Abstract

Reflectrometric measurements provide an objective assessment of the directionality of the photoreceptors in the human retina. Measurements are obtained by imaging the distribution at the pupil plane of light reflected off the human fundus in a bleached condition. We propose that scattering as well as waveguides must be included in a model of the intensity distribution at the pupil plane. For scattering, the cone-photoreceptor array is treated as a random rough surface, characterized by the correlation length T (related to the distance between scatterers, i.e., mean cone spacing) and the roughness standard deviation σ (assuming random length variations of the cone outer-segment lengths that produce random phase differences). For realistic values of T and σ we can use the Kirchhoff approximation for computing the scattering distribution. The scattered component of the distribution can be fitted to a Gaussian function whose width depends only on T and λ. Actual measurements vary with experimental conditions (exposure time, retinal eccentricity, and λ) in a manner consistent with the scattering model. However, photoreceptor directionality must be included in the model to explain the actual location of the peak of the intensity distribution in the pupil plane and the total angular spread of light.

© 1998 Optical Society of America

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    [CrossRef] [PubMed]
  2. A. E. Elsner, S. A. Burns, G. W. Hughes, R. H. Webb, “Reflectometry with a scanning laser ophthalmoscope,” Appl. Opt. 31, 3697–3710 (1992).
    [CrossRef] [PubMed]
  3. A. E. Elsner, S. A. Burns, R. H. Webb, “Mapping cone pigment optical density in humans,” J. Opt. Soc. Am. A 10, 52–58 (1993).
    [CrossRef] [PubMed]
  4. 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, 3123–3135 (1994).
    [CrossRef]
  5. S. Marcos, R. Navarro, P. Artal, “Coherent imaging of the cone mosaic in the living human eye,” J. Opt. Soc. Am. A 13, 897–905 (1996).
    [CrossRef]
  6. S. A. Burns, S. Wu, F. C. Delori, A. E. Elsner, “Direct measurement of human cone-photoreceptor alignment,” J. Opt. Soc. Am. A 12, 2329–2338 (1996).
    [CrossRef]
  7. S. Marcos, R. Navarro, “Determination of the foveal cone spacing by ocular speckle interferometry: limiting factors and acuity predictions,” J. Opt. Soc. Am. A 14, 731–740 (1997).
    [CrossRef]
  8. S. A. Burns, A. E. Elsner, J. M. Gorrand, M. R. Kreitz, F. C. Delori, “Comparison of reflectometric and psychophysical measures of cone orientation,” in Noninvasive Assessment of the Visual System Vol. 1 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 160–163; J. He, S. A. Burns, “Cone photoreceptor directionality assessed using psychophysical and imaging techniques in vivo,” Invest. Ophthalmol. Visual Sci. Suppl. 38, 1015 (1997).
  9. 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]
  10. G. J. Van Blockland, “Directionality and alignment of the foveal photoreceptors assessed with light scattered from the human fundus in vivo,” Vision Res. 26, 495–500 (1986).
    [CrossRef]
  11. J. M. Gorrand, F. C. Delori, “A reflectometric technique for assessing photoreceptor alignment,” Vision Res. 35, 999–1010 (1995).
    [CrossRef] [PubMed]
  12. P. J. Delint, T. T.-J. M. Berendschot, D. van Norren, “Local photoreceptor alignment measured with a scanning laser ophthalmoscope,” Vision Res. 37, 243–248 (1997).
    [CrossRef] [PubMed]
  13. R. A. Applegate, V. Lakshminarayanan, “Parametric representation of Stiles–Crawford functions: normal variation of peak location and directionality,” J. Opt. Soc. Am. A 10, 1611–1623 (1993).
    [CrossRef] [PubMed]
  14. J. M. Gorrand, F. C. Delori, “A model for assessment of cone directionality,” J. Mod. Opt. 44, 473–491 (1997).
    [CrossRef]
  15. The factor of 2 is for the design of Refs. 6 and 9 Burns et al. For their own design the prediction is a factor of 4.
  16. V=πD(nco2-n2)1/2/λ; where D is the diameter of the waveguide, and nco and n are the respective refractive indices of the core and surrounding medium;E. Snitzer, “Cylindrical dielectric waveguide modes,” J. Opt. Soc. Am. 51, 491–498 (1961).
    [CrossRef]
  17. For those larger cones the intensity distribution at the pupil plane becomes the sum of 1296 terms, instead of 81. J. M. Gorrand, Faculté de Medicine, Inserm U.684 BP 38 Clermont-Ferrand 63001, France (personal communication, 1997).
  18. A. W. Snyder, C. L. Pask, “The Stiles–Crawford effect—explanation and consequences,” Vision Res. 13, 1115–1137 (1973).
    [CrossRef] [PubMed]
  19. M. Nieto-Vesperinas, J. C. Dainty, eds., Scattering in Volumes and Surfaces (North-Holland, Amsterdam, 1990).
  20. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 7.
  21. K. A. O’Donnell, E. R. Méndez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
    [CrossRef]
  22. J. M. Bennett, ed., Surface Finish and Its Measurement, Vol. 2 of Collected Works in Optics (Optical Society of America, Washington, D.C., 1992).
