OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 15, Iss. 8 — Aug. 1, 1998
  • pp: 2012–2022

Model for cone directionality reflectometric measurements based on scattering

Susana Marcos, Stephen A. Burns, and Ji Chang He  »View Author Affiliations

JOSA A, Vol. 15, Issue 8, pp. 2012-2022 (1998)

View Full Text Article

Enhanced HTML    Acrobat PDF (3247 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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

OCIS Codes
(110.0110) Imaging systems : Imaging systems
(290.0290) Scattering : Scattering
(330.4300) Vision, color, and visual optics : Vision system - noninvasive assessment
(330.5310) Vision, color, and visual optics : Vision - photoreceptors
(330.5370) Vision, color, and visual optics : Physiological optics

Original Manuscript: November 13, 1997
Revised Manuscript: March 27, 1998
Manuscript Accepted: April 2, 1998
Published: August 1, 1998

Susana Marcos, Stephen A. Burns, and Ji Chang He, "Model for cone directionality reflectometric measurements based on scattering," J. Opt. Soc. Am. A 15, 2012-2022 (1998)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. J. Santamarı́a, 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] [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).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited