Modal integration of Hartmann and Shack&#x2013;Hartmann patterns

Geovanni Hernández-Gómez; Zacarías Malacara-Hernández; Daniel Malacara-Doblado; Rufino Díaz-Uribe; Daniel Malacara-Hernández

doi:10.1364/JOSAA.31.000846

Journal of the Optical Society of America A
Vol. 31,
Issue 4,
pp. 846-851
(2014)
•https://doi.org/10.1364/JOSAA.31.000846

Modal integration of Hartmann and Shack–Hartmann patterns

Geovanni Hernández-Gómez, Zacarías Malacara-Hernández, Daniel Malacara-Doblado, Rufino Díaz-Uribe, and Daniel Malacara-Hernández

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Geovanni Hernández-Gómez, Zacarías Malacara-Hernández, Daniel Malacara-Doblado, Rufino Díaz-Uribe, and Daniel Malacara-Hernández, "Modal integration of Hartmann and Shack–Hartmann patterns," J. Opt. Soc. Am. A 31, 846-851 (2014)

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Abstract

Instead of measuring the wavefront deformations directly, Hartmann and Shack–Hartmann tests measure the wavefront slopes, which are equivalent to the ray transverse aberrations. Numerous different integration methods have been described in the literature to obtain the wavefront deformations from these measurements. Basically, they can be classified in two different categories, i.e., modal and zonal. In this work we describe a modal method to integrate Hartmann and Shack–Hartmann patterns using orthogonal wavefront slope aberration polynomials, instead of the commonly used Zernike polynomials for the wavefront deformations.

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$n$	$m$	$n - 2 m$	$r$	Term	Name
0	0	0	1	1	Piston
1	0	1	2	$ρ \sin θ$	Tilt about $x$ axis
1	1	$- 1$	3	$ρ \cos θ$	Tilt about $y$ axis
2	0	2	4	$ρ^{2} \sin 2 θ$	Astigmatism at $\pm 45 °$
2	1	0	5	$ρ^{2}$	Defocus
2	2	$- 2$	6	$ρ^{2} \cos 2 θ$	Astigmatism at 0° or 90°
3	0	3	7	$ρ^{3} \sin 3 θ$	Triangular astigmatism, base on $y$ axis (trefoil)
3	1	1	8	$ρ^{3} \sin θ$	Primary coma along $y$ axis
3	2	$- 1$	9	$ρ^{3} \cos θ$	Primary coma along $x$ axis
3	3	$- 3$	10	$ρ^{3} \cos 3 θ$	Triangular astigmatism, base on $x$ axis (trefoil)
4	0	4	11	$ρ^{4} \sin 4 θ$	Ashtray 0° (quatrefoil)
4	1	2	12	$ρ^{4} \sin 2 θ$	Secondary astigmatism at $\pm 45 °$
4	2	0	13	$ρ^{4}$	Primary spherical aberration
4	3	$- 2$	14	$ρ^{4} \cos 2 θ$	Secondary astigmatism at 0° or 90°
4	4	$- 4$	15	$ρ^{4} \cos 4 θ$	Ashtray 22.5° (quatrefoil)
5	0	5	16	$ρ^{5} \sin 5 θ$
5	1	3	17	$ρ^{5} \sin 3 θ$
5	2	1	18	$ρ^{5} \sin θ$	Fifth-order coma along $y$ axis
5	3	$- 1$	19	$ρ^{5} \cos θ$	Fifth-order coma along $x$ axis
5	4	$- 3$	20	$ρ^{5} \cos 3 θ$
5	5	$- 5$	21	$ρ^{5} \cos 5 θ$
6	0	6	22	$ρ^{6} \sin 6 θ$
6	1	4	23	$ρ^{6} \sin 4 θ$
6	2	2	24	$ρ^{6} \sin 2 θ$	Tertiary astigmatism at 0°
6	3	0	25	$ρ^{6}$	Fifth-order spherical aberration
6	4	$- 2$	26	$ρ^{6} \cos 2 θ$	Tertiary astigmatism at 45°
6	5	$- 4$	27	$ρ^{6} \cos 4 θ$
6	6	$- 6$	28	$ρ^{6} \cos 6 θ$

