Spherical Harmonic Expansion

Analytical and semi-analytical methods can be used to reveal the general principles in the electromagnetic issues, to catch the characteristic properties of a radiating system, and to predict the applicability boundaries and limitations of the numerical methods. In this chapter, we discuss the spherical harmonic expansion solutions to the fields of a bounded source in free space. We treat the three-dimensional space in vacuum as a two-port radial waveguide The spatial distributing properties of the fields on the transverse spherical surfaces and the propagation properties of the fields in the longitudinal direction are handled separately. This strategy is quite flexible for handling the discontinuities in the waveguide, like the local reflection at the interface of two waveguides. The spherical harmonic expansion expressions for the fields and potentials have been derived both in time domain and in frequency domain. The dyadic Green’s functions are rederived using a method different from that was used by Collin and Tai. The formulation is more straightforward, and the treatment of the discontinuities crossing the source point is simpler. We will show that the expansion of the static fields, the time harmonic fields, and the transient fields can be handled in almost the same manner.