Abstract

A Shack–Hartmann aberrometer was used to measure the monochromatic aberration structure along the primary line of sight of 200 cyclopleged, normal, healthy eyes from 100 individuals. Sphero-cylindrical refractive errors were corrected with ophthalmic spectacle lenses based on the results of a subjective refraction performed immediately prior to experimentation. Zernike expansions of the experimental wave-front aberration functions were used to determine aberration coefficients for a series of pupil diameters. The residual Zernike coefficients for defocus were not zero but varied systematically with pupil diameter and with the Zernike coefficient for spherical aberration in a way that maximizes visual acuity. We infer from these results that subjective best focus occurs when the area of the central, aberration-free region of the pupil is maximized. We found that the population averages of Zernike coefficients were nearly zero for all of the higher-order modes except spherical aberration. This result indicates that a hypothetical average eye representing the central tendency of the population is nearly free of aberrations, suggesting the possible influence of an emmetropization process or evolutionary pressure. However, for any individual eye the aberration coefficients were rarely zero for any Zernike mode. To first approximation, wave-front error fell exponentially with Zernike order and increased linearly with pupil area. On average, the total wave-front variance produced by higher-order aberrations was less than the wave-front variance of residual defocus and astigmatism. For example, the average amount of higher-order aberrations present for a 7.5-mm pupil was equivalent to the wave-front error produced by less than 1/4 diopter (D) of defocus. The largest pupil for which an eye may be considered diffraction-limited was 1.22 mm on average. Correlation of aberrations from the left and right eyes indicated the presence of significant bilateral symmetry. No evidence was found of a universal anatomical feature responsible for third-order optical aberrations. Using the Marechal criterion, we conclude that correction of the 12 largest principal components, or 14 largest Zernike modes, would be required to achieve diffraction-limited performance on average for a 6-mm pupil. Different methods of computing population averages provided upper and lower limits to the mean optical transfer function and mean point-spread function for our population of eyes.

© 2002 Optical Society of America

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    [CrossRef]
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  22. Critical pupil diameter should not be confused with optimum pupil diameter, defined as that pupil size which maximizes image quality. The latter is more difficult to quantify satisfactorily because of the lack of a universally accepted metric of image quality.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  32. R. A. Applegate, V. Lakshminarayanan, “Parametric representation of Stiles–Crawford functions: normal variation of peak location and directionality,” J. Opt. Soc. Am. A 10, 1611–1623 (1993).
    [CrossRef] [PubMed]
  33. D. A. Atchison, A. Joblin, G. Smith, “Influence of Stiles–Crawford effect apodization on spatial visual performance,” J. Opt. Soc. Am. A 15, 2545–2551 (1998).
    [CrossRef]
  34. S. Marcos, E. Moreno, R. Navarro, “The depth-of-field of the human eye from objective and subjective measurements,” Vision Res. 39, 2039–2049 (1999).
    [CrossRef] [PubMed]
  35. S. Marcos, S. A. Burns, “On the symmetry between eyes of wavefront aberration and cone directionality,” Vision Res. 40, 2437–2447 (2000).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  40. Readers unfamiliar with Zernike analysis should beware of the confusing, dual-usage of terms such as “spherical aberration” and “defocus.” The difference between classical Seidel usage and the more modern Zernike usage is analogous to the difference between the spherical power in a conventional prescription of refractive error and the mean spherical equivalent used in power vector notation. If one extracts the mean spherical equivalent from a conventional prescription, the remainder is an orthogonal component called “Jackson crossed cylinder” to avoid confusion with the term “cylinder” used in conventional prescriptions. Similarly, if one extracts mean sphere from Seidel spherical aberration the result is Zernike spherical aberration. Unfortunately, authors sometimes fail to explicitly state which definition they have in mind when using common names for aberrations. Thus readers must rely on context to resolve the ambiguity.
  41. P. Mouroulis, “Aberration and image quality representation for visual optical systems,” in Visual Instrumentation: Optical Design and Engineering Principles, P. Mouroulis, ed. (McGraw-Hill, New York, 1999), pp. 27–68.
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    [CrossRef]
  43. The same argument in a more familiar context would arise in a statistical analysis of the Fourier coefficients obtained from physiological responses to square-wave stimulation. Fourier expansion of a square wave yields a strict relationship between the amplitudes of the various harmonic components. Therefore, a quasi-square-wave-response waveform should still show evidence of a predictable correlation between Fourier coefficients.
  44. S. M. MacRae, R. R. Krueger, R. A. Applegate, Customized Corneal Ablation: The Quest for Super Vision (Slack, Thorofare, N.J., 2001).
  45. L. N. Piotrowski, F. W. Campbell, “A demonstration of the visual importance and flexibility of spatial-frequency amplitude and phase,” Perception 11, 337–346 (1982).
    [CrossRef] [PubMed]
  46. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  47. M. C. Rynders, B. A. Lidkea, W. J. Chisholm, L. N. Thibos, “Statistical distribution of foveal transverse chromatic aberration, pupil centration, and angle psi in a population of young adult eyes,” J. Opt. Soc. Am. A 12, 2348–2357 (1995).
    [CrossRef]

