利用符号计算解析研究光脉冲相互作用的若干问题
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摘要
在光通信系统中,信息以一系列光脉冲的形式进行传输。为了提高系统通信容量,总是尽可能减小相邻光脉冲的间距。当孤子技术应用于光通信系统中时,就涉及到光孤子相互作用问题。光孤子相互作用会使得光孤子波形畸变、传输恶化,从而导致光通信系统的误码率上升,传输距离变短,并最终影响到系统的通信质量和容量。因此,随着传输速率的提高,为了在相同码率下获得尽可能大的传输距离,需要对光孤子相互作用进行研究。
通常利用非线性Schrodinger(NLS)类模型来描述光孤子在光纤中的传输。在光通信系统中,一般采用数值方法研究该类模型,计算处理的数据和输出的结果都是数值的。如果能找到该类模型的解析解,将有助于了解模型所描述的非线性现象的本质特征。为了得到该类模型的解析解,通常需要忽略一些物理效应和降低计算精度等,同时计算过程也繁琐。随着科学技术的发展,符号计算被用于该类模型的解析求解。利用符号计算,通过设计问题的代数算法,可以借助计算机进行复杂的数学演算和推理,从而解析地处理数据和运算,并得到该类模型没有误差的解析解。
在本论文中,作者结合光孤子研究领域的发展现状,借助于符号计算和双线性方法,解析研究光通信系统中NLS类模型的孤子解和光孤子相互作用的若干问题。解决的主要问题包括：如何尽量减小光孤子间距而又使光孤子相互作用减弱?光孤子相互作用时,能否在保持原有光孤子间距的条件下将光孤子压缩,从而减小光孤子相互作用?是否可以实现光孤子的自发放大而不需要周期性地添加集中放大器?是否可以利用传输介质的相关特性对光孤子进行控制?主要研究内容包括四个方面：光纤正常色散区中明、暗孤子的产生及其相互作用；孤子效应脉冲压缩及其相互作用；光孤子放大及其相互作用；孤子控制及其相互作用。要点如下：
(1)光纤正常色散区明、暗孤子的产生及其相互作用研究：借助于符号计算,通过对NLS模型、高阶NLS模型、变系数NLS模型和带损耗的变系数NLS模型这四个非线性模型的解析研究,得到它们的解析单、双孤子解。发现在光纤的正常色散区能够产生明、暗孤子,并得到在光纤的正常色散区产生明、暗孤子的条件。利用渐近分析法分析光孤子相互作用情形,发现在高阶NLS模型中,明、暗孤子间的相互作用是非弹性的；而在另外三个NLS模型中,光孤子相互作用是弹性的。在相互作用前后它们的能量是守恒的。通过改变高阶NLS模型的双孤子解的物理参数可以改变光孤子的传输模式。当传输媒质为非均匀光纤时,可以选择色散递减光纤(DDF)不同的色散曲线函数进行孤子控制。并且这种孤子控制方法对于光孤子相互作用没有影响。另外,对带损耗的变系数NLS模型的孤子解进行了稳定性分析,验证了孤子解的稳定性。
(2)孤子效应脉冲压缩和光孤子相互作用研究：借助于符号计算,通过求解用于描述孤子效应脉冲压缩的高阶NLS模型,得到该模型的解析单、双和三孤子解,从而解析地研究孤子效应脉冲压缩技术。当两个光孤子间距较小时,会发生相互作用,产生孤子效应脉冲压缩现象,并且光孤子在传输过程中呈现周期性的变化。当光孤子间距较大时,可以实现光孤子间无相互作用的传输。同时,利用三孤子解验证所得结论的正确性。发现由于受激Raman散射(SRS)、自陡峭效应(SS)和三阶色散(TOD)的共同作用,三阶孤子效应脉冲压缩的压缩因子更高。但是由于TOD效应的存在,三阶孤子带有基座,会导致光孤子压缩质量下降,光孤子带宽也会被展宽。
(3)光纤反常色散区光孤子放大及其相互作用研究：借助于符号计算,解析研究高阶NLS模型、变系数高阶NLS模型以及带损耗的变系数NLS模型,得到它们的解析单、双孤子解。研究发现利用DDF和色散位移光纤(DSF)能够实现光孤子放大。对于变系数高阶NLS模型,主要借助于具有不同色散曲线函数的DDF实现光孤子放大。DDF的色散曲线函数不同,光孤子放大增益也不相同。对于高阶NLS模型,主要利用DSF研究光孤子放大。光孤子相互作用后,会发生能量交换,从而实现光孤子放大功能。同时,借助带损耗的变系数NLS模型,还研究一种具有新型色散曲线函数的DDF,发现光孤子在较短距离能被放大,并且放大的速度还能够被控制。
(4)孤子控制及其相互作用研究：借助于符号计算,解析研究变系数NLS模型和变系数高阶NLS模型,得到它们的解析单、双和三孤子解。研究发现选择DDF不同的色散曲线函数,光孤子振幅或速度会发生相应的改变,从而达到孤子控制的目的。并且该孤子控制方法不会对光孤子相互作用产生影响。依据DDF的色散曲线函数,当选择不同的群速度色散(GVD)参数时,光孤子振幅会以相应的方式发生改变。在研究暗孤子时,发现当TOD系数函数与DDF的色散曲线函数成比例时,也能够实现孤子控制,即使存在高阶效应对孤子控制也没有影响。在利用Gauss型DDF进行光孤子传输时,在光孤子间距很小的情况下,光孤子也不会发生相互作用,这有利于增大系统的通信容量。另外,利用渐近分析法揭示在系统中光孤子相互作用是弹性的,在相互作用过程中能量是守恒的。
In optical communication systems, the information transmitted are usually some optical pulse trains. Minimizing the distance between two adjacent pulses has been used to improve the capacity of the communication systems. When the soliton technique is applied to the optical communication systems, the problems of the interactions between two solitons have arisen, which cause soliton distortion and transmission deterioration in the optical communication systems, accordingly bring rising bit error rate and shorter transmission distance, and finally negatively affect the quality and capacity of the optical communication systems. With increasing data transfer rate, the interactions between two solitons should be studied in order to obtain the greatest possible transmission distance under the same bit rate.
