Inverse scattering with a non self-adjoint variational formulation

The weak scattering approximation is used when designing optical media that couple fields together, but to account for the interactions of multiple fields in a volume or to achieve the best efficiency, the solution must be consistent with Maxwell's equations. We describe a method based on the variational formulation of Maxwell's equations typically employed in the finite element method (FEM) that finds both the fields and the medium that couples incident and scattered fields together, and so can be considered an extension of the FEM when both the field and the medium are allowed to vary. The method iteratively updates estimates of the field and the medium and can be readily implemented. We demonstrate designs of diffractive and refractive elements that couple fields together using an iteratively updated finite-difference-frequency-domain (FDFD) solution. Such methods that are fully consistent with Maxwell's equations are needed to design metamaterials that fully exploit strongly interacting metamaterial elements.