Abstract

We have used a modified double-pass apparatus with unequal entrance and exit pupil sizes to measure the optical transfer function in the human eye and have applied the technique to three different problems. First, we confirm that in the eye the double-pass spread function is the cross correlation of the input spread function with the output spread function [ J. Opt. Soc. Am. A 12, 195 ( 1995)]. Consequently, when entrance and exit pupil sizes are equal, phase information is lost from the double-pass images. Second, we show that in double-pass measurements the eye behaves like a reversible optical system. That is, when entrance and exit pupils are equal, the double-pass image results from two passes through an optical system having a transfer function that is the same in both directions. To test for reversibility in the living eye we have used a double-pass apparatus with different exit and entrance pupil sizes (one of them small enough to consider the eye diffraction limited), so that the ingoing and the outgoing transfer functions are different. The measured image quality was unchanged when the pupils were interchanged, i.e., when the first-pass entrance pupil size becomes the second-pass exit pupil size, and vice versa. Third, the technique provides a means for inferring the complete optical transfer function of the eye, including the phase transfer function, and the shape of the point-spread function.

© 1995 Optical Society of America

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References

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  1. M. F. Flamant, “Étude de la repartition de lumière dans l’image retinienne d’une fente,” Rev. Opt. 34, 433–459 (1955).
  2. F. W. Campbell, R. W. Gubisch, “Optical image quality of the human eye,” J. Physiol. (London) 186, 558–578 (1966).
  3. J. Santamaría, P. Artal, J. Bescós, “Determination of the point-spread function of human eyes using a hybrid optical–digital method,” J. Opt. Soc. Am. A 4, 1109–1114 (1987).
    [CrossRef]
  4. R. Navarro, P. Artal, D. R. Williams, “Modulation transfer of the human eye as a function of retinal eccentricity,” J. Opt. Soc. Am. A 10, 201–212 (1993).
    [CrossRef] [PubMed]
  5. P. Artal, R. Navarro, “Monochromatic modulation transfer function of the human eye for different pupil diameters: an analytical expression,” J. Opt. Soc. Am. A 11, 246–249 (1994).
    [CrossRef]
  6. P. Artal, S. Marcos, R. Navarro, M. Ferro, I. Miranda, “Through focus image quality with monofocal and multifocal intraocular lenses,” Opt. Eng. 34, 772–779 (1995).
    [CrossRef]
  7. 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]
  8. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  9. G. J. van Blokland, D. van Norren, “Intensity and polarization of light scattered at small angles from the human fovea,” Vision Res. 26, 485–494 (1986).
    [CrossRef] [PubMed]
  10. D. R. Williams, D. Brainard, M. MacHahon, R. Navarro, “Double-pass and interferometric measures of the optical quality of the eye,” J. Opt. Soc. Am. A 11, 3123–3135 (1994).
    [CrossRef]
  11. D. Sliney, M. Wolbarsht, Safety with Laser and Other Optical Sources (Plenum, New York, 1980).
  12. H. C. Howland, B. Howland, “Subjective method for the measurement of monochromatic aberrations of the eye,” J. Opt. Soc. Am. 67, 1508–1518 (1977).
    [CrossRef] [PubMed]
  13. M. C. W. Campbell, E. M. Harrison, P. Simonet, “Psychophysical measurements of the blur on the retina due to the optical aberrations of the eye,” Vision Res. 30, 1587–1602 (1990).
    [CrossRef]
  14. D. G. Green, “Visual resolution when light enters through different parts of the pupil,” J. Physiol. (London) 190, 583–593 (1967).
  15. S. Marcos, P. Artal, D. G. Green, “The effect of decentered small pupils on ocular modulation transfer and contrast sensitivity,” Invest. Ophthalmol. Vis. Sci. Suppl. 35, 1258 (1994).
  16. P. Artal, R. Navarro, “Simultaneous measurement of two-point-spread functions at different locations across the human fovea,” Appl. Opt. 31, 3646–3656 (1992).
    [CrossRef] [PubMed]
  17. P. Artal, J. Santamaría, J. Bescós, “Retrieval of the wave aberration of human eyes from actual point-spread function data,” J. Opt. Soc. Am. A 5, 1201–1206 (1988).
    [CrossRef] [PubMed]

