Kerr-type Media Research Articles

Nonlinear localized states near the interface with nonlinear response between the medium with a parabolic index spatial profile and Kerr-type medium

The interface with nonlinear response separating the parabolic graded-index and the Kerr nonlinear media are considered. Exact solutions to the nonlinear Schrödinger equation with nonlinear short-range potential and a parabolic spatial profile are found applying to the theoretical description of the stationary states localized near interface with nonlinear properties. Localized states with continuous/discrete energy spectrum are described by the Whittaker function/Hermite polynomials in the medium with a parabolic profile of characteristic and the hyperbolic cosine (sine) in the medium with a self-focusing/defocusing Kerr nonlinearity. The field localization length is wider in the case of a self-focusing nonlinearity than in the case of a defocusing one. The maximum of the wave function is located in a nonlinear medium in the case of a self-focusing nonlinearity and at the interface in the case of a defocusing one. It is shown the possibility of a motion closer to the interface (or away from it) the maximum intensity of the localized state by changing the values of the interface response parameters at the fixed localization energy. A growth of the width of the parabolic graded-index layer adduced an increase in the maximum height of localized states of discrete spectrum and theirs localization length in the graded-index layer, but it had almost no effect on the profile of localized states of the continuous spectrum.

Circular dichroism in nonlinear topological Weyl semimetals

In recent years, the field of topological photonics has emerged as a promising area of research due to its potential for the development of new photonic devices with unique properties. Topological Weyl semimetals (TWSs), which are characterized by the presence of Weyl points in their electronic band structure, are one such example of a material with interesting topological properties. In this study, Kerr and Faraday rotations were used to determine the nonlinear characteristics of TWSs. We focused on surfaces where no Fermi arcs are involved, so that Maxwell’s equations would contain some peculiar topological terms. In Weyl semimetals with a specific topology, the distance between Weyl nodes aligned along the z-direction functions as a magnet. This results in a significant polar Kerr/Faraday rotation effect that is proportional to the separation distance, when light is directed onto the surface of the TWS that lacks Fermi arc states. Conversely, when the light is directed onto a surface with Fermi arc states, the Voigt effect is quadratically proportional to the separation distance. We considered electromagnetic wave propagation in a nonlinear Kerr-type medium. We have derived and solved the linear and nonlinear Helmholtz equations for TWSs using the tanh method. Our findings reveal that wave solutions could have some potentially significant implications for the design and optimization of photonic devices based on TWSs.

A nonlocal vibration suppression for a multilayered magneto-viscoelastic nanobeam on a three-parameter-type medium

The free vibrational behavior of the multilayered magneto-viscoelastic nanostructure is analyzed in the present study. Smart plies and viscoelastic layers are used for the vibration control and energy dissipation of the multilayered nanobeam. The simply supported microstructure is located on a medium of Kerr-type. Magnetostrictive material is employed to represent the used smart materials under the constant feedback gain control system. A higher-order shear deformation theory and Hamilton’s principle are employed to describe the linear dynamic responses associated with the presented structure. Kelvin–Voigt viscoelastic model is taken into account to deal with the viscoelastic material in the structure. The study outcomes are displayed tabularly and graphically with various exhaustive discussions to analyze the dynamic characteristics of the presented adaptive system.

Nonlinear guided wave propagation along layers with the exponential index profile, constant index and the Kerr nonlinearity

Layered planar waveguide consisting of the layer with an exponential profile of the dielectric permittivity, optical linear layer, and nonlinear medium of Kerr-type is considered. Exact analytical solutions describing the nonlinear guided waves in such waveguide are found. The profiles of the waves in perpendicular to the layers direction are analyzed. The distinctions between layered waveguide structures with a linear optical substrate and a nonlinear one and the impact of dielectric interlayer separating the exponentially graded-index main waveguide layer and nonlinear material of Kerr-type are discussed. The presence of a dielectric interlayer leads to the possibility of the appearance of field oscillations inside it. The possibility of controlling the field localization length across the layers and its amplitude by changing the interlayer width is shown.

