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A study of the phototermal instabilities in a Fabry-Perot cavity is reported, where one mirror consists of a silicon-nitride membrane coated by the molecular organic semiconductor tris(8-hydroxyquinoline) aluminum and silver layers. We propose a theoretical model to describe the back-action associated with the delayed response of the cavity field to the radiation pressure force and the photothermal force. For the case under investigation, the photothermal force response occurs on a timescale that is comparable to that of mirror oscillations and dominates over the radiation pressure force. A phase diagram analysis has been performed to map the stability of the static solution as a function of the control parameters. The model equations are integrated numerically and the time history is compared to experimental measurements of the transmitted field and displacement of the membrane. In both experimental and theoretical data we observe large amplitude oscillations when the cavity length is scanned at a low speed compared to the growth rate of the instability. The perturbation is found to evolve through three regimes: sinusoidal oscillations, double peaks and single peaks followed by a lethargic regime. When the cavity length is scanned in opposite directions, dynamical hysteresis is observed, whose extension has a power law dependence on the scanning rate

Model and phase-diagram analysis of photothermal instabilities in an optomechanical resonator