Peter S. Lomdahl,et al.  The discrete self-trapping equation , 1985 .

[21]  Malomed,et al.  Internal dynamics of a vector soliton in a nonlinear optical fiber. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[22]  B A Malomed,et al.  Stability of multiple pulses in discrete systems. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  A. Aceves,et al.  Generation of solitons in a nonlinear periodic medium. , 1992, Optics letters.

[24]  A Smerzi,et al.  Discrete solitons and breathers with dilute Bose-Einstein condensates. , 2001, Physical review letters.

[25]  A Bräuer,et al.  Anomalous refraction and diffraction in discrete optical systems. , 2002, Physical review letters.

[26]  Nonlinear Josephson-type oscillations of a driven, two-component Bose-Einstein condensate , 1998, cond-mat/9806337.

[27]  A. R. Bishop,et al.  Propagation failure, universal scalings and Goldstone modes , 2001 .

[28]  B. Malomed,et al.  Internal vibrations of a vector soliton in the coupled nonlinear Schrödinger equations , 1998 .

[29]  Yuri S. Kivshar,et al.  Discreteness-induced oscillatory instabilities of dark solitons , 1998, patt-sol/9811002.

[30]  Aubry,et al.  Growth and decay of discrete nonlinear Schrodinger breathers interacting with internal modes or standing-wave phonons , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[31]  Alan J. Heeger,et al.  Solitons in polyacetylene , 1979 .

[32]  S. Stenholm,et al.  Internal Josephson effect in trapped double condensates , 1999 .

[33]  B. Luther-Davies,et al.  Observation of dipole-mode vector solitons , 2000, Nonlinear Optics: Materials, Fundamentals, and Applications. Technical Digest. Postconference Edition. TOPS Vol.46 (IEEE Cat. No.00CH37174).

[34]  E. V. Zemlyanaya,et al.  Vibrations and Oscillatory Instabilities of Gap Solitons , 1998 .

[35]  M Segev,et al.  Observation of two-dimensional multimode solitons. , 2000, Optics letters.

[36]  Rubenchik,et al.  Energy localization in nonlinear fiber arrays: Collapse-effect compressor. , 1995, Physical review letters.

[37]  M. Grillakis,et al.  Analysis of the linearization around a critical point of an infinite dimensional Hamiltonian system , 1990 .

[38]  U. Peschel,et al.  Oscillation and decay of discrete solitons in modulated waveguide arrays , 2002 .

[39]  P. Kevrekidis,et al.  Stability of waves in discrete systems , 2001 .

[40]  Malomed,et al.  Vibration modes of a gap soliton in a nonlinear optical medium. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[41]  Demetrios N. Christodoulides,et al.  Observation of Multihump Multimode Solitons , 1998 .

[42]  J. C. van der Meer,et al.  Hamiltonian Hopf bifurcation with symmetry , 1990 .

[43]  J. Aitchison,et al.  Kerr spatiotemporal self-focusing in a planar glass waveguide. , 2001, Physical review letters.

[44]  V. Arnold,et al.  Mathematical aspects of classical and celestial mechanics , 1997 .

[45]  D. Christodoulides,et al.  Vector solitons in birefringent nonlinear dispersive media. , 1988, Optics letters.

[46]  Alan R. Bishop,et al.  Comparison of one-dimensional and two-dimensional discrete breathers , 2001 .

[47]  F. Lederer,et al.  Strongly localized vectorial modes in nonlinear waveguide arrays , 1998 .

[48]  Alwyn C. Scott,et al.  Davydov’s Soliton , 1992 .

[49]  M. Ablowitz,et al.  Discrete diffraction managed spatial solitons. , 2001, Physical review letters.

[50]  J. Carr,et al.  Stability of stationary solutions of the discrete self-trapping equation , 1985 .

[51]  Kivshar,et al.  Dipole-mode vector solitons , 1999, Physical review letters.

[52]  Exner,et al.  Local geometry of Fe3+ ions on the potassium sites in KTaO3. , 1993, Physical review. B, Condensed matter.

[53]  Demetrios N. Christodoulides,et al.  Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices , 2003, Nature.

[54]  Bishop,et al.  Two-dimensional discrete breathers: construction, stability, and bifurcations , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[55]  Mordechai Segev,et al.  Discrete solitons in photorefractive optically induced photonic lattices. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[56]  Peng,et al.  Symmetric and asymmetric solitons in twin-core nonlinear optical fibers. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[57]  R. Ballagh,et al.  Theory of an Output Coupler for Bose-Einstein Condensed Atoms , 1997 .

[58]  M. Grillakis,et al.  Linearized instability for nonlinear Schr?odinger and Klein-Gordon equations , 1988 .

[59]  V. Konotop,et al.  Nonlinear excitations in arrays of Bose-Einstein condensates , 2001, cond-mat/0106042.

[60]  Yuri S. Kivshar,et al.  Nonlinear dynamics of the Frenkel—Kontorova model , 1998 .

[61]  Yuri S. Kivshar,et al.  Stability of multihump optical solitons , 1999, patt-sol/9905006.

[62]  Kim Ø. Rasmussen,et al.  THE DISCRETE NONLINEAR SCHRÖDINGER EQUATION: A SURVEY OF RECENT RESULTS , 2001 .

[63]  John E. Sipe,et al.  III Gap Solitons , 1994 .

[64]  Demetrios N. Christodoulides,et al.  Blocking and Routing Discrete Solitons in Two-Dimensional Networks of Nonlinear Waveguide Arrays , 2001 .

[65]  D. Christodoulides,et al.  Discrete self-focusing in nonlinear arrays of coupled waveguides. , 1988, Optics letters.

[66]  D. Christodoulides,et al.  Nonlinear wave propagation in optically active and birefringent media. , 1991, Optics letters.

[67]  Sheppard,et al.  Bifurcation phenomena and multiple soliton-bound states in isotropic Kerr media. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.