Inverse design of time-varying transmission-line filters for broadband spatial signal processing

In this paper, we explore the inverse design of ultrawideband spatial filters by temporal profiles. As opposed to previous works, here we employ a complete nonlinear inversion scheme, which leads to flexibility in the design and to practically reasonable profiles. We express the forward problem using the temporal local impedance and temporal reflection coefficient, akin to those used in the theory of time-harmonic and spatially inhomogeneous transmission lines. For the inversion, we use a straightforward gradient-descent scheme that is applied to a loss function calculated in terms of the temporal reflection coefficient. This calculation yields a piecewise discontinuous temporal profile of the required filter. We further demonstrate, using finite-difference-time-domain simulations, the applicability of the piecewise discontinuous design also for practical soft (gradual) temporal transitions. For the sake of simplification, we assume that the time-dependent medium is instantaneous (i.e., dispersion effects are neglected) and that the temporal transitions can be implemented synchronously over the entire transmission-line structure. While these assumptions are obviously restricting in the general case, structures respecting these constraints have been experimentally demonstrated in radio frequencies using slow-wave metamaterial transmission-line configurations. Since the approach is formulated in terms of transmission-line quantities, it is applicable not only in electromagnetics but also in acoustics.