2001 (2)

J. S. McLellan, S. Marcos, S. A. Burns, “Agerelated changes in monochromatic wave aberrations of the human eye,” Invest. Ophthalmol. Visual Sci. 42, 1390–1395 (2001).

J. Porter, A. Guirao, I. G. Cox, D. R. Williams, “The human eye’s monochromatic aberrations in a large population,” J. Opt. Soc. Am. A 18, 1793–1803 (2001).
[CrossRef]

2000 (1)

S. Marcos, S. A. Burns, “On the symmetry between eyes of wavefront aberration and cone directionality,” Vision Res. 40, 2437–2447 (2000).
[CrossRef] [PubMed]

1999 (4)

S. Marcos, E. Moreno, R. Navarro, “The depth-of-field of the human eye from objective and subjective measurements,” Vision Res. 39, 2039–2049 (1999).
[CrossRef] [PubMed]

A. Guirao, C. Gonzalez, M. Redondo, E. Geraghty, S. Norrby, P. Artal, “Average optical performance of the human eye as a function of age in a normal population,” Invest. Ophthalmol. Visual Sci. 40, 203–213 (1999).

L. N. Thibos, X. Hong, “Clinical applications of the Shack–Hartmann aberrometer,” Optom. Vision Sci. 76, 817–825 (1999).
[CrossRef]

X. Zhang, M. Ye, A. Bradley, L. Thibos, “Apodization by the Stiles–Crawford effect moderates the visual impact of retinal image defocus,” J. Opt. Soc. Am. A 16, 812–820 (1999).
[CrossRef]

1998 (4)

1997 (3)

J. Liang, D. R. Williams, “Aberrations and retinal image quality of the normal human eye,” J. Opt. Soc. Am. A 14, 2873–2883 (1997).
[CrossRef]

L. N. Thibos, W. Wheeler, D. G. Horner, “Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error,” Optom. Vision Sci. 74, 367–375 (1997).
[CrossRef]

D. A. Atchison, W. N. Charman, R. L. Woods, “Subjective depth-of-focus of the eye,” Optom. Vision Sci. 74, 511–520 (1997).
[CrossRef]

1995 (1)

1994 (1)

1993 (1)

1992 (1)

1991 (1)

W. N. Charman, “Wavefront aberrations of the eye: a review,” Optom. Vision Sci. 68, 574–583 (1991).
[CrossRef]

1990 (1)

M. C. W. Campbell, E. M. Harrison, P. Simonet, “Psychophysical measurement of the blur on the retina due to optical aberrations of the eye,” Vision Res. 30, 1587–1602 (1990).
[CrossRef] [PubMed]

1986 (2)

P. A. Howarth, A. Bradley, “The longitudinal chromatic aberration of the human eye, and its correction,” Vision Res. 26, 361–366 (1986).
[CrossRef] [PubMed]

V. N. Mahajan, “Uniform versus Gaussian beams: a comparison of the effects of diffraction, abscuration, and aberrations,” J. Opt. Soc. Am. A 3, 470–485 (1986).
[CrossRef]

1984 (1)

1982 (1)

L. N. Piotrowski, F. W. Campbell, “A demonstration of the visual importance and flexibility of spatial-frequency amplitude and phase,” Perception 11, 337–346 (1982).
[CrossRef] [PubMed]

1978 (1)

W. N. Charman, J. A. M. Jennings, H. Whitefoot, “The refraction of the eye in relation to spherical aberration and pupil size,” Br. J. Physiol. Opt. 32, 78–93 (1978).

