Time-Domain Formulation of Electromagnetic Scattering Based on a Polarization Mode Expansion and the Principle of Least Action

Abstract

A fresh approach to the full wave analysis of time evolution of the polarization induced in the electromagnetic scattering from dispersive particles is presented. It is grounded on the combination of the Hopfield model for the polarization field, the expansion of the polarization field in terms of longitudinal and transverse modes of the particle, the expansion of the radiation field in terms of transverse wave modes of free space, and the principle of least action. The polarization field is linearly coupled to the electromagnetic field. The losses of the matter are provided through a linear coupling of the polarization field to a bath of harmonic oscillators with a continuous range of natural frequencies. The set of linear ordinary differential-integral equations of convolution type of the overall system is reduced by eliminating both the radiation degrees of freedoms and the bath degrees of freedom. The reduced system of equations, which governs the time evolution of longitudinal and transverse mode amplitudes of the polarization, is studied. The principal characteristics of the temporal evolution of the mode amplitudes are found as the particle size varies, including the impulse response. Results are presented for the analytically solvable spherical particle, which are validated in the small size limit. The proposed approach leads to a general method for the analysis of the temporal evolution of the polarization field induced in dispersive particles with arbitrary shape, as well as, for the computation of the transients and the steady states.