3.4 Expansion on Zernike polynomials

It is convenient to express the thermal lensing and the mirror distortions in terms of Zernike polynomials. This synthesizes the algebraic results already obtained and allows one to use the results, for instance, in optical simulation codes like the E2E by Caltech, Finesse at Birmingham (U.K.) and DarkF in Virgo. It is easy to find the coefficients of these polynomials. Recall that the axisymmetric Zernike polynomials are functions of , and that the first ones are
corresponding respectively to the piston, curvature, and spherical aberration. A recurrence relation allows one to compute any order
An important relation is [6]
which allows one to immediately create the Zernike expansion of any thermal lens. Assume a thermal lens of the generic FB form
It can also be represented by a series of Zernike polynomials as
where the coefficients are given by

In the case of Ex1 (, w = 2 cm), we give, in order to be specific, some values of (see Table 7). The number of terms for a good reconstruction of the lenses and of the surface is about 15. The coefficient gives the mean curvature over the whole circular aperture of the mirror. Thus there is no relation between and the curvature averaged and weighted by the intensity profile.

Table 7: Zernike coefficients for w = 2 cm
 lensing: lensing: aberration: aberration: heating heating heating heating by bulk by coating by coating by bulk µm/W µm/W nm/W nm/W 0 0.940 0.873 140.76 25.85 1 0.633 0.599 34.79 17.45 2 0.470 0.447 25.60 12.25 3 0.333 0.319 17.95 7.80 4 0.250 0.239 13.42 5.06 5 0.192 0.185 10.38 3.35 6 0.151 0.146 8.17 2.27 7 0.120 0.116 6.50 1.57 8 0.095 0.092 5.19 1.10 9 0.076 0.074 4.14 0.79 10 0.060 0.059 3.30 0.57 11 0.048 0.047 2.61 0.41 12 0.038 0.037 2.06 0.30 13 0.029 0.029 1.61 0.21 14 0.023 0.022 1.25 0.16

Reconstruction of an mode requires many polynomials due to the sharp and far-from-spherical power profile of the beam. For higher-order modes and a fortiori for a flat or mesa mode, the reconstruction is achieved with fewer polynomials. In Table 8 we give the results of our mode (Ex3), and in Table 9 for the mesa mode (the figures are quite similar for the flat mode).

Table 8: Zernike coefficients for w = 3.5 cm
 lensing: lensing: aberration: aberration: heating heating heating heating by bulk by coating by coating by bulk µm/W µm/W nm/W nm/W 0 0.817 0.865 16.48 24.21 1 0.229 0.242 13.00 6.64 2 0.044 0.046 2.67 1.38 3 0.019 0.019 1.058 0.42 4 0.021 0.022 1.198 0 5 0.002 0.003 0.135 0 6 0.006 0.006 0.314 0 7 0.001 0.001 0.047 0 8 0.002 0.002 0.091 0 9 0.006 0.007 0.356 0 10 0.006 0.006 0.315 0

Table 9: Zernike coefficients for the mesa mode
 lensing: lensing: aberration: aberration: heating heating heating heating by bulk by coating by coating by bulk µm/W µm/W nm/W nm/W 0 0.845 0.895 30.81 25.05 1 0.387 0.408 22.25 11.19 2 0.147 0.155 8.55 4.12 3 0.185 0.020 0 0 4 0.011 0.012 0 0 5 0.008 0.008 0 0