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We present an analytical model for the dynamical self-heating effect in air-cladded optical microring resonators (ORRs). The spatially and time resolved temperature field is calculated by integrating the corresponding boundary value problem of the heat equation. It turns out that the self-heating amplitude is approximately proportional to the total absorbed power and anti-proportional to the thermal conductivity of the cladding material. Further, two-photon absorption plays a major role in the heating process, even for moderate input powers, due to the strong light confinement. Heating times are determined to be in the microsecond range and may limit the response time of ORR devices. The explicit formulas for the temperature fields allow a much faster determination of heating properties compared to elaborate finite element simulations. Thus, our model is predestinated for scanning large parameter spaces. We present such an analytical model for the self-heating effect in ORRs. For this purpose, we solve the heat equation on the ORRs geometrical domain. The heat source is caused by two effects, linear absorption from defect states and quadratic two-photon absorption (TPA). Due to the strong light confinement on resonance, very high light intensities are reachable in the resonator ring and the TPA might become a dominant heat source even for low excitation powers. We utilize insulating Neumann boundary conditions to calculate the temperature increase in the substrate region as a convolution between heat source and the corresponding Greens function. The temperature field in the ring structure is calculated by solving the corresponding eigenvalue problem that arises from a separation ansatz. The result is discussed in terms of maximum self-heating, response time and power dependence for ORRs with very high Q-factors of over 100 000. Finally, we compare the analytical calculations of the self-heating effect with finite element computations.
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