Abstract

In the extrafoveal retina, interference fringes at spatial frequencies higher than the resolution limit look like two-dimensional spatial noise, the origin of which has not been firmly established. We show that over a limited range of high spatial frequencies this noise takes on a striated appearance, with the striations running perpendicular to the true fringe orientation. A model of cone aliasing based on anatomical measurements of extrafoveal cone position predicts that this orientation reversal should occur when the period of the interference fringe roughly equals the spacing between cones, i.e., when the fringe spatial frequency is about twice the cone Nyquist frequency. Psychophysical measurements of the orientation reversal at retinal eccentricities from 0.75 to 10 deg are in quantitative agreement with this prediction. This agreement implies that at least part of the spatial noise observed under these conditions results from aliasing by the cone mosaic. The orientation reversal provides a psychophysical method for estimating spacing in less regular mosaics, complementing another psychophysical technique for measuring spacing in the more regular mosaic of foveal cones [D. R. Williams, Vision Res. 25, 195 (1985); Vision Res. (submitted)].

© 1987 Optical Society of America

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  1. B. Borwein, C. Borwein, J. Medeiros, and J. W. McGowan, "The ultrastructure of monkey foveal photoreceptors, with special reference to the structure, shape, size and spacing of the foveal cones," Am. J. Anat. 159, 125–146 (1980).
  2. J. Hirsch and R. Hylton, "Quality of the primate photoreceptor lattice and limits of spatial vision," Vision Res. 24, 347–356 (1984).
  3. W. H. Miller, "Ocular optical filtering," in Handbook of Sensory Physiology, H. Autrum, ed. (Springer-Verlag, Berlin, 1979), pp. 70–143.
  4. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978).
  5. D. R. Williams, "Aliasing in human foveal vision," Vision Res. 25, 195–205 (1985).
  6. G. M. Byram, "The physical and photochemical basis of visual resolving power. Part II. Visual acuity and the photochemistry of the retina," J. Opt. Soc. Am. 34, 718–738 (1944).
  7. F. W. Campbell and D. G. Green, "Optical and retinal factors affecting visual resolution," J. Physiol. 181, 576–593 (1965).
  8. D. R. Williams, "Topography of the foveal cone mosaic in the living human eye," submitted to Vision Res.
  9. L. N. Thibos and D. J. Walsh, "Detection of high-frequency gratings in the periphery," J. Opt. Soc. Am. A 2(13), P64 (1985).
  10. L. N. Thibos, D. J. Walsh, and F. E. Cheney, "Vision beyond the resolution limit: aliasing in the periphery," submitted to Vision Res.
  11. R. A. Smith and P. F. Cass, "Aliasing in the parafovea with incoherent light," J. Opt. Soc. Am. A 4, 1530–1534 (1987).
  12. J. Hirsch and W. H. Miller, "Irregularity of foveal cone lattice increases with eccentricity," Invest. Ophthalmol. Vis. Sci. Suppl. 26, 10 (1985).
  13. J. Hirsch and W. H. Miller, "Does cone positional disorder limit resolution?" J. Opt. Soc. Am. A 4, 1481–1492 (1987).
  14. V. H. Perry and A. Cowey, "The ganglion cell and cone distributions in the monkey's retina: implications for central magnification factors," Vison Res. 25, 1795–1810 (1985).
  15. H. Wassle and H. J. Riemann, "The mosaic of nerve cells in the mammalian retina," Proc. R. Soc. London Ser. B 200, 441–461 (1978).
  16. J. I. Yellott, "Spectral analysis of spatial sampling by photoreceptors: topological disorder prevents aliasing," Vision Res. 22, 1205–1210 (1982).
  17. J. I. Yellott, "Spectral consequences of photoreceptor sampling in the rhesus retina," Science 221, 382–385 (1983).
  18. D. R. Williams and R. J. Collier, "Consequences of spatial sampling by a human photoreceptor mosaic," Science 221, 385–387 (1983).
  19. D. R. Williams and N. J. Coletta, "Cone spacing and the visual resolution limit," J. Opt. Soc. Am. A 4, 1514–1523 (1987).
  20. This observer did not have so strong an asymmetry between horizontal and vertical fringes as did the observer whose data are shown in Fig. 1, making it easier to select a single spatial frequency.
  21. C. A. Taylor, Images, Wykeham Science Series (Wykeham, London, 1978).
  22. J. G. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  23. We measured the modal frequency of the cone mosaic from a photometric scan of the transform image. This measured frequency and the frequency calculated from the cone density, assuming regular triangular packing, differed by less than 4%; this difference is within our measurement error. Therefore the assumption of triangular packing provides a good approximation of the modal frequency of the irregular parafoveal mosaic.
