Abstract

A usual approximation in Hartmann–Shack aberrometry is that the centroid displacements are proportional to the spatial averages of the wave-front slopes at the sampling subapertures. However, these spatial averages are actually weighted by the local irradiance distribution across each microlens. The irradiance across the eye pupil is not uniform in usual reflectometric aberrometers, which is due to several factors including retinal scattering and cone waveguiding directionality. It is shown that neglecting this fact in usual least-squares reconstruction procedures gives rise to a biased estimation of the aberration coefficients. The magnitude of this bias depends on the actual irradiance distribution across the eye pupil, the mode being estimated, the detailed modal composition of the aberrated wave front, and the geometry of the wave-front sampling array. Order-of-magnitude calculations suggest that this bias may well be in the range 5%–10% for relatively smooth irradiance distributions. The systematic nature of this error makes it advisable to check for its presence and, if required, to compensate for it by an adequate choice of the least-squares reconstruction matrix.

© 2003 Optical Society of America

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References

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2003 (1)

2002 (4)

2001 (1)

E. Moreno-Barriuso, S. Marcos, R. Navarro, S. A. Burns, “Comparing laser ray tracing, spatially resolved refractometer and Hartmann–Shack sensor to measure the ocular wavefront aberration,” Optom. Vision Sci. 78, 152–156 (2001).
[CrossRef]

2000 (4)

1999 (3)

1998 (2)

1997 (4)

1995 (1)

1994 (1)

1992 (1)

1990 (1)

J.-M. Gorrand, F. C. Delori, “A method for assessing the photoreceptor directionality,” Invest. Ophthalmol. Visual Sci. (Suppl.) 31, 425 (1990).

1984 (1)

1982 (1)

1980 (2)

1977 (1)

1961 (1)

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

1904 (1)

J. Hartmann, “Objektivuntersuchungen,” Z. Instrumentenkd. XXIV(1), 1–21 (1904).

Applegate, R. A.

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

Ares, J.

Artal, P.

Bará, S.

Berendschot, T. T. J. M.

Bille, J.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1987), pp. 464–468, 767–772.

Bradley, A.

Burns, S. A.

S. A. Burns, S. Marcos, A. E. Elsner, S. Bará, “Contrast improvement for confocal retinal imaging using phase correcting plates,” Opt. Lett. 27, 400–402 (2002).
[CrossRef]

E. Moreno-Barriuso, S. Marcos, R. Navarro, S. A. Burns, “Comparing laser ray tracing, spatially resolved refractometer and Hartmann–Shack sensor to measure the ocular wavefront aberration,” Optom. Vision Sci. 78, 152–156 (2001).
[CrossRef]

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, S. A. Burns, “Comparison of cone directionality determined by psychophysical and reflectometric techniques,” J. Opt. Soc. Am. A 16, 2363–2369 (1999).
[CrossRef]

S. Marcos, S. A. Burns, “Cone spacing and waveguide properties from cone directionality measurements,” J. Opt. Soc. Am. A 16, 995–1004 (1999).
[CrossRef]

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

S. Marcos, S. A. Burns, J. C. He, “Model for cone directionality reflectometric measurements based on scattering,” J. Opt. Soc. Am. A 15, 2012–2022 (1998).
[CrossRef]

S. A. Burns, S. Wu, J. C. He, A. E. Elsner, “Variations in photoreceptor directionality across the central retina,” J. Opt. Soc. Am. A 14, 2033–2040 (1997).
[CrossRef]

S. A. Burns, S. Wu, F. Delori, A. E. Elsner, “Direct measurement of human-cone-photoreceptor alignment,” J. Opt. Soc. Am. A 12, 2329–2338 (1995).
[CrossRef]

Charman, W. N.

Cheng, X.

Dainty, C.

Delori, F.

Delori, F. C.

J.-M. Gorrand, F. C. Delori, “A method for assessing the photoreceptor directionality,” Invest. Ophthalmol. Visual Sci. (Suppl.) 31, 425 (1990).

DeVore, S. L.

