Abstract

A description of a Hartmann–Shack sensor to measure the aberrations of the human eye is presented. We performed an analysis of the accuracy and limitations of the sensor using experimental results and computer simulations. We compared the ocular modulation transfer function obtained from simultaneously recorded double-pass and Hartmann–Shack images. The following factors affecting the sensor performance were evaluated: the statistical accuracy, the number of modes used to reconstruct the wave front, the size of the microlenses, and the exposure time.

© 2000 Optical Society of America

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References

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  1. M. S. Smirnov, “Measurement of the wave aberration of the human eye,” Biophysics 6, 776–794 (1961).
  2. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1985).
  3. 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]
  4. F. Vargas-Martı́n, P. Prieto, P. Artal, “Correction of the aberrations in the human eye with liquid-crystal spatial light modulators: limits to the performance,” J. Opt. Soc. Am. A 15, 2552–2562 (1998).
    [CrossRef]
  5. D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, New York, 1992).
  6. 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]
  7. J. Liang, B. Grimm, S. Goelz, J. F. Bille, “Objective measurement of WA’s of the human eye with the use of a Hartmann–Shack wave-front sensor,” J. Opt. Soc. Am. A 11, 1949–1957 (1994).
    [CrossRef]
  8. 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]
  9. G. Rousset, “Wavefront sensing,” in Adaptive Optics for Astronomy, D. M. Alloin, J.-M. Mariotti, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1994).
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  14. J. Santamarı́a, P. Artal, J. Bescós, “Determination of the point-spread function of the human eye using a hybrid optical-digital method,” J. Opt. Soc. Am. A 4, 1109–1114 (1987).
    [CrossRef]
  15. American National Standard for the Safe Use of Lasers, (Laser Institute of America, Orlando, Fla., 1993).
  16. F. Vargas-Martı́n, “Optica adaptativa en oftalmoscopia: correción de los aberraciones oculaires con un modulador espacial cristal liquido,” Ph.D. Thesis (University of Murcia, Murcia, Spain, 1999).
  17. P. Artal, S. Marcos, R. Navarro, D. R. Williams , “Odd aberrations and double-pass measurements of retinal image quality,” J. Opt. Soc. Am. A 12, 195–201 (1995).
    [CrossRef]
  18. D. R. Williams, D. H. Brainard, M. J. McMahon, R. Navarro, “Double-pass and interferometric measures of the optical quality of the eye,” J. Opt. Soc. Am. A 11, 3123–3134 (1994).
    [CrossRef]
  19. J. Herrmann, “Cross coupling and aliasing in modal wave-front estimation,” J. Opt. Soc. Am. 71, 989–992 (1981).
    [CrossRef]
  20. H. J. Hofer, P. Artal, J. L. Aragón, D. R. Williams, “Temporal characteristics of the eye’s aberrations,” Invest. Ophthalmol. Visual Sci. Suppl. 40, S365 (1999).
  21. H. J. Hofer, J. Porter, D. R. Williams, “Dynamic measurement of the wave aberration of the human eye,” Invest. Ophthalmol. Visual Sci. Suppl. 39, S209 (1998).

1999 (1)

H. J. Hofer, P. Artal, J. L. Aragón, D. R. Williams, “Temporal characteristics of the eye’s aberrations,” Invest. Ophthalmol. Visual Sci. Suppl. 40, S365 (1999).

1998 (3)

1997 (2)

1995 (1)

1994 (2)

1991 (1)

1987 (1)

1981 (1)

1980 (1)

1979 (1)

1976 (1)

1961 (1)

M. S. Smirnov, “Measurement of the wave aberration of the human eye,” Biophysics 6, 776–794 (1961).

Aragón, J. L.

H. J. Hofer, P. Artal, J. L. Aragón, D. R. Williams, “Temporal characteristics of the eye’s aberrations,” Invest. Ophthalmol. Visual Sci. Suppl. 40, S365 (1999).

Artal, P.

Berrio, E.

Bescós, J.

Bille, J. F.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1985).

Brainard, D. H.

Cubalchini, R.

Goelz, S.

Grimm, B.

Herrmann, J.

Hofer, H. J.

H. J. Hofer, P. Artal, J. L. Aragón, D. R. Williams, “Temporal characteristics of the eye’s aberrations,” Invest. Ophthalmol. Visual Sci. Suppl. 40, S365 (1999).

H. J. Hofer, J. Porter, D. R. Williams, “Dynamic measurement of the wave aberration of the human eye,” Invest. Ophthalmol. Visual Sci. Suppl. 39, S209 (1998).

Iglesias, I.

Liang, J.

Malacara, D.

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, New York, 1992).

Marcos, S.

McMahon, M. J.

Miller, D. T.

Navarro, R.

Noll, R. J.

Porter, J.

H. J. Hofer, J. Porter, D. R. Williams, “Dynamic measurement of the wave aberration of the human eye,” Invest. Ophthalmol. Visual Sci. Suppl. 39, S209 (1998).