The common mathematical model describing the optical soliton transmission in the optical fibers is the nonlinear Schrodinger (NLS)-type model. In optical communication systems, numerical methods have been used to study the model, and the input data and output results all are numerical values. Obtaining the analytic solutions for the model is helpful for the understanding of the nature of the nonlinear phenomena described by the model. Typically, to obtain the analytic solutions for the model, ignoring some physical effects and reducing the computational accuracy would be required, so would be the complicated calculation process. With the development of science and technology, symbolic computation has been used to obtain the analytic solutions for the model. With the symbolic computation and the algebraic algorithm, the complex mathematical calculations and deductions can be conducted with computer to ensure the acquisition of the analytic solutions with absolute accuracy for the model.
Integrating the current development of the optical soliton field, this dissertation an-alytically study the soliton solutions and interactions for the NLS-type models by means of symbolic computation and the bilinear methods. Main problems to be solved include: How the distance between optical solitons be minimized with the optical soliton inter-action diminished? Can optical solitons be compressed while maintaining its original space, thereby reduces the optical soliton interaction? Can spontaneous soliton amplifi-cation be achieved without requiring the periodical focus amplifier? Can optical solitons be controlled with the properties of optical transmission medium? What are under in-vestigation include the following four aspects:the generation of the bright and dark solitons and their interactions in the normal group velocity dispersion (GVD) regime of optical fibers; soli ton-effect pulse compression and their interactions; optical soliton amplification and their interactions; soliton control and their interactions. Specifically, the work is outlined as follows:
(1) Generation of the bright and dark solitons and their interactions in the normal GVD regime of optical fibers:By analytic studies on the NLS model, higher-order NLS (HNLS) model, variable-coefficient NLS (vcNLS) model and vcNLS model with the losses, their one- and two- soliton solutions are obtained with symbolic computation. It concludes that the bright and dark solitons can be obtained in the normal GVD regime of optical fibers and the corresponding conditions for their generation are also given. Making the asymptotic analysis of the interactions between two solitons qualitatively, results indicate that the interactions between bright and dark solitons are inelastic in the HNLS model, while the interactions in the other three types of NLS models are elastic, with the energy conserved before and after the interactions. Changes in the physical parameters of the two-soliton solutions in the HNLS model can alter the optical soliton transmission mode. In the nonuniform fiber transmission medium, the dispersion decreasing fibers with different dispersion profiles can be selected to conduct soliton control, which exerts no effect on the optical soliton interactions. In addition, stability analysis of the soliton solutions for the vcNLS model is conducted to verify their stability.
