Abstract

The influence of the corneal shape on the quality of the retinal image has been studied. We have distinguished between the role of the cornea and that of the crystalline lens; the wave surface behind the cornea was determined. First, the corneal topology was measured by photographic keratometry. Then, for the same subject, the spherical aberration of the eye was determined by the Foucault knife-edge test. The shapes of the anterior and posterior surfaces of the crystalline lens have been determined from experimental data.

© 1973 Optical Society of America

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References

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  1. Y. Le Grand, C.R. Acad. Sci. (Paris) 215, 547 (1942).
  2. G. Jurin, cited in R. Smith, A Compleat System of Optics and Harmonics (Thurlbourn, Cambridge, 1738–1748).
  3. T. Young, A Course of Lectures on Natural Philosophy and the Mechanical Arts (Johnson, London, 1807).
    [Crossref]
  4. A. Volkmann, Neue Beitrage zur Physiologie des Gesichtssinnes (Breitkopf and Hartel, Leipzig, 1836).
  5. L. Matthiessen, C.R. Acad. Sci. (Paris) 24, 875 (1847).
  6. M. H. E. Tscherning, Optique Physiologique (Carré et Naud, Paris, 1898).
  7. A. Gullstrand, Ber. Klin. Mbl. Aug. 13, 183 (1911).
  8. G. Ames and R. Proctor, J. Opt. Soc. Am. 5, 22 (1921).
    [Crossref]
  9. A. Arnulf, F. Flamant, and M. Françon, Rev. Opt. Theor. Instrum. 27, 741 (1948).
  10. G. Von Bahr, Acta Ophthalmol. 23, 1 (1945).
    [Crossref]
  11. A. Ivanoff, Ann. Opt. Ocul. 2, 97 (1953).
  12. R. Jackson, Trans. Am. Ophthalmol. Soc. 5, 141 (1888).
  13. H. T. Pi, Trans. Ophthalmol. Soc. U.K. 45, 393 (1925).
  14. H. Stine, Am. J. Ophthalmol. 13, 101 (1930).
  15. M. Coates, Dioptric Rev. 1, 127 (1940);Dioptric Rev. 2, 102 (1941).
  16. T. Jenkins, Brit. J. Physiol. Opt. 20, 59 (1963).
  17. A. Arnulf and O. Dupuy, Rev. Opt. Theor. Instrum. 39, 5 (1960).
  18. S. G. El Hage, C.R. Acad. Sci. (Paris) 270, 3299 (1970).
  19. F. Berny, Thèse de Docteur-ingenieur, University of Paris (1968).
  20. S. G. El Hage, Am. J. Optom. 49, 5 (1972).
  21. A. Maréchal and M. Françon, Diffraction, Structure des Images; Influence de la Cohérence de la Lumiere (ed. Rev. Opt., Paris, 1960).
  22. A. Angot, Complément de Mathematique a l’Usage des Ingénieurs de l’Electrotechnique et de Télécommunication (ed. Rev. Opt., Paris, 1965).
  23. Y. Le Grand, Optique Physiologique, La Dioptrique de l’Oeil et sa Correction (ed. Rev. Opt., Paris, 1965).
  24. Y. Le Grand, Optique Physiologique, l’Espace Visuel (ed. Rev. Opt., Paris, 1956).

1972 (1)

S. G. El Hage, Am. J. Optom. 49, 5 (1972).

1970 (1)

S. G. El Hage, C.R. Acad. Sci. (Paris) 270, 3299 (1970).

1963 (1)

T. Jenkins, Brit. J. Physiol. Opt. 20, 59 (1963).

1960 (1)

A. Arnulf and O. Dupuy, Rev. Opt. Theor. Instrum. 39, 5 (1960).

1953 (1)

A. Ivanoff, Ann. Opt. Ocul. 2, 97 (1953).

1948 (1)

A. Arnulf, F. Flamant, and M. Françon, Rev. Opt. Theor. Instrum. 27, 741 (1948).

1945 (1)

G. Von Bahr, Acta Ophthalmol. 23, 1 (1945).
[Crossref]

1942 (1)

