Abstract

The magnitude of lateral chromatic aberration and its effect on image contrast were computed for a modified, reduced-eye model of the human eye, using geometrical optics. The results indicate that lateral chromatic aberration is a major factor affecting image quality for obliquely incident rays of polychromatic light. Modulation transfer functions for white sinusoidal gratings decline monotonically with spatial frequency, with eccentricity of the stimulus in the peripheral visual field, with grating orientation relative to the visual meridian, and with decentering of the pupil. Image contrast is largely independent of the color temperature of white light over the range 2800 to 12,000 K, but it improves significantly for the polychromatic green light of the P-31 oscilloscope phosphor. Selective filtering by macular pigment increases image contrast by an amount that grows with spatial frequency to about a factor of 1.5 at the foveal resolution limit. Reduced contrast caused by lateral chromatic aberration accounts for most of the threefold loss of acuity that occurs for foveal viewing through a decentered pupil. The aberration probably has negligible effect on peripheral acuity but may act to limit aliasing of peripheral patterns.

© 1987 Optical Society of America

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References

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  1. W. N. Charman, "The retinal image in the human eye," Prog. Retinal Res. 2, 1–50 (1983).
    [CrossRef]
  2. H. von Helmholtz, Treatise on Physiological Optics (with appendices by A. Gullstrand), J. P. C. Southall, ed. (Optical Society of America, Washington, D.C., 1924), Vol. 1.
  3. H. H. Emsley, Visual Optics, 5th ed. (Hatton, London, 1952).
  4. Y. Le Grand, Form and Space Vision, G. G. Heath and M. Millodot, eds. (Indiana U. Press, Bloomington, Ind., 1967).
  5. P. A. Howarth, "The lateral chromatic aberration of the eye," Ophthalmol. Physiol. Opt. 4, 223–226 (1984).
    [CrossRef]
  6. J. A. Sundet, "The effect of pupil size variations on the colour stereoscopic phenomenon," Vision Res. 12, 1027–1032 (1972). Although this is best demonstrated with eccentric pupils, some individuals observe the illusion spontaneously, presumably because of the natural deviation of the pupillary axis from the optical axis.
    [CrossRef] [PubMed]
  7. R. D. Freeman, "Alignment detection and resolution as a function of retinal location," Am. J. Optom. Physiol. Opt. 43, 812–817 (1966).
    [CrossRef]
  8. G. Westheimer, "The spatial grain of the perifoveal visual field," Vision Res. 22, 157–162 (1982).
    [CrossRef] [PubMed]
  9. A. van Meeteren and C. J. W. Dunnewold, "Image quality of the human eye for eccentric entrance pupils," Vision Res. 23, 573–579 (1983).
    [CrossRef] [PubMed]
  10. D. G. Green, "Visual resolution when light enters the eye through different parts of the pupil," J. Physiol. 190, 583–593 (1967).
    [PubMed]
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    [CrossRef] [PubMed]
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    [PubMed]
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    [PubMed]
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    [PubMed]
  20. L. A. Temme, L. Malcus, and W. K. Noell, "Peripheral visual field is radially organized," Am. J. Optom. Physiol. Opt. 62, 545–554 (1985).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  25. P. L. Pease, A. J. Adams, and E. Nuccio, "Optical density of human macular pigment," Vision Res. (to be published).
    [PubMed]
  26. T. Mandelman and J. G. Sivak, "Longitudinal chromatic aberration of the vertebrate eye," Vision Res. 23, 1555–1559. (1983).
    [CrossRef] [PubMed]
  27. R. A. Weale, "Spectrai sensitivity and wave-length discrimination of the peripheral retina," J. Physiol. 119, 170–190 (1953).
    [PubMed]
  28. B. R. Wooten, K. Fuld, and L. Spillmann, "Photopic spectral sensitivity of the peripheral retina," J. Opt. Soc. Am. 65, 334–342 (1975).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  31. N. Drasdo and C. W. Fowler, "Non-linear projection of the retinal image in a wide-angle schematic eye," Br. J. Ophthalmol. 58, 709–714 (1974).
    [CrossRef] [PubMed]

1987 (1)

1985 (3)

L. A. Temme, L. Malcus, and W. K. Noell, "Peripheral visual field is radially organized," Am. J. Optom. Physiol. Opt. 62, 545–554 (1985).
[CrossRef] [PubMed]

D. R. Williams, "Aliasing in human foveal vision," Vision Res. 25, 195–205 (1985).
[CrossRef] [PubMed]

D. J. Walsh and L. N. Thibos, "Oblique and meridional effects in peripheral vision," J. Opt. Soc. Am. A 2(13), P65 (1985).

