Time-domain constraints for Positive Real functions: Applications to the dielectric response of a passive material

TL;DR

This paper develops a method to derive physical bounds on the time-domain response of passive systems using Positive Real functions, with applications to dielectric responses of materials like gold.

Contribution

It introduces a systematic approach leveraging Cauer's representation and sum rules to establish time-domain bounds for PR functions, applicable to passive material responses.

Findings

Derived explicit bounds for dielectric step responses.

Applied bounds to gold's electric susceptibility models.

Demonstrated bounds depend on plasma frequency.

Abstract

This paper presents a systematic approach to derive physical bounds for Positive Real (PR) functions directly in the Time-Domain (TD). The theory is based on Cauer's representation of an arbitrary PR function together with associated sum rules (moments of the measure) and exploits the unilateral Laplace transform to derive rigorous bounds on the TD response of a passive system. The existence of useful sum rules and related physical bounds relies heavily on an assumption about the PR function having a low- or high-frequency asymptotic expansion at least of odd order 1. As a canonical example, we explore the time-domain dielectric step response of a passive material, either with or without a given pulse raise time. As a particular numerical example, we consider here the electric susceptibility of gold (Au) which is commonly modeled by well established Drude or Brendel Bormann models. An explicit physical bound on the early-time step response of the material is then given in terms of a quadratic function in time which is completely determined by the plasma frequency of the metal.