1977 (1)

1975 (2)

J. A. Van Loo, J. M. Enoch, “The scotopic Stiles–Crawford effect,” Vision Res. 13, 1005–1009 (1975).
[CrossRef]

J. Tucker, W. N. Charman, “The depth-of-focus of the human eye for Snellen letters,” Am. J. Optom. Physiol. Opt. 52, 3–21 (1975).
[CrossRef] [PubMed]

1961 (1)

M. S. Smirnov, “Measurement of the wave aberration of the human eye,” Biofizika 6, 687–703 (1961).

1957 (1)

F. W. Campbell, “The depth-of-field of the human eye,” Opt. Acta 4, 157–164 (1957).
[CrossRef]

1951 (1)

1949 (1)

1947 (1)

A. Marechal, “Etude des effets combinés de la diffraction et des aberrations géométriques sur l’image d’un point lumineux,” Rev. d’Optique 26, 257–277 (1947).

Applegate, R. A.

R. A. Applegate, V. Lakshminarayanan, “Parametric representation of Stiles–Crawford functions: normal variation of peak location and directionality,” J. Opt. Soc. Am. A 10, 1611–1623 (1993).
[CrossRef] [PubMed]

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, “Standards for reporting the optical aberrations of eyes,” in Vision Science and Its Applications, Vol. 35 of Trends in Optics and Photonics Series, V. Lakshminarayanan, ed. (Optical Society of America, Washington, D.C., 2000), pp. 232–244.

S. M. MacRae, R. R. Krueger, R. A. Applegate, Customized Corneal Ablation: The Quest for Super Vision (Slack, Thorofare, N.J., 2001).

Artal, P.

A. Guirao, C. Gonzalez, M. Redondo, E. Geraghty, S. Norrby, P. Artal, “Average optical performance of the human eye as a function of age in a normal population,” Invest. Ophthalmol. Visual Sci. 40, 203–213 (1999).

I. Iglesias, E. Berrio, P. Artal, “Estimates of the ocular wave aberration from pairs of double-pass retinal images,” J. Opt. Soc. Am. A 15, 2466–2476 (1998).
[CrossRef]

Atchison, D. A.

D. A. Atchison, A. Joblin, G. Smith, “Influence of Stiles–Crawford effect apodization on spatial visual performance,” J. Opt. Soc. Am. A 15, 2545–2551 (1998).
[CrossRef]

D. A. Atchison, W. N. Charman, R. L. Woods, “Subjective depth-of-focus of the eye,” Optom. Vision Sci. 74, 511–520 (1997).
[CrossRef]

Berrio, E.

Bille, J.

Bradley, A.

Burns, S. A.

J. S. McLellan, S. Marcos, S. A. Burns, “Agerelated changes in monochromatic wave aberrations of the human eye,” Invest. Ophthalmol. Visual Sci. 42, 1390–1395 (2001).

S. Marcos, S. A. Burns, “On the symmetry between eyes of wavefront aberration and cone directionality,” Vision Res. 40, 2437–2447 (2000).
[CrossRef] [PubMed]

J. C. He, S. Marcos, R. H. Webb, S. A. Burns, “Measurement of the wave-front aberration of the eye by a fast psychophysical procedure,” J. Opt. Soc. Am. A 15, 2449–2456 (1998).
[CrossRef]

Campbell, F. W.

L. N. Piotrowski, F. W. Campbell, “A demonstration of the visual importance and flexibility of spatial-frequency amplitude and phase,” Perception 11, 337–346 (1982).
[CrossRef] [PubMed]

F. W. Campbell, “The depth-of-field of the human eye,” Opt. Acta 4, 157–164 (1957).
[CrossRef]

Campbell, M. C. W.

M. C. W. Campbell, E. M. Harrison, P. Simonet, “Psychophysical measurement of the blur on the retina due to optical aberrations of the eye,” Vision Res. 30, 1587–1602 (1990).
[CrossRef] [PubMed]

Charman, W. N.

D. A. Atchison, W. N. Charman, R. L. Woods, “Subjective depth-of-focus of the eye,” Optom. Vision Sci. 74, 511–520 (1997).
[CrossRef]

W. N. Charman, “Wavefront aberrations of the eye: a review,” Optom. Vision Sci. 68, 574–583 (1991).
[CrossRef]

G. Walsh, W. N. Charman, 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]

W. N. Charman, J. A. M. Jennings, H. Whitefoot, “The refraction of the eye in relation to spherical aberration and pupil size,” Br. J. Physiol. Opt. 32, 78–93 (1978).

J. Tucker, W. N. Charman, “The depth-of-focus of the human eye for Snellen letters,” Am. J. Optom. Physiol. Opt. 52, 3–21 (1975).
[CrossRef] [PubMed]

Chisholm, W. J.

Cox, I. G.

Enoch, J. M.

J. A. Van Loo, J. M. Enoch, “The scotopic Stiles–Crawford effect,” Vision Res. 13, 1005–1009 (1975).
[CrossRef]

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Geraghty, E.

