Nearly non-scattering electromagnetic wave set and its application

For any inhomogeneous compactly supported electromagnetic (EM) medium, it is shown that there exists an infinite set of linearly independent EM 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 perfectly electric conducting or perfectly magnetic conducting 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 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 surgeons operating under the lamp.