非均匀光纤和非均匀波导中光脉冲和光波的传输特性研究
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摘要
光脉冲在光纤中的传输,由于其在光通信中的重要应用,在过去几十年里一直是人们研究的热点之一。而近年来,随着光通信的不断发展,以及色散管理和色散、非线性和增益/损耗的综合管理的提出,光脉冲在非均匀光纤系统中的传输已得到了人们特别的关注。在理论上,光脉冲在非均匀光纤系统中的传输可以由变系数非线性薛定谔方程模型来描述。另一方面,由于空间光孤子是未来实现全光通信及全光器件最理想的途径之一,因而也同样引起了人们的兴趣。
本文主要以变系数非线性薛定谔方程,变系数高阶非线性薛定谔方程,非均匀抛物型阶跃波导中的光波传输方程以及克尔介质中方位极化非傍轴光波的传输方程为模型,通过解析和数值相结合的方法,分别对啁啾暗(灰)孤波在非均匀光纤中的传输,高阶效应下脉冲串的产生、压缩和传输,非均匀波导中自相似光波的非线性隧穿以及克尔介质中的非傍轴环形孤波的传输进行了详细地研究。本文的结果将为进一步研究实际的光孤子控制系统或非均匀光纤系统,实现超高速、大容量的光信息传输提供了一定的理论依据;而且也为全光控制和全光开关的研究提供了一定的理论指导。
本文的主要内容包括以下几个方面:
(1)以描述皮秒光脉冲在非均匀光纤系统中传输的变系数非线性薛定谔方程为模型,运用变量代换,获得了该方程的精确啁啾暗(灰)1-孤子和2-孤子解。依据该精确解,我们运用数值的方法讨论了啁啾暗(灰)孤波在非均匀光纤系统中的传输特性,包括其在偏离可积条件和有限初始扰动影响下的传输稳定性,以及啁啾暗(灰)孤子之间的相互作用。由于在数值模拟中,我们采用了超高斯脉冲作为背景波。因此我们对超高斯型有限宽度背景波、有限宽度背景中的啁啾暗(灰)孤子的传输也进行了详细地研究。结果表明,通过选取适宜的初始啁啾参量,啁啾暗(灰)孤子可以有效地被压缩。并且啁啾暗(灰)孤子可以在非均匀光纤中有效地抑制偏离限制性条件和有限初始扰动的影响而稳定地传输。同时,我们还发现有限宽度背景波也可以在非均匀光纤中不受负载啁啾暗(灰)孤子的影响而稳定地传输。当取背景波脉宽与啁啾暗(灰)孤子的初始脉宽比例较大时,有限宽度背景中啁啾暗(灰)孤子的数值结果基本与其精确解相吻合。即使选取的背景波脉宽不够宽,有限宽度背景中的啁啾暗(灰)脉冲仍可以很好地保持其孤子性质。这些结果将为进一步的实验验证提供了一定的理论依据。
(2)以描述亚皮秒或飞秒光脉冲在非均匀光纤系统中传输的变系数高阶非线性薛定谔方程为模型,在一般的Hirota条件(该条件是强加在变系数高阶非线性薛定谔方程系数上的一个限制关系)下,运用Darboux变换,获得了该方程精确的连续波背景上的孤子解。在一般情况下,这个解可以精确地描述由无限的周期扰动引起连续波的调制不稳定性过程。这个不稳定性的增长导致了连续波背景上序列脉冲串的形成。在实际应用中,为了得到能够稳定传输的脉冲串,我们从精确解中去掉背景波。然后运用这个没有背景波的脉冲串作为数值模拟的初始条件,对其在非均匀光纤系统中的传输进行讨论。结果表明,只要脉冲串的能量足够的大,该脉冲串就可以在非均匀光纤系统中有效地抑制有限的初始扰动和偏离一般Hirota条件的影响而稳定地长距离传输。这一结果将有利于提高其光孤子通信中的信号比特率,增加其信道容量。
(3)以非均匀抛物型阶跃波导中的光波传输方程为模型,运用变量代换,给出该模型的精确亮暗空间自相似解。作为应用,我们讨论了自相似光波在非均匀抛物型阶跃波导中的非线性隧穿效应,以及它们之间的相互作用。结果显示在可积条件下,自相似光波可以自相似的通过非线性势垒(势阱),并且两相邻亮自相似光波在一定范围内呈现出弹性碰撞的相互作用。而在非可积条件下,当非线性势垒高度相对小时,自相似光波通过非线性势垒时可以有效地被压缩。但是当垒高参数足够大时,自相似光波将被分裂成一些细小的光波。这一结果,将有可能为全光控制和全光开关的研究提供一定的理论指导。
(4)从Maxwell方程出发导出描述在克尔介质中方位极化非傍轴光波传输的非线性方程,并以此为模型,运用数值的方法求解出该方程的一系列非傍轴孤波解。随后,以此解为初始条件,利用保守定律和有限差分相结合的方法,数值模拟非傍轴暗孤波和环形孤波的传输。结果显示,我们所采用的方法,虽然可以有效的重现非傍轴暗孤波的稳定传输。但是在模拟环形孤波的传输时,该孤波只可以在短距离内传输,随后便出现了不稳定。
The transmission of optical pulses in fibers has been an attractive topic of research because of their important applications to optical communication in past decades.Recently,with the progress of optical communication,and the appearance of dispersion management and the comprehensive management of dispersion,nonlinearity and gain/loss,the transmission of optical pulses in inhomogenenous fibers has been obtained the particular attentions.The transmission of optical pulses in inhomogeneous fibers can be described by the generalized nonlinear Schr(o|¨)dinger equation with variable coefficients.Otherwise,because it may provide an effective means for all-optical circuits and control,spatial soliton has also attracted more interest.
