4.2 Simple model of a radiator

Now assume a compensation system based on a ring radiator of radius bC, at a distance DC from the face, placed behind the rear face of the mirror. Assume a total radiated power of PR. A small element of length dl of the ring (assumed very thin) radiates the elementary power dP = PC × dl∕2πbC. The radiated intensity at a distance ρ is
dP PRdl dI = ---2-= --2---2. (4.6 ) 4πρ 8π bCρ
Now the distance of the considered element, having the angular coordinate ϕ on the ring to any point A on the mirror surface of coordinates A (r,Φ), is such that
ρ2 = b2 + D2 + r2 − 2b rcos(ϕ − Φ ), (4.7 ) C C C
so that the global intensity at A, integrated on the ring, is
P ∫ 2π dϕ I(r) = --R2- -2-----2----2------------, (4.8 ) 8π 0 bC + D C + r + 2bC rcos ϕ
which gives
I (r ) = PR-∘---------------1-----------------. (4.9 ) 4π (b2C + D2C)2 − 2(b2C − D2C)r2 + r4
The fraction of the total radiated power, which is absorbed by the rear face of the mirror, is
( ⌊ ⌋) ∫ a { ∘ (-------2----2-)2----- 2 2 2 } ΔP = 2π I (r )rdr = 1P ln bC--⌈ a2 +-D-C-−-bC- + 1 + a--+-D-C-−-bC-⌉ , (4.10 ) 0 4 R ( DC 2bCDC 2bCDC )
(provided that bc > DC) This allows one to normalize the intensity distribution to PC integrated on the mirror:
I0 I(r) = PC ∘---2-----2-2------2-----2--2----4 (4.11 ) (bC + D C) − 2(bC − D C)r + r
( ⌊∘ ---------------------- ⌋) { ( 2 2 2 )2 2 2 2 } 1-= πln bC--⌈ a-+-D-C-−-bC-- + 1 + a--+-D-C-−-bC-⌉ (4.12 ) I0 ( DC 2bCDC 2bCDC )
See Figure 27View Image to get an idea of the intensity profile.
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Figure 27: Normalized intensity I(r) on the mirror rear face from a ring radiator

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

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Figure 28: Thermal lensing from a ring radiator. Red dashed curve: nearest paraboloid (weighted by the readout beam intensity). The readout beam is TEM00 with w = 2 cm. The curvature is weakly dependent on the beam width: 87 km W for w = 2 cm, 95 km W for w = 6.65 cm. Green dashed curve: Zernike expansion of the lens

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 29View Image in which one tries to compensate for the thermal lens caused by an LG00 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.

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Figure 29: Thermal compensation with a ring radiator: minimization of coupling losses

Table 10: Thermal compensation with a heating ring: LG0,0 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 30View Image 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:

c0 = 0.759 μm ∕W , c1 = 0.016μm ∕W , c2 = − 0.044 μm ∕W , c3 = − 0.011μm ∕W , c4 = 0.002 μm ∕W , c5 = 0.004μm ∕W . (4.13 )
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Figure 30: Ring radiator: correction of the thermal lensing caused by a TEM 00 beam of width w = 2 cm dissipating 100 mW on the coating. Dashed line: nearest paraboloid (flat for the optimal TCS power of 26.5 W)

It is probably possible to enhance these results up to a certain extent by tuning the parameters bC and DC, 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: LG5,5 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 ∞

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