Abstract

By utilizing existing data on refractive indices of the ocular media I developed methods to convert refractive indices within the visible spectrum. The calculated values have approximately the same accuracy as measurements with standard methods. An independent test confirmed the validity of the formulas.

© 1992 Optical Society of America

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References

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  1. R. S. Longhurst, Geometrical and Physical Optics, 3rd ed. (Longman, London, 1973).
  2. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  3. Y. Le Grand, L’Espace Visuel, Vol. 3 of Optique Physiologique, Rev. Opt. Theor. Instrum. (1956).
  4. J. G. Sivak, T. Mandelman, “Chromatic dispersion of the ocular media,” Vision Res. 22, 997–1003 (1982).
    [CrossRef] [PubMed]
  5. R. K. Luneburg, Mathematical Theory of Optics (Brown U. Press, Providence, R.I., 1944).
  6. L. Matthiessen, “Über den physikalisch-optischen Bau des Auges der Cetaceen und der Fische,” Pfluegers Arch. 38, 521–528 (1886).
    [CrossRef]
  7. R. H. H. Kröger, “Dioptrik, Funktion der Pupille und Akkommodation bei Zahnwalen,” Ph.D. dissertation (Universität Tübingen, Tübingen, Germany, 1989).
  8. A. M. Mass, A. Y. Supin, A. N. Severtsov, “Topographic distribution of sizes and density of ganglion cells in the retina of the porpoise, Phocoena phocoena,” Aquat. Mammals 12, 95–102 (1986).
  9. G. Westheimer, “Optical properties of vertebrate eyes,” in Physiology of Photoreceptor Organs. Handbook of Sensory Physiology VII/2, M. G. F. Fuortes, ed. (Springer, Berlin, 1972).
    [CrossRef]
  10. G. Westheimer, “Visual acuity and spatial modulation thresholds,” in Visual Psychophysics. Handbook of Sensory Physiology VII/4, D. Jameson, L. M. Hurvich, eds. (Springer, Berlin, 1972).
    [CrossRef]
  11. A. W. Snyder, W. H. Miller, “Photoreceptor diameter and spacing for highest resolving power,”J. Opt. Soc. Am. 67, 696–698 (1977).
    [CrossRef] [PubMed]
  12. A. Chaudhuri, P. E. Hallet, J. A. Parker, “Aspheric curvatures, refractive indices and chromatic aberration for the rat eye,” Vision Res. 23, 1351–1361 (1983).
    [CrossRef] [PubMed]
  13. P. McIntyre, K. Kirschfeld, “Chromatic aberration of a dipteran corneal lens,”J. Comp. Physiol. 146, 493–500 (1982).
    [CrossRef]

1986 (1)

A. M. Mass, A. Y. Supin, A. N. Severtsov, “Topographic distribution of sizes and density of ganglion cells in the retina of the porpoise, Phocoena phocoena,” Aquat. Mammals 12, 95–102 (1986).

1983 (1)

A. Chaudhuri, P. E. Hallet, J. A. Parker, “Aspheric curvatures, refractive indices and chromatic aberration for the rat eye,” Vision Res. 23, 1351–1361 (1983).
[CrossRef] [PubMed]

1982 (2)

P. McIntyre, K. Kirschfeld, “Chromatic aberration of a dipteran corneal lens,”J. Comp. Physiol. 146, 493–500 (1982).
[CrossRef]

J. G. Sivak, T. Mandelman, “Chromatic dispersion of the ocular media,” Vision Res. 22, 997–1003 (1982).
[CrossRef] [PubMed]

1977 (1)

1956 (1)

Y. Le Grand, L’Espace Visuel, Vol. 3 of Optique Physiologique, Rev. Opt. Theor. Instrum. (1956).

1886 (1)

L. Matthiessen, “Über den physikalisch-optischen Bau des Auges der Cetaceen und der Fische,” Pfluegers Arch. 38, 521–528 (1886).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Chaudhuri, A.

A. Chaudhuri, P. E. Hallet, J. A. Parker, “Aspheric curvatures, refractive indices and chromatic aberration for the rat eye,” Vision Res. 23, 1351–1361 (1983).
[CrossRef] [PubMed]

Hallet, P. E.

A. Chaudhuri, P. E. Hallet, J. A. Parker, “Aspheric curvatures, refractive indices and chromatic aberration for the rat eye,” Vision Res. 23, 1351–1361 (1983).
[CrossRef] [PubMed]

Kirschfeld, K.

P. McIntyre, K. Kirschfeld, “Chromatic aberration of a dipteran corneal lens,”J. Comp. Physiol. 146, 493–500 (1982).
[CrossRef]

Kröger, R. H. H.