1984 (1)

P. A. Howarth, "The lateral chromatic aberration of the eye," Ophthalmol. Physiol. Opt. 4, 223–226 (1984).
[CrossRef]

1983 (3)

W. N. Charman, "The retinal image in the human eye," Prog. Retinal Res. 2, 1–50 (1983).
[CrossRef]

A. van Meeteren and C. J. W. Dunnewold, "Image quality of the human eye for eccentric entrance pupils," Vision Res. 23, 573–579 (1983).
[CrossRef] [PubMed]

T. Mandelman and J. G. Sivak, "Longitudinal chromatic aberration of the vertebrate eye," Vision Res. 23, 1555–1559. (1983).
[CrossRef] [PubMed]

1982 (2)

J. Rovamo, V. Virsu, P. Laurinen, and L. Hyvarinen, "Resolution of gratings oriented along and across meridians in peripheral vision," Invest. Ophthalmol. Vis. Sci. 23, 666–670 (1982).
[PubMed]

G. Westheimer, "The spatial grain of the perifoveal visual field," Vision Res. 22, 157–162 (1982).
[CrossRef] [PubMed]

1980 (1)

1977 (1)

1975 (2)

B. R. Wooten, K. Fuld, and L. Spillmann, "Photopic spectral sensitivity of the peripheral retina," J. Opt. Soc. Am. 65, 334–342 (1975).
[CrossRef] [PubMed]

L. Frisen and A. Glansholm, "Optical and neural resolution in peripheral vision," Invest. Ophthalmol. 14, 528–536 (1975).
[PubMed]

1974 (2)

N. Drasdo and C. W. Fowler, "Non-linear projection of the retinal image in a wide-angle schematic eye," Br. J. Ophthalmol. 58, 709–714 (1974).
[CrossRef] [PubMed]

V. M. Reading and R. A. Weale, "Macular pigment and chromatic aberration," J. Opt. Soc. Am. 64, 231–234 (1974).
[CrossRef] [PubMed]

1972 (1)

J. A. Sundet, "The effect of pupil size variations on the colour stereoscopic phenomenon," Vision Res. 12, 1027–1032 (1972). Although this is best demonstrated with eccentric pupils, some individuals observe the illusion spontaneously, presumably because of the natural deviation of the pupillary axis from the optical axis.
[CrossRef] [PubMed]

1967 (2)

D. G. Green, "Visual resolution when light enters the eye through different parts of the pupil," J. Physiol. 190, 583–593 (1967).
[PubMed]

F. W. Campbell and R. W. Gubisch, "The effect of chromatic aberration on visual acuity," J. Physiol. 192, 345–358 (1967).
[PubMed]

1966 (1)

R. D. Freeman, "Alignment detection and resolution as a function of retinal location," Am. J. Optom. Physiol. Opt. 43, 812–817 (1966).
[CrossRef]

1953 (1)

R. A. Weale, "Spectrai sensitivity and wave-length discrimination of the peripheral retina," J. Physiol. 119, 170–190 (1953).
[PubMed]

Dunnewold, C. J. W.

A. van Meeteren and C. J. W. Dunnewold, "Image quality of the human eye for eccentric entrance pupils," Vision Res. 23, 573–579 (1983).
[CrossRef] [PubMed]

Meeteren, A.

A. van Meeteren and C. J. W. Dunnewold, "Image quality of the human eye for eccentric entrance pupils," Vision Res. 23, 573–579 (1983).
[CrossRef] [PubMed]

von Helmholtz, H.

H. von Helmholtz, Treatise on Physiological Optics (with appendices by A. Gullstrand), J. P. C. Southall, ed. (Optical Society of America, Washington, D.C., 1924), Vol. 1.

Abramov, I.

Adams, A. J.

P. L. Pease, A. J. Adams, and E. Nuccio, "Optical density of human macular pigment," Vision Res. (to be published).
[PubMed]

Campbell, F. W.

F. W. Campbell and R. W. Gubisch, "The effect of chromatic aberration on visual acuity," J. Physiol. 192, 345–358 (1967).
[PubMed]

Charman, W. N.

