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 epsilon(omega) and permeability mu(omega). 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 kappa(omega 0) = epsilon(omega 0) / epsilon 0 = mu(omega 0) / mu 0 = (-1), the single-frequency (omega 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 kappa(omega 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 = (-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 kappa(omega) about omega = omega 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 --> 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 kappa(omega 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 = (-infinity) 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.