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The first 15 Zernike polynomials

j $n$ $m$ Name Formula Picture
1 0 0 Piston $Z_1 = 1$ Z1
2 1 1 Tip $Z_2 = 2r \cos \theta$ Z2
3 1 1 Tilt $Z_3 = 2r \sin \theta$ Z3
4 2 0 Defocus $Z_4 = \sqrt{3} (2r^2 -1)$ Z4
5 2 2 Astigmatism $Z_5 = \sqrt{6}r^2 \sin 2 \theta$ Z5
6 2 2 Astigmatism $Z_6 = \sqrt{6}r^2 \cos 2 \theta$ Z6
7 3 1 Coma $Z_7 = \sqrt{8} (3r^3-2r) \sin \theta$ Z7
8 3 1 Coma $Z_8 = \sqrt{8} (3r^3-2r) \cos \theta$ Z8
9 3 3 Trefoil $Z_9 = \sqrt{8} 3r^3 \sin 3 \theta $ Z9
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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