OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 13 — Jun. 21, 2010
  • pp: 13851–13862

Robust and fast computation for the polynomials of optics

G. W. Forbes  »View Author Affiliations


Optics Express, Vol. 18, Issue 13, pp. 13851-13862 (2010)
http://dx.doi.org/10.1364/OE.18.013851


View Full Text Article

Enhanced HTML    Acrobat PDF (827 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

Mathematical methods that are poorly known in the field of optics are adapted and shown to have striking significance. Orthogonal polynomials are common tools in physics and optics, but problems are encountered when they are used to higher orders. Applications to arbitrarily high orders are shown to be enabled by remarkably simple and robust algorithms that are derived from well known recurrence relations. Such methods are demonstrated for a couple of familiar optical applications where, just as in other areas, there is a clear trend to higher orders.

© 2010 OSA

OCIS Codes
(000.3860) General : Mathematical methods in physics
(220.0220) Optical design and fabrication : Optical design and fabrication
(220.1250) Optical design and fabrication : Aspherics
(260.1960) Physical optics : Diffraction theory

ToC Category:
Optical Design and Fabrication

History
Original Manuscript: May 12, 2010
Revised Manuscript: June 2, 2010
Manuscript Accepted: June 3, 2010
Published: June 11, 2010

Citation
G. W. Forbes, "Robust and fast computation for the polynomials of optics," Opt. Express 18, 13851-13862 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-13851


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. M. Born, and E. Wolf, Principles of Optics (Cambridge, 1999), see Sec. 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). [CrossRef]
  5. D. R. Myrick, “A generalization of the radial polynomials of F. Zernike,” J. Soc. Ind. Appl. Math. 14(3), 476–492 (1966). [CrossRef]
  6. E. C. Kintner, “On the mathematical properties of the Zernike polynomials,” Opt. Acta (Lond.) 23, 679–680 (1976). [CrossRef]
  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). [CrossRef]
  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). [CrossRef]
  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). [CrossRef]
  12. H. E. Salzer, “A Recurrence Scheme for Converting from One Orthogonal Expansion into Another,” Commun. ACM 16(11), 705–707 (1973). [CrossRef]
  13. E. H. Doha, “On the coefficients of differentiated expansions and derivatives of Jacobi polynomials,” J. Phys. Math. Gen. 35(15), 3467–3478 (2002). [CrossRef]
  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). [CrossRef]
  15. B. Y. Ting and Y. L. Luke, “Conversion of Polynomials between Different Polynomial Bases,” IMA J. Numer. Anal. 1(2), 229–234 (1981). [CrossRef]
  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). [CrossRef]
  17. J. Schwiegerling, “Scaling Zernike expansion coefficients to different pupil sizes,” J. Opt. Soc. Am. A 19(10), 1937–1945 (2002). [CrossRef]
  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). [CrossRef]
  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). [CrossRef]
  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. Soc. Am. A 23(8), 1960–1966 (2006). [CrossRef]
  21. A. J. E. M. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlith. Microfab Microsyst. 5(3), 030501–3 (2006). [CrossRef]
  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 . [CrossRef] [PubMed]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited