Accurate simulation of a spherical reflector antenna by the complex point source-dual series approach

In a previous paper (see IEEE Trans. Antennas and Propagation, vol.43, no.8, p.793-801, 1995), we considered a circular-cylindrical reflector fed by complex-point E and H-polarized line sources. Dual series equations (DSE) were formulated in terms of the trigonometric functions, with the static part analytically inverted based on the Riemann-Hilbert problem solution. The resulting matrix equation was of the Fredholm 2nd kind. A remarkable feature of this equation is that it can be solved numerically with a guaranteed accuracy and small CPU time expenditures. The present study is similar but deals with a more practical model of a 3-D spherical reflector fed by a complex point source. The solution is obtained, via the DSE, in terms of the associated Legendre functions, with the static part inversion based on the Abel integral equation technique. We consider here two scalar problems of the hard and soft reflector surface, but keep in mind the vector electromagnetic problem as an ultimate goal.