In the dissertation,based on the generalized nonlinear Schr(o|¨)dinger equation with variable coefficients,the generalized higher-order nonlinear Schr(o|¨)dinger equation with variable coefficients,the nonlinear wave equation governing the transmission of optical waves in the inhomogeneous parabolic-index waveguides,and the nonlinear wave equation describing the transmission of azimuthally polarized nonparaxial optical waves in Kerr media,by analytical and numerical methods,we in detail discuss the transmission of chirped dark(gray) solitons in inhomogeneous fibers, generation,compression and transmission of pulse trains under higher-order effects,nonlinear tunneling of optical similaritons in inhomogeneous waveguides,nonparaxial ring solitary waves in Kerr media.Here,the results obtained and the methods used may be helpful to provide some theoretical analysis for studying the stable transmission of optical pulses in real optical soliton control systems or inhomogeneous fiber systems,and studying all-optical switches and logic.The main contents are as follows:
(1) Based on the nonlinear Schr(o|¨)dinger equation with variable coefficients,governing the transmission of picosecond optical pulses in inhomogeneous fibers,and by using direct transformation of variables and functions,the explicit chirped dark(gray) soliton solutions are presented.By employing the exact solutions,we in detail analyze the propagation characteristics of the chirped dark(gray) soliton,including the stability against either the deviation from integrable condition or the initial perturbation,and the interaction between the chirped dark(gray) solitons. Because we use a super-Gaussian pulse as a background wave in our numerical simulation,it is necessary to analyze the propagation of finite-width super-Gaussian background waves and chirped dark(gray) solitons superimposed upon finite-width background waves.The results show that the dark(gray) solitons can be effectively compressed by choosing the appropriate initial chirp.And the chirped dark(gray) pulses are stable against the deviation from integrable condition,as well as the initial perturbation. The super-Gaussian background waves can stably propagate in inhomogeneous fibers,even though chirped dark(gray) solitons are superimposed upon them.When the ratio of the width of the background wave to the initial width of chirped dark(gray) soliton is large enough,the numerical solutions of chirped dark(gray) solitons can be in agreement with the exact solutions.Even if the width of background waves is not enough broad,chirped dark(gray) pulses can also maintain their soliton characteristics.These results can provide some theoretical analysis for experimental verification.
