Incompleteness of the Periodic Moment Method

Introduction: The periodic moment method, developed by Munk and colleagues at Ohio State University, is one of the most important and powerful contributions to numerical analysis of antenna arrays, frequency selective surfaces and gratings. PMM, with its extensions and improvements, can handle multiple dielectric layers and slotted metallic sheets, with wires partially inside a dielectric or air layer or both [I-61. In PMM, the mutual impedance is determined between a wire expansion segment with sinusoidal distribution and a wire test segment with sinusoidal distribution; the latter in an infinite regular array. Thus the number of unknowns in a moment method formulation is just equal to the number of expansion dipoles, and the method is Galerkin, which is intrinsically efficient. The mutual impedance is written as a double spectral (Floquet) sum over x and y wavenumbers, where the wavenumbers contain the scan angles and the lattice dimensions. Spacings between the expansion and test segments are incorporated in exponential factors that also contain the wavenumbers. Segment lengths are represented by far-field pattern functions. A rectangular array of test dipoles along the x-axis, and in the x-y plane, with lattice spacings d,, d , has mutual impedance to an expansion dipole separated by xo, yo: