  In this paper, we present a method for computing bounds for a variety of
efficiency metrics in photonics, such as the focusing efficiency or the mode
purity. 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 Julia
package for basic simulations and bounds.