(2) A generalized higher-order nonlinear Schr(o|¨)dinger equation with variable coefficients,describing the transmission of subpicosecond and femtosecond optical pulses in inhomogeneous fibers,is considered.Imposing generalized Hirota conditions on the variable coefficients,we obtain exact solutions for a soliton sitting on top of a continuous-wave(CW) background by means of the Darboux transform.In the general form,the same solution provides for an exact description of the development of the modulational instability of a CW state,initiated by an infinitesimal periodic perturbation and leading to formation of a periodic array of solitons with a residual CW background.To obtain a more practically relevant solution for a soliton array without the CW component,we subtract it from the exact solution,and use the result as an initial approximation,to generate solutions in direct simulations.As a result,if only the energy of pulse trains is large enough,we can obtain robust pulse trains,which are stable against arbitrary perturbations, as well as against violations of the Hirota conditions.It is useful for raising the signal bit-rate and increasing the capacity in optical communications.
(3) The nonlinear wave equation,governing the transmission of optical waves in the inhomogeneous parabolic-index waveguide,is considered.By using the direct transformation of variables and functions,we present the exact general bright and dark spatial self-similar solutions.As an application, we discuss the nonlinear tunneling of optical similaritons and their interactions.The results show that under integrable condition,the optical waves can similarliy pass through the nonlinear barrier or well.And the interaction between the neighboring waves is elastic collision in certain distance.Under nonintegrable condition,when they pass through the nonlinear barrier,the optical waves can be effectively compressed for the relatively small value of height parameter of nonlinear barrier.However, when the parameter is large enough,the wave splits into some filaments. These results may be helpful to provide some theoretical analysis for studying all-optical switches and logic.
(4) Finally,the nonlinear wave equation,which is obtained directly from the Maxwell equations,describing the transmission of azimuthally polarized nonparaxial optical waves in Kerr media,is considered.By numerical method, we present a set of nonparaxial solitary wave solutions.Then using these results as initial approximations,by the conservation law and finite-difference methods,we discuss the propagation of the nonparaxial dark and ring solitary waves.The results show that these methods can effectively simulate the stable propagation of nonparaxial dark solitary waves.But by these methods,the nonparaxial ring solitary waves only stably propagate in finite distance,then they become unstable.
