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. Previously, we have developed a basis of functions generated from gradients of Zernike polynomials. Here, we complete the basis by adding a complementary set of functions with zero divergence – those which are defined locally as a rotation or curl.

© 2008 Optical Society of America

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References

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  1. C. Zhao and J. H. Burge, "Orthonormal vector polynomials in a unit circle, Part I: basis set derived from gradients of Zernike polynomials," Opt. Express 15, 18014-18024 (2007).
    [CrossRef] [PubMed]
  2. C. Zhao,  et al, "Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique," Proc. SPIE 6293, 62930k (2006).
    [CrossRef]
  3. R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-211(1976).
    [CrossRef]
  4. H. F. Davis and A. D. Snider, Introduction to Vector Analysis, (Wm. C. Brown Publisher, 1986).

2007

2006

C. Zhao,  et al, "Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique," Proc. SPIE 6293, 62930k (2006).
[CrossRef]

1976

J. Opt. Soc. Am.

Opt. Express

Proc. SPIE

C. Zhao,  et al, "Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique," Proc. SPIE 6293, 62930k (2006).
[CrossRef]

Other

H. F. Davis and A. D. Snider, Introduction to Vector Analysis, (Wm. C. Brown Publisher, 1986).

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Figures (1)

Fig. 1.
Fig. 1.

Relations between the S⃗ and T⃗ polynomials. The Laplacian vector fields are the overlap between S⃗ and T⃗. The dashed circles and associated solid arrows illustrate the local behaviors of the vectors in different sets after subtracting the local constant vector.

Tables (3)

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Table 1. List of the first 14 non-trivial S⃗ polynomials as linear combinations of Zernike polynomials.

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Table 2. Analytical expressions of the first 15 T⃗ polynomials.

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Table 3. Plots of the first 12 non-trivial T⃗ polynomials.

Equations (14)

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S j = ϕ j = i ̂ ϕ j x + j ̂ ϕ j y .
ϕ j = 1 2 n ( n + 1 ) Z j , for all j with n = m ,
ϕ j = 1 4 n ( n + 1 ) ( Z j n + 1 n 1 Z j ( n = n 2 , m = m ) ) , for all j with n m .
( A , B ) = 1 π ( A B ) dxdy ,
( S i , S j ) = 1 π ( ( ϕ i ) ( ϕ j ) ) dxdy = δ ij .
v = ϕ + × P ,
T = × P = [ i ̂ j ̂ k ̂ x y z P x P y P z ] .
P = ψ k , ̂
T i = × ( ψ i k ̂ ) = i ̂ ψ i y j ̂ ψ i x .
( T i , T j ) = ( ( ψ i y ) ( ψ j y ) + ( ψ i x ) ( ψ j x ) ) dxdy
= ( ( ψ i ) ( ψ j ) ) dxdy .
2 ϕ j 2 Z j ( 1 r r ( r r ) + 1 r 2 2 θ 2 ) [ r n ( cos n θ sin n θ ) ] = 0 ,
S d l = 0 .
T n ̂ d l = 0 ,

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