A. Guirao, C. Gonzalez, M. Redondo, E. Geraghty, S. Norrby, P. Artal, “Average optical performance of the human eye as a function of age in a normal population,” Invest. Ophthalmol. Visual Sci. 40, 203–213 (1999).

Goelz, S.

Gonzalez, C.

A. Guirao, C. Gonzalez, M. Redondo, E. Geraghty, S. Norrby, P. Artal, “Average optical performance of the human eye as a function of age in a normal population,” Invest. Ophthalmol. Visual Sci. 40, 203–213 (1999).

Grimm, B.

Guirao, A.

J. Porter, A. Guirao, I. G. Cox, D. R. Williams, “The human eye’s monochromatic aberrations in a large population,” J. Opt. Soc. Am. A 18, 1793–1803 (2001).
[CrossRef]

A. Guirao, C. Gonzalez, M. Redondo, E. Geraghty, S. Norrby, P. Artal, “Average optical performance of the human eye as a function of age in a normal population,” Invest. Ophthalmol. Visual Sci. 40, 203–213 (1999).

Harrison, E. M.

M. C. W. Campbell, E. M. Harrison, P. Simonet, “Psychophysical measurement of the blur on the retina due to optical aberrations of the eye,” Vision Res. 30, 1587–1602 (1990).
[CrossRef] [PubMed]

He, J. C.

Hong, X.

L. N. Thibos, X. Hong, “Clinical applications of the Shack–Hartmann aberrometer,” Optom. Vision Sci. 76, 817–825 (1999).
[CrossRef]

Horner, D. G.

L. N. Thibos, W. Wheeler, D. G. Horner, “Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error,” Optom. Vision Sci. 74, 367–375 (1997).
[CrossRef]

Howarth, P. A.

P. A. Howarth, A. Bradley, “The longitudinal chromatic aberration of the human eye, and its correction,” Vision Res. 26, 361–366 (1986).
[CrossRef] [PubMed]

Howland, B.

Howland, H. C.

Iglesias, I.

Jackson, J. E.

J. E. Jackson, A User’s Guide To Principal Components (Wiley, New York, 1991).

Jennings, J. A. M.

W. N. Charman, J. A. M. Jennings, H. Whitefoot, “The refraction of the eye in relation to spherical aberration and pupil size,” Br. J. Physiol. Opt. 32, 78–93 (1978).

Joblin, A.

Koomen, M.

Krueger, R. R.

S. M. MacRae, R. R. Krueger, R. A. Applegate, Customized Corneal Ablation: The Quest for Super Vision (Slack, Thorofare, N.J., 2001).

Lakshminarayanan, V.

Liang, J.

Lidkea, B. A.

MacRae, S. M.

S. M. MacRae, R. R. Krueger, R. A. Applegate, Customized Corneal Ablation: The Quest for Super Vision (Slack, Thorofare, N.J., 2001).

Mahajan, V. N.

Marcos, S.

J. S. McLellan, S. Marcos, S. A. Burns, “Agerelated changes in monochromatic wave aberrations of the human eye,” Invest. Ophthalmol. Visual Sci. 42, 1390–1395 (2001).

S. Marcos, S. A. Burns, “On the symmetry between eyes of wavefront aberration and cone directionality,” Vision Res. 40, 2437–2447 (2000).
[CrossRef] [PubMed]

S. Marcos, E. Moreno, R. Navarro, “The depth-of-field of the human eye from objective and subjective measurements,” Vision Res. 39, 2039–2049 (1999).
[CrossRef] [PubMed]

J. C. He, S. Marcos, R. H. Webb, S. A. Burns, “Measurement of the wave-front aberration of the eye by a fast psychophysical procedure,” J. Opt. Soc. Am. A 15, 2449–2456 (1998).
[CrossRef]

Marechal, A.

A. Marechal, “Etude des effets combinés de la diffraction et des aberrations géométriques sur l’image d’un point lumineux,” Rev. d’Optique 26, 257–277 (1947).

McLellan, J. S.

J. S. McLellan, S. Marcos, S. A. Burns, “Agerelated changes in monochromatic wave aberrations of the human eye,” Invest. Ophthalmol. Visual Sci. 42, 1390–1395 (2001).

Moreno, E.

S. Marcos, E. Moreno, R. Navarro, “The depth-of-field of the human eye from objective and subjective measurements,” Vision Res. 39, 2039–2049 (1999).
[CrossRef] [PubMed]

Mouroulis, P.

