Poles of the Scattering Matrix: An Inverse Method for Designing Photonic Resonators

We develop and implement a new mathematical and computational framework for designing photonic elements with one or more high-$Q$ scattering resonances. The approach relies on solving for the poles of the scattering matrix, which mathematically amounts to minimizing the determinant of the Fredholm integral operator of the electric field with respect to the permittivity profile of the scattering element. We apply the method to design subwavelength gradient-permittivity structures with multiple scattering resonances and quality factors exceeding 500. We also find the spectral scattering cross sections are consistent with Fano lineshapes. The compact form and computational efficiency of our formalism suggest it can be a useful tool for designing Fano-resonant structures with multiple high-$Q$ resonances for applications such as frequency mixing and conversion.