Cubic-quintic Nonlinear Media Research Articles

A finite difference scheme for (2+1)D cubic-quintic nonlinear Schrödinger equations with nonlinear damping

Solitons of the purely cubic nonlinear Schrödinger equation in a space dimension of n≥2 suffer critical and supercritical collapses. These solitons can be stabilized in a cubic-quintic nonlinear medium. In this paper, we analyze the Crank-Nicolson finite difference scheme for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping. We show that both the discrete solution, in the discrete L2-norm, and discrete energy are bounded. By using appropriate settings and estimations, the existence and the uniqueness of the numerical solution are proved. In addition, the error estimations are established in terms of second order for both space and time in discrete L2-norm and H1-norm. Numerical simulations for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping are conducted to validate the convergence.

Quadruple Gaussian laser beam in cubic-quintic nonlinear media: effect of nonlinear absorption

Quadruple Gaussian laser beam in cubic-quintic nonlinear media: effect of nonlinear absorption

Fisher Information on the Phase Transition from Regular to Flat Top Optical Pulse in Cubic-Quintic Nonlinear Media

Fisher Information on the Phase Transition from Regular to Flat Top Optical Pulse in Cubic-Quintic Nonlinear Media

Stable higher-charge vortex solitons in the cubic-quintic medium with a ring potential.

We put forward a model for trapping stable optical vortex solitons (VSs) with high topological charges m. The cubic-quintic nonlinear medium with an imprinted ring-shaped modulation of the refractive index is shown to support two branches of VSs, which are controlled by the radius, width, and depth of the modulation profile. While the lower-branch VSs are unstable in their nearly whole existence domain, the upper branch is completely stable. Vortex solitons with m ≤ 12 obey the anti-Vakhitov-Kolokolov stability criterion. The results suggest possibilities for the creation of stable narrow optical VSs with a low power, carrying higher vorticities.

Fast soliton interactions in cubic-quintic nonlinear media with weak dissipation

We derive the expressions for the collision-induced amplitude dynamics of two flat-top solitons in spatial dimensions of 1, 2, and 3 caused by the generic weak nonlinear loss under the framework of coupled cubic-quintic (n+1)D nonlinear Schrödinger equations for n=1,2,3. We develop a new perturbative technique which is mainly based on the calculations for the fast collision-induced changes in the soliton envelope and the use of the single perturbed soliton solution for the calculations. These results quantify the energy dropdown due to a fast collision of two pulses (n=1), or two optical beams (n=2), or two light bullets (n=3) in cubic-quintic nonlinear media with dissipation. The theoretical calculations are then confirmed by numerical simulations with the corresponding coupled nonlinear Schrödinger equations. Our approach can be used for studying the effects of dissipation on colliding solitons described by the coupled nonlinear Schrödinger models, where the unperturbed equation is nonintegrable.

Dynamics of Compressed Optical Pulse in Cubic-quintic Media

ABSTRACT We consider optical pulse propagation in cubic-quintic nonlinear media and analyze the dynamics of pulse compression within the framework of variational approach. Based on a potential model we find effective pulse width for stable propagation. We see that a pulse propagating in cubic media gets compressed due to quintic nonlinearity. The value of width of a compressed pulse decreases with the increase of power. However, its minimum value is limited by the growing instability above the critical power. We explicitly examine that the compressed pulse is dynamically stable due to the competition between cubic and quintic nonlinearities.

Electromagnetic Guided Waves in a Lossless Cubic-Quintic Nonlinear Waveguide

Paper focuses on the problem of transverse-electric wave propagation in a lossless cubic-quintic nonlinear waveguide. Using an original tool, we study the problem in detail avoiding the use of special functions. It is shown that a waveguide filled with cubic-quintic nonlinear medium supports infinitely many guided waves in the focusing case. The found solutions are split into a finite number of waves having linear counterparts and an infinite number of waves that stay away from any solution to the corresponding linear problem.

Soliton–soliton interactions in a grating-assisted coupler with cubic-quintic nonlinearity

ABSTRACTWe systematically investigate the interaction of co-propagating Bragg grating solitons in a dual-core fibre in which one of the cores has a Bragg grating written a cubic-quintic nonlinear medium and the other one is linear and uniform. In a previous work, it has been shown that this structure supports two disjoint families of Bragg grating solitons (called Type 1 and Type 2). Type 1 solitons can be viewed as the generalization of Bragg grating solitons in the semilinear coupled system with cubic nonlinearity. On the other hand, Type 2 solitons exist where quintic nonlinearity is more pronounced. It has also been shown that Type 2 solitons are always unstable. However, Type 1 solitons can be stable in a portion of the bandgap. Interactions between stable in-phase Type 1 quiescent solitons may result in the destruction of one or both solitons, symmetric or asymmetric separation, merger into a quiescent soliton and transformation. The effects of the coupling coefficient, group velocity mismatch, initial phase difference and the separation between the solitons on the outcome of interactions are analysed.

