Temporal equivalent of the Sommerfeld half-plane problem
Sommerfeld's half-plane problem, describing diffraction of a monochromatic wave by a semi-infinite perfectly electric conducting (PEC) infinitesimally thin plate, is one of the cornerstones of classical electromagnetic scattering theory. Here we study its temporal analogue: a semi-infinite medium with the Drude dispersion, adjacent to air, whose plasma frequency is rapidly reduced to zero, thus turning it into air, while a monochromatic wave with frequency below the plasma frequency illuminates it under normal incidence. Unlike the PEC plate, the Drude medium supports evanescent penetration of the field prior to its temporal transition; the temporal interface here acts as the analogue of the sharp spatial boundary in the Sommerfeld problem, but now separates two time intervals rather than two spatial half-planes. We show that this transition forces the standing wave in air in front of the Drude medium and the evanescent fields inside it to propagate in both directions, while preserving their spatial field structure. In addition, we observe the excitation of additional electric and magnetic field waves generated by a "DC" current density associated with the time interface. Our findings offer insight into the electromagnetic wave behavior at temporal transitions in dispersive media and may find applications in time- and space-time metamaterials