Abstract

The photoreceptors of the living human eye are known to exhibit waveguide-characteristic features. This is evidenced by the Stiles–Crawford effect observed for light incident near the pupil rim, and by the directional component of light reflected off the retina in the related optical Stiles–Crawford effect. We describe a model for the coupling of light to/from photoreceptors on the basis of waveguide theory that includes diffraction between the eye pupil and the photoreceptor apertures, and we show that valuable insight can be gained from a Gaussian approximation to the mode field. We apply this knowledge to a detailed study of the relationship between the Stiles–Crawford effect and its optical counterpart.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. S. Stiles, B. H. Crawford, “The luminous efficiency of rays entering the eye pupil at different points,” Proc. R. Soc. London 112, 428–450 (1933).
    [CrossRef]
  2. D. A. Atchison, D. H. Scott, N. C. Strang, P. Artal, “Influence of Stiles–Crawford apodization on visual acuity,” J. Opt. Soc. Am. A 19, 1073–1083 (2002).
    [CrossRef]
  3. G. Toraldo di Francia, L. Ronchi, “Directional scattering of light by the human retina,” J. Opt. Soc. Am. 42, 782–783 (1952).
  4. 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 (1996).
    [CrossRef]
  5. B. O’Brien, “A theory of the Stiles and Crawford effect,” J. Opt. Soc. Am. 36, 506–509 (1946).
    [CrossRef] [PubMed]
  6. J. M. Enoch, “Optical properties of the retinal receptors,” J. Opt. Soc. Am. 53, 71–85 (1963).
    [CrossRef]
  7. A. W. Snyder, C. Pask, “The Stiles–Crawford effect—explanation and consequences,” Vision Res. 13, 1115–1137 (1973).
    [CrossRef] [PubMed]
  8. A. Roorda, D. R. Williams, “Optical fiber properties of individual human cones,” J. Vision 2, 404–412 (2002).
    [CrossRef]
  9. A. M. Laties, J. M. Enoch, “An analysis of retinal receptor orientation,” Invest. Ophthalmol. Visual Sci. 10, 69–77 (1971).
  10. S. A. Burns, S. Wu, F. Delori, A. E. Elsner, “Direct measurement of human-cone-photoreceptor alignment,” J. Opt. Soc. Am. A 12, 2329–2338 (1995).
    [CrossRef]
  11. 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]
  12. J. C. He, S. Marcos, S. A. Burns, “Comparison of cone directionality determined by psychophysical and reflectometric techniques,” J. Opt. Soc. Am. A 16, 2363–2369 (1999).
    [CrossRef]
  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. A. Safir, L. Hyams, “Distribution of cone orientations as an explanation of the Stiles–Crawford effect,” J. Opt. Soc. Am. 59, 757–765 (1969).
    [CrossRef] [PubMed]
  16. G. J. van Blokland, “Directionality and alignment of the foveal receptors, assessed with light scattered from the human fundus in vivo ,” Vision Res. 26, 495–500 (1986).
    [CrossRef] [PubMed]
  17. S. Marcos, S. A. Burns, J. C. He, “Model for cone directionality reflectometric measurements based on scattering,” J. Opt. Soc. Am. A 15, 2012–2022 (1998).
    [CrossRef]
  18. S. Marcos, S. A. Burns, “Cone spacing and waveguide properties from cone directionality measurements,” J. Opt. Soc. Am. A 16, 995–1004 (1999).
    [CrossRef]
  19. W. S. Stiles, “The luminous efficiency of monochromatic rays entering the eye pupil at different points and a new colour effect,” Proc. R. Soc. London, Ser. B 123, 90–118 (1937).
    [CrossRef]
  20. N. P. A. Zagers, J. van de Kraats, T. T. J. M. Berendschot, D. van Norren, “Simultaneous measurement of foveal spectral reflectance and cone-photoreceptor directionality,” Appl. Opt. 41, 4686–4696 (2002).
    [CrossRef] [PubMed]
  21. N. P. A. Zagers, T. T. J. M. Berendschot, D. van Norren, “Wavelength dependence of reflectometric cone photoreceptor directionality,” J. Opt. Soc. Am. A 20, 18–23 (2003).
    [CrossRef]
  22. A. W. Snyder, “Excitation of waveguide modes in retinal receptors,” J. Opt. Soc. Am. 56, 705–706 (1966).
    [CrossRef] [PubMed]
  23. M. J. Piket-May, A. Taflove, J. B. Troy, “Electrodynamics of visible-light interactions with the vertebrate retinal rod,” Opt. Lett. 18, 568–570 (1993).
    [CrossRef] [PubMed]
  24. B. Vohnsen, I. Iglesias, P. Artal, “Directional imaging of the retinal cone mosaic,” Opt. Lett. 29, 968–970 (2004).
    [CrossRef] [PubMed]