Y. Le Grand, C.R. Acad. Sci. (Paris) 215, 547 (1942).

1940 (1)

M. Coates, Dioptric Rev. 1, 127 (1940);Dioptric Rev. 2, 102 (1941).

1930 (1)

H. Stine, Am. J. Ophthalmol. 13, 101 (1930).

1925 (1)

H. T. Pi, Trans. Ophthalmol. Soc. U.K. 45, 393 (1925).

1921 (1)

1911 (1)

A. Gullstrand, Ber. Klin. Mbl. Aug. 13, 183 (1911).

1888 (1)

R. Jackson, Trans. Am. Ophthalmol. Soc. 5, 141 (1888).

1847 (1)

L. Matthiessen, C.R. Acad. Sci. (Paris) 24, 875 (1847).

Ames, G.

Angot, A.

A. Angot, Complément de Mathematique a l’Usage des Ingénieurs de l’Electrotechnique et de Télécommunication (ed. Rev. Opt., Paris, 1965).

Arnulf, A.

A. Arnulf and O. Dupuy, Rev. Opt. Theor. Instrum. 39, 5 (1960).

A. Arnulf, F. Flamant, and M. Françon, Rev. Opt. Theor. Instrum. 27, 741 (1948).

Berny, F.

F. Berny, Thèse de Docteur-ingenieur, University of Paris (1968).

Coates, M.

M. Coates, Dioptric Rev. 1, 127 (1940);Dioptric Rev. 2, 102 (1941).

Dupuy, O.

A. Arnulf and O. Dupuy, Rev. Opt. Theor. Instrum. 39, 5 (1960).

El Hage, S. G.

S. G. El Hage, Am. J. Optom. 49, 5 (1972).

S. G. El Hage, C.R. Acad. Sci. (Paris) 270, 3299 (1970).

Flamant, F.

A. Arnulf, F. Flamant, and M. Françon, Rev. Opt. Theor. Instrum. 27, 741 (1948).

Françon, M.

A. Arnulf, F. Flamant, and M. Françon, Rev. Opt. Theor. Instrum. 27, 741 (1948).

A. Maréchal and M. Françon, Diffraction, Structure des Images; Influence de la Cohérence de la Lumiere (ed. Rev. Opt., Paris, 1960).

Gullstrand, A.

A. Gullstrand, Ber. Klin. Mbl. Aug. 13, 183 (1911).

Ivanoff, A.

A. Ivanoff, Ann. Opt. Ocul. 2, 97 (1953).

Jackson, R.

R. Jackson, Trans. Am. Ophthalmol. Soc. 5, 141 (1888).

Jenkins, T.

T. Jenkins, Brit. J. Physiol. Opt. 20, 59 (1963).

Jurin, G.

G. Jurin, cited in R. Smith, A Compleat System of Optics and Harmonics (Thurlbourn, Cambridge, 1738–1748).

Le Grand, Y.

Y. Le Grand, C.R. Acad. Sci. (Paris) 215, 547 (1942).

Y. Le Grand, Optique Physiologique, La Dioptrique de l’Oeil et sa Correction (ed. Rev. Opt., Paris, 1965).

Y. Le Grand, Optique Physiologique, l’Espace Visuel (ed. Rev. Opt., Paris, 1956).

Maréchal, A.

A. Maréchal and M. Françon, Diffraction, Structure des Images; Influence de la Cohérence de la Lumiere (ed. Rev. Opt., Paris, 1960).

Matthiessen, L.

L. Matthiessen, C.R. Acad. Sci. (Paris) 24, 875 (1847).

Pi, H. T.

H. T. Pi, Trans. Ophthalmol. Soc. U.K. 45, 393 (1925).

Proctor, R.

Smith, R.

G. Jurin, cited in R. Smith, A Compleat System of Optics and Harmonics (Thurlbourn, Cambridge, 1738–1748).

Stine, H.

H. Stine, Am. J. Ophthalmol. 13, 101 (1930).

Tscherning, M. H. E.

