  For any inhomogeneous compactly supported electromagnetic (EM) medium, it is
shown that there exists an infinite set of linearly independent electromagnetic
waves which generate nearly vanishing scattered wave fields. If the
inhomogeneous medium is coated with a layer of properly chosen conducting
medium, then the wave set is generated from the Maxwell-Herglotz approximation
to the interior PEC or PMC eigenfunctions and depends only on the shape of the
inhomogeneous medium. If no such a conducting coating is used, then the wave
set is generated from the Maxwell-Herglotz approximation to the generalised
interior transmission eigenfunctions and depends on both the content and the
shape of the inhomogeneous medium. We characterise the nearly non-scattering
wave sets in both cases with sharp estimates. The results can be used to give a
conceptual design of a novel shadowless lamp. The crucial ingredient is to
properly choose the source of the lamp so that nearly no shadow will be
produced by the surgeons operating under the lamp.