Prieto, P.

Roddier, C.

Roddier, F.

Rousset, G.

G. Rousset, “Wavefront sensing,” in Adaptive Optics for Astronomy, D. M. Alloin, J.-M. Mariotti, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1994).

Santamari´a, J.

Smirnov, M. S.

M. S. Smirnov, “Measurement of the wave aberration of the human eye,” Biophysics 6, 776–794 (1961).

Southwell, W. H.

Vargas-Marti´n, F.

F. Vargas-Martı́n, P. Prieto, P. Artal, “Correction of the aberrations in the human eye with liquid-crystal spatial light modulators: limits to the performance,” J. Opt. Soc. Am. A 15, 2552–2562 (1998).
[CrossRef]

F. Vargas-Martı́n, “Optica adaptativa en oftalmoscopia: correción de los aberraciones oculaires con un modulador espacial cristal liquido,” Ph.D. Thesis (University of Murcia, Murcia, Spain, 1999).

Williams, D. R.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1985).

Appl. Opt. (1)

Biophysics (1)

M. S. Smirnov, “Measurement of the wave aberration of the human eye,” Biophysics 6, 776–794 (1961).

Invest. Ophthalmol. Visual Sci. Suppl. (2)

H. J. Hofer, P. Artal, J. L. Aragón, D. R. Williams, “Temporal characteristics of the eye’s aberrations,” Invest. Ophthalmol. Visual Sci. Suppl. 40, S365 (1999).

H. J. Hofer, J. Porter, D. R. Williams, “Dynamic measurement of the wave aberration of the human eye,” Invest. Ophthalmol. Visual Sci. Suppl. 39, S209 (1998).

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (8)

J. Santamarı́a, P. Artal, J. Bescós, “Determination of the point-spread function of the human eye using a hybrid optical-digital method,” J. Opt. Soc. Am. A 4, 1109–1114 (1987).
[CrossRef]

J. Liang, B. Grimm, S. Goelz, J. F. Bille, “Objective measurement of WA’s of the human eye with the use of a Hartmann–Shack wave-front sensor,” J. Opt. Soc. Am. A 11, 1949–1957 (1994).
[CrossRef]

D. R. Williams, D. H. Brainard, M. J. McMahon, R. Navarro, “Double-pass and interferometric measures of the optical quality of the eye,” J. Opt. Soc. Am. A 11, 3123–3134 (1994).
[CrossRef]

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]

F. Vargas-Martı́n, P. Prieto, P. Artal, “Correction of the aberrations in the human eye with liquid-crystal spatial light modulators: limits to the performance,” J. Opt. Soc. Am. A 15, 2552–2562 (1998).
[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]

P. Artal, S. Marcos, R. Navarro, D. R. Williams , “Odd aberrations and double-pass measurements of retinal image quality,” J. Opt. Soc. Am. A 12, 195–201 (1995).
[CrossRef]

Other (5)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1985).

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, New York, 1992).

G. Rousset, “Wavefront sensing,” in Adaptive Optics for Astronomy, D. M. Alloin, J.-M. Mariotti, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1994).

American National Standard for the Safe Use of Lasers, (Laser Institute of America, Orlando, Fla., 1993).

F. Vargas-Martı́n, “Optica adaptativa en oftalmoscopia: correción de los aberraciones oculaires con un modulador espacial cristal liquido,” Ph.D. Thesis (University of Murcia, Murcia, Spain, 1999).

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Figures (12)

Fig. 1
Fig. 1

Experimental setup for simultaneous recording of HS and DP images. See text for a description of the components.

Fig. 2
Fig. 2

Comparison between radially averaged MTF estimates obtained from DP images (DP MTF, dashed curves) and from WA estimates provided by HS (HSMTF, solid curves) for subject PA at two focuses.

Fig. 3
Fig. 3

Comparison between radially averaged MTF estimates obtained from DP images (DP MTF, dashed curves) and from WA estimates provided by HS (HS MTF, solid curves) for subject IH at two focuses.

Fig. 4
Fig. 4

Gray-scale image representing the statistical fluctuations of the WA estimation algorithm when a series of random x and y displacements (equiprobable distribution between [-1, 1] pixel) are introduced. The gray level at each point represents the standard deviation of the WA value in a set of 10,000 simulations.

Fig. 5
Fig. 5

Radial average of the standard deviation of the WA estimates provided by the algorithm when a series of equiprobable random x and y displacements are introduced for different fluctuation amplitudes. Solid curve, [-1, 1] pixel; long-dashed curve, [-2, 2] pixels; short-dashed curve, [-3, 3] pixels; dashed–dotted curve, [-5, 5] pixels; dashed–double-dot curve, [-10, 10] pixels.

