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Recent experiments have demonstrated the generation of widely spaced parametric sidebands that can evolve into "clustered" optical frequency combs in Kerr microresonators. Here we describe the physics that underpins the formation of such clustered comb states. In particular, we show that the phase matching required for the initial sideband generation is such that (at least) one of the sidebands experiences anomalous dispersion, enabling the sideband to drive frequency comb formation via degenerate and non-degenerate four-wave mixing. We validate our proposal through a combination of experimental observations made in a magnesium-fluoride microresonator and corresponding numerical simulations. We also investigate the coherence properties of the resulting clustered frequency combs. Our findings provide valuable insights on the generation and dynamics of widely spaced parametric sidebands and clustered frequency combs in Kerr microresonators.

Efficient evaluation of the potential field is an essential requirement for simulation of large ensembles of particles in many applications, including astrophysics, plasma physics, molecular dynamics, very large-scale integration systems and micro-electromechanical systems. Current methods use multipole expansion of spherical harmonics for the potential field, which is computationally expensive in terms of running time and memory requirements when a high degree of accuracy is desired. In this paper, a new approach is presented for efficient and rapid evaluation of the potential field in three dimensions. The mathematical background for the proposed approach stems from an exponential integral representation of Green's function, 1/r, and an approximation to the integral by using Gauss quadratures, which distinctively differs from the theory of spherical harmonics. The translations are simple in structure, error-free and independent of the approximation, which enables the overall accuracy and computational performance to be controlled externally via the approximation. In addition, the gradient of the potential can be readily retrieved as a by-product of the computational process. More importantly, the memory requirement is independent of the desired degree of accuracy. The technique presented here opens new possibilities for efficient distributed computing and parallel processing of large-scale simulation of particle systems.