  24. In order to clarify the flanking annuli for illustrative purposes, the photographs in Figs. 5C and 5D were obtained by translating the diffraction pattern of the mosaic alone across the film plane. In the actual spectrum, the two side rings (first-order functions) have half of the intensity of the central one (a zero-order function) and thus would appear dimmer in the figure. However, the reconstructed images were obtained from the actual diffraction pattern, not from this simulated version.
  25. G. østerberg, "Topography of the layer of rods and cones in the human retina," Acta Ophthalmol. Suppl. 6, 11–103 (1935).
  26. The mean orientation-reversal frequency did not critically depend upon the target size, however. At 1.25 deg of eccentricity in the nasal retina, the reversal frequency was measured on one observer for three different target sizes, 20 min, 40 min, and 1.25 deg. These test fields produced reversal frequencies of 64, 62, and 63 cycles/deg, respectively. Thus scaling the target size by a constant number of cones was merely convenient and was not by itself responsible for the good agreement between our data and the anatomical estimates of cone spacing.
  27. The anatomical data of østerberg and Curcio et al. are provided in terms of density (cones per square millimeter). In order to transform these data into cone spacing in minutes of arc, the following assumptions were made. First, we assumed a factor of 0.291 mm/deg to convert the data from retinal distances to angular measures. Second, we assumed that the cones were arranged in a crystalline triangular mosaic, for which the spacing between rows of cones, r (minutes of arc), can be calculated according to the following formula: r = (60/0.291) (√3/2D)½, where D equals the cone density in cones per square millimeter. This formula should be considered a close approximation only, since we lack an analytic expression relating spacing and density for mosaics with the irregularity of the extrafovea. However, we have empirically confirmed that the assumption of perfect regularity yields an estimate of spacing that differs less than 4% from the modal cone spacing calculated from the power spectrum of the mosaic.
  28. C. A. Curcio, K. R. Sloan, Jr., O. Packer, A. E. Hendrickson, and R. E. Kalina, "Distribution of cones in human and monkey retina: individual variability and radial asymmetry," Science 236, 579–582 (1987).
  29. L. Maffei and F. W. Campbell, "Neurophysiological localization of the vertical and horizontal, visual coordinates in man," Science 167, 386–387 (1970).
  30. A. J. Ahumada, Jr., and A. Poirson, "Cone sampling array models," J. Opt. Soc. Am. A 4, 1493–1502 (1987).
  31. Our main interest in adding a spatial filter after the initial cone sampling was to determine whether the low-spatial-frequency energy was responsible for the perceived striations. We used a circular aperture, or window, for simplicity; however, some of our psychophysical observations might be modeled by an asymmetric spatial filter. For example, the oblique effect in the resolution limit is consistent with a cross-shaped window. If the cross is oriented with the arms vertical and horizontal, the window would pass energy at all orientations at low spatial frequencies but would pass only energy along the horizontal and vertical meridians at high spatial frequencies. This might explain why the orientation of an aliased oblique grating was more difficult to judge than that of a vertical or horizontal grating (Fig. 2). For oblique gratings, the aliased energy would be more isotropic, lacking a predominant orientation. At spatial frequencies near the limits of the reversal range, observers occasionally saw a plaid pattern. At these frequencies, presumably, there is aliased energy present in both the vertical and horizontal directions. This could result from the sampling operation if the annuli in the power spectrum overlap in such a way that they surround the central delta function (imagine sliding the outer rings in Fig. 5C closer together). A cross-shaped window would then filter out the oblique components of the aliased energy, leaving only the vertical and horizontal components. In some cases we also found unequal resolution limits for vertical and horizontal gratings. There was a tendency for the reversal frequency range to be smaller for the grating orientation with the higher resolution limit. This is consistent with a window that is elongated in one of the primary meridians (i.e., the one corresponding to the higher resolution limit). These and other orientational asymmetries will be more rigorously addressed in a forthcoming study.

