Abstract

The light capturing properties of cone photoreceptors create the elementary signals that form the basis of vision. Variation in the amplitude of individual cone signals has been found physiologically as part of normal retinal circuit processing. Less well characterized is how cone signals may vary due to purely optical properties. We present a model of light propagation in cones using a finite difference beam propagation method to simulate how light from a small stimulus travels through a cone plus its immediate neighbors. The model calculates the amount of light absorbed in the cone outer segments, from which an estimate of the photoresponse can be made. We apply the method to adaptive optics microstimulation to find the optimum optical conditions that will confine the most light into a single cone in the human retina. We found that light capture is especially sensitive to beam size at the pupil and to the cone diameter itself, with the two factors having a complex relationship leading to sizable variation in light capture. Model predictions were validated with two types of psychophysical data. The model can be employed with arbitrary stimuli and photoreceptor parameters, making it a useful tool for studying photoreceptor function in normal or diseased conditions.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  56. S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14(1), 29–49 (1975).
    [Crossref] [PubMed]
  57. D. I. A. MacLeod, D. R. Williams, and W. Makous, “A visual nonlinearity fed by single cones,” Vision Res. 32(2), 347–363 (1992).
    [Crossref] [PubMed]
  58. H. Rao, R. Scarmozzino, and R. M. Osgood, “Bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photonics Technol. Lett. 11(7), 830–832 (1999).
    [Crossref]
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2017 (2)

L. Sawides, A. de Castro, and S. A. Burns, “The organization of the cone photoreceptor mosaic measured in the living human retina,” Vision Res. 132, 34–44 (2017).
[Crossref] [PubMed]

B. Vohnsen, A. Carmichael, N. Sharmin, S. Qaysi, and D. Valente, “Volumetric integration model of the Stiles-Crawford effect of the first kind and its experimental verification,” J. Vis. 17(12), 18 (2017).
[Crossref] [PubMed]

2015 (5)

Q. Wang, W. S. Tuten, B. J. Lujan, J. Holland, P. S. Bernstein, S. D. Schwartz, J. L. Duncan, and A. Roorda, “Adaptive optics microperimetry and OCT images show preserved function and recovery of cone visibility in macular telangiectasia type 2 retinal lesions,” Invest. Ophthalmol. Vis. Sci. 56(2), 778–786 (2015).
[Crossref] [PubMed]

T. Zhang, P. Godara, E. R. Blanco, R. L. Griffin, X. Wang, C. A. Curcio, and Y. Zhang, “Variability in human cone topography assessed by adaptive optics scanning laser ophthalmoscopy,” Am. J. Ophthalmol. 160(2), 290–300 (2015).
[Crossref] [PubMed]

M. B. Toomey, P. Olsson, R. Frederiksen, M. C. Cornwall, R. Oulton, A. Kelber, J. C. Corbo, N. W. Roberts, and D. Wilby, “Optics of cone photoreceptors in the chicken (Gallus gallus domesticus),” J. R. Soc. Interface 12(111), 20150591 (2015).
[Crossref] [PubMed]

K. S. Bruce, W. M. Harmening, B. R. Langston, W. S. Tuten, A. Roorda, and L. C. Sincich, “Normal perceptual sensitivity arising from weakly reflective cone photoreceptors,” Invest. Ophthalmol. Vis. Sci. 56(8), 4431–4438 (2015).
[Crossref] [PubMed]

Z. Liu, O. P. Kocaoglu, T. L. Turner, and D. T. Miller, “Modal content of living human cone photoreceptors,” Biomed. Opt. Express 6(9), 3378–3404 (2015).
[Crossref] [PubMed]

2014 (3)

P. H. Li, G. D. Field, M. Greschner, D. Ahn, D. E. Gunning, K. Mathieson, A. Sher, A. M. Litke, and E. J. Chichilnisky, “Retinal representation of the elementary visual signal,” Neuron 81(1), 130–139 (2014).
[Crossref] [PubMed]

W. M. Harmening, W. S. Tuten, A. Roorda, and L. C. Sincich, “Mapping the perceptual grain of the human retina,” J. Neurosci. 34(16), 5667–5677 (2014).
[Crossref] [PubMed]

B. Vohnsen, “Directional sensitivity of the retina: A layered scattering model of outer-segment photoreceptor pigments,” Biomed. Opt. Express 5(5), 1569–1587 (2014).
[Crossref] [PubMed]

2013 (1)

T. J. T. P. van den Berg, L. Franssen, B. Kruijt, and J. E. Coppens, “History of ocular straylight measurement: A review,” Z. Med. Phys. 23(1), 6–20 (2013).
[Crossref] [PubMed]

2012 (2)

2011 (2)

H. Song, T. Y. P. Chui, Z. Zhong, A. E. Elsner, and S. A. Burns, “Variation of cone photoreceptor packing density with retinal eccentricity and age,” Invest. Ophthalmol. Vis. Sci. 52(10), 7376–7384 (2011).
[Crossref] [PubMed]

R. F. Spaide and C. A. Curcio, “Anatomical correlates to the bands seen in the outer retina by optical coherence tomography: literature review and model,” Retina 31(8), 1609–1619 (2011).
[Crossref] [PubMed]

2010 (3)

G. D. Field, J. L. Gauthier, A. Sher, M. Greschner, T. A. Machado, L. H. Jepson, J. Shlens, D. E. Gunning, K. Mathieson, W. Dabrowski, L. Paninski, A. M. Litke, and E. J. Chichilnisky, “Functional connectivity in the retina at the resolution of photoreceptors,” Nature 467(7316), 673–677 (2010).
[Crossref] [PubMed]

L. Fischer, A. Zvyagin, T. Plakhotnik, and M. Vorobyev, “Numerical modeling of light propagation in a hexagonal array of dielectric cylinders,” J. Opt. Soc. Am. A 27(4), 865–872 (2010).
[Crossref] [PubMed]

Q. Yang, D. W. Arathorn, P. Tiruveedhula, C. R. Vogel, and A. Roorda, “Design of an integrated hardware interface for AOSLO image capture and cone-targeted stimulus delivery,” Opt. Express 18(17), 3283–3291 (2010).