Fig. 6
Fig. 6

Simulation of the performance of HS sensors with different spatial sampling. The black bar represents the first 15 randomly generated coefficients for the Zernike expansion of a WA (100 coefficients were simulated with Gaussian distribution to obtain realistic values). The other bars represent the result of a sixth-order fitting when the WA is sampled with microlens arrays of decreasing resolution: forward-slashed bar, 24×24 microlenses inside the pupil; dark-gray bar, 16×16; horizontally dashed bar; 12×12; light-gray bar, 8×8; back-slashed bar, 6×6. Only the first 15 of 35 coefficients are plotted.

Fig. 7
Fig. 7

Results of processing the spot image obtained for subject PA at best focus with different numbers of Zernike polynomials. Diamonds, fitting up to second order; squares, fitting up to fourth order; triangles, fitting up to sixth order; circles, fitting up to eighth order. Vertical gray lines show the last coefficient in each fitting.

Fig. 8
Fig. 8

Contour plots of the successive estimates of the WA of subject PA at best focus, each one obtained from the corresponding set of coefficients shown in Fig. (7). (a) Fitting up to second order, (b) fitting up to fourth order, (c) fitting up to sixth order, (d) fitting up to seventh order.

Fig. 9
Fig. 9

Comparison between MTF estimates for subject PA at best focus, obtained from DP images (DP MTF, solid black curve) and from WA estimates provided by HS with different numbers of Zernike coefficients (HS MTF from WA up to second order, solid gray curve; up to fourth order, dotted curve; up to sixth order, long-dashed curve; up to eighth order, short-dashed curve).

Fig. 10
Fig. 10

Comparison between MTF estimates for subject PA in a defocused case, obtained from DP images (DP MTF, solid black curve) and from WA estimates provided by HS with different numbers of Zernike coefficients (HS MTF from WA up to second order, gray curve; up to fourth order, dotted curve).

Fig. 11
Fig. 11

Comparison between radial average of the MTF estimates obtained from DP images (DP MTF, dashed curve) and from WA estimates provided by HS (HS MTF, solid curve) for subject PA in best focus and for different exposure times: (a) 2 s, (b) 1 s, (c) 0.5 s, (d) 0.25 s, (e) 0.1 s.

Fig. 12
Fig. 12

Comparison between HS and DP images for different exposure times. For each exposure time studied we display the total 768×512 pixel HS image, a zoomed 178×178 pixel area (corresponding to 4×4 spots, i.e., 1.6×1.6 mm), and a 100×100 section of the DP image (corresponding to a field of 23 arc min).

Tables (1)

Tables Icon

Table 1 Mean Value and Standard Deviation in the First 15 Zernike Modes for the Simulation Testing the Statistical Accuracy of the HS Sensor for 1000 Events with a Random Probability of [-1, 1] Pixels in the Centroid Positioning

Equations (18)

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

Xj=AjxI(x, y)dxdyAjI(x, y)dxdy,Yj=AjyI(x, y)dxdyAjI(x, y)dxdy,
Δxj=fAAjW(x, y)xdxdy,
Δyj=fAAjW(x, y)ydxdy,
W(x, y)=kkmaxζkZk(x, y),
AiW(x, y)xdxdy=kkmaxζkAiZk(x, y)xdxdy,
AiW(x, y)ydxdy=kkmaxζkAiZk(x, y)ydxdy.
Δ=Bζ,
P(x, y)=p(x, y)exp[iW(x, y)],
p(x, y)=1πR2ifx2+y2R20otherwise.
OTF(x, y)=P(x, y)P*(x-x, y-y)dxdy.
X˜j=(nx,ny)Aj nxI(nx, ny)(nx,ny)Aj I(nx, ny),
Y˜j=(nx,ny)Aj nyI(nx, ny)(nx,ny)Aj I(nx, ny).
n=Int-1+8k-72,
m=2 Intk2-n(n+1)4ifniseven1+2 Intk-12-n(n+1)4ifnisodd,
Zk(r, θ)=n+1Rn0(r)ifm=0n+1Rnm(r)2cos(mθ)ifm0andkiseven,n+1Rnm(r)2sin(mθ)ifm0andkisodd
Rnm(r)
=s=0(n-m)/2(-1)s(n-s)!s![(n+m)/2-s]![(n-m)/2-s]! rn-2s.
Zk(x,y)={n+1b=0n/2c=0n/2-b(-1)b(n-b)!b!(n/2-b)!(n/2-b-c)!c!xn-2b-2cy2cif   m=02(n+1)a=0Int(m/2)b=0(n-m)/2c=0(n-m)/2-b(-1)a+b(m2a)×(n-b)!b![(n+m)/2-b]![(n-m)/2-b-c]!c!xn-2a-2b-2cy2a+2cif   m0   and   k   even.2(n+1)a=0Int[(m-1)/2]b=0(n-m)/2c=0(n-m)/2-b(-1)a+b(m2a+1)×(n-b)!b![(n+m)/2-b]![(n-m)/2-b-c]!c!xn-2a-2b-2c-1y2a+2c+1if   m0   and   k   odd

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