$n$	$m$	$n - 2 m$	$r$	Term
1	0	1	1	$\sin θ$
1	1	$- 1$	2	$\cos θ$
2	0	2	3	$2 ρ \sin 2 θ$
2	1	0	4	$2 ρ$
2	2	$- 2$	5	$2 ρ \cos 2 θ$
3	0	3	6	$3 ρ^{2} \sin 3 θ$
3	1	1	7	$3 ρ^{2} \sin θ$
3	2	$- 1$	8	$3 ρ^{2} \cos θ$
3	3	$- 3$	9	$3 ρ^{2} \cos 3 θ$
4	0	4	10	$4 ρ^{3} \sin 4 θ$
4	1	2	11	$4 ρ^{3} \sin 2 θ$
4	2	0	12	$4 ρ^{3}$
4	3	$- 2$	13	$4 ρ^{3} \cos 2 θ$
4	4	$- 4$	14	$4 ρ^{3} \cos 4 θ$
5	0	5	15	$5 ρ^{4} \sin 5 θ$
5	1	3	16	$5 ρ^{4} \sin 3 θ$
5	2	1	17	$5 ρ^{4} \sin θ$
5	3	$- 1$	18	$5 ρ^{4} \cos θ$
5	4	$- 3$	19	$5 ρ^{4} \cos 3 θ$
5	5	$- 5$	20	$5 ρ^{4} \cos 5 θ$
6	0	6	21	$6 ρ^{5} \sin 6 θ$
6	1	4	22	$6 ρ^{5} \sin 4 θ$
6	2	2	23	$6 ρ^{5} \sin 2 θ$
6	3	0	24	$6 ρ^{5}$
6	4	$- 2$	25	$6 ρ^{5} \cos 2 θ$
6	5	$- 4$	26	$6 ρ^{5} \cos 4 θ$
6	6	$- 6$	27	$6 ρ^{5} \cos 6 θ$

