  Strong light-matter coupling has recently been demonstrated in sub-wavelength
volumes by coupling engineered optical transitions in semiconductor
heterostructures (e.g., quantum wells) to metasurface resonances via near
fields. It has also been shown that different resonator shapes may lead to
different Rabi splittings, though this has not yet been well explained. In this
paper, our aim is to understand the correlation between resonator shape and
Rabi splitting, and in particular determine and quantify the physical
parameters that affect strong coupling by developing an equivalent circuit
network model whose elements describe energy and dissipation. Because of the
subwavelength dimension of each metasurface element, we resort to the
quasi-static (electrostatic) description of the near-field and hence define an
equivalent capacitance associated to each dipolar element of a flat
metasurface, and we show that this is also able to accurately model the
phenomenology involved in strong coupling between the metasurface and the
intersubband transitions in quantum wells. We show that the spectral properties
and stored energy of a metasurface/quantum-well system obtained using our model
are in good agreement with both full-wave simulation and experimental results.
We then analyze metasurfaces made of three different resonator geometries and
observe that the magnitude of the Rabi splitting increases with the resonator
capacitance in agreement with our theory, providing a phenomenological
explanation for the resonator shape dependence of the strong coupling process.