1987 (5)

1985 (4)

D. R. Williams, "Aliasing in human foveal vision," Vision Res. 25, 195–205 (1985).

L. N. Thibos and D. J. Walsh, "Detection of high-frequency gratings in the periphery," J. Opt. Soc. Am. A 2(13), P64 (1985).

J. Hirsch and W. H. Miller, "Irregularity of foveal cone lattice increases with eccentricity," Invest. Ophthalmol. Vis. Sci. Suppl. 26, 10 (1985).

V. H. Perry and A. Cowey, "The ganglion cell and cone distributions in the monkey's retina: implications for central magnification factors," Vison Res. 25, 1795–1810 (1985).

1984 (1)

J. Hirsch and R. Hylton, "Quality of the primate photoreceptor lattice and limits of spatial vision," Vision Res. 24, 347–356 (1984).

1983 (2)

J. I. Yellott, "Spectral consequences of photoreceptor sampling in the rhesus retina," Science 221, 382–385 (1983).

D. R. Williams and R. J. Collier, "Consequences of spatial sampling by a human photoreceptor mosaic," Science 221, 385–387 (1983).

1982 (1)

J. I. Yellott, "Spectral analysis of spatial sampling by photoreceptors: topological disorder prevents aliasing," Vision Res. 22, 1205–1210 (1982).

1980 (1)

B. Borwein, C. Borwein, J. Medeiros, and J. W. McGowan, "The ultrastructure of monkey foveal photoreceptors, with special reference to the structure, shape, size and spacing of the foveal cones," Am. J. Anat. 159, 125–146 (1980).

1978 (1)

H. Wassle and H. J. Riemann, "The mosaic of nerve cells in the mammalian retina," Proc. R. Soc. London Ser. B 200, 441–461 (1978).

1970 (1)

L. Maffei and F. W. Campbell, "Neurophysiological localization of the vertical and horizontal, visual coordinates in man," Science 167, 386–387 (1970).

1965 (1)

F. W. Campbell and D. G. Green, "Optical and retinal factors affecting visual resolution," J. Physiol. 181, 576–593 (1965).

1944 (1)

1935 (1)

G. østerberg, "Topography of the layer of rods and cones in the human retina," Acta Ophthalmol. Suppl. 6, 11–103 (1935).

Ahumada, Jr., A. J.

Borwein, B.

B. Borwein, C. Borwein, J. Medeiros, and J. W. McGowan, "The ultrastructure of monkey foveal photoreceptors, with special reference to the structure, shape, size and spacing of the foveal cones," Am. J. Anat. 159, 125–146 (1980).

Borwein, C.

B. Borwein, C. Borwein, J. Medeiros, and J. W. McGowan, "The ultrastructure of monkey foveal photoreceptors, with special reference to the structure, shape, size and spacing of the foveal cones," Am. J. Anat. 159, 125–146 (1980).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978).

Byram, G. M.

Campbell, F. W.

L. Maffei and F. W. Campbell, "Neurophysiological localization of the vertical and horizontal, visual coordinates in man," Science 167, 386–387 (1970).

F. W. Campbell and D. G. Green, "Optical and retinal factors affecting visual resolution," J. Physiol. 181, 576–593 (1965).

