Abstract

A model of the human eye is presented with the crystalline lens treated as having a gradient-index structure. By defining an accommodation index I ranging from 0 (unaccommodated) to 1 (accommodated), the optical parameters of the eye in various states of accommodation may be found. The results are in agreement with experimental values.

© 1980 Optical Society of America

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References

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  1. H. von Helmholtz, Treatise on Physiological Optics, 3rd ed., translated by J. P. C. Southall (Optical Society of America, New York, 1924).
  2. Clinical Ophthalmology, edited by T. D. Duane (Harper and Row, New York, 1978), Vol. I, Chap. 68.
  3. W. M. Rosenblum and J. L. Christensen, “Objective and subjective spherical aberration measurements in the human eye,” in Progress in Optics XIII, edited by E. Wolf (North-Holland, Amsterdam, 1976), Chap. III.
    [CrossRef]
  4. For an historical overview, see S. Duke-Elder and D. Abrams, System of Ophthalmology V (Mosby, St. Louis, 1970).
  5. D. T. Moore, “Design of singlets with continuously varying indices of refraction,” J. Opt. Soc. Am. 61, 886–894 (1971).
    [CrossRef]
  6. S. Stenstrom, “Investigation of the variation and correlation of the optical elements of the human eye. III,” Am. J. Optom. Arch. Am. Acad. Optom. 25, 340–387 (1948).
    [CrossRef]
  7. Reference 1, p. 340.
  8. E. F. Fincham, “The Mechanism of Accommodation,” Br. J. Opthalmol., Monogr. 8, London (1937).
  9. N. Brown, “The change in shape and internal form of the lens of the eye on accommodation,” Exp. Eye Res. 15, 441–459 (1973).
    [CrossRef] [PubMed]
  10. B. Patnaik, “A photographic study of accommodative mechanisms: changes in the lens nucleus during accommodation,” Invest. Ophthalmol. 6, 601–611 (1967).
    [PubMed]
  11. R. D. Binkhorst (private communication).

1973 (1)

N. Brown, “The change in shape and internal form of the lens of the eye on accommodation,” Exp. Eye Res. 15, 441–459 (1973).
[CrossRef] [PubMed]

1971 (1)

1967 (1)

B. Patnaik, “A photographic study of accommodative mechanisms: changes in the lens nucleus during accommodation,” Invest. Ophthalmol. 6, 601–611 (1967).
[PubMed]

1948 (1)

S. Stenstrom, “Investigation of the variation and correlation of the optical elements of the human eye. III,” Am. J. Optom. Arch. Am. Acad. Optom. 25, 340–387 (1948).
[CrossRef]

Abrams, D.

For an historical overview, see S. Duke-Elder and D. Abrams, System of Ophthalmology V (Mosby, St. Louis, 1970).

Binkhorst, R. D.

R. D. Binkhorst (private communication).

Brown, N.

N. Brown, “The change in shape and internal form of the lens of the eye on accommodation,” Exp. Eye Res. 15, 441–459 (1973).
[CrossRef] [PubMed]

Christensen, J. L.

W. M. Rosenblum and J. L. Christensen, “Objective and subjective spherical aberration measurements in the human eye,” in Progress in Optics XIII, edited by E. Wolf (North-Holland, Amsterdam, 1976), Chap. III.
[CrossRef]

Duke-Elder, S.

For an historical overview, see S. Duke-Elder and D. Abrams, System of Ophthalmology V (Mosby, St. Louis, 1970).

Fincham, E. F.

E. F. Fincham, “The Mechanism of Accommodation,” Br. J. Opthalmol., Monogr. 8, London (1937).

Moore, D. T.

Patnaik, B.

B. Patnaik, “A photographic study of accommodative mechanisms: changes in the lens nucleus during accommodation,” Invest. Ophthalmol. 6, 601–611 (1967).
[PubMed]

Rosenblum, W. M.

W. M. Rosenblum and J. L. Christensen, “Objective and subjective spherical aberration measurements in the human eye,” in Progress in Optics XIII, edited by E. Wolf (North-Holland, Amsterdam, 1976), Chap. III.
[CrossRef]

Stenstrom, S.

