Abstract

A model for the human eye is proposed, similar to Gullstrand’s well-known 4-radius model, however with the front surface of the cornea and the back surface of the crystalline lens taken to be rotationally symmetric aspherics. Whereas for the cornea a polynomial is used based on experimental data of Bonnet, a second-order parabola was tentatively adopted for the back surface of the lens. This model results in slight spherical undercorrection, in agreement with experimental findings. On the other hand, the sine condition is not well satisfied, probably due to neglect of the shell structure of the lens. By ray tracing, astigmatism and coma as well as the meridional and sagittal focal lengths were computed up to a visual angle of 90°. Calculations were also made for the same model preceded by a plano-concave contact lens (Goldmann 3-mirror contact glass), showing that this combination results in considerably reduced astigmatism.

© 1971 Optical Society of America

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References

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  1. F. Fankhauser and W. Lotmar, Arch. Ophthalmol. 77, 320 (1967).
    [Crossref] [PubMed]
  2. W. Lotmar, F. Fankhauser, and A. Roulier, Arch. Ophthalmol. 82, 314 (1969).
    [Crossref] [PubMed]
  3. T. Schmidt, Ophthalmologica 129, 303 (1955).
    [Crossref] [PubMed]
  4. F. Fankhauser and J. M. Enoch, Arch. Ophthalmol. 68, 240 (1962).
    [Crossref] [PubMed]
  5. C. E. Ferree and G. Rand, in Report of a Joint Discussion on Vision (Cambridge U. P., New York, 1932), p. 244.
  6. F. Rempt, J. Hoogerheide, and W. P. H. Hoogenboom, Ophthalmologica 162, 1 (1971).
    [Crossref]
  7. F. Fankhauser and W. Lotmar, Acta Ophthalmol. 48, 253 (1970).
    [Crossref]
  8. P. Lee, O. Pomerantzeff, and C. L. Schepens, Arch. Ophthalmol. 84, 650 (1970).
    [Crossref] [PubMed]
  9. Y. Le Grand, Optique Physiologique I (Ed. Rev. Opt., Paris, 1953), p. 52.
  10. S. Nakao, S. Fujimoto, R. Nagata, and K. Iwata, J. Opt. Soc. Am. 58, 1125 (1968).
    [Crossref] [PubMed]
  11. G. van den Brink, Vision Res. 2, 233 (1962).
    [Crossref]
  12. H. Schober, H. Munker, and F. Zolleis, Opt. Acta 15, 47 (1968)
    [Crossref]
  13. F. Berny, Vision Res. 9, 977 (1969).
    [Crossref] [PubMed]
  14. R. Bonnet, La Topographie Cornéenne (Desroches, Paris, 1964).
  15. R. Bonnet and P. Cochet, Bull. Soc. Franç. Ophtalmol. 73, 688 (1960).
  16. Visual angle α (Fig. 1) is different from the angle α in Fig. 2 and Eq. (1) as used by Bonnet.
  17. W. Merté, in Handbuch der Physik XVIII, edited by H. Geiger and K. Scheel (Springer, Berlin, 1927), p. 73.
  18. S. Nakao, personal communication.
  19. H. Goldmann and T. Schmidt, Ophthalmologica 149, 481 (1965).
    [Crossref]

1971 (1)

F. Rempt, J. Hoogerheide, and W. P. H. Hoogenboom, Ophthalmologica 162, 1 (1971).
[Crossref]

1970 (2)

F. Fankhauser and W. Lotmar, Acta Ophthalmol. 48, 253 (1970).
[Crossref]

P. Lee, O. Pomerantzeff, and C. L. Schepens, Arch. Ophthalmol. 84, 650 (1970).
[Crossref] [PubMed]

1969 (2)

W. Lotmar, F. Fankhauser, and A. Roulier, Arch. Ophthalmol. 82, 314 (1969).
[Crossref] [PubMed]

F. Berny, Vision Res. 9, 977 (1969).
[Crossref] [PubMed]

1968 (2)

1967 (1)

F. Fankhauser and W. Lotmar, Arch. Ophthalmol. 77, 320 (1967).
[Crossref] [PubMed]

1965 (1)

H. Goldmann and T. Schmidt, Ophthalmologica 149, 481 (1965).
[Crossref]

1962 (2)

F. Fankhauser and J. M. Enoch, Arch. Ophthalmol. 68, 240 (1962).
[Crossref] [PubMed]

G. van den Brink, Vision Res. 2, 233 (1962).
[Crossref]

1960 (1)

R. Bonnet and P. Cochet, Bull. Soc. Franç. Ophtalmol. 73, 688 (1960).

1955 (1)

T. Schmidt, Ophthalmologica 129, 303 (1955).
[Crossref] [PubMed]

Berny, F.

F. Berny, Vision Res. 9, 977 (1969).
[Crossref] [PubMed]

Bonnet, R.

R. Bonnet and P. Cochet, Bull. Soc. Franç. Ophtalmol. 73, 688 (1960).

R. Bonnet, La Topographie Cornéenne (Desroches, Paris, 1964).

Cochet, P.

