Representation-induced superposition breakdown in linear physics

Abstract

The superposition principle is fundamental to linear wave systems, ensuring that their physical behaviour is independent of the chosen basis representation. While this principle underpins many analytical techniques, including modal decompositions and scattering formulations, we show that superposition expansion can fail in multilayered media when fields are expressed as infinite series of evanescent and inhomogeneous waves. Using the Airy formula and the scattering-matrix formalism, we identify conditions under which the superposition of partial waves diverges, particularly in systems with three or more interfaces. This divergence occurs because evanescent wave components cannot be normalised within the conventional basis and is not a numerical artefact. To address this, we introduce power flux modes corresponding to orthonormal basis wave solutions that preserve energy conservation in scattering events and consequently restore convergence. We prove that in the flux-orthonormal basis, interface scattering is unitary and propagation eigenvalues are bounded guaranteeing convergence. Our approach generalises to scalar, electromagnetic, and elastic wave systems, providing a robust framework for eliminating evanescent mode divergence without regularisation or renormalisation.