Resonant-state expansion and approximations for treating perturbations in the medium surrounding an optical system

Abstract

Developing a rigorous perturbation theory to treat changes in open optical systems may present a challenging task due to the leaky nature of such systems, in which the conventional conditions of Hermitian systems become inapplicable. Definitions of orthonormality and completeness have been reformulated for a special class of resonances, popularly known as resonant states or quasi-normal modes. Such resonances have been employed to develop modal expansion techniques for treating perturbations in open systems, such as the resonant-state expansion method (RSE). The RSE transforms Maxwell's equations into a linear eigenvalue problem, producing the perturbed modes with high accuracy and quick convergence. A major limitation of the RSE is that perturbations must be taken within the system where the resonant states are complete. In this thesis, we develop a rigorous theory based on the RSE to treat perturbations in the surrounding medium. This is achieved by performing a transformation to Maxwell's equations, which maps perturbations of the surrounding medium onto effective perturbations within the system where the resonant states are complete. By treating such perturbations rigorously with the RSE, we find the perturbed modes of the system for arbitrary homogeneous perturbations of the surrounding medium with any desired accuracy. We further develop approximations beyond the first-order accuracy using only one mode. We apply the developed approaches to accurately predict the resonance shift of highly responsive modes due to changes in the surrounding medium. We provide a special treatment to chiral systems that have a weak impact on the resonance shift. Such an impact is enhanced by deriving a linear splitting of quasi-degenerate modes in chirality. We illustrate such a splitting using quasi-degenerate modes deduced from our analysis of spherical systems.