OFFSET
0,2
COMMENTS
The natural arrangement of the indices n (radial index) and m (azimuthal index) of the Zernike polynomial Z(n,m) is a triangle with row index n, in each row m ranging from -n to n in steps of 2:
(0,0)
(1,-1) (1,1)
(2,-2) (2,0) (2,2)
(3,-3) (3,-1) (3,1) (3,3)
(4,-4) (4,-2) (4,0) (4,2) (4,4)
(5,-5) (5,-3) (5,-1) (5,1) (5,3) (5,5)
(6,-6) (6,-4) (6,-2) (6,0) (6,2) (6,4) (6,6)
(7,-7) (7,-5) (7,-3) (7,-1) (7,1) (7,3) (7,5) (7,7)
For uses in linear algebra related to beam optics, a standard scheme of assigning a single index j>=1 to each double-index (n,m) has become a de-facto standard, proposed by Noll. The triangle of the j at the equivalent positions reads
1,
3,2,
5,4,6,
9,7,8,10,
15,13,11,12,14,
21,19,17,16,18,20,
27,25,23,22,24,26,28,
35,33,31,29,30,32,34,36,
which defines the OEIS entries. The rule of translation is that odd j are assigned to m<0, even j to m>=0, and smaller j to smaller |m|.
LINKS
N. Chetty, D. J. Griffith, Zernike-basis expansion of the fractional and radial Hilbert phase masks, Current Applied Physics, 15 (2015) 739-747
R. J. Noll, Zernike polynomials and atmospheric turbulence, J. Opt. Soc. Am 66 (1976) 207.
Thomas Risse, Least Square Approximation with Zernike Polynomials Using SAGE, (2011).
Wikipedia, Zernike Polynomials
CROSSREFS
KEYWORD
AUTHOR
R. J. Mathar, Dec 08 2010
STATUS
approved