Localized source above a time-modulated dielectric half-space: Green's function theory

In this paper, we explore the two-dimensional Green's function problem above an instantaneous time-modulated dielectric half-space. In particular, we focus on studying the excitation dynamic with modulation that is carried out near the critical angle and the total internal reflection regime, where an impinging spectrum of waves expects the so-called lateral wave excitation on the interface. We start by analyzing the reflection of a plane wave, with detailed attention given to the difference between two cases: a slow quasistatic modulation versus a relatively fast modulation that leads to substantial excitation of intermodulation frequencies. Next, we provide a space-time spectral representation to the space-domain Green's function, and later, we move to evaluate the spectral integral both in a brute-force numerically exact manner and using integration along the steepest descent path and the branch cuts, which unravel the distinction between different wave phenomena associated with the excitation problem. Thus, we identify different wave species that can be associated with reflected rays and waves that resemble the known head wave in the classical problem of stationary stratified media. We examine the unique time-dependent behavior of each wave species that arises as a consequence of the interface between the time-harmonics of the reflected space-time plane wave spectrum. Lastly, we demonstrate the broader validity of our analytical predictions, also in the case of a dispersive and finite time-modulated medium. To that end, we use a full-wave simulation of a source above a finite time-modulated thick layer that is implemented using a time-varying wire medium. These full-wave results are compared with our current analytical model with effective (homogenized) parameters of the time-modulated wire medium taken from Phys. Rev. Appl. 16, 054003 (2021).