  25. In the case that the incoming field at the retina ψr and the mode field ψm have not been normalized before performing the integration in Eq. (3), the normalization can be performed on the entire expression by replacing the equation withPmPr=∣∫∫ψrψm*dxdy∣2∫∫∣ψr∣2dxdy∫∫∣ψm∣2dxdy. Just as before this expression gives the percentage of incident power coupled to the guided mode m. The total amount of guided power is found by adding up the contributions Pm from each allowed mode. Clearly, any single photoreceptor will carry only a tiny fraction of the total incident light power Pr unless the incident field is strongly confined at the photoreceptor. In turn, the guided modes are strongly localized at each waveguide and their area of normalization therefore need only to be slightly larger than the waveguide diameter.
  26. Note that the mode nomenclature of a weakly guiding step-index fiber has been chosen. Here the fundamental mode LP01 is rotationally symmetric and can be represented by zeroth order Bessel functions J0(2U01r∕di) in the core and K0(2V2−U012r∕di) in the cladding (U01 is the fundamental solution of the mode equation). Generally, the modes are found by solving the equationUlmJl+1(Ulm)Jl(Ulm)=V2−Ulm2Kl+1(V2−Ulm2)Kl(V2−Ulm2) numerically for the unknown Ulm, where Jl is the Bessel function of the first kind and Kl is the modified Bessel function of the second kind both of order l (see also pp. 135–137 in Ref. [27]). The fundamental mode profile (l=0,m=1) strongly resembles a Gaussian distribution function exp(−r2∕wm2) and the same holds true for the fundamental mode of fibers with other index distributions (see Fig. 8.12 in Ref. [27]).
  27. A. K. Ghatak, K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, 1998), pp. 149–156.
  28. B. Chen, W. Makous, “Light capture by human cones,” J. Physiol. (London) 414, 89–109 (1989).
  29. J. van de Kraats, T. T. J. M. Berendschot, D. Van Norren, “The pathways of light measured in fundus reflectometry,” Vision Res. 36, 2229–2247 (1996).
    [CrossRef] [PubMed]
  30. 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]
  31. J. A. Van Loo, J. M. Enoch, “The scotopic Stiles–Crawford effect,” Vision Res. 15, 1005–1009 (1975).
    [CrossRef] [PubMed]
  32. O. Packer, D. G. Bensinger, D. R. Williams, “ In vitro angular tuning of single primate rods and cones and the Stiles–Crawford effect [abstract],” Invest. Ophthalmol. Visual Sci. 35, 1572 (1994).
  33. C. Pask, A. Stacey, “Optical properties of retinal photoreceptors and the Campbell effect,” Vision Res. 38, 953–961 (1998).
    [CrossRef] [PubMed]
  34. T. T. J. M. Berendschot, J. van de Kraats, D. Van Norren, “Wavelength dependence of the Stiles–Crawford effect explained by perception of backscattered light from the choroid,” J. Opt. Soc. Am. A 18, 1445–1451 (2001).
    [CrossRef]
  35. G. Westheimer, “Dependence of the magnitude of the Stiles–Crawford effect on retinal location,” J. Physiol. (London) 192, 309–315 (1967).
  36. J. M. Enoch, G. M. Hope, “Directional sensitivity of the foveal and parafoveal retina,” Invest. Ophthalmol. Visual Sci. 12, 497–503 (1973).
  37. G. Toraldo di Francia, “Retina cones as dielectric antennas,” J. Opt. Soc. Am. 39, 324 (1949).
    [CrossRef]
  38. C. A. Curcio, K. R. Sloan, R. E. Kalina, A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292, 497–523 (1990).
    [CrossRef] [PubMed]
  39. M. P. Rowe, N. Engheta, S. S. Easter, E. N. Pugh, “Graded-index model of a fish double cone exhibits differential polarization sensitivity,” J. Opt. Soc. Am. A 11, 55–70 (1994).
    [CrossRef]
  40. L. J. Bour, J. C. M. Verhoosel, “Directional sensitivity of photoreceptors for different degrees of coherence and directions of polarization of the incident light,” Vision Res. 19, 717–719 (1979).
    [CrossRef] [PubMed]
  41. J. M. Enoch, “Summated response of the retina to light entering different parts of the pupil,” J. Opt. Soc. Am. 48, 392–405 (1958).
    [CrossRef] [PubMed]
  42. B. Drum, “Additivity of the Stiles–Crawford effect for a Fraunhofer image,” Vision Res. 15, 291–298 (1975).
    [CrossRef] [PubMed]
  43. B. Vohnsen, I. Iglesias, P. Artal, “Directional light scanning laser ophthalmoscope,” J. Opt. Soc. Am. A 22(12) (2005).
    [CrossRef]

