澤爾尼克多項式

澤爾尼克多項式是一個以1953年獲諾貝爾物理學獎荷蘭物理學家弗里茨·澤爾尼克命名的正交多項式，分為奇、偶兩類

頭15個澤爾尼克多項式

20個澤爾尼克多項式 以Noll序列表示

奇多項式：

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

偶多項式

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

其中${\displaystyle n\geq m}$ 為非負整數，

${\displaystyle \phi }$為方位角

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

如果 n-m為偶數則

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

如果n-m為奇數，則

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

澤爾尼克多項式的超幾何函數表示

澤爾尼克多項式也可以表示為超幾何函數

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

Noll 序列

Noll 用一個J數字表示 [n,m]:如下表

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

澤爾尼克多項式

由於

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

其中${\displaystyle k_{j}}$因j而異，

${\displaystyle k_{1}=1}$

${\displaystyle k_{2}={\frac {1}{4}}}$

${\displaystyle k_{3}={\frac {1}{4}}}$

${\displaystyle k_{4}={\frac {1}{3}}}$

${\displaystyle k_{5}={\frac {1}{6}}}$

必須先歸一化

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

使得

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

歸一化澤爾尼克多項式以Noll序列排列如下：

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

徑向正交性

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

角度正交性

${\displaystyle \int _{0}^{2\pi }\cos(m\varphi )\cos(m'\varphi )\,d\varphi =\epsilon _{m}\pi \delta _{|m|,|m'|},}$

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

${\displaystyle \int _{0}^{2\pi }\cos(m\varphi )\sin(m'\varphi )\,d\varphi =0,}$

其中 ${\displaystyle \epsilon _{m}}$ 稱為Neumann因子，其數值為 2 如果滿足 ${\displaystyle m=0}$ ，數值為 1，如果 ${\displaystyle m\neq 0}$.

徑向與角度正交性

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

其中 ${\displaystyle d^{2}r=\rho \,d\rho \,d\varphi }$ 為 雅可比矩陣

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

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