Nonparabolic solar concentrators matching the parabola

We consider the limit of geometric concentration for a focusing concave mirror, e.g., a parabolic trough or dish, designed to collect all radiation within a finite acceptance angle and direct it to a receiver with a flat or circular cross-section. While a concentrator with a parabolic cross-section indeed achieves this limit, it is not the only geometry capable of doing so. We demonstrate that there are infinitely many solutions. The significance of this finding is that geometries which can be more easily constructed than the parabola can be utilized without loss of concentration, thus presenting new avenues for reducing the cost of solar collectors. In particular, we investigate a low-cost trough mirror profile which can be constructed by inflating a stack of thin polymer membranes and show how it can always be designed to match the geometric concentration of a parabola of similar form.