On two types of convergence in the method of auxiliary sources

The method of auxiliary sources (MAS) is often applied to problems involving an externally illuminated, smooth, perfect electric conductor (PEC). One seeks to approximately satisfy the boundary condition on the PEC surface using auxiliary sources located inside the PEC surface. Usually, the underlying auxiliary surface is also smooth and closed. The currents on the auxiliary sources ("MAS currents") are the initial unknowns; once they have been found, one can easily determine the field due to them ("MAS field") at all points external to the auxiliary surface and, in particular, at all points external to the PEC scatterer. We show that, in the limit N → ∞, it is possible to have a convergent MAS field together with divergent MAS currents, and that this phenomenon is accompanied by an abrupt behavior of the limiting value of the MAS field. We show this possibility through an analytical study of a two-dimensional scattering problem involving a circular cylinder, in which MAS fields and currents can be determined explicitly. The analytical study proceeds from first principles; it involves fundamental electromagnetics and relatively simple mathematical manipulations. Numerical results supplement the analytical study and demonstrate the nature of the aforementioned divergence. Our study sheds light on other aspects of MAS; in particular, it establishes interesting similarities and differences between MAS and its "continuous version," and reveals many similarities between MAS currents and numerical solutions of Hallen's integral equation with the approximate kernel.