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 0 R 2n(ρ) are functions of ρ = r∕a, and that the first ones are
R00(ρ ) = 1, R02(ρ) = 2ρ2 − 1, R04(ρ) = 6ρ4 − 6ρ2 + 1 (3.166 )
corresponding respectively to the piston, curvature, and spherical aberration. A recurrence relation allows one to compute any order
0 -1[ 2 0 0 ] R2n(ρ) = n (2n − 1 )(2ρ − 1)R 2n−2 − (n − 1)R2n−2 . (3.167 )
An important relation is [6]
∫ 1 J (ζ ) R02n(ρ )J0 (ζ ρ) = (− )n--2n+1----, (3.168 ) 0 ζ
which allows one to immediately create the Zernike expansion of any thermal lens. Assume a thermal lens of the generic FB form
Z (r) = ∑ Z J (ζ r∕a). (3.169 ) s 0 s s
It can also be represented by a series of Zernike polynomials as
∑ Z(r) = cnR02n(r∕a ), (3.170 ) n
where the coefficients cn, n = 0,1,2... are given by
n ∑ J2n+1(ζs) cn = (− ) 2(2n + 1) Zs ---------. (3.171 ) s ζs

In the case of Ex1 (LG00, w = 2 cm), we give, in order to be specific, some values of cn (see Table 7). The number of terms for a good reconstruction of the lenses and of the surface is about 15. The c1 coefficient gives the mean curvature over the whole circular aperture of the mirror. Thus there is no relation between c1 and the curvature averaged and weighted by the intensity profile.


Table 7: Zernike coefficients cn for LG00 w = 2 cm
lensing: lensing: aberration: aberration:
heating heating heating heating
by bulk by coating by coating by bulk
n 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 LG00 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 LG55 mode (Ex3), and in Table 9 for the mesa mode (the figures are quite similar for the flat mode).


Table 8: Zernike coefficients cn for LG55 w = 3.5 cm
lensing: lensing: aberration: aberration:
heating heating heating heating
by bulk by coating by coating by bulk
n 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 cn for the mesa mode
lensing: lensing: aberration: aberration:
heating heating heating heating
by bulk by coating by coating by bulk
n 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


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