Manipulating the quasi-normal modes of radially symmetric resonators

The frequency response of a resonator is governed by the locations of its quasi-normal modes in the complex frequency plane. The real part of the quasi-normal mode determines the resonance frequency and the imaginary part determines the width of the resonance. For applications such as energy harvesting and sensing, the ability to manipulate the frequency, linewidth and multipolar nature of resonances is key. Here, we derive two methods for simultaneously controlling the resonance frequency, linewidth and multipolar nature of the resonances of radially symmetric structures. Firstly, we formulate an eigenvalue problem for a global shift in the permittivity of the structure to place a resonance at a particular complex frequency. Next, we employ quasi-normal mode perturbation theory to design radially graded structures with resonances at desired frequencies.