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,

(2) |

where
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
,
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
are given in columns 4 and 5 in the table below
(TTF values are 50% longer - need to update).

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 |