A plane wave expansion of spherical wave functions for modal analysis of guided wave structures and scatterers

A new finite plane wave series expansion for spherical wave functions of the first kind is presented in this paper. The formulation converts the spherical wave function described in the spherical coordinate system into a series of plane wave functions represented in the Cartesian coordinate system. The series expansion will be very useful in modal analysis of three dimensional guided wave structures and scatterers containing planar boundary surfaces. For a given range of orders m and degree n and for a region with |r| < R, the same set of plane waves can be used. The theory is numerically verified for a wide range of parameters, showing its fast convergence characteristics. The plane wave expansions of the vector multipole fields can also be obtained.