2005

B. Vohnsen, I. Iglesias, P. Artal, “Directional light scanning laser ophthalmoscope,” J. Opt. Soc. Am. A 22(12) (2005).
[CrossRef]

2004

2003

2002

2001

1999

1998

S. Marcos, S. A. Burns, J. C. He, “Model for cone directionality reflectometric measurements based on scattering,” J. Opt. Soc. Am. A 15, 2012–2022 (1998).
[CrossRef]

C. Pask, A. Stacey, “Optical properties of retinal photoreceptors and the Campbell effect,” Vision Res. 38, 953–961 (1998).
[CrossRef] [PubMed]

1997

1996

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 (1996).
[CrossRef]

J. van de Kraats, T. T. J. M. Berendschot, D. Van Norren, “The pathways of light measured in fundus reflectometry,” Vision Res. 36, 2229–2247 (1996).
[CrossRef] [PubMed]

1995

1994

M. P. Rowe, N. Engheta, S. S. Easter, E. N. Pugh, “Graded-index model of a fish double cone exhibits differential polarization sensitivity,” J. Opt. Soc. Am. A 11, 55–70 (1994).
[CrossRef]

O. Packer, D. G. Bensinger, D. R. Williams, “ In vitro angular tuning of single primate rods and cones and the Stiles–Crawford effect [abstract],” Invest. Ophthalmol. Visual Sci. 35, 1572 (1994).

1993

1990

C. A. Curcio, K. R. Sloan, R. E. Kalina, A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292, 497–523 (1990).
[CrossRef] [PubMed]

1989

B. Chen, W. Makous, “Light capture by human cones,” J. Physiol. (London) 414, 89–109 (1989).

1986

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

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]

1979

L. J. Bour, J. C. M. Verhoosel, “Directional sensitivity of photoreceptors for different degrees of coherence and directions of polarization of the incident light,” Vision Res. 19, 717–719 (1979).
[CrossRef] [PubMed]

1975

B. Drum, “Additivity of the Stiles–Crawford effect for a Fraunhofer image,” Vision Res. 15, 291–298 (1975).
[CrossRef] [PubMed]

J. A. Van Loo, J. M. Enoch, “The scotopic Stiles–Crawford effect,” Vision Res. 15, 1005–1009 (1975).
[CrossRef] [PubMed]

1973

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

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

1971

A. M. Laties, J. M. Enoch, “An analysis of retinal receptor orientation,” Invest. Ophthalmol. Visual Sci. 10, 69–77 (1971).