Cass, P. F.

Cheney, F. E.

L. N. Thibos, D. J. Walsh, and F. E. Cheney, "Vision beyond the resolution limit: aliasing in the periphery," submitted to Vision Res.

Coletta, N. J.

Collier, R. J.

D. R. Williams and R. J. Collier, "Consequences of spatial sampling by a human photoreceptor mosaic," Science 221, 385–387 (1983).

Cowey, A.

V. H. Perry and A. Cowey, "The ganglion cell and cone distributions in the monkey's retina: implications for central magnification factors," Vison Res. 25, 1795–1810 (1985).

Curcio, C. A.

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

Goodman, J. G.

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

Green, D. G.

F. W. Campbell and D. G. Green, "Optical and retinal factors affecting visual resolution," J. Physiol. 181, 576–593 (1965).

Hendrickson, A. E.

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

Hirsch, J.

J. Hirsch and W. H. Miller, "Does cone positional disorder limit resolution?" J. Opt. Soc. Am. A 4, 1481–1492 (1987).

J. Hirsch and W. H. Miller, "Irregularity of foveal cone lattice increases with eccentricity," Invest. Ophthalmol. Vis. Sci. Suppl. 26, 10 (1985).

J. Hirsch and R. Hylton, "Quality of the primate photoreceptor lattice and limits of spatial vision," Vision Res. 24, 347–356 (1984).

Hylton, R.

J. Hirsch and R. Hylton, "Quality of the primate photoreceptor lattice and limits of spatial vision," Vision Res. 24, 347–356 (1984).

Kalina, R. E.

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

Maffei, L.

L. Maffei and F. W. Campbell, "Neurophysiological localization of the vertical and horizontal, visual coordinates in man," Science 167, 386–387 (1970).

McGowan, J. W.

B. Borwein, C. Borwein, J. Medeiros, and J. W. McGowan, "The ultrastructure of monkey foveal photoreceptors, with special reference to the structure, shape, size and spacing of the foveal cones," Am. J. Anat. 159, 125–146 (1980).

Medeiros, J.

B. Borwein, C. Borwein, J. Medeiros, and J. W. McGowan, "The ultrastructure of monkey foveal photoreceptors, with special reference to the structure, shape, size and spacing of the foveal cones," Am. J. Anat. 159, 125–146 (1980).

Miller, W. H.

J. Hirsch and W. H. Miller, "Does cone positional disorder limit resolution?" J. Opt. Soc. Am. A 4, 1481–1492 (1987).

J. Hirsch and W. H. Miller, "Irregularity of foveal cone lattice increases with eccentricity," Invest. Ophthalmol. Vis. Sci. Suppl. 26, 10 (1985).

W. H. Miller, "Ocular optical filtering," in Handbook of Sensory Physiology, H. Autrum, ed. (Springer-Verlag, Berlin, 1979), pp. 70–143.

østerberg, G.

G. østerberg, "Topography of the layer of rods and cones in the human retina," Acta Ophthalmol. Suppl. 6, 11–103 (1935).

Packer, O.

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

Perry, V. H.

V. H. Perry and A. Cowey, "The ganglion cell and cone distributions in the monkey's retina: implications for central magnification factors," Vison Res. 25, 1795–1810 (1985).

Poirson, A.

Riemann, H. J.

H. Wassle and H. J. Riemann, "The mosaic of nerve cells in the mammalian retina," Proc. R. Soc. London Ser. B 200, 441–461 (1978).

Sloan, Jr., K. R.

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

Smith, R. A.

Taylor, C. A.

C. A. Taylor, Images, Wykeham Science Series (Wykeham, London, 1978).

Thibos, L. N.

L. N. Thibos and D. J. Walsh, "Detection of high-frequency gratings in the periphery," J. Opt. Soc. Am. A 2(13), P64 (1985).

L. N. Thibos, D. J. Walsh, and F. E. Cheney, "Vision beyond the resolution limit: aliasing in the periphery," submitted to Vision Res.

