Thermo-optic bistability in 2D all-dielectric resonators

We consider thermo-optic bistability in resonant excitation of high-quality modes in two-dimensional dielectric resonators. We develop a coupled-mode theory approach which account for the frequency shift due to a temperature dependent dielectric permittivity. The model is applied to rectangular and hexagonal resonators supporting an isolated high-quality resonant mode. The results are verified in comparison with straightforward finite-element simulations. It is shown that the model accurately describes the effect bistabily which occurs under variation of the angle of incidence or the intensity of the incident wave. In particular, it is demonstrated that variation of the incident angle can optimize the coupling between the resonator and the incident waves leading to bistabily with low intensity incident waves $W_0 = 0.35 {\rm \mu W/\mu m}^2$. The bistability threshold is shown to be extremely sensitive to the imaginary part of the dielectric permittivity $\epsilon''$.