OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Stephen A. Burns
  • Vol. 23, Iss. 7 — Jul. 1, 2006
  • pp: 1657–1668

Wavefront expansion basis functions and their relationships

Guang-ming Dai  »View Author Affiliations

JOSA A, Vol. 23, Issue 7, pp. 1657-1668 (2006)

View Full Text Article

Enhanced HTML    Acrobat PDF (228 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



Wavefront expansion basis functions are important in representing ocular aberrations and phase perturbations due to atmospheric turbulence. A general discussion is presented for the conversions of the coefficients between any two sets of basis functions. Several popular sets of basis functions, namely, Zernike polynomials, Fourier series, and Taylor monomials, are discussed and the conversion matrices between any two of these basis functions are derived. Some analytical and numerical examples are given to demonstrate conversion of coefficients of different basis function sets.

© 2006 Optical Society of America

OCIS Codes
(000.3870) General : Mathematics
(010.1080) Atmospheric and oceanic optics : Active or adaptive optics
(010.1290) Atmospheric and oceanic optics : Atmospheric optics
(220.1010) Optical design and fabrication : Aberrations (global)
(330.4460) Vision, color, and visual optics : Ophthalmic optics and devices

ToC Category:
Atmospheric and Oceanic Optics

Original Manuscript: September 30, 2005
Manuscript Accepted: January 3, 2006

Virtual Issues
Vol. 1, Iss. 8 Virtual Journal for Biomedical Optics

Guang-ming Dai, "Wavefront expansion basis functions and their relationships," J. Opt. Soc. Am. A 23, 1657-1668 (2006)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. J. W. Hardy, J. E. Leferbre, and C. L. Koliopoulos, 'Real time atmospheric compensation,' J. Opt. Soc. Am. 67, 360-369 (1977). [CrossRef]
  2. R. Foy and A. Labeyrie, 'Feasibility of adaptive optics telescope with laser probe,' Astron. Astrophys. 152, L29-L31 (1985).
  3. F. Roddier, 'Curvature sensing and compensation: a new concept in adaptive optics,' Appl. Opt. 27, 1223-1225 (1988). [CrossRef] [PubMed]
  4. L. A. Thompson and C. S. Gardner, 'Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,' Nature 328, 229-231 (1987). [CrossRef]
  5. G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, 'First diffraction-limited astronomical images with adaptive optics,' Astron. Astrophys. 230, 29-32 (1990).
  6. J. Liang, B. Grimm, S. Goelz, and J. Bille, 'Objective measurement of the wave aberrations of the human eye with the use of a Hartmann-Shack wavefront sensor,' J. Opt. Soc. Am. A 11, 1949-1957 (1994). [CrossRef]
  7. J. Liang and D. R. Williams, 'Aberrations and retinal image quality of the normal human eye,' J. Opt. Soc. Am. A 14, 2873-2883 (1997). [CrossRef]
  8. J. Liang, D. R. Williams, and D. T. Miller, 'Supernormal vision and high-resolution retinal imaging through adaptive optics,' J. Opt. Soc. Am. A 14, 2884-2892 (1997). [CrossRef]
  9. A. Roorda and D. R. Williams, 'The arrangement of the three cone classes in the living human eye,' Nature 397, 520-522 (1999). [CrossRef] [PubMed]
  10. H. W. Babcock, 'The possibility of compensating astronomical seeing,' Publ. Astron. Soc. Pac. 65, 229-236 (1953). [CrossRef]
  11. W. H. Southwell, 'Wavefront estimation from wavefront slope measurements,' J. Opt. Soc. Am. 70, 998-1006 (1980). [CrossRef]
  12. E. P. Wallner, 'Optimal wavefront correction using slope measurement,' J. Opt. Soc. Am. 73, 1771-1776 (1983). [CrossRef]
  13. B. M. Welsh and C. S. Gardner, 'Performance analysis of adaptive-optics systems using laser guide stars and slope sensors,' J. Opt. Soc. Am. A 6, 1913-1923 (1989). [CrossRef]
  14. G.-m. Dai, 'Modal wavefront reconstruction with Zernike polynomials and Karhunen-Loève functions,' J. Opt. Soc. Am. A 13, 1218-1225 (1996). [CrossRef]
  15. A. J. E. M. Janssen, 'Extended Nijboer-Zernike approach for the computation of optical point-spread functions,' J. Opt. Soc. Am. A 19, 849-857 (2002). [CrossRef]
  16. J. Braat, P. Dirksen, and A. J. E. M. Janssen, 'Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions,' J. Opt. Soc. Am. A 19, 858-870 (2002). [CrossRef]
