AO tutorial 1: turbulence

All credit to Dr. Andrei A. Tokovinin at Cerro Tololo Inter-American Observatory, La Serena, Chile

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The first 15 Zernike polynomials

j Name Formula Picture

1 0 0 Piston

2 1 1 Tip $Z_2 = 2r \cos \theta$

3 1 1 Tilt $Z_3 = 2r \sin \theta$

4 2 0 Defocus $Z_4 = \sqrt{3} (2r^2 -1)$

5 2 2 Astigmatism $Z_5 = \sqrt{6}r^2 \sin 2 \theta$

6 2 2 Astigmatism $Z_6 = \sqrt{6}r^2 \cos 2 \theta$

7 3 1 Coma $Z_7 = \sqrt{8} (3r^3-2r) \sin \theta$

8 3 1 Coma $Z_8 = \sqrt{8} (3r^3-2r) \cos \theta$

9 3 3 Trefoil $Z_9 = \sqrt{8} 3r^3 \sin 3 \theta$

10 3 3 Trefoil $Z_{10} = \sqrt{8} 3r^3 \cos 3 \theta$ Z10

11 4 0 Spherical $Z_{11}=\sqrt{5} (6r^4 -6r^2 +1)$ Z11

12 4 2 Astigm. 5th $Z_{12} = \sqrt{10}( 10r^4 -3r^3) \cos 2\theta$ Z12

13 4 2 Astigm. 5th $Z_{13} = \sqrt{10}( 10r^4 -3r^3) \sin 2\theta$ Z13

14 4 4 Ashtray $Z_{14} = \sqrt{10}r^4 \cos 4\theta$ Z14

15 4 4 Ashtray $Z_{15} = \sqrt{10}r^4 \sin 4\theta$ Z15

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j			Name	Formula	Picture
1	0	0	Piston
2	1	1	Tip	$Z_2 = 2r \cos \theta$
3	1	1	Tilt	$Z_3 = 2r \sin \theta$
4	2	0	Defocus	$Z_4 = \sqrt{3} (2r^2 -1)$
5	2	2	Astigmatism	$Z_5 = \sqrt{6}r^2 \sin 2 \theta$
6	2	2	Astigmatism	$Z_6 = \sqrt{6}r^2 \cos 2 \theta$
7	3	1	Coma	$Z_7 = \sqrt{8} (3r^3-2r) \sin \theta$
8	3	1	Coma	$Z_8 = \sqrt{8} (3r^3-2r) \cos \theta$
9	3	3	Trefoil	$Z_9 = \sqrt{8} 3r^3 \sin 3 \theta$
10	3	3	Trefoil	$Z_{10} = \sqrt{8} 3r^3 \cos 3 \theta$
11	4	0	Spherical	$Z_{11}=\sqrt{5} (6r^4 -6r^2 +1)$
12	4	2	Astigm. 5th	$Z_{12} = \sqrt{10}( 10r^4 -3r^3) \cos 2\theta$
13	4	2	Astigm. 5th	$Z_{13} = \sqrt{10}( 10r^4 -3r^3) \sin 2\theta$
14	4	4	Ashtray	$Z_{14} = \sqrt{10}r^4 \cos 4\theta$
15	4	4	Ashtray	$Z_{15} = \sqrt{10}r^4 \sin 4\theta$