1969

1967

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

1966

1963

1958

1952

1949

1946

1937

W. S. Stiles, “The luminous efficiency of monochromatic rays entering the eye pupil at different points and a new colour effect,” Proc. R. Soc. London, Ser. B 123, 90–118 (1937).
[CrossRef]

1933

W. S. Stiles, B. H. Crawford, “The luminous efficiency of rays entering the eye pupil at different points,” Proc. R. Soc. London 112, 428–450 (1933).
[CrossRef]

Applegate, R. A.

Artal, P.

Atchison, D. A.

Bensinger, D. G.

O. Packer, D. G. Bensinger, D. R. Williams, “ In vitro angular tuning of single primate rods and cones and the Stiles–Crawford effect [abstract],” Invest. Ophthalmol. Visual Sci. 35, 1572 (1994).

Berendschot, T. T. J. M.

Bour, L. J.

L. J. Bour, J. C. M. Verhoosel, “Directional sensitivity of photoreceptors for different degrees of coherence and directions of polarization of the incident light,” Vision Res. 19, 717–719 (1979).
[CrossRef] [PubMed]

Burns, S. A.

Chen, B.

B. Chen, W. Makous, “Light capture by human cones,” J. Physiol. (London) 414, 89–109 (1989).

Crawford, B. H.

W. S. Stiles, B. H. Crawford, “The luminous efficiency of rays entering the eye pupil at different points,” Proc. R. Soc. London 112, 428–450 (1933).
[CrossRef]

Curcio, C. A.

C. A. Curcio, K. R. Sloan, R. E. Kalina, A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292, 497–523 (1990).
[CrossRef] [PubMed]

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 (1996).
[CrossRef]

Delori, F.

Delori, F. C.

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

Drum, B.

B. Drum, “Additivity of the Stiles–Crawford effect for a Fraunhofer image,” Vision Res. 15, 291–298 (1975).
[CrossRef] [PubMed]

Easter, S. S.

Elsner, A. E.

Engheta, N.

Enoch, J. M.

J. A. Van Loo, J. M. Enoch, “The scotopic Stiles–Crawford effect,” Vision Res. 15, 1005–1009 (1975).
[CrossRef] [PubMed]

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

A. M. Laties, J. M. Enoch, “An analysis of retinal receptor orientation,” Invest. Ophthalmol. Visual Sci. 10, 69–77 (1971).

J. M. Enoch, “Optical properties of the retinal receptors,” J. Opt. Soc. Am. 53, 71–85 (1963).
[CrossRef]

J. M. Enoch, “Summated response of the retina to light entering different parts of the pupil,” J. Opt. Soc. Am. 48, 392–405 (1958).
[CrossRef] [PubMed]

Ghatak, A. K.

A. K. Ghatak, K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, 1998), pp. 149–156.

Gorrand, J.-M.

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

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 (1990).
[CrossRef] [PubMed]

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).

Hyams, L.

Iglesias, I.

B. Vohnsen, I. Iglesias, P. Artal, “Directional light scanning laser ophthalmoscope,” J. Opt. Soc. Am. A 22(12) (2005).
[CrossRef]

B. Vohnsen, I. Iglesias, P. Artal, “Directional imaging of the retinal cone mosaic,” Opt. Lett. 29, 968–970 (2004).
[CrossRef] [PubMed]

Kalina, R. E.

C. A. Curcio, K. R. Sloan, R. E. Kalina, A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292, 497–523 (1990).
[CrossRef] [PubMed]

Lakshminarayanan, V.

Laties, A. M.

A. M. Laties, J. M. Enoch, “An analysis of retinal receptor orientation,” Invest. Ophthalmol. Visual Sci. 10, 69–77 (1971).

Makous, W.

B. Chen, W. Makous, “Light capture by human cones,” J. Physiol. (London) 414, 89–109 (1989).

Marcos, S.

O’Brien, B.

Packer, O.