W. N. Charman, "The retinal image in the human eye," Prog. Retinal Res. 2, 1–50 (1983).
[CrossRef]

Cheney, F. E.

L. N. Thibos, F. E. Cheney, and D. J. Walsh, "Retinal limits to the detection and resolution of gratings," J. Opt. Soc. Am. A 4, 1524–1529(1987).
[CrossRef] [PubMed]

L. N. Thibos, D. J. Walsh, and F. E. Cheney, "Vision beyond the resolution limit: aliasing in the periphery," submitted to Vision Res.

F. E. Cheney, "The effect of lateral chromatic aberration on the detection of gratings in peripheral vision," M.S. thesis (Indiana University, Bloomington, Ind., 1987).

Drasdo, N.

N. Drasdo and C. W. Fowler, "Non-linear projection of the retinal image in a wide-angle schematic eye," Br. J. Ophthalmol. 58, 709–714 (1974).
[CrossRef] [PubMed]

Emsley, H. H.

H. H. Emsley, Visual Optics, 5th ed. (Hatton, London, 1952).

Fowler, C. W.

N. Drasdo and C. W. Fowler, "Non-linear projection of the retinal image in a wide-angle schematic eye," Br. J. Ophthalmol. 58, 709–714 (1974).
[CrossRef] [PubMed]

Freeman, R. D.

R. D. Freeman, "Alignment detection and resolution as a function of retinal location," Am. J. Optom. Physiol. Opt. 43, 812–817 (1966).
[CrossRef]

Frisen, L.

L. Frisen and A. Glansholm, "Optical and neural resolution in peripheral vision," Invest. Ophthalmol. 14, 528–536 (1975).
[PubMed]

Fuld, K.

Glansholm, A.

L. Frisen and A. Glansholm, "Optical and neural resolution in peripheral vision," Invest. Ophthalmol. 14, 528–536 (1975).
[PubMed]

Gordon, J.

Grand, Y. Le

Y. Le Grand, Form and Space Vision, G. G. Heath and M. Millodot, eds. (Indiana U. Press, Bloomington, Ind., 1967).

Green, D. G.

D. G. Green, "Visual resolution when light enters the eye through different parts of the pupil," J. Physiol. 190, 583–593 (1967).
[PubMed]

Gubisch, R. W.

F. W. Campbell and R. W. Gubisch, "The effect of chromatic aberration on visual acuity," J. Physiol. 192, 345–358 (1967).
[PubMed]

Houstoun, R. A.

R. A. Houstoun, A Treatise on Light, 4th ed. (Longmans, Green, London, 1925).

Howarth, P. A.

P. A. Howarth, "The lateral chromatic aberration of the eye," Ophthalmol. Physiol. Opt. 4, 223–226 (1984).
[CrossRef]

Hyvarinen, L.

J. Rovamo, V. Virsu, P. Laurinen, and L. Hyvarinen, "Resolution of gratings oriented along and across meridians in peripheral vision," Invest. Ophthalmol. Vis. Sci. 23, 666–670 (1982).
[PubMed]

Laurinen, P.

J. Rovamo, V. Virsu, P. Laurinen, and L. Hyvarinen, "Resolution of gratings oriented along and across meridians in peripheral vision," Invest. Ophthalmol. Vis. Sci. 23, 666–670 (1982).
[PubMed]

Malcus, L.

L. A. Temme, L. Malcus, and W. K. Noell, "Peripheral visual field is radially organized," Am. J. Optom. Physiol. Opt. 62, 545–554 (1985).
[CrossRef] [PubMed]

Mandelman, T.

T. Mandelman and J. G. Sivak, "Longitudinal chromatic aberration of the vertebrate eye," Vision Res. 23, 1555–1559. (1983).
[CrossRef] [PubMed]

Moon, P.

P. Moon, The Scientific Basis of Illumination Engineering (Dover, New York, 1961).

Noell, W. K.

L. A. Temme, L. Malcus, and W. K. Noell, "Peripheral visual field is radially organized," Am. J. Optom. Physiol. Opt. 62, 545–554 (1985).
[CrossRef] [PubMed]

Nuccio, E.

P. L. Pease, A. J. Adams, and E. Nuccio, "Optical density of human macular pigment," Vision Res. (to be published).
[PubMed]

Pease, P. L.