H1	$\sin θ$
H2	$\cos θ$
H3	$2 ρ \sin 2 θ - 1.103 \sin θ$
H4	$2 ρ - 1.728 \cos θ$
H5	$2 ρ \cos 2 θ - 2.1341 \cos θ + 1.7467 ρ$
H6	$3 ρ^{2} \sin 3 θ - 3.3694 ρ \sin 2 θ + 1.9270 \sin θ$
H7	$3 ρ^{2} + 1.8385 ρ^{2} \sin 3 θ - 2.8303 ρ \sin 2 θ + 0.0196 \sin θ$
H8	$3 ρ^{2} \cos θ - 1.4854 ρ \cos 2 θ + 0.7123 \cos θ - 2.1199 ρ$
H9	$3 ρ^{3} \cos 3 θ + 8.0367 ρ^{2} \cos θ - 6.4359 ρ 2 θ + 3.0891 \cos θ - 6.2368 ρ$
H10	$4 ρ^{3} \sin 4 θ - 3.3483 ρ^{2} \sin θ - 6.3228 ρ^{2} \sin 3 θ + 6.5325 ρ \sin 2 θ - 1.4623 \sin θ$
H11	$4 ρ^{3} \sin 2 θ + 1.4806 ρ^{3} \sin 4 θ - 4.0220 ρ^{2} \sin θ - 3.6224 ρ^{2} \sin 3 θ + 1.4959 ρ \sin 2 θ + 0.0373 \sin θ$
H12	$- 0.5752 ρ^{3} \cos 3 θ - 5.3259 ρ^{2} \cos θ + 1.9356 ρ \cos 2 θ + 0.5754 \cos θ + 4 ρ^{3} - 0.6009 ρ$
H13	$4 ρ^{3} \cos 2 θ - 2.0358 ρ^{3} \cos 3 θ - 10.6236 ρ^{2} \cos θ + 2.5292 ρ \cos 2 θ - 0.5742 \cos θ + 3.7666 ρ^{3} + 3.0172 ρ$
H14	$14.75 ρ^{3} \cos 2 θ - 13.2587 ρ^{3} \cos 3 θ + 4 ρ^{3} \cos 4 θ - 38.2849 ρ^{2} \cos θ + 13.4296 ρ \cos 2 θ - 3.8770 \cos θ + 10.8755 ρ^{3} + 13.2406 ρ$
H15	$5 ρ^{4} \sin 5 θ - 9.7004 ρ^{3} \sin 2 θ - 12.2999 ρ^{3} \sin 4 θ + 10.9475 ρ^{2} \sin θ + 17.9201 ρ^{2} \sin 3 θ - 9.6163 ρ \sin 2 θ + 1.5440 \sin θ$
H16	$5 ρ^{4} \sin 3 θ + 2.0298 ρ^{4} \sin 5 θ - 10.0800 ρ^{3} \sin 2 θ - 6.1248 ρ^{3} \sin 4 θ + 8.7904 ρ^{2} \sin θ + 6.1310 ρ^{2} \sin 3 θ - 2.7557 ρ \sin 2 θ - 0.2512 \sin θ$
H17	$5 ρ^{4} \sin θ + 6.082 ρ^{4} \sin 3 θ + 1.6824 ρ^{4} \sin 5 θ - 12.8811 ρ^{3} \sin 2 θ - 5.8479 ρ^{3} \sin 4 θ + 5.0123 ρ^{2} \sin θ + 5.4439 ρ^{2} \sin 3 θ - 1.1577 ρ \sin 2 θ + 0.0404 \sin θ$
H18	$5 ρ^{4} \cos θ - 4.1973 ρ^{3} \cos 2 θ + 1.5088 ρ^{3} \cos 3 θ - 0.2776 ρ^{3} \cos 4 θ + 2.8223 ρ^{2} \cos θ - 0.2440 ρ \cos 2 θ - 0.1147 \cos θ - 4.7402 ρ^{2} + 0.2606 ρ$
H19	$13.5064 ρ^{4} \cos θ + 5 ρ^{4} \cos 3 θ - 19.5239 ρ^{3} \cos 2 θ + 5.4786 ρ^{3} \cos 3 θ - 2.6952 ρ^{2} \cos 4 θ + 21.2041 ρ^{2} \cos θ - 3.2730 ρ \cos 2 θ + 0.4694 \cos θ - 16.6607 ρ^{3} - 3.4571 ρ$
H20	$48.7130 ρ^{4} \cos θ + 24.6885 ρ^{4} \cos 3 θ + 5 ρ^{4} \cos 5 θ - 90.9424 ρ^{3} \cos 2 θ + 36.5296 ρ^{3} \cos 3 θ - 22.9665 ρ^{3} \cos 4 θ + 109.3130 ρ^{2} \cos θ - 23.3788 ρ \cos 2 θ + 4.8610 \cos θ - 67.9522 ρ^{3} - 23.3780 ρ$
H21	$6 ρ^{5} \sin 6 θ - 7.3045 ρ^{4} \sin θ - 20.7271 ρ^{4} \sin 3 θ - 18.5639 ρ^{4} \sin 5 θ + 39.5496 ρ^{3} \sin 2 θ + 34.8922 ρ^{3} \sin 4 θ - 23.7184 ρ^{2} \sin θ - 29.0512 ρ^{2} \sin 3 θ + 10.8290 ρ \sin 2 θ - 0.8141 \sin θ$
H22	$6 ρ^{5} \sin 4 θ + 2.163 ρ^{5} \sin 6 θ - 8.8982 ρ^{4} \sin θ - 19.6797 ρ^{4} \sin 3 θ - 8.9644 ρ^{4} 5 θ + 31.4107 ρ^{3} \sin 2 θ + 14.6595 ρ^{3} \sin 4 θ - 13.9049 ρ^{2} \sin θ - 12.2907 ρ^{2} \sin 3 θ + 2.7483 ρ \sin 2 θ + 0.1678 \sin θ$
H23	$6 ρ^{5} \sin 2 θ + 4.4299 ρ^{5} \sin 4 θ + 0.9179 ρ^{5} \sin 6 θ - 10.1906 ρ^{4} \sin θ - 14.6174 ρ^{4} \sin 3 θ - 4.4472 ρ^{4} \sin 5 θ + 14.2913 ρ^{3} \sin 2 θ + 6.8629 ρ^{3} \sin 4 θ - 4.0213 ρ^{2} \sin θ - 4.0300 ρ^{2} \sin 3 θ + 0.7319 ρ \sin 2 θ + 0.0004 \sin θ$
H24	$- 15.4781 ρ^{4} \cos θ - 3.7120 ρ^{4} \cos 3 θ - 0.3084 ρ^{4} \cos 5 θ + 14.9338 ρ^{3} \cos 2 θ - 3.9985 ρ^{3} \cos 3 θ + 2.3519 ρ^{3} \cos 4 θ - 8.4754 ρ^{2} \cos θ + 1.768 ρ \cos 2 θ - 0.6171 \cos θ + 6 ρ^{5} + 5.6206 ρ^{2} + 2.5959 ρ$
H25	$6 ρ^{5} \cos 2 θ - 21.1546 ρ^{4} \cos θ - 5.9988 ρ^{4} \cos 3 θ - 0.4382 ρ^{4} \cos 5 θ + 10.2438 ρ^{3} \cos 2 θ - 2.0657 ρ^{3} \cos 3 θ + 2.5386 ρ^{3} \cos 4 θ - 3.5074 ρ^{2} \cos θ + 0.3061 ρ \cos 2 θ - 0.0197 \cos θ + 5.8511 ρ^{5} + 8.1821 ρ^{3} + 0.0712 ρ$
H26	$23.16 ρ^{5} \cos 2 θ + 6 ρ^{5} \cos 4 θ - 84.9650 ρ^{4} \cos θ - 34.3058 ρ^{4} \cos 3 θ - 3.3804 ρ^{4} \cos 5 θ + 59.0707 ρ^{3} \cos 2 θ - 11.4670 ρ^{3} \cos 3 θ + 10.8027 ρ^{3} \cos 4 θ - 38.9913 ρ^{2} \cos θ + 4.6347 ρ \cos 2 θ - 0.5912 \cos θ + 17.3481 ρ^{5} + 47.9832 ρ^{3} + 4.7359 ρ$
H27	$90 ρ^{5} \cos 2 θ + 36 ρ^{5} \cos 4 θ + 6 ρ^{5} \cos 6 θ - 345.6064 ρ^{4} \cos θ - 172.8038 ρ^{4} \cos 3 θ - 34.5600 ρ^{4} \cos 5 θ + 300.8283 ρ^{3} \cos 2 θ - 76.4499 ρ^{3} \cos 3 θ + 75.2048 ρ^{3} \cos 4 θ - 229.3398 ρ^{2} \cos θ + 35.7451 ρ \cos 2 θ - 6.0076 \cos θ + 60.0003 ρ^{5} + 225.6175 ρ^{3} + 76.4499 ρ^{3} \cos 3 θ + 75.2048 ρ^{3} \cos 4 θ - 229.3398 ρ^{2} \cos θ + 35.7451 ρ \cos 2 θ - 6.0076 \cos θ + 60.0003 ρ^{5} + 225.6175 ρ^{3} + 35.7454 ρ$