  23. D. Miller, G. Bennedek, Intraocular Light Scattering: Theory and Clinical Applications (Thomas, Springfield, Ill., 1973); R. Navarro, “Incorporation of intraocular scattering in schematic eye models,” J. Opt. Soc. Am. A 2, 1981–1984 (1985). R. Navarro, J. Méndez-Morales, J. Santamarı́a, “Optical quality of the eye lens surfaces from roughness and diffusion measurements,” J. Opt. Soc. Am. A 3, 228–234 (1996).
    [CrossRef]
  24. P. Beckmann, A. Spizzino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  25. M. Nieto-Vesperinas, “Depolarization of electromagnetic waves scattered from slightly rough random surfaces: a study by means of the extinction theorem,” J. Opt. Soc. Am. 72, 539–547 (1982).
    [CrossRef]
  26. J. A. Sánchez-Gil, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
    [CrossRef]
  27. M. Nieto-Vesperinas, N. Garcı́a, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
    [CrossRef]
  28. J. M. Soto-Crespo, M. Nieto-Vesperinas, “Electromagnetic scattering from very rough surfaces and deep reflection gratings,” J. Opt. Soc. Am. A 6, 367–384 (1989).
    [CrossRef]
  29. L. N. Deruyugin, “Equations for the coefficients of reflection of waves from a periodically rough surface,” Dokl. Akad. Nauk SSSR 87, 913–916 (1952).
  30. The diffused component can be approximated by exp(-Q2T2/4), where Q is the momentum transfer of the surface (see Refs. 24 and 25). At normal incidence Q=(2π/λ)sin θ, where θ is the angle of scattering (called, the observation angle), and sin θ=r/z.
  31. S. Marcos, R. Navarro, “Imaging the foveal cones in vivo through ocular speckle interferometry: theory and nu-merical simulations,” J. Opt. Soc. Am. A 13, 2329–2340 (1996).
    [CrossRef]
  32. C. A. Curcio, K. R. Sloan, R. E. Kalina, A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292, 497–523 (1992).
    [CrossRef]
  33. R. Young, “The renewal of rod and cone outer segments in the rhesus monkey,” J. Cell Biol. 39, 303–318 (1971).
    [CrossRef]
  34. A. Madrazo, M. Nieto-Vesperinas, “Scattering of light and other electromagnetic waves from a body buried beneath a highly rough random surface,” J. Opt. Soc. Am. A 14, 1859–1866 (1997).
    [CrossRef]
  35. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  36. J. J. Yellot, “Spectral analysis of spatial sampling of photoreceptors: topological disorder prevents aliasing,” Vision Res. 22, 1205–1210 (1982).
    [CrossRef]
  37. N. D. Drasdo, C. W. Fowler, “Non-linear projection of a retinal image in a wide-angle schematic eye,” Br. J. Ophthamol. 58, 709–714 (1974).
    [CrossRef]
  38. There are no predictions available from the waveguide model for reflectometric ρ as a function of λ, since Gorrand and Delori14 used only 543 nm. Snyder and Pask18 studied the variation of ρ as a function of wavelength and showed (for their specific values of cone diameters, do=1 µm and di=3.2 µm for the outer and inner segments, respectively, and refractive indices, ni=1.353,no=1.430, and nipm=1.340 for the inner and outer segment, and the interphotoreceptor matrix, respectively) that ρ increased from wavelengths between 543 nm and 650 nm. Nevertheless, we have computed that for the same diameters and slightly different indices of refraction (those used by Gorrand and Delori: ni=1.361,no=1.419, and nipm=1.347) the behavior is reversed and ρ decreases with wavelength between 555 and 650 nm, thus indicating that the variation of ρ with λ in the waveguide models is non-systematic and strongly dependent on the choice of parameters.
  39. A. E. Elsner, S. A. Burns, J. J. Weiter, F. C. Delori, “Infrared imaging of subretinal structures in the human ocular fundus,” Vision Res. 36, 191–205 (1996).
    [CrossRef] [PubMed]
  40. The multiplication arises from the fact that at each pupil location the contribution from each retinal location is attenuated by the same waveguide component. That is,  Ip(x, y)∫ξ∫ζOi(ξ, ζ)10-ρwg22(x2+y2)×exp-i2πλz(ξx+ζy)dξdζ2, where Ip(x, y) is the intensity distribution at the plane of the pupil, Oi(ξ, ζ) is the complex amplitude of the retinal eccentricity, (x,y) are pupil coordinates, and (ξ, ζ) are retinal coordinates. Since the waveguide component 10-ρwg2(x2+y2) does not depend on the retinal coordinates, it can be moved out of the integral.
  41. C. A. Curcio, “Diameters of presumed cone apertures in human retina,” in Annual Meeting, Vol. 20 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), p. 83.
  42. D. I. MacLeod, “Directionally selective light adaptation: a visual consequence of receptor disarray?” Vision Res. 14, 369–378 (1974).
    [CrossRef] [PubMed]
  43. G. Westheimer, “Dependence of the magnitude of the Stiles–Crawford effect on retinal location,” J. Physiol. (London) 192, 309–315 (1967).
  44. J. M. Enoch, G. M. Hope, “Directional sensitivity of the foveal and parafoveal retina,” Invest. Ophthalmol. Visual Sci. 12, 497–503 (1973).