2009 (1)

L. C. Sincich, Y. Zhang, P. Tiruveedhula, J. C. Horton, and A. Roorda, “Resolving single cone inputs to visual receptive fields,” Nat. Neurosci. 12(8), 967–969 (2009).
[Crossref] [PubMed]

2008 (1)

V. J. Srinivasan, B. K. Monson, M. Wojtkowski, R. A. Bilonick, I. Gorczynska, R. Chen, J. S. Duker, J. S. Schuman, and J. G. Fujimoto, “Characterization of outer retinal morphology with high-speed, ultrahigh-resolution optical coherence tomography,” Invest. Ophthalmol. Vis. Sci. 49(4), 1571–1579 (2008).
[Crossref] [PubMed]

2007 (3)

2006 (1)

J. Xiao, X. Liu, C. Cai, H. Fan, and X. Sun, “An improved three-dimensional full-vectorial finite-difference imaginary-distance beam propagation method,” Sci. China, Ser. F Inf. Sci. 49(4), 516–532 (2006).

2005 (2)

B. Vohnsen, I. Iglesias, and P. Artal, “Guided light and diffraction model of human-eye photoreceptors,” J. Opt. Soc. Am. A 22(11), 2318–2328 (2005).
[Crossref] [PubMed]

A. M. Pozo, F. Pérez-Ocón, and J. R. Jiménez, “FDTD analysis of the light propagation in the cones of the human retina: an approach to the Stiles-Crawford effect of the first kind,” J. Opt. A, Pure Appl. Opt. 7(8), 357–363 (2005).
[Crossref]

2003 (1)

T. T. J. M. Berendschot, P. J. DeLint, and D. van Norren, “Fundus reflectance–historical and present ideas,” Prog. Retin. Eye Res. 22(2), 171–200 (2003).
[Crossref] [PubMed]

2002 (1)

Q. V. Hoang, R. A. Linsenmeier, C. K. Chung, and C. A. Curcio, “Photoreceptor inner segments in monkey and human retina: mitochondrial density, optics, and regional variation,” Vis. Neurosci. 19(4), 395–407 (2002).
[Crossref] [PubMed]

2001 (1)

1999 (1)

H. Rao, R. Scarmozzino, and R. M. Osgood, “Bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photonics Technol. Lett. 11(7), 830–832 (1999).
[Crossref]

1997 (2)

J. A. Besley and J. D. Love, “Supermode analysis of fibre transmission,” IEE Proc., Optoelectron. 144(6), 411–419 (1997).
[Crossref]

J.-M. Gorrand and F. C. Delori, “A model for the assessment of cone directionality,” J. Mod. Opt. 44(3), 473–491 (1997).
[Crossref]

1993 (2)

M. J. Piket-May, A. Taflove, and J. B. Troy, “Electrodynamics of visible-light interactions with the vertebrate retinal rod,” Opt. Lett. 18(8), 568–570 (1993).
[Crossref] [PubMed]

W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29(10), 2639–2649 (1993).
[Crossref]

1992 (2)

W. P. Huang, C. L. Xu, and S. K. Chaudhuri, “A finite-difference vector beam propagation method for three-dimensional waveguide structures,” IEEE Photonics Technol. Lett. 4(2), 148–151 (1992).
[Crossref]

D. I. A. MacLeod, D. R. Williams, and W. Makous, “A visual nonlinearity fed by single cones,” Vision Res. 32(2), 347–363 (1992).
[Crossref] [PubMed]

1991 (1)

W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, “A vector beam propagation method for guided-wave optics,” IEEE Photonics Technol. Lett. 3(10), 910–913 (1991).
[Crossref]

1990 (1)

C. A. Curcio, K. R. Sloan, R. E. Kalina, and A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292(4), 497–523 (1990).
[Crossref] [PubMed]

1987 (2)

C. A. Curcio, K. R. Sloan, O. Packer, A. E. Hendrickson, and R. E. Kalina, “Distribution of cones in human and monkey retina: individual variability and radial asymmetry,” Science 236(4801), 579–582 (1987).
[Crossref] [PubMed]

F. I. Hárosi, “Application of Fourier transform smoothing and statistical techniques to the determination of spectral parameters,” J. Gen. Physiol. 89(5), 717–743 (1987).
[Crossref] [PubMed]

1984 (2)

D. A. Baylor, B. J. Nunn, and J. L. Schnapf, “The photocurrent, noise and spectral sensitivity of rods of the monkey Macaca fascicularis,” J. Physiol. 357(1), 575–607 (1984).
[Crossref] [PubMed]

J. Hirsch and R. Hylton, “Quality of the primate photoreceptor lattice,” Vision Res. 24(4), 347–355 (1984).
[Crossref] [PubMed]

1983 (1)

W. H. Miller and G. D. Bernard, “Averaging over the foveal receptor aperture curtails aliasing,” Vision Res. 23(12), 1365–1369 (1983).
[Crossref] [PubMed]

1980 (1)

R. H. Steinberg, S. K. Fisher, and D. H. Anderson, “Disc morphogenesis in vertebrate photoreceptors,” J. Comp. Neurol. 190(3), 501–518 (1980).
[Crossref] [PubMed]

1978 (1)

1975 (2)

S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14(1), 29–49 (1975).
[Crossref] [PubMed]

D. G. Stavenga, “Waveguide modes and refractive index in photoreceptors of invertebrates,” Vision Res. 15(3), 323–330 (1975).
[Crossref] [PubMed]

1974 (1)

1973 (1)

A. W. Snyder and C. Pask, “The Stiles-Crawford effect-explanation and consequences,” Vision Res. 13(6), 1115–1137 (1973).
[Crossref] [PubMed]

1972 (1)

A. W. Snyder and M. Hamer, “The light-capture area of a photoreceptor,” Vision Res. 12(10), 1749–1753 (1972).
[Crossref] [PubMed]

1971 (1)

1970 (1)

A. W. Snyder, “Coupling of modes on a tapered dielectric cylinder,” IEEE Trans. Microw. Theory Tech. 18(7), 383–392 (1970).
[Crossref]

1969 (1)

B. B. Boycott, J. E. Dowling, and H. Kolb, “Organization of the primate retina: light microscopy,” Proc. R. Soc. London, Ser. B, Biol. Sci. 255(799), 109– 184 (1969).