$n$	$m$	$n - 2 m$	$r$	Term	Name
0	0	0	1	1	Piston
1	0	1	2	$ρ \sin θ$	Tilt about $x$ axis
1	1	$- 1$	3	$ρ \cos θ$	Tilt about $y$ axis
2	0	2	4	$ρ^{2} \sin 2 θ$	Astigmatism at $\pm 45 °$
2	1	0	5	$ρ^{2}$	Defocus
2	2	$- 2$	6	$ρ^{2} \cos 2 θ$	Astigmatism at 0° or 90°
3	0	3	7	$ρ^{3} \sin 3 θ$	Triangular astigmatism, base on $y$ axis (trefoil)
3	1	1	8	$ρ^{3} \sin θ$	Primary coma along $y$ axis
3	2	$- 1$	9	$ρ^{3} \cos θ$	Primary coma along $x$ axis
3	3	$- 3$	10	$ρ^{3} \cos 3 θ$	Triangular astigmatism, base on $x$ axis (trefoil)
4	0	4	11	$ρ^{4} \sin 4 θ$	Ashtray 0° (quatrefoil)
4	1	2	12	$ρ^{4} \sin 2 θ$	Secondary astigmatism at $\pm 45 °$
4	2	0	13	$ρ^{4}$	Primary spherical aberration
4	3	$- 2$	14	$ρ^{4} \cos 2 θ$	Secondary astigmatism at 0° or 90°
4	4	$- 4$	15	$ρ^{4} \cos 4 θ$	Ashtray 22.5° (quatrefoil)
5	0	5	16	$ρ^{5} \sin 5 θ$
5	1	3	17	$ρ^{5} \sin 3 θ$
5	2	1	18	$ρ^{5} \sin θ$	Fifth-order coma along $y$ axis
5	3	$- 1$	19	$ρ^{5} \cos θ$	Fifth-order coma along $x$ axis
5	4	$- 3$	20	$ρ^{5} \cos 3 θ$
5	5	$- 5$	21	$ρ^{5} \cos 5 θ$
6	0	6	22	$ρ^{6} \sin 6 θ$
6	1	4	23	$ρ^{6} \sin 4 θ$
6	2	2	24	$ρ^{6} \sin 2 θ$	Tertiary astigmatism at 0°
6	3	0	25	$ρ^{6}$	Fifth-order spherical aberration
6	4	$- 2$	26	$ρ^{6} \cos 2 θ$	Tertiary astigmatism at 45°
6	5	$- 4$	27	$ρ^{6} \cos 4 θ$
6	6	$- 6$	28	$ρ^{6} \cos 6 θ$

