Kerr-type Nonlinear Media Research Articles

Localization features near the interface with nonlinear properties separating the Kerr-type nonlinear medium and a linear graded-index medium

Abstract Interface with nonlinear response between nonlinear medium and a linear graded-index medium is considered. Exact solutions to the nonlinear Schrödinger equation with the nonlinear delta-function potential and the linear spatial term are found. The solutions describe the localized states in the self-focusing and defocusing nonlinear media separated from linear graded-index medium by interface with nonlinear response. Localization features in dependence on signs of defect parameters are analyzed. It is shown that nonlinearity of the defect leads to the possibility of localization with the different signs of defect parameters. Localized states arise for all combinations of repulsing and attracting defect, and self-focusing and defocusing nonlinear response of the defect. Localized states characterized by an asymmetric distribution with two maxima appear due to the presence of a nonlinear response of the defect. Controlling the defect parameters allows adjusting the height of the field intensity at the interface between the media and the depth of localization.

An unconditional stable modified finite element methods for Maxwell’s equation in Kerr-type nonlinear media

The purpose of this paper is to construct a newly efficient finite element method which named as modified finite element method, for Maxwell’s equation in Kerr-type nonlinear media. Because the efficient unconditional stable alternative direction implicit method can’t extend to the Maxwell’s equations in Kerr-type nonlinear media (the difficulty will be discussed in section 3.1), we design the modified method to construct an efficient unconditional stable scheme for Maxwell’s equations in Kerr-type nonlinear media. Furthermore, combined with the direct method, we construct a more efficient direct modified scheme. Comparisons of the efficiency, stability, convergence rate for our modified method and classical explicit leapfrog method were done by theoretical analysis and numerical experiments. The modified method is unconditional stable and its efficiency is not less than leapfrog method.

Dynamics of fractional optical solitary waves to the cubic–quintic coupled nonlinear Helmholtz equation

This work investigates the dynamics of optical waves to the generalized coupled nonlinear fractional Helmholtz equation with quintic and cubic nonlinear effects. The evolution of broad multicomponent self-trapping beams in Kerr-type nonlinear media is described by the coupled Helmholtz equation, which also take into account spatial dispersion due to non-paraxial effects. This phenomena is particularly relevant to the progressive miniaturization of optics, where the optical wavelength is similar to the beam width. It is essential to integrate non-Kerr terms, such as the self-steepening and the self frequency shift, into the coupled Helmholtz system in order to investigate the propagation of ultrashort optical pulses in the non-paraxial domain. For optical wave solutions, nonlinear ordinary equation of the governing model is achieved by applying the fractional transformation. The different types of the solutions like bright, dark, singular and mixed type solitons are extracted by applying advanced integration methods, namely modified Sardar subequation method and new Kudryashov method. These solutions provide very useful information on how the system operates. The applied approaches are highly efficient and have significant computational capability to efficiently tackle the nonlinear systems. Additionally, we include a diverse array of graphs to demonstrate the physical interpretation of the obtained solutions in relation to a number of significant parameters, thereby highlighting the impact of fractional derivatives. In the context of the proposed model, these visualizations assist with a comprehensive understanding of the solution’s behavior and characteristics. It is anticipated that these solutions may be significant in the study of wave propagation and related fields.

Wave packet dynamics of entangled q-deformed states

This paper explores the wave packet dynamics of a math-type q-deformed field interacting with atoms in a Kerr-type nonlinear medium. The primary focus is on the generation and dynamics of entanglement using the q-deformed field, with the quantification of entanglement accomplished through the von Neumann entropy. Two distinct initial q-deformed states, the q-deformed Fock state, and the q-deformed coherent state, are investigated. The entanglement dynamics reveal characteristics of periodic, quasi-periodic, and chaotic behaviour. Non-deformed initial states display wave packet near revivals and fractional revivals in entanglement dynamics while introducing q-deformation eliminates these features. The q-deformation significantly influences wave packet revivals and fractional revivals, with even a slight introduction causing their disappearance. For large values of q, the entanglement dynamics exhibit a chaotic nature. In the case of a beam splitter-type interaction applied to the initial deformed Fock state, an optimal deformation parameter q is identified, leading to maximum entanglement exceeding the non-deformed scenario.

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.

Numerical analysis of the direct method for 3D Maxwell’s equation in Kerr-type nonlinear media

Numerical analysis of the direct method for 3D Maxwell’s equation in Kerr-type nonlinear media

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.

Interaction of a Spatial Soliton on an Interface between Two Nonlinear Media

Spatial solitons are the solutions of nonlinear partial differential equations describing the propagation of optical beams in Kerr type nonlinear medium. This paper studies the scattering of a spatial solitons of the Cubic-Quintic Nonlinear Schrodinger Equation (C-Q NLSE) on an interface between two nonlinear media. The scattering process will be investigated by variational approximation method and by direct numerical solution of C-Q NLSE. This variational approximation method has been used to analyse the dynamic of the width and center of mass position of a soliton during the scattering process. Meanwhile, a direct numerical simulation of C-Q NLSE is run to check the accuracy of the approximation by using the same range of parameters and initial condition. The results for direct numerical simulation of CQNLSE for soliton parameters are quite similar with the variational equation. The studies showed that soliton can be reflected by or transmitted through the interface, also the nonlinear surface wave can be formed depending on the parameters of interface and initial soliton.

Soliton content of wave packets with strong phase modulation: The Wentzel–Kramers–Brillouin approach

The WKB approximation for the Zakharov–Shabat scattering problem is used for the analysis of decay of wave packets stimulated by a strong phase modulation in Kerr-type nonlinear medium. It is shown, that for moderate nonlinearity, solitons tend to propagate in the directions of interference fringes formed in the linear limit, and soliton parameters are reasonably well approximated with the WKB method. Simple analytical approximations for distributions of soliton parameters are reported.

