### 4.2 Simple model of a radiator

Now assume a compensation system based on a ring radiator of radius , at a distance from the face, placed behind the rear face of the mirror. Assume a total radiated power of . A small element of length of the ring (assumed very thin) radiates the elementary power . The radiated intensity at a distance is
Now the distance of the considered element, having the angular coordinate on the ring to any point A on the mirror surface of coordinates , is such that
so that the global intensity at A, integrated on the ring, is
which gives
The fraction of the total radiated power, which is absorbed by the rear face of the mirror, is
(provided that ) This allows one to normalize the intensity distribution to integrated on the mirror:
with
See Figure 27 to get an idea of the intensity profile.

The FB coefficients of can easily be numerically computed. The resulting profile of the thermal lens is shown for a particular case in Figure 28.

The power in the compensation lens must be adjusted in order to get zero curvature when added to the heat source coming from the readout beam. For small readout-beam dissipation (less than a few mW), this minimizes the matching losses. An example is given in Figure 29 in which one tries to compensate for the thermal lens caused by an mode of width w = 2 cm dissipating either 10, 20 or 30 mW on the coating. One sees that, by increasing the compensation power, it is possible to reduce the coupling losses from their initial (uncompensated) values by a factor of about 15 (see Table 10). However, it can be seen that the residual loss is proportional to the dissipated power, which means that, in the case of cavities storing about 1 MW, even with thermalization rates on the order of 1 ppm, the system is useless. For 100 mW dissipated, one can see that the TCS is able to reduce the initial losses of almost 3.5% to about 0.24 %, which is still much too high, and at the price of 26.5 W TCS power dissipated.

Table 10: Thermal compensation with a heating ring: mode of w = 2 cm.
 dissipated power initial losses compensation power minimal losses wavefront curvature 10 mW 350 ppm 2.7 W 24 ppm 20 mW 1,400 ppm 5.3 W 96 ppm 30 mW 3,100 ppm 8.0 W 220 ppm 100 mW 34,300 ppm 26.5 W 2392 ppm

In the latter case, we see in Figure 30 that even if the mean curvature radius is infinite, it does not mean that the lens is exactly flat, so that, even if the focusing effect is suppressed, some losses are to be expected due to departure of the lens from a plane (or from a sphere having a large radius). Only six Zernike polynomials are required to retrieve this special TCS lens:

It is probably possible to enhance these results up to a certain extent by tuning the parameters and , but not to seriously change the orders of magnitude. However, results are better with higher-order modes (see Table 11).

Table 11: Thermal compensation with a heating ring: mode of w = 3.5 cm.
 dissipated power initial losses compensation power minimal losses wavefront curvature 10 mW 56 ppm 50 mW 6 ppm 20 mW 213 ppm 100 mW 11 ppm 30 mW 474 ppm 130 mW 15 ppm 100 mW 5,218 ppm 450 mW 122 ppm