Dynamics of solutions of nonlinear functional differential equation of parabolic type

Purpose of this work is to study the initial-boundary value problem for a parabolic functional-differential equation in an annular region, which describes the dynamics of phase modulation of a light wave passing through a thin layer of a nonlinear Kerr-type medium in an optical system with a feedback loop, with a rotation transformation (corresponds the involution operator) and the Neumann conditions on the boundary in the class of periodic functions. A more detailed study is made of spatially inhomogeneous stationary solutions bifurcating from a spatially homogeneous stationary solution as a result of a bifurcation of the “fork” type and time-periodic solutions of the “traveling wave” type. Methods. To represent the original equation in the form of nonlinear integral equations, the Green’s function is used. The method of central manifolds is used to prove the theorem on the existence of solutions of the indicated equation in a neighborhood of the bifurcation parameter and to study their asymptotic form. Numerical modeling of spatially inhomogeneous solutions and traveling waves was carried out using the Galerkin method. Results. Integral representations of the considered problem are obtained depending on the form of the linearized operator. Using the method of central manifolds, a theorem on the existence and asymptotic form of solutions of the initial-boundary value problem for a functional-differential equation of parabolic type with an involution operator on an annulus is proved. As a result of numerical modeling based on Galerkin approximations, in the problem under consideration, approximate spatially inhomogeneous stationary solutions and time-periodic solutions of the traveling wave type are constructed. Conclusion. The proposed scheme is applicable not only to involutive rotation operators and Neumann conditions on the boundary of the ring, but also to other boundary conditions and circular domains. The representation of the initial-boundary value problem in the form of nonlinear integral equations of the second kind allows one to more simply find the coefficients of asymptotic expansions, prove existence and uniqueness theorems, and also use a different number of expansion coefficients of the nonlinear component in the right-hand side of the original equation in the neighborhood of the selected solution (for example, stationary). Visualization of the numerical solution confirms the theoretical calculations and shows the possibility of forming complex phase structures.

Fundamental and vortex gap solitons in quasiperiodic photonic lattices.

We address the existence and stability of fundamental, single-charged vortex, and double-charged vortex gap solitons in two-dimensional quasiperiodic photonic lattices imprinted in a Kerr-type medium. Fundamental and vortex gap solitons can bifurcate from linear localized states or their combination supported by quasiperiodic lattices for both defocusing and focusing nonlinearities. We find that the three types of solitons mentioned above are stable in the entire existence domain for defocusing nonlinearities, and that they can also be stable at a lower power level for focusing nonlinearities. At higher power, unstable solitons are characterized by a ring-shaped symmetry-breaking distribution, and the unique spot profile formed is repeatedly observed with changes in propagation distance.

Frequency control of cross-ply magnetostrictive viscoelastic plates resting on Kerr-type elastic medium

Frequency control of cross-ply magnetostrictive viscoelastic plates resting on Kerr-type elastic medium

Kerr-nonlinearity induced bistable-like parity-time phase transition in coupled waveguides.

We analyze the parity-time (PT) symmetric phase in coupled two waveguides with a Kerr-type medium in between. Paying attention to the emitted field from a dipole source inside, we show that when the strength of the dipole increases, the optical Kerr effect can render a phase transition from the exact PT phase to the broken PT phase. Furthermore, a salient phenomenon of bistable-like PT phase is observed, in which the emitted field possesses a paradox between the two kinds of PT phases. We show that the physical mechanism of this bistable-like phenomenon is a globally inhomogeneous PT phase, in which different spatial regions of the whole structure can possess different PT phases (broken or exact). This study highlights the potential to manipulate the PT phase transition by using optical nonlinearity for many interesting applications.

Dark-managed solitons in inhomogeneous cubic–quintic–septimal nonlinear media

We investigate the inhomogeneous higher-order nonlinear Schrodinger (INHLS) equation including cubic–quintic–septic (CQS) nonlinear terms and gain or loss with variable coefficients. The exact analytic solution that describes dark soliton-type pulse propagation is found for the model by employing the ansatz method. Unlike the traditional $$\text {tanh}$$ dark soliton in Kerr-type media, the functional form of this novel dark-managed soliton structure takes a $$\text {sech}^{2/3}$$ profile. Parametric conditions are presented in which these optical solitons exist. We also investigated the stability of these dark-managed solitons under some initial perturbations by employing the numerical simulation methods. Finally, the interaction dynamics of two and three dark-managed solitons has been numerically explored.

