A Generalization of the Kramers-Kronig Relations for Linear Time-Varying Media

TL;DR

This paper extends the classical Kramers-Kronig relations to media with linear time-varying dispersive properties, providing a rigorous mathematical framework for understanding their electromagnetic response even under rapid temporal changes.

Contribution

It introduces a generalized set of Kramers-Kronig relations for time-varying media and develops circuit analogs for their differential equations, advancing the theoretical understanding of dynamic dispersive systems.

Findings

Generalized Kramers-Kronig relations for time-varying media

Circuit models for time-dependent dielectric responses

Analysis of Lorentzian dielectric with varying atomic density

Abstract

We explore the mathematical theory to rigorously describe the response of media with linear time-varying, generally dispersive, electromagnetic constitutive parameters. We show that even when the temporal inhomogeneity takes place on a time scale comparable or shorter than the driving fields' time period, one can still define a physically meaningful time-varying dispersion. Accordingly, a generalized set of Kramers-Kronig relations is investigated to link the real and imaginary parts of the time-varying frequency-dispersive spectra characterizing the medium's constitutive response. Among others, we study the case of a Lorentzian dielectric response with time-varying volumetric density of polarizable atoms and present the varying circuital equivalents of the governing differential equation, which in turn allow us to use the notion of generalized time-varying impedances/admittances of a time-dependent resistor, inductor and capacitor.