P. L. Pease, A. J. Adams, and E. Nuccio, "Optical density of human macular pigment," Vision Res. (to be published).
[PubMed]

Reading, V. M.

Rovamo, J.

J. Rovamo, V. Virsu, P. Laurinen, and L. Hyvarinen, "Resolution of gratings oriented along and across meridians in peripheral vision," Invest. Ophthalmol. Vis. Sci. 23, 666–670 (1982).
[PubMed]

Sivak, J. G.

T. Mandelman and J. G. Sivak, "Longitudinal chromatic aberration of the vertebrate eye," Vision Res. 23, 1555–1559. (1983).
[CrossRef] [PubMed]

Spillmann, L.

Stabell, B.

Stabell, U.

Sundet, J. A.

J. A. Sundet, "The effect of pupil size variations on the colour stereoscopic phenomenon," Vision Res. 12, 1027–1032 (1972). Although this is best demonstrated with eccentric pupils, some individuals observe the illusion spontaneously, presumably because of the natural deviation of the pupillary axis from the optical axis.
[CrossRef] [PubMed]

Temme, L. A.

L. A. Temme, L. Malcus, and W. K. Noell, "Peripheral visual field is radially organized," Am. J. Optom. Physiol. Opt. 62, 545–554 (1985).
[CrossRef] [PubMed]

Thibos, L. N.

L. N. Thibos, F. E. Cheney, and D. J. Walsh, "Retinal limits to the detection and resolution of gratings," J. Opt. Soc. Am. A 4, 1524–1529(1987).
[CrossRef] [PubMed]

D. J. Walsh and L. N. Thibos, "Oblique and meridional effects in peripheral vision," J. Opt. Soc. Am. A 2(13), P65 (1985).

L. N. Thibos, D. J. Walsh, and F. E. Cheney, "Vision beyond the resolution limit: aliasing in the periphery," submitted to Vision Res.

Virsu, V.

J. Rovamo, V. Virsu, P. Laurinen, and L. Hyvarinen, "Resolution of gratings oriented along and across meridians in peripheral vision," Invest. Ophthalmol. Vis. Sci. 23, 666–670 (1982).
[PubMed]

Walls, G.

G. Walls, The Vertebrate Eye and Its Adaptive Radiation (Hafner, London, 1963).

Walsh, D. J.

L. N. Thibos, F. E. Cheney, and D. J. Walsh, "Retinal limits to the detection and resolution of gratings," J. Opt. Soc. Am. A 4, 1524–1529(1987).
[CrossRef] [PubMed]

D. J. Walsh and L. N. Thibos, "Oblique and meridional effects in peripheral vision," J. Opt. Soc. Am. A 2(13), P65 (1985).

L. N. Thibos, D. J. Walsh, and F. E. Cheney, "Vision beyond the resolution limit: aliasing in the periphery," submitted to Vision Res.

Walsh, J. W. T.

J. W. T. Walsh, The Science of Daylight (Macdonald, London, 1961).

Weale, R. A.

V. M. Reading and R. A. Weale, "Macular pigment and chromatic aberration," J. Opt. Soc. Am. 64, 231–234 (1974).
[CrossRef] [PubMed]

R. A. Weale, "Spectrai sensitivity and wave-length discrimination of the peripheral retina," J. Physiol. 119, 170–190 (1953).
[PubMed]

Westheimer, G.

G. Westheimer, "The spatial grain of the perifoveal visual field," Vision Res. 22, 157–162 (1982).
[CrossRef] [PubMed]

Williams, D. R.

D. R. Williams, "Aliasing in human foveal vision," Vision Res. 25, 195–205 (1985).
[CrossRef] [PubMed]

Wooten, B. R.

Am. J. Optom. Physiol. Opt. (2)

R. D. Freeman, "Alignment detection and resolution as a function of retinal location," Am. J. Optom. Physiol. Opt. 43, 812–817 (1966).
[CrossRef]

L. A. Temme, L. Malcus, and W. K. Noell, "Peripheral visual field is radially organized," Am. J. Optom. Physiol. Opt. 62, 545–554 (1985).
[CrossRef] [PubMed]

Br. J. Ophthalmol. (1)

N. Drasdo and C. W. Fowler, "Non-linear projection of the retinal image in a wide-angle schematic eye," Br. J. Ophthalmol. 58, 709–714 (1974).
[CrossRef] [PubMed]

