Multipole Theory in the Time Domain

Spherical outgoing waves of arbitrary time dependence are first written in the usual way as a Fourier integral of a sinusoidally time‐varying multipole expansion. It is then shown that the integrals over ω of the r‐ and t‐dependent part of the multipole terms can be replaced by differential operators operating on arbitrary functions of retarded time. Thus a form of the multipole expansion is obtained that does not explicitly contain the frequency spectrum of the multipoles. Given the value of Er (for electric multipoles, or Br for magnetic multipoles) as a function of time on the surface of a sphere, expressions for the multipole expansions of all the spherical field components are derived. The method employs a convolution integral and is useful in problems involving a very broad frequency spectrum.