Multi-pole dark solitons in nonlocal and cubic-quintic nonlinear medium

In this paper, we mainly simulate the characteristics of the ground state dark soliton and the multipole dark soliton in the nonlocal and cubic-quintic nonlinear medium. Firstly, the influences of the degree of nonlocality on the amplitude and the width of the dark soliton in the self-defocusing cubic-and self-focusing quantic-nonlinear medium are studied. Secondly, we find the nonlinear parameters affecting the amplitude values of solitons, but the refractive index induced by the light beam is always a fixed value. The numerical results show that the ground state dark soliton can be propagated stably alone the z axis, and the stable states of the dipole soliton and the dark tri-pole and quadru-pole solitons are stable. However, some quadru-pole dark soliton is unstable after propagating the remote distance. Furthermore, we also discuss the characteristics of the ground state dark soliton and the dark dipole soliton in the local cubic-nonlinear and nonlocal quantic nonlinear media. Both the amplitude and the beam width of the dark ground state soliton and dark dipole soliton are also affected by the degree of nonlocality and nonlinearity. Two boundary values of the induced refractive index change with the variations of the three nonlinear parameters. The dark soliton and the dipole dark soliton are more stable in the self-focusing cubic nonlinear and the nonlocal self-defocusing quantic nonlinear medium than those in the self defocusing cubic nonlinear and nonlocal self-focusing quantic nonlinear medium. The powers of single dark soliton and dark tri-pole soliton decrease monotonically with the increase of propagation constant when the cubic-quintic nonlinearities are certain values and these degrees of nonlocalities are taken different values. Furthermore, we also analyze linear stabilities of various nonlocal spatial dark solitons. And we find that the dipole dark soliton is unstable when the propagation constant is in the region[-0.9,-1.0]. These properties of linear stabilities of other multi-pole dark solitons are the same as their propagation properties.

Elastic collision and breather formation of spatiotemporal vortex light bullets in a cubic-quintic nonlinear medium

The statics and dynamics of a stable, mobile three-dimensional (3D) spatiotemporal vortex light bullet in a cubic-quintic nonlinear medium with a focusing cubic nonlinearity above a critical value and any defocusing quintic nonlinearity are considered. The present study is based on an analytic variational approximation and a full numerical solution of the 3D nonlinear Schrödinger equation. The 3D vortex bullet can propagate with constant velocity. Stability of the vortex bullet is established numerically and variationally. Collision between two vortex bullets moving along the angular momentum axis is considered. At large velocities the collision is quasi-elastic, with the bullets emerging after collision with practically no distortion. At small velocities two bullets coalesce to form a single entity called a breather.

Periodic waves and their stability in competing cubic-quintic nonlinearity

Exact expressions are obtained for stationary periodic waves in cubic-quintic nonlinear media with opposite signs of nonlinearity. It is shown that defocussing/focussing quintic nonlinearity can lead to stabilization of snoidal/(cnoidal and dnoidal) periodic waves depending on whether the Kerr medium is focusing/defocusing. Direct numerical simulations confirm results of the linear stability analysis. Different stable patterns e.g. bright solitons and kinks are also obtained under certain conditions.

Propagating properties of spatial solitons in the competing nonlocal cubic-quintic nonlinear media

