Dipole and Quadrupole Solitons in Optically Induced Two‐Dimensional Photonic Lattices: Theory and Experiment

Yang, Jianke; Makasyuk, Igor; Bezryadina, Anna; Chen, Zhigang

doi:10.1111/j.0022-2526.2004.01539.x

Cited by 102 publications

(75 citation statements)

References 43 publications

(41 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We note in passing that stable dipole states on the infinite lattice were predicted in Ref. [17], and later observed experimentally in a photorefractive crystal [18]. All the above seed configurations are real, and the horseshoe may, in principle, also be regarded as a truncated quadrupole, which is a real solution as well [19].…”

Section: Perturbation Analysismentioning

confidence: 53%

Discrete surface solitons in two dimensions

et al. 2007

View full text Add to dashboard Cite

We investigate fundamental localized modes in 2D lattices with an edge (surface). Interaction with the edge expands the stability area for ordinary solitons, and induces a difference between perpendicular and parallel dipoles; on the contrary, lattice vortices cannot exist too close to the border. Furthermore, we show analytically and numerically that the edge stabilizes a novel wave species, which is entirely unstable in the uniform lattice, namely, a "horseshoe" soliton, consisting of 3 sites. Unstable horseshoes transform themselves into a pair of ordinary solitons.

show abstract

Section: Perturbation Analysismentioning

confidence: 53%

Discrete surface solitons in two dimensions

et al. 2007

View full text Add to dashboard Cite

show abstract

“…This is the natural outcome of the topological instability [42]. When a dc field is applied along the crystalline c-axis, the SBN crystal turns into a self-focusing medium [5,9,10,11,12,13]. Under a proper strength of the nonlinearity, the vortex beam evolves into a S = 2 vortex soliton.…”

Section: Resultsmentioning

confidence: 99%

“…For our consideration of the continuum problem, we use the non-dimensionalized version of the photorefractive model with saturable nonlinearity, as developed in detail in [12,40], in the following form:…”

Section: Continuum Models: Bifurcation Analysis and Dynamical Evolutionmentioning

confidence: 99%

Geometric stabilization of extendedS=2vortices in two-dimensional photonic lattices: Theoretical analysis, numerical computation, and experimental results

et al. 2009

Self Cite

View full text Add to dashboard Cite

In this work, we focus our studies on the subject of nonlinear discrete self-trapping of S = 2 (doubly-charged) vortices in two-dimensional photonic lattices, including theoretical analysis, numerical computation and experimental demonstration. We revisit earlier findings about S = 2 vortices with a discrete model, and find that S = 2 vortices extended over eight lattice sites can indeed be stable (or only weakly unstable) under certain conditions, not only for the cubic nonlinearity previously used, but also for a saturable nonlinearity more relevant to our experiment with a biased photorefractive nonlinear crystal. We then use the discrete analysis as a guide towards numerically identifying stable (and unstable) vortex solutions in a more realistic continuum model with a periodic potential. Finally, we present our experimental observation of such geometrically extended S = 2 vortex solitons in optically induced lattices under both self-focusing and self-defocusing nonlinearities, and show clearly that the S = 2 vortex singularities are preserved during nonlinear propagation.

show abstract

“…Due to the two inequalities (27) and (28) on eigenvalues Λ M for the two cases (25) and (26), we see that all eigenvalues of L M are non-positive, and the kernel of L M is the same as the kernel of L 1 . In addition, the convergence condition |1 + Λ M ∆t| < 1 for non-zero eigenvalues Λ M of L M will be satisfied if…”

Section: Theorem 2 Let Assumption 1 Be Valid Andmentioning

confidence: 94%

Universally‐Convergent Squared‐Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations

2007

Self Cite

View full text Add to dashboard Cite

Summary:Three new iteration methods, namely the squared-operator method, the modified squaredoperator method, and the power-conserving squared-operator method, for solitary waves in general scalar and vector nonlinear wave equations are proposed. These methods are based on iterating new differential equations whose linearization operators are squares of those for the original equations, together with acceleration techniques. The first two methods keep the propagation constants fixed, while the third method keeps the powers (or other arbitrary functionals) of the solution fixed. It is proved that all these methods are guaranteed to converge to any solitary wave (either ground state or not) as long as the initial condition is sufficiently close to the corresponding exact solution, and the time step in the iteration schemes is below a certain threshold value. Furthermore, these schemes are fast-converging, highly accurate, and easy to implement. If the solitary wave exists only at isolated propagation constant values, the corresponding squared-operator methods are developed as well. These methods are applied to various solitary wave problems of physical interest, such as higher-gap vortex solitons in the two-dimensional nonlinear Schrödinger equations with periodic potentials, and isolated solitons in Ginzburg-Landau equations, and some new types of solitary wave solutions are obtained. It is also demonstrated that the modified squared-operator method delivers the best performance among the methods proposed in this article.

show abstract

Dipole and Quadrupole Solitons in Optically Induced Two‐Dimensional Photonic Lattices: Theory and Experiment

Cited by 102 publications

References 43 publications

Discrete surface solitons in two dimensions

Discrete surface solitons in two dimensions

Geometric stabilization of extendedS=2vortices in two-dimensional photonic lattices: Theoretical analysis, numerical computation, and experimental results

Universally‐Convergent Squared‐Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations

Contact Info

Product

Resources

About