1966 (1)

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. 14(3), 302–307 (1966).
[Crossref]

1961 (1)

J. M. Enoch, “Wave-guide modes in retinal receptors,” Science 133(3461), 1353–1354 (1961).
[Crossref] [PubMed]

1957 (2)

R. Barer, “Refractometry and interferometry of living cells,” J. Opt. Soc. Am. 47(6), 545–556 (1957).
[Crossref] [PubMed]

R. L. Sidman, “The structure and concentration of solids in photoreceptor cells studied by refractometry and interference microscopy,” J. Biophys. Biochem. Cytol. 3(1), 15–30 (1957).
[Crossref] [PubMed]

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V. J. Srinivasan, B. K. Monson, M. Wojtkowski, R. A. Bilonick, I. Gorczynska, R. Chen, J. S. Duker, J. S. Schuman, and J. G. Fujimoto, “Characterization of outer retinal morphology with high-speed, ultrahigh-resolution optical coherence tomography,” Invest. Ophthalmol. Vis. Sci. 49(4), 1571–1579 (2008).
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B. B. Boycott, J. E. Dowling, and H. Kolb, “Organization of the primate retina: light microscopy,” Proc. R. Soc. London, Ser. B, Biol. Sci. 255(799), 109– 184 (1969).

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K. S. Bruce, W. M. Harmening, B. R. Langston, W. S. Tuten, A. Roorda, and L. C. Sincich, “Normal perceptual sensitivity arising from weakly reflective cone photoreceptors,” Invest. Ophthalmol. Vis. Sci. 56(8), 4431–4438 (2015).
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J. Xiao, X. Liu, C. Cai, H. Fan, and X. Sun, “An improved three-dimensional full-vectorial finite-difference imaginary-distance beam propagation method,” Sci. China, Ser. F Inf. Sci. 49(4), 516–532 (2006).

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W. P. Huang, C. L. Xu, and S. K. Chaudhuri, “A finite-difference vector beam propagation method for three-dimensional waveguide structures,” IEEE Photonics Technol. Lett. 4(2), 148–151 (1992).
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Chen, R.

V. J. Srinivasan, B. K. Monson, M. Wojtkowski, R. A. Bilonick, I. Gorczynska, R. Chen, J. S. Duker, J. S. Schuman, and J. G. Fujimoto, “Characterization of outer retinal morphology with high-speed, ultrahigh-resolution optical coherence tomography,” Invest. Ophthalmol. Vis. Sci. 49(4), 1571–1579 (2008).
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P. H. Li, G. D. Field, M. Greschner, D. Ahn, D. E. Gunning, K. Mathieson, A. Sher, A. M. Litke, and E. J. Chichilnisky, “Retinal representation of the elementary visual signal,” Neuron 81(1), 130–139 (2014).
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W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, “A vector beam propagation method for guided-wave optics,” IEEE Photonics Technol. Lett. 3(10), 910–913 (1991).
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Chui, T. Y. P.

H. Song, T. Y. P. Chui, Z. Zhong, A. E. Elsner, and S. A. Burns, “Variation of cone photoreceptor packing density with retinal eccentricity and age,” Invest. Ophthalmol. Vis. Sci. 52(10), 7376–7384 (2011).
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Q. V. Hoang, R. A. Linsenmeier, C. K. Chung, and C. A. Curcio, “Photoreceptor inner segments in monkey and human retina: mitochondrial density, optics, and regional variation,” Vis. Neurosci. 19(4), 395–407 (2002).
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Coppens, J. E.

T. J. T. P. van den Berg, L. Franssen, B. Kruijt, and J. E. Coppens, “History of ocular straylight measurement: A review,” Z. Med. Phys. 23(1), 6–20 (2013).
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L. Franssen, J. Tabernero, J. E. Coppens, and T. J. T. P. van den Berg, “Pupil size and retinal straylight in the normal eye,” Invest. Ophthalmol. Vis. Sci. 48(5), 2375–2382 (2007).
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Corbo, J. C.

M. B. Toomey, P. Olsson, R. Frederiksen, M. C. Cornwall, R. Oulton, A. Kelber, J. C. Corbo, N. W. Roberts, and D. Wilby, “Optics of cone photoreceptors in the chicken (Gallus gallus domesticus),” J. R. Soc. Interface 12(111), 20150591 (2015).
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M. B. Toomey, P. Olsson, R. Frederiksen, M. C. Cornwall, R. Oulton, A. Kelber, J. C. Corbo, N. W. Roberts, and D. Wilby, “Optics of cone photoreceptors in the chicken (Gallus gallus domesticus),” J. R. Soc. Interface 12(111), 20150591 (2015).
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Curcio, C. A.

T. Zhang, P. Godara, E. R. Blanco, R. L. Griffin, X. Wang, C. A. Curcio, and Y. Zhang, “Variability in human cone topography assessed by adaptive optics scanning laser ophthalmoscopy,” Am. J. Ophthalmol. 160(2), 290–300 (2015).
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Q. V. Hoang, R. A. Linsenmeier, C. K. Chung, and C. A. Curcio, “Photoreceptor inner segments in monkey and human retina: mitochondrial density, optics, and regional variation,” Vis. Neurosci. 19(4), 395–407 (2002).
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C. A. Curcio, K. R. Sloan, R. E. Kalina, and A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292(4), 497–523 (1990).
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C. A. Curcio, K. R. Sloan, O. Packer, A. E. Hendrickson, and R. E. Kalina, “Distribution of cones in human and monkey retina: individual variability and radial asymmetry,” Science 236(4801), 579–582 (1987).
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Dabrowski, W.

G. D. Field, J. L. Gauthier, A. Sher, M. Greschner, T. A. Machado, L. H. Jepson, J. Shlens, D. E. Gunning, K. Mathieson, W. Dabrowski, L. Paninski, A. M. Litke, and E. J. Chichilnisky, “Functional connectivity in the retina at the resolution of photoreceptors,” Nature 467(7316), 673–677 (2010).
[Crossref] [PubMed]

de Castro, A.

L. Sawides, A. de Castro, and S. A. Burns, “The organization of the cone photoreceptor mosaic measured in the living human retina,” Vision Res. 132, 34–44 (2017).
[Crossref] [PubMed]

DeLint, P. J.

T. T. J. M. Berendschot, P. J. DeLint, and D. van Norren, “Fundus reflectance–historical and present ideas,” Prog. Retin. Eye Res. 22(2), 171–200 (2003).
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Delori, F. C.

J.-M. Gorrand and F. C. Delori, “A model for the assessment of cone directionality,” J. Mod. Opt. 44(3), 473–491 (1997).
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Dowling, J. E.

B. B. Boycott, J. E. Dowling, and H. Kolb, “Organization of the primate retina: light microscopy,” Proc. R. Soc. London, Ser. B, Biol. Sci. 255(799), 109– 184 (1969).

Duker, J. S.

