In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike, laureate of the 1953 Nobel Prize in Physics and the inventor of phase-contrast microscopy, they play important roles in various optics branches such as beam optics and imaging.^[1][2]

Definitions

There are even and odd Zernike polynomials. The even Zernike polynomials are defined as

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

(even function over the azimuthal angle ${\displaystyle \varphi }$), and the odd Zernike polynomials are defined as

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

(odd function over the azimuthal angle ${\displaystyle \varphi }$) where m and n are nonnegative integers with n ≥ m ≥ 0 (m = 0 for spherical Zernike polynomials), ${\displaystyle \varphi }$ is the azimuthal angle, ρ is the radial distance ${\displaystyle 0\leq \rho \leq 1}$, and ${\displaystyle R_{n}^{m}}$ are the radial polynomials defined below. Zernike polynomials have the property of being limited to a range of −1 to +1, i.e. ${\displaystyle |Z_{n}^{m}(\rho ,\varphi )|\leq 1}$. The radial polynomials ${\displaystyle R_{n}^{m}}$ are defined as

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

for an even number of n − m, while it is 0 for an odd number of n − m. A special value is

${\displaystyle R_{n}^{m}(1)=1.}$

Other representations

Rewriting the ratios of factorials in the radial part as products of binomials shows that the coefficients are integer numbers:

${\displaystyle R_{n}^{m}(\rho )=\sum _{k=0}^{\tfrac {n-m}{2}}(-1)^{k}{\binom {n-k}{k}}{\binom {n-2k}{{\tfrac {n-m}{2}}-k}}\rho ^{n-2k}}$.

A notation as terminating Gaussian hypergeometric functions is useful to reveal recurrences, to demonstrate that they are special cases of Jacobi polynomials, to write down the differential equations, etc.:

Zdroj:https://en.wikipedia.org?pojem=Zernike_polynomials
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