空间光孤子
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摘要
当一束很窄的光束在各向同性线性介质中传播时，由于自然衍射效应，其在横向方向会展宽，激光束越窄，展宽的速率就越快，这对于光作为信息的载体是不利的。在非线性介质中，当介质的非线性引起的光束自聚焦效应和衍射效应相平衡时，激光束的横向尺寸(微米量级)不再随传播距离而变化，非线性介质中这种自限的光束便是空间光孤子，它们可以存在于很多对光的非线性响应机制不同的介质中，并且在运动碰撞过程中可以分裂、融合、湮灭及旋转等。空间光孤子这种具有固定空间形状的波包作为最基本信息单元在未来信息传输及处理领域有着巨大的潜在应用价值。另一方面，空间光孤子是实现未来全光通信及全光器件的最理想途径之一。
本论文的工作就是围绕空间光孤子展开的，所取得的主要成果如下：
1、更加普适的求解偏微分方程初值问题的频域龙格-库塔算法
描述空间光孤子运动的动力学方程是非线性薛定谔方程(NLS)。在本论文第二章中，我们给出了非线性薛定谔方程和离散非线性薛定谔方程的详细推导过程。一般来讲，NLS方程在数学上都是不可积的，因此必须用数值方法求解。在第五章和附录中，我们介绍了一些求解NLS方程及稳态孤子解的数值算法，包括我们自己提出的频域龙格-库塔算法，通过和分步傅立叶算法的比较，我们发现这种算法有着良好的计算精度和效率。
2、矢量光学涡流孤子角向调制不稳定的抑制及其在角向微扰作用下的分裂
矢量光学孤子是由两个或两个以上的光孤子组成的复合孤子，其形成机制可理解为：多束激光在非线性介质中传播，它们通过介质的非线性响应改变了介质的局部折射率分布，在介质中形成一个光波导，当每束光都是这个波导的一个本征模时，便形成了由多孤子组成的矢量孤子。在第三章中，我们研究了饱和非线性介质中由两个涡流分量构成的双分量及有两个涡流分量加一个基本孤子分量构成的三分量矢量光学孤子的形成及稳定性，这两个涡流分量分别携带大小相同而方向相同或相反的光子轨道角动量。通过线性稳定性分析，我们发现这种含涡流分量的矢量孤子在传播过程中受到“对称破缺不稳定”不稳定的影响，发生分
When very narrow optical beams propagate in linear isotropic media, they undergo natural diffraction and broaden with distance. The narrower the beam is, the faster it diverges (diffracts). Such phenomenon is disadvantageous for light as the information carrier. In nonlinear materials, when self-focusing effect induced by the nonlinear response of the media to the light balances the diffraction, the beam becomes self-trapped at a very narrow width (~um) and does not spread in the transverse direction with the propagation distance, and is called a spatial optical soliton (SOS). SOS has been demonstrated to exist by virtue of a variety of nonlinear self-trapping mechanisms. They exhibit a richness of phenomena such as fusion, fission, annihilation, and stable orbiting etc. Their well-defined shape and robustness makes them attractive as fundamental bits in future data transmission and processing schemes, and on the other hand, SOS may provide a powerful means for creating reconfigurable all-optical circuits where light is guided and controlled by light itself.
The work of this thesis mainly concentrates on spatial optical solitons, and the primary achievements obtained are as the following:
1. More universal algorithm—Frequency Domain Runger-Kutta method for solving the initial problem of partial differential equation
The dynamical equation of spatial optical solitons is Nonlinear Schrodinger equation (NLS). In chapter 2, we gave the detailed process of how to derive the NLS equation, including the discrete nonlinear Schrodinger equation. Generally speaking, NLS equation is non-integral, therefore, it must be solved numerically. Some numerical methods including our new method—Frequency Domain Runger-Kutta method for solving NLS equation and the stationary soliton solution were introduced in chapter 5 and in the appendix. By comparing with Split-step Fourier
本论文的工作就是围绕空间光孤子展开的，所取得的主要成果如下：
1、更加普适的求解偏微分方程初值问题的频域龙格-库塔算法
描述空间光孤子运动的动力学方程是非线性薛定谔方程(NLS)。在本论文第二章中，我们给出了非线性薛定谔方程和离散非线性薛定谔方程的详细推导过程。一般来讲，NLS方程在数学上都是不可积的，因此必须用数值方法求解。在第五章和附录中，我们介绍了一些求解NLS方程及稳态孤子解的数值算法，包括我们自己提出的频域龙格-库塔算法，通过和分步傅立叶算法的比较，我们发现这种算法有着良好的计算精度和效率。
2、矢量光学涡流孤子角向调制不稳定的抑制及其在角向微扰作用下的分裂
矢量光学孤子是由两个或两个以上的光孤子组成的复合孤子，其形成机制可理解为：多束激光在非线性介质中传播，它们通过介质的非线性响应改变了介质的局部折射率分布，在介质中形成一个光波导，当每束光都是这个波导的一个本征模时，便形成了由多孤子组成的矢量孤子。在第三章中，我们研究了饱和非线性介质中由两个涡流分量构成的双分量及有两个涡流分量加一个基本孤子分量构成的三分量矢量光学孤子的形成及稳定性，这两个涡流分量分别携带大小相同而方向相同或相反的光子轨道角动量。通过线性稳定性分析，我们发现这种含涡流分量的矢量孤子在传播过程中受到“对称破缺不稳定”不稳定的影响，发生分
When very narrow optical beams propagate in linear isotropic media, they undergo natural diffraction and broaden with distance. The narrower the beam is, the faster it diverges (diffracts). Such phenomenon is disadvantageous for light as the information carrier. In nonlinear materials, when self-focusing effect induced by the nonlinear response of the media to the light balances the diffraction, the beam becomes self-trapped at a very narrow width (~um) and does not spread in the transverse direction with the propagation distance, and is called a spatial optical soliton (SOS). SOS has been demonstrated to exist by virtue of a variety of nonlinear self-trapping mechanisms. They exhibit a richness of phenomena such as fusion, fission, annihilation, and stable orbiting etc. Their well-defined shape and robustness makes them attractive as fundamental bits in future data transmission and processing schemes, and on the other hand, SOS may provide a powerful means for creating reconfigurable all-optical circuits where light is guided and controlled by light itself.
The work of this thesis mainly concentrates on spatial optical solitons, and the primary achievements obtained are as the following:
1. More universal algorithm—Frequency Domain Runger-Kutta method for solving the initial problem of partial differential equation
The dynamical equation of spatial optical solitons is Nonlinear Schrodinger equation (NLS). In chapter 2, we gave the detailed process of how to derive the NLS equation, including the discrete nonlinear Schrodinger equation. Generally speaking, NLS equation is non-integral, therefore, it must be solved numerically. Some numerical methods including our new method—Frequency Domain Runger-Kutta method for solving NLS equation and the stationary soliton solution were introduced in chapter 5 and in the appendix. By comparing with Split-step Fourier
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