Time-varying equivalent circuit for dynamically modulated dielectric slabs and its implications for stability analysis

Equivalent circuits (ECs) have played an instrumental role in analyzing and modeling distributed electromagnetic structures for many decades, owing to their intrinsic simplicity and the valuable intuition and physical insights they provide. Here, we envision what we believe to be a novel class of ECs featuring linear time-varying (LTV) circuit elements to harness the power of EC analysis for studying LTV structures, particularly LTV dielectric slabs. Our time-varying equivalent circuit (TVEC)-consisting of infinitely many time-varying LC resonators interconnected in either series or parallel-is rigorously derived and exact, with closed-form expressions for the circuit elements. Its convenient form, enabled by a judicious use of Mittag-Leffler expansion, simplifies the analysis of LTV slabs experiencing various resonant and harmonic phenomena. We demonstrate this point by applying our TVEC to the problem of parametric instability in linear time-periodic slabs. Despite its immense importance, instability has been largely neglected in the literature on periodically modulated structures, mainly due to the inherent complexity of problems involving partial differential equations with time-periodic coefficients. Time-periodic circuits, on the other hand, enjoy an abundance of mathematical tools and methods in the form of the theory of Hill's equation. We utilize this rich theory to investigate the nature of parametric instability in linear time-periodic slabs. Numerical results are then presented to verify these findings. Our TVEC is simple, applicable to any form of temporal modulation (not just periodic), and potentially extensible to other geometries, making it likely to have a significant impact on a multitude of hotly researched LTV topics.