First-order perturbation theory for material changes in the surrounding of open optical resonators

The single-mode approximation of the resonant state expansion has proven to give accurate first-order approximations of resonance shifts and linewidth changes when modifying the material properties inside open optical resonators. Here, we extend this first-order perturbation theory to modifications of the material properties in the surrounding medium. As a side product of our derivations, we retrieve the already known analytical normalization condition for resonant states. We apply our theory to two example systems: A metallic nanosphere and a one-dimensional photonic crystal slab.