Time-Domain Green's function for a vertical dipole above a graphene sheet

The three-dimensional potential time-domain Green’s function for a vertical electric dipole radiating in vacuum in the presence of a graphene sheet is derived in a semi-analytical form, by assuming a local Drude-like model for the graphene conductivity in the intraband regime, through a modified Cagniard-de Hoop approach. The derived expression is simple and requires only a straightforward evaluation of a finite smooth one-dimensional integral. Moreover, for the particular and practical case of source and observer along the same vertical axis, the result is provided in a closed form thus allowing for an extremely efficient calculation of the pulsed time-domain field. Numerical results are provided that validate the proposed formulation and assess its computational efficiency against a direct numerical inversion of the relevant spectral-domain potential.