Relations between cylindrical and spherical vector wavefunctions

By comparing the expansions of the plane wave in terms of cylindrical and spherical vector wavefunctions, a simple method of expressing cylindrical vector wavefunctions in terms of spherical vector wavefunctions is presented. For the expressions of the latter in terms of the former, a theoretical procedure is developed and the explicit relations are given. The results and approaches provided in general are useful for interpretation of such electromagnetic (EM) problems as arbitrary shaped beam scattering by small particles, EM radiations with complex boundaries, elastic or inelastic, and so on. As an application example, the scattering fields of cylindrical waves by spherical obstacles, both homogeneous and coated spheres, are discussed.