Provable Routing Analysis of Programmable Photonics

Programmable photonic integrated circuits (PPICs) are an emerging technology recently proposed as an alternative to custom-designed application-specific integrated photonics. Light routing is one of the most important functions that need to be realized on a PPIC. Previous literature has investigated the light routing problem from an algorithmic or experimental perspective, e.g., adopting graph theory to route an optical signal. In this paper, we also focus on the light routing problem, but from a complementary and theoretical perspective, to answer questions about what is possible to be routed. Specifically, we demonstrate that not all path lengths (defined as the number of tunable basic units that an optical signal traverses) can be realized on a square-mesh PPIC, and a rigorous realizability condition is proposed and proved. We further consider multi-path routing, where we provide an analytical expression on path length sum, upper bounds on path length mean/variance, and the maximum number of realizable paths. All of our conclusions are proven mathematically. Illustrative potential optical applications using our observations are also presented.