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： 我们报告了一个普适恒等式：一维传输或散射矩阵中任意一层的矩阵元等于代表具有预定义振幅和相位演化的波模式的 2**(N-1) 条有向路径的相干叠加，其中 N 是层数。 将这些路径视为解析的基本构件，我们在不借助矩阵乘积或递归的情况下，得到了任意分层介质（光学、声学、弹性或电子）的闭合形式结果。 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，通过一个设计实例进行了说明，该实例重新诠释了四分之一波长原理。 我们还量化了路径方法相对于经典传输矩阵链式法则的计算加速，并展示了两个跨学科应用（场地响应地震学和量子超晶格）。 这一范式用物理透明的路径构建规则代替了使用传输矩阵法的数值扫描；其适用范围跨越多个物理学科和尺度：从小于纳米级的光学涂层到千米级的地震地层。 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.