Temperature effects in narrow-linewidth optical cavity control with a surrogate quasi-second-harmonic field

Fabry-Perot cavities are widely used in precision interferometric applications. Various techniques have been developed to achieve the resonance condition via the direct interrogation of the cavity with the main laser field of interest. Some use cases, however, require a surrogate field for cavity control. In this study, we construct a bichromatic cavity to study the surrogate control approach, where the main and the surrogate fields are related by the second-harmonic generation with nonlinear optics. We experimentally verify the temperature dependence of the differential reflection phase of a dielectric coating design optimized for the surrogate control approach of the optical cavities of the light-shining-through-a-wall experiment Any Light Particle Search II and develop a comprehensive cavity model for quasi-second-harmonic resonances that considers also other important factors, such as the Gouy phase shift, for a detailed analysis of the surrogate control approach.