4.1 Heating the rear face of a mirror

Technical issues make it difficult to install any device in front of the cavity input mirrors. Thus, it has generally been proposed that one heat the rear face of these mirrors, which is more accessible from the central part of the vacuum tank. The heating is due to infrared radiation produced either by a hot material or by a CO2 auxiliary laser, but in both cases, the wavelength of the heating radiator is such that absorption by the silica substrate occurs in a thin layer. If we assume another heat source located on the rear face of a mirror, we must modify the model developed above. The extension is straightforward. For instance, let us consider the case in which the thermal lens to be compensated for is caused by thermalization on the coating of the intracavity-stored power P. As usual, we denote by ps the FB coefficients related to the main readout beam and by P and pC,s the FB coefficients of the power distribution of the compensation system, radiating an integrated power PC . We get the following temperature field
∑ T (r,z) = T (z)J (k r) (4.1 ) s 0 s s
with
{ [ ] Ts(z) = --1--------------1------------ × e− γs (ζs − χ)eksz−2γs + (ζs + χ)e−ksz P ps πaK (ζs + χ)2 − (ζs + χ)2e−4γs +e −3γs [(ζs + χ )eksz+2γs + (ζs − χ)e− ksz]PC pC,s} . (4.2 )
(For brevity, we have set ks ≡ ζs∕a and γs ≡ ksh∕2, with ζs defined as before). By integrating Equation (4.2View Equation) on the thickness, we get the thermal lens
dn ∫ h∕2 Z(r) = --- T (r,z)dz, (4.3 ) dT −h∕2
which yields
∑ Z(r) = ZsJ0(ksr) (4.4 ) s
with
Z = dn- -1-sinh-γsP-ps +-PC-pC,s (4.5 ) s dT πK ζs d1,s
with d1,s = ζssinh γs + χ coshγs. As expected, this shows that the global thermal lens is simply the sum of the primary lens (caused by the stored light) plus the compensation lens. Thus, it is possible to imagine power profiles compensating for the primary lens. This is the starting point for thermal compensation systems.
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