O. Packer, D. G. Bensinger, D. R. Williams, “ In vitro angular tuning of single primate rods and cones and the Stiles–Crawford effect [abstract],” Invest. Ophthalmol. Visual Sci. 35, 1572 (1994).

Pask, C.

C. Pask, A. Stacey, “Optical properties of retinal photoreceptors and the Campbell effect,” Vision Res. 38, 953–961 (1998).
[CrossRef] [PubMed]

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

Piket-May, M. J.

Pugh, E. N.

Ronchi, L.

Roorda, A.

A. Roorda, D. R. Williams, “Optical fiber properties of individual human cones,” J. Vision 2, 404–412 (2002).
[CrossRef]

Rowe, M. P.

Safir, A.

Scott, D. H.

Sloan, K. R.

C. A. Curcio, K. R. Sloan, R. E. Kalina, A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292, 497–523 (1990).
[CrossRef] [PubMed]

Snyder, A. W.

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

A. W. Snyder, “Excitation of waveguide modes in retinal receptors,” J. Opt. Soc. Am. 56, 705–706 (1966).
[CrossRef] [PubMed]

Stacey, A.

C. Pask, A. Stacey, “Optical properties of retinal photoreceptors and the Campbell effect,” Vision Res. 38, 953–961 (1998).
[CrossRef] [PubMed]

Stiles, W. S.

W. S. Stiles, “The luminous efficiency of monochromatic rays entering the eye pupil at different points and a new colour effect,” Proc. R. Soc. London, Ser. B 123, 90–118 (1937).
[CrossRef]

W. S. Stiles, B. H. Crawford, “The luminous efficiency of rays entering the eye pupil at different points,” Proc. R. Soc. London 112, 428–450 (1933).
[CrossRef]

Strang, N. C.

Taflove, A.

Thyagarajan, K.

A. K. Ghatak, K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, 1998), pp. 149–156.

Toraldo di Francia, G.

Troy, J. B.

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]

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

van de Kraats, J.

Van Loo, J. A.

J. A. Van Loo, J. M. Enoch, “The scotopic Stiles–Crawford effect,” Vision Res. 15, 1005–1009 (1975).
[CrossRef] [PubMed]

van Norren, D.

N. P. A. Zagers, T. T. J. M. Berendschot, D. van Norren, “Wavelength dependence of reflectometric cone photoreceptor directionality,” J. Opt. Soc. Am. A 20, 18–23 (2003).
[CrossRef]

N. P. A. Zagers, J. van de Kraats, T. T. J. M. Berendschot, D. van Norren, “Simultaneous measurement of foveal spectral reflectance and cone-photoreceptor directionality,” Appl. Opt. 41, 4686–4696 (2002).
[CrossRef] [PubMed]

T. T. J. M. Berendschot, J. van de Kraats, D. Van Norren, “Wavelength dependence of the Stiles–Crawford effect explained by perception of backscattered light from the choroid,” J. Opt. Soc. Am. A 18, 1445–1451 (2001).
[CrossRef]

J. van de Kraats, T. T. J. M. Berendschot, D. Van Norren, “The pathways of light measured in fundus reflectometry,” Vision Res. 36, 2229–2247 (1996).
[CrossRef] [PubMed]

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 (1996).
[CrossRef]

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]

Verhoosel, J. C. M.

L. J. Bour, J. C. M. Verhoosel, “Directional sensitivity of photoreceptors for different degrees of coherence and directions of polarization of the incident light,” Vision Res. 19, 717–719 (1979).
[CrossRef] [PubMed]

Vohnsen, B.

B. Vohnsen, I. Iglesias, P. Artal, “Directional light scanning laser ophthalmoscope,” J. Opt. Soc. Am. A 22(12) (2005).
[CrossRef]

B. Vohnsen, I. Iglesias, P. Artal, “Directional imaging of the retinal cone mosaic,” Opt. Lett. 29, 968–970 (2004).
[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.

