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 where they are used to describe optical aberrations.
Definitions There are even and odd Zernike polynomials. The even Zernike polynomials of radial degree n ≥ 0 {\displaystyle n\geq 0} and azimuthal degree l = m {\displaystyle l=m} with 0 ≤ m ≤ n {\displaystyle 0\leq m\leq n} are defined as
Z n m ( ρ , φ ) = R n m ( ρ ) cos ( m φ ) {\displaystyle Z_{n}^{m}(\rho ,\varphi )=R_{n}^{m}(\rho )\,\cos(m\,\varphi )\!}
and are even function over the azimuthal angle φ {\displaystyle \varphi } . The odd Zernike polynomials with (negative) azimuthal degree l = − m {\displaystyle l=-m} , where 0 < m ≤ n {\displaystyle 0<m\leq n} , are defined as
Z n − m ( ρ , φ ) = R n m ( ρ ) sin ( m φ ) {\displaystyle Z_{n}^{-m}(\rho ,\varphi )=R_{n}^{m}(\rho )\,\sin(m\,\varphi )\!}
and are odd function over the azimuthal angle φ {\displaystyle \varphi } . Here, ρ {\displaystyle \rho } is the radial distance 0 ≤ ρ ≤ 1 {\displaystyle 0\leq \rho \leq 1} , and R n m {\displaystyle R_{n}^{m}} are the radial polynomials defined below. The Zernike polynomials are sometimes represented in a way explicitly showing o (odd) and e (even): e U n m ( ρ , φ ) = Z n m ( ρ , φ ) = R n m ( ρ ) cos ( m φ ) o U n m ( ρ , φ ) = Z n − m ( ρ , φ ) = R n m ( ρ ) sin ( m φ ) {\displaystyle {\begin{array}{lcl}^{e}\,U_{n}^{m}(\rho ,\varphi )&=&Z_{n}^{m}(\rho ,\varphi )&=&R_{n}^{m}(\rho )\,\cos(m\,\varphi )\\^{o}\,U_{n}^{m}(\rho ,\varphi )&=&Z_{n}^{-m}(\rho ,\varphi )&=&R_{n}^{m}(\rho )\,\sin(m\,\varphi )\end{array}}} Zernike polynomials have the property of being limited to a range of −1 to +1 in the unit disk, i.e. | Z n m ( ρ , φ ) | ≤ 1 {\displaystyle |Z_{n}^{m}(\rho ,\varphi )|\leq 1} if ρ ≤ 1 {\displaystyle \rho \leq 1} . The radial polynomials R n m {\displaystyle R_{n}^{m}} are defined as
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