Implications in the interpretation of plane-wave expansions in lossy media and the need for a generalized definition

Plane-wave expansions (PWEs) based on Fourier transform and their physical interpretation are discussed for the case of homogeneous and isotropic lossy media. Albeit being mathematically correct, standard Fourier-based definition leads to nonphysical properties, such as the absence of homogeneous plane waves, lack of dissipation along transversal directions and inaccurate identification of single plane waves. Generalizing the PWE definition using Laplace transform, which amounts to switching to complex spectral variables, is shown to solve these issues, reinstating physical consistency. This approach no longer leads to a unique PWE for a field distribution, as it allows an infinite number of equivalent definitions, implying that the interpretation of the individual components of a PWE as physical plane waves does not appear as justified. The multiplicity of the generalized definitions is illustrated by applying it to the near-field radiation of an elementary electric dipole, for different choices of Laplace cuts, showing the main differences in the generalized PWEs.