Least-squares fitting of Hartmann or Shack&#x2013;Hartmann data with a circular array of sampling points

Zacarías Malacara-Hernández; Daniel Malacara-Doblado; Daniel Malacara-Hernández

doi:10.1364/AO.54.00E113

Applied Optics
Vol. 54,
Issue 28,
pp. E113-E122
(2015)
•https://doi.org/10.1364/AO.54.00E113

Least-squares fitting of Hartmann or Shack–Hartmann data with a circular array of sampling points

Zacarías Malacara-Hernández, Daniel Malacara-Doblado, and Daniel Malacara-Hernández

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Zacarías Malacara-Hernández, Daniel Malacara-Doblado, and Daniel Malacara-Hernández, "Least-squares fitting of Hartmann or Shack–Hartmann data with a circular array of sampling points," Appl. Opt. 54, E113-E122 (2015)

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Abstract

A least-squares procedure to find the tilts, curvature, astigmatism, coma, and triangular astigmatism by means of measurements of the transverse aberrations using a Hartmann or Shack–Hartmann test is described. The sampling points are distributed in a ring centered on the pupil of the optical system. The properties and characteristics of rings with three, four, five, six, or more sampling points are analyzed with more detail and better mathematical analysis than in previous publications.

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$n$	$m$	$r$	$L$	Polar Coordinates	Name
0	0	0	0	1	Piston
1	0	1	1	$ρ \cos θ$	Tilt about $y$ -axis
	1	2	1	$ρ \sin θ$	Tilt about $x$ -axis
2	0	3	0	$ρ^{2}$	Defocusing
	1	4	2	$ρ^{2} \cos 2 θ$	Astigmatism, axis at 0° or 90°
	2	5	2	$ρ^{2} \sin 2 θ$	Astigmatism, axis at $\pm 45^{0}$
3	0	6	1	$ρ^{3} \cos θ$	Coma, along $x$ -axis
	1	7	1	$ρ^{3} \sin θ$	Coma, along $y$ -axis
	2	8	3	$ρ^{3} \cos 3 θ$	Triangular astigmatism, axes at 0°, 120°, 240°
	3	9	3	$ρ^{3} \sin 3 θ$	Triangular astigmatism, axes at 30°, 150°, 270°

Name	1st Wavefront	2nd Wavefront
Piston	0.0	0.0
Tilt about $y$ -axis	$1.0 \times 10^{- 5}$	$1.0 \times 10^{- 5}$
Tilt about $x$ -axis	$- 2.0 \times 10^{- 5}$	$- 2.0 \times 10^{- 5}$
Defocusing	$- 1.0 \times 10^{- 5}$	$- 1.0 \times 10^{- 5}$
Astigmatism, axis at 0° or 90°	$3.0 \times 10^{- 7}$	$3.0 \times 10^{- 7}$
Astigmatism, axis at $\pm 45 °$	$2.0 \times 10^{- 7}$	$2.0 \times 10^{- 7}$
Coma, along $x$ -axis	0.0	$7.0 \times 10^{- 9}$
Coma, along $y$ -axis	0.0	$3.0 \times 10^{- 9}$
Triangular astigmatism, axes at 0°,120°, 240°	0.0	$4.0 \times 10^{- 9}$
Triangular astigmatism, axes at 30°,150°, 270°	0.0	$6.0 \times 10^{- 9}$

$n$	$m$	$r$	$L$	Polar Coordinates	Name
0	0	0	0	1	Piston
1	0	1	1	$ρ \cos θ$	Tilt about $y$ -axis
	1	2	1	$ρ \sin θ$	Tilt about $x$ -axis
2	0	3	0	$ρ^{2}$	Defocusing
	1	4	2	$ρ^{2} \cos 2 θ$	Astigmatism, axis at 0° or 90°
	2	5	2	$ρ^{2} \sin 2 θ$	Astigmatism, axis at $\pm 45^{0}$
3	0	6	1	$ρ^{3} \cos θ$	Coma, along $x$ -axis
	1	7	1	$ρ^{3} \sin θ$	Coma, along $y$ -axis
	2	8	3	$ρ^{3} \cos 3 θ$	Triangular astigmatism, axes at 0°, 120°, 240°
	3	9	3	$ρ^{3} \sin 3 θ$	Triangular astigmatism, axes at 30°, 150°, 270°

Name	1st Wavefront	2nd Wavefront
Piston	0.0	0.0
Tilt about $y$ -axis	$1.0 \times 10^{- 5}$	$1.0 \times 10^{- 5}$
Tilt about $x$ -axis	$- 2.0 \times 10^{- 5}$	$- 2.0 \times 10^{- 5}$
Defocusing	$- 1.0 \times 10^{- 5}$	$- 1.0 \times 10^{- 5}$
Astigmatism, axis at 0° or 90°	$3.0 \times 10^{- 7}$	$3.0 \times 10^{- 7}$
Astigmatism, axis at $\pm 45 °$	$2.0 \times 10^{- 7}$	$2.0 \times 10^{- 7}$
Coma, along $x$ -axis	0.0	$7.0 \times 10^{- 9}$
Coma, along $y$ -axis	0.0	$3.0 \times 10^{- 9}$
Triangular astigmatism, axes at 0°,120°, 240°	0.0	$4.0 \times 10^{- 9}$
Triangular astigmatism, axes at 30°,150°, 270°	0.0	$6.0 \times 10^{- 9}$

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Equations (71)

Applied Optics