(2) Soliton-effect pulse compression and soliton interaction:The HNLS model is used to describe the soliton-effect pulse compression. The one-, two- and three-soliton solutions for the model are derived with symbolic computation, and the soliton-effect pulse compression mechanism is studied analytically. When the spacing between two optical solitons is comparatively small, the interaction will occur, which results in the phenomena of the soliton-effect pulse compression with optical solitons exhibiting pe-riodic changes. When the spacing between two optical solitons is comparatively large, optical solitons propagate without interactions. The above conclusions are verified by the three-soliton solution. Third-order soliton-effect pulse compression is confirmed to manifest higher compression factor being influenced by the combined effects of third- order dispersion (TOD), self-steepening (SS) and stimulated Raman scattering (SRS). However, due to the TOD effect, the third-order optical soliton is attached with an os-cillating structure, which leads to the lower-quality compressed optical solitons with the optical soliton width broadening.
(3) Optical soliton amplification and their interactions in the anomalous GVD regime of optical fibers:Analytic studies on the three models(the vcHNLS model, HNLS model and vcNLS model with losses) are performed to acquire the one- and two- soliton solutions with symbolic computation. The dispersion-shifted fiber (DSF) and dispersion-decreasing fiber (DDF) are verified to realize the optical soliton amplification. For the vcHNLS model, the DDF with different dispersion profiles can be used to achieve the optical soliton amplification, with different dispersion profiles corresponding to different optical soliton amplification gain. While the HNLS model is mainly utilized in the DSF to investigate the optical soliton amplification, which can be realized via the energy exchange after the optical soliton interaction. In the vcNLS model with losses, the in-vestigation on the DDF with a new kind of dispersion profile suggests that the optical soliton can be amplified within a comparatively short distance with the amplification speed being controlled.
(4) Soliton control and interaction:The vcNLS model and vcHNLS model are ana-lytically studied and their one-, two- and three-soliton analytical solutions are obtained via symbolic computation. With the DDF of different dispersion profiles conducting the change of the optical soliton amplitude or speed, soliton control can be realized and have no impact on optical soliton interaction. According to the dispersion profiles of the DDF, the optical soliton amplitude varies with the GVD parameter. Research on the dark soliton indicates that dark solitons can be controlled when the TOD coefficient function is proportional to the dispersion profiles of the DDF, even if there is higher-order effect. When the optical solitons propagate in the Gaussian DDF, provided that the optical soliton spacing is comparatively small, the optical solitons will not interact, and the capacity of the optical communication system will be improved. In addition, it is found via asymptotic analysis that the optical soliton interactions are elastic in the systems with the energy conserved.
通常利用非线性Schrodinger(NLS)类模型来描述光孤子在光纤中的传输。在光通信系统中,一般采用数值方法研究该类模型,计算处理的数据和输出的结果都是数值的。如果能找到该类模型的解析解,将有助于了解模型所描述的非线性现象的本质特征。为了得到该类模型的解析解,通常需要忽略一些物理效应和降低计算精度等,同时计算过程也繁琐。随着科学技术的发展,符号计算被用于该类模型的解析求解。利用符号计算,通过设计问题的代数算法,可以借助计算机进行复杂的数学演算和推理,从而解析地处理数据和运算,并得到该类模型没有误差的解析解。
