Use of Zernike polynomials for efficient estimation of orthonormal aberration coefficients over variable noncircular pupils

Hanshin Lee

doi:10.1364/OL.35.002173

Optics Letters
Vol. 35,
Issue 13,
pp. 2173-2175
(2010)
•https://doi.org/10.1364/OL.35.002173

Use of Zernike polynomials for efficient estimation of orthonormal aberration coefficients over variable noncircular pupils

Hanshin Lee

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Hanshin Lee, "Use of Zernike polynomials for efficient estimation of orthonormal aberration coefficients over variable noncircular pupils," Opt. Lett. 35, 2173-2175 (2010)

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Abstract

An efficient way of estimating orthonormal aberration coefficients on variable noncircular pupils is proposed. The method is based on the fact that all necessary pieces of information for constructing orthonormal polynomials (via the Gram–Schmidt process) can be numerically obtained during a routine least-squares fit of Zernike polynomials to wavefront data. This allows the method to use the usual Zernike polynomial fitting with an additional procedure that swiftly estimates the desired orthonormal aberration coefficients without having to use the functional forms of orthonormal polynomials. It is also shown that the method naturally accounts for the pixelation effect of pupil geometries, intrinsic to recording wavefront data on imaging sensors (e.g., CCDs), making the coefficient estimate optimal over a given pixelated pupil geometry. With these features, the method can be ideal for real-time wavefront analysis over dynamically changing pupils, such as in the Hobby–Eberly Telescope (HET), which is otherwise inefficient with analytic methods used in past studies.

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Equations (9)

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i	True $β_{i}$	Method X	Method Y	$\| X - Y \|$
1	$- 0.0076$	$- 0.0076$	$- 0.0072$	$0.434 \times 10^{- 3}$
2	$- 0.1133$	$- 0.1134$	$- 0.1134$	$0.051 \times 10^{- 3}$
3	$- 0.0233$	$- 0.0234$	$- 0.0235$	$0.055 \times 10^{- 3}$
4	0.2741	0.2740	0.2741	$0.103 \times 10^{- 3}$
5	$- 0.0501$	$- 0.0502$	$- 0.0502$	$0.012 \times 10^{- 3}$
6	0.1706	0.1706	0.1706	$0.023 \times 10^{- 3}$
7	$- 0.1108$	$- 0.1110$	$- 0.1107$	$0.247 \times 10^{- 3}$
8	$- 0.0409$	$- 0.0411$	$- 0.0411$	$0.025 \times 10^{- 3}$
9	$- 0.1064$	$- 0.1062$	$- 0.1062$	$0.070 \times 10^{- 3}$
10	$- 0.0369$	$- 0.0365$	$- 0.0365$	$0.001 \times 10^{- 3}$
11	$- 0.0648$	$- 0.0648$	$- 0.0645$	$0.310 \times 10^{- 3}$
12	$- 0.0057$	$- 0.0058$	$- 0.0058$	$0.063 \times 10^{- 3}$
13	$- 0.1070$	$- 0.1071$	$- 0.1072$	$0.109 \times 10^{- 3}$
14	$- 0.0750$	$- 0.0748$	$- 0.0749$	$0.111 \times 10^{- 3}$
15	0.0464	0.0466	0.0466	$0.018 \times 10^{- 3}$
$σ_{W}^{2}$	0.1704	0.1703	0.1704	$0.034 \times 10^{- 3}$

$i, j$	$D_{i, j}$	${\hat{D}}_{i j}$	$i, j$	$D_{i j}$	${\hat{D}}_{i j}$
01,01	1.000	1.000	08,08	1.214	1.214
02,02	1.025	1.025	09,02	$- 0.006$	$- 0.003$
03,03	1.022	1.022	09,07	$- 0.006$	$- 0.003$
04,01	0.109	0.108	09,09	1.068	1.068
04,04	1.355	1.355	10,03	$- 0.008$	$- 0.011$
05,05	1.106	1.106	10,08	$- 0.007$	$- 0.009$
06,01	$- 0.004$	$- 0.004$	10,10	1.385	1.385
06,04	$- 0.009$	$- 0.008$	11,01	0.827	0.827
06,06	1.105	1.104	11,04	0.098	0.096
07,02	0.495	0.493	11,06	$- 0.007$	$- 0.006$
07,07	1.221	1.221	11,11	1.632	1.634
08,03	0.484	0.484	—	—	—

i	True $β_{i}$	Method X	Method Y	$\| X - Y \|$
1	$- 0.0076$	$- 0.0076$	$- 0.0072$	$0.434 \times 10^{- 3}$
2	$- 0.1133$	$- 0.1134$	$- 0.1134$	$0.051 \times 10^{- 3}$
3	$- 0.0233$	$- 0.0234$	$- 0.0235$	$0.055 \times 10^{- 3}$
4	0.2741	0.2740	0.2741	$0.103 \times 10^{- 3}$
5	$- 0.0501$	$- 0.0502$	$- 0.0502$	$0.012 \times 10^{- 3}$
6	0.1706	0.1706	0.1706	$0.023 \times 10^{- 3}$
7	$- 0.1108$	$- 0.1110$	$- 0.1107$	$0.247 \times 10^{- 3}$
8	$- 0.0409$	$- 0.0411$	$- 0.0411$	$0.025 \times 10^{- 3}$
9	$- 0.1064$	$- 0.1062$	$- 0.1062$	$0.070 \times 10^{- 3}$
10	$- 0.0369$	$- 0.0365$	$- 0.0365$	$0.001 \times 10^{- 3}$
11	$- 0.0648$	$- 0.0648$	$- 0.0645$	$0.310 \times 10^{- 3}$
12	$- 0.0057$	$- 0.0058$	$- 0.0058$	$0.063 \times 10^{- 3}$
13	$- 0.1070$	$- 0.1071$	$- 0.1072$	$0.109 \times 10^{- 3}$
14	$- 0.0750$	$- 0.0748$	$- 0.0749$	$0.111 \times 10^{- 3}$
15	0.0464	0.0466	0.0466	$0.018 \times 10^{- 3}$
$σ_{W}^{2}$	0.1704	0.1703	0.1704	$0.034 \times 10^{- 3}$

$i, j$	$D_{i, j}$	${\hat{D}}_{i j}$	$i, j$	$D_{i j}$	${\hat{D}}_{i j}$
01,01	1.000	1.000	08,08	1.214	1.214
02,02	1.025	1.025	09,02	$- 0.006$	$- 0.003$
03,03	1.022	1.022	09,07	$- 0.006$	$- 0.003$
04,01	0.109	0.108	09,09	1.068	1.068
04,04	1.355	1.355	10,03	$- 0.008$	$- 0.011$
05,05	1.106	1.106	10,08	$- 0.007$	$- 0.009$
06,01	$- 0.004$	$- 0.004$	10,10	1.385	1.385
06,04	$- 0.009$	$- 0.008$	11,01	0.827	0.827
06,06	1.105	1.104	11,04	0.098	0.096
07,02	0.495	0.493	11,06	$- 0.007$	$- 0.006$
07,07	1.221	1.221	11,11	1.632	1.634
08,03	0.484	0.484	—	—	—

Abstract

Supplementary Material (1)

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Equations (9)

Optics Letters