Helmholtz bright spatial solitons and surface waves at power-law optical interfaces

We consider arbitrary-angle interactions between spatial solitons and the planar boundary between two optical materials with a single power-law nonlinear refractive index. Extensive analysis has uncovered a wide range of new qualitative phenomena in non-Kerr regimes. A universal Helmholtz-Snell law describing soliton refraction is derived using exact solutions to the governing equation as a nonlinear basis. New predictions are tested through exhaustive computations, which have uncovered substantially enhanced Goos-Hänchen shifts at some non-Kerr interfaces. Helmholtz nonlinear surface waves are analyzed theoretically, and their stability properties are investigated numerically for the first time. Interactions between surface waves and obliquely-incident solitons are also considered. Novel solution behaviours have been uncovered, which depend upon a complex interplay between incidence angle, medium mismatch parameters, and the power-law nonlinearity exponent

Variational approximations of soliton dynamics in the Ablowitz-Musslimani nonlinear Schrödinger equation

We study the integrable nonlocal nonlinear Schrödinger equation proposed by Ablowitz and Musslimani, that is considered as a particular example of equations with parity-time (PT) symmetric self-induced potential. We consider dynamics (including collisions) of moving solitons. Analytically we develop a collective coordinate approach based on variational methods and examine its applicability in the system. We show numerically that a single moving soliton can pass the origin and decay or be trapped at the origin and blows up at a finite time. Using a standard soliton ansatz, the variational approximation can capture the dynamics well, including the finite-time blow up, even though the ansatz is relatively far from the actual blowing-up soliton solution. In the case of two solitons moving towards each other, we show that there can be a mass transfer between them, in addition to wave scattering. We also demonstrate that defocussing nonlinearity can support bright solitons

On vortex and dark solitons in the cubic–quintic nonlinear Schrödinger equation

Financiado para publicación en acceso aberto: Universidade de Vigo/CISUGWe study topologically charged propagation-invariant eigenstates of the 1+2-dimensional Schrödinger equation with a cubic (focusing)–quintic (defocusing) nonlinear term. First, we revisit the self-trapped vortex soliton solutions. Using a variational ansatz that allows us to describe the solutions as a liquid with a surface tension, we derive a simple formula relating the inner and outer radii of the bright vortex ring. Then, using numerical and variational techniques, we analyse dark soliton solutions for which the wave function density asymptotes to a non-vanishing value. We find an eigenvalue cutoff for the propagation constant that depends on the topological charge l. The variational profile provides simple and very accurate results for l ≥ 2. We also study the azimuthal stability of the eigenstates by a linear analysis finding that they are stable for all values of the propagation constant, at least for small l.Ministerio de Ciencia e Innovación | Ref. PID2020-118613GB-I00Xunta de Galicia | Ref. ED431B 2021/2

ABC Method and Fractional Momentum Layer for the FDTD Method to Solve the Schrödinger Equation on Unbounded Domains

The finite­difference time­domain (FDTD) method and its generalized variant (G­FDTD) are efficient numerical tools for solving the linear and nonlinear Schrödinger equations because not only are they explicit, allowing parallelization, but they also provide high­order accuracy with relatively inexpensive computational costs. In addition, the G­FDTD method has a relaxed stability condition when compared to the original FDTD method. It is important to note that the existing simulations of the G­FDTD scheme employed analytical solutions to obtain function values at the points along the boundary; however, in simulations for which the analytical solution is unknown, theoretical approximations for values at points along the boundary are desperately needed. Hence, the objective of this dissertation research is to develop absorbing boundary conditions (ABCs) so that the G­FDTD method can be used to solve the nonlinear Schrödinger equation when the analytical solution is unknown. To create the ABCs for the nonlinear Schrödinger equation, we initially determine the associated Engquist­Majda one­way wave equations and then proceed to develop a finite difference scheme for them. These ABCs are made to be adaptive using a windowed Fourier transform to estimate a value of the wavenumber of the carrier wave. These ABCs were tested using the nonlinear Schrödinger equation for 1D and 2D soliton propagation as well as Gaussian packet collision and dipole radiation. Results show that these ABCs perform well, but they have three key limitations. First, there are inherent reflections at the interface of the interior and boundary domains due to the different schemes used the two regions; second, to use the ABCs, one needs to estimate a value for the carrier wavenumber and poor estimates can cause even more reflection at the interface; and finally, the ABCs require different schemes in different regions of the boundary, and this domain decomposition makes the ABCs tedious both to develop and to implement. To address these limitations for the FDTD method, we employ the fractional­order derivative concept to unify the Schrödinger equation with its one­way wave equation over an interval where the fractional order is allowed to vary. Through careful construction of a variable­order fractional momentum operator, outgoing waves may enter the fractionalorder region with little to no reflection and, inside this region, any reflected portions of the wave will decay exponentially with time. The fractional momentum operator is then used to create a fractional­order FDTD scheme. Importantly, this single scheme can be used for the entire computational domain, and the scheme smooths the abrupt transition between the FDTD method and the ABCs. Furthermore, the fractional FDTD scheme relaxes the precision needed for the estimated carrier wavenumber. This fractional FDTD scheme is tested for both the linear and nonlinear Schrödinger equations. Example cases include a 1D Gaussian packet scattering off of a potential, a 1D soliton propagating to the right, as well as 2D soliton propagation, and the collision of Gaussian packets. Results show that the fractional FDTD method outperforms the FDTD method with ABCs

