Plane-Wave Solutions to Frequency-Domain and Time-Domain Scattering from Magnetodielectric Slabs

TL;DR

This paper derives exact solutions for electromagnetic scattering from magnetodielectric slabs in both frequency and time domains, revealing how finite-time excitation leads to finite, imperfectly focused fields, unlike the divergent fields predicted by idealized single-frequency models.

Contribution

It provides a rigorous analysis of time-dependent scattering in DNG slabs, demonstrating the impact of causality and finite excitation duration on field focusing and divergence.

Findings

Finite-time excitation yields finite, imperfectly focused fields.

Single-frequency solutions exhibit divergence due to infinite energy buildup.

Field focusing becomes perfect only as time approaches infinity.

Abstract

Plane-wave representations are used to formulate the exact solutions to frequency-domain and time-domain sources illuminating a magnetodielectric slab with complex permittivity and permeability. In the special case of a line source at z=0 a distance d<L in front of an L wide lossless double negative (DNG) slab with permittivity and permeability equal to -1, the single-frequency solution exhibits not only "perfectly focused" fields for z>2L but also divergent infinite fields in the region 2d<z<2L. In contrast, the solution to the same lossless -1 DNG slab illuminated by a sinusoidal wave that begins at some initial time t =0 (and thus has a nonzero bandwidth, unlike the single-frequency excitation that begins at t=-infinity) is proven to have imperfectly focused fields and convergent finite fields everywhere for all finite time t. The proof hinges on the variation of permittivity and permeability having a lower bound imposed by causality and energy conservation. The minimum time found to produce a given resolution is proportional to the estimate obtained by [Gomez-Santos, Phys. Rev. Lett., 90, 077401 (2003)]. Only as t approaches infinity do the fields become perfectly focused in the region z>2L and divergent in the region 2d<z<2L. These theoretical results, which are confirmed by numerical examples, imply that divergent fields of the single-frequency solution are not caused by an inherent inconsistency in assuming an ideal lossless -1 DNG material, but are the result of the continuous single-frequency wave (which contains infinite energy) building up infinite reactive fields during the infinite duration of time from t=-infinity to the present time t that the single-frequency excitation has been applied.