Universal path decomposition of transfer and scattering matrices

Abstract: 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.