P. Mouroulis, “Aberration and image quality representation for visual optical systems,” in Visual Instrumentation: Optical Design and Engineering Principles, P. Mouroulis, ed. (McGraw-Hill, New York, 1999), pp. 27–68.

Navarro, R.

S. Marcos, E. Moreno, R. Navarro, “The depth-of-field of the human eye from objective and subjective measurements,” Vision Res. 39, 2039–2049 (1999).
[CrossRef] [PubMed]

Norrby, S.

A. Guirao, C. Gonzalez, M. Redondo, E. Geraghty, S. Norrby, P. Artal, “Average optical performance of the human eye as a function of age in a normal population,” Invest. Ophthalmol. Visual Sci. 40, 203–213 (1999).

Piotrowski, L. N.

L. N. Piotrowski, F. W. Campbell, “A demonstration of the visual importance and flexibility of spatial-frequency amplitude and phase,” Perception 11, 337–346 (1982).
[CrossRef] [PubMed]

Porter, J.

Redondo, M.

A. Guirao, C. Gonzalez, M. Redondo, E. Geraghty, S. Norrby, P. Artal, “Average optical performance of the human eye as a function of age in a normal population,” Invest. Ophthalmol. Visual Sci. 40, 203–213 (1999).

Rynders, M. C.

Salmon, T. O.

Schwiegerling, J. T.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, “Standards for reporting the optical aberrations of eyes,” in Vision Science and Its Applications, Vol. 35 of Trends in Optics and Photonics Series, V. Lakshminarayanan, ed. (Optical Society of America, Washington, D.C., 2000), pp. 232–244.

Scolnik, R.

Simonet, P.

M. C. W. Campbell, E. M. Harrison, P. Simonet, “Psychophysical measurement of the blur on the retina due to optical aberrations of the eye,” Vision Res. 30, 1587–1602 (1990).
[CrossRef] [PubMed]

Sliney, D.

D. Sliney, M. Wolbarsht, Safety with Lasers and Other Optical Sources (Plenum, New York, 1980).

Smirnov, M. S.

M. S. Smirnov, “Measurement of the wave aberration of the human eye,” Biofizika 6, 687–703 (1961).

Smith, G.

Thibos, L.

Thibos, L. N.

L. N. Thibos, X. Hong, “Clinical applications of the Shack–Hartmann aberrometer,” Optom. Vision Sci. 76, 817–825 (1999).
[CrossRef]

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[CrossRef]

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[CrossRef]

M. C. Rynders, B. A. Lidkea, W. J. Chisholm, L. N. Thibos, “Statistical distribution of foveal transverse chromatic aberration, pupil centration, and angle psi in a population of young adult eyes,” J. Opt. Soc. Am. A 12, 2348–2357 (1995).
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L. N. Thibos, A. Bradley, “Modeling the refractive and neuro-sensor systems of the eye,” in Visual Instrumentation: Optical Design and Engineering Principles, P. Mouroulis, ed. (McGraw-Hill, New York, 1999), pp. 101–159.

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L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, “Standards for reporting the optical aberrations of eyes,” in Vision Science and Its Applications, Vol. 35 of Trends in Optics and Photonics Series, V. Lakshminarayanan, ed. (Optical Society of America, Washington, D.C., 2000), pp. 232–244.

Webb, R. H.

Wheeler, W.

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W. N. Charman, J. A. M. Jennings, H. Whitefoot, “The refraction of the eye in relation to spherical aberration and pupil size,” Br. J. Physiol. Opt. 32, 78–93 (1978).

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[CrossRef]

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[CrossRef]

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[CrossRef]

Other (14)

Readers unfamiliar with Zernike analysis should beware of the confusing, dual-usage of terms such as “spherical aberration” and “defocus.” The difference between classical Seidel usage and the more modern Zernike usage is analogous to the difference between the spherical power in a conventional prescription of refractive error and the mean spherical equivalent used in power vector notation. If one extracts the mean spherical equivalent from a conventional prescription, the remainder is an orthogonal component called “Jackson crossed cylinder” to avoid confusion with the term “cylinder” used in conventional prescriptions. Similarly, if one extracts mean sphere from Seidel spherical aberration the result is Zernike spherical aberration. Unfortunately, authors sometimes fail to explicitly state which definition they have in mind when using common names for aberrations. Thus readers must rely on context to resolve the ambiguity.

P. Mouroulis, “Aberration and image quality representation for visual optical systems,” in Visual Instrumentation: Optical Design and Engineering Principles, P. Mouroulis, ed. (McGraw-Hill, New York, 1999), pp. 27–68.

