Thermal noise computation in gravitational wave interferometers from first principles

We propose a universal method of computation of thermal noise in mirrors of gravitational wave interferometers based on first principles. We imagine a situation where a mirror is part of a Fabry-Perot cavity. The movement of the mirror's surface produces variation of the eigen frequency of the cavity, which is computed by evaluating the variation of the energy stored in cavity. We consider two particular examples: first, the thermal noise from a dielectric slab inside the Fabry-Perot cavity, and second, the polarization-dependent thermal noise in the folded cavity.