Tunable transverse-electric surface plasmon polariton waves in a parallel-plate graphene-based waveguide

In this study, a three-layered system consisting of a Kerr-type nonlinear medium confined between two semi-infinite linear media is considered. The nonlinear medium is a parallel-plate graphene waveguide with a thickness of d. Analytical expressions for dispersion relation and the field profiles in terms of Jacobi elliptic functions are performed and solved numerically. Plasmonic properties such as the effective mode index (), localization length () are studied. It is found that and of the system can be tuned by chemical potential of graphene, arising from its tunable optical conductivity. Localization of transverse electric (TE) surface waves increases with increasing the medium nonlinearity, providing deep-subwavelength confinement. Moreover, increasing d from 1 to 130 nm enhances the mode confinement. We also apply our scheme to a bilayer graphene (BLG) waveguide, allowing for predicting the existence of TE modes. Our results show that the TE modes in BLG are more pronounced than those in single-layer graphene. Therefore, the BLG structure is found to be promising in the fabrication of optical devices with TE plasmons.

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.

Digital signal processing in coupled photonic crystal waveguides and its application to an all-optical AND logic gate

The realization of all-optical AND logic gates for pulsed signal operation based on the photonic bandgap transmission phenomenon is proposed. We are using realistic planar air-hole type coupled photonic crystal waveguides (C-PCWs) with Kerr-type nonlinear background medium. The novelty of our analysis is that the proposed AND logic gate operates with the temporal solitons, which maintain a stable envelope propagating in the nonlinear C-PCWs, enabling true ultrafast full-optical digital signal processing in the time-domain. The bandgap transmission takes place when the operating frequency is chosen at the very edge of the dispersion curve of one of the supermodes in the C-PCWs. In this regard, our original fast and accurate method is used to efficiently calculate the supermodes of the C-PCW system. The underlying semi-analytical full-wave modal analysis is based on the evaluation of the lattice sums for complex wavenumbers using the transition-matrix method in combination with the generalized reflection-matrix approach. As a proof of concept successful pulse operation of the all-optical AND logic gate is demonstrated in the framework of extensive full-wave finite-difference time-domain electromagnetics analysis.

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.

Quantum synchronization of two mechanical oscillators in coupled optomechanical systems with Kerr nonlinearity

We investigate the quantum synchronization phenomena of two mechanical oscillators of different frequencies in two optomechanical systems under periodically modulating cavity detunings or driving amplitudes, which can interact mutually through an optical fiber or a phonon tunneling. The cavities are filled with Kerr-type nonlinear medium. It is found that, no matter which the coupling and periodically modulation we choose, both of the quantum synchronization of nonlinear optomechanical system are more appealing than the linear optomechanical system. It is easier to observe greatly enhanced quantum synchronization with Kerr nonlinearity. In addition, the different influences on the quantum synchronization between the two coupling ways and the two modulating ways are compared and discussed.

A new FDTD scheme for Maxwell’s equations in Kerr-type nonlinear media

In this paper, we develop a totally new direct finite difference solver for solving the Maxwell’s equations in Kerr-type nonlinear media. The direct method is free of iteration error and more efficient than the classic iteration method. We also prove the continuous stability for the Kerr model and the discrete stability for the proposed scheme. Extensive numerical results are presented to validate our analysis.

Solutions for ultra-broad beam propagation in a planar waveguide with Kerr-like nonlinearity

We consider nonlinear optical systems describing the propagation of beams in Kerr-type nonlinear media, experiencing diffraction in transverse and longitudinal directions. The first model investigated, under nonparaxial approximation, is the complex nonlinear Helmholtz equation, recast into a coupled, real system of partial differential equations. We construct and apply its conserved vectors to determine exact solutions. This approach of a double reduction combines a point symmetry with a particular conservation law to enact a reduction and derive solutions. A second model is also studied, that is related to the Maxwell’s equations under paraxial approximation — a version of the nonlinear Schrödinger equation. We show that when the paraxial effect vanishes, a number of additional exact solutions and conservation laws are admitted.

Effects of Kerr medium in coupled cavities on quantum state transfer

We study the effect of Kerr type nonlinear medium in quantum state transfer (QST). We have investigated the effect of different coupling schemes and Kerr medium parameters [Formula: see text] and [Formula: see text]. We found that the Kerr medium introduced in the connection channel can act like a controller for QST. The numerical simulations are performed without taking the adiabatic approximation. Rotating wave approximation is used in the atom–cavity interaction only in the lower coupling regime.

Effect of dissipative environment on collapses and revivals of a non-linear quantum oscillator

We study the dissipative dynamics of a wave packet passing through two different non-linear media. The effect of dissipation on the phenomenon of collapses and revivals of a wave packet as it evolves in a Kerr-type non-linear medium (represented by the Hamiltonian $({a}^{\dag} a)^2$) is investigated. We find that partial revivals do take place when dissipation values are moderate. For a certain regime of parameters we find a solution where revivals do not die even in the presence of dissipation and the non-linearity appears to compensate for the energy and coherence loss. We consider the next order non-linearity, represented by the Hamiltonian $({a}^{\dag} a)^3$, where we observe the phenomena of super revivals. The effect of dissipation in this case has an additional feature of number dependence for the displaced number states. While our simulations explore the degree to which the phenomena of collapses and revivals degrades in a dissipative environment, we also discovered the presence of a situation where degradation is minimal.