Resonant states and their role in nanophotonics

Resonant phenomena have been extensively used in micro- and nanophotonics. Mathematically, these phenomena originate in a discrete set of basis functions known as resonant states or quasi-normal modes. Therefore, it is extremely beneficial to develop theoretical approaches that use these resonant states as a physically meaningful basis in order to describe the light-matter interaction in micro- and nanoresonators. However, the question of how to normalize resonant states correctly for such an expansion initially hampered many theoretical attempts. Only recently, this problem of normalization has been solved via different approaches, providing a completely rigorous basis for not only explaining but also quantifying a large variety of resonant phenomena. This review article provides an overview of the related activities in the field and typical applications. We compare the different approaches with a focus on formulations via the Mittag-Leffler expansion of the Green's dyadic on the complex frequency plane and an analytic normalization scheme for the resonant states. Specifically, we discuss the pole expansion of the near and far field and outline related theoretical tools such as the resonant-state expansion and first-order perturbation theories. These approaches allow for efficiently describing light-matter interaction between local emitters and resonators, scattering of light at nanoparticles, and resonantly-enhanced optical sensing. Moreover, the resulting equations provide insight into the underlying physical mechanisms, which can be used to tailor the light-matter interaction and to predict new phenomena such as the recently observed complex-valued mode volumes. Since the Mittag-Leffler theorem is valid beyond the continuation of physical quantities to the complex frequency plane, an introduction to alternative modal approaches, namely those based on permittivity eigenmodes and propagating modes, is included here as well. While the link of these approaches to resonant phenomena is less obvious, they can be advantageous in some cases. Finally, we show that modal theories can be even applied in nonlinear optics. Hence, the theory of resonant states provides a general theoretical framework in micro- and nanophotonics.