Abstract

We investigated the formation of the aerial image in the double-pass method to measure the optical quality of the human eye. We show theoretically and empirically that the double pass through the eye’s optics forces the light distribution in the aerial image to be an even-symmetric function even if the single-pass point-spread function is asymmetric as a result of odd aberrations in the eye. The reason for this is that the double-pass imaging process is described by the autocorrelation rather than the autoconvolution of the single-pass point-spread functions, as has been previously assumed. This implies that although the modulation transfer function can be computed from the double-pass aerial image, the phase transfer function cannot. We also show that the lateral chromatic aberration of the eye cannot be measured with the double-pass procedure because it is canceled by the second pass through the eye’s optics.

© 1995 Optical Society of America

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References

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  1. W. N. Charman, “Optics of the human eye,” in Visual Optics and Instrumentation, Vol. 1 of Vision and Visual Dysfunction, J. R. Cronly-Dillion, ed. (Macmillan, London, 1991), pp. 1–26.
  2. M. F. Flamant, “Étude de la repartition de lumière dans l’image retinienne d’une fente,” Rev. Opt. 34, 433–459 (1955).
  3. J. Krauskpof, “Light distribution in human retinal images,”J. Opt. Soc. Am. 52, 1046–1050 (1962).
    [CrossRef]
  4. F. W. Campbell, R. W. Gubisch, “Optical image quality of the human eye,”J. Physiol. (London) 186, 558–578 (1966).
  5. R. W. Gubisch, “Optical performance of the human eye,”J. Opt. Soc. Am. 57, 407–415 (1967).
    [CrossRef]
  6. R. Rohler, U. Miller, M. Aberl, “Zur Messung der Modulatonsubertragungsfunktion des Lebenden menschlichen Auges in reflektierten Licht,” Vision Res. 9, 407–428 (1969).
    [CrossRef]
  7. J. A. M. Jennings, W. N. Charman, “Off-axis image quality in the human eye,” Vision Res. 21, 445–454 (1981).
    [CrossRef] [PubMed]
  8. 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]
  9. 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]
  10. 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]
  11. J. G. van Blockland, D. van Norren, “Intensity and polarization of light scattered at small angles from the human fovea,” Vision Res. 26, 485–494 (1986).
    [CrossRef]
  12. J. M. Gorrand, “Reflection characteristics of the human fovea assessed by reflectomodulometry,” Ophthalmol. Physiol. Opt. 9, 53–60 (1989).
    [CrossRef]
  13. 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 (to be published).
  14. J. F. Simon, P. Denieul, “Influence of the size of the test employed in measurements of modulation transfer function of the eye,”J. Opt. Soc. Am. 63, 894–896 (1973).
    [CrossRef] [PubMed]
  15. J. J. Vos, J. Walraven, A. Meeteren, “Light profiles of the foveal images of a point test,” Vision Res. 16, 215–219 (1976).
    [CrossRef]
  16. H. C. Howland, B. Howland, “A subjective method for the measurement of monochromatic aberrations of the eye,”J. Opt. Soc. Am. 67, 1508–1518 (1977).
    [CrossRef] [PubMed]
  17. 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]
  18. S. Marcos, P. Artal, D. G. Green, “The effect of decentered small pupils on optical modulation transfer and contrast sensitivity,” Invest. Ophthalmol. Vis. Sci. Suppl. 35, 1258 (1994).
  19. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  20. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).
  21. P. Artal, J. Santamaría, J. Bescós, “Phase-transfer function of the human eye and its influence on point-spread function and wave aberration,” J. Opt. Soc. Am. A 5, 1791–1795 (1988).
    [CrossRef] [PubMed]
  22. P. Artal, R. Navarro, D. H. Brainard, S. J. Galvin, D. R. Williams, “Off-axis optical quality of the eye and retinal sampling,” Invest. Ophthalmol. Vis. Sci. Suppl. 33, 3241 (1992).
  23. P. Artal, M. Ferro, I. Miranda, R. Navarro, “Effects of aging in retinal image quality,” J. Opt. Soc. Am. A 10, 1656–1662 (1993).
    [CrossRef] [PubMed]
  24. M. A. Losada, R. Navarro, J. Santamaría, “Relative contribution of optical and neural limitations to human contrast sensitivity at different luminance levels,” Vision Res. 33, 2321–2336 (1993).
    [CrossRef] [PubMed]
  25. N. Sekiguchi, D. R. Williams, D. H. Brainard, “Efficiency in detection of isoluminant and isochomatic interference fringes,” J. Opt. Soc. Am. A 10, 2118–2133 (1993).
    [CrossRef]
  26. P. Artal, J. Santamaría, J. Bescós, “Retrieval of wave aberration of human eyes from actual point-spread-function data,” J. Opt. Soc. Am. A 5, 1201–1206 (1988).
    [CrossRef] [PubMed]
  27. J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–273.
  28. 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]
  29. 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]

