Nonlinear Helmholtz Equation Research Articles

Dual variational methods and nonvanishing for the nonlinear Helmholtz equation

We set up a dual variational framework to detect real standing wave solutions of the nonlinear Helmholtz equation−Δu−k2u=Q(x)|u|p−2u,u∈W2,p(RN) with N≥3, 2(N+1)(N−1)<p<2NN−2 and nonnegative Q∈L∞(RN). We prove the existence of nontrivial solutions for periodic Q as well as in the case where Q(x)→0 as |x|→∞. In the periodic case, a key ingredient of the approach is a new nonvanishing theorem related to an associated integral equation. The solutions we study are superpositions of outgoing and incoming waves and are characterized by a nonlinear far field relation.

Analytic Studies on the Helmholtz Spatial Solitons in Power-Law Optical Materials

In order to describe refraction phenomena, the propagation of a spatial soliton in power-law optical materials is considered. Hereby, we work on the behavior of solitons described by a generalized nonlinear Helmholtz equation in the power-law optical materials. Via the Hirota method, analytic one- and two-soliton solutions are obtained. We study the spatial solitons in two adjoining power-law materials with dissimilar medium coefficients. Based on the choice of the Heaviside function in the equation, we distinguish the two different optical materials. One-soliton dynamics with different choices of the nonlinear and linear refractive properties along the propagation direction of the carrier wave are discussed. In the first medium, with an increase of the inverse width, the soliton will be amplified. In the second medium, the soliton amplitude will increase when the linear refractive property increases, or the nonlinear refractive property decreases. Asymptotic analysis is carried out on the two-soliton solutions. Through a graphic and asymptotic analysis, we find that there exist elastic and inelastic collisions between two solitons, as well as soliton fusion.

Efficient parallel implementation of the nonparaxial beam propagation method

An efficient parallel implementation of a nonparaxial beam propagation method for the numerical study of the nonlinear Helmholtz equation is presented. Our solution focuses on minimizing communication and computational demands of the method which are dependent on a nonparaxiality parameter. Performance tests carried out on different types of parallel systems behave according theoretical predictions and show that our proposal exhibits a better behavior than those solutions based on the use of conventional parallel fast Fourier transform implementations. The application of our design is illustrated in a particularly demanding scenario: the study of dark solitons at interfaces separating two defocusing Kerr media, where it is shown to play a key role.

Widely varying giant Goos–Hänchen shifts from Airy beams at nonlinear interfaces

We present a numerical study of the giant Goos-Hänchen shifts (GHSs) obtained from an Airy beam impinging on a nonlinear interface. To avoid any angular restriction associated with the paraxial approximation, the analysis is based on the nonlinear Helmholtz equation. We report the existence of nonstandard nonlinear GHSs displaying an extreme sensitivity to the input intensity and the existence of multiple critical values. These intermittent and oscillatory regimes can be explained in terms of competition between critical coupling to a surface mode and soliton emission from the refracted beam component and how this interplay varies with localization of the initial Airy beam.

Real Solutions to the Nonlinear Helmholtz Equation with Local Nonlinearity

In this paper, we study real solutions of the nonlinear Helmholtz equation $$ - \Delta u - k^2 u = f(x,u),\qquad x\in \R^N $$ satisfying the asymptotic conditions $$ u(x)=O(|x|^{\frac{1-N}{2}}) \quad \text{and} \quad \frac{\partial^2 u}{\partial r^2}(x)+k^2 u(x)) =o(|x|^{\frac{1-N}{2}}) \qquad \text{as $r=|x| \to \infty$.} $$ We develop the variational framework to prove the existence of nontrivial solutions for compactly supported nonlinearities without any symmetry assumptions. In addition, we consider the radial case in which, for a larger class of nonlinearities, infinitely many solutions are shown to exist. Our results give rise to the existence of standing wave solutions of corresponding nonlinear Klein-Gordon equations with arbitrarily large frequency.

Propagation of TM modes in a circular dielectric waveguide filled with a medium with Kerr nonlinearity

Eigenmodes propagating in a cylindrical dielectric waveguide of circular cross section filled with a nonlinear medium whose nonlinearity is described by the Kerr law are analyzed. The nonlinear eigenvalue boundary problem for the nonlinear Helmholtz equation is considered. An iterative numerical method for obtaining the approximate solutions is developed. Numerical results are presented.

Nonreciprocal bistability in nonlinear fractal Fibonacci photonic crystals

It is investigated that the interplay of fractal resonant states and Kerr nonlinearity in the presence of the intrinsic asymmetry of fractal Fibonacci photonic crystals (FFPC) leads to nonreciprocal bistability. Finite element analysis of the 1D nonlinear Helmholtz equation and the nonlinear finite difference time domain are used to study both stable and dynamic electromagnetic behaviors of nonlinear FFPC and nonreciprocal transmission is interpreted through coupled mode theory.