V. J. Srinivasan, B. K. Monson, M. Wojtkowski, R. A. Bilonick, I. Gorczynska, R. Chen, J. S. Duker, J. S. Schuman, and J. G. Fujimoto, “Characterization of outer retinal morphology with high-speed, ultrahigh-resolution optical coherence tomography,” Invest. Ophthalmol. Vis. Sci. 49(4), 1571–1579 (2008).
[Crossref] [PubMed]

Duncan, J. L.

Q. Wang, W. S. Tuten, B. J. Lujan, J. Holland, P. S. Bernstein, S. D. Schwartz, J. L. Duncan, and A. Roorda, “Adaptive optics microperimetry and OCT images show preserved function and recovery of cone visibility in macular telangiectasia type 2 retinal lesions,” Invest. Ophthalmol. Vis. Sci. 56(2), 778–786 (2015).
[Crossref] [PubMed]

Elsner, A. E.

H. Song, T. Y. P. Chui, Z. Zhong, A. E. Elsner, and S. A. Burns, “Variation of cone photoreceptor packing density with retinal eccentricity and age,” Invest. Ophthalmol. Vis. Sci. 52(10), 7376–7384 (2011).
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J. M. Enoch, “Wave-guide modes in retinal receptors,” Science 133(3461), 1353–1354 (1961).
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Fan, H.

J. Xiao, X. Liu, C. Cai, H. Fan, and X. Sun, “An improved three-dimensional full-vectorial finite-difference imaginary-distance beam propagation method,” Sci. China, Ser. F Inf. Sci. 49(4), 516–532 (2006).

Field, G. D.

P. H. Li, G. D. Field, M. Greschner, D. Ahn, D. E. Gunning, K. Mathieson, A. Sher, A. M. Litke, and E. J. Chichilnisky, “Retinal representation of the elementary visual signal,” Neuron 81(1), 130–139 (2014).
[Crossref] [PubMed]

G. D. Field, J. L. Gauthier, A. Sher, M. Greschner, T. A. Machado, L. H. Jepson, J. Shlens, D. E. Gunning, K. Mathieson, W. Dabrowski, L. Paninski, A. M. Litke, and E. J. Chichilnisky, “Functional connectivity in the retina at the resolution of photoreceptors,” Nature 467(7316), 673–677 (2010).
[Crossref] [PubMed]

Fischer, L.

Fisher, S. K.

R. H. Steinberg, S. K. Fisher, and D. H. Anderson, “Disc morphogenesis in vertebrate photoreceptors,” J. Comp. Neurol. 190(3), 501–518 (1980).
[Crossref] [PubMed]

Franssen, L.

T. J. T. P. van den Berg, L. Franssen, B. Kruijt, and J. E. Coppens, “History of ocular straylight measurement: A review,” Z. Med. Phys. 23(1), 6–20 (2013).
[Crossref] [PubMed]

L. Franssen, J. Tabernero, J. E. Coppens, and T. J. T. P. van den Berg, “Pupil size and retinal straylight in the normal eye,” Invest. Ophthalmol. Vis. Sci. 48(5), 2375–2382 (2007).
[Crossref] [PubMed]

Frederiksen, R.

M. B. Toomey, P. Olsson, R. Frederiksen, M. C. Cornwall, R. Oulton, A. Kelber, J. C. Corbo, N. W. Roberts, and D. Wilby, “Optics of cone photoreceptors in the chicken (Gallus gallus domesticus),” J. R. Soc. Interface 12(111), 20150591 (2015).
[Crossref] [PubMed]

Fujimoto, J. G.

V. J. Srinivasan, B. K. Monson, M. Wojtkowski, R. A. Bilonick, I. Gorczynska, R. Chen, J. S. Duker, J. S. Schuman, and J. G. Fujimoto, “Characterization of outer retinal morphology with high-speed, ultrahigh-resolution optical coherence tomography,” Invest. Ophthalmol. Vis. Sci. 49(4), 1571–1579 (2008).
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Gauthier, J. L.

G. D. Field, J. L. Gauthier, A. Sher, M. Greschner, T. A. Machado, L. H. Jepson, J. Shlens, D. E. Gunning, K. Mathieson, W. Dabrowski, L. Paninski, A. M. Litke, and E. J. Chichilnisky, “Functional connectivity in the retina at the resolution of photoreceptors,” Nature 467(7316), 673–677 (2010).
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Gloge, D.

Godara, P.

T. Zhang, P. Godara, E. R. Blanco, R. L. Griffin, X. Wang, C. A. Curcio, and Y. Zhang, “Variability in human cone topography assessed by adaptive optics scanning laser ophthalmoscopy,” Am. J. Ophthalmol. 160(2), 290–300 (2015).
[Crossref] [PubMed]

Gorczynska, I.

V. J. Srinivasan, B. K. Monson, M. Wojtkowski, R. A. Bilonick, I. Gorczynska, R. Chen, J. S. Duker, J. S. Schuman, and J. G. Fujimoto, “Characterization of outer retinal morphology with high-speed, ultrahigh-resolution optical coherence tomography,” Invest. Ophthalmol. Vis. Sci. 49(4), 1571–1579 (2008).
[Crossref] [PubMed]

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J.-M. Gorrand and F. C. Delori, “A model for the assessment of cone directionality,” J. Mod. Opt. 44(3), 473–491 (1997).
[Crossref]

Greschner, M.

P. H. Li, G. D. Field, M. Greschner, D. Ahn, D. E. Gunning, K. Mathieson, A. Sher, A. M. Litke, and E. J. Chichilnisky, “Retinal representation of the elementary visual signal,” Neuron 81(1), 130–139 (2014).
[Crossref] [PubMed]

G. D. Field, J. L. Gauthier, A. Sher, M. Greschner, T. A. Machado, L. H. Jepson, J. Shlens, D. E. Gunning, K. Mathieson, W. Dabrowski, L. Paninski, A. M. Litke, and E. J. Chichilnisky, “Functional connectivity in the retina at the resolution of photoreceptors,” Nature 467(7316), 673–677 (2010).
[Crossref] [PubMed]

Griffin, R. L.

T. Zhang, P. Godara, E. R. Blanco, R. L. Griffin, X. Wang, C. A. Curcio, and Y. Zhang, “Variability in human cone topography assessed by adaptive optics scanning laser ophthalmoscopy,” Am. J. Ophthalmol. 160(2), 290–300 (2015).
[Crossref] [PubMed]

Gunning, D. E.