1997 (5)

1996 (4)

1995 (1)

J. M. Gorrand, F. C. Delori, “A reflectometric technique for assessing photoreceptor alignment,” Vision Res. 35, 999–1010 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (2)

1992 (2)

A. E. Elsner, S. A. Burns, G. W. Hughes, R. H. Webb, “Reflectometry with a scanning laser ophthalmoscope,” Appl. Opt. 31, 3697–3710 (1992).
[CrossRef] [PubMed]

C. A. Curcio, K. R. Sloan, R. E. Kalina, A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292, 497–523 (1992).
[CrossRef]

1991 (1)

1989 (1)

1987 (2)

1986 (1)

G. J. Van Blockland, “Directionality and alignment of the foveal photoreceptors assessed with light scattered from the human fundus in vivo,” Vision Res. 26, 495–500 (1986).
[CrossRef]

1982 (2)

J. J. Yellot, “Spectral analysis of spatial sampling of photoreceptors: topological disorder prevents aliasing,” Vision Res. 22, 1205–1210 (1982).
[CrossRef]

M. Nieto-Vesperinas, “Depolarization of electromagnetic waves scattered from slightly rough random surfaces: a study by means of the extinction theorem,” J. Opt. Soc. Am. 72, 539–547 (1982).
[CrossRef]

1981 (1)

M. Nieto-Vesperinas, N. Garcı́a, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

1974 (2)

N. D. Drasdo, C. W. Fowler, “Non-linear projection of a retinal image in a wide-angle schematic eye,” Br. J. Ophthamol. 58, 709–714 (1974).
[CrossRef]

D. I. MacLeod, “Directionally selective light adaptation: a visual consequence of receptor disarray?” Vision Res. 14, 369–378 (1974).
[CrossRef] [PubMed]

1973 (2)

A. W. Snyder, C. L. Pask, “The Stiles–Crawford effect—explanation and consequences,” Vision Res. 13, 1115–1137 (1973).
[CrossRef] [PubMed]

J. M. Enoch, G. M. Hope, “Directional sensitivity of the foveal and parafoveal retina,” Invest. Ophthalmol. Visual Sci. 12, 497–503 (1973).

1971 (1)

R. Young, “The renewal of rod and cone outer segments in the rhesus monkey,” J. Cell Biol. 39, 303–318 (1971).
[CrossRef]

1967 (1)

G. Westheimer, “Dependence of the magnitude of the Stiles–Crawford effect on retinal location,” J. Physiol. (London) 192, 309–315 (1967).