A. Roorda, D. R. Williams, “Optical fiber properties of individual human cones,” J. Vision 2, 404–412 (2002).
[CrossRef]

O. Packer, D. G. Bensinger, D. R. Williams, “ In vitro angular tuning of single primate rods and cones and the Stiles–Crawford effect [abstract],” Invest. Ophthalmol. Visual Sci. 35, 1572 (1994).

Wu, S.

Zagers, N. P. A.

Appl. Opt.

Invest. Ophthalmol. Visual Sci.

A. M. Laties, J. M. Enoch, “An analysis of retinal receptor orientation,” Invest. Ophthalmol. Visual Sci. 10, 69–77 (1971).

O. Packer, D. G. Bensinger, D. R. Williams, “ In vitro angular tuning of single primate rods and cones and the Stiles–Crawford effect [abstract],” Invest. Ophthalmol. Visual Sci. 35, 1572 (1994).

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

J. Comp. Neurol.

C. A. Curcio, K. R. Sloan, R. E. Kalina, A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292, 497–523 (1990).
[CrossRef] [PubMed]

J. Mod. Opt.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

N. P. A. Zagers, T. T. J. M. Berendschot, D. van Norren, “Wavelength dependence of reflectometric cone photoreceptor directionality,” J. Opt. Soc. Am. A 20, 18–23 (2003).
[CrossRef]

T. T. J. M. Berendschot, J. van de Kraats, D. Van Norren, “Wavelength dependence of the Stiles–Crawford effect explained by perception of backscattered light from the choroid,” J. Opt. Soc. Am. A 18, 1445–1451 (2001).
[CrossRef]

D. A. Atchison, D. H. Scott, N. C. Strang, P. Artal, “Influence of Stiles–Crawford apodization on visual acuity,” J. Opt. Soc. Am. A 19, 1073–1083 (2002).
[CrossRef]

S. A. Burns, S. Wu, F. Delori, A. E. Elsner, “Direct measurement of human-cone-photoreceptor alignment,” J. Opt. Soc. Am. A 12, 2329–2338 (1995).
[CrossRef]

B. Vohnsen, I. Iglesias, P. Artal, “Directional light scanning laser ophthalmoscope,” J. Opt. Soc. Am. A 22(12) (2005).
[CrossRef]

M. P. Rowe, N. Engheta, S. S. Easter, E. N. Pugh, “Graded-index model of a fish double cone exhibits differential polarization sensitivity,” J. Opt. Soc. Am. A 11, 55–70 (1994).
[CrossRef]

S. Marcos, S. A. Burns, “Cone spacing and waveguide properties from cone directionality measurements,” J. Opt. Soc. Am. A 16, 995–1004 (1999).
[CrossRef]

J. C. He, S. Marcos, S. A. Burns, “Comparison of cone directionality determined by psychophysical and reflectometric techniques,” J. Opt. Soc. Am. A 16, 2363–2369 (1999).
[CrossRef]

S. Marcos, S. A. Burns, J. C. He, “Model for cone directionality reflectometric measurements based on scattering,” J. Opt. Soc. Am. A 15, 2012–2022 (1998).
[CrossRef]

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]

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]

J. Physiol. (London)

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

B. Chen, W. Makous, “Light capture by human cones,” J. Physiol. (London) 414, 89–109 (1989).

J. Vision

A. Roorda, D. R. Williams, “Optical fiber properties of individual human cones,” J. Vision 2, 404–412 (2002).
[CrossRef]

Opt. Lett.

Proc. R. Soc. London

W. S. Stiles, B. H. Crawford, “The luminous efficiency of rays entering the eye pupil at different points,” Proc. R. Soc. London 112, 428–450 (1933).
[CrossRef]

Proc. R. Soc. London, Ser. B

W. S. Stiles, “The luminous efficiency of monochromatic rays entering the eye pupil at different points and a new colour effect,” Proc. R. Soc. London, Ser. B 123, 90–118 (1937).
[CrossRef]

Vision Res.

