Advances in computing power and optimization algorithms have revolutionized photonics engineering. Traditionally, the design of a photonic structure is selected from a collection of geometric motifs based on intuitive physical principles (e.g., bowtie antennae, photonic crystals, ring resonators) with a handful of tunable parameters. Modern inverse design techniques are capable of treating every computational pixel as a design degree of freedom, leading to drastic improvements in device performance. However, the resulting structures are often highly non-intuitive. This leads to a natural question: what are the fundamental limits to device performance, and how close are we to achieving them?

In this talk, I will detail a computational framework we have developed for evaluating such limits, taking as input only the natural specifications of the material used (susceptibility) and device footprint (a design region that encompasses the structure). The framework combines physical conservation laws with mathematical optimization theory, and is broadly applicable to any photonics design objective that can be expressed as a quadratic function of the fields. Concrete results pertaining to canonical photonics design problems such as thermal emission, scattering cross sections, Purcell enhancement, and wavefront engineering will be presented, where the limits often come to within an order of magnitude of inverse design performance. Finally, on-going work on further generalizations of the framework will be discussed, such as the inclusion of fabrication constraints and application to non-linear photonics problems.