Time-dependent plane wave and multipole expansions of the electromagnetic field

Abstract

A new and conceptually simple derivation is presented of the time-dependent multipole expansion of the electromagnetic field radiated by a time-varying, localized, volume charge-current distribution. The analysis is based on a new time-dependent plane wave representation of the electromagnetic field (i.e., an “angular spectrum expansion” in the time domain), also derived in the paper. Expressions are given for the time-dependent plane wave spectra in terms of a fourfold Radon transform representation of the transverse current distribution. Two alternative expressions for the time-dependent multipole moments are derived; the first gives them in terms of the spectral amplitude vectors of the corresponding time-dependent plane wave representation while the second gives them in terms of a weighted radial-temporal average of the current distribution. Thus the analysis also sheds light on the relationship between the time-dependent plane wave and multipole expansions in their common domains of validity.