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 (1996).
[CrossRef]

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

C. Pask, A. Stacey, “Optical properties of retinal photoreceptors and the Campbell effect,” Vision Res. 38, 953–961 (1998).
[CrossRef] [PubMed]

L. J. Bour, J. C. M. Verhoosel, “Directional sensitivity of photoreceptors for different degrees of coherence and directions of polarization of the incident light,” Vision Res. 19, 717–719 (1979).
[CrossRef] [PubMed]

B. Drum, “Additivity of the Stiles–Crawford effect for a Fraunhofer image,” Vision Res. 15, 291–298 (1975).
[CrossRef] [PubMed]

J. van de Kraats, T. T. J. M. Berendschot, D. Van Norren, “The pathways of light measured in fundus reflectometry,” Vision Res. 36, 2229–2247 (1996).
[CrossRef] [PubMed]

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]

J. A. Van Loo, J. M. Enoch, “The scotopic Stiles–Crawford effect,” Vision Res. 15, 1005–1009 (1975).
[CrossRef] [PubMed]

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

Other

In the case that the incoming field at the retina ψr and the mode field ψm have not been normalized before performing the integration in Eq. (3), the normalization can be performed on the entire expression by replacing the equation withPmPr=∣∫∫ψrψm*dxdy∣2∫∫∣ψr∣2dxdy∫∫∣ψm∣2dxdy. Just as before this expression gives the percentage of incident power coupled to the guided mode m. The total amount of guided power is found by adding up the contributions Pm from each allowed mode. Clearly, any single photoreceptor will carry only a tiny fraction of the total incident light power Pr unless the incident field is strongly confined at the photoreceptor. In turn, the guided modes are strongly localized at each waveguide and their area of normalization therefore need only to be slightly larger than the waveguide diameter.

Note that the mode nomenclature of a weakly guiding step-index fiber has been chosen. Here the fundamental mode LP01 is rotationally symmetric and can be represented by zeroth order Bessel functions J0(2U01r∕di) in the core and K0(2V2−U012r∕di) in the cladding (U01 is the fundamental solution of the mode equation). Generally, the modes are found by solving the equationUlmJl+1(Ulm)Jl(Ulm)=V2−Ulm2Kl+1(V2−Ulm2)Kl(V2−Ulm2) numerically for the unknown Ulm, where Jl is the Bessel function of the first kind and Kl is the modified Bessel function of the second kind both of order l (see also pp. 135–137 in Ref. [27]). The fundamental mode profile (l=0,m=1) strongly resembles a Gaussian distribution function exp(−r2∕wm2) and the same holds true for the fundamental mode of fibers with other index distributions (see Fig. 8.12 in Ref. [27]).

A. K. Ghatak, K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, 1998), pp. 149–156.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Schematic of the configuration considered for light coupling to and from a photoreceptor. (a) The Gaussian beam incident on the eye is focused at the retina slightly off axis from the photoreceptor axis and coupled to mode m. (b) The incident Gaussian beam is displaced in the pupil a distance r p with the result of intersecting the photoreceptor at an angle θ. (c) The back-reflected beam diffracts from the photoreceptor and produces a Gaussian intensity distribution at the eye pupil.

Fig. 2
Fig. 2

Calculated coupling efficiency T ( θ ) (thick solid curve) at λ = 0.543 μ m for inner segments (a) 2.5 μ m and (b) 5.0 μ m including the contributions from individual modes: LP 01 (thin solid curve), LP 11 (dashed curve), LP 21 (dashed–dotted curve), and LP 02 (dotted curve). Best-fitted Gaussian distributions versus angle (in radians): (a) T 01 ( θ ) = 0.8605 exp [ 2 θ 2 0.1318 2 ] , T sum ( θ ) = 0.8764 exp [ 2 θ 2 0.1918 2 ] , (b) T 01 ( θ ) = 0.6980 exp [ 2 θ 2 0.0734 2 ] , T sum ( θ ) = 0.7831 exp [ 2 θ 2 0.1416 2 ] , shown as a series of crosses ( LP 01 ) and circles (sum), respectively.

