Inhomogeneous broadening in the time domain: Gauss-Lorentz, Gauss-Drude and Gauss-Debye material models

Forty-five years after the initial attempts - first by Efimov-Khitrov in 1979, then by Brendel-Bormann in 1992 - we present a comprehensive, causal, and physically consistent framework for modeling the dielectric function with inhomogeneous (non-Lorentzian) broadening, where scattering becomes frequency- or time-dependent. This theoretical framework is based on spectral diffusion, described in the frequency domain by a complex probability density function and in the time domain by a matching characteristic function. The proposed approach accurately models the lineshapes resulting from multiple broadening mechanisms and enables the retrieval of intrinsic homogeneous linewidths as well as inhomogeneous disorder-controlled material dispersion features. To implement the new general dispersion function in time-domain Maxwell solvers, we have designed a constrained minimax-based semi-analytical approximation method (MiMOSA) that generates the shortest possible numerical stencils for a given approximation error. Application examples of exact and approximate MiMOSA models include the Gauss-Lorentz oscillator, Gauss-Debye relaxation, and Gauss-Drude conductivity. Although this study primarily focuses on the optical domain, the resulting models, which account for the Doppler shift, are equally applicable to other wave propagation phenomena in disordered dispersive media in a broad range of areas, including acoustics, magnonics, astrophysics, seismology, plasma, and quantum technologies.