【作者】 王丽丽；

【导师】 刘文军；

【作者基本信息】 北京邮电大学 ， 数学， 2021， 硕士

【摘要】 光纤通信以其传输容量大、传输距离长、保密性好等优点已经成为当今通信领域中一种重要的通讯方式[1]。凭借其在传输过程中保持波形、速度、幅度等不变的特性[2-4],光孤子成为了光纤通信中最具前景的介质。目前,光孤子研究的主要实验平台是锁模光纤激光器,因此对于光纤激光器的一些理论研究也就非常重要。在理论方面,光孤子在光纤中的传输可以用非线性薛定谔方程来建模[5],孤子解是在研究非线性模型中的一个重要方面。本文对光纤激光器中的广义非线性薛定谔方程展开理论研究,借助Hirota方法求得方程的孤子解并对其进行理论及应用分析,具体的研究内容如下:(1)传统非线性薛定谔方程的解析研究:选择三阶变系数非线性薛定谔方程作为研究模型,通过Hirota方法求得双孤子和三孤子解并对孤子的传输特性进行理论分析。研究表明,调整三阶色散的取值可以改变孤子的幅度,此发现可以被应用光放大器中;调整三阶色散的函数类型可以改变孤子包络的形状,这一性质可以被应用在光开关的设计中;调整群速度色散的函数类型及相关系数,可以控制孤子之间产生相互作用的位置及程度,这一性质为改善光孤子传输性能提供理论指导,最终提高通信质量。(2)耦合方程的解析研究:选择(2+1)维耦合非线性薛定谔方程作为研究模型,借助Hirota方法求得单双明孤子解并探究了四波混频效应和自由参数对于孤子传输的影响。研究发现,四波混频效应会影响孤子的幅度,调整自由参数的取值可以实现对孤子传输方向、速度和孤子之间相互作用的控制;讨论了孤子在碰撞后再分开时的相互作用过程。得到的结果可以应用于多模光纤,实现孤子的放大、传输方向的控制以及通信质量的提高。(3)金兹堡-朗道方程的解析研究:选取描述耗散系统的金兹堡-朗道方程作为研究模型。在求解时选择修正的双线性方法,此种方法对方程进行线性化时所使用的变换形式不同。在解的基础上讨论了方程中的参数对孤子传输的影响。分析可知,方程中的自由参数会影响孤子的传输方向以及幅度大小。另外,色散项的取值可以决定孤子的包络形状以及放大程度,可以应用于孤子放大和孤子整形。得到的结果可能会对光孤子在多模光纤中的放大和方向控制等有所帮助。更多还原

【Abstract】 Due to its advantages of large transmission capacity,long transmission distance,and good confidentiality,optical fiber communication has become an important method in today’s communication field[1].Keeping the waveform,velocity and amplitude unchanged during transmission,optical soliton has become the most promising medium in optical fiber communication[2-4].The main experimental platform of optical soliton research is mode-locked fiber laser,so it is necessary to study the theory of fiber laser.Theoretically,optical soliton propagation in optical fiber can be modeled by nonlinear Schrodinger equation[5].Soliton solution is an important aspect in the study of nonlinear model.In this paper,the generalized nonlinear Schrodinger equation in fiber laser is studied theoretically.The soliton solution of the equation is obtained by Hirota method,and its theory and application are both analyzed.The specific research contents are as follows:(1)The analytical study of traditional nonlinear Schrodinger equation:The third-order nonlinear Schrodinger equation is selected as the research model.The two-soliton and three-soliton solutions are obtained by virtue of Hirota method,and the transmission characteristics of solitons are analyzed based on the solutions obtained.The results show that the amplitude of soliton can be controlled by adjusting the value of third-order dispersion,which can be used in optical amplifiers.The shape of soliton envelope can also be changed by adjusting the function type of third-order dispersion,which may be helpful in the design of optical switch.In addition,by adjusting the group velocity dispersion and correlation coefficient,we can control the position and degree of interaction between solitons,which can provide theoretical guidance for improving the transmission performance of optical solitons,and ultimately improves the communication quality.(2)The analytical study of coupled equations:the(2+1)-dimensional coupled nonlinear Schrodinger equation is selected as the research model.Its one-and two-bright soliton solutions are obtained by Hirota method.The effects of four wave mixing parameters and free parameters on soliton transmission are explored.It is found that four wave mixing parameters can affect the amplitude of solitons,and one can control the direction and velocity of solitons by adjusting the value of free parameters.The energy transfer phenomenon when solitons are separated after collision is also discussed.The results given here may be applied to the multimode fiber and realize the control of soliton transmission direction and the improvement of communication quality.(3)The analytical study of Ginzburg-Landau equation:The Ginzburg-Landau equation is selected as the research model.Because the composition of this kind of equation is different from that of the traditional nonlinear Schrodinger equation,the modified bilinear method is chosen to solve the equation,and the transformation form is different.Based on the solution obtained,the influence of the parameters in the equation on soliton propagation is discussed.The analysis shows that the free parameter in the equation will affect the propagation direction and amplitude of the soliton.In addition,the value of dispersion can determine the envelope shape and amplification degree of soliton,which can be applied to soliton amplification and reshaping.The results may be helpful for the amplification and direction control of optical solitons in multimode fibers.更多还原