Xian-Jing Lai(來嫻靜), Xiao-Ou Cai(蔡曉鷗),and Jie-Fang Zhang(張解放)

1College of Basic Science,Zhejiang Shuren University,Hangzhou 310015,China

2Zhejiang University of Media and Communications,Hangzhou 310018,China

1 Introduction

Vortex is spatial soliton characterized by screw phase dislocations and orbital angular momentum.It has arrested the attention in many physics fields since the vortex concept was first proposed by Volyar,[1]such as nonlinear optical systems, fluid dynamics,plasma physics,Bose–Einstein condensation,and so on.The relevant theories have been developed.[2?4]So far,theoretical analysis on the vortex was putted into practice governed by a nonlinear Schr?dinger(NLS)equation.[5?12]As we know,it is not a thorough presentation of a photoamplifier,though the features of vortices are well conformed with the results of the experiment.[13?14]For instance,in the applications of device for high power levels,nonlinear gain saturation should be considered.[15?16]In a uniform medium,it is widely considered that self-focusing medium supports bright solitons,while self-defocusing supports dark solitons.Changing this situation needs appropriate inhomogeneous medium.[13,17?18]In optical systems,vortices with new features have been also predicted and demonstrated with spatial modulation of the nonlinearity along the transverse axis.Multiple devices have been exploited to stabilize vortices in conservative systems,which involve optical lattices,nonlocal nonlinear response,and competing nonlinearities.[16]

At the same time,dissipative solitons,have also received widespread attention in varieties of physical systems.[19?21]Due to losses in the medium have to be made up for gains,stability is a basic question for such solutions.While,uniform linear gain causes the background around the localized excitations unstable to make it unsteadily.Several methods to solve this problem have been designed,such as the nonlinear gain and higher order absorption,[22]localized linear gain,[23]a localized cubic gain as well as without higher order losses.[15]Notice that the delicate balances between the parameters of medium to insure the presence of these analytical dissipative solitons.Vortices also occur in dissipative physical systems.Stable dissipative vortices were detected in laser amplifi ers in frame work of the complex nonlinear Schr?dinger equation.[16,24]Recently,practice has proven that a spatially modulated linear gain impact remarkably on the evolution of solitons in nonlinear dissipative system with two-photon absorption.[16]Constitution of nonlinear excitations in spatially modulated linear gain was analysed in various systems,such as optical waveguides,[25]photonic crystals with periodical index modulations,[26]optical system with Kerr nonlinearity and two-photon absorption,[15]Bose-Einstein condensates,[5]and so on.In these systems,nonlinear excitations take shape in the domain of maximum gain,therefore the symmetry of soliton is determined by linear gain topology.The nonlinear loss working on the setting possibly restrains instability,although spatially localized gain guarantees the background stability.The method of improving the stability of dissipative soliton mentioned above has been displayed in two-dimensional systems due to the interaction e ff ect among localized gain and nonlinear loss.Instead of the common belief,Ref.[15]reveals that novel nonlinear excitations can also be supported by the spatially modulated absorption,while none of them is in existence in homogeneous dissipative system,as far as we know.In particular,stable dissipative solitons can be excited in system in the existence of uniform linear gain and inhomogeneous tow-photon absorption.

In this paper,with the help of a universal ansatz method and variational approach,we firstly study a dissipative optical setting in framework of a complex nonlinear Schr?dinger equation.The radially symmetric dissipative vortices are obtained analytically.Secondly,by use of split-step Fourier method,we simulate the evolution of these vortex solitons and test their stabilities.Evolution analysis reveals that they,propagating in selfdefocusing nonlinear dissipative media,could be stabilized in presence of spatially inhomogeneous nonlinear absorption.Some samples of initial condition for the support of vortex solitons are presented.

2 Complex Nonlinear Schr?dinger Equation and Dissipative Vortex Solitons

Consider a beam propagates through a medium described by the following complex nonlinear Schr?dinger(CNLS)equation:[15]

In optics,such setting can be established along with a spatial constant linear gain and modulated material with doped two-photon absorption.Hereu(z,r,?)is envelope of the electric field changing slowly,?2is the transverse Laplacian operator,ris the radial coordinate,zis the propagation distance,g0andχare the coefficients in connection with the linear gain and cubic nonlinearity respectively,meanwhile the termrepresents the nonlinear loss e ff ect with coefficientκrelated to two-photon absorption.As far as we know,two-photon absorption has been found to have important functions in alloptical switching.[27?28]In undoped optical media,it is also a known fact that dynamics of higher-order solitons are affected by nonlinear absorption and inclined to collapse.The two-photon absorption e ff ect is especially important for new kinds of media made of lead silicate glasses or semiconductor microcrystal doped glasses[29]appearing considerably high nonlinearities.Without nonlinear gain saturation,we can simplify Eq.(1)to NLS equation with uniform gain,corresponding to the optical amplifier with normal group-velocity dispersion.It is natural for us to ask questions,does the vortices still exist in the CNLS equation?If so,how do the nonlinear absorption a ff ect behavior of these dissipative vortices?

