The deployment of bottom-up and top-down nanotechnologies has led to a recent proliferation of new generations of optical micro/nano-resonators, achieving an unprecedented level of control on light-matter interaction on micro/nanometer scales. Optical resonances now play a leading role in nanophotonics and impact many areas of modern photonics, from quantum information processing, optical metamaterials to optical sensors. Classically, the resonant interaction of light with optical resonances is modeled via continuum scattering theory with Maxwellâ€™s equation solvers, operating either in the real frequency domain, or in the time domain. The analysis requires many repetitions: it must be repeated for every individual frequency in the frequency-domain, and for every instance of the driving field in the time-domain, e.g. the pulse duration, polarization, incidence angle [book]. Additionally, interpretation is not straight since the physics at hand, often the excitation of a few dominant resonance modes, is only indirectly accessed from the computation. The project RESONANCE elaborate a theoretical and numerical formalism that is expected to lift usual deficiencies encountered with general purpose methods for analyzing optical resonators. By placing the natural modes of the resonator at the heart of the analysis, the formalism puts the emphasis on analyticity, offers modelling capabilities with unprecedented computational performance, and highlights the physics. It is expected to bring valuable input, comparable to that brought by waveguide mode theory to integrated devices and circuits, in further creative designs in resonance optics.

First, **we will challenge the frontiers of the theoretical knowledge about resonators**.
We will try to answer fundamental questions about the nature of the radiation QNMs belonging to the continuum,
try to understand mathematically under what condition leaky and radiation QNMs may form a complete set, and explore the critical factors limiting their calculation by discretizing finite spaces.

**Second, we will elaborate rigorous formulations based on finite-element methods (FEM) to calculate
and normalize QNMs for arbitrary geometries**.
The entire effect of material dispersion will be taken into account by auxiliary fields and all the QNMs will be calculated with a single diagonalization, just like for non-dispersive media. We will implement the FEM formulations and release efficient open-access QNM Codes to calculate QNMs for the most general case of materials with absorption, frequency dispersionâ€¦

**Finally, we will promote the QNM Codes by benchmarking the software on generic examples of nanophotonics against well-established fully-vectorial approaches, such as FDTD and FEM**.
Due to analyticity, we expect to reach computational speeds that are several orders of magnitude faster than those of current solvers, while maintaining a sufficient accuracy. We will also develop post-processing toolboxes, which will help the users become familiar with the released software.