M. H. E. Tscherning, Optique Physiologique (Carré et Naud, Paris, 1898).

Volkmann, A.

A. Volkmann, Neue Beitrage zur Physiologie des Gesichtssinnes (Breitkopf and Hartel, Leipzig, 1836).

Von Bahr, G.

G. Von Bahr, Acta Ophthalmol. 23, 1 (1945).
[Crossref]

Young, T.

T. Young, A Course of Lectures on Natural Philosophy and the Mechanical Arts (Johnson, London, 1807).
[Crossref]

Acta Ophthalmol. (1)

G. Von Bahr, Acta Ophthalmol. 23, 1 (1945).
[Crossref]

Am. J. Ophthalmol. (1)

H. Stine, Am. J. Ophthalmol. 13, 101 (1930).

Am. J. Optom. (1)

S. G. El Hage, Am. J. Optom. 49, 5 (1972).

Ann. Opt. Ocul. (1)

A. Ivanoff, Ann. Opt. Ocul. 2, 97 (1953).

Ber. Klin. Mbl. Aug. (1)

A. Gullstrand, Ber. Klin. Mbl. Aug. 13, 183 (1911).

Brit. J. Physiol. Opt. (1)

T. Jenkins, Brit. J. Physiol. Opt. 20, 59 (1963).

C.R. Acad. Sci. (Paris) (3)

L. Matthiessen, C.R. Acad. Sci. (Paris) 24, 875 (1847).

Y. Le Grand, C.R. Acad. Sci. (Paris) 215, 547 (1942).

S. G. El Hage, C.R. Acad. Sci. (Paris) 270, 3299 (1970).

Dioptric Rev. (1)

M. Coates, Dioptric Rev. 1, 127 (1940);Dioptric Rev. 2, 102 (1941).

J. Opt. Soc. Am. (1)

Rev. Opt. Theor. Instrum. (2)

A. Arnulf, F. Flamant, and M. Françon, Rev. Opt. Theor. Instrum. 27, 741 (1948).

A. Arnulf and O. Dupuy, Rev. Opt. Theor. Instrum. 39, 5 (1960).

Trans. Am. Ophthalmol. Soc. (1)

R. Jackson, Trans. Am. Ophthalmol. Soc. 5, 141 (1888).

Trans. Ophthalmol. Soc. U.K. (1)

H. T. Pi, Trans. Ophthalmol. Soc. U.K. 45, 393 (1925).

Other (9)

M. H. E. Tscherning, Optique Physiologique (Carré et Naud, Paris, 1898).

G. Jurin, cited in R. Smith, A Compleat System of Optics and Harmonics (Thurlbourn, Cambridge, 1738–1748).

T. Young, A Course of Lectures on Natural Philosophy and the Mechanical Arts (Johnson, London, 1807).
[Crossref]

A. Volkmann, Neue Beitrage zur Physiologie des Gesichtssinnes (Breitkopf and Hartel, Leipzig, 1836).

F. Berny, Thèse de Docteur-ingenieur, University of Paris (1968).

A. Maréchal and M. Françon, Diffraction, Structure des Images; Influence de la Cohérence de la Lumiere (ed. Rev. Opt., Paris, 1960).

A. Angot, Complément de Mathematique a l’Usage des Ingénieurs de l’Electrotechnique et de Télécommunication (ed. Rev. Opt., Paris, 1965).

Y. Le Grand, Optique Physiologique, La Dioptrique de l’Oeil et sa Correction (ed. Rev. Opt., Paris, 1965).

Y. Le Grand, Optique Physiologique, l’Espace Visuel (ed. Rev. Opt., Paris, 1956).

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Figures (16)

F. 1
F. 1

Spherical aberration. Light rays from point A pass through different heights h above or below the optic axis and are focused at A0′ and A′. A0′A′ represents the longitudinal spherical aberration for rays penetrating the refractive surface at a distance h from the axis.

F. 2
F. 2

Experimental apparatus. W = source of light, W′=Filter, M = slit, L, L′ = semi-transparent mirror, (beam splitter), L1, L2 =identical lenses, E = knife edge, R = retina.