1961 (1)

1952 (1)

L. N. Deruyugin, “Equations for the coefficients of reflection of waves from a periodically rough surface,” Dokl. Akad. Nauk SSSR 87, 913–916 (1952).

Applegate, R. A.

Artal, P.

Beckmann, P.

P. Beckmann, A. Spizzino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Bennedek, G.

D. Miller, G. Bennedek, Intraocular Light Scattering: Theory and Clinical Applications (Thomas, Springfield, Ill., 1973); R. Navarro, “Incorporation of intraocular scattering in schematic eye models,” J. Opt. Soc. Am. A 2, 1981–1984 (1985). R. Navarro, J. Méndez-Morales, J. Santamarı́a, “Optical quality of the eye lens surfaces from roughness and diffusion measurements,” J. Opt. Soc. Am. A 3, 228–234 (1996).
[CrossRef]

Berendschot, T. T.-J. M.

P. J. Delint, T. T.-J. M. Berendschot, D. van Norren, “Local photoreceptor alignment measured with a scanning laser ophthalmoscope,” Vision Res. 37, 243–248 (1997).
[CrossRef] [PubMed]

Bescós, J.

Brainard, D. H.

Burns, S. A.

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. C. Delori, A. E. Elsner, “Direct measurement of human cone-photoreceptor alignment,” J. Opt. Soc. Am. A 12, 2329–2338 (1996).
[CrossRef]

A. E. Elsner, S. A. Burns, J. J. Weiter, F. C. Delori, “Infrared imaging of subretinal structures in the human ocular fundus,” Vision Res. 36, 191–205 (1996).
[CrossRef] [PubMed]

A. E. Elsner, S. A. Burns, R. H. Webb, “Mapping cone pigment optical density in humans,” J. Opt. Soc. Am. A 10, 52–58 (1993).
[CrossRef] [PubMed]

A. E. Elsner, S. A. Burns, G. W. Hughes, R. H. Webb, “Reflectometry with a scanning laser ophthalmoscope,” Appl. Opt. 31, 3697–3710 (1992).
[CrossRef] [PubMed]

S. A. Burns, A. E. Elsner, J. M. Gorrand, M. R. Kreitz, F. C. Delori, “Comparison of reflectometric and psychophysical measures of cone orientation,” in Noninvasive Assessment of the Visual System Vol. 1 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 160–163; J. He, S. A. Burns, “Cone photoreceptor directionality assessed using psychophysical and imaging techniques in vivo,” Invest. Ophthalmol. Visual Sci. Suppl. 38, 1015 (1997).

Curcio, C. A.

C. A. Curcio, K. R. Sloan, R. E. Kalina, A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292, 497–523 (1992).
[CrossRef]

C. A. Curcio, “Diameters of presumed cone apertures in human retina,” in Annual Meeting, Vol. 20 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), p. 83.

Delint, P. J.

P. J. Delint, T. T.-J. M. Berendschot, D. van Norren, “Local photoreceptor alignment measured with a scanning laser ophthalmoscope,” Vision Res. 37, 243–248 (1997).
[CrossRef] [PubMed]

Delori, F. C.

J. M. Gorrand, F. C. Delori, “A model for assessment of cone directionality,” J. Mod. Opt. 44, 473–491 (1997).
[CrossRef]

S. A. Burns, S. Wu, F. C. Delori, A. E. Elsner, “Direct measurement of human cone-photoreceptor alignment,” J. Opt. Soc. Am. A 12, 2329–2338 (1996).
[CrossRef]

A. E. Elsner, S. A. Burns, J. J. Weiter, F. C. Delori, “Infrared imaging of subretinal structures in the human ocular fundus,” Vision Res. 36, 191–205 (1996).
[CrossRef] [PubMed]

J. M. Gorrand, F. C. Delori, “A reflectometric technique for assessing photoreceptor alignment,” Vision Res. 35, 999–1010 (1995).
[CrossRef] [PubMed]

S. A. Burns, A. E. Elsner, J. M. Gorrand, M. R. Kreitz, F. C. Delori, “Comparison of reflectometric and psychophysical measures of cone orientation,” in Noninvasive Assessment of the Visual System Vol. 1 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 160–163; J. He, S. A. Burns, “Cone photoreceptor directionality assessed using psychophysical and imaging techniques in vivo,” Invest. Ophthalmol. Visual Sci. Suppl. 38, 1015 (1997).

