标题： 周期稳态脉冲在短脉冲光纤激光器中的Floquet稳定性 显示英文标题

标题： Floquet stability of periodically stationary pulses in a short-pulse fiber laser

摘要： 现代短脉冲光纤激光器的定量建模和设计无法使用平均模型，因为每个往返过程中脉冲参数有较大的变化。 相反，需要使用通过连接激光器各个组件的模型而得到的集中模型。 由于集中模型中的光脉冲是周期性的，因此使用单值化算子来研究其线性稳定性，该算子是关于脉冲的往返算子的线性化。 开发了一种基于梯度的优化方法来发现周期性脉冲。 目标函数梯度的计算涉及往返算子和单值化算子的伴随算子作用的数值计算。 引入了一种新的傅里叶分裂步法来计算非线性、非局部、刚性方程的线性化方程的解，该方程模拟光纤放大器中的光传播。 该方法是通过对非线性方程的分裂步法中的两个求解算子进行线性化得到的。 单值化算子的谱包括本质谱，对于本质谱有一个解析公式，以及特征值。 在$\lambda=1$处有一个重数为二的特征值，这是由于相位和平移不变性引起的。 其余的特征值由单值化算子的矩阵离散化确定。 仿真结果验证了数值方法的准确性，展示了周期稳态脉冲的例子，它们的谱和本征函数，并讨论了它们的稳定性。 显示英文摘要

摘要： The quantitative modeling and design of modern short-pulse fiber lasers cannot be performed with averaged models because of large variations in the pulse parameters within each round trip. Instead, lumped models obtained by concatenating models for the various components of the laser are required. Since the optical pulses in lumped models are periodic, their linear stability is investigated using the monodromy operator, which is the linearization of the roundtrip operator about the pulse. A gradient-based optimization method is developed to discover periodic pulses. The computation of the gradient of the objective function involves numerical computation of the action of both the round trip operator and the adjoint of the monodromy operator. A novel Fourier split-step method is introduced to compute solutions of the linearization of the nonlinear, nonlocal, stiff equation that models optical propagation in the fiber amplifier. This method is derived by linearizing the two solution operators in a split-step method for the nonlinear equation. The spectrum of the monodromy operator consists of the essential spectrum, for which there is an analytical formula, and the eigenvalues. There is a multiplicity two eigenvalue at $\lambda=1$, which is due to phase and translation invariance. The remaining eigenvalues are determined from a matrix discretization of the monodromy operator. Simulation results verify the accuracy of the numerical methods, show examples of periodically stationary pulses, their spectra and eigenfunctions, and discuss their stability.