Lightning-Fast Simulation of Open System: an Effortless Modal Approach
A key goal of nanoscale quantum optics is enhancing the weak interaction of light with matter, enabling applications such as spectroscopy on individual molecules, generating single photons for quantum computing, solar energy harvesting, passive radiative cooling, and nanoscale optical forces. Progress is driven by research on optical nanoresonators, such as metallic nanoparticles. These structures induce intense field hotspots able to enhance light-matter interactions by 10 orders of magnitude, but simultaneously leading to a challenging and laborious simulation and design process. Typically, the nanoresonator response needs to be simulated and integrated over a volume of source positions and orientations, requiring a prohibitive number of simulations. Simulations are typically performed on COMSOL Multiphysics® via Frequency Domain studies using the RF Module, repeating the simulation for each source position and orientation.
A well-known alternative is first to use an Eigenfrequency Study in the RF Module, to find the natural modes of the structure without any excitation. This provides a set of basis modes capable of representing the fields excited by any arbitrary collection of sources, generating each subsequent simulation almost instantaneously. The initial time investment for finding the modes is often more than repaid. This powerful technique is known as normal mode expansion and is applicable to many different partial differential equations, not just Maxwell's equations. In fact, it is already implemented in COMSOL Multiphysics® as the Frequency Domain Modal study of the RF Module. However, normal mode expansion is valid only for closed resonators, such as microwave cavities, and not for open systems such as photonic nanoresonators, where energy is constantly radiating to infinity.
We present a successful generalization of normal mode expansion to either lossy or lossless resonators in open systems, while retaining all of its simplicity and rigor. For example, we do not experience the exponentially diverging fields usually associated with modal methods in open systems, and it is valid everywhere over infinite space. Our key innovation is remarkably simple: instead of expanding using eigenfrequency modes, we employ eigenpermittivity modes, corresponding to modes with a range of potential depths. Finding these modes is simple on COMSOL Multiphysics® using an eigenfrequency study. Although this study was designed for eigenfrequency modes, we repurpose it with one simple substitution trick, requiring less than two minutes of implementation. To complete the modal expansion, minor post-processing is required, such as weighting each mode by its eigenvalue, evaluating each mode at source locations, and summing, tasks that we perform using LiveLink™ for MATLAB®. The computed results agree well with a direct simulation using a Frequency Domain study, and against Mie scattering theory. One additional advantage is that the method is capable of generating modes of interacting nanoresonators without any further COMSOL Multiphysics® simulation, by applying post-processing to the previously calculated modes of the individual resonators.
Our modal expansion method has attracted significant research interest, and we are actively collaborating with other research groups, simulating coupled resonators for quantum optical effects such as two-photon emission, scattering from anisotropic hyperbolic metamaterial rods, and thermal emission from non-reciprocal resonators.