Helmholtz algebraic solitons

We report, to the best of our knowledge, the first exact analytical algebraic solitons of a generalized cubic-quintic Helmholtz equation. This class of governing equation plays a key role in photonics modelling, allowing a full description of the propagation and interaction of broad scalar beams. New conservation laws are presented, and the recovery of paraxial results is discussed in detail. The stability properties of the new solitons are investigated by combining semi-analytical methods and computer simulations. In particular, new general stability regimes are reported for algebraic bright solitons

Roadmap on optical rogue waves and extreme events

The pioneering paper 'Optical rogue waves' by Solli et al (2007 Nature 450 1054) started the new subfield in optics. This work launched a great deal of activity on this novel subject. As a result, the initial concept has expanded and has been enriched by new ideas. Various approaches have been suggested since then. A fresh look at the older results and new discoveries has been undertaken, stimulated by the concept of 'optical rogue waves'. Presently, there may not by a unique view on how this new scientific term should be used and developed. There is nothing surprising when the opinion of the experts diverge in any new field of research. After all, rogue waves may appear for a multiplicity of reasons and not necessarily only in optical fibers and not only in the process of supercontinuum generation. We know by now that rogue waves may be generated by lasers, appear in wide aperture cavities, in plasmas and in a variety of other optical systems. Theorists, in turn, have suggested many other situations when rogue waves may be observed. The strict definition of a rogue wave is still an open question. For example, it has been suggested that it is defined as 'an optical pulse whose amplitude or intensity is much higher than that of the surrounding pulses'. This definition (as suggested by a peer reviewer) is clear at the intuitive level and can be easily extended to the case of spatial beams although additional clarifications are still needed. An extended definition has been presented earlier by N Akhmediev and E Pelinovsky (2010 Eur. Phys. J. Spec. Top. 185 1-4). Discussions along these lines are always useful and all new approaches stimulate research and encourage discoveries of new phenomena. Despite the potentially existing disagreements, the scientific terms 'optical rogue waves' and 'extreme events' do exist. Therefore coordination of our efforts in either unifying the concept or in introducing alternative definitions must be continued. From this point of view, a number of the scientists who work in this area of research have come together to present their research in a single review article that will greatly benefit all interested parties of this research direction. Whether the authors of this 'roadmap' have similar views or different from the original concept, the potential reader of the review will enrich their knowledge by encountering most of the existing views on the subject. Previously, a special issue on optical rogue waves (2013 J. Opt. 15 060201) was successful in achieving this goal but over two years have passed and more material has been published in this quickly emerging subject. Thus, it is time for a roadmap that may stimulate and encourage further research.Peer ReviewedPostprint (author's final draft

Analytical and Numerical Investigation of Soliton Solutions of Coupled Equations in Quadratic Media: A Novel Approach

Solitons have been hitherto studied in Kerr media. However, we show that the optical solitons are not only limited to non-linear Kerr media. In this paper, a new method is devised to achieve the analytical approximate solutions to the soliton packages in one and two dimensions. Inputting (1) and (2) to Maxwell equations and using second-harmonic generation of type I, the coupled motion equations are obtained. Some states of spatial bright solitons are studied for large values of . The variational method is briefly reviewed and analytical approximate solution to quadratic-soliton coupled motion equations is investigated in one and two dimensions and also numerically solved. The comparison of analytical and numerical results indicates that our method is more accurate than the variational method.Key words Bright and dark solitons; Spatial optical soliton; Quadratic Kerr media; Variational method; Secondharmonic generatio