Direct and inverse problems for the nonlinear time-harmonic Maxwell equations in Kerr-type media

In the current paper we consider an inverse boundary value problem of electromagnetism in a nonlinear Kerr medium. We show the unique determination of the electromagnetic material parameters and the nonlinear susceptibility parameters of the medium by making electromagnetic measurements on the boundary. We are interested in the case of the time-harmonic Maxwell equations.

Scattering for the 𝐿² supercritical point NLS

We consider the 1D nonlinear Schrödinger equation with focusing point nonlinearity. “Point” means that the pure-power nonlinearity has an inhomogeneous potential and the potential is the delta function supported at the origin. This equation is used to model a Kerr-type medium with a narrow strip in the optic fibre. There are several mathematical studies on this equation and the local/global existence of a solution, blow-up occurrence, and blow-up profile have been investigated. In this paper we focus on the asymptotic behavior of the global solution, i.e., we show that the global solution scatters as t → ± ∞ t\to \pm \infty in the L 2 L^2 supercritical case. The main argument we use is due to Kenig-Merle, but it is required to make use of an appropriate function space (not Strichartz space) according to the smoothing properties of the associated integral equation.

Light-Matter Interaction and Hybrid Vector Breather

The nonlinear coherent interaction of light with the dispersive and Kerr-type third-order susceptibility medium containing optical impurity atoms or semiconductor quantum dots is considered. Using the generalized perturbation reduction method, the nonlinear wave equation is reduced to the coupled nonlinear Schr\"odinger equations. It is shown that the second-order derivatives play a key role in the description of the process of formation of the bound state of two breathers oscillating with the sum and the difference of frequencies and wave numbers. The resonant, nonresonant and hybrid mechanisms of the formation of the two-component nonlinear pulse -- the vector breather are realized depending on the light and medium parameters. Explicit analytical expressions for the profile and parameters of the nonlinear pulse are presented. The conditions of the excitation of resonant, nonresonant and hybrid nonlinear waves are discussed. In the particular case, the resonant vector breather coincides with the vector $0\pi$ pulse of self-induced transparency.

Recovery time of matter Airy beams using the path integral quantum trajectory model

Abstract Following their discovery in the late 1970s, optical Airy beams have found numerous applications in technologies such as microscopy and optical trapping, many of which are based on the wave packets’ unique features such as zero or minimal diffraction, self-acceleration, and self-healing. Recent advancements have shown that Airy beams can also be produced using matter waves with many of the same unique characteristics of their optical counterparts. We present here a study of the recovery time of damaged matter Airy wave packets in free space and a nonlinear Kerr-type medium. We show that in free space the recovery time increases approximately linearly with mass and is independent of other kinematical parameters such as momentum, velocity, and spatial width. In the Kerr-type medium, recovery time is decreased compared to free space and does not scale linearly with mass. In order to study matter Airy beams, we introduce the Path Integral Quantum Trajectory model as a new computational tool for the study of non-relativistic, quantum mechanical wave packets and demonstrate its effectiveness in dealing with heavy particle dynamics.

Elliptical Gaussian beam propagation in nonlinear fibres with focusing and defocusing refractive profiles

The paper shows systematic analysis of elliptical Gaussian beam (GB) propagating in self-focusing/self-defocusing media of Kerr type with focusing/defocusing linear refractive profiles. Instead of solving commonly accepted Nonlinear Schrödinger Equation (NLS) we propose equations of geometrical optics: complex eikonal equation and complex transport one. Eikonal equation let us derive immediately the ordinary differential equations for principle widths omitting this way complicated variational process and virial theory used in nonlinear optics. From the transport equation we obtain first order ordinary differential equation for complex amplitude evolution and conservation principle for energy flux in the elliptical GB cross-section. For combined effects of diffraction, nonlinear refraction and focusing/defocusing graded-index factor we observe three types of the solutions for the elliptical radius of principle beam widths: self-focusing or self-defocusing together with diffraction widening, stationary and oscillatory ones under and without collapse effect. We prove here that principle curvatures of the wave front can introduce into our description the additional ellipticity of the beam, which can effectively limit collapse effect. Moreover, we discuss the evolution of elliptical Gaussian beam in nonlinear self-defocusing fibre with focusing refractive profile presenting conditions for existence of elliptical spatial solitons. We discuss here also the influence of initial wave front curvatures on elliptical Gaussian beam evolution in nonlinear inhomogeneous fibres of Kerr-type.

