Resonant state expansion for relativistic and non-relativistic wave equations

The resonant state expansion (RSE) is a rigorous perturbation method in electrodynamics which we recently invented and implemented.

The RSE uses the resonances of a system. These are a fundamental and powerful concept in physics, dealing with a countable number of states and thus offering a natural discretisation of the properties of the system. This is in contrast to the artificial grid used in other methods in order to discretise the material continuously distributed in space.

In optical systems, resonant states (RSs) are eigen-solutions of the Maxwell equation having outgoing wave boundary conditions. Their energies are generally complex, reflecting the fact that the excitations of the system decay in time, leaking to the environment. As a consequence of this leakage, RSs are characterised by exponentially growing tails outside the system that requires a modified normalisation.

Activities

We have applied the RSE to calculate the RSs in finite 1D, 2D and 3D systems, such as perturbed planar, cylindrical and spherical resonators. The method was shown to be particularly suited for the calculation of sharp resonances, such as WGMs in microcylinders and microspheres. This exact method is at least 2 orders of magnitude faster than the existing approximate electromagnetic solvers, such as COMSOL.

The RSE works for any finite strength of the perturbation and is capable to treat even huge ones. However, it becomes particularly efficient and quick once a relatively small perturbation is being treated. This is obvious from Fig.1 showing the contribution of different unperturbed RSs to a particular perturbed mode of a dielectric cylinder - the full spectrum is shown in Fig.2. This contribution decreases quickly with the distance to the spectral position of the perturbed mode, with the dominant component coming from the few nearest unperturbed RS. This demonstrates that if we are interested in the modes within a small spectral region, the RSE can work very efficiently also as a local perturbation theory which requires a very limited basis of RSs of that region.