D. Malacara, S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), Chap. 13, pp. 455–499.

Di´az-Santana, L.

Elsner, A. E.

Goelz, S.

Gorrand, J.-M.

J.-M. Gorrand, F. C. Delori, “A method for assessing the photoreceptor directionality,” Invest. Ophthalmol. Visual Sci. (Suppl.) 31, 425 (1990).

Grimm, B.

Hartmann, J.

J. Hartmann, “Objektivuntersuchungen,” Z. Instrumentenkd. XXIV(1), 1–21 (1904).

He, J. C.

Herrmann, J.

Hong, X.

Howland, B.

Howland, H. C.

Iglesias, I.

Julien, Y.

Liang, J.

Llorente, L.

Losada, M. A.

R. Navarro, M. A. Losada, “Aberrations and relative efficiency of light pencils in the living human eye,” Optom. Vision Sci. 74, 540–547 (1997).
[CrossRef]

Malacara, D.

D. Malacara, S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), Chap. 13, pp. 455–499.

Mancebo, T.

Marcos, S.

S. A. Burns, S. Marcos, A. E. Elsner, S. Bará, “Contrast improvement for confocal retinal imaging using phase correcting plates,” Opt. Lett. 27, 400–402 (2002).
[CrossRef]

S. Marcos, L. Dı́az-Santana, L. Llorente, C. Dainty, “Ocular aberrations with ray tracing and Shack–Hartmann wave-front sensors: does polarization play a role?” J. Opt. Soc. Am. A 19, 1063–1072 (2002).
[CrossRef]

E. Moreno-Barriuso, S. Marcos, R. Navarro, S. A. Burns, “Comparing laser ray tracing, spatially resolved refractometer and Hartmann–Shack sensor to measure the ocular wavefront aberration,” Optom. Vision Sci. 78, 152–156 (2001).
[CrossRef]

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, S. A. Burns, “Comparison of cone directionality determined by psychophysical and reflectometric techniques,” J. Opt. Soc. Am. A 16, 2363–2369 (1999).
[CrossRef]

S. Marcos, S. A. Burns, “Cone spacing and waveguide properties from cone directionality measurements,” J. Opt. Soc. Am. A 16, 995–1004 (1999).
[CrossRef]

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

S. Marcos, S. A. Burns, J. C. He, “Model for cone directionality reflectometric measurements based on scattering,” J. Opt. Soc. Am. A 15, 2012–2022 (1998).
[CrossRef]

Miller, D. T.

Moreno-Barriuso, E.

E. Moreno-Barriuso, S. Marcos, R. Navarro, S. A. Burns, “Comparing laser ray tracing, spatially resolved refractometer and Hartmann–Shack sensor to measure the ocular wavefront aberration,” Optom. Vision Sci. 78, 152–156 (2001).
[CrossRef]

E. Moreno-Barriuso, R. Navarro, “Laser ray tracing versus Hartmann–Shack sensor for measuring optical aberrations in the human eye,” J. Opt. Soc. Am. A 17, 974–985 (2000).
[CrossRef]

R. Navarro, E. Moreno-Barriuso, “Laser ray-tracing method for optical testing,” Opt. Lett. 24, 951–953 (1999).
[CrossRef]

Navarro, R.

E. Moreno-Barriuso, S. Marcos, R. Navarro, S. A. Burns, “Comparing laser ray tracing, spatially resolved refractometer and Hartmann–Shack sensor to measure the ocular wavefront aberration,” Optom. Vision Sci. 78, 152–156 (2001).
[CrossRef]

E. Moreno-Barriuso, R. Navarro, “Laser ray tracing versus Hartmann–Shack sensor for measuring optical aberrations in the human eye,” J. Opt. Soc. Am. A 17, 974–985 (2000).
[CrossRef]

R. Navarro, E. Moreno-Barriuso, “Laser ray-tracing method for optical testing,” Opt. Lett. 24, 951–953 (1999).
[CrossRef]

R. Navarro, M. A. Losada, “Aberrations and relative efficiency of light pencils in the living human eye,” Optom. Vision Sci. 74, 540–547 (1997).
[CrossRef]

Penney, C. M.

