Time Domain Weyl's Identity and the Causality Trick Based Formulation of the Time Domain Periodic Green's Function

Time domain Floquet modes for analyzing scattering (radiation) from periodically arranged planar scatterers (sources) have been presented recently in a series of papers by Felsen and Capolino. In these papers, they have investigated phenomenological details of transient scattering (radiation) by such structures. Typically, time domain Floquet modes are derived either by Fourier transforming their frequency domain counterparts or using spatial synthesis using Poisson summation. This paper proposes an alternative means to develop similar expressions and is motivated by the possibility of easily extending the method to the analysis of periodic structures in a layered medium. As will be shown, the time domain Weyl's identity can be used to construct the periodic Green's function using both homogeneous and inhomogeneous plane waves. We will also derive expressions using the Causality Trick. This technique involves using time-symmetric (non-causal) signals; however, it yields much simpler expressions that involve only homogeneous time domain plane waves and the results are identical to those obtained using the time domain Weyl's identity for the time region of interest. Finally, we will use the Whittaker formulation, which relies on real signals, to derive time domain Floquet modes. This formulation is applicable for simultaneous excitation only.