1995 (2)

P. Artal, S. Marcos, R. Navarro, M. Ferro, I. Miranda, “Through focus image quality with monofocal and multifocal intraocular lenses,” Opt. Eng. 34, 772–779 (1995).
[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]

1994 (3)

1993 (1)

1992 (1)

1990 (1)

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

1988 (1)

1987 (1)

1986 (1)

G. J. van Blokland, D. van Norren, “Intensity and polarization of light scattered at small angles from the human fovea,” Vision Res. 26, 485–494 (1986).
[CrossRef] [PubMed]

1977 (1)

1967 (1)

D. G. Green, “Visual resolution when light enters through different parts of the pupil,” J. Physiol. (London) 190, 583–593 (1967).

1966 (1)

F. W. Campbell, R. W. Gubisch, “Optical image quality of the human eye,” J. Physiol. (London) 186, 558–578 (1966).

1955 (1)

M. F. Flamant, “Étude de la repartition de lumière dans l’image retinienne d’une fente,” Rev. Opt. 34, 433–459 (1955).

Artal, P.

Bescós, J.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Brainard, D.

Campbell, F. W.

F. W. Campbell, R. W. Gubisch, “Optical image quality of the human eye,” J. Physiol. (London) 186, 558–578 (1966).

Campbell, M. C. W.

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

Ferro, M.

P. Artal, S. Marcos, R. Navarro, M. Ferro, I. Miranda, “Through focus image quality with monofocal and multifocal intraocular lenses,” Opt. Eng. 34, 772–779 (1995).
[CrossRef]

Flamant, M. F.

M. F. Flamant, “Étude de la repartition de lumière dans l’image retinienne d’une fente,” Rev. Opt. 34, 433–459 (1955).

Green, D. G.

S. Marcos, P. Artal, D. G. Green, “The effect of decentered small pupils on ocular modulation transfer and contrast sensitivity,” Invest. Ophthalmol. Vis. Sci. Suppl. 35, 1258 (1994).

D. G. Green, “Visual resolution when light enters through different parts of the pupil,” J. Physiol. (London) 190, 583–593 (1967).

Gubisch, R. W.

F. W. Campbell, R. W. Gubisch, “Optical image quality of the human eye,” J. Physiol. (London) 186, 558–578 (1966).

Harrison, E. M.

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

Howland, B.

Howland, H. C.

MacHahon, M.

Marcos, S.

P. Artal, S. Marcos, R. Navarro, M. Ferro, I. Miranda, “Through focus image quality with monofocal and multifocal intraocular lenses,” Opt. Eng. 34, 772–779 (1995).
[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]

S. Marcos, P. Artal, D. G. Green, “The effect of decentered small pupils on ocular modulation transfer and contrast sensitivity,” Invest. Ophthalmol. Vis. Sci. Suppl. 35, 1258 (1994).

Miranda, I.

P. Artal, S. Marcos, R. Navarro, M. Ferro, I. Miranda, “Through focus image quality with monofocal and multifocal intraocular lenses,” Opt. Eng. 34, 772–779 (1995).
[CrossRef]

Navarro, R.

Santamaría, J.

Simonet, P.

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

Sliney, D.

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

van Blokland, G. J.

G. J. van Blokland, D. van Norren, “Intensity and polarization of light scattered at small angles from the human fovea,” Vision Res. 26, 485–494 (1986).
[CrossRef] [PubMed]

van Norren, D.

G. J. van Blokland, D. van Norren, “Intensity and polarization of light scattered at small angles from the human fovea,” Vision Res. 26, 485–494 (1986).
[CrossRef] [PubMed]

Williams, D. R.

Wolbarsht, M.

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

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Appl. Opt. (1)

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

S. Marcos, P. Artal, D. G. Green, “The effect of decentered small pupils on ocular modulation transfer and contrast sensitivity,” Invest. Ophthalmol. Vis. Sci. Suppl. 35, 1258 (1994).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

J. Physiol. (London) (2)

D. G. Green, “Visual resolution when light enters through different parts of the pupil,” J. Physiol. (London) 190, 583–593 (1967).