Deruyugin, L. N.

L. N. Deruyugin, “Equations for the coefficients of reflection of waves from a periodically rough surface,” Dokl. Akad. Nauk SSSR 87, 913–916 (1952).

Drasdo, N. D.

N. D. Drasdo, C. W. Fowler, “Non-linear projection of a retinal image in a wide-angle schematic eye,” Br. J. Ophthamol. 58, 709–714 (1974).
[CrossRef]

Elsner, A. E.

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. C. Delori, A. E. Elsner, “Direct measurement of human cone-photoreceptor alignment,” J. Opt. Soc. Am. A 12, 2329–2338 (1996).
[CrossRef]

A. E. Elsner, S. A. Burns, J. J. Weiter, F. C. Delori, “Infrared imaging of subretinal structures in the human ocular fundus,” Vision Res. 36, 191–205 (1996).
[CrossRef] [PubMed]

A. E. Elsner, S. A. Burns, R. H. Webb, “Mapping cone pigment optical density in humans,” J. Opt. Soc. Am. A 10, 52–58 (1993).
[CrossRef] [PubMed]

A. E. Elsner, S. A. Burns, G. W. Hughes, R. H. Webb, “Reflectometry with a scanning laser ophthalmoscope,” Appl. Opt. 31, 3697–3710 (1992).
[CrossRef] [PubMed]

S. A. Burns, A. E. Elsner, J. M. Gorrand, M. R. Kreitz, F. C. Delori, “Comparison of reflectometric and psychophysical measures of cone orientation,” in Noninvasive Assessment of the Visual System Vol. 1 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 160–163; J. He, S. A. Burns, “Cone photoreceptor directionality assessed using psychophysical and imaging techniques in vivo,” Invest. Ophthalmol. Visual Sci. Suppl. 38, 1015 (1997).

Enoch, J. M.

J. M. Enoch, G. M. Hope, “Directional sensitivity of the foveal and parafoveal retina,” Invest. Ophthalmol. Visual Sci. 12, 497–503 (1973).

Fowler, C. W.

N. D. Drasdo, C. W. Fowler, “Non-linear projection of a retinal image in a wide-angle schematic eye,” Br. J. Ophthamol. 58, 709–714 (1974).
[CrossRef]

Garci´a, N.

M. Nieto-Vesperinas, N. Garcı́a, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

Goodman, J. W.

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

Gorrand, J. M.

J. M. Gorrand, F. C. Delori, “A model for assessment of cone directionality,” J. Mod. Opt. 44, 473–491 (1997).
[CrossRef]

J. M. Gorrand, F. C. Delori, “A reflectometric technique for assessing photoreceptor alignment,” Vision Res. 35, 999–1010 (1995).
[CrossRef] [PubMed]

S. A. Burns, A. E. Elsner, J. M. Gorrand, M. R. Kreitz, F. C. Delori, “Comparison of reflectometric and psychophysical measures of cone orientation,” in Noninvasive Assessment of the Visual System Vol. 1 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 160–163; J. He, S. A. Burns, “Cone photoreceptor directionality assessed using psychophysical and imaging techniques in vivo,” Invest. Ophthalmol. Visual Sci. Suppl. 38, 1015 (1997).

For those larger cones the intensity distribution at the pupil plane becomes the sum of 1296 terms, instead of 81. J. M. Gorrand, Faculté de Medicine, Inserm U.684 BP 38 Clermont-Ferrand 63001, France (personal communication, 1997).

He, J. C.

Hendrickson, A. E.

C. A. Curcio, K. R. Sloan, R. E. Kalina, A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292, 497–523 (1992).
[CrossRef]

Hope, G. M.

J. M. Enoch, G. M. Hope, “Directional sensitivity of the foveal and parafoveal retina,” Invest. Ophthalmol. Visual Sci. 12, 497–503 (1973).

Hughes, G. W.

Kalina, R. E.

C. A. Curcio, K. R. Sloan, R. E. Kalina, A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292, 497–523 (1992).
[CrossRef]

Kreitz, M. R.

