Thermal time constant for the TTF

All of the etalon parts in Fig. 2 are essentially cylindrical optical flats. For each component, the heat transfer as a function of position in cylindrical coordinates (r,z) is given by the diffusion equation,

$${{\partial^2 T}\over{\partial r^2}} + {{1}\over{r}}{{\partial... ...ial z^2}} = {{1}\over{\alpha}} {{\partial T}\over{\partial t}}$$ (2)

where $\alpha$ is the ratio of the thermal conductivity to the thermal capacity. For purely radial heat flow, i.e., ignoring the z dependence, the thermal equation with the most elementary boundary conditions has a complicated solution involving Bessel functions (Heisler 1947). The cylindrical plates exhibit an exponential cooling rate with a thermal time constant $\tau \approx L^2/\alpha$, where L is a characteristic length given by the ratio of the cylindrical volume to the surface area. A crude finite-difference solution to equation (A1) also exhibits exponential behavior with a similar time constant. The estimated values of L and $\tau$ are given in columns 4 and 5 in the table below (TTF values are 50% longer - need to update).

Table 1. Thermal time constants of a 50mm aperture etalon assembly

optical	radius	thickness	characteristic	time
flat	(mm)	(mm)	length	constant (min)
capacitor	7	15	4.8	7
central pillar	25	15	9.4	24
outer plate	40.5	19	12.9	50
pillar+plate	25-40	34	24.8	184