We study the new spatial optical solitons and their propagating properties in the one-dimensional nonlocal cubic-quintic (C-Q) nonlinear model by the numerical method. We obtain multi-bright solitons and multipole soliton solutions in the one-dimensional nonlocal C-Q nonlinear model. The propagation of bright solitons is stable in the competing nonlocal cubic self-defocusing and quintic self-focusing nonlinear media when these nonlocal and nonlinear parameters are in the appropriate value domain. Considering the different nonlinear cubic effects, the interaction between two optical solitons with the same phase in the general nonlocal media displays the attraction or the repulsion for different nonlocal and nonlinear parameters. We find that the interval of two solitons affects the interaction between them. The refractive index is changed with the propagating constant when the nonlocal constant d3 is 10. Moreover, the triplepole, quadrupole and pentapole solitons can propagate steadily when the nonlocal parameters are appropriate, but hexa-pole (or above) solitons propagate unsteadily for any nonlocal parameter. Furthermore, we investigate the multi-pole solitons and their propagation stabilities by the Newton difference method and the Fourier split step method, obtain the stable propagation conditions for dipole, triplepole and quadrupole solitons, and find that the propagation of the pentapole and higher-order pole solitons is unstable. We also discuss the interactions of multi-pole solitons when they propagate along the axis z. The interactions are attraction or repulsion when the nonlocal and the nonlinear parameters are different. Meanwhile, we simulate the evolution of the refractive index along the axis z when the spatial optical solitons are multi-pole solitons. Finally, we study the relation between the power of soliton and the propagation constant under different degree of nonlocality. The power of the single bright soliton does not monotonically increase with the increasing propagation constant when the degree of nonlocality d3 is 10. We also derive the relation between the power of dipole bright solitons with the cubic nonlinearity parameter and the propagation constant under different degree of nonlocality. The power decreases monotonically with the increasing propagation constant when the cubic nonlinearity is a certain value or with the increasing cubic nonlinearity when the propagation constant is a certain value.

Elastic collision and molecule formation of spatiotemporal light bullets in a cubic-quintic nonlinear medium.

We consider the statics and dynamics of a stable, mobile three-dimensional (3D) spatiotemporal light bullet in a cubic-quintic nonlinear medium with a focusing cubic nonlinearity above a critical value and any defocusing quintic nonlinearity. The 3D light bullet can propagate with a constant velocity in any direction. Stability of the light bullet under a small perturbation is established numerically. We consider frontal collision between two light bullets with different relative velocities. At large velocities the collision is elastic with the bullets emerge after collision with practically no distortion. At small velocities two bullets coalesce to form a bullet molecule. At a small range of intermediate velocities the localized bullets could form a single entity which expands indefinitely, leading to a destruction of the bullets after collision. The present study is based on an analytic Lagrange variational approximation and a full numerical solution of the 3D nonlinear Schrödinger equation.

Compression of optical similaritons induced by cubic-quintic nonlinear media in a graded-index waveguide

We employ the similarity reductions in two steps to obtain a family of bright and dark similaritons for the variable coefficient cubic–quintic nonlinear Schrödinger equation. Also, parameter domains are delineated in which kink and double-kink similaritons exist for this model. This methodology introduces a free parameter through cubic nonlinearity coefficient which gives us freedom to tune the amplitude and the propagation distance of similaritons in a tapered graded-index waveguide. Furthermore, we observe rapid beam compression of these similaritons for varying detuning parameter and the coefficient of cubic nonlinearity.

(2+1)-Dimensional Spatial Localized Modes in Cubic-Quintic Nonlinear Media with the -Symmetric Potentials* *Supported by the Project of Technology Office in Zhejiang Province under Grant No. 2014C32006, the Special Foundation for theoretical physics Research Program of China under Grant No. 11447124, National Natural Science Foundation of China under Grant No. 11374254 and the Higher School Visiting Scholar Development under Grant No.

We obtain exact spatial localized mode solutions of a (2+1)-dimensional nonlinear Schrödinger equation with constant diffraction and cubic-quintic nonlinearity in -symmetric potential, and study the linear stability of these solutions. Based on these results, we further derive exact spatial localized mode solutions in a cubic-quintic medium with harmonic and -symmetric potentials. Moreover, the dynamical behaviors of spatial localized modes in the exponential diffraction decreasing waveguide and the periodic distributed amplification system are investigated.

Bistable dissipative soliton in cubic-quintic nonlinear medium with multiphoton absorption and gain dispersion

We studied the propagation of optical pulse in a dissipative cubic-quintic nonlinear medium with multiphoton absorption and gain dispersion. Variational method in conjugation with Rayleigh dissipative function is used to analytically solve the governing complex cubic-quintic Ginzburg–Landau equation. Three cases of multiphoton absorption are studied: (i) two-photon absorption (TPA), (ii) three-photon absorption (3PA) and (iii) both TPA and 3PA. For all three cases, the pulse intensity decays and width broadens. The greater the gain dispersion, the wider the pulse. Increasing quintic nonlinearity marginally affects pulse amplitude but significantly reduces pulse broadening. TPA and 3PA, which are functions of positive imaginary parts of, respectively, third- and fifth-order susceptibility, individually cause pulse decay and pulse broadening. Thus, their combination only expedites pulse degradation. Conversely, negative imaginary part of fifth-order susceptibility may lead to an effect that impedes pulse degradation. Matching numerical results are obtained that validates the entire analytical outcome. A suitable gain can arrest both pulse broadening and decay. Such dissipative solitonic pulses, bistable owing to quintic nonlinearity, are found for all cases.

