Felsen, Frequency-domain Green’s function for a planar periodic semi-infinite phased array—part I: Truncated Floquet wave formulation

Abstract—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 radi-ators is restructured into a double series of wavenumber spectral integrals whose asymptotic reduction yields a series encompassing propagating and evanescent Floquet waves (FW’s) 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 (Part I). They involve a sequence of manipulations in the complex spectral wavenumber planes that prepare the integrands for subsequent efficient and physically inci-sive 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 inte-grals, performed by the Van der Waerden procedure and yielding a variety of edge-diffracted fields, completes the formal treatment. The companion paper (Part II [1]) deals with the phenomenology of these local diffracted waves based on the present formal solu-tion. The phenomenology encompasses all possible contributions of propagating and evanescent edge diffractions excited by propa-gating and evanescent FW’s. The outcome is a physically appealing and accurate high-frequency algorithm, which is numerically effi-cient due to the rapid convergence of both the FW series and the series of relevant diffracted fields. This is demonstrated by numer-ical examples for radiation from a strip array in Part II