Plane-wave solutions to frequency-domain and time-domain scattering from magnetodielectric slabs

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 µ(). 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 (₀)=(₀)/₀=µ(₀)/µ₀=–1, the single-frequency (₀) 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 (₀)=–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=–) is proven to have imperfectly focused fields and convergent finite fields everywhere for all finite time t. The proof hinges on the variation of () about =₀ 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 G. Gómez-Santos, [Phys. Rev. Lett. 90, 077401 (2003)]. Only as t 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 (₀)=–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=– to the present time t that the single-frequency excitation has been applied. An analogous situation occurs at the resonant frequencies of a lossless cavity. A single-frequency (zero-bandwidth) source inside the cavity produces infinite fields at a resonant frequency, whereas the same source turned on at time t=0 (so that it has a nonzero bandwidth) produces finite fields.

©2006 The American Physical Society

See Also

Erratum: Plane-wave solutions to frequency-domain and time-domain scattering from magnetodielectric slabs [Phys. Rev. E 73, 046608 (2006)]
Arthur D. Yaghjian and Thorkild B. Hansen
Phys. Rev. E 76, 049903 (E) (2007)