  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.
