Table 1.

Zernike modes 1 to 15 and numbering scheme. The modes are expressed over the unit disk as functions of r and θ with 0 < r < 1 and 0 < θ < 2π.

Index, z	Zernike mode	Name
1	1	Piston
2	2r cosθ	Tip
3	2r sinθ	Tilt
4	$\sqrt{3}\left(2r^2-1\right)$	Defocus
5	$\sqrt{6}{r}^2\text{cos}2\theta$	Astigmatism, 1st order
6	$\sqrt{6}{r}^2\text{sin}2\theta$	Astigmatism, 1st order
7	$2\sqrt{2}\left(3r^3-2r\right)\text{cos}\theta$	Coma
8	$2\sqrt{2}\left(3r^3-2r\right)\text{sin}\theta$	Coma
9	$2\sqrt{2}{r}^3\text{cos}3\theta$	Trefoil
10	$2\sqrt{2}{r}^3\text{sin}3\theta$	Trefoil
11	$\sqrt{5}\left(6r^4-6r^2+1\right)$	Spherical aberration
12	$\sqrt{10}\left(4r^4-3r^2\right)\text{cos}2\theta$	Astigmatism, 2nd order
13	$\sqrt{10}\left(4r^4-3r^2\right)\text{sin}2\theta$	Astigmatism, 2nd order
14	$\sqrt{10}{r}^4\text{cos}4\theta$	Tetrafoil
15	$\sqrt{10}{r}^4\text{sin}4\theta$	Tetrafoil