本文主要以变系数非线性薛定谔方程,变系数高阶非线性薛定谔方程,非均匀抛物型阶跃波导中的光波传输方程以及克尔介质中方位极化非傍轴光波的传输方程为模型,通过解析和数值相结合的方法,分别对啁啾暗(灰)孤波在非均匀光纤中的传输,高阶效应下脉冲串的产生、压缩和传输,非均匀波导中自相似光波的非线性隧穿以及克尔介质中的非傍轴环形孤波的传输进行了详细地研究。本文的结果将为进一步研究实际的光孤子控制系统或非均匀光纤系统,实现超高速、大容量的光信息传输提供了一定的理论依据;而且也为全光控制和全光开关的研究提供了一定的理论指导。
本文的主要内容包括以下几个方面:
(1)以描述皮秒光脉冲在非均匀光纤系统中传输的变系数非线性薛定谔方程为模型,运用变量代换,获得了该方程的精确啁啾暗(灰)1-孤子和2-孤子解。依据该精确解,我们运用数值的方法讨论了啁啾暗(灰)孤波在非均匀光纤系统中的传输特性,包括其在偏离可积条件和有限初始扰动影响下的传输稳定性,以及啁啾暗(灰)孤子之间的相互作用。由于在数值模拟中,我们采用了超高斯脉冲作为背景波。因此我们对超高斯型有限宽度背景波、有限宽度背景中的啁啾暗(灰)孤子的传输也进行了详细地研究。结果表明,通过选取适宜的初始啁啾参量,啁啾暗(灰)孤子可以有效地被压缩。并且啁啾暗(灰)孤子可以在非均匀光纤中有效地抑制偏离限制性条件和有限初始扰动的影响而稳定地传输。同时,我们还发现有限宽度背景波也可以在非均匀光纤中不受负载啁啾暗(灰)孤子的影响而稳定地传输。当取背景波脉宽与啁啾暗(灰)孤子的初始脉宽比例较大时,有限宽度背景中啁啾暗(灰)孤子的数值结果基本与其精确解相吻合。即使选取的背景波脉宽不够宽,有限宽度背景中的啁啾暗(灰)脉冲仍可以很好地保持其孤子性质。这些结果将为进一步的实验验证提供了一定的理论依据。
(2)以描述亚皮秒或飞秒光脉冲在非均匀光纤系统中传输的变系数高阶非线性薛定谔方程为模型,在一般的Hirota条件(该条件是强加在变系数高阶非线性薛定谔方程系数上的一个限制关系)下,运用Darboux变换,获得了该方程精确的连续波背景上的孤子解。在一般情况下,这个解可以精确地描述由无限的周期扰动引起连续波的调制不稳定性过程。这个不稳定性的增长导致了连续波背景上序列脉冲串的形成。在实际应用中,为了得到能够稳定传输的脉冲串,我们从精确解中去掉背景波。然后运用这个没有背景波的脉冲串作为数值模拟的初始条件,对其在非均匀光纤系统中的传输进行讨论。结果表明,只要脉冲串的能量足够的大,该脉冲串就可以在非均匀光纤系统中有效地抑制有限的初始扰动和偏离一般Hirota条件的影响而稳定地长距离传输。这一结果将有利于提高其光孤子通信中的信号比特率,增加其信道容量。
(3)以非均匀抛物型阶跃波导中的光波传输方程为模型,运用变量代换,给出该模型的精确亮暗空间自相似解。作为应用,我们讨论了自相似光波在非均匀抛物型阶跃波导中的非线性隧穿效应,以及它们之间的相互作用。结果显示在可积条件下,自相似光波可以自相似的通过非线性势垒(势阱),并且两相邻亮自相似光波在一定范围内呈现出弹性碰撞的相互作用。而在非可积条件下,当非线性势垒高度相对小时,自相似光波通过非线性势垒时可以有效地被压缩。但是当垒高参数足够大时,自相似光波将被分裂成一些细小的光波。这一结果,将有可能为全光控制和全光开关的研究提供一定的理论指导。
(4)从Maxwell方程出发导出描述在克尔介质中方位极化非傍轴光波传输的非线性方程,并以此为模型,运用数值的方法求解出该方程的一系列非傍轴孤波解。随后,以此解为初始条件,利用保守定律和有限差分相结合的方法,数值模拟非傍轴暗孤波和环形孤波的传输。结果显示,我们所采用的方法,虽然可以有效的重现非傍轴暗孤波的稳定传输。但是在模拟环形孤波的传输时,该孤波只可以在短距离内传输,随后便出现了不稳定。
The transmission of optical pulses in fibers has been an attractive topic of research because of their important applications to optical communication in past decades.Recently,with the progress of optical communication,and the appearance of dispersion management and the comprehensive management of dispersion,nonlinearity and gain/loss,the transmission of optical pulses in inhomogenenous fibers has been obtained the particular attentions.The transmission of optical pulses in inhomogeneous fibers can be described by the generalized nonlinear Schr(o|¨)dinger equation with variable coefficients.Otherwise,because it may provide an effective means for all-optical circuits and control,spatial soliton has also attracted more interest.
