Thermal detuning of a bichromatic narrow linewidth optical cavity

In the Advanced $\mathrm{Virgo}+$ interferometric gravitational-wave detector, the length control of the Fabry-Pérot cavities in the arms and of the detuned filter cavity, used for generating frequency-dependent squeezing, uses an auxiliary green beam at half of the operation laser wavelength (1064 nm). While operating the filter cavity with such a bichromatic control scheme for tens of hours, we observed that the mirror reflection phase shift of the fields at the two wavelengths responds differently to temperature changes in the mirrors, causing a change in the relative resonance condition of the two beams. In this paper we show that this thermal detuning effect can be explained by considering the thermomechanical properties of the mirror coating. Our experimental measurements are in good agreement with the theoretical predictions and allow us to drive requirements on the bicolor coating design and mirror temperature stability for long-term stable cavity control.