Analysis of non-paraxial solitons using a collective variable approach

Based on the nonlinear Helmholtz equation, we consider a model of the propagation of non-paraxial optical solitons, namely the non-paraxial nonlinear Schrödinger equation. Using this model, we present a theoretical treatment leading to dynamical equations of motion by means of the collective variable (CV) approach. We show that the non-paraxiality parameter leads to a more complicated analysis of the dynamics of the relevant parameters, called CVs, associated with the soliton in comparison with those behaviors taking into account the approximation of the slowly varying amplitude of the propagating beam. We provide some propagation results of the analysis on a typical semi-infinite Kerr medium for some coefficient values of the model. Also, we find that stable propagation of non-paraxial solitons can be achieved for an appropriate choice of parameters.

A simple model of ultrasound propagation in a cavitating liquid. Part I: Theory, nonlinear attenuation and traveling wave generation

The bubbles involved in sonochemistry and other applications of cavitation oscillate inertially. A correct estimation of the wave attenuation in such bubbly media requires a realistic estimation of the power dissipated by the oscillation of each bubble, by thermal diffusion in the gas and viscous friction in the liquid. Both quantities and calculated numerically for a single inertial bubble driven at 20 kHz, and are found to be several orders of magnitude larger than the linear prediction. Viscous dissipation is found to be the predominant cause of energy loss for bubbles small enough. Then, the classical nonlinear Caflish equations describing the propagation of acoustic waves in a bubbly liquid are recast and simplified conveniently. The main harmonic part of the sound field is found to fulfill a nonlinear Helmholtz equation, where the imaginary part of the squared wave number is directly correlated with the energy lost by a single bubble. For low acoustic driving, linear theory is recovered, but for larger drivings, namely above the Blake threshold, the attenuation coefficient is found to be more than 3 orders of magnitude larger then the linear prediction. A huge attenuation of the wave is thus expected in regions where inertial bubbles are present, which is confirmed by numerical simulations of the nonlinear Helmholtz equation in a 1D standing wave configuration. The expected strong attenuation is not only observed but furthermore, the examination of the phase between the pressure field and its gradient clearly demonstrates that a traveling wave appears in the medium.

A split-step finite difference method for nonparaxial nonlinear Schrödinger equation at critical dimension

The critical nonlinear Schrödinger equation (NLS) is the model equation for propagation of laser beam in bulk Kerr medium. One of the final stages in the derivation of NLS from the nonlinear Helmholtz equation (NLH) is to apply paraxial approximation. However, there is numerical evidence suggesting nonparaxiality prevents singularity formation in the solutions of NLS. Therefore, it is important to develop numerical methods for solving nonparaxial NLS. Split-step methods are widely used for finding numerical solutions of NLS equation. Nevertheless, these methods cannot be applied to nonparaxial NLS directly. In this study, we extend the applicability of split-step methods to nonparaxial NLS by using Padé approximant operators. In particular, split-step Crank–Nicolson (SSCN) method is used in conjunction with Padé approximants to provide examples of numerical solutions of nonparaxial NLS.

A numerical scheme for nonlinear Helmholtz equations with strong nonlinear optical effects

A numerical scheme is presented to solve the nonlinear Helmholtz (NLH) equation modeling second-harmonic generation (SHG) in photonic bandgap material doped with a nonlinear χ((2)) effect and the NLH equation modeling wave propagation in Kerr type gratings with a nonlinear χ((3)) effect in the one-dimensional case. Both of these nonlinear phenomena arise as a result of the combination of high electromagnetic mode density and nonlinear reaction from the medium. When the mode intensity of the incident wave is significantly strong, which makes the nonlinear effect non-negligible, numerical methods based on the linearization of the essentially nonlinear problem will become inadequate. In this work, a robust, stable numerical scheme is designed to simulate the NLH equations with strong nonlinearity.

HELMHOLTZ SOLITONS IN OPTICAL MATERIALS WITH A DUAL POWER-LAW REFRACTIVE INDEX

A nonlinear Helmholtz equation is proposed for modelling scalar optical beams in uniform planar waveguides whose refractive index exhibits a purely-focusing dual power-law dependence on the electric field amplitude. Two families of exact analytical solitons, describing forward- and backward-propagating beams, are derived. These solutions are physically and mathematically distinct from those recently discovered for related nonlinearities. The geometry of the new solitons is examined, conservation laws are reported, and classic paraxial predictions are recovered in a simultaneous multiple limit. Conventional semi-analytical techniques assist in studying the stability of these nonparaxial solitons, whose propagation properties are investigated through extensive simulations.

Bistable dark solitons of a cubic-quintic Helmholtz equation

We provide a report on exact analytical bistable dark spatial solitons of a nonlinear Helmholtz equation with a cubic-quintic refractive-index model. Our analysis begins with an investigation of the modulational instability characteristics of Helmholtz plane waves. We then derive a dark soliton by mapping the desired asymptotic form onto a uniform background field and obtain a more general solution by deploying rotational invariance laws in the laboratory frame. The geometry of the new soliton is explored in detail, and a range of new physical predictions is uncovered. Particular attention is paid to the unified phenomena of arbitrary-angle off-axis propagation and nondegenerate bistability. Crucially, the corresponding solution of paraxial theory emerges in a simultaneous multiple limit. We conclude with a set of computer simulations that examine the role of Helmholtz dark solitons as robust attractors.

