Table 1.

Coefficient conversion relationships for Zernike polynomial expansions up to order 8

n	m	New expansion coefficients bn,m
0	0	$\begin{array}{l}{b}_{0,m}=a_{0,m}-\sqrt{3}\left(1-\frac{1}{{\lambda }^2}\right)a_{2,m}+\sqrt{5}\left(1-\frac{3}{{\lambda }^2}+\frac{2}{{\lambda }^4}\right)a_{4,m}\\ -\sqrt{7}\left(1-\frac{6}{{\lambda }^2}+\frac{10}{{\lambda }^4}-\frac{5}{{\lambda }^6}\right)a_{6,m}+3\left(1-\frac{10}{{\lambda }^2}+\frac{30}{{\lambda }^4}-\frac{35}{{\lambda }^6}+\frac{14}{{\lambda }^8}\right)a_{8,m}\end{array}$
1	−1, 1	$b_{1,m}=\frac{1}{\lambda }\left[a_{1,m}-2\sqrt{2}\left(1-\frac{1}{{\lambda }^2}\right)a_{3,m}+\sqrt{3}\left(3-\frac{8}{{\lambda }^2}+\frac{5}{{\lambda }^4}\right)a_{5,m}-4\left(2-\frac{10}{{\lambda }^2}+\frac{15}{{\lambda }^4}-\frac{7}{{\lambda }^6}\right)a_{7,m}\right]$
2	−2, 0, 2	$b_{2,m}=\frac{1}{{\lambda }^2}\left[a_{2,m}-\sqrt{15}\left(1-\frac{1}{{\lambda }^2}\right)a_{4,m}+\sqrt{21}\left(2-\frac{5}{{\lambda }^2}+\frac{3}{{\lambda }^4}\right)a_{6,m}-\sqrt{3}\left(10-\frac{45}{{\lambda }^2}+\frac{63}{{\lambda }^4}-\frac{28}{{\lambda }^6}\right)a_{8,m}\right]$
3	−3, −1, 1, 3	$b_{3,m}=\frac{1}{{\lambda }^3}\left[a_{3,m}-2\sqrt{6}\left(1-\frac{1}{{\lambda }^2}\right)a_{5,m}+2\sqrt{2}\left(5-\frac{12}{{\lambda }^2}+\frac{7}{{\lambda }^4}\right)a_{7,m}\right]$
4	−4 −2, 0, 2, 4	$b_{4,m}=\frac{1}{{\lambda }^4}\left[a_{4,m}-\sqrt{35}\left(1-\frac{1}{{\lambda }^2}\right)a_{6,m}+3\sqrt{5}\left(3-\frac{7}{{\lambda }^2}+\frac{4}{{\lambda }^4}\right)a_{8,m}\right]$
5	−5, −3, −1, 1, 3, 5	$b_{5,m}=\frac{1}{{\lambda }^5}\left[a_{5,m}-4\sqrt{3}\left(1-\frac{1}{{\lambda }^2}\right)a_{7,m}\right]$
6	−6, −4 −2, 0, 2, 4, 6	$b_{6,m}=\frac{1}{{\lambda }^6}\left[a_{6,m}-3\sqrt{7}\left(1-\frac{1}{{\lambda }^2}\right)a_{8,m}\right]$
7	−7, −5, −3, −1, 1, 3, 5, 7	$b_{7,m}=\frac{1}{{\lambda }^7}{a}_{7,m}$
8	−8 −6, −4, −2, 0, 2, 4, 6, 8	$b_{8,m}=\frac{1}{{\lambda }^8}{a}_{8,m}$