Towards nanophotonic optical isolation via inverse design of energy transfer in non-reciprocal media

In this work we generalise the adjoint method of inverse design to non-reciprocal media. As a test case, we use three-dimensional topology optimisation via the level-set method to optimise one-way energy transfer for point-like source and observation points. To achieve this we introduce a suite of tools, chiefly what we term the `Faraday-adjoint' method which allows for efficient shape optimisation in the presence of magneto-optical media. We carry out an optimisation based on a very general equation that we derive for energy transfer in a non-reciprocal medium, and link finite-different time-domain numerics to analytics via a modified Born series generalised to a tensor permittivity. This work represents a stepping stone towards practical nanophotonic optical isolation, often regarded as the `holy grail' of integrated photonics.