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Applied Optics

Applied Optics


  • Editor: Jospeh N. Mait
  • Vol. 48, Iss. 3 — Jan. 20, 2009
  • pp: 477–488

Wavefront propagation from one plane to another with the use of Zernike polynomials and Taylor monomials

Guang-ming Dai, Charles E. Campbell, Li Chen, Huawei Zhao, and Dimitri Chernyak  »View Author Affiliations

Applied Optics, Vol. 48, Issue 3, pp. 477-488 (2009)

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In wavefront-driven vision correction, ocular aberrations are often measured on the pupil plane and the correction is applied on a different plane. The problem with this practice is that any changes undergone by the wavefront as it propagates between planes are not currently included in devising customized vision correction. With some valid approximations, we have developed an analytical foundation based on geometric optics in which Zernike polynomials are used to characterize the propagation of the wavefront from one plane to another. Both the boundary and the magnitude of the wavefront change after the propagation. Taylor monomials were used to realize the propagation because of their simple form for this purpose. The method we developed to identify changes in low-order aberrations was verified with the classical vertex correction formula. The method we developed to identify changes in high-order aberrations was verified with ZEMAX ray-tracing software. Although the method may not be valid for highly irregular wavefronts and it was only proven for wavefronts with low-order or high-order aberrations, our analysis showed that changes in the propagating wavefront are significant and should, therefore, be included in calculating vision correction. This new approach could be of major significance in calculating wavefront-driven vision correction whether by refractive surgery, contact lenses, intraocular lenses, or spectacles.

© 2009 Optical Society of America

OCIS Codes
(120.3890) Instrumentation, measurement, and metrology : Medical optics instrumentation
(170.1020) Medical optics and biotechnology : Ablation of tissue
(170.4460) Medical optics and biotechnology : Ophthalmic optics and devices
(220.2740) Optical design and fabrication : Geometric optical design
(330.4460) Vision, color, and visual optics : Ophthalmic optics and devices

ToC Category:
Medical Optics and Biotechnology

Original Manuscript: June 10, 2008
Revised Manuscript: September 15, 2008
Manuscript Accepted: October 29, 2008
Published: January 13, 2009

Virtual Issues
Vol. 4, Iss. 3 Virtual Journal for Biomedical Optics

Guang-ming Dai, Charles E. Campbell, Li Chen, Huawei Zhao, and Dimitri Chernyak, "Wavefront propagation from one plane to another with the use of Zernike polynomials and Taylor monomials," Appl. Opt. 48, 477-488 (2009)

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  1. J. Liang, W. Grimm, S. Goelz, and J. F. Bille, “Objective measurement of wave aberrations of the human eye with the use of a Hartmann-Shack wave-front sensor,” J. Opt. Soc. Am. A 11, 1949-1957 (1994). [CrossRef]
  2. J. Liang and D. R. Williams, “Aberrations and retinal image quality of the normal human eyes,” J. Opt. Soc. Am. A 14, 2873-2883 (1997). [CrossRef]
  3. 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]
  4. 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]
  5. G. Walsh, “The effect of mydriasis on the pupillary centration of the human eye,” Ophthal. Physiol. Opt. 8, 178-182 (1988). [CrossRef]
  6. M. A. Wilson, M. C. W. Campbell, and P. Simonet, “Change of pupil centration with change of illumination and pupil size,” Optom. Vis. Sci. 69, 129-136 (1992). [CrossRef] [PubMed]
  7. E. Donnenfeld, “The pupil is a moving target: centration, repeatability, and registration,” J. Refract. Surg. 20, 593-596 (2004).
  8. D. A. Chernyak, “Cyclotorsional eye motion occurring between wavefront measurement and refractive surgery,” J. Cataract Refract. Surg. 30, 633-638 (2004). [CrossRef] [PubMed]
  9. A. Guirao, D. Williams, and I. Cox, “Effect of the rotation and translation on the expected benefit of an ideal method to correct the eye's high-order aberrations,” J. Opt. Soc. Am. A 18, 1003-1015 (2001). [CrossRef]
  10. K. A. Goldberg and K. Geary, “Wave-front measurement errors from restricted concentric subdomains,” J. Opt. Soc. Am. A 18, 2146-2152 (2001). [CrossRef]
  11. J. Schwiegerling, “Scaling Zernike expansion coefficients to different pupil sizes,” J. Opt. Soc. Am. A 19, 1937-1945 (2002). [CrossRef]
  12. 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]
  13. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207-211 (1976). [CrossRef]
  14. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  15. G.-M. Dai, “Scaling Zernike expansion coefficients to smaller pupil sizes: a simpler formula,” J. Opt. Soc. Am. A 23, 539-543(2006). [CrossRef]
  16. H. Shu, L. Luo, and G. Han, “General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes,” J. Opt. Soc. Am. A 23, 1960-1968 (2006). [CrossRef]
  17. A. J. E. M. Janssen and P. Dirksen, “A concise formula for the Zernike coefficients of scaled pupils,” J. Microlith. Microfab. Microsyst. 5, 030501 (2006). [CrossRef]
  18. S. Bará, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated and/or displaced pupils,” J. Opt. Soc. Am. A 23, 2061-2066 (2006). [CrossRef]
  19. L. Lundström and P. Unsbo, “Transformation of Zernike coefficients: scaled, translated, and rotated wavefronts with circular and elliptical pupils,” J. Opt. Soc. Am. A 24, 569-577 (2007). [CrossRef]
  20. S. Bará, T. Mancebo, and E. Moreno-Barriuso, “Positioning tolerances for phase plates compensating aberrations of the human eye,” Appl. Opt. 39, 3413-3420 (2000). [CrossRef]
  21. W. F. Harris, “Wavefronts and their propagation in astigmatic systems,” Optom. Vis. Sci. 73, 606-612 (1996). [CrossRef] [PubMed]
  22. L. N. Thibos, “Propagation of astigmatic wavefronts using power vectors,” S. Afr. Optom. 62, 111-113 (2003).
  23. P. R. Riera, G. S. Pankretz, and D. M. Topa, “Efficient computation with special functions like the circle polynomials of Zernike,” Proc. SPIE 4769, 130-144 (2002). [CrossRef]
  24. G.-M. Dai, “Wavefront expansion basis functions and their relationships,” J. Opt. Soc. Am. A 23, 1657-1666 (2006). [CrossRef]
  25. G.-M. Dai, “Wavefront expansion basis functions and their relationships: errata,” J. Opt. Soc. Am. A 23, 2970-2971(2006). [CrossRef]
  26. American National Standards Institute, “Methods for reporting optical aberrations of eyes,” ANSI Z80.28-2004 (Optical Laboratories Association, 2004), Annex B, pp. 19-28.
  27. D. A. Atchison, D. H. Scott, and W. N. Charman, “Hartmann-Shack technique and refraction across the horizontal visual field,” J. Opt. Soc. Am. A 20, 965-973 (2003). [CrossRef]

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