Time-domain constraints for passive materials: The Brendel-Bormann model revisited

Abstract

This paper presents a systematic approach to derive physical bounds for passive systems, or equivalently for positive real (PR) functions, directly in the time-domain (TD). As a generic, canonical example we explore the TD dielectric response of a passive material. We will furthermore revisit the theoretical foundation regarding the Brendel-Bormann (BB) oscillator model which is reportedly very suitable for the modeling of thin metallic films in high-speed optoelectronic devices. To this end, an important result here is to re-establish the physical realizability of the BB model by showing that it represents a passive and causal system. 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. Similar bounds have recently been reported for more general casual systems with other a priori assumptions. To this end, it is important to note here that the existence of useful sum rules and related physical bounds rely heavily on an assumption about the PR functions having a low- or high-frequency asymptotic expansion at least of odd order 1. As a particular numerical example, we consider here the electric susceptibility of gold (Au) which is commonly modeled by well established Drude or BB models. Explicit physical bounds are given as well as an efficient fast-Fourier transform-based numerical procedure to compute the TD impulse response associated with the nonrational BB model.This paper presents a systematic approach to derive physical bounds for passive systems, or equivalently for positive real (PR) functions, directly in the time-domain (TD). As a generic, canonical example we explore the TD dielectric response of a passive material. We will furthermore revisit the theoretical foundation regarding the Brendel-Bormann (BB) oscillator model which is reportedly very suitable for the modeling of thin metallic films in high-speed optoelectronic devices. To this end, an important result here is to re-establish the physical realizability of the BB model by showing that it represents a passive and causal system. 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. Similar bounds have recently been reported for more general casual systems with other a priori assumptions. To this end, it is important to note here that the existence of useful sum rules and related physical bounds rely heavily on an assumption about the PR functions having a low- or high-frequency asymptotic expansion at least of odd order 1. As a particular numerical example, we consider here the electric susceptibility of gold (Au) which is commonly modeled by well established Drude or BB models. Explicit physical bounds are given as well as an efficient fast-Fourier transform-based numerical procedure to compute the TD impulse response associated with the nonrational BB model.