在本论文中,作者结合光孤子研究领域的发展现状,借助于符号计算和双线性方法,解析研究光通信系统中NLS类模型的孤子解和光孤子相互作用的若干问题。解决的主要问题包括：如何尽量减小光孤子间距而又使光孤子相互作用减弱?光孤子相互作用时,能否在保持原有光孤子间距的条件下将光孤子压缩,从而减小光孤子相互作用?是否可以实现光孤子的自发放大而不需要周期性地添加集中放大器?是否可以利用传输介质的相关特性对光孤子进行控制?主要研究内容包括四个方面：光纤正常色散区中明、暗孤子的产生及其相互作用；孤子效应脉冲压缩及其相互作用；光孤子放大及其相互作用；孤子控制及其相互作用。要点如下：
(1)光纤正常色散区明、暗孤子的产生及其相互作用研究：借助于符号计算,通过对NLS模型、高阶NLS模型、变系数NLS模型和带损耗的变系数NLS模型这四个非线性模型的解析研究,得到它们的解析单、双孤子解。发现在光纤的正常色散区能够产生明、暗孤子,并得到在光纤的正常色散区产生明、暗孤子的条件。利用渐近分析法分析光孤子相互作用情形,发现在高阶NLS模型中,明、暗孤子间的相互作用是非弹性的；而在另外三个NLS模型中,光孤子相互作用是弹性的。在相互作用前后它们的能量是守恒的。通过改变高阶NLS模型的双孤子解的物理参数可以改变光孤子的传输模式。当传输媒质为非均匀光纤时,可以选择色散递减光纤(DDF)不同的色散曲线函数进行孤子控制。并且这种孤子控制方法对于光孤子相互作用没有影响。另外,对带损耗的变系数NLS模型的孤子解进行了稳定性分析,验证了孤子解的稳定性。
(2)孤子效应脉冲压缩和光孤子相互作用研究：借助于符号计算,通过求解用于描述孤子效应脉冲压缩的高阶NLS模型,得到该模型的解析单、双和三孤子解,从而解析地研究孤子效应脉冲压缩技术。当两个光孤子间距较小时,会发生相互作用,产生孤子效应脉冲压缩现象,并且光孤子在传输过程中呈现周期性的变化。当光孤子间距较大时,可以实现光孤子间无相互作用的传输。同时,利用三孤子解验证所得结论的正确性。发现由于受激Raman散射(SRS)、自陡峭效应(SS)和三阶色散(TOD)的共同作用,三阶孤子效应脉冲压缩的压缩因子更高。但是由于TOD效应的存在,三阶孤子带有基座,会导致光孤子压缩质量下降,光孤子带宽也会被展宽。
(3)光纤反常色散区光孤子放大及其相互作用研究：借助于符号计算,解析研究高阶NLS模型、变系数高阶NLS模型以及带损耗的变系数NLS模型,得到它们的解析单、双孤子解。研究发现利用DDF和色散位移光纤(DSF)能够实现光孤子放大。对于变系数高阶NLS模型,主要借助于具有不同色散曲线函数的DDF实现光孤子放大。DDF的色散曲线函数不同,光孤子放大增益也不相同。对于高阶NLS模型,主要利用DSF研究光孤子放大。光孤子相互作用后,会发生能量交换,从而实现光孤子放大功能。同时,借助带损耗的变系数NLS模型,还研究一种具有新型色散曲线函数的DDF,发现光孤子在较短距离能被放大,并且放大的速度还能够被控制。
(4)孤子控制及其相互作用研究：借助于符号计算,解析研究变系数NLS模型和变系数高阶NLS模型,得到它们的解析单、双和三孤子解。研究发现选择DDF不同的色散曲线函数,光孤子振幅或速度会发生相应的改变,从而达到孤子控制的目的。并且该孤子控制方法不会对光孤子相互作用产生影响。依据DDF的色散曲线函数,当选择不同的群速度色散(GVD)参数时,光孤子振幅会以相应的方式发生改变。在研究暗孤子时,发现当TOD系数函数与DDF的色散曲线函数成比例时,也能够实现孤子控制,即使存在高阶效应对孤子控制也没有影响。在利用Gauss型DDF进行光孤子传输时,在光孤子间距很小的情况下,光孤子也不会发生相互作用,这有利于增大系统的通信容量。另外,利用渐近分析法揭示在系统中光孤子相互作用是弹性的,在相互作用过程中能量是守恒的。
In optical communication systems, the information transmitted are usually some optical pulse trains. Minimizing the distance between two adjacent pulses has been used to improve the capacity of the communication systems. When the soliton technique is applied to the optical communication systems, the problems of the interactions between two solitons have arisen, which cause soliton distortion and transmission deterioration in the optical communication systems, accordingly bring rising bit error rate and shorter transmission distance, and finally negatively affect the quality and capacity of the optical communication systems. With increasing data transfer rate, the interactions between two solitons should be studied in order to obtain the greatest possible transmission distance under the same bit rate.
The common mathematical model describing the optical soliton transmission in the optical fibers is the nonlinear Schrodinger (NLS)-type model. In optical communication systems, numerical methods have been used to study the model, and the input data and output results all are numerical values. Obtaining the analytic solutions for the model is helpful for the understanding of the nature of the nonlinear phenomena described by the model. Typically, to obtain the analytic solutions for the model, ignoring some physical effects and reducing the computational accuracy would be required, so would be the complicated calculation process. With the development of science and technology, symbolic computation has been used to obtain the analytic solutions for the model. With the symbolic computation and the algebraic algorithm, the complex mathematical calculations and deductions can be conducted with computer to ensure the acquisition of the analytic solutions with absolute accuracy for the model.