Walsh, D. J.

L. N. Thibos and D. J. Walsh, "Detection of high-frequency gratings in the periphery," J. Opt. Soc. Am. A 2(13), P64 (1985).

L. N. Thibos, D. J. Walsh, and F. E. Cheney, "Vision beyond the resolution limit: aliasing in the periphery," submitted to Vision Res.

Wassle, H.

H. Wassle and H. J. Riemann, "The mosaic of nerve cells in the mammalian retina," Proc. R. Soc. London Ser. B 200, 441–461 (1978).

Williams, D. R.

D. R. Williams and N. J. Coletta, "Cone spacing and the visual resolution limit," J. Opt. Soc. Am. A 4, 1514–1523 (1987).

D. R. Williams, "Aliasing in human foveal vision," Vision Res. 25, 195–205 (1985).

D. R. Williams and R. J. Collier, "Consequences of spatial sampling by a human photoreceptor mosaic," Science 221, 385–387 (1983).

D. R. Williams, "Topography of the foveal cone mosaic in the living human eye," submitted to Vision Res.

Yellott, J. I.

J. I. Yellott, "Spectral consequences of photoreceptor sampling in the rhesus retina," Science 221, 382–385 (1983).

J. I. Yellott, "Spectral analysis of spatial sampling by photoreceptors: topological disorder prevents aliasing," Vision Res. 22, 1205–1210 (1982).

Acta Ophthalmol. Suppl. (1)

G. østerberg, "Topography of the layer of rods and cones in the human retina," Acta Ophthalmol. Suppl. 6, 11–103 (1935).

Am. J. Anat. (1)

B. Borwein, C. Borwein, J. Medeiros, and J. W. McGowan, "The ultrastructure of monkey foveal photoreceptors, with special reference to the structure, shape, size and spacing of the foveal cones," Am. J. Anat. 159, 125–146 (1980).

Invest. Ophthalmol. Vis. Sci. Suppl. (1)

J. Hirsch and W. H. Miller, "Irregularity of foveal cone lattice increases with eccentricity," Invest. Ophthalmol. Vis. Sci. Suppl. 26, 10 (1985).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

J. Physiol. (1)

F. W. Campbell and D. G. Green, "Optical and retinal factors affecting visual resolution," J. Physiol. 181, 576–593 (1965).

Proc. R. Soc. London Ser. B (1)

H. Wassle and H. J. Riemann, "The mosaic of nerve cells in the mammalian retina," Proc. R. Soc. London Ser. B 200, 441–461 (1978).

Science (4)

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

L. Maffei and F. W. Campbell, "Neurophysiological localization of the vertical and horizontal, visual coordinates in man," Science 167, 386–387 (1970).

J. I. Yellott, "Spectral consequences of photoreceptor sampling in the rhesus retina," Science 221, 382–385 (1983).

D. R. Williams and R. J. Collier, "Consequences of spatial sampling by a human photoreceptor mosaic," Science 221, 385–387 (1983).

Vision Res. (3)

D. R. Williams, "Aliasing in human foveal vision," Vision Res. 25, 195–205 (1985).

J. I. Yellott, "Spectral analysis of spatial sampling by photoreceptors: topological disorder prevents aliasing," Vision Res. 22, 1205–1210 (1982).

J. Hirsch and R. Hylton, "Quality of the primate photoreceptor lattice and limits of spatial vision," Vision Res. 24, 347–356 (1984).

Vison Res. (1)

V. H. Perry and A. Cowey, "The ganglion cell and cone distributions in the monkey's retina: implications for central magnification factors," Vison Res. 25, 1795–1810 (1985).

