Investigation of the TE$-z$/TM$-z$ Alternative Spherical Wavefunction Expansion

Spherical wave function expansions involving a TM- $R$ and TE-$R$ field decomposition are widely used. Here we examine a spherical wavefunction expansion based on a TM- $z$ and TE-z field decomposition. The impetus is the straightforward relationship between the TM-z and TE-z spherical wavefunctions and the cartesian multipoles which, in turn, have contemporary application in wireless power transfer. However, there are potentially other useful features of such a TM-$z$ and TE- $z$ spherical wavefunction expansion. For example, the TE/TM-z decomposition results in the $\theta$ -polarized far electric field being associated exclusively with the TM-$z$ wavefunctions while the $\phi-$ polarized far electric field is associated exclusively with the TE-z spherical wavefunctions. Unfortunately, from the analysis given here, it appears that it is not possible to represent a generalized finite source in TE/TM-z spherical wavefunctions exclusively. One succinct way to show this is by attempting to expand the far field of an $x$ or $y$ directed electric or magnetic dipole in TE/TM- $z$ spherical wavefunctions. The dipole can be represented by a single TE/TM-$R$ spherical wavefunction, but the expansion in TE/TM-z wavefunctions is problematic.