The same argument in a more familiar context would arise in a statistical analysis of the Fourier coefficients obtained from physiological responses to square-wave stimulation. Fourier expansion of a square wave yields a strict relationship between the amplitudes of the various harmonic components. Therefore, a quasi-square-wave-response waveform should still show evidence of a predictable correlation between Fourier coefficients.

S. M. MacRae, R. R. Krueger, R. A. Applegate, Customized Corneal Ablation: The Quest for Super Vision (Slack, Thorofare, N.J., 2001).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

L. N. Thibos, A. Bradley, “Modeling the refractive and neuro-sensor systems of the eye,” in Visual Instrumentation: Optical Design and Engineering Principles, P. Mouroulis, ed. (McGraw-Hill, New York, 1999), pp. 101–159.

J. E. Jackson, A User’s Guide To Principal Components (Wiley, New York, 1991).

A. J. Thomasian, The Structure of Probability Theory with Applications (McGraw-Hill, New York, 1969).

One implication of this result is that the mean magnitudes of Gaussian aberration coefficients are virtually ensured to be statistical significant. A t-test of the null hypothesis that the mean magnitude is zero would be rejected if the observed mean were more than twice the standard error of the mean. Since standard error=standarddeviation/n,rejection is ensured for n>2π-4=2.3.Thus a population of three or more individuals is enough to cause rejection of the null hypothesis for Gaussian variables.

Critical pupil diameter should not be confused with optimum pupil diameter, defined as that pupil size which maximizes image quality. The latter is more difficult to quantify satisfactorily because of the lack of a universally accepted metric of image quality.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, “Standards for reporting the optical aberrations of eyes,” in Vision Science and Its Applications, Vol. 35 of Trends in Optics and Photonics Series, V. Lakshminarayanan, ed. (Optical Society of America, Washington, D.C., 2000), pp. 232–244.

D. Sliney, M. Wolbarsht, Safety with Lasers and Other Optical Sources (Plenum, New York, 1980).

X. Hong, L. N. Thibos, K. M. Haggerty, “Shack–Hartmann data analysis software for MATLAB” (2000), http://research.opt.indiana.edu .

The array of spots formed in a Shack–Hartmann aberrometer provides only a crude estimate of pupil center and pupil diameter because the spots are displaced by the eye’s aberrations. A more accurate estimate may be obtained from the locations of the lenslets that produced the spots.

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Figures (22)

Fig. 1
Fig. 1

Frequency distributions for A, age; B, pupil diameter determined by aberrometry; C, spherical equivalent of refractive error determined by subjective refraction, and D, orthogonal components of astigmatism determined subjectively. Positive J0 indicates “with the rule” astigmatism, and negative J0 indicates “against the rule.” Note quantization of data in 1/8-D steps, which corresponds to 0.25-D steps in power of cylindrical trial lenses. N=200 eyes.

Fig. 2
Fig. 2

Visualizations of Zernike polynomials as gray-scale images (light areas=positive aberrations=phase advance, dark areas=negative aberrations=phase lag). The periodic table of Zernike functions fits naturally into a pyramid arrangement, with each row indicating the order and each column indicating the meridional frequency. A single-index numbering scheme (small numbers in upper left corner of each function) labels the functions from left to right, top to bottom. Order 0 (a constant) is not shown. All aberration functions in this figure have the same wave-front variance.

Fig. 3
Fig. 3

Examples of wave-front aberration functions. Top row, data from an eye with weak aberrations; bottom row, data from an eye with strong aberrations. Left column, data image from the aberrometer overlaid with a circle denoting pupil boundary; the cross indicates pupil center. Middle column, derived wave-front aberration function shown as a contour map, with zero and first-order coefficients set to zero for display purposes. The contour passing through the center of the pupil corresponds to zero aberration. Light areas in the contour maps indicate positive aberrations; dark areas indicate negative aberrations. Right column, graphical display of the spectrum of Zernike aberration coefficients using the pyramidal organization established in Fig. 2. Positive coefficients are encoded by a rectangle lighter than the background; negative coefficients are encoded by rectangles darker than the background. The intensity scale in the pyramid display assigns white and black to maximum (+1.2 μm) and minimum (-1.2 μm) Zernike coefficients, respectively. Pupil diameter=6.0 mm; contour intervals=0.633 μm=1 wavelength.

