Helmholtz solitons in power-law optical materials

Christian, J. M.; McDonald, GS; Potton, R J; Chamorro‐Posada, Pedro

doi:10.1103/physreva.76.033834

Cited by 28 publications

(39 citation statements)

References 54 publications

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“…Simulations employed a nonparaxial beam propagation method [81], that has been crucial in the development and validation of Helmholtz soliton theory [68][69][70][71][74][75][76]. All of the numerical results presented have exploited the rotational symmetry that a Helmholtz framework allows [66].…”

Section: Numerical Considerationsmentioning

confidence: 99%

“…The model equation is the full Nonlinear Helmholtz (NLH) equation [62,66] without any further approximation. Not only bright Kerr, but also dark Kerr [68], two-component [69], boundary [70], bistable [71] and algebraic [72] Helmholtz solitons have been found to display non-trivial Helmholtz corrections. This latter framework has permitted, for instance, description of collisions of Kerr bright solitons at arbitrary angles [73].…”

Section: Introductionmentioning

confidence: 99%

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Black and gray Helmholtz-Kerr soliton refraction

2011

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Refraction of black and gray solitons at boundaries separating different defocusing Kerr media is analyzed within a Helmholtz framework. A universal nonlinear Snell's law is derived that describes gray soliton refraction, in addition to capturing the behavior of bright and black Kerr solitons at interfaces. Key regimes, defined by beam and interface characteristics, are identified and predictions are verified by full numerical simulations. The existence of a unique total non-refraction angle for gray solitons is reported; both internal and external refraction at a single interface is shown possible (dependent only on incidence angle). This, in turn, leads to the proposal of positive or negative lensing operations on soliton arrays at planar boundaries.

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Section: Numerical Considerationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Black and gray Helmholtz-Kerr soliton refraction

2011

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“…'2 nd -order-in-z') character, it supports both forward-and backward-propagating solutions [31,34,35]. By convention, we consider here only the forward solutions and note that the corresponding paraxial model [16] has no counterpart to the backward solutions.…”

Section: Soliton Solutionsmentioning

confidence: 99%

“…Exact analytical Helmholtz soliton solutions are known for focusing [27] and defocusing [30] Kerr media, and also for power-law [31] media. Vector generalizations of the Kerr solitons have also recently been derived [32].…”

Section: Paraxial Versus Non-paraxial Solitonsmentioning

confidence: 99%

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Helmholtz bright and boundary solitons

Christian¹,

McDonald²,

Chamorro‐Posada³

2007

J. Phys. A: Math. Theor.

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Abstract. We report, for the first time, exact analytical boundary solitons of a generalized cubic-quintic Non-Linear Helmholtz (NLH) equation. These solutions have a linked-plateau topology that is distinct from conventional dark soliton solutions; their amplitude and intensity distributions are spatially delocalized and connect regions of finite and zero wave-field disturbances (suggesting also the classification as "edge solitons"). Extensive numerical simulations compare the stability properties of recentlyreported Helmholtz bright solitons, for this type of polynomial non-linearity, to those of the new boundary solitons. The latter are found to possess a remarkable stability characteristic, exhibiting robustness against perturbations that would otherwise lead to the destabilizing of their bright-soliton counterparts. Helmholtz bright and boundary solitons 2 IntroductionSolitons are ubiquitous entities in nature. Whenever linear effects (such as dispersion, diffraction or diffusion) are balanced exactly by non-linearity (self-phase modulation, self-focusing or reactionkinetic properties, respectively), robust self-trapped structures -solitons -can emerge as dominant modes of the system dynamics. These localized self-stabilizing non-linear waves arise widely in nature since quite different physical systems are governed by a relatively small set of universal equations, at least to first approximation. Solitons are often sech ("bell")-or tanh ("S")-shaped structures. The latter class are sometimes referred to as kink solitons, and they generally possess topologically non-trivial phase distributions. Phase-topological kink solitons appear in a range of physically diverse systems, and play the role of "fronts" and "domain walls". In classical mechanics, for example, they describe collective long-wave excitations on a line of weakly-coupled pendula. In condensed matter, kink solitons arise in simple models of one-dimensional lattice-dynamics when studying the motion of dislocations and domain walls in ferromagnetic crystals, and they also play a key role in the phenomenological understanding of phase transitions. In chemical kinetics, kink solitons appear as solutions to reaction-diffusion equations. They also occur in hydrodynamics, plasma physics, quantum field theory and cosmology. Comprehensive reviews of these systems can be found in Refs. [1][2][3][4].Our principle concern in this paper is with spatial soliton beams found in non-linear optics [5,6]. These types of soliton can arise when the tendency of a collimated light beam to diffract is opposed by the non-linear properties of the optical medium. When these two effects (diffractive broadening, and narrowing due to self-focusing) become comparable, then a stationary beam can exist whose transverse intensity distribution is invariant along the propagation direction. Spatial solitons are of theoretical interest as particular solutions to generic non-linear evolution equations, but they are also the subject of considerable experimental investigation. The robu...

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Bifurcation and Stability for Nonlinear Schrödinger Equations with Double Well Potential in the Semiclassical Limit

Fukuizumi

Sacchetti

2011

J Stat Phys

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We consider the stationary solutions for a class of Schrödinger equations with a symmetric double-well potential and a nonlinear perturbation. Here, in the semiclassical limit we prove that the reduction to a finitemode approximation give the stationary solutions, up to an exponentially small term, and that symmetry-breaking bifurcation occurs at a given value for the strength of the nonlinear term. The kind of bifurcation picture only depends on the non-linearity power. We then discuss the stability/instability properties of each branch of the stationary solutions. Finally, we consider an explicit one-dimensional toy model where the double well potential is given by means of a couple of attractive Dirac's delta pointwise interactions.

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Helmholtz solitons in power-law optical materials

Abstract: 2.

Cited by 28 publications

References 54 publications

Black and gray Helmholtz-Kerr soliton refraction

Black and gray Helmholtz-Kerr soliton refraction

Helmholtz bright and boundary solitons

Bifurcation and Stability for Nonlinear Schrödinger Equations with Double Well Potential in the Semiclassical Limit

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