### 4.4 CO2 laser compensation by scanning

By scanning the rear face with a powerful CO2 laser, it is possible, in principle, to obtain any given intensity profile. In particular, it is possible to obtain an intensity distribution giving a flat thermal lens over a large central part of the mirror. Consider an intensity profile of the form
depending on the only parameter and normalized to W (see Figure 33 for the case of  = 16.9 cm).

The resulting thermal lens has a perfect flatness in the central region. The goal is to create a profile such that, combined with the readout beam heat source, it gives that ideal profile. Consider, for instance, an intensity mask of the form

see the profile on Figure 34. This is nothing but the complement to a source of heat corresponding to a mode dissipating W on the coating. The resulting global thermal lens (thus corrected) can be seen in Figure 35.

The price to pay is to provide the correcting power. According to Equation (4.15), the integrated power of the mask is

Table 13: Thermal compensation with a scanning CO2 beam for mode (w = 2 cm)
 dissipated power initial losses compensation power minimal losses wavefront curvature 10 mW 350 ppm 1.9 W 0.7 ppm 20 mW 1,400 ppm 3.8 W 3 ppm 30 mW 3,100 ppm 5.6 W 6.4 ppm 100 mW 34,300 ppm 18.8 W 71 ppm

We see in this rather academic case (Table 13) that the residual losses are much less than in the preceding case (by a factor of about 20), but at the price of higher TCS power. The expansion in terms of Zernike polynomials is given in Table 14.

Table 14: Zernike coefficients for three TCS systems compensating mode (w = 2 cm)
 heating ring Axicon CO2 scan µm/W µm/W µm/W 0 0.759 0.018 0.774 1 0.016 –0.008 –0.058 2 –0.044 0.003 –0.016 3 –0.012 0.001 0.001 4 0.002 –0.001 –0.002 5 0.004 0.001 0.002 6 0.002 0 –0.001 7 0 0 0.001 8 –0.001 0 –0.001