Reciprocity and Normality in the Scattering Matrix of Disordered Media

Abstract

The scattering matrix formalism provides a practical characterization of wave transport in linear, source-free systems by relating a set of operationally defined input and output spatial channels. The matrix is structured as a block operator, with diagonal blocks encoding same-side reflection matrices (RMs) and off-diagonal blocks encoding transmission matrices (TMs) in opposing propagation directions. Under Helmholtz reciprocity, symmetry relations are imposed: RMs are symmetric, and forward and reverse TMs are mathematical transposes of each other. These relations were employed as constraints to correct system-induced aberrations in measured scattering matrices of complex optical media via a matrix-based gradient descent procedure. Resulting phase corrections corresponded closely with classical aberration modes without heuristic parameterizations, suggesting that these modes naturally arise to restore reciprocity-induced symmetry. Vectorial TMs were measured for single- and double-pass propagation through step-index MMFs and scattering samples, with corrected phase terms showing agreement across sample types. Furthermore, matrix normality was introduced as a descriptor of stable modal transport. Normal matrices admit unitary diagonalization, reflecting orthogonal eigenchannels and spectrally coherent propagation. Near-normal behavior was observed in fiber TMs, while RMs of scattering slabs remained strongly non-normal, as quantified by a normalized Henrici departure. Sufficient conditions for normality were identified in terms of the system Green’s function and its bi-compression onto the measurement basis. A complementary dispersion experiment investigated two regimes: nearly-normal MMFs, where the Wigner–Smith time-delay operator was jointly diagonalizable and supported accurate first-order spectral models; and mechanically compressed fibers, where loss of normality produced noncommuting operators and collapse of model fidelity. These results suggest that normality captures well-behaved modal transport, underpinning the validity of parametric models and other operator-based analyses of disordered media. Together, reciprocity and normality impose complementary constraints on wave transport: reciprocity governs global symmetry, while normality captures internal coherence of modal propagation. Relevance is noted for matrix-based imaging, inverse scattering theory, and non-Hermitian wave physics, where symmetry and modal stability remain central.