P. H. Li, G. D. Field, M. Greschner, D. Ahn, D. E. Gunning, K. Mathieson, A. Sher, A. M. Litke, and E. J. Chichilnisky, “Retinal representation of the elementary visual signal,” Neuron 81(1), 130–139 (2014).
[Crossref] [PubMed]

G. D. Field, J. L. Gauthier, A. Sher, M. Greschner, T. A. Machado, L. H. Jepson, J. Shlens, D. E. Gunning, K. Mathieson, W. Dabrowski, L. Paninski, A. M. Litke, and E. J. Chichilnisky, “Functional connectivity in the retina at the resolution of photoreceptors,” Nature 467(7316), 673–677 (2010).
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A. W. Snyder and M. Hamer, “The light-capture area of a photoreceptor,” Vision Res. 12(10), 1749–1753 (1972).
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K. S. Bruce, W. M. Harmening, B. R. Langston, W. S. Tuten, A. Roorda, and L. C. Sincich, “Normal perceptual sensitivity arising from weakly reflective cone photoreceptors,” Invest. Ophthalmol. Vis. Sci. 56(8), 4431–4438 (2015).
[Crossref] [PubMed]

W. M. Harmening, W. S. Tuten, A. Roorda, and L. C. Sincich, “Mapping the perceptual grain of the human retina,” J. Neurosci. 34(16), 5667–5677 (2014).
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W. M. Harmening, P. Tiruveedhula, A. Roorda, and L. C. Sincich, “Measurement and correction of transverse chromatic offsets for multi-wavelength retinal microscopy in the living eye,” Biomed. Opt. Express 3(9), 2066–2077 (2012).
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Hendrickson, A. E.

C. A. Curcio, K. R. Sloan, R. E. Kalina, and A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292(4), 497–523 (1990).
[Crossref] [PubMed]

C. A. Curcio, K. R. Sloan, O. Packer, A. E. Hendrickson, and R. E. Kalina, “Distribution of cones in human and monkey retina: individual variability and radial asymmetry,” Science 236(4801), 579–582 (1987).
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Hirsch, J.

J. Hirsch and R. Hylton, “Quality of the primate photoreceptor lattice,” Vision Res. 24(4), 347–355 (1984).
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Hoang, Q. V.

Q. V. Hoang, R. A. Linsenmeier, C. K. Chung, and C. A. Curcio, “Photoreceptor inner segments in monkey and human retina: mitochondrial density, optics, and regional variation,” Vis. Neurosci. 19(4), 395–407 (2002).
[Crossref] [PubMed]

Holland, J.

Q. Wang, W. S. Tuten, B. J. Lujan, J. Holland, P. S. Bernstein, S. D. Schwartz, J. L. Duncan, and A. Roorda, “Adaptive optics microperimetry and OCT images show preserved function and recovery of cone visibility in macular telangiectasia type 2 retinal lesions,” Invest. Ophthalmol. Vis. Sci. 56(2), 778–786 (2015).
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L. C. Sincich, Y. Zhang, P. Tiruveedhula, J. C. Horton, and A. Roorda, “Resolving single cone inputs to visual receptive fields,” Nat. Neurosci. 12(8), 967–969 (2009).
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W. P. Huang, C. L. Xu, and S. K. Chaudhuri, “A finite-difference vector beam propagation method for three-dimensional waveguide structures,” IEEE Photonics Technol. Lett. 4(2), 148–151 (1992).
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W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, “A vector beam propagation method for guided-wave optics,” IEEE Photonics Technol. Lett. 3(10), 910–913 (1991).
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Hylton, R.

J. Hirsch and R. Hylton, “Quality of the primate photoreceptor lattice,” Vision Res. 24(4), 347–355 (1984).
[Crossref] [PubMed]

Iglesias, I.

Jepson, L. H.

G. D. Field, J. L. Gauthier, A. Sher, M. Greschner, T. A. Machado, L. H. Jepson, J. Shlens, D. E. Gunning, K. Mathieson, W. Dabrowski, L. Paninski, A. M. Litke, and E. J. Chichilnisky, “Functional connectivity in the retina at the resolution of photoreceptors,” Nature 467(7316), 673–677 (2010).
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A. M. Pozo, F. Pérez-Ocón, and J. R. Jiménez, “FDTD analysis of the light propagation in the cones of the human retina: an approach to the Stiles-Crawford effect of the first kind,” J. Opt. A, Pure Appl. Opt. 7(8), 357–363 (2005).
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Kalina, R. E.

C. A. Curcio, K. R. Sloan, R. E. Kalina, and A. E. Hendrickson, “Human photoreceptor topography,” J. Comp. Neurol. 292(4), 497–523 (1990).
[Crossref] [PubMed]

C. A. Curcio, K. R. Sloan, O. Packer, A. E. Hendrickson, and R. E. Kalina, “Distribution of cones in human and monkey retina: individual variability and radial asymmetry,” Science 236(4801), 579–582 (1987).
[Crossref] [PubMed]

Kelber, A.

M. B. Toomey, P. Olsson, R. Frederiksen, M. C. Cornwall, R. Oulton, A. Kelber, J. C. Corbo, N. W. Roberts, and D. Wilby, “Optics of cone photoreceptors in the chicken (Gallus gallus domesticus),” J. R. Soc. Interface 12(111), 20150591 (2015).
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Kocaoglu, O. P.

Kolb, H.

B. B. Boycott, J. E. Dowling, and H. Kolb, “Organization of the primate retina: light microscopy,” Proc. R. Soc. London, Ser. B, Biol. Sci. 255(799), 109– 184 (1969).

Kruijt, B.

T. J. T. P. van den Berg, L. Franssen, B. Kruijt, and J. E. Coppens, “History of ocular straylight measurement: A review,” Z. Med. Phys. 23(1), 6–20 (2013).
[Crossref] [PubMed]

Langston, B. R.

K. S. Bruce, W. M. Harmening, B. R. Langston, W. S. Tuten, A. Roorda, and L. C. Sincich, “Normal perceptual sensitivity arising from weakly reflective cone photoreceptors,” Invest. Ophthalmol. Vis. Sci. 56(8), 4431–4438 (2015).
[Crossref] [PubMed]

Li, P. H.

P. H. Li, G. D. Field, M. Greschner, D. Ahn, D. E. Gunning, K. Mathieson, A. Sher, A. M. Litke, and E. J. Chichilnisky, “Retinal representation of the elementary visual signal,” Neuron 81(1), 130–139 (2014).
[Crossref] [PubMed]

Linsenmeier, R. A.

