When filament light sources are used in nonimaging luminaires that are intended to ensure prescribed angular control and high optical efficiency, a 3-D back reflector must often be incorporated, so as to redirect rays emitted by the filaments that would otherwise miss the luminaire entrance aperture. The glass envelope precludes the filaments from filling the luminaire entrance aperture. The introduction of a back reflector can serve to create a virtual light source, away from the actual grid of filaments, which mitigates the problem. The back reflector should be compact, with high radiative efficiency, namely, a small number of reflections while accommodating all rays. We develop a new nonimaging back reflector design comprised of multiple rotated elliptical arcs in a rotationally symmetric device that is a good compromise.