We are currently wondering why the 1550 nm light is not co-resonant with the 775 nm light, used to lock the GQuEST Output Filter Cavities, over time. Some ideas are the AOM frequency drifts (unlikely), but a potential culperate is differential length changes of the cavity due to thermooptic effects (thermal expansion and thermo-refraction). For light that hits the surface of the coating, Evans (2008) (Eq. 3) gives the sensitivity of the sensed position of the mirror \Delta T to temperature fluctuations T as
\[ \frac{\partial \Delta z}{\partial T} = \bar{\alpha}_c h_c - \bar{\beta}\lambda \]
Where \bar{\alpha}_c is the effective coefficient of thermal expansion, h_c is the coating thickness, \bar{\beta} is the effective thermo-refractive coefficient, and \lambda is the wavelength. This expresion doesn't work for the 775 nm light since it is below a layer of 1550 nm coating. Assume a fraction (\gamma) of the coating is for 1550 nm light and (1-\gamma) of the coating is for 775 nm light. \gamma \approx 0.5, although the 775 nm light coating is probably thinner since the fractional wavelength stacks are thinner. This analysis might also not be exact for the mirrors since none of them are HR for both wavelengths.
\[ \frac{\partial \Delta z_{775~\text{nm}}}{\partial T} = (1-\gamma)\bar{\alpha}_c h_c - \gamma\bar{\alpha}_c h_c(n-1) - \bar{\beta}( \gamma h_c+\lambda) \]
Where n is the index of refraction of 775 nm light in the 1550 nm coating. There is a minus sign in front of it because a higher index causes more phase to be accumulated than in air.
Now consider the differential sensitivity of the sensed position of the mirror, assuming the thermorefractive coefficient is very roughly the same for both wavelengths.
\[ \Delta \frac{\partial \Delta z}{\partial T} \equiv \frac{\partial \Delta z_{1550~\text{nm}}}{\partial T} - \frac{\partial \Delta z_{775~\text{nm}}}{\partial T}\]
\[ \Delta \frac{\partial \Delta z}{\partial T} = \gamma h_c (\bar{\alpha}_c n + \bar{\beta}) \]
Plugging in numbers from our GQuEST Paper Table II,
\[ \Delta \frac{\partial \Delta z}{\partial T} \approx 2 \cdot 10^{-10}~\text{m/K} \]
I'm not sure if ~1/1000 of a wavelength can explain the drift in the co-resonance over time.
Using Delta f /f = Delta l / L, where f is the frequency of the light and L is the length of the cavity, we get 16 kHz / K, not enough to explain the overnight drift of ~ 1 MHz. Lee points out that this temperature dependence might be enhanced by the quarter and half wavelength stacks.