Q. V. Hoang, R. A. Linsenmeier, C. K. Chung, and C. A. Curcio, “Photoreceptor inner segments in monkey and human retina: mitochondrial density, optics, and regional variation,” Vis. Neurosci. 19(4), 395–407 (2002).
[Crossref] [PubMed]

Litke, A. M.

P. H. Li, G. D. Field, M. Greschner, D. Ahn, D. E. Gunning, K. Mathieson, A. Sher, A. M. Litke, and E. J. Chichilnisky, “Retinal representation of the elementary visual signal,” Neuron 81(1), 130–139 (2014).
[Crossref] [PubMed]

G. D. Field, J. L. Gauthier, A. Sher, M. Greschner, T. A. Machado, L. H. Jepson, J. Shlens, D. E. Gunning, K. Mathieson, W. Dabrowski, L. Paninski, A. M. Litke, and E. J. Chichilnisky, “Functional connectivity in the retina at the resolution of photoreceptors,” Nature 467(7316), 673–677 (2010).
[Crossref] [PubMed]

Liu, X.

J. Xiao, X. Liu, C. Cai, H. Fan, and X. Sun, “An improved three-dimensional full-vectorial finite-difference imaginary-distance beam propagation method,” Sci. China, Ser. F Inf. Sci. 49(4), 516–532 (2006).

Liu, Z.

Love, J. D.

J. A. Besley and J. D. Love, “Supermode analysis of fibre transmission,” IEE Proc., Optoelectron. 144(6), 411–419 (1997).
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W. M. Harmening, P. Tiruveedhula, A. Roorda, and L. C. Sincich, “Measurement and correction of transverse chromatic offsets for multi-wavelength retinal microscopy in the living eye,” Biomed. Opt. Express 3(9), 2066–2077 (2012).
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L. C. Sincich, Y. Zhang, P. Tiruveedhula, J. C. Horton, and A. Roorda, “Resolving single cone inputs to visual receptive fields,” Nat. Neurosci. 12(8), 967–969 (2009).
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D. A. Baylor, B. J. Nunn, and J. L. Schnapf, “The photocurrent, noise and spectral sensitivity of rods of the monkey Macaca fascicularis,” J. Physiol. 357(1), 575–607 (1984).
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V. J. Srinivasan, B. K. Monson, M. Wojtkowski, R. A. Bilonick, I. Gorczynska, R. Chen, J. S. Duker, J. S. Schuman, and J. G. Fujimoto, “Characterization of outer retinal morphology with high-speed, ultrahigh-resolution optical coherence tomography,” Invest. Ophthalmol. Vis. Sci. 49(4), 1571–1579 (2008).
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B. Vohnsen, A. Carmichael, N. Sharmin, S. Qaysi, and D. Valente, “Volumetric integration model of the Stiles-Crawford effect of the first kind and its experimental verification,” J. Vis. 17(12), 18 (2017).
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P. H. Li, G. D. Field, M. Greschner, D. Ahn, D. E. Gunning, K. Mathieson, A. Sher, A. M. Litke, and E. J. Chichilnisky, “Retinal representation of the elementary visual signal,” Neuron 81(1), 130–139 (2014).
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L. C. Sincich, Y. Zhang, P. Tiruveedhula, J. C. Horton, and A. Roorda, “Resolving single cone inputs to visual receptive fields,” Nat. Neurosci. 12(8), 967–969 (2009).
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L. Franssen, J. Tabernero, J. E. Coppens, and T. J. T. P. van den Berg, “Pupil size and retinal straylight in the normal eye,” Invest. Ophthalmol. Vis. Sci. 48(5), 2375–2382 (2007).
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Tiruveedhula, P.

W. M. Harmening, P. Tiruveedhula, A. Roorda, and L. C. Sincich, “Measurement and correction of transverse chromatic offsets for multi-wavelength retinal microscopy in the living eye,” Biomed. Opt. Express 3(9), 2066–2077 (2012).
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L. C. Sincich, Y. Zhang, P. Tiruveedhula, J. C. Horton, and A. Roorda, “Resolving single cone inputs to visual receptive fields,” Nat. Neurosci. 12(8), 967–969 (2009).
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M. B. Toomey, P. Olsson, R. Frederiksen, M. C. Cornwall, R. Oulton, A. Kelber, J. C. Corbo, N. W. Roberts, and D. Wilby, “Optics of cone photoreceptors in the chicken (Gallus gallus domesticus),” J. R. Soc. Interface 12(111), 20150591 (2015).
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Turner, T. L.

Tuten, W. S.

K. S. Bruce, W. M. Harmening, B. R. Langston, W. S. Tuten, A. Roorda, and L. C. Sincich, “Normal perceptual sensitivity arising from weakly reflective cone photoreceptors,” Invest. Ophthalmol. Vis. Sci. 56(8), 4431–4438 (2015).
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B. Vohnsen, A. Carmichael, N. Sharmin, S. Qaysi, and D. Valente, “Volumetric integration model of the Stiles-Crawford effect of the first kind and its experimental verification,” J. Vis. 17(12), 18 (2017).
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Vogel, C. R.

Q. Yang, D. W. Arathorn, P. Tiruveedhula, C. R. Vogel, and A. Roorda, “Design of an integrated hardware interface for AOSLO image capture and cone-targeted stimulus delivery,” Opt. Express 18(17), 3283–3291 (2010).

D. W. Arathorn, Q. Yang, C. R. Vogel, Y. Zhang, P. Tiruveedhula, and A. Roorda, “Retinally stabilized cone-targeted stimulus delivery,” Opt. Express 15(21), 13731–13744 (2007).
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Q. Wang, W. S. Tuten, B. J. Lujan, J. Holland, P. S. Bernstein, S. D. Schwartz, J. L. Duncan, and A. Roorda, “Adaptive optics microperimetry and OCT images show preserved function and recovery of cone visibility in macular telangiectasia type 2 retinal lesions,” Invest. Ophthalmol. Vis. Sci. 56(2), 778–786 (2015).
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M. B. Toomey, P. Olsson, R. Frederiksen, M. C. Cornwall, R. Oulton, A. Kelber, J. C. Corbo, N. W. Roberts, and D. Wilby, “Optics of cone photoreceptors in the chicken (Gallus gallus domesticus),” J. R. Soc. Interface 12(111), 20150591 (2015).
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J. Xiao, X. Liu, C. Cai, H. Fan, and X. Sun, “An improved three-dimensional full-vectorial finite-difference imaginary-distance beam propagation method,” Sci. China, Ser. F Inf. Sci. 49(4), 516–532 (2006).