Prieto, P. M.

Ragazzoni, R.

Schwiegerling, J. T.

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

Silva, D. E.

Smirnov, M. S.

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

Tatarskii, V. I.

V. I. Tatarskii, The Propagation of Waves in the Turbulent Atmosphere (Nauka, Moscow, 1967), pp. 385–390 (in Russian).

Teague, M. R.

Thibos, L. N.

L. N. Thibos, X. Hong, A. Bradley, X. Cheng, “Statistical variation of aberration structure and image quality in a normal population of healthy eyes,” J. Opt. Soc. Am. A 19, 2329–2348 (2002).
[CrossRef]

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

Thompson, K. P.

van Norren, D.

Vargas-Martin, F.

Walsh, G.

Wang, J. Y.

Webb, R.

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

Webb, R. H.

Williams, D. R.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1987), pp. 464–468, 767–772.

Wu, S.

Zagers, N. P. A.

Appl. Opt. (3)

Biofizika (1)

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

Invest. Ophthalmol. Visual Sci. (Suppl.) (1)

J.-M. Gorrand, F. C. Delori, “A method for assessing the photoreceptor directionality,” Invest. Ophthalmol. Visual Sci. (Suppl.) 31, 425 (1990).

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (15)

P. M. Prieto, F. Vargas-Martin, S. Goelz, P. Artal, “Analysis of the performance of the Hartmann–Shack sensor in the human eye,” J. Opt. Soc. Am. A 17, 1388–1398 (2000).
[CrossRef]

S. A. Burns, S. Wu, F. Delori, A. E. Elsner, “Direct measurement of human-cone-photoreceptor alignment,” J. Opt. Soc. Am. A 12, 2329–2338 (1995).
[CrossRef]

S. Marcos, L. Dı́az-Santana, L. Llorente, C. Dainty, “Ocular aberrations with ray tracing and Shack–Hartmann wave-front sensors: does polarization play a role?” J. Opt. Soc. Am. A 19, 1063–1072 (2002).
[CrossRef]

L. N. Thibos, X. Hong, A. Bradley, X. Cheng, “Statistical variation of aberration structure and image quality in a normal population of healthy eyes,” J. Opt. Soc. Am. A 19, 2329–2348 (2002).
[CrossRef]

N. P. A. Zagers, T. T. J. M. Berendschot, D. van Norren, “Wavelength dependence of reflectometric cone photoreceptor directionality,” J. Opt. Soc. Am. A 20, 18–23 (2003).
[CrossRef]

E. Moreno-Barriuso, R. Navarro, “Laser ray tracing versus Hartmann–Shack sensor for measuring optical aberrations in the human eye,” J. Opt. Soc. Am. A 17, 974–985 (2000).
[CrossRef]

J. Liang, B. Grimm, S. Goelz, J. 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]

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]

S. Marcos, S. A. Burns, “Cone spacing and waveguide properties from cone directionality measurements,” J. Opt. Soc. Am. A 16, 995–1004 (1999).
[CrossRef]

J. C. He, S. Marcos, S. A. Burns, “Comparison of cone directionality determined by psychophysical and reflectometric techniques,” J. Opt. Soc. Am. A 16, 2363–2369 (1999).
[CrossRef]

S. Marcos, S. A. Burns, J. C. He, “Model for cone directionality reflectometric measurements based on scattering,” J. Opt. Soc. Am. A 15, 2012–2022 (1998).
[CrossRef]

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

S. A. Burns, S. Wu, J. C. He, A. E. Elsner, “Variations in photoreceptor directionality across the central retina,” J. Opt. Soc. Am. A 14, 2033–2040 (1997).
[CrossRef]

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]

J. Liang, D. R. Williams, D. T. Miller, “Supernormal vision and high-resolution retinal imaging through adaptive optics,” J. Opt. Soc. Am. A 14, 2884–2892 (1997).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Optom. Vision Sci. (2)

