Frequency-domain Green's function for a planar periodic semi-infinite phased array .I. Truncated floquet wave formulation

This two-part sequence deals with the derivation and physical interpretation of a uniform high-frequency solution for the field radiated at finite distance by a planar semi-infinite phased array of parallel elementary electric dipoles. The field obtained by direct summation over the contributions from the individual radiators is restructured into a double series of wavenumber spectral integrals whose asymptotic reduction yields a series encompassing propagating and evanescent Floquet waves (FWs) together with corresponding diffracted rays, which arise from scattering of the FW at the edge of the array. The formal aspects of the solution are treated in the present paper. They involve a sequence of manipulations in the complex spectral wavenumber planes that prepare the integrands for subsequent efficient and physically incisive asymptotics based on the method of steepest descent. Different species of spectral poles define the various species of propagating and evanescent FW. Their interception by the steepest descent path (SDP) determines the variety of shadow boundaries for the edge truncated FW. The uniform asymptotic reduction of the SDP integrals, performed by the Van der Waerden (1951) procedure and yielding a variety of edge-diffracted fields, completes the formal treatment.