Tapered wave with a least squared error property

The problem of constructing a 3D tapered wave as a superposition of plane waves is considered. The use of the simple Gaussian plane wave spectrum is recommended in order to avoid problems near the grazing incidence. The introduced special choice for the polarization vectors removes the problems of losing a dominant polarization state and degradation of tapering near the normal incidence. Mathematically speaking, the proposed polarization vectors are analytic at the origin of the 2D wavenumber space. Moreover, the choice of polarization vectors is shown to lead to an exact solution of the Maxwell equations which is an optimal approximation of an ideal but non-Maxwellian tapered field that is constructed by multiplying a scalar tapered wave with a constant polarization vector. The result is a reliably tapered wave with a dominant polarization state that can be used uniformly for all angles of incidence