In the dissertation,based on the generalized nonlinear Schr(o|¨)dinger equation with variable coefficients,the generalized higher-order nonlinear Schr(o|¨)dinger equation with variable coefficients,the nonlinear wave equation governing the transmission of optical waves in the inhomogeneous parabolic-index waveguides,and the nonlinear wave equation describing the transmission of azimuthally polarized nonparaxial optical waves in Kerr media,by analytical and numerical methods,we in detail discuss the transmission of chirped dark(gray) solitons in inhomogeneous fibers, generation,compression and transmission of pulse trains under higher-order effects,nonlinear tunneling of optical similaritons in inhomogeneous waveguides,nonparaxial ring solitary waves in Kerr media.Here,the results obtained and the methods used may be helpful to provide some theoretical analysis for studying the stable transmission of optical pulses in real optical soliton control systems or inhomogeneous fiber systems,and studying all-optical switches and logic.The main contents are as follows:
(1) Based on the nonlinear Schr(o|¨)dinger equation with variable coefficients,governing the transmission of picosecond optical pulses in inhomogeneous fibers,and by using direct transformation of variables and functions,the explicit chirped dark(gray) soliton solutions are presented.By employing the exact solutions,we in detail analyze the propagation characteristics of the chirped dark(gray) soliton,including the stability against either the deviation from integrable condition or the initial perturbation,and the interaction between the chirped dark(gray) solitons. Because we use a super-Gaussian pulse as a background wave in our numerical simulation,it is necessary to analyze the propagation of finite-width super-Gaussian background waves and chirped dark(gray) solitons superimposed upon finite-width background waves.The results show that the dark(gray) solitons can be effectively compressed by choosing the appropriate initial chirp.And the chirped dark(gray) pulses are stable against the deviation from integrable condition,as well as the initial perturbation. The super-Gaussian background waves can stably propagate in inhomogeneous fibers,even though chirped dark(gray) solitons are superimposed upon them.When the ratio of the width of the background wave to the initial width of chirped dark(gray) soliton is large enough,the numerical solutions of chirped dark(gray) solitons can be in agreement with the exact solutions.Even if the width of background waves is not enough broad,chirped dark(gray) pulses can also maintain their soliton characteristics.These results can provide some theoretical analysis for experimental verification.
(2) A generalized higher-order nonlinear Schr(o|¨)dinger equation with variable coefficients,describing the transmission of subpicosecond and femtosecond optical pulses in inhomogeneous fibers,is considered.Imposing generalized Hirota conditions on the variable coefficients,we obtain exact solutions for a soliton sitting on top of a continuous-wave(CW) background by means of the Darboux transform.In the general form,the same solution provides for an exact description of the development of the modulational instability of a CW state,initiated by an infinitesimal periodic perturbation and leading to formation of a periodic array of solitons with a residual CW background.To obtain a more practically relevant solution for a soliton array without the CW component,we subtract it from the exact solution,and use the result as an initial approximation,to generate solutions in direct simulations.As a result,if only the energy of pulse trains is large enough,we can obtain robust pulse trains,which are stable against arbitrary perturbations, as well as against violations of the Hirota conditions.It is useful for raising the signal bit-rate and increasing the capacity in optical communications.
(3) The nonlinear wave equation,governing the transmission of optical waves in the inhomogeneous parabolic-index waveguide,is considered.By using the direct transformation of variables and functions,we present the exact general bright and dark spatial self-similar solutions.As an application, we discuss the nonlinear tunneling of optical similaritons and their interactions.The results show that under integrable condition,the optical waves can similarliy pass through the nonlinear barrier or well.And the interaction between the neighboring waves is elastic collision in certain distance.Under nonintegrable condition,when they pass through the nonlinear barrier,the optical waves can be effectively compressed for the relatively small value of height parameter of nonlinear barrier.However, when the parameter is large enough,the wave splits into some filaments. These results may be helpful to provide some theoretical analysis for studying all-optical switches and logic.
(4) Finally,the nonlinear wave equation,which is obtained directly from the Maxwell equations,describing the transmission of azimuthally polarized nonparaxial optical waves in Kerr media,is considered.By numerical method, we present a set of nonparaxial solitary wave solutions.Then using these results as initial approximations,by the conservation law and finite-difference methods,we discuss the propagation of the nonparaxial dark and ring solitary waves.The results show that these methods can effectively simulate the stable propagation of nonparaxial dark solitary waves.But by these methods,the nonparaxial ring solitary waves only stably propagate in finite distance,then they become unstable.
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