Effects of finite terms on the truncation error for the addition theorem of spherical vector wave functions

A finite sum adopted for the translational Addition Theorem (AT) of the spherical vector wave functions is a key-point for addressing a well known concept regarding the classical electromagnetic scattering of an incoming wave across a generic dispositions of spheres. The AT is essential for the translation of the vector spherical wave function expressed with respect to a coordinates system toward a different reference. According to its general formulation, the dependence of the translation coefficients on the relative direction of displacement linking two distinct references is the core study to be addressed. The inherent analytical aspects have been intensively studied during the last decades, but the respective numerical encoding has been limited to few basic configurations which provided a partial validity of the theory with quite approximate solutions. In literature there are many authors reporting different sets of the vector translation coefficients, among which we mention those calculated by Stein, Cruzan and Mackowski as the most prominent of them. We have selected the Cruzan formulation of the vector translation coefficients for its structure based on the Wigner 3-j function. We have developed the criteria of truncating the inherent infinite series to achieve convergence with a finite version of the same, which leads to an extremely negligible error. The procedure relies on the different parameters of the geometrical system, such as the direction angles and norm of the displacement vector, the angular indexes of spherical wave vector function, the propagation constant, and so on. During our numerical tests, we have deeply investigated generic truncation errors and outlined a repeatable procedure to get an acceptable convergence. Our method allows to calculate a suitable truncation limit which ensures acceptable results in terms of contained AT truncation errors and simultaneously a onvergent series able to efficiently reconstruct the generic displaced vector wave function. We are sure that our improved approach will help researchers in developing efficient codes for the problem of light scattering across general distributions of spheres.