Robust and fast computation for the polynomials of optics

Optics Express

Volumn 18, Issue 13, 2010, Pages 13851-13862

  Forbes, G W ^a  

^a QED TECHNOLOGIES   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

FAST COMPUTATION; HIGH ORDER; HIGHER ORDER; MATHEMATICAL METHOD; OPTICAL APPLICATIONS; ORTHOGONAL POLYNOMIAL; RECURRENCE RELATIONS; ROBUST ALGORITHM;

ALGORITHM; ARTICLE; METHODOLOGY; OPTICS; PHYSICS; THEORETICAL MODEL;

ALGORITHMS; MODELS, THEORETICAL; OPTICS AND PHOTONICS; PHYSICS;

EID: 77953815431     PISSN: None     EISSN: 10944087     Source Type: Journal    
DOI: 10.1364/OE.18.013851     Document Type: Article

References (22)

1. M. Born, and E. Wolf, Principles of Optics (Cambridge, 1999), see Seo. 9.2 and Appendix VII.
2. A. E. Siegman, Lasers (University Science Books, 1986), Chaps. 16-17.
3. M. Abramowitz, and I. Stegun, Handbook of Mathematical Functions (Dover, 1978), Chap. 22.
4. A. B. Bhatia, E. Wolf, and M. Born, "On the circle polynomials of Zernike and related orthogonal sets," Proc. Camb. Philos. Soc. 50(01), 40-48 (1954).
5. D. R. Myrick, "A generalization of the radial polynomials of F. Zernike," J. Soc. Ind. Appl. Math. 14(3), 476-492 (1966).
6. E. C. Kintner, "On the mathematical properties of the Zernike polynomials," Opt. Aota (Lond.) 23, 679-680 (1976).
7. R. Barakat, "Optimum balanced wave-front aberrations for radially symmetric amplitude distributions: generalizations of Zernike polynomials," J. Opt. Soc. Am. 70(6), 739-742 (1980).
8. C.-W. Chong, P. Raveendran, and R. Mukundan, "A comparative analysis of algorithms for fast computation of Zernike moments," Pattern Recognit. 36(3), 731-742 (2003).
9. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1992) Section 5.5.
10. C. W. Clenshaw, "A Note on the Summation of Chebyshev Series," Math. Tables Other Aids Comput. 9, 118-120 (1955), http://www.jstor.org/stable/ 2002068.
11. F.J. Smith, "An Algorithm for Summing Orthogonal Polynomial Series and their Derivatives with Applications to Curve-Fitting and Interpolation," Math. Comput. 19(89), 33-36 (1965).
12. H. E. Salzer, "A Recurrence Scheme for Converting from One Orthogonal Expansion into Another," Commun. ACM 16(11), 705-707 (1973).
13. E. H. Doha, "On the coefficients of differentiated expansions and derivatives of Jacobi polynomials," J. Phys. Math. Gen. 35(15), 3467-3478 (2002).
14. R. Barrio, and J. M. Peña, "Numerical evaluation of the p'th derivative of Jacobi series," Appl. Numer. Math. 43(4), 335-357 (2002).
15. B. Y. Ting, and Y. L. Luke, "Conversion of Polynomials between Different Polynomial Bases," IMA J. Numer. Anal. 1(2), 229-234 (1981).
16. K. A. Goldberg, and K. Geary, "Wave-front measurement errors from restricted concentric subdomains," J. Opt. Soc. Am. A 18(9), 2146-2152 (2001).
17. J. Schwiegerling, "Scaling Zernike expansion coefficients to different pupil sizes," J. Opt. Soc. Am. A 19(10), 1937-1945 (2002).
18. C. E. Campbell, "Matrix method to find a new set of Zernike coefficients from an original set when the aperture radius is changed," J. Opt. Soc. Am. A 20(2), 209-217 (2003).
19. G. M. Dai, "Scaling Zernike expansion coefficients to smaller pupil sizes: a simpler formula," J. Opt. Soc. Am. A 23(3), 539-543 (2006).
20. H. Shu, L. Luo, G. Han, and J.-L. Coatrieux, "General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes," J. Opt. Soo. Am. A 23(8), 1960-1966 (2006).
21. A. J. E. M. Janssen, and P. Dirksen, "Concise formula for the Zernike coefficients of scaled pupils," J. Microlith. Microfab Microsyst. 5(3), 030501-30503 (2006).
22. G. W. Forbes, "Shape specification for axially symmetric optical surfaces," Opt. Express 15(8), 5218-5226 (2007), http://www.opticsinfobase. org/oe/abstract.cfm?URI=oe-15-8-5218.