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

Sven Nordebo, Martin Stumpf

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.

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

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.
Paper Structure (13 sections, 59 equations, 4 figures, 1 table)

This paper contains 13 sections, 59 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Time-domain constraints for Positive Real functions
  3. Early-time bounds
  4. Late-time bounds
  5. Step response of a dielectric material
  6. Standard conductivity model
  7. Debye model
  8. Lorentz model
  9. Drude model
  10. The Brendel Bormann model
  11. Summary and conclusions
  12. Basic properties of Positive Real functions
  13. Sum rules for positive real functions

Figures (4)

  • Figure 1: Early- and late-time bounds for the unit step response of a dielectric constant with first order asymptotics given by \ref{['eq:pLorentz']} and a comparison with the actual response of the Lorentz model \ref{['eq:TDLorentz']} with $\epsilon_\infty=1$, $\omega_0=\omega_\mathrm{p}=1$ and $\nu\in\{0, 0.5, 1.95\}$. The bounds are in black color and the Lorentz responses are in blue color.
  • Figure 2: Early-time bounds for the generalized step response \ref{['eq:earlytimeDrude2']} with pulse raise time $\tau$ and asymptotics given by \ref{['eq:pDrude']} and a comparison with the actual response of the Drude model \ref{['eq:Druderesponse1']} for gold (Au) with plasma frequency $\omega_\mathrm{p}$ and losses $\nu$ according to the free electron model of Olmon et alOlmon+etal2012. The plot is made in femtoseconds for $t\in[0,50/\omega_\mathrm{p}]$. The bounds are in black color and the Drude responses are in blue color.
  • Figure 3: Same plot as in Fig. \ref{['fig:BBfig3']}, only that the plot is made here for the larger time range $t\in[0,500/\omega_\mathrm{p}]$. The bounds are in black color and the Drude responses are in blue color.
  • Figure 4: Early-time bounds for the generalized step response \ref{['eq:earlytimeDrude2']} with pulse raise time $\tau$ and asymptotics given by \ref{['eq:pBB']}. Here, the equivalent plasma frequency $\omega_\mathrm{p}$ is for gold (Au) according to the Brendel Bormann model Rakic+etal1998 summarized in Tab. \ref{['tab:metals']}. The plot is made in femtoseconds for $t\in[0,100/\omega_\mathrm{p}]$.