分数阶衍射蜂窝晶格中带隙涡旋孤子的传输与控制

Transmission and control of band gap vortex solitons in fractional-order diffraction honeycomb lattices

基于分数阶非线性薛定谔方程,研究分数阶衍射效应下蜂窝晶格中带隙涡旋光孤子的存在性与传输特性.首先采用平面波展开法得到蜂窝晶格能带结构,其次在带隙结构中分别采用改进的平方算子迭代法、分步傅里叶法和傅里叶配置法研究含有蜂窝晶格势的分数阶非线性薛定谔方程中带隙涡旋孤子的模式及其传输特性.研究结果发现带隙涡旋孤子的传输特性受Lévy指数和传播常数的影响.在稳定区间,带隙涡旋孤子可以稳定传输,而在非稳定区间,带隙涡旋孤子会随着传输距离的增加而逐渐汇聚,失去环状结构演变为基孤子.且Lévy指数越大,带隙涡旋孤子能够稳定传输的距离越长,功率越低.此外,相邻晶格同相位两个带隙涡旋孤子与旁瓣能量相叠加,反相位两带隙涡旋孤子与旁瓣能量相抵消,传输过程中逐渐失去环状结构,演化为类偶极子模式,且受方位角调制影响而周期性旋转.在非相邻晶格处两带隙涡旋孤子,由于旁瓣影响较小,带隙涡旋孤子在传输过程中能较好地保持环状结构.

In this paper,the existence and transmission characteristics of gap vortex optical solitons in a honeycomb lattice are investigated based on the fractional nonlinear Schrödinger equation.Firstly,the band-gap structure of honeycomb lattice is obtained by the plane wave expansion method.Then the gap vortex soliton modes and their transmission properties in the fractional nonlinear Schrödinger equation with the honeycomb lattice potential are investigated by the modified squared-operator method,the split-step Fourier method and the Fourier collocation method,respectively.The results show that the transmission of gap vortex solitons is influenced by the Levy index and the propagation constant.The stable transmission region of gap vortex soliton can be obtained through power graphs.In the stable region,the gap vortex soliton can transmit stably without being disturbed.However,in the unstable region,the gap vortex soliton will gradually lose ring structure and evolves into a fundamental soliton with the transmission distance increasing.And the larger the Levy index,the longer the stable transmission distance and the lower the power of the bandgap vortex soliton.When multiple vortex solitons transmit in the lattice,the interaction between them is influenced by the lattice position and phase.Two vortex solitons that are in phase and located at adjacent lattices,are superimposed with sidelobe energy,while two vortex solitonsthat are out of phase are cancelled with sidelobe energy.These vortex solitons will gradually lose ring structure and evolve into dipole modes in the transmission process.And they are periodic rotation under the azimuth angle modulating.When two vortex solitons located at non-adjacent lattice,vortex solitons can maintain a ring-shaped structure due to the small influence of sidelobes.When three gap vortex solitons are located at non-adjacent lattices,the solitons can also maintain their ring-like structures.However,when there are more than three gap vortex solitons,the intensity distribution of vortex solitons are uneven due to the sidelobe energy superimposed.These vortex solitons will form dipole modes and rotate under the azimuthal angle modulating in the transmission process.These results can offer theoretical guidance for transmitting and controlling the gap vortex solitons in the lattice.