The Excitation And Control Of The Nonautonomous Vortex Beams And Vortex Solitons

In this paper,the nonlinear Schr ¨odinger equation with variable coefficients from a variety of nonuniform optical medium have been studied using analytic and numerical simulation.The effects of parametric diffraction and nonlinearity on vortex amplitude,phase,wave width of transmission characteristics are discussed.Put forward the feasible scheme for the practical research to activate and maintain vortex rotating speed and shape.Provide a theoretical basis for control of optical vortex in parametric optical system.Research work here has the potential application value of study on the dynamics of matter wave solitons and vortex plasma vortices in a solitary wave and other physical fields.The main contents are as follows:1.Azimuthal modulated Laguerre-Gauss type vortex beam in nonlocal medi-umWith the help of similarity transformation and variational approach,we have presented a systematic analysis of the two-dimensional generalized nonlocal nonlinear equation.Twodimensional azimuthal modulated Laguerre-Gauss type vortex beams(AMLG)are introduced and investigated analytically and numerically.In addition,the AMLG includes the soliton cluster and vortex soliton as particular members,with zero and maximum modulation depths,respectively.We demonstrate that the AMLG of critical power in the strongly nonlocal limit are more stable than the one with lower nonlocality.Our major attentions have been paid on a periodic distributed amplification system.Remarkably,these AMLG have the angle frequency modulated by the distributed coefficient,apart from the beam width and intensity changing self-similarly as the breather.2.Research on Whitaker vortex solitons in strongly nonlocal nonlinear mediaWe derive analytical Whitaker vortex solitons(WVS)of the strongly nonlocal nonlinear media with space-dependent diffractive,gain(attenuation)coefficient based on the similarity transformation and variational approach.These WVS can be reduced to the rotating vectornecklace-ring solitons,and non-rotating radially symmetric optical soliton,such as Gaussian solitons,vortex solitons,soliton cluster,under certain conditions.Remarkably,in addition to the movement of the beam center,the important feature is the appearance of the distributed coefficient and the beam width in the azimuthal function,which strongly affect the rotating behavior of the azimuthons with the non-zero angular frequency.Therefore,the rotation angle of the WVS can be modulated in the certain area.Further on,with the zero gain coefficient and constant beam width,we discuss the interaction of the two asymmetric and non-coplanar WVS theoretically,and gain the trajectories of beam center in the transversal plane.3.Nonautonomous vortices in(2+1)-dimensional graded-index waveguideWe demonstrate nonautonomous vortex beams in a isotropic inhomogeneous refractiveindex waveguide with self-focusing nonlinearity.The results show that the nonautonomous vortex beams can self-similarly evolve during a short-term propagation when delicate balance is obtained between the diffraction,nonlinearity and gain,although these vortices are approximate.We are concerned with some special but interesting situations,and discussing the changes of the height,width,energy,and central position of the vortices as the increase of propagation distance.In addition,we are also interested in the azimuthal modulational instability of the system,and compare our predictions for the modulational instability growth rates to numerical results.4.Study of spatial modulated vortex solitons under external electric fieldBy means of the variational approach,three classes of azimuthally modulated vortex solitons are constructed for the(2+1)-dimensional nonlinear Schr¨odinger equation with a type of transverse nonperiodic modulation.These solutions include numerical single layer azimuthal vortex solitons in the medium with Gaussian type modulation and analytical azimuthons in medium with the anharmonic potential.We present the results for two types of nonlinearities which are common in optical materials.The properties of vortex solitons,which are characteristic of the topological charge,propagation constant,angular velocity and radial quantum number,are studied.The numerical simulations are used to verify the propagation properties of these vortex solitons.We address the possibilities to support stable vortex in 2D media with the spatially inhomogeneous quintic nonlinearity,including the 2D analytical vortex soliton and numerically constructed vortex soliton in the absence of external potential or in the anharmotic potential.We focus on 2D models with modulation profiles in the form of Bessel or Whittaker functions,for which some vortex solutions can be obtained in an exact analytical forms.Exact vortex solitons in both cases are constructed by using the similarity transformation.The radial profiles of these vortex solitons show that there are one or several bright ring surrounding the vortex core.In addition to the exact ones,a number of vortex solitons are found numerically.We give an additional argument,based on linear static analysis estimates.For the one layer ring vortex solitons of the self-defocusing media,the values of nonlinearity coefficients are zero near the center and increase rapidly toward the periphery.It is shown that exact vortex solitons are stable,while the numerically constructed ones exhibit an interesting pattern,switching between stable and unstable as the soliton eigenvalue increases.For the multilayer ring vortex solitons of the self-focusing media,with the increase of the propagation constant,all the vortex solitons only exists in the narrow parameter regions,either numerical or analytical one.