In the absence of random mode mixing (RMM), linearization of the cross-mode modulation (XMM) under pump-probe configuration is in general complicated because the evolution of the pump light depends on the local mode decomposition of itself. In this work, general derivation of the Hamiltonian of the XMM system without RMM is carried out and discussed under two interesting scenarios where the pump evolution can be effectively linearized, leading to spatial dependent Hamiltonian of the probe light. Representative evolutions of pump and probe light over transmission are investigated with the assistance of Poincaré sphere based on the Hamiltonian approach. Our results are benchmarked against the results given by precession equations examined by Lin and Agrawal and numerical simulations using nonlinear coupled mode equations (CMEs). By highlighting the eigenstates as well as eigenvalues of the system, our approach provides an intuitive yet powerful approach to understand the XMM nonlinear problems and to dynamically manipulate the spatial profiles of the probe light.

中文翻译：

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交叉模式调制的空间相关哈密顿公式

在没有随机模式混合 (RMM) 的情况下，泵浦探针配置下的交叉模式调制 (XMM) 的线性化通常很复杂，因为泵浦光的演变取决于自身的局部模式分解。在这项工作中，没有 RMM 的 XMM 系统的哈密顿量的一般推导在两个有趣的场景下进行并讨论，其中泵浦演化可以有效地线性化，导致探测光的空间相关哈密顿量。在基于哈密顿方法的庞加莱球的帮助下，研究了泵浦光和探测光在传输过程中的代表性演变。我们的结果以 Lin 和 Agrawal 检验的进动方程以及使用非线性耦合模式方程 (CME) 的数值模拟给出的结果为基准。