Fig. 4
Fig. 4

Frequency distribution of residual refractive errors determined by aberrometry following spectacle correction of 100 right eyes and 100 left eyes on the basis of subjective refraction. (A) Distribution of M, the spherical equivalent (solid bars, right eyes; open bars, left eyes), and (B) joint distribution of J0 and J45, the orthogonal components of astigmatism (triangles, right eyes; circles, left eyes). Solid curve, 95% probability ellipse for a bivariate Gaussian distribution with the same mean and variance as the experimental data. Pupil diameter=6.0 mm.

Fig. 5
Fig. 5

Frequency histograms of Zernike coefficients, arranged according to the pyramid structure of Fig. 2. For each histogram the abscissa is the value of the Zernike coefficients in micrometers; the ordinate is the number of eyes in each bin. All ordinates have the same scale. All abscissas in the same row have the same scale. Open bars, left eyes; solid bars, right eyes. Pupil diameter=6.0 mm.

Fig. 6
Fig. 6

Statistical summaries of Zernike coefficients displayed in Fig. 5. A, Mean values of signed aberration coefficients are indicated by squares for right eyes and circles for left eyes, with error bars indicating ±1 standard deviation of the population. A solid symbol indicates that the coefficient was found to be significantly different from zero according to both statistical tests administered (nonparametric sign test and Student’s t-test). All aberration coefficients are in micrometers. The dioptric scale shown for second-order aberrations applies also to higher-order modes according to the concept of equivalent defocus [see Eq. (3)]. B, Statistical tests of significance, orders 2–7. Modes found to be statistically significant for both tests on both eyes are indicated by a white rectangle. The arrangement of data in parts A and B follows the pyramid structure of Fig. 2. Pupil diameter=6.0 mm.

Fig. 7
Fig. 7

Daily variation in total wave-front error of the higher-order aberrations determined for three subjects in an auxiliary experiment. Symbols indicate the mean of five measurements on each of five successive days; error bars indicate ±1 standard error of the mean. The symbol at the far right in each graph shows the mean across days, with error bars showing the standard error of the mean computed from the five daily means. Open symbols, left eyes; solid symbols, right eyes. The analyzed pupil diameter was different for each eye as indicated. Solid vertical bar, 95% probability range for the full study population of RMS values measured for 4.5-mm pupils; open vertical bar, same range for 6-mm pupils.

Fig. 8
Fig. 8

A, Statistical summaries of the absolute values of Zernike coefficients from 100 subjects. Plotting conventions are the same as for Fig. 6. B, Scatter graph of population averages of unsigned aberration coefficients (abscissa) versus population standard deviation of unsigned aberration coefficients (ordinate) for higher-order modes. Linear prediction (slope=0.76) is based on the assumption that signed coefficients are Gaussian random variables with zero mean.

Fig. 9
Fig. 9

Variation of wave-front variance with radial order of Zernike modes for different pupil diameters. A, Symbols indicate mean values, and error bars indicate standard deviation of the population (standard errors are smaller than symbols in all cases). Data sets are parameterized by pupil diameter, which was varied by applying the appropriately scaled mask to the array of centroids before computing Zernike coefficients. Data from left and right eyes are pooled (N=140 for pupil diameter of 7.5 mm; N=200 for other pupil diameters). Regression lines on semilog coordinates indicate exponential decline. B, The same data are displayed as RMS error versus pupil area, with data sets parameterized by radial order. Symbols show results obtained by the centroid masking method; solid curves show results obtained by the Taylor series method for scaling pupil diameters.

Fig. 10
Fig. 10

Frequency distribution of the equivalent defocus of higher order (n>2) aberrations, computed for four pupil sizes.

Fig. 11
Fig. 11

Visualization of the correlation matrix for Zernike coefficients. Light squares indicate positive correlation coefficients; dark squares indicate negative correlation coefficients (see calibration bar on the right). Only those correlation coefficients that are statistically significant (F-test, p=0.05) are shown. The linear sequence of mode numbers corresponding to rows and columns of the correlation matrix is defined in Fig. 2.

Fig. 12
Fig. 12

A, Distribution of critical pupil diameter, defined as the largest pupil that achieves diffraction-limited performance according to Marechal’s criterion RMS=λ/14. Zernike orders 1–7 were included in the calculations. B, Variation of mean critical diameter (symbols) with the lowest Zernike order used in the calculation. Error bars indicate ±1 standard deviation across 200 eyes.

Fig. 13
Fig. 13

Correlation of higher-order aberrations (n>2) between the two eyes. Each symbol shows the value of aberration coefficients for a given Zernike mode determined for left and right eyes of the same individual. Bilateral symmetry predicts that data will fall along the positive diagonal (dashed line) for even-symmetric modes, or along the negative diagonal for odd-symmetric modes.

