Zernike polynomials

In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike, winner of the 1953 Nobel Prize in Physics and the inventor of phase-contrast microscopy, they play an important role in beam optics.

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

and the odd ones as

where m and n are nonnegative integers with n ≥ m, φ is the azimuthal angle, ρ is the radial distance ${\displaystyle 0\leq \rho \leq 1}$, and R^m_n 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 R^m_n are defined as

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