R. Bonnet and P. Cochet, Bull. Soc. Franç. Ophtalmol. 73, 688 (1960).

Enoch, J. M.

F. Fankhauser and J. M. Enoch, Arch. Ophthalmol. 68, 240 (1962).
[Crossref] [PubMed]

Fankhauser, F.

F. Fankhauser and W. Lotmar, Acta Ophthalmol. 48, 253 (1970).
[Crossref]

W. Lotmar, F. Fankhauser, and A. Roulier, Arch. Ophthalmol. 82, 314 (1969).
[Crossref] [PubMed]

F. Fankhauser and W. Lotmar, Arch. Ophthalmol. 77, 320 (1967).
[Crossref] [PubMed]

F. Fankhauser and J. M. Enoch, Arch. Ophthalmol. 68, 240 (1962).
[Crossref] [PubMed]

Ferree, C. E.

C. E. Ferree and G. Rand, in Report of a Joint Discussion on Vision (Cambridge U. P., New York, 1932), p. 244.

Fujimoto, S.

Goldmann, H.

H. Goldmann and T. Schmidt, Ophthalmologica 149, 481 (1965).
[Crossref]

Hoogenboom, W. P. H.

F. Rempt, J. Hoogerheide, and W. P. H. Hoogenboom, Ophthalmologica 162, 1 (1971).
[Crossref]

Hoogerheide, J.

F. Rempt, J. Hoogerheide, and W. P. H. Hoogenboom, Ophthalmologica 162, 1 (1971).
[Crossref]

Iwata, K.

Le Grand, Y.

Y. Le Grand, Optique Physiologique I (Ed. Rev. Opt., Paris, 1953), p. 52.

Lee, P.

P. Lee, O. Pomerantzeff, and C. L. Schepens, Arch. Ophthalmol. 84, 650 (1970).
[Crossref] [PubMed]

Lotmar, W.

F. Fankhauser and W. Lotmar, Acta Ophthalmol. 48, 253 (1970).
[Crossref]

W. Lotmar, F. Fankhauser, and A. Roulier, Arch. Ophthalmol. 82, 314 (1969).
[Crossref] [PubMed]

F. Fankhauser and W. Lotmar, Arch. Ophthalmol. 77, 320 (1967).
[Crossref] [PubMed]

Merté, W.

W. Merté, in Handbuch der Physik XVIII, edited by H. Geiger and K. Scheel (Springer, Berlin, 1927), p. 73.

Munker, H.

H. Schober, H. Munker, and F. Zolleis, Opt. Acta 15, 47 (1968)
[Crossref]

Nagata, R.

Nakao, S.

Pomerantzeff, O.

P. Lee, O. Pomerantzeff, and C. L. Schepens, Arch. Ophthalmol. 84, 650 (1970).
[Crossref] [PubMed]

Rand, G.

C. E. Ferree and G. Rand, in Report of a Joint Discussion on Vision (Cambridge U. P., New York, 1932), p. 244.

Rempt, F.

F. Rempt, J. Hoogerheide, and W. P. H. Hoogenboom, Ophthalmologica 162, 1 (1971).
[Crossref]

Roulier, A.

W. Lotmar, F. Fankhauser, and A. Roulier, Arch. Ophthalmol. 82, 314 (1969).
[Crossref] [PubMed]

Schepens, C. L.

P. Lee, O. Pomerantzeff, and C. L. Schepens, Arch. Ophthalmol. 84, 650 (1970).
[Crossref] [PubMed]

Schmidt, T.

H. Goldmann and T. Schmidt, Ophthalmologica 149, 481 (1965).
[Crossref]

T. Schmidt, Ophthalmologica 129, 303 (1955).
[Crossref] [PubMed]

Schober, H.

H. Schober, H. Munker, and F. Zolleis, Opt. Acta 15, 47 (1968)
[Crossref]

van den Brink, G.

G. van den Brink, Vision Res. 2, 233 (1962).
[Crossref]

Zolleis, F.

H. Schober, H. Munker, and F. Zolleis, Opt. Acta 15, 47 (1968)
[Crossref]

Acta Ophthalmol. (1)

F. Fankhauser and W. Lotmar, Acta Ophthalmol. 48, 253 (1970).
[Crossref]

Arch. Ophthalmol. (4)

P. Lee, O. Pomerantzeff, and C. L. Schepens, Arch. Ophthalmol. 84, 650 (1970).
[Crossref] [PubMed]

F. Fankhauser and W. Lotmar, Arch. Ophthalmol. 77, 320 (1967).
[Crossref] [PubMed]

W. Lotmar, F. Fankhauser, and A. Roulier, Arch. Ophthalmol. 82, 314 (1969).
[Crossref] [PubMed]

F. Fankhauser and J. M. Enoch, Arch. Ophthalmol. 68, 240 (1962).
[Crossref] [PubMed]

Bull. Soc. Franç. Ophtalmol. (1)

R. Bonnet and P. Cochet, Bull. Soc. Franç. Ophtalmol. 73, 688 (1960).

J. Opt. Soc. Am. (1)

Ophthalmologica (3)

