Abstract

Zernike polynomials provide a well known, orthogonal set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. A related set of orthogonal functions is given here which represent vector quantities, such as mapping distortion or wavefront gradient. These functions are generated from gradients of Zernike polynomials, made orthonormal using the Gram-Schmidt technique. This set provides a complete basis for representing vector fields that can be defined as a gradient of some scalar function. It is then efficient to transform from the coefficients of the vector functions to the scalar Zernike polynomials that represent the function whose gradient was fit. These new vector functions have immediate application for fitting data from a Shack-Hartmann wavefront sensor or for fitting mapping distortion for optical testing. A subsequent paper gives an additional set of vector functions consisting only of rotational terms with zero divergence. The two sets together provide a complete basis that can represent all vector distributions in a circular domain.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]

2007

P. C. V. Mallik, C. Zhao, J. H. Burge, "Measurement of a 2-meter flat using a pentaprism scanning system," Opt. Eng. 46, 023602 (2007).
[CrossRef]

2004

2000

1996

1995

1992

1982

1976

Acosta, E.

Bara, S.

Brown, T. G.

Burge, J. H.

P. C. V. Mallik, C. Zhao, J. H. Burge, "Measurement of a 2-meter flat using a pentaprism scanning system," Opt. Eng. 46, 023602 (2007).
[CrossRef]

Ellerbroek, B.

Gavrielides, A.

Harbers, G.

Kunst, P. J.

Lane, R. G.

Leibbrandt, G. W. R.

Mallik, P. C. V.

P. C. V. Mallik, C. Zhao, J. H. Burge, "Measurement of a 2-meter flat using a pentaprism scanning system," Opt. Eng. 46, 023602 (2007).
[CrossRef]

Moore, D. T.

Murphy, P. E.

Noll, R. J.

Rama, M. A.

Rios, S.

Tallon, M.

Upton, R.

Zhao, C.

P. C. V. Mallik, C. Zhao, J. H. Burge, "Measurement of a 2-meter flat using a pentaprism scanning system," Opt. Eng. 46, 023602 (2007).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

Opt. Eng.

P. C. V. Mallik, C. Zhao, J. H. Burge, "Measurement of a 2-meter flat using a pentaprism scanning system," Opt. Eng. 46, 023602 (2007).
[CrossRef]

Opt. Lett.

Other

M. Born and E. Wolf, Principles of Optics, (Pergamon Press, 1980) pg. 464-468.

See http://wyant.opt-sci.arizona.edu/zernikes/zernikes.htm.

J. H. Burge, Advanced Techniques for Measuring Primary Mirrors for Astronomical Telescopes, Ph. D. Dissertation, Optical Sciences, University of Arizona (1993).

DurangoTM Interferometry Software, Diffraction International, Minnetonka, MN.

C. Zhao and J. H. Burge, "Orthonormal vector polynomials in a unit circle, Part II: completing the basis set," to be submitted to Optics Express (2007).

T. M. Apostol, Linear Algebra: A First Course, with Applications to Differential Equations (John Wiley & Sons, 1997), pg. 111-114.

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Tables (7)

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Table 1. Gradient of Zernike polynomials

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Table 2. List of the inner products of the first 13 Zernike gradients

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Table 3. List of first 36 orthonormal vector polynomials S i as functions of Zernike gradients.

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Table 4. List of S polynomials expressed as linear combinations of Zernike polynomials.

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Table 5. The rules for writing S in terms of linear combinations of Zernikes.

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Table 6. Plots of first 12 S polynomials in a unit circle.

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Table 7 The first 37 Zernike polynomials according to Noll’s numbering:

Equations (16)

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Z even j = n + 1 R n m ( r ) 2 cos ( m θ ) Z odd j = n + 1 R n m ( r ) 2 sin ( m θ ) } m 0
Z j = n + 1 R n 0 ( r ) , m = 0
R n m ( r ) = s = 0 ( n m ) 2 ( 1 ) s ( n s ) ! s ! [ ( n + m ) 2 s ] ! [ ( n m ) 2 s ] ! r n 2 s
( A , B ) = 1 π ( A · B ) dx dy
S j = 1 2 n ( n + 1 ) Z j .
S j = 1 4 n ( n + 1 ) ( Z j n + 1 n 1 Z j ( n = n 2 , m = m ) )
S j = i ̂ S jx + j ̂ S jy
S jx = C a Z ja ( n 1 , m 1 ) + C a Z ja ( n 1 , m + 1 ) ,
S jy = C b Z jb ( n 1 , m 1 ) + C b Z jb ( n 1 , m + 1 ) .
ϕ j = 1 2 n ( n + 1 ) Z j .
ϕ j = 1 4 n ( n + 1 ) ( Z j n + 1 n 1 Z j ( n = n 2 , m = m ) ) ,
V = α i S i .
Φ = α i ϕ i ,
Φ = α i ϕ i = γ i Z i
γ j = α j ( n , m ) 2 n ( n + 1 ) n = m
γ j = α j ( n , m ) 4 n ( n + 1 ) α j ( n + 2 , m ) 4 ( n + 1 ) ( n + 2 ) n m

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