The Zernike polynomials are a set of orthogonal polynomials that arise in the expansion of a wavefront function for optical systems with circular pupils. The odd and even Zernike polynomials are given by
(1)
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where the radial function is defined for and integers with by
(2)
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Here, is the azimuthal angle with and is the radial distance with (Prata and Rusch 1989). The even and odd polynomials are sometimes also denoted
(3)
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(4)
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Zernike polynomials are implemented in the Wolfram Language as ZernikeR[n, m, rho].
Other closed forms for include
(5)
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for odd and , where is the gamma function and is a hypergeometric function. This can also be written in terms of the Jacobi polynomial as
(6)
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The first few nonzero radial polynomials are
(7)
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(8)
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(9)
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(10)
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(11)
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(12)
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(13)
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(14)
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(15)
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(Born and Wolf 1989, p. 465).
The radial functions satisfy the orthogonality relation
(16)
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where is the Kronecker delta, and are related to the Bessel function of the first kind by
(17)
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(Born and Wolf 1989, p. 466). The radial Zernike polynomials have the generating function
(18)
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(correcting the typo of Born and Wolf) and are normalized so that
(19)
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(Born and Wolf 1989, p. 465).
The Zernike polynomials also satisfy the recurrence relations
(20)
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(Prata and Rusch 1989). The coefficients and in the expansion of an arbitrary radial function in terms of Zernike polynomials
(21)
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are given by
(22)
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where
(23)
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Let a "primary" aberration be given by
(24)
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with and where is the complex conjugate of , and define
(25)
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giving
(26)
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Then the types of primary aberrations are given in the following table (Born and Wolf 1989, p. 470).
aberration | |||||
spherical aberration | 0 | 4 | 0 | ||
coma | 0 | 3 | 1 | ||
astigmatism | 0 | 2 | 2 | ||
field curvature | 1 | 2 | 0 | ||
distortion | 1 | 1 | 1 |