Since we have to review the calculation for Thermo-Optic noise (TO), I'll sketch an outline and some remarks here.
**==TO noise overview==**
To calculate TO noise, we have to calculate** temperature fluctuations**, then multiply by Thermoelastic (**TE) **and Thermorefractive (**TR)** coefficients to convert temperature fluctuation to displacement noise. Usually, in the frequency of interest, thermal length is much larger than coating thickness. Thermal fluctuations in coatings are uniform making the whole coatings expand/ contract uniformly. This assumption is important for cancellation between TE and TR. As TE effect comes from the whole coating thickness, while TR comes from only the first few layers (most of the power is reflected from these top layers). Modifying the first few layers can change TR effect significantly.
**==Temperature fluctuations**==
can be obtained from direct method (Levin 2008), by injecting heat with Gaussian beam profile. Example are done in Levin 2008, Evans etal 2008.
A few issues about these calculations:
**heat flow in 1-D**, under the assumption that temperature gradient is mostly in z direction coating thickness d << thermal length << beam radius. Where thermal length is ~ sqrt ( kappa/ (rho*C* 2pif) ) This is not true for AlGaAs coatings where kappa is ~ 60 W/mK which gives thermal length to be~ 2370 um [sqrt (1Hz/f)], beam radius is ~ 200um. Cerdonio 2001, and Mike Martin's thesis have the calculation in 3-D, however,** heat diffusion** in coatings is not taken into account.
**Heat diffusion in coatings**, is done in Fejer 2004, Somiya2009. (It is ignored in BGV1999/Liuthorne2000/cerdonio 2001)
At this point, I think Somiya paper is very good for us to look through. The calculation includes TE and TR. However, I don't quite get it yet. The calculation solve heat equation in 1-D, but has results for finite test mass. I need to spend more time on the paper.
Heinert 2011, has calculation for TR in finite size substrate. I'm not sure how to connect the results to our setup yet. Plus, for our setup, the actual coatings will be ~ 8mm in diameter, with 1" diameter substrate, the boundary conditions will be non-trivial for us.
** ==TE +TR coefficients==**
[ coming soon] |