[Paper Review] Inhomogeneous Spatially Dispersive Electromagnetic Media

Jonathan Gratus, Matthew McCormack|arXiv (Cornell University)|Sep 3, 2013

Electromagnetic Scattering and Analysis15 references3 citations

TL;DR

This paper investigates wave propagation and boundary behavior in inhomogeneous spatially dispersive electromagnetic media using a generalized single-resonance model with position-dependent parameters. It derives two distinct sets of boundary conditions via Lagrangian and distributional methods, showing they diverge when propagation speed β differs between regions, and provides approximate analytic and numerical solutions for wave packets in periodic media with small inhomogeneity.

ABSTRACT

Two key types of inhomogeneous spatially dispersive media are described, both based on a spatially dispersive generalisation of the single resonance model of permittivity. The boundary conditions for two such media with different properties are investigated using Lagrangian and distributional methods. Wave packet solutions to Maxwell's equations, where the permittivity varies and is periodic in the medium, are then found.

Motivation & Objective

• To understand wave packet propagation in inhomogeneous, spatially dispersive media where permittivity varies periodically.
• To derive and compare boundary conditions for spatially dispersive media at interfaces between two regions with different spatial dispersion properties.
• To determine whether Lagrangian-based natural boundary conditions or distributional (pill-box) methods yield consistent results when β is non-zero and spatially varying.
• To provide approximate analytic solutions and a numerical method for finding permitted frequencies and mode profiles in periodic spatially dispersive media.
• To assess the implications of differing boundary condition methods for modeling electromagnetic wave transmission in advanced dielectric wakefield accelerators.

Proposed method

• Generalizes the single-resonance permittivity model to include spatial dispersion by introducing a finite propagation speed β, leading to a second-order PDE in space and time (Eq. 2).
• Uses a Lagrangian formulation to derive natural boundary conditions for the electromagnetic fields at interfaces between two spatially dispersive media.
• Applies the distributional (pill-box) method to derive boundary conditions, comparing results with the Lagrangian approach.
• For periodic media, assumes small inhomogeneity (α(x) = α₀ + α₁cos x, with α₁ ≪ α₀) and seeks wave packet solutions of the form e²πiωt P̂(x).
• Derives approximate analytic solutions for even and odd modes using perturbation theory up to O(Λ⁴) in the small parameter Λ.
• Develops a numerical method based on truncating the infinite system of equations to a finite matrix eigenvalue problem (M b = 0) to compute exact solutions for arbitrary Λ.

Experimental results

Research questions

• RQ1 How do boundary conditions for spatially dispersive media differ when derived via Lagrangian versus distributional methods?
• RQ2 Under what conditions do the two boundary condition methods agree or diverge, particularly when the propagation speed β varies across the interface?
• RQ3 What are the approximate analytic solutions for wave packets in a periodic, spatially dispersive medium with small inhomogeneity?
• RQ4 How do the numerical solutions for the dispersion relation and mode profiles compare with the approximate analytic solutions?
• RQ5 Can the higher-frequency modes in a periodic spatially dispersive medium be understood as resembling those in homogeneous media?

Key findings

• The Lagrangian and distributional methods yield identical boundary conditions for the fields Ê, Ê′, and P̂ when β is constant, but differ in the discontinuity of P̂′ when β varies between regions.
• In the limit β → 0, both methods reduce to Pekar’s standard boundary conditions, making it impossible to distinguish them via this limit alone.
• Two approximate analytic solutions are derived: an even mode (ω^(e), P^(e)) and an odd mode (ω^(o), P^(o)), valid for small Λ, with errors O(Λ⁴) at k = ±1 and O(Λ^{|m|+1}) for |m| ≥ 2.
• The even mode is found to be damped and unsupported in the medium, while the odd mode shows stable behavior under the same conditions.
• A numerical method is implemented by truncating the infinite system of equations to a finite matrix eigenvalue problem, enabling computation of exact solutions for arbitrary Λ.
• Numerical results suggest that higher-frequency modes in the periodic medium resemble those in homogeneous media, a behavior currently under further investigation.