T. Schmidt, Ophthalmologica 129, 303 (1955).
[Crossref] [PubMed]

F. Rempt, J. Hoogerheide, and W. P. H. Hoogenboom, Ophthalmologica 162, 1 (1971).
[Crossref]

H. Goldmann and T. Schmidt, Ophthalmologica 149, 481 (1965).
[Crossref]

Opt. Acta (1)

H. Schober, H. Munker, and F. Zolleis, Opt. Acta 15, 47 (1968)
[Crossref]

Vision Res. (2)

F. Berny, Vision Res. 9, 977 (1969).
[Crossref] [PubMed]

G. van den Brink, Vision Res. 2, 233 (1962).
[Crossref]

Other (6)

Y. Le Grand, Optique Physiologique I (Ed. Rev. Opt., Paris, 1953), p. 52.

C. E. Ferree and G. Rand, in Report of a Joint Discussion on Vision (Cambridge U. P., New York, 1932), p. 244.

R. Bonnet, La Topographie Cornéenne (Desroches, Paris, 1964).

Visual angle α (Fig. 1) is different from the angle α in Fig. 2 and Eq. (1) as used by Bonnet.

W. Merté, in Handbuch der Physik XVIII, edited by H. Geiger and K. Scheel (Springer, Berlin, 1927), p. 73.

S. Nakao, personal communication.

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Figures (21)

Fig. 1
Fig. 1

Theoretical eye model of Gullstrand–Le Grand. α, visual angle; , internal angle; γ, angle of acceptance.

Fig. 2
Fig. 2

Definition of angles in Eq. (1). B, profile curve of Bonnet; C, osculating circle.

Fig. 3
Fig. 3

B, Bonnet’s profile curve for the cornea according to Eq. (3); C, osculating circle; E, ellipse; P, parabola.

Fig. 4
Fig. 4

Meridional and sagittal radii of curvature ρm and ρs, respectively, of body of revolution corresponding to polynomial of Eq. (3).

Fig. 5
Fig. 5

Spherical aberration of eye model according to data of Table I with aspherics. Δs, aberration of intersection length; Δf, aberration of focal length.

Fig. 6
Fig. 6

Le Grand’s eye model. Course of principal rays. R, schematized retina (r5 of Table I).

Fig. 7
Fig. 7

Plot of internal angle vs visual angle α.

Fig. 8
Fig. 8

Angle of acceptance γ vs visual angle α. (a) for model retina, (b) for Le Grand’s cross section (Fig. 6).

Fig. 9
Fig. 9

Astigmatism of eye model of Table I with aspherics. Ordinate axis represents reference image sphere r5. R, position of retina according to Le Grand.

Fig. 10
Fig. 10

Definition of focal length for oblique incidence of a parallel meridional bundle, fm=d/u.

Fig. 11
Fig. 11

Meridional and sagittal focal lengths, fm and fs, respectively, vs visual angle for eye model of Table I with aspherics.

Fig. 12
Fig. 12

Convergence ratio of individual refracting surfaces of eye model with aspherics. r1 (sph), curve if the front surface of cornea is spherical.

Fig. 13
Fig. 13

Longitudinal comatic aberration of eye model with aspherics. Reference image sphere is r5 of Table I. Plotted curves are projections of true curves on h axis along the principal rays of the corresponding bundles.

Fig. 14
Fig. 14

Spherical aberration of model with lens consisting of seven shells.

Fig. 15
Fig. 15

Astigmatism of emmetropic human eyes according to Ferree and Rand. Classes (a) and (b).

Fig. 16
Fig. 16

Eye model with aspherics in combination with plano-concave contact glass (ECG system). For ray tracing, the angle between the principal ray and the mirror was assumed to be /2, so that incidence on the front surface of the contact glass is at right angles.

Fig. 17
Fig. 17

Spherical aberration of ECG system. Scale of sine-condition aberration is comparable to that of Fig. 5.

Fig. 18
Fig. 18

Astigmatism of ECG system.

Fig. 19
Fig. 19

Longitudinal comatic aberration of ECG system. See remarks to Fig. 13.

Fig. 20
Fig. 20

Meridional and sagittal convergence ratios of ECG system.

Fig. 21
Fig. 21

Sturm’s interval, Δ, for eye model with aspherics (a), and ECG system (b).

Tables (1)

Tables Icon

Table I The eye model of Gullstrand–Le Grand.

Equations (8)

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log 10 β = k α + b ,
log 10 β = 3 ( α - 1 ) ,
x = h 2 2 r o [ 1 + 5 28 ( h r o ) 2 - 1 12 ( h r o ) 4 ] ,
tan α = 1 / x = d h / d x
sin β = ( x + h x - r o ) sin α / r o .
ρ m r o [ 1 + 8 ( h / 10 ) 4 ]
ρ s r o ( 1 + 1.66 × 10 - 3 h 2.6 ) .
f m = f 0 · ( f m 1 / f 0 ) · ( 1 / γ 2 ) · ( 1 / γ 3 ) · ( 1 / γ 4 ) .