3 Heating and Thermal Effects in the Steady State

We consider here the direct effects resulting from absorption of light in the mirrors (either on the reflecting surface or in the bulk material) and the indirect effects, such as thermal lensing and thermal distortions. First, we discuss the steady state (the principles are given in [20]), then the quasi-static case (heating from an initial uniform temperature [21]), and finally the general dynamical case. We consider the case of cavity mirrors storing large optical power, heated partially by thermalization of light at the coated face. There is also heating by propagation losses inside the substrate. We assume thermal equilibrium by thermal radiation; the mirror being suspended by thin wires in a vacuum, there is no convection loss and we neglect conduction loss. We further assume a small relative excess of temperature, justified by the good quality of the coatings and of the bulk silica. With these assumptions, the problem becomes linear and we can treat separately the contribution to heating caused by the coating and the bulk substrate. We consider a cylindrical mirror of diameter 2a and of thickness h (see Figure 7View Image). The coordinates are radial 0 ≤ r ≤ a, azimuthal 0 ≤ φ ≤ 2π and longitudinal − h∕2 ≤ z ≤ h∕2.

View Image

Figure 7: Notations for a cylindrical mirror
 3.1 Steady temperature field
  3.1.1 Coating absorption
  3.1.2 Bulk absorption
  3.1.3 Fourier–Bessel expansion of the readout beam intensity
  3.1.4 Numerical results on temperature fields
 3.2 Steady thermal lensing
  3.2.1 Thermal lensing from coating absorption
  3.2.2 Thermal lens from bulk absorption
  3.2.3 Equivalent paraboloid
  3.2.4 Coupling losses
  3.2.5 Numerical results
 3.3 Thermal distortions in the steady state
  3.3.1 Thermal expansion from thermalization on the coating
  3.3.2 Thermal expansion from internal absorption
 3.4 Expansion on Zernike polynomials

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