Fig. 14
Fig. 14

Visualization of the four principal components of wave-front aberration functions that account for 82% of the variance between individuals. Component #1 is the most significant, component #4 the least significant. Pupil diameter=6.0 mm.

Fig. 15
Fig. 15

Tests of the hypothesis that residual defocus reported in Fig. 4 is due to pupil apodization by the SCE. (A) Variation of Zernike defocus (solid symbols) and paraxial (Seidel) defocus (open symbols) with pupil diameter. Symbols show mean values (140 eyes for 7.5-mm pupil, 200 otherwise); error bars show ±2 standard errors of the mean. (B) Covariation of Zernike coefficients for defocus (ordinate) and spherical aberration (abscissa). Symbols show the aberrations of individual eyes for 6-mm pupil diameter. Dashed line, prediction of text Eq. (9); solid line, least-squares regression line (R=0.36).

Fig. 16
Fig. 16

Tests of the hypothesis that residual Zernike defocus is needed to balance spherical aberration to flatten the central portion of the wave-front aberration function. A, Variation of RMS wave-front error with pupil radius for three optical models: spherical aberration only (dashed curve), defocus only (dotted curve), or both (solid curve). Shaded area indicates diffraction-limited performance according to Marechal’s criterion. B, Simulated retinal images for the three optical models shown in A, plus the diffraction limited case.

Fig. 17
Fig. 17

Equivalent defocus of higher-order aberrations for four pupil sizes as a function of radial order (A) and meridional frequency (B) of the Zernike modes. Wave-front errors were converted to equivalent defocus by using Eq. (3).

Fig. 18
Fig. 18

Covariation of the orthogonal components of coma (A, left eyes; B, right eyes) and trefoil (C, left eyes; D, right eyes). Each symbol represents the mean of three measurements on a given eye, and the solid curves are the 95% probability ellipses for each data set. In a polar coordinate reference frame, the aberration magnitude for a given eye is determined by the radial distance of the corresponding symbol from the origin, and the axis of the aberration is determined by the polar angle of the symbol. Pupil diameter=6.0 mm.

Fig. 19
Fig. 19

Computed reduction of wave-front variance by correcting higher-order aberrations, ranked according to magnitude of aberration coefficient. Circles, results for a Zernike expansion of the wave front; triangles, results for an expansion based on principal component analysis. The inset shows the same data plotted on logarithmic axes. The arrow shows the Marechal criterion for diffraction-limited performance. Pupil diameter=6 mm.

Fig. 20
Fig. 20

Magnitude portion of average OTFs computed four ways. Parts A, B, C, and D correspond to methods A, B, C, and D respectively, as defined by Eq. (10). Lower-order aberrations (n2) were omitted from calculations. Pupil diameter=6 mm. Units of frequency axis are in cycles per degree.

Fig. 21
Fig. 21

Average PSFs computed as the Fourier transform of the corresponding OTFs of Fig. 19. Intensity (z) axis is scaled such that a diffraction-limited system would have a peak value of unity. Thus the peak value of experimental functions is equal to the Strehl ratio. Units of spatial axis are in arc minutes.

Fig. 22
Fig. 22

Comparison of radial MTFs of individual eyes with population means. Thick solid curves are the radial averages of the four OTFs in Fig. 19. Thin curves are individual radial MTFs for 200 eyes. The dashed curve is the prediction for a diffraction-limited optical system with the same pupil size.

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

W(x, y)=n,fcnfZnf,
M=-c2043r2,J0=-c2+226r2
J45=-c2-226r2,J=J02+J452,
M=4π3RMSerrorpupilarea.
RMSerror=λ/14=(Merc2)/43.
rc=0.56/Me.
p(x)=2σ2πexp(-x2/2σ2)ifx>00ifx0
PC1=-0.88Z3-1+0.44Z3-3+0.13Z40,
PC2=0.80Z31+0.42Z40+0.39Z33,
PC3=-0.63Z40+0.56Z31+0.42Z3-3,
PC4=0.73Z33-0.57Z40-0.25Z3-3.
W(r)=ar4=c00+c203(2r2-1)+c405(6r4-6r2+1).
c40=a65,c20=a23,c20=c4015.
MS=(-43c20+125c40)/(pupilradius)2.
OTF(Z¯)MethodA:averagethe
Zernikecoefficients.
OTF(|Z|¯)MethodB:averagethe
magnitudeofZcoefficients.
OTF(Z)¯ MethodC:averagethe
complex-valuedOTFs.
|OTF(Z)|¯ MethodD:averagethe
real-valuedMTFs.

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