  Dielectric metasurfaces supporting quasi-bound states in the continuum
(qBICs) enable high field enhancement with narrow-linewidth resonances in the
visible and near-infrared ranges. The resonance emerges when distorting the
meta-atom's geometry away from a symmetry-protected BIC condition and, usually,
a given design can sustain one or two of these states. In this work, we
introduce a silicon-on-silica metasurface that simultaneously supports up to
four qBIC resonances in the near-infrared region. This is achieved by combining
multiple symmetry-breaking distortions on an elliptical cylinder array. By
pumping two of these resonances, the nonlinear process of degenerate four-wave
mixing is experimentally realized. By comparing the nonlinear response with
that of an unpatterned silicon film, the near-field enhancement inside the
nanostructured dielectric is revealed. The presented results demonstrate
independent geometric control of multiple qBICs and their interaction trough
wave mixing processes, opening new research pathways in nanophotonics, with
potential applications in information multiplexing, multi-wavelength sensing
and nonlinear imaging.
