Generalizing normal mode expansion of electromagnetic Green's tensor to lossy resonators in open systems
We generalize normal mode expansion of Green's tensor $\bar{\bar{G}}(\bf{r},\bf{r}')$ to lossy resonators in open systems, resolving a longstanding open challenge. We obtain a simple yet robust formulation, whereby radiation of energy to infinity is captured by a complete, discrete set of modes, rather than a continuum. This enables rapid simulations by providing the spatial variation of $\bar{\bar{G}}(\bf{r},\bf{r}')$ over both $\bf{r}$ and $\bf{r}'$ in one simulation. Few eigenmodes are often necessary for nanostructures, facilitating both analytic calculations and unified insight into computationally intensive phenomena such as Purcell enhancement, radiative heat transfer, van der Waals forces, and Förster resonance energy transfer. We bypass all implementation and completeness issues associated with the alternative quasinormal eigenmode methods, by defining modes with permittivity rather than frequency as the eigenvalue. We obtain true stationary modes that decay rather than diverge at infinity, and are trivially normalized. Completeness is achieved both for sources located within the inclusion and the background through use of the Lippmann-Schwinger equation. Modes are defined by a linear eigenvalue problem, readily implemented using any numerical method. We demonstrate its simple implementation on COMSOL Multiphysics, using the default inbuilt tools. Results were validated against direct scattering simulations, including analytical Mie theory, attaining arbitrarily accurate agreement regardless of source location or detuning from resonance.