Zernike olivary polynomials for applications with olivary pupils

Yi Zheng; Shanshan Sun; Ying Li

doi:10.1364/AO.55.003116

Applied Optics
Vol. 55,
Issue 12,
pp. 3116-3125
(2016)
•https://doi.org/10.1364/AO.55.003116

Zernike olivary polynomials for applications with olivary pupils

Yi Zheng, Shanshan Sun, and Ying Li

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Yi Zheng, Shanshan Sun, and Ying Li, "Zernike olivary polynomials for applications with olivary pupils," Appl. Opt. 55, 3116-3125 (2016)

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Abstract

Orthonormal polynomials have been extensively applied in optical image systems. One important optical pupil, which is widely processed in lateral shearing interferometers (LSI) and subaperture stitch tests (SST), is the overlap region of two circular wavefronts that are displaced from each other. We call it an olivary pupil. In this paper, the normalized process of an olivary pupil in a unit circle is first presented. Then, using a nonrecursive matrix method, Zernike olivary polynomials (ZOPs) are obtained. Previously, Zernike elliptical polynomials (ZEPs) have been considered as an approximation over an olivary pupil. We compare ZOPs with their ZEPs counterparts. Results show that they share the same components but are in different proportions. For some low-order aberrations such as defocus, coma, and spherical, the differences are considerable and may lead to deviations. Using a least-squares method to fit coefficient curves, we present a power-series expansion form for the first 15 ZOPs, which can be used conveniently with less than 0.1% error. The applications of ZOP are demonstrated in wavefront decomposition, LSI interferogram reconstruction, and SST overlap domain evaluation.

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Corrections

4 May 2016: Corrections were made to Refs.6, 16, 17, and 20–23.

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$j$	$n$	$m$	$Z_{j} (ρ, θ)$	Aberration
1	0	0	1	Piston
2	1	1	$2 ρ \cos θ$	Tip (x tilt)
3	1	1	$2 ρ \sin θ$	Tilt (y tilt)
4	2	0	$\sqrt{3} (2 ρ^{2} - 1)$	Defocus
5	2	2	$\sqrt{6} ρ^{2} \sin 2 θ$	Oblique astigmatism
6	2	2	$\sqrt{6} ρ^{2} \cos 2 θ$	Vertical astigmatism
7	3	1	$\sqrt{8} (ρ^{3} - 2 ρ) \sin θ$	Vertical coma
8	3	1	$\sqrt{8} (ρ^{3} - 2 ρ) \cos θ$	Horizontal coma
9	3	3	$\sqrt{8} ρ^{3} \sin 3 θ$	Vertical trefoil
10	3	3	$\sqrt{8} ρ^{3} \sin 3 θ$	Oblique trefoil
11	4	0	$\sqrt{5} (6 ρ^{4} - 6 ρ^{2} + 1)$	Primary spherical

	Olivary Coefficients			Elliptical Coefficients
$j$	$J = 4$	$J = 11$	$J = 15$	$J = 4$	$J = 11$	$J = 15$
1	1.2675	1.2676	1.2676	1.3888	1.4423	1.4096
2	1.4166	1.4166	1.4166	1.4343	1.4667	1.4667
3	−0.1901	−0.1901	−0.1901	−0.1976	−0.2133	−0.2133
4	1.2433	1.2433	1.2433	1.3244	1.4290	1.3802
5		0.1246	0.1246		0.1346	0.1764
6		−0.2815	−0.2815		−0.3099	−0.3287
7		0.0000	0.0000		−0.0079	−0.0079
8		0.1440	0.1440		0.1865	0.1865
9		−0.2218	−0.2218		−0.2384	−0.2384
10		−0.2207	−0.2207		−0.2326	−0.2326
11		0.3971	0.3971		0.4436	0.4030
12			−0.1554			−0.2050
13			0.1592			0.2085
14			0.3331			0.3584
15			−0.3330			−0.3616
$σ$	1.8944	1.9899	2.0570	1.9622	2.1669	2.2122
$σ$ Error	−0.1626	−0.0671	0.0000	−0.0948	0.1099	−0.1552

$j$	$n$	$m$	$Z_{j} (ρ, θ)$	Aberration
1	0	0	1	Piston
2	1	1	$2 ρ \cos θ$	Tip (x tilt)
3	1	1	$2 ρ \sin θ$	Tilt (y tilt)
4	2	0	$\sqrt{3} (2 ρ^{2} - 1)$	Defocus
5	2	2	$\sqrt{6} ρ^{2} \sin 2 θ$	Oblique astigmatism
6	2	2	$\sqrt{6} ρ^{2} \cos 2 θ$	Vertical astigmatism
7	3	1	$\sqrt{8} (ρ^{3} - 2 ρ) \sin θ$	Vertical coma
8	3	1	$\sqrt{8} (ρ^{3} - 2 ρ) \cos θ$	Horizontal coma
9	3	3	$\sqrt{8} ρ^{3} \sin 3 θ$	Vertical trefoil
10	3	3	$\sqrt{8} ρ^{3} \sin 3 θ$	Oblique trefoil
11	4	0	$\sqrt{5} (6 ρ^{4} - 6 ρ^{2} + 1)$	Primary spherical

	Olivary Coefficients			Elliptical Coefficients
$j$	$J = 4$	$J = 11$	$J = 15$	$J = 4$	$J = 11$	$J = 15$
1	1.2675	1.2676	1.2676	1.3888	1.4423	1.4096
2	1.4166	1.4166	1.4166	1.4343	1.4667	1.4667
3	−0.1901	−0.1901	−0.1901	−0.1976	−0.2133	−0.2133
4	1.2433	1.2433	1.2433	1.3244	1.4290	1.3802
5		0.1246	0.1246		0.1346	0.1764
6		−0.2815	−0.2815		−0.3099	−0.3287
7		0.0000	0.0000		−0.0079	−0.0079
8		0.1440	0.1440		0.1865	0.1865
9		−0.2218	−0.2218		−0.2384	−0.2384
10		−0.2207	−0.2207		−0.2326	−0.2326
11		0.3971	0.3971		0.4436	0.4030
12			−0.1554			−0.2050
13			0.1592			0.2085
14			0.3331			0.3584
15			−0.3330			−0.3616
$σ$	1.8944	1.9899	2.0570	1.9622	2.1669	2.2122
$σ$ Error	−0.1626	−0.0671	0.0000	−0.0948	0.1099	−0.1552

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Applied Optics