$n$	$m$	$n - 2 m$	$r$	Term
1	0	1	1	$\sin θ$
1	1	$- 1$	2	$\cos θ$
2	0	2	3	$2 ρ \sin 2 θ$
2	1	0	4	$2 ρ$
2	2	$- 2$	5	$2 ρ \cos 2 θ$
3	0	3	6	$3 ρ^{2} \sin 3 θ$
3	1	1	7	$3 ρ^{2} \sin θ$
3	2	$- 1$	8	$3 ρ^{2} \cos θ$
3	3	$- 3$	9	$3 ρ^{2} \cos 3 θ$
4	0	4	10	$4 ρ^{3} \sin 4 θ$
4	1	2	11	$4 ρ^{3} \sin 2 θ$
4	2	0	12	$4 ρ^{3}$
4	3	$- 2$	13	$4 ρ^{3} \cos 2 θ$
4	4	$- 4$	14	$4 ρ^{3} \cos 4 θ$
5	0	5	15	$5 ρ^{4} \sin 5 θ$
5	1	3	16	$5 ρ^{4} \sin 3 θ$
5	2	1	17	$5 ρ^{4} \sin θ$
5	3	$- 1$	18	$5 ρ^{4} \cos θ$
5	4	$- 3$	19	$5 ρ^{4} \cos 3 θ$
5	5	$- 5$	20	$5 ρ^{4} \cos 5 θ$
6	0	6	21	$6 ρ^{5} \sin 6 θ$
6	1	4	22	$6 ρ^{5} \sin 4 θ$
6	2	2	23	$6 ρ^{5} \sin 2 θ$
6	3	0	24	$6 ρ^{5}$
6	4	$- 2$	25	$6 ρ^{5} \cos 2 θ$
6	5	$- 4$	26	$6 ρ^{5} \cos 4 θ$
6	6	$- 6$	27	$6 ρ^{5} \cos 6 θ$

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Equations (24)

Journal of the Optical Society of America A