Application of resonant-state expansion to inhomogeneous and non-spherical optical resonators

Resonances determine the optical properties of an object, such as its transmittance, scattering cross-section, or local field enhancement. Resonant states (RSs) provide a physically intuitive way to describe these features and a range of related physical phenomena. In this project both the resonances of dielectric optical systems and the method called resonant-state expansion (RSE) is studied. The RSE is a perturbative approach to calculate RSs, capable of treating perturbations of arbitrary strength. It expresses the RSs of the perturbed system in the basis of the RSs of a known, unperturbed system, transforming the problem of solving Maxwell's equation to find the resonances into a linear matrix eigenvalue problem.

The following concepts are studied in detail: the completeness of the eigenmodes, the convergence rate of the Mittag-Leffler (ML) expansion of the Green's function (GF), as well as the convergence of sum rules that the eigenmodes satisfy, and limits of validity are established. The modes of spherically symmetric, radially inhomogeneous resonators are studied, and found that by appropriately engineering the permittivity gradient one can achieve quasi-degeneracy of the transverse-electric and transverse-magnetic whispering gallery modes. Based on the RSE, a first-order perturbation theory is derived that can treat material and shape changes of arbitrary resonators with the same formalism, and a remarkable phenomenon is discovered when all orders of the standard perturbation series can contribute linearly in the change of a small parameter. The RSE is applied to non-spherical systems, and additional divergent terms in the ML expansion of the GF are revealed affecting the RSE convergence. Finally, a RSE based approach, which links the GF to the scattering matrix, is applied for calculation of the cross-section of cylindrical dielectric resonators. This method can potentially supersede the computational efficiency of many other existing methods, as it does not require to solve Maxwell's equation across all space, and it also does not require overlap volume integrals between the mode fields and the excitation wave, which are used in other methods.