Локализация возбуждений вблизи тонкого дефектного слоя с нелинейными свойствами, разделяющего линейный и нелинейный кристаллы

It is shown that the localized and quasi-local stationary states exist near a thin defect layer with nonlinear properties separating a linear medium from a non-linear medium of Kerr type. Localized states are characterized by a monotonically decreasing field amplitude on both sides of the interface. Quasi-local states are described by a field in the form of a standing wave in a linear medium and monotonously decreasing in a nonlinear medium. The contacts with nonlinear self-focusing and defocusing media are analyzed. The mathematical formulation of the proposed model is a system of linear and nonlinear Schrödinger equations with a potential that is nonlinear with respect to the field and which simulates a thin defect layer with nonlinear properties. Dispersion relations determining the energy of local and quasi-local states are obtained. The expressions for energies were obtained explicitly in limiting cases and the conditions for their existence were indicated.

Estudio de las Funciones de Coherencia de la Luz en un Sistema Jaynes Cummings no lineal

En este trabajo se consideró un punto cuántico de dos niveles dentro de una cavidad con un medio no lineal tipo kerr y un solo modo del campo electromagnético cuantizado. Se construyó la ecuación maestra considerando procesos disipativos, el término no lineal Kerr y se solucionó numéricamente para el estado estacionario teniendo en cuenta la temperatura. A partir de estos resultados se analizó la influencia que tiene el medio no lineal en la evolución temporal del número medio de fotones, la inversión de población, el espectro de fotoluminiscencia y se determinaron las características cuánticas-clásicas del estado de la luz mediante el cálculo de las funciones de coherencia de fotones de segundo orden.

Tunable surface waves in nonlinear graphene-based one-dimensional-photonic crystal

The nonlinear TE surface waves guided by an interface of a semi-infinite Kerr-type dielectric medium and a one-dimensional graphene-based photonic crystal have been investigated using the transfer-matrix method. It is shown that increasing the intensity of the incident beam drastically changes dispersion behavior inside the graphene-induced photonic band gap as a result of nonlinearity. Also regarding the simulation results, it reveals that the peculiar dispersion that is governed by a nonlinear structure appeared to be tuned by graphene conductivity via an external electric field. Applying such an electric field makes it possible to easily tune the nonlinear surface waves at the desired frequency only by adjusting the chemical potential of the graphene. Also, it is shown that for application purposes, the results of studies can be applied to the case in which graphene nanolayers are embedded by identical dielectric constant, i.e., zero permittivity contrast of dielectric.

Dual-switching behavior of nonlocal interfaces

Nonlinear interfaces separating two diffusive Kerr-type media exhibit dual switching between total internal reflection and transmission. This property is found within a weakly nonlocal regime when both a nonparaxial treatment of the problem and a full two-dimensional model for carrier diffusion are assumed. The theoretical model is shown to predict an effective cubic-quintic nonlinearity with competing terms that produces such property. The validity of the analysis is contrasted with a full set of numerical simulations.

Characterization of spatiotemporal chaos in a Kerr optical frequency comb and in all fiber cavities.

Complex spatiotemporal dynamics have been a subject of recent experimental investigations in optical frequency comb microresonators and in driven fiber cavities with Kerr-type media. We show that this complex behavior has a spatiotemporal chaotic nature. We determine numerically the Lyapunov spectra, allowing us to characterize different dynamical behavior occurring in these simple devices. The Yorke-Kaplan dimension is used as an order parameter to characterize the bifurcation diagram. We identify a wide regime of parameters where the system exhibits a coexistence between the spatiotemporal chaos, the oscillatory localized structure, and the homogeneous steady state. The destabilization of an oscillatory localized state through radiation of counter-propagating fronts between the homogeneous and the spatiotemporal chaotic states is analyzed. To characterize better the spatiotemporal chaos, we estimate the front speed as a function of the pump intensity.

Cnoidal waves, solitons and vortices in the flow of polaritons

We theoretically have investigated the properties of the phonon-polariton waves propagating in Kerr-type dielectric medium, and examined the conditions of appearance of the spatial solitons, cnoidal waves and polariton vortices. It is shown that the polariton flow can propagate in the form of spatial soliton, or can be split into several flows in the form of filaments depending on the field density and parameters of the nonlinear medium. In the polariton flow the vortices may appear at the presence of local inhomogeneities of medium refractive index causing the wavefront rotation.