R. H. H. Kröger, “Dioptrik, Funktion der Pupille und Akkommodation bei Zahnwalen,” Ph.D. dissertation (Universität Tübingen, Tübingen, Germany, 1989).

Le Grand, Y.

Y. Le Grand, L’Espace Visuel, Vol. 3 of Optique Physiologique, Rev. Opt. Theor. Instrum. (1956).

Longhurst, R. S.

R. S. Longhurst, Geometrical and Physical Optics, 3rd ed. (Longman, London, 1973).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (Brown U. Press, Providence, R.I., 1944).

Mandelman, T.

J. G. Sivak, T. Mandelman, “Chromatic dispersion of the ocular media,” Vision Res. 22, 997–1003 (1982).
[CrossRef] [PubMed]

Mass, A. M.

A. M. Mass, A. Y. Supin, A. N. Severtsov, “Topographic distribution of sizes and density of ganglion cells in the retina of the porpoise, Phocoena phocoena,” Aquat. Mammals 12, 95–102 (1986).

Matthiessen, L.

L. Matthiessen, “Über den physikalisch-optischen Bau des Auges der Cetaceen und der Fische,” Pfluegers Arch. 38, 521–528 (1886).
[CrossRef]

McIntyre, P.

P. McIntyre, K. Kirschfeld, “Chromatic aberration of a dipteran corneal lens,”J. Comp. Physiol. 146, 493–500 (1982).
[CrossRef]

Miller, W. H.

Parker, J. A.

A. Chaudhuri, P. E. Hallet, J. A. Parker, “Aspheric curvatures, refractive indices and chromatic aberration for the rat eye,” Vision Res. 23, 1351–1361 (1983).
[CrossRef] [PubMed]

Severtsov, A. N.

A. M. Mass, A. Y. Supin, A. N. Severtsov, “Topographic distribution of sizes and density of ganglion cells in the retina of the porpoise, Phocoena phocoena,” Aquat. Mammals 12, 95–102 (1986).

Sivak, J. G.

J. G. Sivak, T. Mandelman, “Chromatic dispersion of the ocular media,” Vision Res. 22, 997–1003 (1982).
[CrossRef] [PubMed]

Snyder, A. W.

Supin, A. Y.

A. M. Mass, A. Y. Supin, A. N. Severtsov, “Topographic distribution of sizes and density of ganglion cells in the retina of the porpoise, Phocoena phocoena,” Aquat. Mammals 12, 95–102 (1986).

Westheimer, G.

G. Westheimer, “Optical properties of vertebrate eyes,” in Physiology of Photoreceptor Organs. Handbook of Sensory Physiology VII/2, M. G. F. Fuortes, ed. (Springer, Berlin, 1972).
[CrossRef]

G. Westheimer, “Visual acuity and spatial modulation thresholds,” in Visual Psychophysics. Handbook of Sensory Physiology VII/4, D. Jameson, L. M. Hurvich, eds. (Springer, Berlin, 1972).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Aquat. Mammals (1)

A. M. Mass, A. Y. Supin, A. N. Severtsov, “Topographic distribution of sizes and density of ganglion cells in the retina of the porpoise, Phocoena phocoena,” Aquat. Mammals 12, 95–102 (1986).

J. Comp. Physiol. (1)

P. McIntyre, K. Kirschfeld, “Chromatic aberration of a dipteran corneal lens,”J. Comp. Physiol. 146, 493–500 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

L’Espace Visuel (1)

Y. Le Grand, L’Espace Visuel, Vol. 3 of Optique Physiologique, Rev. Opt. Theor. Instrum. (1956).

Pfluegers Arch. (1)

L. Matthiessen, “Über den physikalisch-optischen Bau des Auges der Cetaceen und der Fische,” Pfluegers Arch. 38, 521–528 (1886).
[CrossRef]

Vision Res. (2)

J. G. Sivak, T. Mandelman, “Chromatic dispersion of the ocular media,” Vision Res. 22, 997–1003 (1982).
[CrossRef] [PubMed]

A. Chaudhuri, P. E. Hallet, J. A. Parker, “Aspheric curvatures, refractive indices and chromatic aberration for the rat eye,” Vision Res. 23, 1351–1361 (1983).
[CrossRef] [PubMed]

Other (6)

R. K. Luneburg, Mathematical Theory of Optics (Brown U. Press, Providence, R.I., 1944).

R. S. Longhurst, Geometrical and Physical Optics, 3rd ed. (Longman, London, 1973).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

R. H. H. Kröger, “Dioptrik, Funktion der Pupille und Akkommodation bei Zahnwalen,” Ph.D. dissertation (Universität Tübingen, Tübingen, Germany, 1989).