The vortex of Eq.(1)can be acquired by using ansatz method.[5?12]Substituting the ansatz

whereA≡A(r)is real,Sis the topological charge,?is the azimuthal angle,andμis the propagation constant.As we know,the winding number of the input phase determines the value of chargeS.

Substitute ansatz Eq.(6)into Eq.(4),and we get the following set of equations

whereis the solution of first excited state,is the effective external potential.Equation(11)shows that the Kerr nonlinearity is related to the nonlinear gain saturation and effective external potential.

According to Eqs.(12)–(13),we have following set of solutions

whereJα(η)andYα(η)withare Bessel functions of first and second kind,a,candLare positive constants.Therefore the interplay of?χand?κrelated to nonlinearity pro fi le and two-photon absorption respectively is

This implies that the two-photon absorbtion effect should be adopted to make up for the kerr nonlinear effect.

In the middle of area,has a monotone increasing in the condition ofb=0 anda=c,whilea=corb=0,there is oscillating variation of the nonlinearity outline.

Therefore,together with Eqs.(6),(14),and(16),we have acquired dissipative vortices for the CNLS equation directly as

with Eqs.(14)–(15),which are characterized by the topological chargeSand parametrized by the propagation constantμ.

Fig.1 (Color on line)Example of vortex solitons of Eq.(19)with topological charge(a)S=1;(b)S=8.Other relative parameters areμ=10,P0=1,b=0,a=c=1.Intensities and phase pro files are shown at z=0.

Fig.2 The density pro files of vortex solitons of Eq.(19)with different(a)topological charge S=1,2,3,4;(b)propagation constantμ=2,4,6,10,respectively.Other parameters are the same as that in Fig.1.

Two sample pro files of stable radially symmetric vortex withS=1,8 are shown in Fig.1 and Fig.2.It is clearly that there is one bright ring around the vortex core in this case.For fixed propagation constantμ,increase of chargeSresults in increase of the width but decrease of the amplitude of the vortex.For fixed chargeS,although there are vortices with different nonlocality they have the constant amplitude.Besides,as shown,amplitude of vortex ofS=1 is not zero asr→0.However,the vortex acquires asymptotic conditions with rises in topological chargeS.In addition,vortices extend outside,which in turn increase their angular momentum.

3 Numerical Analysis

To study stability,we did numerical simulation and analysis for the propagation and evolution mechanism of Eq.(1),by split-step Fourier method[34]and perturbation theory.We choose a suitable experimental apparatus,[35]in whichn0=2.7,with the characteristic transverse scaleW0=34μm,the di ff raction length isLD=3.7 cm,a frequency-doubled Nd:YAG laser nearλ=532 nm.The initial seed is

The distancezis measured in units of di ff raction lengthLD.And we simulated the propagation of the vortex with a white noise of fi ve percent.Besides,enhancing the experimental possibilities,the energy of vortex could be expected to leak away and can be realized by using numerical simulation combined with absorbing boundary conditions.

Fig.3 (Color on line)(a)The initial seed of Eq.(20)with the topological charge S=2 at z=0;(b)–(d)Three typical time evolutions of the initial seed at z=2 with uniform nonlinear absorption κ = ?0.01 and different uniform linear gains(b)g0=0.1,μ=0.5;(c)g0=1.8,μ=0.5;and(d)g0=0.1,μ=10.White noise of five percent is added.Other parameters are the same as that in Fig.1.

We discuss two different scenarios.First,we focus on existence of vortex solitons in the presence of a spatially uniform two-photon absorbtion.For all we know,κcould be adjusted to?1,?10?1and?10?2for semiconductor microcrystal doped glass,As2S3chalcogenide glass,and lead silicate glass,respectively.[24]Furthermore,there are also some unfavorable factors in practical applications of vortex beams.For example,there are some secondary bright rings besides the inner main bright ring of the vortex beam,i.e.the sidelobes of optical vortex solitons.Here,a similar situation has also occurred during the course of evolution.Figures 3(b)–3(d)depict the numerical simulation at the same propagation distance with different uniform linear gaing0and propagation constantμ.As shown in the fi gure,in general,the vortices are subject to instability and transform themselves into some weaken ones with lots of sidelobes,while the lower value ofg0can suppress the range of sidelobes efficiently,as shown in Fig.3(b).Although only the conclusions ofS=2 are explained exactly,the results of other values of charge are resemble.It should be noted that,in the intermediate region,κandχmust vary monotonically ofr,i.e.a=candb=0 to ensure the instability.

The evolutions of vortex soliton families are characterized by linear gaing0,propagation constantμ,and the integer vorticityS,Fig.4 displays the energyE=of vortices under the in fl uence of these different factors.In order to explore the in fl uence on the solution ofg0,μ,S,respectively,we set the other parameters same as that in Fig.1.

Fig.4 The energy profiles of vortex solitons of Eq.(19)with the uniform nonlinear absorption κ = ?0.01 for different uniform linear gain g0,propagation constantμand topological charge S,respectively.Parameters are(a)S=2,μ=0.5;(b)S=2,g0=0.1 and(c)g0=0.1,μ=1.Other parameters are the same as that in Fig.1.

