In this paper, we present a method for computing bounds for a variety of
efficiency metrics in photonics. We focus on the special case where the
objective function can be written as the ratio of two quadratic functions of
the field and show that there exists a simple semidefinite programming
relaxation for this problem. We provide a numerical example of bounding the
maximal mode conversion purity for a device of given size. This paper is
accompanied by an open source implementation