The Study Of Beam Propagation And Its Interaction Under Fractional Schr?dinger Equation

By studying the standard Schr?dinger equation,two of the functional forms with stable solutions are obtained.There are hyperbolic secant function and Airy function of wave packet.The former is a stable conventional symmetric beam,and the latter is an asymmetric self-accelerating beam.The propagation characteristics of soliton beams and Airy beams based on the standard Schr?dinger equation have been widely studied.With the development of quantum mechanics,it is found that the fractional Schr?dinger equation is a branch extension in the field of quantum mechanics.The fractional Schr?dinger equation with spatial derivative is a generalization of standard Schr?dinger equation.However,the characteristics of the propagation of soliton and Airy beam based on the fractional Schr?dinger equation have a few studied so far.The fractional Schr?dinger equation may provide more free choice space for beam control.The transmission of super-Gaussian beam based on the fractional Schr?dinger equation is fundamental different from that in the standard Schr?dinger equation.In this paper,we study the interaction of soliton beams,the interaction of Airy beams and the propagation characteristics of super-Gaussian beam in linear and nonlinear regimes,which is based on fractional Schr?dinger equation,so as to further understand the fractional Schr?dinger equation.The main contents of this paper and the results of the experiment are as follows:Firstly,by changing the initial separation distance,initial phase difference and relative amplitude parameters of the soliton beam,we observe the transmission effect of different Lévy index on the beam interactions which is based on the fractional Schr?dinger equation.The initial relative interval parameters will affect the intensity of the interaction of the soliton beam.The Lévy index can control the range of the beam interaction more,and control the position of the first fusion point of the inphase beam interaction.For the interaction of the out-of-phase beams,the rejection phenomenon of the beam can be weakened.Beam attraction and repulsion mutually depends on the initial phase difference and Lévy parameter values.The energy distribution and the peak intensity of beams can be controlled by the relative amplitude and Lévy index parameters.Secondly,the influence of different Lévy index on the interaction of Airy beams is studied based on the fractional Schr?dinger equation,and the general characteristics of beam propagation are obtained by changing the initial separation,relative amplitude and initial phase difference respectively.In the process of beams interaction,a single soliton,a breathing soliton,and a pair of solitons are produced with different periods and pulse width.The formation of soliton shapes depends upon the initial separation,relative amplitude,phase difference parameter and Lévy value.The nonlinear effects can be tuned by changing the value of the Lévy index,which will provide a new freedom parameter for controlling the pulse width and intensity of the beams.The last but not the least,we have investigated the propagation dynamics of super-Gaussian optical beams in fractional Schr?dinger equation.We have identified the difference between the propagation dynamics of super-Gaussian beams and that of Gaussian beams.We show that,the linear propagation dynamics of the super-Gaussian beams with order m(29)1 undergo an initial compression phase before they split into two sub-beams.In the nonlinear regime,the super-Gaussian beams evolve to become a single soliton,breathing soliton or soliton pair depending on the order of super-Gaussian beams,nonlinearity,as well as the Lévy index.In two dimensions,the linear evolution of super-Gaussian beams is similar to that for one dimension case,but the initial compression of the input super-Gaussian beams and the diffraction of the splitting beams are much stronger than that for one dimension case.Our results show the nonlinear effects can be tuned by varying the Lévy index in the fractional Schr?dinger equation for a fixed input power.The results related to this study are published on Optics Express.