Invest. Ophthalmol. (1)

L. Frisen and A. Glansholm, "Optical and neural resolution in peripheral vision," Invest. Ophthalmol. 14, 528–536 (1975).
[PubMed]

Invest. Ophthalmol. Vis. Sci. (1)

J. Rovamo, V. Virsu, P. Laurinen, and L. Hyvarinen, "Resolution of gratings oriented along and across meridians in peripheral vision," Invest. Ophthalmol. Vis. Sci. 23, 666–670 (1982).
[PubMed]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

D. J. Walsh and L. N. Thibos, "Oblique and meridional effects in peripheral vision," J. Opt. Soc. Am. A 2(13), P65 (1985).

L. N. Thibos, F. E. Cheney, and D. J. Walsh, "Retinal limits to the detection and resolution of gratings," J. Opt. Soc. Am. A 4, 1524–1529(1987).
[CrossRef] [PubMed]

J. Physiol. (3)

D. G. Green, "Visual resolution when light enters the eye through different parts of the pupil," J. Physiol. 190, 583–593 (1967).
[PubMed]

F. W. Campbell and R. W. Gubisch, "The effect of chromatic aberration on visual acuity," J. Physiol. 192, 345–358 (1967).
[PubMed]

R. A. Weale, "Spectrai sensitivity and wave-length discrimination of the peripheral retina," J. Physiol. 119, 170–190 (1953).
[PubMed]

Ophthalmol. Physiol. Opt. (1)

P. A. Howarth, "The lateral chromatic aberration of the eye," Ophthalmol. Physiol. Opt. 4, 223–226 (1984).
[CrossRef]

Prog. Retinal Res. (1)

W. N. Charman, "The retinal image in the human eye," Prog. Retinal Res. 2, 1–50 (1983).
[CrossRef]

Vision Res. (5)

J. A. Sundet, "The effect of pupil size variations on the colour stereoscopic phenomenon," Vision Res. 12, 1027–1032 (1972). Although this is best demonstrated with eccentric pupils, some individuals observe the illusion spontaneously, presumably because of the natural deviation of the pupillary axis from the optical axis.
[CrossRef] [PubMed]

G. Westheimer, "The spatial grain of the perifoveal visual field," Vision Res. 22, 157–162 (1982).
[CrossRef] [PubMed]

A. van Meeteren and C. J. W. Dunnewold, "Image quality of the human eye for eccentric entrance pupils," Vision Res. 23, 573–579 (1983).
[CrossRef] [PubMed]

T. Mandelman and J. G. Sivak, "Longitudinal chromatic aberration of the vertebrate eye," Vision Res. 23, 1555–1559. (1983).
[CrossRef] [PubMed]

D. R. Williams, "Aliasing in human foveal vision," Vision Res. 25, 195–205 (1985).
[CrossRef] [PubMed]

Other (10)

F. E. Cheney, "The effect of lateral chromatic aberration on the detection of gratings in peripheral vision," M.S. thesis (Indiana University, Bloomington, Ind., 1987).

L. N. Thibos, D. J. Walsh, and F. E. Cheney, "Vision beyond the resolution limit: aliasing in the periphery," submitted to Vision Res.

P. L. Pease, A. J. Adams, and E. Nuccio, "Optical density of human macular pigment," Vision Res. (to be published).
[PubMed]

H. von Helmholtz, Treatise on Physiological Optics (with appendices by A. Gullstrand), J. P. C. Southall, ed. (Optical Society of America, Washington, D.C., 1924), Vol. 1.

H. H. Emsley, Visual Optics, 5th ed. (Hatton, London, 1952).

Y. Le Grand, Form and Space Vision, G. G. Heath and M. Millodot, eds. (Indiana U. Press, Bloomington, Ind., 1967).

R. A. Houstoun, A Treatise on Light, 4th ed. (Longmans, Green, London, 1925).

P. Moon, The Scientific Basis of Illumination Engineering (Dover, New York, 1961).

J. W. T. Walsh, The Science of Daylight (Macdonald, London, 1961).

G. Walls, The Vertebrate Eye and Its Adaptive Radiation (Hafner, London, 1963).

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Figures (7)

Fig. 1
Fig. 1

Geometrical optics of the reduced-eye modal of the human eye as modified to include the entrance and exit pupils. An incident chief ray of eccentricity is directed toward the center of the entrance pupil P and emerges from the center of the exit pupil at Q. The exit angle γ varies with wavelength, thus inducing a change in image location and therefore a variable phase shift in the image of a polychromatic grating.