R. Navarro, M. A. Losada, “Aberrations and relative efficiency of light pencils in the living human eye,” Optom. Vision Sci. 74, 540–547 (1997).
[CrossRef]

E. Moreno-Barriuso, S. Marcos, R. Navarro, S. A. Burns, “Comparing laser ray tracing, spatially resolved refractometer and Hartmann–Shack sensor to measure the ocular wavefront aberration,” Optom. Vision Sci. 78, 152–156 (2001).
[CrossRef]

Vision Res. (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]

Z. Instrumentenkd. (1)

J. Hartmann, “Objektivuntersuchungen,” Z. Instrumentenkd. XXIV(1), 1–21 (1904).

Other (4)

V. I. Tatarskii, The Propagation of Waves in the Turbulent Atmosphere (Nauka, Moscow, 1967), pp. 385–390 (in Russian).

D. Malacara, S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), Chap. 13, pp. 455–499.

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

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1987), pp. 464–468, 767–772.

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Figures (5)

Fig. 1
Fig. 1

Elements of the main diagonal of the coupling matrices MB (constant irradiance assumed in the measurement model; solid curves) and MC (centroid positions taken as proportional to the wave-front slopes at the center of the subpupils; dotted curves) for the irradiance model of Eq. (19) with σ=0.64 in length units normalized to the pupil radius and Γ=0.5, 0.7, and 0.9.

Fig. 2
Fig. 2

Logarithmic gray-scale display of the coupling matrix MB for σ=0.64 and Γ=0.7.

Fig. 3
Fig. 3

Logarithmic gray-scale display of the coupling matrix MC for σ=0.64 and Γ=0.7.

Fig. 4
Fig. 4

Histograms of the absolute values of elements of MB (upper panel) and MC (lower panel) for Γ=0.7 and σ=0.64. Horizontal axis: absolute values in logarithmic scale; vertical axis: number of elements in each interval. The vertical bar close to 0 corresponds to the main-diagonal elements. Most of the 1225 elements of the coupling matrices have negligible values: Only the elements with absolute value higher than 0.0001 have been included in these plots.

Fig. 5
Fig. 5

Elements of the main diagonal of the coupling matrices MB (constant irradiance assumed in the measurement model) for wave-front sampling arrays composed of 37, 57, 89, and 121 square subpupils and actual irradiance given by Eq. (19) with σ=0.64 and Γ=0.9.

Equations (25)

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

u0(ρ)=(1/iλz)exp(ikz)u(r)expik2z|r-ρ|2d2r,
ρc(z)=ρI0(ρ)d2ρI0(ρ)d2ρ,
ρc(z)=ρc(0)+z(1/E)I(r)W(r)d2r,
W(r)=Wa(r)+WL(r),
ρc(z)=ρc(0)1-zf+z1ES SI(r)Wa(r)d2r.
ρc(f)f=1ES SI(r)Wa(r)d2r.
ρc(f)f=1S SWa(r)d2r.
msρc(f)f+νs=1ES SI(r)Wa(r)d2r+νs,
Wa(r)=i=1MaiZi(r),
m=Aa+ν,
Aki=1ES SI(r)kZi(r)d2r,
aˆ=Rm,
R=(ATA)-1AT.
aˆ=Rm=(ATA)-1ATAa+ν=a,
RB=(BTB)-1BT,
Bki=1S SkZi(r)d2r.
aˆ=(BTB)-1BTAa.
I(r)=B+A×10-γ|r-r0|2,
I(r)=(1-Γ)+Γ exp[-(r/σ)2],
I0(ρ)=1(λz)2 rru(r)u*(r)expik2z(r2-r2)exp-ikz(r-r)·ρd2rd2r.
δ(x)=12π - exp(-iαx)dα,
δ(n)(x)dnδ(x)dxn=12π(-i)n-αn exp(-iαx)dα,
-αn exp(-iαx)dα=2π(-i)-nδ(n)(x).
-f(x)δ(n)(x)dx=(-1)nf(n)(0),
-f(x)δ(n)[c(x-x)]dx=(-1)nc-n-1f(n)(x),

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