具有竞争非线性损耗介质中非局域光孤子的研究

Nonlocal Optical Solitons in Loss Media with Competing Nonlineariti

利用JUNG P S等提出的竞争非局域模型,研究了高斯光束在竞争非局域非线性损耗介质中的传输特性。利用变分法得到孤子参数之间的解析关系式,在此基础上给出无损耗情况下亮孤子临界功率和亮孤子势能函数。利用势能函数分析了无损耗情况下亮孤子宽度和入射功率的关系。当损耗较小时,入射功率在小于、等于和大于临界功率情况下,亮孤子均以准呼吸子形式传输,在传输过程中光束宽度逐渐增大。该变分结论与数值结论相符。最后,利用平方算子迭代法求出无损耗时的孤子解,并把该孤子解作为分步傅里叶算法的初始输入仿真了小损耗和小增益时的光束传输特性。当有小增益时,亮孤子也以准呼吸子形式传输,传输过程中光束宽度逐渐减小。研究结果表明,损耗或增益的存在对光束传输影响的效果很明显,可以利用材料的损耗或增益对光束整形。

To date,there are almost no literature reports on the propagation properties of beam propagation in loss liquid crystals with competing re-orientational and thermal nonlocal nonlinearities.Related research indicates that the presence of losses significantly affect the propagation of beams in media compared to the lossless case.Therefore,it is crucial to study the beam propagation in lossy media.In this paper,we investigate the propagation properties of beams in competing nonlocal nonlinear lossy media using the model proposed by JUNG P S in 2017.The key difference between this model and others is how the refractive index change induced by the beam is expressed.In the traditional models,the refractive index change induced by the light beam is expressed asΔn=Δn1+Δn2,whereΔn1 andΔn2 denote the refractive index change due to two independent nonlinear effects,respectively.In contrast,JUNG P S´s model expresses it asΔn=Δn1Δn2,acknowledging that the two competing nonlinear effects may be closely related and tightly connected rather than independent.Based on this model,we analytically and numerically investigate the beam propagation in loss media with competing nonlinearities.Firstly,the Lagrangian density equation for beam propagation in the competing nonlocal nonlinear lossy medium is derived based on the Schrödinger´s equation.Then the average Lagrangian function is obtained through integration the Lagrangian density equation with respect to x.According to the variational principle,the analytical relationships between soliton parameters are obtained by the variational method.The critical power of soliton and potential V are obtained in the absence of the loss.Furthermore,the potential V was utilized to analyze the propagation properties of bright solitons.Through the variational analysis,it is found that the bright soliton propagates in the form of quasi-breathers when the loss is small.When the losses are small,bright solitons with input powers below,equal to and above the critical power are all propagate in the form of quasi-breathers.The width of the beam increases with the increase of losses.Finally,we focus on the numerical method to discuss the influence of the gain coefficient on beam propagation.We first obtain the stationary solutions by the modified squared-operator iteration method based on the beam propagation equation under the condition of G=0.And then use these solutions as the initial conditions to investigate the beam propagation by a split-step Fourier scheme.When the incident power equals the critical power,the beam width in a lossless or gainless medium does not vary with the propagation distance.However,in the presence of loss,the beam width increases with the increase of loss,whereas the beam width decreases with the increase of gain.In the case of gain,the beam width decreases with the increase of propagation distance,and the larger the gain(absolute value of G),the greater the tendency of beam width compression for the same propagation distance.Additionally,it was found that when the incident power exceeds the critical power,the solitons have good stability in media with either losses or gains.When the incident power is less than the critical power,the beam width initially increases monotonically with propagation distance,reaches a maximum value,and then decreases monotonically with the propagationdistance.Conversely,when the incident power exceeds the critical power,the beam width first decreases,reaches a minimum value,and then oscillates periodically with the propagation distance.Through numerous numerical simulations,it has been demonstrated that the material´s loss and gain result in a decrease or increase in the maximum amplitude of the beam.When G is greater than zero,the peak intensity of the beam monotonically decreases with the increase of propagation distance.Conversely,in the presence of material gain,the peak intensity of the beam monotonically increases with the increase of propagation distance.The study results indicate that the presence of loss or gain significantly impacts on beam propagation.Our research may provide theoretical guidance for long-distance beam transmission and has potential implications for realizing all-optical devices.