S. A. Burns, A. E. Elsner, J. M. Gorrand, M. R. Kreitz, F. C. Delori, “Comparison of reflectometric and psychophysical measures of cone orientation,” in Noninvasive Assessment of the Visual System Vol. 1 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 160–163; J. He, S. A. Burns, “Cone photoreceptor directionality assessed using psychophysical and imaging techniques in vivo,” Invest. Ophthalmol. Visual Sci. Suppl. 38, 1015 (1997).

Lakshminarayanan, V.

MacLeod, D. I.

D. I. MacLeod, “Directionally selective light adaptation: a visual consequence of receptor disarray?” Vision Res. 14, 369–378 (1974).
[CrossRef] [PubMed]

Madrazo, A.

Marcos, S.

McMahon, M. J.

Méndez, E. R.

Miller, D.

D. Miller, G. Bennedek, Intraocular Light Scattering: Theory and Clinical Applications (Thomas, Springfield, Ill., 1973); R. Navarro, “Incorporation of intraocular scattering in schematic eye models,” J. Opt. Soc. Am. A 2, 1981–1984 (1985). R. Navarro, J. Méndez-Morales, J. Santamarı́a, “Optical quality of the eye lens surfaces from roughness and diffusion measurements,” J. Opt. Soc. Am. A 3, 228–234 (1996).
[CrossRef]

Navarro, R.

Nieto-Vesperinas, M.

O’Donnell, K. A.

Pask, C. L.

A. W. Snyder, C. L. Pask, “The Stiles–Crawford effect—explanation and consequences,” Vision Res. 13, 1115–1137 (1973).
[CrossRef] [PubMed]

Sánchez-Gil, J. A.

Santamari´a, J.

Sloan, K. R.

C. A. Curcio, K. R. Sloan, R. E. Kalina, A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292, 497–523 (1992).
[CrossRef]

Snitzer, E.

Snyder, A. W.

A. W. Snyder, C. L. Pask, “The Stiles–Crawford effect—explanation and consequences,” Vision Res. 13, 1115–1137 (1973).
[CrossRef] [PubMed]

Soto-Crespo, J. M.

Spizzino, A.

P. Beckmann, A. Spizzino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Van Blockland, G. J.

G. J. Van Blockland, “Directionality and alignment of the foveal photoreceptors assessed with light scattered from the human fundus in vivo,” Vision Res. 26, 495–500 (1986).
[CrossRef]

van Norren, D.

P. J. Delint, T. T.-J. M. Berendschot, D. van Norren, “Local photoreceptor alignment measured with a scanning laser ophthalmoscope,” Vision Res. 37, 243–248 (1997).
[CrossRef] [PubMed]

Webb, R. H.

Weiter, J. J.

A. E. Elsner, S. A. Burns, J. J. Weiter, F. C. Delori, “Infrared imaging of subretinal structures in the human ocular fundus,” Vision Res. 36, 191–205 (1996).
[CrossRef] [PubMed]

Westheimer, G.

G. Westheimer, “Dependence of the magnitude of the Stiles–Crawford effect on retinal location,” J. Physiol. (London) 192, 309–315 (1967).

Williams, D. R.

Wu, S.

Yellot, J. J.

J. J. Yellot, “Spectral analysis of spatial sampling of photoreceptors: topological disorder prevents aliasing,” Vision Res. 22, 1205–1210 (1982).
[CrossRef]

Young, R.

R. Young, “The renewal of rod and cone outer segments in the rhesus monkey,” J. Cell Biol. 39, 303–318 (1971).
[CrossRef]

Appl. Opt. (1)

Br. J. Ophthamol. (1)

N. D. Drasdo, C. W. Fowler, “Non-linear projection of a retinal image in a wide-angle schematic eye,” Br. J. Ophthamol. 58, 709–714 (1974).
[CrossRef]

Dokl. Akad. Nauk SSSR (1)

L. N. Deruyugin, “Equations for the coefficients of reflection of waves from a periodically rough surface,” Dokl. Akad. Nauk SSSR 87, 913–916 (1952).

Invest. Ophthalmol. Visual Sci. (1)

J. M. Enoch, G. M. Hope, “Directional sensitivity of the foveal and parafoveal retina,” Invest. Ophthalmol. Visual Sci. 12, 497–503 (1973).