Other (12)

D. R. Williams, "Topography of the foveal cone mosaic in the living human eye," submitted to Vision Res.

The mean orientation-reversal frequency did not critically depend upon the target size, however. At 1.25 deg of eccentricity in the nasal retina, the reversal frequency was measured on one observer for three different target sizes, 20 min, 40 min, and 1.25 deg. These test fields produced reversal frequencies of 64, 62, and 63 cycles/deg, respectively. Thus scaling the target size by a constant number of cones was merely convenient and was not by itself responsible for the good agreement between our data and the anatomical estimates of cone spacing.

The anatomical data of østerberg and Curcio et al. are provided in terms of density (cones per square millimeter). In order to transform these data into cone spacing in minutes of arc, the following assumptions were made. First, we assumed a factor of 0.291 mm/deg to convert the data from retinal distances to angular measures. Second, we assumed that the cones were arranged in a crystalline triangular mosaic, for which the spacing between rows of cones, r (minutes of arc), can be calculated according to the following formula: r = (60/0.291) (√3/2D)½, where D equals the cone density in cones per square millimeter. This formula should be considered a close approximation only, since we lack an analytic expression relating spacing and density for mosaics with the irregularity of the extrafovea. However, we have empirically confirmed that the assumption of perfect regularity yields an estimate of spacing that differs less than 4% from the modal cone spacing calculated from the power spectrum of the mosaic.

W. H. Miller, "Ocular optical filtering," in Handbook of Sensory Physiology, H. Autrum, ed. (Springer-Verlag, Berlin, 1979), pp. 70–143.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978).

L. N. Thibos, D. J. Walsh, and F. E. Cheney, "Vision beyond the resolution limit: aliasing in the periphery," submitted to Vision Res.

This observer did not have so strong an asymmetry between horizontal and vertical fringes as did the observer whose data are shown in Fig. 1, making it easier to select a single spatial frequency.

C. A. Taylor, Images, Wykeham Science Series (Wykeham, London, 1978).

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

We measured the modal frequency of the cone mosaic from a photometric scan of the transform image. This measured frequency and the frequency calculated from the cone density, assuming regular triangular packing, differed by less than 4%; this difference is within our measurement error. Therefore the assumption of triangular packing provides a good approximation of the modal frequency of the irregular parafoveal mosaic.

In order to clarify the flanking annuli for illustrative purposes, the photographs in Figs. 5C and 5D were obtained by translating the diffraction pattern of the mosaic alone across the film plane. In the actual spectrum, the two side rings (first-order functions) have half of the intensity of the central one (a zero-order function) and thus would appear dimmer in the figure. However, the reconstructed images were obtained from the actual diffraction pattern, not from this simulated version.

Our main interest in adding a spatial filter after the initial cone sampling was to determine whether the low-spatial-frequency energy was responsible for the perceived striations. We used a circular aperture, or window, for simplicity; however, some of our psychophysical observations might be modeled by an asymmetric spatial filter. For example, the oblique effect in the resolution limit is consistent with a cross-shaped window. If the cross is oriented with the arms vertical and horizontal, the window would pass energy at all orientations at low spatial frequencies but would pass only energy along the horizontal and vertical meridians at high spatial frequencies. This might explain why the orientation of an aliased oblique grating was more difficult to judge than that of a vertical or horizontal grating (Fig. 2). For oblique gratings, the aliased energy would be more isotropic, lacking a predominant orientation. At spatial frequencies near the limits of the reversal range, observers occasionally saw a plaid pattern. At these frequencies, presumably, there is aliased energy present in both the vertical and horizontal directions. This could result from the sampling operation if the annuli in the power spectrum overlap in such a way that they surround the central delta function (imagine sliding the outer rings in Fig. 5C closer together). A cross-shaped window would then filter out the oblique components of the aliased energy, leaving only the vertical and horizontal components. In some cases we also found unequal resolution limits for vertical and horizontal gratings. There was a tendency for the reversal frequency range to be smaller for the grating orientation with the higher resolution limit. This is consistent with a window that is elongated in one of the primary meridians (i.e., the one corresponding to the higher resolution limit). These and other orientational asymmetries will be more rigorously addressed in a forthcoming study.

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