Green's function theory of spatiotemporally modulated loaded wire surface

In this paper we explore the two-dimensional Green's function problem above a spatiotemporally modulated loaded wire media. We suggest a spatiotemporal spectral representation to the space-time domain Green's function. Then, we evaluate the spectral representation both by a brute-force numerical integration as well as in a physics-guided fashion by deforming the integration path along the steepest descent path and encircling singular points in the complex spectral plane. The latter approach leads in general to three isolated wave contributions: a refracted ray due to a saddle point contribution, a head-wave-like due to the branch points, and leaky modes that are associated with pole singularities. This way, we identify different wave species that are all subject to a synthetic motion that is effectively caused by the spatiotemporal modulation and give rise to unique nonreciprocal behavior. Lastly, we provide asymptotic closed form expressions for the various waves that comprise the Green's function, as well as conditions under which they contribute.