标题： 传递和散射矩阵的通用路径分解 显示英文标题

标题： Universal path decomposition of transfer and scattering matrices

摘要： 我们报告了一个通用恒等式：由N层组成的二维传输或散射矩阵的任意条目等于2^(N-1)个有向路径的相干求和，这些路径表示具有预定义振幅和相位演化的波模式。 将这些路径视为解析构建模块，我们推导出任意分层介质（光学、声学、弹性或电子）的闭式结果，而无需使用矩阵乘积或递归。 路径的组合构造将分层系统设计转化为基于规则的路径工程，通过一个设计示例进行说明，该示例对四分之一波长原理进行了重新解释。 我们还量化了路径方法相对于经典传输矩阵链的计算速度提升，并展示了两个更多跨学科的应用（站点响应地震学和量子超晶格）。 这一范式用物理透明的路径构造规则取代了使用传输矩阵方法的数值扫描；其适用性跨越多个物理学科和尺度：从纳米级光学涂层到千米级地震地层。 显示英文摘要

摘要： We report a universal identity: any entry of a one-dimensional transfer or scattering matrix comprising N layers equals a coherent sum of 2**(N-1) directed paths representing wave patterns with pre-defined amplitude and phase evolutions. Treating those paths as analytic building blocks, we derive closed-form results for arbitrary stratified media - optical, acoustic, elastic, or electronic - without resorting to matrix products or recursion. The combinatorial construction of paths turns layered system design into rule-based path engineering, illustrated with a design example that offers a reinterpretation of the quarter-wavelength principle. We also quantify the computational speed-up of the path method over classical transfer-matrix chaining and showcase two more cross-disciplinary applications (site-response seismology and quantum superlattices). This paradigm replaces numerical sweeps that employ the transfer matrix method with physically-transparent path-construction rules; its applicability spans across physical disciplines and scales: from nanometer optical coatings to kilometer-scale seismic strata.