# 泽尔尼克多项式

${\displaystyle Z_{n}^{m}(\rho ,\varphi )=R_{n}^{m}(\rho )\,\cos(m\,\varphi )\!}$

${\displaystyle Z_{n}^{-m}(\rho ,\varphi )=R_{n}^{m}(\rho )\,\sin(m\,\varphi ),\!}$

${\displaystyle \phi }$方位角

${\displaystyle 0\leq \rho \leq 1}$ 为径向距离


${\displaystyle R_{n}^{m}(\rho )=\sum _{k=0}^{\tfrac {n-m}{2}}{\frac {(-1)^{k}\,(n-k)!}{k!\left({\tfrac {n+m}{2}}-k\right)!\left({\tfrac {n-m}{2}}-k\right)!}}\;\rho ^{n-2\,k}}$

${\displaystyle R_{n}^{m}(\rho )=0}$

## 泽尔尼克多项式的超几何函数表示

{\displaystyle {\begin{aligned}R_{n}^{m}(\rho )&={\binom {n}{\tfrac {n+m}{2}}}\rho ^{n}\ {}_{2}F_{1}\left(-{\tfrac {n+m}{2}},-{\tfrac {n-m}{2}};-n;\rho ^{-2}\right)\\&=(-1)^{\tfrac {n+m}{2}}{\binom {\tfrac {n+m}{2}}{\tfrac {n-m}{2}}}\rho ^{m}\ {}_{2}F_{1}\left(1+n,1-{\tfrac {n-m}{2}};1+{\tfrac {n+m}{2}};\rho ^{2}\right)\end{aligned}}}

## Noll 序列

Noll 用一个J数字表示 [n,m]:如下表

 n,m j n,m j 0,0 1,1 1,−1 2,0 2,−2 2,2 3,−1 3,1 3,−3 3,3 1 2 3 4 5 6 7 8 9 10 4,0 4,2 4,−2 4,4 4,−4 5,1 5,−1 5,3 5,−3 5,5 11 12 13 14 15 16 17 18 19 20

## 泽尔尼克多项式

${\displaystyle I_{j}=\int _{0}^{2\pi }\int _{0}^{1}Z_{j}^{2}\,\rho \,d\rho \,d\theta =k_{j}*\pi .}$

${\displaystyle k_{1}=1}$
${\displaystyle k_{2}={\frac {1}{4}}}$
${\displaystyle k_{3}={\frac {1}{4}}}$
${\displaystyle k_{4}={\frac {1}{3}}}$
${\displaystyle k_{5}={\frac {1}{6}}}$

${\displaystyle Z_{j}=Z_{j}/{\sqrt {(}}k_{j})}$

${\displaystyle I_{j}=\int _{0}^{2\pi }\int _{0}^{1}Z_{j}^{2}\,\rho \,d\rho \,d\theta =\pi .}$

Noll index (${\displaystyle j}$ ) Radial degree (${\displaystyle n}$ ) Azimuthal degree (${\displaystyle m}$ ) ${\displaystyle Z_{j}}$  Classical name
1 0 0 ${\displaystyle 1}$  Piston
2 1 1 ${\displaystyle 2\rho \cos \theta }$  Tip (lateral position) (X-Tilt)
3 1 −1 ${\displaystyle 2\rho \sin \theta }$  Tilt (lateral position) (Y-Tilt)
4 2 0 ${\displaystyle {\sqrt {3}}(2\rho ^{2}-1)}$  Defocus (longitudinal position)
5 2 −2 ${\displaystyle {\sqrt {6}}\rho ^{2}\sin 2\theta }$  Astigmatism
6 2 2 ${\displaystyle {\sqrt {6}}\rho ^{2}\cos 2\theta }$  Astigmatism
7 3 −1 ${\displaystyle {\sqrt {8}}(3\rho ^{3}-2\rho )\sin \theta }$  Coma
8 3 1 ${\displaystyle {\sqrt {8}}(3\rho ^{3}-2\rho )\cos \theta }$  Coma
9 3 −3 ${\displaystyle {\sqrt {8}}\rho ^{3}\sin 3\theta }$  Trefoil
10 3 3 ${\displaystyle {\sqrt {8}}\rho ^{3}\cos 3\theta }$  Trefoil
11 4 0 ${\displaystyle {\sqrt {5}}(6\rho ^{4}-6\rho ^{2}+1)}$  Third-order spherical
12 4 2 ${\displaystyle {\sqrt {10}}(4\rho ^{4}-3\rho ^{2})\cos 2\theta }$
13 4 −2 ${\displaystyle {\sqrt {10}}(4\rho ^{4}-3\rho ^{2})\sin 2\theta }$
14 4 4 ${\displaystyle {\sqrt {10}}\rho ^{4}\cos 4\theta }$
15 4 −4 ${\displaystyle {\sqrt {10}}\rho ^{4}\sin 4\theta }$

## 正交性

${\displaystyle \int _{0}^{1}\rho {\sqrt {2n+2}}R_{n}^{m}(\rho )\,{\sqrt {2n'+2}}R_{n'}^{m}(\rho )\,d\rho =\delta _{n,n'}.}$

${\displaystyle \int _{0}^{2\pi }\cos(m\varphi )\cos(m'\varphi )\,d\varphi =\epsilon _{m}\pi \delta _{|m|,|m'|},}$
${\displaystyle \int _{0}^{2\pi }\sin(m\varphi )\sin(m'\varphi )\,d\varphi =(-1)^{m+m'}\pi \delta _{|m|,|m'|};\quad m\neq 0,}$
${\displaystyle \int _{0}^{2\pi }\cos(m\varphi )\sin(m'\varphi )\,d\varphi =0,}$

${\displaystyle \int Z_{n}^{m}(\rho ,\varphi )Z_{n'}^{m'}(\rho ,\varphi )\,d^{2}r={\frac {\epsilon _{m}\pi }{2n+2}}\delta _{n,n'}\delta _{m,m'},}$

${\displaystyle n-m}$ ${\displaystyle n'-m'}$  都是偶数.