1994 (2)

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

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]

1993 (4)

1992 (2)

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]

P. Artal, R. Navarro, D. H. Brainard, S. J. Galvin, D. R. Williams, “Off-axis optical quality of the eye and retinal sampling,” Invest. Ophthalmol. Vis. Sci. Suppl. 33, 3241 (1992).

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]

1989 (1)

J. M. Gorrand, “Reflection characteristics of the human fovea assessed by reflectomodulometry,” Ophthalmol. Physiol. Opt. 9, 53–60 (1989).
[CrossRef]

1988 (2)

1987 (1)

1986 (1)

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

1984 (1)

1981 (1)

J. A. M. Jennings, W. N. Charman, “Off-axis image quality in the human eye,” Vision Res. 21, 445–454 (1981).
[CrossRef] [PubMed]

1977 (1)

1976 (1)

J. J. Vos, J. Walraven, A. Meeteren, “Light profiles of the foveal images of a point test,” Vision Res. 16, 215–219 (1976).
[CrossRef]

1973 (1)

1969 (1)

R. Rohler, U. Miller, M. Aberl, “Zur Messung der Modulatonsubertragungsfunktion des Lebenden menschlichen Auges in reflektierten Licht,” Vision Res. 9, 407–428 (1969).
[CrossRef]

1967 (1)

1966 (1)

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

1962 (1)

1955 (1)

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

Aberl, M.

R. Rohler, U. Miller, M. Aberl, “Zur Messung der Modulatonsubertragungsfunktion des Lebenden menschlichen Auges in reflektierten Licht,” Vision Res. 9, 407–428 (1969).
[CrossRef]

Artal, P.

Bescós, J.

Bille, J.

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

Brainard, D.

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 (to be published).

Brainard, D. H.

N. Sekiguchi, D. R. Williams, D. H. Brainard, “Efficiency in detection of isoluminant and isochomatic interference fringes,” J. Opt. Soc. Am. A 10, 2118–2133 (1993).
[CrossRef]

P. Artal, R. Navarro, D. H. Brainard, S. J. Galvin, D. R. Williams, “Off-axis optical quality of the eye and retinal sampling,” Invest. Ophthalmol. Vis. Sci. Suppl. 33, 3241 (1992).

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]

Charman, W. N.

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]

J. A. M. Jennings, W. N. Charman, “Off-axis image quality in the human eye,” Vision Res. 21, 445–454 (1981).
[CrossRef] [PubMed]

W. N. Charman, “Optics of the human eye,” in Visual Optics and Instrumentation, Vol. 1 of Vision and Visual Dysfunction, J. R. Cronly-Dillion, ed. (Macmillan, London, 1991), pp. 1–26.

Dainty, J. C.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–273.

Denieul, P.

Ferro, M.

Fienup, J. R.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–273.

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).

Galvin, S. J.

P. Artal, R. Navarro, D. H. Brainard, S. J. Galvin, D. R. Williams, “Off-axis optical quality of the eye and retinal sampling,” Invest. Ophthalmol. Vis. Sci. Suppl. 33, 3241 (1992).

Goelz, S.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gorrand, J. M.

J. M. Gorrand, “Reflection characteristics of the human fovea assessed by reflectomodulometry,” Ophthalmol. Physiol. Opt. 9, 53–60 (1989).
[CrossRef]

Green, D. G.

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

Grimm, B.

Gubisch, R. W.

R. W. Gubisch, “Optical performance of the human eye,”J. Opt. Soc. Am. 57, 407–415 (1967).
[CrossRef]

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.

Jennings, J. A. M.

J. A. M. Jennings, W. N. Charman, “Off-axis image quality in the human eye,” Vision Res. 21, 445–454 (1981).
[CrossRef] [PubMed]

Krauskpof, J.

Liang, J.

Losada, M. A.

M. A. Losada, R. Navarro, J. Santamaría, “Relative contribution of optical and neural limitations to human contrast sensitivity at different luminance levels,” Vision Res. 33, 2321–2336 (1993).
[CrossRef] [PubMed]

MacHahon, M.

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 (to be published).

Marcos, S.