Fig.5 (Color online)Four typical of time evolutions for the vortex solitons(19)with the variable nonlinear absorption(22)(?=1).White noise of fi ve percent is added.Parameters are(a)S=1,g0=3.8,μ =10;(b)S=2,g0=3.8,μ=10;(c)S=3,g0=3.8,μ=10;(d)S=8,g0=1,μ=0.5,respectively.Other parameters are the same as that in Fig.1.

Next,we address vortex solitons supported by inhomogeneous material with doped two-photon absorbtion.Laboratory techniques on aspects of the realization of suitable inhomogeneous dopant concentration pro files have been provided detailed in Ref.[35].Another preferred technique is to induce an efficient spatially modulated nonlinearity by using the non-uniform fields which influences the detuning of dopants of evenly distributed.[37]In particular,here we begin with an optical setting in which?κin Eq.(4)increases to the periphery As discussed earlier,the evolution must be reconfirmed that it still meets condition ofa=candb=0 for stability.

Our numerical results demonstrate that,fortunately,there is no more sidelobes during the courses of evolution in this case.The exponential absorption landscape withg0below a certain value can support stable dissipative vortex solitons.In general,the change rate ofκneeded for the instability of reduction of a high charged vortex much smaller than theκfor a vortex of lowerS.Wheng0exceeds the permitted values,radially symmetric vortices become unstable except situation ofS=1.AndSincreases,and it stability worsens.Figure 5 shows the evolution comparison after a certain propagation distance.As shown in Figs.5(b)–5(c),at fixedg0=3.8,vortex withS=2 orS=3 features the spontaneous symmetry breaking in the medium,the density of vortex component is not radially symmetric,and it destructs the ring structure.There is a process of subsidence in the center of vortex,in general,the number of sink holes is related to the value of chargeS.However,pulse splitting is also observed forg0=1 with high charge.Actually,vortices with high charge only stable at small values of the gain,but the decrease of propagation constantμcan also delay the appearance of instability,as shown in Fig.5(d).

Fig.6 The energy pro files of vortex solitons(S=2)of Eq.(19)with the variable nonlinear absorption(22)for different uniform linear gain(a)g0=1,(b)g0=1.5,and(c)g0=2,respectively.Other parameters just like ones set in Fig.1.

Fig.7 Three typical pro files of χ related to Kerr nonlinearity:(a)uniform coefficient κ = ?0.01 of two-photon absorption;(b)spatially inhomogeneous two-photon absorption of Eq.(22)with b=0;(c)spatially inhomogeneous two-photon absorption of Eq.(22)with b=0.6.Other parameters are S=2,μ=0.5,P0=1,a=c=1.

The dependency of vortex energy withS=2 to the uniform linear gaing0are shown in Fig.6.We set the other parameters the same value.

In general,these examples demonstrate that the spatially modulated nonlinear absorption e ff ect in Eq.(1)is essential for the existence of stable vortex solitons considering the uniform gain exist.And the uniform linear gain indeed a ff ects the stability of vortex in this media.In addition,the studies demonstrate that the solution with lower values ofg0,μ,Sis more stable than the solutions with high values.With appropriate conditions,the vortex solitons can propagate stably and feature no symmetry breaking,although the beams exhibit radical compression and amplification as they propagate.These performances a ff ord an available approach for the experimental manipulation of high charge vortices in a fixed parameter model.

It is worth stressing that,no matter two-photon absorbtion is uniform or spatially modulated,χrelated to Kerr nonlinearity is positive,i.e.χ>0,as shown in Fig.7,whose intensity increases rapidly from the center to the edge.It suggests that the cases discussed above all travel inside the nonlinear Kerr self-defocusing medium.It means that the set can support bright vortex solitons when the cubic nonlinearity is self-defocusing,as previ-ously mentioned in Ref.[7].

4 Conclusions

In conclusion,with the help of a universal ansatz method and variational approach,exact vortex solitons were obtained.Our analysis has revealed that these vortex solitons,propagating in self-defocusing nonlinear dissipative media,could be stabilized in presence of spatially inhomogeneous nonlinear absorption.

The evolution was based on a CNLS equation that involved with the uniform linear gain and spatially inhomogeneous nonlinear loss.Firstly,we verify the existence of vortex solitons in such system.We have a large freedom to choose parameters in Eq.(20)to obtain physical meaningful vortex.It is relevant to mention that Eq.(1)admits a stable dissipative vortex solitons with the suitable gain saturation modulation pro file.Some sample of initial condition for the support of vortex solitons were presented.In homogeneous media with localized nonlinearities,such vortices are almost tend to suffer azimuthal instability or to generate sidelobes.We show that stable dissipative vortex solitons can exist,provided that the uniform gain interplay with inhomogeneous nonlinear absorption,whose intensity increase rapidly from the center to the edge.Note that the uniform linear gain indeed affects the stability of vortices in this media.In addition,the studies demonstrate that the solution with lower values ofg0,μ,Sis more stable than the solutions with high values.With appropriate conditions,the vortex solitons can propagate stably and feature no symmetry breaking,although the beams exhibit radical compression and amplification as they propagate.

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