Optical bistability by two laser beams in photorefractive crystals

For a nonlinear medium, a given incident laser beam may produce different transmitted light waves. This phenomenon (the so-called optical bistability) corresponds to multiple solutions of the boundary value problems of the nonlinear Helmholtz equation. Optical bistability can be useful in the design of optical switches. Devices that display this behavior could potentially play a major role in the development of optical communication systems and computing. In this article we present experimental results concerning the optical bistability in photorefractive BaTiO 3:Fe crystal. Two laser beams were used to interact with the photorefractive crystal which resulted in the bistability of the intensity of transmitted wave. This was achieved without the application of any optical resonator. High contrast optical bistability is found experimentally in the pump-ratio dependence of the output intensity.

Self-organization of dissipationless solitons in positive- and negative-refractive-index materials

General nonlinear and nonparaxial dissipative complex Helmholtz equations for magnetic and electric fields propagating in negative refractive index materials (NIMs) are derived ab initio from Maxwell equations. In order to describe nonconservative soliton dynamics in NIMs, such coupled equations are reduced into generalized Ginzburg-Landau equation. Cross-compensation between the excess of saturating nonlinearity, losses, and gain renders these self-organized solitons dissipationless and exceptionally robust. The presence of such solitons makes NIMs effectively dissipationless.

Dark solitons at nonlinear interfaces

The refraction of dark solitons at a planar boundary separating two defocusing Kerr media is simulated and analyzed, for the first time (to our knowledge). Analysis is based on the nonlinear Helmholtz equation and is thus valid for any angle of incidence. A new law, governing refraction of black solitons, is combined with one describing bright soliton refraction to yield a generalized Snell's law whose validity is verified numerically. The complexity of gray soliton refraction is also analyzed, and illustrated by a change from external to internal refraction on varying the soliton contrast parameter.

A fourth-order numerical method for the one-dimensional nonlinear Helmholtz equation with multilayered material

Based on a compact approximation of finite element type, a fourth-order numerical scheme is developed for the one-dimensional nonlinear Helmholtz equation with multilayered material. The Hermite–Birkhoff interpolation is employed to approximate the field between the grid nodes with fourth-order accuracy. The second-order derivatives that appear in the Hermite–Birkhoff interpolation are evaluated by the original differential equation. The resulting nonlinear system of equations is solved by using the Newton’s iterations combined with a continuation method. Numerical experiments are conducted to demonstrate that the scheme possesses the anticipated order of accuracy. To validate the numerical method, we compute several numerical experiments about optical bistability phenomenon with the developed scheme.

Bistable Helmholtz bright solitons in saturable materials

We present, to the best of our knowledge, the first exact analytical solitons of a nonlinear Helmholtz equation with a saturable refractive-index model. These new two-dimensional spatial solitons have a bistable characteristic in some parameter regimes, and they capture oblique (arbitrary-angle) beam propagation in both the forward and backward directions. New conservation laws are reported, and the classic paraxial solution is recovered in an appropriate multiple limit. Analysis and simulations examine the stability of both solution branches, and stationary Helmholtz solitons are found to emerge from a range of perturbed input beams.

Nonlinear interfaces: intrinsically nonparaxial regimes and effects

The behaviour of optical solitons at planar nonlinear boundaries is a problem rich in intrinsically nonparaxial regimes that cannot be fully addressed by theories based on the nonlinear Schrodinger equation. For instance, large propagation angles are typically involved in external refraction at interfaces. Using a recently proposed generalized Snell's law for Helmholtz solitons, we analyse two such effects: nonlinear external refraction and total internal reflection at interfaces where internal and external refraction, respectively, would be found in the absence of nonlinearity. The solutions obtained from the full numerical integration of the nonlinear Helmholtz equation show excellent agreement with the theoretical predictions.

A high-order numerical method for the nonlinear Helmholtz equation in multidimensional layered media

We present a novel computational methodology for solving the scalar nonlinear Helmholtz equation (NLH) that governs the propagation of laser light in Kerr dielectrics.The methodology addresses two well-known challenges in nonlinear optics: Singular behavior of solutions when the scattering in the medium is assumed predominantly forward (paraxial regime), and the presence of discontinuities in the optical properties of the medium. Specifically, we consider a slab of nonlinear material which may be grated in the direction of propagation and which is immersed in a linear medium as a whole. The key components of the methodology are a semi-compact high-order finite-difference scheme that maintains accuracy across the discontinuities and enables sub-wavelength resolution on large domains at a tolerable cost, a nonlocal two-way artificial boundary condition (ABC) that simultaneously facilitates the reflectionless propagation of the outgoing waves and forward propagation of the given incoming waves, and a nonlinear solver based on Newton’s method.The proposed methodology combines and substantially extends the capabilities of our previous techniques built for 1D and for multi-D. It facilitates a direct numerical study of nonparaxial propagation and goes well beyond the approaches in the literature based on the “augmented” paraxial models. In particular, it provides the first ever evidence that the singularity of the solution indeed disappears in the scalar NLH model that includes the nonparaxial effects. It also enables simulation of the wavelength-width spatial solitons, as well as of the counter-propagating solitons.