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

Meeteren, A.

J. J. Vos, J. Walraven, A. Meeteren, “Light profiles of the foveal images of a point test,” Vision Res. 16, 215–219 (1976).
[CrossRef]

Miller, U.

R. Rohler, U. Miller, M. Aberl, “Zur Messung der Modulatonsubertragungsfunktion des Lebenden menschlichen Auges in reflektierten Licht,” Vision Res. 9, 407–428 (1969).
[CrossRef]

Miranda, I.

Navarro, R.

P. Artal, M. Ferro, I. Miranda, R. Navarro, “Effects of aging in retinal image quality,” J. Opt. Soc. Am. A 10, 1656–1662 (1993).
[CrossRef] [PubMed]

M. A. Losada, R. Navarro, J. Santamaría, “Relative contribution of optical and neural limitations to human contrast sensitivity at different luminance levels,” Vision Res. 33, 2321–2336 (1993).
[CrossRef] [PubMed]

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]

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]

P. Artal, R. Navarro, D. H. Brainard, S. J. Galvin, D. R. Williams, “Off-axis optical quality of the eye and retinal sampling,” Invest. Ophthalmol. Vis. Sci. Suppl. 33, 3241 (1992).

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 (to be published).

Rohler, R.

R. Rohler, U. Miller, M. Aberl, “Zur Messung der Modulatonsubertragungsfunktion des Lebenden menschlichen Auges in reflektierten Licht,” Vision Res. 9, 407–428 (1969).
[CrossRef]

Santamaría, J.

Sekiguchi, N.

Simon, J. F.

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]

van Blockland, J. G.

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

van Norren, D.

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

Vos, J. J.

J. J. Vos, J. Walraven, A. Meeteren, “Light profiles of the foveal images of a point test,” Vision Res. 16, 215–219 (1976).
[CrossRef]

Walraven, J.

J. J. Vos, J. Walraven, A. Meeteren, “Light profiles of the foveal images of a point test,” Vision Res. 16, 215–219 (1976).
[CrossRef]

Walsh, G.

Williams, D. R.

N. Sekiguchi, D. R. Williams, D. H. Brainard, “Efficiency in detection of isoluminant and isochomatic interference fringes,” J. Opt. Soc. Am. A 10, 2118–2133 (1993).
[CrossRef]

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]

P. Artal, R. Navarro, D. H. Brainard, S. J. Galvin, D. R. Williams, “Off-axis optical quality of the eye and retinal sampling,” Invest. Ophthalmol. Vis. Sci. Suppl. 33, 3241 (1992).

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 (to be published).

Appl. Opt. (1)

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

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

P. Artal, R. Navarro, D. H. Brainard, S. J. Galvin, D. R. Williams, “Off-axis optical quality of the eye and retinal sampling,” Invest. Ophthalmol. Vis. Sci. Suppl. 33, 3241 (1992).

J. Opt. Soc. Am. (4)

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

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]

P. Artal, M. Ferro, I. Miranda, R. Navarro, “Effects of aging in retinal image quality,” J. Opt. Soc. Am. A 10, 1656–1662 (1993).
[CrossRef] [PubMed]

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]

P. Artal, J. Santamaría, J. Bescós, “Phase-transfer function of the human eye and its influence on point-spread function and wave aberration,” J. Opt. Soc. Am. A 5, 1791–1795 (1988).
[CrossRef] [PubMed]

N. Sekiguchi, D. R. Williams, D. H. Brainard, “Efficiency in detection of isoluminant and isochomatic interference fringes,” J. Opt. Soc. Am. A 10, 2118–2133 (1993).
[CrossRef]

P. Artal, J. Santamaría, J. Bescós, “Retrieval of wave aberration of human eyes from actual point-spread-function data,” J. Opt. Soc. Am. A 5, 1201–1206 (1988).
[CrossRef] [PubMed]

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]

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]

J. Physiol. (London) (1)

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

Ophthalmol. Physiol. Opt. (1)

J. M. Gorrand, “Reflection characteristics of the human fovea assessed by reflectomodulometry,” Ophthalmol. Physiol. Opt. 9, 53–60 (1989).
[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. (6)

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

R. Rohler, U. Miller, M. Aberl, “Zur Messung der Modulatonsubertragungsfunktion des Lebenden menschlichen Auges in reflektierten Licht,” Vision Res. 9, 407–428 (1969).
[CrossRef]

J. A. M. Jennings, W. N. Charman, “Off-axis image quality in the human eye,” Vision Res. 21, 445–454 (1981).
[CrossRef] [PubMed]

J. J. Vos, J. Walraven, A. Meeteren, “Light profiles of the foveal images of a point test,” Vision Res. 16, 215–219 (1976).
[CrossRef]

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]

M. A. Losada, R. Navarro, J. Santamaría, “Relative contribution of optical and neural limitations to human contrast sensitivity at different luminance levels,” Vision Res. 33, 2321–2336 (1993).
[CrossRef] [PubMed]

Other (5)

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–273.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

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 (to be published).