F. W. Campbell, R. W. Gubisch, “Optical image quality of the human eye,” J. Physiol. (London) 186, 558–578 (1966).

Opt. Eng. (1)

P. Artal, S. Marcos, R. Navarro, M. Ferro, I. Miranda, “Through focus image quality with monofocal and multifocal intraocular lenses,” Opt. Eng. 34, 772–779 (1995).
[CrossRef]

Rev. Opt. (1)

M. F. Flamant, “Étude de la repartition de lumière dans l’image retinienne d’une fente,” Rev. Opt. 34, 433–459 (1955).

Vision Res. (2)

G. J. van Blokland, D. van Norren, “Intensity and polarization of light scattered at small angles from the human fovea,” Vision Res. 26, 485–494 (1986).
[CrossRef] [PubMed]

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

Other (2)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

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

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of the double-pass setup with unequal entrance and exit pupil sizes. DF, variable neutral-density (ND) filter (ND, 0.5–2.5); LP, linear polarizer; AP1, artificial entrance pupil; AP2, artificial exit pupil; BS, pellicle beam splitter; L1–L3, achromatic lenses. The solid line indicates the first passage; the dashed line, the second passage. To test the eye’s reversibility, two retinal images are recorded with the entrance and the exit pupils interchanged as shown in the scheme.

Fig. 2
Fig. 2

Scheme of the calculations of the eye’s optical performance from double-pass (D-P) measurements. I(x, y) is the equal pupil sizes’ double-pass image; Id(x, y) is the unequal pupil sizes’ double-pass image; P(x, y) is the ocular point-spread function; Pd(x, y) is the diffraction-limited point-spread function. The images shown here as example correspond to data obtained from subject II.

Fig. 3
Fig. 3

Examples of different stages in the MTF calculations with 4 mm. The curve with squares is the Fourier transform of the retinal images (with the dc peak; raw data). The dashed curve with circles is the function after peak removal [MTF (4 mm) × MTF (1.5 mm)], and the dashed curve with triangles is the final MTF (4 mm). The solid curve represents the diffraction-limited MTF (1.5 mm).

Fig. 4
Fig. 4

Double-pass retinal images obtained with different entrance and exit pupil sizes for subject NL. (a) 4-mm pupil diameter, (b) 6-mm pupil diameter.

Fig. 5
Fig. 5

Radial profile MTF’s for 1.5-mm pupil diameter in subject NL. We computed the MTF shown by the solid curve by square root and the MTF shown by the curve with circles on it by dividing by the diffraction-limited MTF. The dashed curve represents the diffraction-limited MTF for 1.5 mm.

Fig. 6
Fig. 6

Comparison of first-pass and second-pass MTF’s for subjects NL and II. The solid curve with circles represents the diffraction-limited MTF times the second-passage MTF. The dashed curve with triangles represents the diffraction-limited MTF times the first-passage MTF. (a) Subject NL with 4-mm pupil diameter, (b) subject NL with 6-mm pupil diameter, (c) subject II with 4-mm pupil diameter, (d) subject II with 6-mm pupil diameter.

Fig. 7
Fig. 7

MTF’s for subject II and for a 4-mm effective pupil calculated from the equal and the unequal entrance and exit pupils. We calculated the MTF for unequal pupils by dividing by the diffraction-limited 1.5-mm MTF. D-P, double pass.

Fig. 8
Fig. 8

(a) Horizontal and (b) vertical sections of the PTF for subjects NL and II, respectively, for different entrance and exit pupil sizes (1.5–6, 6–1.5, and 6–6 mm).

Fig. 9
Fig. 9

Two-dimensional plots of the PTF (in deg) for (a) subject NL and (b) subject II. The PTF’s were computed from the 1.5–6-mm pupils’ configuration. The gray-level images shown in the corners are the corresponding unequal-pupil double-pass images.

Equations (3)

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I ( x , y ) = P ( x , y ) P ( - x , - y ) .
I d ( x , y ) = P d ( x , y ) P ( - x , - y ) ,
O f ( u , v ) = tan - 1 ( Im { FT [ I d ( x , y ) ] } Re { FT [ I d ( x , y ) ] } ) .

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