Fig. 3
Fig. 3

Calculated coupling efficiency T ( θ ) (thick solid curve) at λ = 0.785 μ m for inner segments (a) 2.5 μ m and (b) 5.0 μ m including the contributions from individual modes: LP 01 (thin solid curve) and LP 11 (dashed curve). Best-fitted Gaussian distributions versus angle (in radians): (a) T 01 ( θ ) = T sum ( θ ) = 0.9310 exp [ 2 θ 2 0.1832 2 ] , (b) T 01 ( θ ) = 0.7833 exp [ 2 θ 2 0.1001 2 ] , T sum ( θ ) = 0.8062 exp [ 2 θ 2 0.1425 2 ] , shown as a series of crosses ( LP 01 ) and circles (sum), respectively.

Fig. 4
Fig. 4

Simulated OSCE at an illumination wavelength of 0.543 μ m of (a) 1° uniformly illuminated retinal patch with 1221 randomly located contributing cones (i.e., a density of 20,000   cones mm 2 ) and fundamental mode width w m = 1.35 μ m (for simplicity partial cone overlap has not been prohibited in the model). The coherent plane-wave contribution [i.e., the unweighted sum of Eq. (18)] is shown in (b) at the pupil plane [ 6 × 6 mm 2 ] . Corresponding intensity distributions for a 6 mm pupil are shown for coherent light with (c) 1, (d), 10, (e) 100 averaged frames and compared with the case of (f) incoherent light.

Tables (2)

Tables Icon

Table 1 Solutions U l m of the Mode Equation (Ref. [26]) for Different Modes LP l m at a Wavelength of 0.543 μ m (Upper Half) and 0.785 μ m (Lower Half)

Tables Icon

Table 2 Coupling Efficiency of a Plane Wave to Modes LP l m in the Inner and Outer Photoreceptor Segments Calculated at Incident Angles of 0, 5, and 10 deg (in Ascending Order) for a Wavelength of 0.543 μ m and 0.785 μ m . a

Equations (20)

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

f ( r = x 2 + y 2 ) = C + A × 10 ρ r 2 = C + A exp ( α r 2 ) ,
w r = λ f eye π n eye w 0
P m P r = ψ r ψ m * d x d y 2 .
T ( u ) = [ 2 w r w m w r 2 + w m 2 ] 2 exp [ 2 u 2 w r 2 + w m 2 ] ,
T ( θ ) = [ 2 w r w m w r 2 + w m 2 ] 2 exp [ 2 ( π n eye w r w m ) 2 θ 2 λ 2 ( w r 2 + w m 2 ) ] .
T ( u ) = exp [ u 2 w 2 ] ,
T ( θ ) = exp [ ( π n eye w λ ) 2 θ 2 ] .
I dir ( r ) = I 0 T R exp [ 2 ( r w p ) 2 ] ,
w p = λ f eye π n eye w m .
ρ SCE = log ( e ) ( π n eye w λ f eye ) 2 .
ρ OSCE = log ( e ) 2 w p 2 = 2 ρ SCE ,
ρ OSCE = 3 ρ SCE .
T ( u ) = ( 2 w m w r ) 2 exp [ 2 u 2 w r 2 ] ,
T ( θ ) = ( 2 w m w r ) 2 exp [ 2 ( π n eye w m ) 2 θ 2 λ 2 ] .
ρ SCE = log ( e ) 2 w p 2 = ρ OSCE .
T total ( θ ) = m = 0 M A exp ( i 2 π λ n eye θ x ) ψ m * d x d y 2 ,
θ l m U l m λ π n i d i .
I dir ( x , y ) = I 0 T R exp [ 2 ( x 2 + y 2 ) w p 2 ] j = 1 N exp [ i ( k x , j x + k y , j y + φ j ) ] 2 ,
P m P r = ψ r ψ m * d x d y 2 ψ r 2 d x d y ψ m 2 d x d y .
U l m J l + 1 ( U l m ) J l ( U l m ) = V 2 U l m 2 K l + 1 ( V 2 U l m 2 ) K l ( V 2 U l m 2 )

Metrics