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Q. Yang, D. W. Arathorn, P. Tiruveedhula, C. R. Vogel, and A. Roorda, “Design of an integrated hardware interface for AOSLO image capture and cone-targeted stimulus delivery,” Opt. Express 18(17), 3283–3291 (2010).

D. W. Arathorn, Q. Yang, C. R. Vogel, Y. Zhang, P. Tiruveedhula, and A. Roorda, “Retinally stabilized cone-targeted stimulus delivery,” Opt. Express 15(21), 13731–13744 (2007).
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Zhong, Z.

H. Song, T. Y. P. Chui, Z. Zhong, A. E. Elsner, and S. A. Burns, “Variation of cone photoreceptor packing density with retinal eccentricity and age,” Invest. Ophthalmol. Vis. Sci. 52(10), 7376–7384 (2011).
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Am. J. Ophthalmol. (1)

T. Zhang, P. Godara, E. R. Blanco, R. L. Griffin, X. Wang, C. A. Curcio, and Y. Zhang, “Variability in human cone topography assessed by adaptive optics scanning laser ophthalmoscopy,” Am. J. Ophthalmol. 160(2), 290–300 (2015).
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Biomed. Opt. Express (3)

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W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, “A vector beam propagation method for guided-wave optics,” IEEE Photonics Technol. Lett. 3(10), 910–913 (1991).
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Figures (9)

Fig. 1
Fig. 1 Refractive index traits of the cone photoreceptor model. The photoreceptor shape, defined by Eq. (31), is shown in 2-D and 3-D representations. Color code depicts the different refractive indices (n) of the compartments: red = myoid, green = ellipsoid, blue = outer segment and white = surrounding media. Dimension variables are d0 = 5 μm, dos = 1.75 μm, lm = 13 μm, le = 16 μm, and los = 35 μm.
Fig. 2
Fig. 2 Shape of the initial electric field ψ0. Modulus of the electric field PSF calculated by Fourier transform of the pupil using a uniform beam profile (A) or with a truncated Gaussian beam (B). (C) Field profiles at y = 0 through the uniform (black line) and truncated Gaussian pupil fields (red line). For both, λ = 543 nm and D = 5.85 mm. The Gaussian function of the truncated field had a 1/e width of 4.75 mm (1/e2 width for the intensity). Both fields are normalized to their peak value. The intensity full-width half-maxima were 1.6 µm for the uniform beam and 2.0 µm for the truncated Gaussian beam.
Fig. 3
Fig. 3 Examples of light propagation in cone photoreceptors. Propagation in water (n = 1.33) using a uniform field at the pupil (A) or with a truncated Gaussian (B). When light is incident centrally on a photoreceptor (C) beating of the field is evident in the outer segment. (D) Propagation when beam lands between photoreceptors (shifted 0.61d0). (E) Diminished light intensity occurs when the beam size is set to 2.0 mm. (F) Propagation down larger photoreceptors (d0 = 5 μm) leads to the largest peak intensity in these examples. (G) Propagation with light entering at an angle of 7.5°. (H) Example with a Zernike coefficient corresponding to coma of 0.1 μm. Modulus of the electric field for the initial fields (right) and the fields in the x-z plane for x-polarized light (left) are normalized to the peak electric field, which occurred in panel F. Light in the negative z direction was calculated by propagating the complex conjugate of ψ0 through a refractive index of 1.33. For all examples, unless stated, beam diameter = 5.85 mm, 1/e width of Gaussian = 4.75 mm, λ = 543 nm, d0 = 4 μm, dos = 1.75 μm, dx = dy = 0.1 μm, dz = 0.2 μm. Values for refractive index and lengths of segments are given in the Methods.
Fig. 4
Fig. 4 Power propagated and absorbed along photoreceptor length. (A) Fraction of power present along length of central photoreceptor. (B) Fraction of power absorbed in outer segment of central photoreceptor. Values calculated from the propagation conditions in Fig. 3(C)-3(H), shown color coded here.
Fig. 5
Fig. 5 Power absorbed by cones for d0 = 4 μm (A-C) and 6 μm (D-F) using a square stimulus. Absorption is based on the initial field having unit power. Grayscale values are interpolated (bicubic) from data computed at intervals of 0.25 mm for beam size and 0.9 μm for stimulus size. Absorbed power contours are in steps of 0.01 (thin lines) and 0.05 (thick lines).
Fig. 6
Fig. 6 Fraction of total power absorbed by the central cone for d0 = 4 µm (A) and 6 µm (B). Absorbed power contours indicated on color bar.
Fig. 7
Fig. 7 Optimum beam size for most efficient light absorption in all 7 cones, for a range of cone sizes. Solid line follows the peak power absorption for a given cone size with a point stimulus. A secondary peak exists (dashed line) because of absorption in surrounding cones.
Fig. 8
Fig. 8 Effect of beam and stimulus size on light capture. (A) Retinal site tested for increment thresholds with 7.92 μm stimulus outlined over targeted cone at 3° eccentricity in Subject 1. (B) Psychometric data and fits for two different beam sizes. Threshold is defined at the power level yielding a 0.5 probability of detection (dashed lines). Data points are scaled to the number of successfully delivered trials at each intensity (out of 20). (C) Map of predicted change in threshold ratio from a 5.85 mm pupil for a square stimulus. Superimposed on the map are the experiment conditions illustrated as red→green dot pairs, representing a shift from large to small beam sizes. The top red-green pair is for the cone tested in panels A-B. (D) Data as in C, but with a circular stimulus. Color coded threshold ratios are interpolated (bicubic) from data computed at intervals of 0.2 mm for pupil size and 0.9 μm for stimulus size. Stimulus size is the edge length of a square or diameter of a circle. Contours have steps of 0.01 normalized units.
Fig. 9
Fig. 9 Gap/cone threshold ratio model of light absorption. (A) Schematic of stimulus conditions from [24], with light either targeted on a cone or in a gap. (B) Modeled fraction of light absorbed when a 3 × 3 pixel stimulus is centered on a cone at varying eccentricities and defocus. (C) As in B, when the same stimulus is placed in a gap between cones. (D) Gap/cone threshold ratio as a function of eccentricity and defocus. (E) Comparison of modeled versus observed ratios (circles from Fig. 5(C) in [24], red line shows linear fit). Threshold ratios for different defocus values (gray lines) showed that the best fit occurred at −0.026 D.

