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.

Figures (15)

• Figure 1: Internal field magnitude: (a) Single layer with two interfaces and (c) two layer system calculated using the same single layer formalism where the reflection and transmission coefficients of the second interface are replaced ones calculated from the second (gray) layer. (b,d) Magnitude of the round-trip coefficient $|r_{12}^2p_2^2|$ corresponding respectively to the structures considered in figures (a) and (c).
• Figure 2: Two layer structure $(0|1|2|3)$. In the top figure the arrows indicating partial waves and their coefficients. The bottom figure shows the multiple scattering and propagation events as a state machine.
• Figure 3: Maximum eigenvalue magnitude $|\lambda|$ of the event scattering matrix for (a) single- and (b) two-layer structures. The insets show the eigenmodes for the four cases considered. The arrows represent propagating waves while the triangles evanescent waves. Where red zones correspond to divergent domains. The shade of the arrows and triangles are signifying the amplitudes.
• Figure 4: Pulse reflected from the double layer structure at different event number using (a) the standard propagation mode definition and (b) the power mode definition. The finite pulse is defined in the evanescent case and its transversal wavevector and frequency spectrum is shown in the inset as a contour plot.
• Figure 5: Reflected field coefficient for the structure discussed in figure (Fig. \ref{['fig:2']}). Solid curves: real (red) and imaginary (blue) parts of the Abel sum $\mathbf{R}(x)$ for $x<x_{max}$. Dashed curves: rational fit (Padé-like) providing analytic continuation toward $x=1$. Crosses: fixed-point solution $(\mathbf{I}-\mathbf{P\cdot S})^{-1}\mathbf{s}_0$ at $x=1$.