Fig. 2
Fig. 2

Variation of lateral chromatic aberration with eccentricity. Solid curve shows the separation of images for wavelengths of 0.43 μm (Fraunhofer G line) and 0.77 μm (Fraunhofer A line) for the modified reduced eye of Fig. 1. Dotted curve shows this image separation for Gullstrand’s reduced eye, which has the entrance and exit pupils at the refracting surface. The aberration is characterized in image space and referenced to the center of the exit pupil.

Fig. 3
Fig. 3

(a) Image displacement and spatial phase shift across the visible spectrum of objects located at various eccentricities from the optical axis. Image position was calculated at 0.01-μm intervals, and the results are displayed relative to image position for the reference wavelength 0.57 μm. Numbers next to curves indicate object eccentricity () from the optical axis in degrees. (b) The effect of image displacement on spatial phase of gratings. Solid curve is the = 30° curve from (b) and is with reference to the right-hand ordinate. Dashed curves show the corresponding phase shift of gratings of various spatial frequencies and refer to the left-hand ordinate. Gratings are oriented orthogonal to visual meridian. c/d, Cycles per degree.

Fig. 4
Fig. 4

The effect of lateral chromatic aberration on image contrast varies with grating orientation relative to visual meridian. σ = 30°. The spatial frequency of each grating is indicated by a number near the curve [cycles per degree (c/d) in object space].

Fig. 5
Fig. 5

Comparison of modulation transfer functions for various blackbody radiators and the P-31 oscilloscope phosphor, with = 20° and a relative orientation of 90°. The dotted curve shows the effect of removing macular pigment (tungsten source).

Fig. 6
Fig. 6

Effect of lateral chromatic aberration on the modulation transfer function at different eccentricities. (a) Modulation transfer functions for selected eccentricities ( is the number near each curve). The relative orientation is 90°, and the source color temperature is 2800 K. The dotted curve and the dashed curve show the loss of image contrast that is due to longitudinal chromatic aberration for 1.5- and 2.5-mm apertures, respectively (data are from Fig. 3 of Ref. 16). The dotted–dashed curve is for a pigment-free observer, with = 5°. (b) The same results are plotted as a function of eccentricity. The number near each curve indicates the spatial frequency in cycles per degree (c/d).

Fig. 7
Fig. 7

Comparison of published calculations based on wave optics, psychophysical results for eccentric pupils, and present calculations based on geometrical optics. Symbols show the combined effects of lateral and longitudinal chromatic aberration for 1-mm (diamonds), 2-mm (circles) and 3-mm (triangles) translations of the entrance pupil computed by Van Meeteren and Dunnewold.9 Solid curves are present results computed for lateral chromatic aberration only, assuming the spectrum of the P-31 phosphor, for the same three values of translation. The dashed curve is the contrast threshold function measured by Green10 for foveal viewing of interference fringes. The intersections of dashed and solid curves give the predicted cutoff spatial frequencies for foveal viewing through artificial pupils decentered by 1, 2, and 3 mm. Arrows show the measured cutoff spatial frequencies determined psychophysically by Green for these same three values of pupil decentering.

Equations (13)

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

sin ( α ) = NP sin ( 180 ° - ) / R = 3.98 sin ( ) / 5.55 ,
sin ( β ) = sin ( α ) / n ,
n = 1.31848 + 0.0066620 / ( λ - 0.1292 ) .
γ = - α + β .
ϕ ( λ ) = ( γ - γ 0 ) f sin ( θ ) / 0.82 ,
L ( x , y ) = S ( λ ) { 1 + M cos 2 π [ f x - ϕ ( λ ) ] } ,
L ( x ) = L ( x , λ ) d λ = S ( λ ) d λ + M S ( λ ) cos 2 π [ f x - ϕ ( λ ) ] d λ .
L ( x ) = L 0 [ 1 + M A 2 + B 2 cos ( 2 π f x - Q ) ] ,
A = 1 L 0 S ( λ ) cos [ ϕ ( λ ) ] d λ ,
B = 1 L 0 S ( λ ) sin [ ϕ ( λ ) ] d λ ,
Q = arctan ( B / A ) .
modulation transfer = A 2 + B 2 .
d = NP sin ( )

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