  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.