  17. W. J. Donnelly III and A. Roorda, 'The optimal pupil size in the eye for axial resolution,' J. Opt. Soc. Am. A 20, 2010-2015 (2003). [CrossRef]
  18. N. Roddier, 'Atmospheric wavefront simulation using Zernike polynomials,' Opt. Eng. 29, 1174-1180 (1990). [CrossRef]
  19. G.-m. Dai, 'Wavefront simulation for atmospheric turbulence,' in Image Reconstruction and Restoration, T.J.Schulz and D.-L.Snyder, eds., Proc. SPIE 2302, 62-72 (1994).
  20. R. C. Cannon, 'Optimal bases for wavefront simulation and reconstruction on annular apertures,' J. Opt. Soc. Am. A 13, 862-867 (1996). [CrossRef]
  21. H. Jakobsson, 'Simulations of time series of atmospherically distorted wave fronts,' Appl. Opt. 35, 1561-1565 (1996). [CrossRef] [PubMed]
  22. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1965), Chap 9.
  23. R. J. Noll, 'Zernike polynomials and atmospheric turbulence,' J. Opt. Soc. Am. 66, 207-211 (1976). [CrossRef]
  24. D. Malacara, Optical Shop Testing (Wiley, 1992), Chap. 13.
  25. G.-m. Dai, 'Modal compensation of atmospheric turbulence with the use of Zernike polynomials and Karhunen-Loève functions,' J. Opt. Soc. Am. A 12, 2182-2193 (1995). [CrossRef]
  26. V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffractive Optics (SPIE, 2002), Chap 5.
  27. H. C. Howland and B. Howland, 'A subjective method for the measurement of monochromatic aberrations of the eye,' J. Opt. Soc. Am. 67, 1508-1518 (1977). [CrossRef] [PubMed]
  28. G. Walsh, W. N. Charman, and H. C. Howland, 'Objective technique for the determination of monochromatic aberrations of the human eye,' J. Opt. Soc. Am. A 1, 987-992 (1984). [CrossRef] [PubMed]
  29. C. Cui and V. Laskshminarayanan, 'Choice of reference axis in ocular wavefront aberration measurement,' J. Opt. Soc. Am. A 15, 2488-2496 (1998). [CrossRef]
  30. K. R. Freischlad and C. L. Koliopoulos, 'Modal estimation of wave front difference measurements using the discrete Fourier transform,' J. Opt. Soc. Am. A 3, 1852-1861 (1986). [CrossRef]
  31. F. Roddier and C. Roddier, 'Wavefront reconstruction using iterative Fourier transform,' Appl. Opt. 30, 1325-1327 (1991). [CrossRef] [PubMed]
  32. L. A. Poyneer, D. T. Gavel, and J. M. Brase, 'Fast wavefront reconstruction in large adaptive optics systems with use of the Fourier transform,' J. Opt. Soc. Am. A 19, 2100-2111 (2002). [CrossRef]
  33. L. A. Poyneer and J.-P. Véran, 'Optimal modal Fourier-transform wavefront control,' J. Opt. Soc. Am. A 22, 1515-1526 (2005). [CrossRef]
  34. G.-m. Dai, 'Zernike aberration coefficients transformed to and from Fourier series coefficients for wavefront representation,' Opt. Lett. 31, 501-503 (2006). [CrossRef] [PubMed]
  35. L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, 'Standards for reporting the optical aberrations of eyes,' Vision Science and Its Applications, Vol. 35 of Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.
  36. R. K. Tyson, 'Zernike aberration coefficients from Seidel and higher-order power series aberration coefficients,' Opt. Lett. 7, 262-264 (1982). [CrossRef] [PubMed]
  37. G. Conforti, 'Zernike aberration coefficients from Seidel and higher-order power series coefficients,' Opt. Lett. 8, 390-391 (1983). [CrossRef]
  38. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Formulas, Graphs, and Mathematical Tables (Dover, 1972).
  39. P. R. Riera, G. S. Pankretz, and D. M. Topa, 'Efficient computation with special functions like the circle polynomials of Zernike,' in Optical Design and Analysis Software II, R.C.Juergens, ed., Proc. SPIE 4769, 130-144 (2002).
  40. J. Schwiegerling, 'Scaling Zernike expansion coefficients to different pupil sizes,' J. Opt. Soc. Am. A 19, 1937-1945 (2002). [CrossRef]
  41. 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, 209-217 (2003). [CrossRef]
  42. G.-m. Dai, 'Scaling Zernike expansion coefficients to smaller pupil sizes: a simpler formula,' J. Opt. Soc. Am. A 23, 539-543 (2006). [CrossRef]
  43. Equation of Ref. ; the notation in Ref. is slightly different from that used in this paper, but the two formulas are equivalent.

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.


Fig. 1

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited