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Correlation between the lens extension size and the broadside directivity of a hemielliptic dielectric lens antenna (DLA) fed by a primary feed with variable radiation pattern is studied in accurate manner. The problem is considered in two-dimensional formulation and solved numerically using in-house software based on the Muller boundary integral equations (MBIE). Our results highlight the key role of the edge taper which can be defined for DLAs similarly to the theory of reflector antennas. A new feature revealed is the relation between the optimal edge taper needed to achieve the highest possible directivity and the permittivity of the lens material.