W. N. Charman, “Optics of the human eye,” in Visual Optics and Instrumentation, Vol. 1 of Vision and Visual Dysfunction, J. R. Cronly-Dillion, ed. (Macmillan, London, 1991), pp. 1–26.

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of the image-formation process in the double pass. x, y, object coordinates; x′, y′, image-plane (retinal) coordinates; x″, y″, double-pass coordinates; d, d′, object and image distances, respectively. Object O is a point test, and P(x′, y′) is the single-pass PSF.

Fig. 2
Fig. 2

Experimental setups for recording the (a) single-pass and (b) double-pass PSF’s in an artificial eye, LT. ND, neutral-density filter; M1, M2, microscope objectives (10×); O, 10-μm pinhole (object test); L1, collimator lens; L2, L3, Badal system lenses (f′ = 120 mm); L, lens (f′ = 200 mm); RD, rotating diffuser.

Fig. 3
Fig. 3

Logarithm of the single-pass PSF [P(x′, y′)] in a gray-level image and a 1-D horizontal section.

Fig. 4
Fig. 4

Logarithm of the single-pass (aerial) image [I″(x″, y″)]: gray-level image and 1-D horizontal section.

Fig. 5
Fig. 5

Logarithm of the autocorrelation of the single-pass PSF: gray-level image and 1-D horizontal section.

Fig. 6
Fig. 6

Logarithm of the autoconvolution of the single-pass PSF: gray-level image and 1-D horizontal section.

Fig. 7
Fig. 7

1-D sections of the MTF’s computed from the single-pass PSF (solid curve) and from the double-pass image (dotted curve).

Fig. 8
Fig. 8

(a) Single-pass PSF’s and (b) double-pass images for red (632-nm) and blue (488-nm) light.

Equations (8)

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

h ( x , y ; x , y ) = 1 λ 2 d d exp [ i W ( ξ 1 , η 1 ) ] × exp { - i 2 π λ d [ ( x + m x ) ξ 1 + ( y + m y ) η 1 ] } d ξ 1 d η 1 = h 1 ( x + m x , y + m y ) ,
h ( x , y ; x , y ) = 1 λ 2 d d exp [ - i W ( ξ 2 , η 2 ) ] × exp { - i 2 π λ d [ ( x + 1 m x ) ξ 2 + ( y + 1 m y ) η 2 ] } d ξ 2 d η 2 = h 2 * ( x + 1 m x , y + 1 m y ) ,
h 1 ( x + m x , y + m y ) = h 2 * ( x + 1 m x , y + 1 m y ) .
O i ( x , y ) = [ h 2 * ( x + 1 m x , y + 1 m y ) ] × [ h 1 ( x , y ) R i ( x , y ) ] d x d y ,
O i ( x , y ) = m 2 h 1 ( m x - m x ¯ , m y - m y ¯ ) × [ h 1 ( - m x ¯ , - m y ¯ ) R i ( - m x ¯ , - m y ¯ ) ] d x ¯ d y ¯ = m 2 [ h 1 ( m x , m y ) ] [ h 1 ( - m x , - m y ) R i ( - m x , - m y ) ] ,
I i ( x , y ) = O i ( x , y ) 2 = m 4 h 1 ( m x , m y ) [ h 1 ( - m x , - m y ) R i ( - m x , - m y ) ] 2 .
I = I ( x , y ) = 1 N i = 1 N I i ( x , y ) h 1 ( m x , m y ) 2 h 1 ( - m x , - m y ) ] 2 = P ( m x , m y ) P ( - m x , - m y ) ,
M ( u , v ) = { FT [ I ( x , y ) ] } 1 / 2 = { FT [ P ( m x , m y ) ] P ( - m x , - m y ) ] } 1 / 2 = { FT [ P ( m x , m y ) FT [ P ( - m x , - m y ) ] } 1 / 2 = [ H ( u , v ) H ( - u , - v ) ] 1 / 2 = [ H ( u , v ) H * ( u , v ) ] 1 / 2 ,

Metrics