S. Stenstrom, “Investigation of the variation and correlation of the optical elements of the human eye. III,” Am. J. Optom. Arch. Am. Acad. Optom. 25, 340–387 (1948).
[CrossRef]

von Helmholtz, H.

H. von Helmholtz, Treatise on Physiological Optics, 3rd ed., translated by J. P. C. Southall (Optical Society of America, New York, 1924).

Am. J. Optom. Arch. Am. Acad. Optom. (1)

S. Stenstrom, “Investigation of the variation and correlation of the optical elements of the human eye. III,” Am. J. Optom. Arch. Am. Acad. Optom. 25, 340–387 (1948).
[CrossRef]

Exp. Eye Res. (1)

N. Brown, “The change in shape and internal form of the lens of the eye on accommodation,” Exp. Eye Res. 15, 441–459 (1973).
[CrossRef] [PubMed]

Invest. Ophthalmol. (1)

B. Patnaik, “A photographic study of accommodative mechanisms: changes in the lens nucleus during accommodation,” Invest. Ophthalmol. 6, 601–611 (1967).
[PubMed]

J. Opt. Soc. Am. (1)

Other (7)

R. D. Binkhorst (private communication).

Reference 1, p. 340.

E. F. Fincham, “The Mechanism of Accommodation,” Br. J. Opthalmol., Monogr. 8, London (1937).

H. von Helmholtz, Treatise on Physiological Optics, 3rd ed., translated by J. P. C. Southall (Optical Society of America, New York, 1924).

Clinical Ophthalmology, edited by T. D. Duane (Harper and Row, New York, 1978), Vol. I, Chap. 68.

W. M. Rosenblum and J. L. Christensen, “Objective and subjective spherical aberration measurements in the human eye,” in Progress in Optics XIII, edited by E. Wolf (North-Holland, Amsterdam, 1976), Chap. III.
[CrossRef]

For an historical overview, see S. Duke-Elder and D. Abrams, System of Ophthalmology V (Mosby, St. Louis, 1970).

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Figures (3)

FIG. 1
FIG. 1

First surface radius of the crystalline lens as measured by Fincham for two 20-year-old eyes approximately 1 D hypermetropic.8

FIG. 2
FIG. 2

Isoindicial surfaces for the gradient-index model lens. The upper half-plane represents the unaccommodated lens and the lower the accommodated.

FIG. 3
FIG. 3

Principal points and nodal points for the gradient-index, model eye as a function of the accommodation index. The values predicted by Gullstrand for I = 0 (unaccommodated) and I = 1 (accommodated) eyes are shown.

Tables (6)

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TABLE I Gullstrand schematic eye.

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TABLE II Optical parameters of the Gullstrand eye measured from the corneal pole.

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TABLE III Selected values of the crystalline lens parameters.

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TABLE IV Radii of the crystalline lens surfaces in this model.

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TABLE V Index coefficients for the crystalline lens.

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TABLE VI Gradient model eye parameters.

Equations (5)

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N = N 0 ( x ) + N 1 ( x ) ξ + N 2 ( x ) ξ 2 + ,
N 0 ( x ) = N 00 + N 01 x + N 02 x 2 + , N i ( x ) = N i 0 + N i 1 x + N i 2 x 2 + , ξ = Y 2 + Z 2 ,
y ( x ) = n = 0 A n x n ,
A m = { n = 2 m 1 [ 2 A n 2 N 1 , m n n ( n 1 ) A n N 0 m n ( m n + 1 ) ( n 1 ) A m 1 N 0 , m n + 1 ] ( m 1 ) A m 1 N 01 + 2 A m 2 N 10 } / ( m ) ( m 1 ) N 00 .
R 1 = 11.0 6 I , R 2 = 5.5 + 0.5 I , N 00 = 1.387 , N 01 = 0.014 + 0.0055 I , N 02 = 0.00384 0.00066 I , N 10 = 0.0012 0.0005 I .