In the aberration analysis of a circular wavefront, Zernike circle
polynomials are used to obtain its wave aberration coefficients. To
obtain these coefficients from the wavefront slope data, we need
vector functions that are orthogonal to the gradients of the Zernike
polynomials, and are irrotational so as to propagate minimum
uncorrelated random noise from the data to the coefficients. In this
paper, we derive such vector functions, which happen to be
polynomials.
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The indices ,
, and
are called the polynomial
number, the radial degree, and the azimuthal frequency,
respectively. The polynomials are ordered such that an
even corresponds to a
symmetric polynomial varying as , whereas an odd
corresponds to an
antisymmetric polynomial varying as
. A polynomial with a
lower value of is ordered first, and for
a given value of , a polynomial with a
lower value of is ordered first.
The words “orthonormal Zernike” are to be
associated with these names, e.g., orthonormal
Zernike 0° primary astigmatism.
Table 2.
Vector Polynomials Orthogonal to the
Gradient of Zernike Polynomials for
The indices ,
, and
are called the polynomial
number, the radial degree, and the azimuthal frequency,
respectively. The polynomials are ordered such that an
even corresponds to a
symmetric polynomial varying as , whereas an odd
corresponds to an
antisymmetric polynomial varying as
. A polynomial with a
lower value of is ordered first, and for
a given value of , a polynomial with a
lower value of is ordered first.
The words “orthonormal Zernike” are to be
associated with these names, e.g., orthonormal
Zernike 0° primary astigmatism.
Table 2.
Vector Polynomials Orthogonal to the
Gradient of Zernike Polynomials for