标题： 如何一次一个波构建转移矩阵 显示英文标题

标题： How to build transfer matrices, one wave at a time

摘要： 我们展示了如何构建层状介质中波传播的传递矩阵的闭合形式表达式。关键在于将穿过分段常数介质的传播表示为有限数量路径的叠加（对于具有$N$层的介质，有$2^{N-1}$条路径），每条路径贡献一定的相位变化（对应于每个单独层中的相位变化的带符号和）和振幅变化（对应于与每条路径相关的传输和/或反射模式）。该方法本质上是组合性的：它从已知基本解的频率域线性控制方程开始，然后枚举整个系统的有限数量路径，接着计算它们相关的相位和振幅变化，最后将所有可能的路径相加以得到最终结果。除了提供物理洞察力外，这种“逐路径”的构造还可以在许多实际应用中绕过传递矩阵数值乘法的需求，从而可能实现显著的计算节省。 显示英文摘要

摘要： We show how to build the closed-form expression of transfer matrices for wave propagation in layered media. The key is to represent the propagation across the piece-wise constant medium as a superposition of a finite number of paths ($2^{N-1}$ paths for a medium with $N$ layers), each one of them contributing a certain phase change (corresponding to signed sums of the phase change in each individual layer) and amplitude change (corresponding to the pattern of transmission and/or reflection associated to each path). The outlined technique is combinatorial in nature: it begins with the linear governing equations in frequency domain, whose fundamental solution is known, then it enumerates the finite number of paths across the overall system, then computes their associated phase and amplitude change, and finally adds all the possible paths to find the final result. Beyond providing physical insight, this ``path-by-path'' construction can also circumvent the need for transfer matrix numerical multiplication in many practical applications, potentially enabling substantial computational savings.