Time-domain Green´s function for an infinite sequentially excited periodic planar array of dipoles

The present paper is a continuation of previous explorations by the authors, aimed at gaining a basic understanding of the time domain (TD) behavior of large periodic phased (i.e., sequentially turned-on) array antennas and related configurations. Our systematic investigation of the relevant canonical TD dipole-excited Green\´s functions has so far included those for infinite and truncated sequentially pulsed line periodic arrays, parameterized in terms of radiating (propagating) and nonradiating (evanescent) conical TD Floquet waves (FW) and truncation-induced TD FW-modulated tip diffractions. The present contribution extends these investigations to an infinite periodic sequentially pulsed planar array, which generates pulsed plane propagating and evanescent FW. Starting from the familiar frequency domain (FD) transformation of the linearly phased element-by-element summation synthesis into summations of propagating and evanescent FWs, we access the time domain by Fourier inversion. The inversion integrals are manipulated in a unified fashion into exact closed forms, which are parameterized by the single nondimensional quantity η=c/v^(p)_u1, where v^(p)_u1 and c are the excitation phase speed along a preferred phasing direction u₁ in the array plane and the ambient wave speed, respectively. The present study deals with the practically relevant rapidly phased propagating case η<1, reserving the more intricate slowly phased η>1 regime for a future manuscript. Numerical reference data generated via element-by-element summation over the fields radiated by the individual dipoles with ultrawide band-limited excitation are compared with results obtained much more efficiently by inclusion of a few TD-FWs. Physical interpretation of the formal TD-FW solutions is obtained by recourse to asymptotics, instantaneous frequencies and wavenumbers, and related constructs. Of special interest is the demonstration that the TD-FWs emerge along "equal-delay" ellipses from the array plane; this furnishes a novel and physically appealing interpretation of the planar array TD-FW phenomenology.