Integrating the current development of the optical soliton field, this dissertation an-alytically study the soliton solutions and interactions for the NLS-type models by means of symbolic computation and the bilinear methods. Main problems to be solved include: How the distance between optical solitons be minimized with the optical soliton inter-action diminished? Can optical solitons be compressed while maintaining its original space, thereby reduces the optical soliton interaction? Can spontaneous soliton amplifi-cation be achieved without requiring the periodical focus amplifier? Can optical solitons be controlled with the properties of optical transmission medium? What are under in-vestigation include the following four aspects:the generation of the bright and dark solitons and their interactions in the normal group velocity dispersion (GVD) regime of optical fibers; soli ton-effect pulse compression and their interactions; optical soliton amplification and their interactions; soliton control and their interactions. Specifically, the work is outlined as follows:
(1) Generation of the bright and dark solitons and their interactions in the normal GVD regime of optical fibers:By analytic studies on the NLS model, higher-order NLS (HNLS) model, variable-coefficient NLS (vcNLS) model and vcNLS model with the losses, their one- and two- soliton solutions are obtained with symbolic computation. It concludes that the bright and dark solitons can be obtained in the normal GVD regime of optical fibers and the corresponding conditions for their generation are also given. Making the asymptotic analysis of the interactions between two solitons qualitatively, results indicate that the interactions between bright and dark solitons are inelastic in the HNLS model, while the interactions in the other three types of NLS models are elastic, with the energy conserved before and after the interactions. Changes in the physical parameters of the two-soliton solutions in the HNLS model can alter the optical soliton transmission mode. In the nonuniform fiber transmission medium, the dispersion decreasing fibers with different dispersion profiles can be selected to conduct soliton control, which exerts no effect on the optical soliton interactions. In addition, stability analysis of the soliton solutions for the vcNLS model is conducted to verify their stability.
(2) Soliton-effect pulse compression and soliton interaction:The HNLS model is used to describe the soliton-effect pulse compression. The one-, two- and three-soliton solutions for the model are derived with symbolic computation, and the soliton-effect pulse compression mechanism is studied analytically. When the spacing between two optical solitons is comparatively small, the interaction will occur, which results in the phenomena of the soliton-effect pulse compression with optical solitons exhibiting pe-riodic changes. When the spacing between two optical solitons is comparatively large, optical solitons propagate without interactions. The above conclusions are verified by the three-soliton solution. Third-order soliton-effect pulse compression is confirmed to manifest higher compression factor being influenced by the combined effects of third- order dispersion (TOD), self-steepening (SS) and stimulated Raman scattering (SRS). However, due to the TOD effect, the third-order optical soliton is attached with an os-cillating structure, which leads to the lower-quality compressed optical solitons with the optical soliton width broadening.
(3) Optical soliton amplification and their interactions in the anomalous GVD regime of optical fibers:Analytic studies on the three models(the vcHNLS model, HNLS model and vcNLS model with losses) are performed to acquire the one- and two- soliton solutions with symbolic computation. The dispersion-shifted fiber (DSF) and dispersion-decreasing fiber (DDF) are verified to realize the optical soliton amplification. For the vcHNLS model, the DDF with different dispersion profiles can be used to achieve the optical soliton amplification, with different dispersion profiles corresponding to different optical soliton amplification gain. While the HNLS model is mainly utilized in the DSF to investigate the optical soliton amplification, which can be realized via the energy exchange after the optical soliton interaction. In the vcNLS model with losses, the in-vestigation on the DDF with a new kind of dispersion profile suggests that the optical soliton can be amplified within a comparatively short distance with the amplification speed being controlled.
(4) Soliton control and interaction:The vcNLS model and vcHNLS model are ana-lytically studied and their one-, two- and three-soliton analytical solutions are obtained via symbolic computation. With the DDF of different dispersion profiles conducting the change of the optical soliton amplitude or speed, soliton control can be realized and have no impact on optical soliton interaction. According to the dispersion profiles of the DDF, the optical soliton amplitude varies with the GVD parameter. Research on the dark soliton indicates that dark solitons can be controlled when the TOD coefficient function is proportional to the dispersion profiles of the DDF, even if there is higher-order effect. When the optical solitons propagate in the Gaussian DDF, provided that the optical soliton spacing is comparatively small, the optical solitons will not interact, and the capacity of the optical communication system will be improved. In addition, it is found via asymptotic analysis that the optical soliton interactions are elastic in the systems with the energy conserved.
引文
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