J. Cell Biol. (1)

R. Young, “The renewal of rod and cone outer segments in the rhesus monkey,” J. Cell Biol. 39, 303–318 (1971).
[CrossRef]

J. Comp. Neurol. (1)

C. A. Curcio, K. R. Sloan, R. E. Kalina, A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292, 497–523 (1992).
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Other (13)

The multiplication arises from the fact that at each pupil location the contribution from each retinal location is attenuated by the same waveguide component. That is,  Ip(x, y)∫ξ∫ζOi(ξ, ζ)10-ρwg22(x2+y2)×exp-i2πλz(ξx+ζy)dξdζ2, where Ip(x, y) is the intensity distribution at the plane of the pupil, Oi(ξ, ζ) is the complex amplitude of the retinal eccentricity, (x,y) are pupil coordinates, and (ξ, ζ) are retinal coordinates. Since the waveguide component 10-ρwg2(x2+y2) does not depend on the retinal coordinates, it can be moved out of the integral.

C. A. Curcio, “Diameters of presumed cone apertures in human retina,” in Annual Meeting, Vol. 20 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), p. 83.

The diffused component can be approximated by exp(-Q2T2/4), where Q is the momentum transfer of the surface (see Refs. 24 and 25). At normal incidence Q=(2π/λ)sin θ, where θ is the angle of scattering (called, the observation angle), and sin θ=r/z.

M. Nieto-Vesperinas, J. C. Dainty, eds., Scattering in Volumes and Surfaces (North-Holland, Amsterdam, 1990).

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 7.

J. M. Bennett, ed., Surface Finish and Its Measurement, Vol. 2 of Collected Works in Optics (Optical Society of America, Washington, D.C., 1992).

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[CrossRef]

P. Beckmann, A. Spizzino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

For those larger cones the intensity distribution at the pupil plane becomes the sum of 1296 terms, instead of 81. J. M. Gorrand, Faculté de Medicine, Inserm U.684 BP 38 Clermont-Ferrand 63001, France (personal communication, 1997).

S. A. Burns, A. E. Elsner, J. M. Gorrand, M. R. Kreitz, F. C. Delori, “Comparison of reflectometric and psychophysical measures of cone orientation,” in Noninvasive Assessment of the Visual System Vol. 1 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 160–163; J. He, S. A. Burns, “Cone photoreceptor directionality assessed using psychophysical and imaging techniques in vivo,” Invest. Ophthalmol. Visual Sci. Suppl. 38, 1015 (1997).

There are no predictions available from the waveguide model for reflectometric ρ as a function of λ, since Gorrand and Delori14 used only 543 nm. Snyder and Pask18 studied the variation of ρ as a function of wavelength and showed (for their specific values of cone diameters, do=1 µm and di=3.2 µm for the outer and inner segments, respectively, and refractive indices, ni=1.353,no=1.430, and nipm=1.340 for the inner and outer segment, and the interphotoreceptor matrix, respectively) that ρ increased from wavelengths between 543 nm and 650 nm. Nevertheless, we have computed that for the same diameters and slightly different indices of refraction (those used by Gorrand and Delori: ni=1.361,no=1.419, and nipm=1.347) the behavior is reversed and ρ decreases with wavelength between 555 and 650 nm, thus indicating that the variation of ρ with λ in the waveguide models is non-systematic and strongly dependent on the choice of parameters.

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

The factor of 2 is for the design of Refs. 6 and 9 Burns et al. For their own design the prediction is a factor of 4.

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Figures (12)

Fig. 1
Fig. 1

Schematic representation the reflectometric technique. A small portion of the retina is illuminated in Maxwellian view (top left panel). Part of the light is multiply scattered and reflected back toward the pupil; part is guided along the cones and reradiated back toward the pupil (left panels). The image of the pupil is formed on a cooled CCD camera (top right panel). After the corneal reflexes are blocked, the brightest part of the image corresponds to light reflected back from the cones. The intensity distribution at the pupil plane can be fitted to a constant (background) plus a Gaussian (guided or directed component). The parameters of the fit are B (constant background), Imax (maximum intensity), x0 and y0 (coordinates of the peak of the distribution), and ρ (shape factor, proportional to the inverse square root of the width of the distribution).

Fig. 2
Fig. 2

Examples of simulated cone distributions for four different values of mean cone spacing.