G. Westheimer, “Optical properties of vertebrate eyes,” in Physiology of Photoreceptor Organs. Handbook of Sensory Physiology VII/2, M. G. F. Fuortes, ed. (Springer, Berlin, 1972).
[CrossRef]

G. Westheimer, “Visual acuity and spatial modulation thresholds,” in Visual Psychophysics. Handbook of Sensory Physiology VII/4, D. Jameson, L. M. Hurvich, eds. (Springer, Berlin, 1972).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

N and K in Eq. (2) plotted against the refractive indices of various vertebrate ocular media at 590 nm. The values for N and K were obtained from the data in Table 1 by curve fitting.

Fig. 2
Fig. 2

Quotients Q in Eq. (6) calculated from the data in Table 1 at 440 and 650 nm and the line of regression as described by Eq. (7).

Fig. 3
Fig. 3

m( λ ¯) and b( λ ¯) in Eqs. (8) and (9), respectively, calculated from the data in Table 1. The lines of regression were used to derive a method to convert the refractive indices of ocular media between wavelengths in the visible spectrum.

Fig. 4
Fig. 4

Refractive indices of the ocular media of the frog Rana pipiens at various wavelengths of light. Crosses mark the values measured by Sivak and Mandelman4 (Table 1). Equation (13) was used to convert the refractive indices between all possible pairs of wavelengths in Table 1, e.g., n(440), n(486), and n(590) were calculated from n(650). The refractive indices at 390 nm were calculated from all the measured data points. The calculated values are so similar that all of them fall within the boxes that mark their averages. C, cornea; A/V, aqueous or vitreous humor; L1–L4, samples from the lens at increasing distance from its center.

Fig. 5
Fig. 5

Refraction of thin bundles of red and blue light by the excised lens of a harbor porpoise (Phocoena phocoena). Circles (red) and squares (blue) mark the midlines of the entrance and the exit beams digitized from photographs. Superimposed are 20 rays of each color (curves) that were traced through the lens with a computer program that used calculated indices to determine the refraction of the rays. Photographic and ray-tracing results were aligned with a least-squares algorithm, using digitized outlines of the lens and orientation marks on the photographs as guidelines. Equation (13) was used to convert the refractive indices in the lens from 590 to 390 nm (blue bundle) and to 640 nm (red bundle).

Tables (3)

Tables Icon

Table 1 Refractive Indices of Various Ocular Media of Vertebrate Species as Measured by Sivak and Mandelmana

Tables Icon

Table 2 Deviation of Calculated Refractive Indices from Measurements for the Frog Rana pipiens

Tables Icon

Table 3 Refractive Indices in the Lens of a Harbor Porpoise (Phocoena phocoena)a

Equations (17)

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n ( λ ) = N + K / ( λ - D ) ,
n ( λ ) = N + K / ( λ - 130 nm ) .
N = m n n + b n ,
K = m k n + b k ,
m n = 0.8991 ± 0.0173 , b n = 0.1613 ± 0.0047 ( scalar ) , m k = 58.64 ± 5.63 , b k = - 72.24 ± 1.53 ( nm ) .
n ( λ ) = m n n ( 590 ) + b n + [ m k n ( 590 ) + b k ] / ( λ - 130 nm ) .
Q ( λ 1 , λ 2 ) = [ n ( λ 1 ) - n ( λ 2 ) ] / ( λ 2 - λ 1 ) .
Q ( n ¯ ) = n ¯ * m ( λ ¯ ) + b ( λ ¯ ) .
m ( λ ¯ ) = λ ¯ M m + B m ,
b ( λ ¯ ) = λ ¯ M b + B b .
Q = ( n - n 0 ) / ( λ 0 - λ ) = n ¯ m + b .
n = [ 2 n 0 + ( λ 0 - λ ) ( m n 0 + 2 b ) ] / [ 2 - m ( λ 0 - λ ) ] .
n = - n 0 + 2 [ ( λ 0 2 - λ 2 ) M b + 2 ( λ 0 - λ ) B b + 4 n 0 ] / [ 4 - ( λ 0 2 - λ 2 ) M m - 2 ( λ 0 - λ ) B m ] .
m m = - M m / 4 , m b = M b / 4 , b m = B m / 2 , b b = - B b / 2.
n = 2 [ n 0 + ( λ 0 2 - λ 2 ) m b - ( λ 0 - λ ) b b ] / [ 1 + ( λ 0 2 - λ 2 ) m m - ( λ 0 - λ ) b m ] - n 0 ,
m m = 0.605 ± 0.047 , m b = 0.754 ± 0.048             ( 10 - 6 nm - 2 ) , b m = 0.834 ± 0.011 , b b = 1.037 ± 0.012             ( 10 - 3 nm - 1 ) .
v = [ n ( 589.3 ) - 1 ] / [ n ( 486.1 ) - n ( 656.3 ) ] .

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