Moving Bragg grating solitons in a cubic-quintic nonlinear medium with dispersive reflectivity

The stability and collision dynamics of moving solitons in Bragg gratings with cubic-quintic nonlinearity and dispersive reflectivity are investigated. Two disjoint families of solitons are found on the plane of the coefficient of quintic nonlinearity versus the normalized frequency (η,Ω(norm)). Through numerical stability analysis, we have identified stability regions on the (η,Ω(norm)) plane for various values of dispersive reflectivity parameter (m) and velocity (v). The size of stability regions is found to be dependent on m and v. Collisions of counterpropgating Type 1 and Type 2 solitons have been systematically investigated. It is found that for low to moderate values of dispersive reflectivity, the collisions of Type 1 solitons can result in various outcomes such as separation of solitons with reduced, increased, unchanged, or asymmetric velocities and generation of a quiescent soliton by merger or formation of three solitons. For strong dispersive reflectivity (e.g., m=0.5), the collisions of low-velocity in-phase Type 1 solitons may lead to repulsion of solitons, asymmetric separation, merger into a single soliton, or formation of three solitons (one quiescent and two moving solitons). At higher velocities collisions predominantly lead to the formation of three solitons. For m=0.5, in-phase Type 2 solitons may repel or form a temporary bound state of quiescent Type 1 solitons that subsequently splits into two asymmetrically separating Type 1 solitons. π-out-of-phase Type 2 solitons may also merge to form a quiescent Type 1 soliton.

Stability and collisions of moving Bragg grating solitons in a cubic-quintic nonlinear medium

The existence, stability, and collisions of moving solitons in Bragg gratings (BGs) in a cubic-quintic nonlinear medium are investigated. Two disjoint families of solitons that are separated by a border are identified. One family (Type 1) can be considered as the generalization of the moving solitons in BGs written in a cubic nonlinear medium. The other family (Type 2) occurs in regions where quintic nonlinearity dominates. Through systematic numerical stability analysis, the stability regions in the plane of quintic nonlinearity versus frequency have been determined. It is found that the stability regions are dependent on the velocity of solitons. The collisions of counterpropagating solitons have been systematically investigated. Collisions of in-phase Type 1 solitons can result in a variety of outcomes including forming two asymmetrically separating solitons and passing through each other and separating symmetrically with reduced, unchanged, or increased velocities. In certain parameter regions, solitons merge to form a quiescent one. An outcome that has not been reported previously for uniform gratings is the formation of a quiescent soliton and two symmetrically separating solitons. This outcome is found to be more robust than the merger.

Nonautonomous Solitons in the (3+1)-Dimensional Inhomogeneous Cubic-Quintic Nonlinear Medium

We have constructed explicit nonautonomous soliton solutions of the generalized nonlinear Schrödinger equation in the (3+1)-dimensional inhomogeneous cubic-quintic nonlinear medium. The gain parameter has no effects on the motion of the soliton's phase or their velocities, and it affects just the evolution of their peaks. As two examples, we discuss the propagation of nonautonomous solitons in the periodic distributed amplification system and the exponential dispersion decreasing system. Results show that the presence of the chirp not only makes the intensity of solitons weaken more promptly, but also broadens their width.

Nonlinear tunneling effect in the (2+1)-dimensional cubic-quintic nonlinear Schrödinger equation with variable coefficients

By means of the similarity transformation, we obtain exact solutions of the (2+1)-dimensional generalized nonlinear Schrodinger equation, which describes the propagation of optical beams in a cubic-quintic nonlinear medium with inhomogeneous dispersion and gain. A one-to-one correspondence between such exact solutions and solutions of the constant-coefficient cubic-quintic nonlinear Schrodinger equation exists when two certain compatibility conditions are satisfied. Under these conditions, we discuss nonlinear tunneling effect of self-similar solutions. Considering the fluctuation of the fiber parameter in real application, the exact balance conditions do not satisfy, and then we perform direct numerical analysis with initial 5% white noise for the bright similariton passing through the diffraction barrier and well. Numerical calculations indicate stable propagation of the bright similariton over tens of diffraction lengths.