Arbitrary Rigid Translation of the Spherical Vector Wave Functions

The need to rigidly translate the solutions of the vector wave equation in spherical coordinates towards an arbitrary direction arises in a plethora of antenna measurement applications. Translating the spherical vector wave functions is a necessary step for the derivation of a spherical near-field transmission formula. It can also be used in the estimation of uncertainty errors in spherical near-field antenna measurements, as well as in scattering applications. There are primarily three methods to achieve such translations, with all of them employing the so-called "addition theorems". In this paper, we analyze and compare these methods, discussing their advantages and disadvantages and suggest a modification that can render the whole process faster and more efficient.