F. 3
F. 3

Longitudinal spherical aberration. The ordinate shows the pupillary heights y of the paraxial ray and the abscissa presents the longitudinal spherical aberration l as a function of the reference image point and the paraxial image.

F. 4
F. 4

Sign convention. The direction of light entering the eye is the positive direction.

F. 5
F. 5

Photokeratoscopic data for the horizontal meridian of the cornea. The ordinate y represents the height of the measured area from the optical axis. The abscissa represents the difference between the corneal profile and the oscillatory circle. (○) The data of the temporal side; (●) the data of nasal side; and (+) are the calculated points for an ellipse of A = 1.16 (A is the conic parameter).

F. 6
F. 6

Transverse spherical aberration t′ as a function of pupillary height y. Δ(ym = 4.00 mm); ○(ym = 3.25 mm); + (ym = 2.50mm).

F. 7
F. 7

Experimentally determined light distribution of the retinal image nΔ/λ as a function of pupillary diameters of 3.25 and 2.50 mm.

F. 8
F. 8

nΔ/λ values for the cornea (+) and for the whole eye (●) for two different pupillary radii, ym; 1.0 mm; and 1.5 mm.

F. 9
F. 9

nΔ/λ values for the cornea (+) and for the whole eye (●) for a pupillary radius, ym = 2.50 mm.

F. 10
F. 10

nΔ/λ values for the cornea (+) and for the whole eye (●) for a pupillary radius, ym=3.25 mm.

F. 11
F. 11

Comparison of the whole eye (○) and the corneal (+) retinal illuminance with that expected from Airy’s disk (●) for pupillary radius of 1.0 mm. The curve represents I as a function of z.

F. 12
F. 12

Comparison of the whole eye (○) and the corneal (+) retinal illuminance with that expected from Airy’s disk (●) for pupillary radius of 2.0 mm. The curve represents I as a function of z.

F. 13
F. 13

Comparison of the whole eye (○) and the corneal (+) retinal illuminance with that expected from Airy’s disk (●) for pupillary radius of 2.5 mm. The curve represents I as a function of z.

F. 14
F. 14

Comparison of the whole eye (○) and the corneal (+) retinal illuminance with that expected from Airy’s disk (●) for pupillary radius of 3.25 mm. The curve represents I as a function of z.

F. 15
F. 15

Calculated and experimental values of transverse spherical aberration (t) for rays penetrating the cornea at various nasal (●) and temporal (○) distances (ym) from the axis. (+) Calculated curve reported to the reference plane of abscissa 296 mm.

F. 16
F. 16

The shape of the crystalline lens determined from spherical aberration data with a pupillary radius of 3.1 mm.

Tables (1)

Tables Icon

Table I Transverse spherical aberration.

Equations (12)

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y 2 2 R x + A x 2 = 0 ,
Δ λ = l λ ( SP 0 ) 0 y m t d y .
exp [ j 2 π n λ Δ ( y ) ] = exp [ j k Δ ( y ) ] = F ( y ) .
ξ ( r ) = π 0 y m F ( y ) J 0 ( k r y SP 0 ) y d y ,
H = y y m , X = H 2 , z = 2 π r y m n λ ( SP 0 ) ,
ξ ( z ) = π y m 2 0 1 F ( X ) J 0 ( z X ) d X = π y m 2 [ ( z ) + j Γ ( z ) ] .
g ( z ) = ξ ξ * = π 2 y m 4 ( 2 + Γ 2 ) ,
( z ) = 0 1 J 0 ( z X ) cos k Δ ( X ) d X , Γ ( z ) = 0 1 J 0 ( z X ) sin k Δ ( X ) d X ,
I = g g 0 = 2 ( Z ) + Γ 2 ( Z ) .
y 2 + 16 x 16 x 2 = 0 .
x = 0.0833 y 2 + 0.00058 y 4 .
x = 0.083333 y 2 + 0.00058 y 4 0.0002 y 6 + 0.000002 y 8 + 10 7 y 10 + 10 8 y 12 .