Tables (2)

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Table 1 Power absorbed in outer segments for propagations in Fig. 3.

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Table 2 Measured vs modeled results for change in threshold from a 5.85 mm pupil

Equations (32)

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2i n 0 k 0 z [ ψ x ψ y ]=[ H xx H xy H yx H yy ][ ψ x ψ y ],
H xx ψ x x ( 1 n 2 x ( n 2 ψ x ) )+ 2 ψ x y 2 + k 0 2 ( n 2 n 0 2 ) ψ x
H yy ψ y 2 ψ y x 2 + y ( 1 n 2 y ( n 2 ψ y ) )+ k 0 2 ( n 2 n 0 2 ) ψ y
2i n 0 k 0 z [ ψ x ψ y ]=[ A x + A y 0 0 B x + B y ][ ψ x ψ y ].
A x ψ x x ( 1 n 2 x ( n 2 ψ x ) )+ 1 2 k 0 2 ( n 2 n 0 2 ) ψ x
A y ψ x 2 ψ x y 2 + 1 2 k 0 2 ( n 2 n 0 2 ) ψ x .
B x ψ y 2 ψ y x 2 + 1 2 k 0 2 ( n 2 n 0 2 ) ψ y
B y ψ y y ( 1 n 2 y ( n 2 ψ y ) )+ 1 2 k 0 2 ( n 2 n 0 2 ) ψ y .
2i n 0 k 0 u m+1 u m Δz =( A x + A y ) u m+1 + u m 2 ,
( 4i n 0 k 0 Δz A y ) u m+½ =( 4i n 0 k 0 Δz + A y ) u m
( 4i n 0 k 0 Δz A x ) u m+1 =( 4i n 0 k 0 Δz + A x ) u m+½ .
4i n 0 k 0 Δz u i,j m+½ u i,j1 m+½ 2 u i,j m+½ + u i,j+1 m+½ Δ y 2 1 2 k 0 2 [ ( n i,j m+½ ) 2 n 0 2 ] u i,j m+½ = 4i n 0 k 0 Δz u i,j m + u i,j1 m 2 u i,j m + u i,j+1 m Δ y 2 1 2 k 0 2 [ ( n i,j m ) 2 n 0 2 ] u i,j m
a j u i,j1 m+½ + b j u i,j m+½ + c j u i,j+1 m+½ = q j m ,
a j = 1 Δ y 2 , b j = 4i n 0 k 0 Δz + 2 Δ y 2 1 2 k 0 2 [ ( n i,j m+½ ) 2 n 0 2 ], c j = 1 Δ y 2 , q j = 1 Δ y 2 [ u i,j1 m + u i,j+1 m ]+{ 4i n 0 k 0 Δz 2 Δ y 2 + 1 2 k 0 2 [ ( n i,j m ) 2 n 0 2 ] } u i,j m
[ b 1 c 1 0 0 . . 0 0 0 a 2 b 2 c 2 0 . . 0 0 0 0 a 3 b 3 c 3 . . 0 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 0 . . a N1 b N1 c N1 0 0 0 0 . . 0 a N b N ][ u 1 m+½ u 2 m+½ u 3 m+½ . . u N1 m+½ u N m+½ ]=[ q 1 q 2 q 3 . . q N1 q N ]
M y u i,j m+½ = q y ,
4i n 0 k 0 Δz u i,j m+1 T i1,j m+1 u i1,j m+1 2 R i,j m+1 u i,j m+1 + T i+1,j m+1 u i+1,j m+1 Δ x 2 1 2 k 0 2 [ ( n i,j m+1 ) 2 n 0 2 ] u i,j m+1 = 4i n 0 k 0 Δz u i,j m+½ + T i1,j m+½ u i1,j m+½ 2 R i,j m+½ u i,j m+½ + T i+1,j m+½ u i+1,j m+½ Δ x 2 1 2 k 0 2 [ ( n i,j m ) 2 n 0 2 ] u i,j m+½
T i±1,j 2 n i±1,j 2 n i+1,j 2 + n i,j 2
R i,j 2 T i+1,j + T i1,j 2 .
M x u i,j m+1 = q x ,
a i = 1 Δ x 2 T i1,j m+1 , b i = 4i n 0 k 0 Δz + 2 Δ x 2 R i,j 1 2 k 0 2 [ ( n i,j m+1 ) 2 n 0 2 ], c i = 1 Δ x 2 T i+1,j m+1 , q i = 1 Δ x 2 [ T i1,j m+½ u i,j1 m+½ + T i+1,j m+½ u i,j+1 m+½ ]+{ 4i n 0 k 0 Δz 2 R i,j m+1/2 Δ x 2 + 1 2 k 0 2 [ ( n i,j m+1/2 ) 2 n 0 2 ] } u i,j m+1/2 .
M y v i,j m+½ = q y ,
a j = 1 Δ y 2 T i,j1 m+½ , b j = 4i n 0 k 0 Δz + 2 Δ y 2 R i,j m+½ 1 2 k 0 2 [ ( n i,j m+½ ) 2 n 0 2 ], c j = 1 Δ y 2 T i,j+1 m+½ , q j = 1 Δ y 2 [ T i,j1 m v i,j1 m + T i,j1 m v i,j+1 m ]+{ 4i n 0 k 0 Δz 2 R i,j m Δ y 2 + 1 2 k 0 2 [ ( n i,j m ) 2 n 0 2 ] } v i,j m ,
T i,j±1 2 n i,j±1 2 n i,j 2 + n i,j±1 2
R i,j 2 T i,j+1 + T i,j1 2 .
M x v i,j m = q x ,
a i = 1 Δ x 2 , b i = 4i n 0 k 0 Δz + 2 Δ x 2 1 2 k 0 2 [ ( n i,j m+1 ) 2 n 0 2 ], c i = 1 Δ x 2 , q i = 1 Δ x 2 [ v i1,j m+½ + v i+1,j m+½ ]+{ 4i n 0 k 0 Δz 2 Δ x 2 + 1 2 k 0 2 [ ( n i,j m+½ ) 2 n 0 2 ] } v i,j m+½ .
P( z )= 0 r=d/2 | ψ( z ) | 2 dr ,
P abs ( z )= P real ( z ) P complex ( z ).
P total = 0 z=L P abs ( z ) ,
d( z )={ d os d 0 l is 2 z 2 + d 0 for 0z l is d os for z l is
d 0 =0.049 φ 2 +1.22φ+2.68

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