Fig. 3
Fig. 3

Histograms of the distribution of effective path-length differences in a simulated cone mosaic in comparison with a Gaussian distribution of the same variance. The roughness is taken as the standard deviation of the distribution. (a) σ=0.04887 µm, (b) σ=0.0733 µm, (c) σ=0.152 µm, (d) σ=0.2715 µm.

Fig. 4
Fig. 4

Far-field intensity, coherent component, and diffused component for a simulated 0.68-deg cone mosaic with 3.76 µm of mean cone spacing for different degrees of roughness. First column, average of intensities; second column, square modulus of the average of complex amplitudes; third column, difference of the first and the second (diffused component). Averages are performed over 50 realizations. The specular peak has been removed in the far-field component and the coherent component in the three first cases. Roughness increases, from top to bottom: (a) σ=0.04887 µm, (b) σ=0.0733 µm, (c) σ=0.152 µm, (d) σ=0.2715 µm. The coherent component is prominent for slight roughness and is masked by the diffused component for higher roughness. The normalized diffused component (third column) is independent of roughness.

Fig. 5
Fig. 5

Simulated diffused component for different values of row-to-row cone spacing. The distribution becomes narrower for increasing cone spacing.

Fig. 6
Fig. 6

Fit of the diffused (scattered) component to a Gaussian of the form 10-ρr2. Dotted curve, radial profile of the simulated diffused component from a cone mosaic with a mean row-to-row cone spacing of 3.76 µm; solid curve, fit to a Gaussian (ρ=0.11 mm-2).

Fig. 7
Fig. 7

ρ (obtained by fitting the simulated diffused component as shown in Fig. 6) as a function of row-to-row cone spacing (circles). The dashed curve represents the prediction from scattering theory (Kirchhoff approximation): ρ=0.00844 s2 (mm-2).

Fig. 8
Fig. 8

Short-exposure images at the pupil plane. (a) Experimental (subject SM, 0 deg eccentricity; spot size 0.5 deg). (b) Simulation (cone spacing 2.25 µm; roughness 0.271 µm; illumination spot size: 0.5 deg).

Fig. 9
Fig. 9

Experimental intensity distribution at the pupil plane for different retinal eccentricities in the nasal retina, for subject SM. The corneal reflex has been removed. The distribution narrows as retinal eccentricity (or, equivalently, cone spacing) increases.

Fig. 10
Fig. 10

ρ as a function of retinal eccentricity. Squares, diamonds, and circles represent experimental measurements in a single session for subjects JH, SM, and SB respectively. Triangles are the predictions from the simulation (fitted to Kirchhoff approximation) when a conversion between cone spacing and retinal eccentricity is considered. The illumination wavelength was 543 nm.

Fig. 11
Fig. 11

(a) Variation of ρ with retinal eccentricity for three different wavelengths (543, 632, and 670 nm) for subject JH (average across three sessions). (b) Ratios ρ632/ρ543 (dashed curve, circles) and ρ670/ρ543 (dotted curve, diamonds), averaging across the three subjects and all sessions. The dashed horizontal line represents the scattering prediction for ρ632/ρ543 and the dotted line for ρ670/ρ543.

Fig. 12
Fig. 12

Predictions of different models for the variation of ρ as a function of retinal eccentricity in comparison with the experimental measurements. Long-dashed curves, prediction of waveguide theory (which we have derived as an extension of the Snyder and Pask model17); Short-dashed curve, prediction from scattering theory; Dotted curve, sum of the two previous curves (scattering model including the waveguide properties of the photoreceptors); solid curve diffraction limit for apertures of the size of the cone inner segments (see Burns et al.9); circles, average experimental measurements (across subjects and sessions). Data have been shifted to the eccentricity equivalent to the mean cone spacing within the sample (the shift is appreciable only for the lower eccentricity: 0 deg is equivalent to 0.3 deg). Error bars represent the standard deviation. The illumination wavelength for measurements and models was 543 nm.

Tables (1)

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Table 1 Ratios ρ632/ρ543 and ρ670/ρ543 for Three Subjects and Two Retinal Eccentricitiesa

Equations (2)

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ρscatt=π2(0.4s)2f2λ2 ln 10,
Ip(x, y)ξζOi(ξ, ζ)10-ρwg22(x2+y2)×exp-i2πλz(ξx+ζy)dξdζ2,

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