6 Heating and Thermal Effects in the Dynamic Regime: Transfer Functions

We address here the case of fluctuations of the incident power due to either fluctuations of the laser itself, fluctuations of the locking system, or even fluctuations equivalent to shot noise. The incident power flux P (ω) is assumed to be of Fourier frequency f = ω ∕2π. The Fourier transform of the Heat equation (assuming a time dependence of all quantities in exp(iωt)) is

[ ] 2 1- 2 ρC-ω- ∂r + r∂r + ∂z − i K T(ω, r,z) = 0 (6.1 )
in the case of a heat source on the coating. We assume a separate solution in the form an FB expansion of the form
∑ T (ω,r,z ) = Ts(ω, z)J0(ksr), (6.2 ) s
where ks has the usual definition ks ≡ ζs∕a, in order to identically satisfy the radiation condition on the edge of the mirror. Then, the longitudinal function obeys
[ ] ∂2 − k2 − iρC-ω- T (ω, z) = 0 (6.3 ) z s K s
so that the general solution is of the form
(1) (2) Ts(ω,z ) = T s (ω )cosh(κsz) + Ts (ω )sinh(κsz) (6.4 )
∘ ----------- 2 ρC-ω- κs = κs(ω) ≡ k s + i K . (6.5 )

We also adopt the new dimensionless parameters ∘ ---------------- ξs ≡ κsa ≡ ζ2s + iρC ωa2 ∕K and ηs ≡ κsh ∕2. With this notation, and by setting the (unchanged since the beginning) boundary conditions, we can specialize the solution to be entirely analogous in form to the static solution.

 6.1 Temperature fields and thermal lensing
  6.1.1 Coating absorption
  6.1.2 Bulk absorption
 6.2 Equivalent displacement noise
  Asymptotic regime

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