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A variation on the classical polygon illumination problem was introduced in [Aichholzer et al. EuroCG'09]. In this variant light sources are replaced by wireless devices called -modems, which can penetrate a fixed number , of 'walls'. A point in the interior of a polygon is 'illuminated' by a -modem if the line segment joining them intersects at most edges of the polygon. It is easy to construct polygons of vertices where the number of -modems required to illuminate all interior points is . However, no non-trivial upper bound is known. In this paper we prove that the number of kmodems required to illuminate any polygon of vertices is . For the cases of illuminating an orthogonal polygon or a set of disjoint orthogonal segments, we give a tighter bound of . Moreover, we present an time algorithm to achieve this bound. [ABSTRACT FROM AUTHOR]