. 2018 Feb 7;7:e32671. doi: 10.7554/eLife.32671

Table 1. Expression of Zernike polynomials and Zernike coefficients in Equation (1).

Defocus
Zernike polynomials	$Z_{2}^{0} (u, v) = \sqrt{3} [2 (u^{2} + v^{2}) - 1]$
Zernike coefficients	$C_{2}^{0} (z) = \frac{n k z \sin^{2} α}{8 π \sqrt{3}} (1 + \frac{1}{4} \sin^{2} α + \frac{9}{80} \sin^{4} α + \frac{1}{16} \sin^{6} α + \cdot \cdot \cdot)$
First-order spherical aberration
Zernike polynomials	$Z_{4}^{0} (u, v) = \sqrt{5} [6 {(u^{2} + v^{2})}^{2} - 6 (u^{2} + v^{2}) + 1]$
Zernike coefficients	$C_{4}^{0} (z) = \frac{n k z \sin^{4} α}{96 π \sqrt{5}} (1 + \frac{3}{4} \sin^{2} α + \frac{15}{18} \sin^{4} α + \cdot \cdot \cdot)$
Second-order spherical aberration
Zernike polynomials	$Z_{6}^{0} (u, v) = \sqrt{7} [20 {(u^{2} + v^{2})}^{3} - 30 {(u^{2} + v^{2})}^{2} + 12 (u^{2} + v^{2}) - 1]$
Zernike coefficients	$C_{6}^{0} (z) = \frac{n k z \sin^{6} α}{640 π \sqrt{7}} (1 + \frac{5}{4} \sin^{2} α + \cdot \cdot \cdot)$

n, refractive index of media between the objective and sample; k, the wavenumber; z, the axial shift of the focus plane in the sample; u, v, coordinates on the SLM phase mask; nsinα, the NA of the objective.