Abstract

We have designed an aberroscope that differs from Tscherning’s classical instrument in that it makes use of an artificial astigmatism rather than an artificial myopia to defocus the image of a point source of light. A subject views the source through a ±5 D crossed cylinder lens with axes at 45° to the principal axes of an intercalated grid and sees a shadow image of the grid. The distortions of this grid image are quantitatively related to the wave aberration of the eye. Using this device we have obtained drawings for more than 50 subjects. These drawings of the grid pattern have been analyzed by means of a two-dimensional polynomial curve Fitting technique that computes Taylor polynomial terms to the fourth order. From the Taylor coefficients it is possible to reconstruct the wave aberration surface. Examination of the Taylor terms so obtained shows that the monochromatic aberrations of the eye are dominated by third-order Taylor terms within the range of physiological pupil sizes, and that spherical aberration frequently appears predominantly about one axis only, a condition that we have termed “cylindrical” aberration. We have computed the optical MTF of our subjects’ eyes and find that the role of aberrations in degrading the MTF may be greater than generally believed.

© 1977 Optical Society of America

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References

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  1. A. van Meeteren, “Calculations on the Optical Modulation Transfer function of the Human Eye for White Light,” Opt. Acta 21, 395–412 (1974).
    [CrossRef]
  2. H. S. Smirnov, “Measurement of Wave Aberration in the Human Eye,” Biophys. 6, 52–66 (1961).
  3. B. Howland, “Use of Crossed Cylinder Lens in Photographic Lens Evaluation,” Appl. Opt. 7, 1587–1599 (1968).
    [CrossRef] [PubMed]
  4. M. Tscherning, “Die Monochromatischen abberationen des Menschlichen Auges,” Z. Psychol. Physiol. Sinn. 6, 456–471 (1894).
  5. R. Barakat and A. Houston, “The aberrations of non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1–30 (1966).
    [CrossRef]
  6. R. T. Hennessy, “Instrument myopia,” J. Opt. Soc. Am. 65, 1114–1120 (1975).
    [CrossRef] [PubMed]
  7. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Sec. 9. 2.
  8. H. H. Hopkins, “The application of frequency response techniques in optics,” Proc. Phys. Soc. 79, 889–919 (1962).
    [CrossRef]
  9. W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966).
  10. B. Howland, “Electronic aberration synthesizer for measurements of the eye,” J. Opt. Soc. Am. 66, 1121 (1976).
  11. B. Howland and H. C. Howland, “Subjective measurement of high-order aberrations of the eye,” Science 193, 580–582 (1976).
    [CrossRef] [PubMed]
  12. J. Krasukopf, “Further measurements of human retinal images,” J. Opt. Soc. Am. 54, 715–716 (1964).
  13. H. Schober, H. Münker, and F. Zolleis, “Die aberration des menschlichen auges und ihre messung,” Opt. Acta 15, 47 (1968).
    [CrossRef]
  14. M. Koomen, R. Tousey, and R. Scolnik, “The spherical aberration of the eye,” J. Opt. Soc. Am. 55, 370–376 (1949).
    [CrossRef]
  15. F. W. Campbell and R. W. Gubisch, “Optical quality of the Human eye,” J. Physiol. 186, 558–578 (1966).
  16. F. W. Campbell and D. G. Green, “Optical and retinal factors affecting visual resolution,” J. Physiol. 181, 576–593 (1965).
  17. D. G. Green and F. W. Campbell, “Effect of focus on the visual response to a sinusoidally modulated spatial stimulus,” J. Opt. Soc. Am. 55, 1154–1157 (1965).
    [CrossRef]

1976 (2)

B. Howland, “Electronic aberration synthesizer for measurements of the eye,” J. Opt. Soc. Am. 66, 1121 (1976).

B. Howland and H. C. Howland, “Subjective measurement of high-order aberrations of the eye,” Science 193, 580–582 (1976).
[CrossRef] [PubMed]

1975 (1)

1974 (1)

A. van Meeteren, “Calculations on the Optical Modulation Transfer function of the Human Eye for White Light,” Opt. Acta 21, 395–412 (1974).
[CrossRef]

1968 (2)

H. Schober, H. Münker, and F. Zolleis, “Die aberration des menschlichen auges und ihre messung,” Opt. Acta 15, 47 (1968).
[CrossRef]

B. Howland, “Use of Crossed Cylinder Lens in Photographic Lens Evaluation,” Appl. Opt. 7, 1587–1599 (1968).
[CrossRef] [PubMed]

1966 (2)

F. W. Campbell and R. W. Gubisch, “Optical quality of the Human eye,” J. Physiol. 186, 558–578 (1966).

R. Barakat and A. Houston, “The aberrations of non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1–30 (1966).
[CrossRef]

1965 (2)

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

D. G. Green and F. W. Campbell, “Effect of focus on the visual response to a sinusoidally modulated spatial stimulus,” J. Opt. Soc. Am. 55, 1154–1157 (1965).
[CrossRef]

1964 (1)

1962 (1)

H. H. Hopkins, “The application of frequency response techniques in optics,” Proc. Phys. Soc. 79, 889–919 (1962).
[CrossRef]

1961 (1)

H. S. Smirnov, “Measurement of Wave Aberration in the Human Eye,” Biophys. 6, 52–66 (1961).

1949 (1)

M. Koomen, R. Tousey, and R. Scolnik, “The spherical aberration of the eye,” J. Opt. Soc. Am. 55, 370–376 (1949).
[CrossRef]

1894 (1)

M. Tscherning, “Die Monochromatischen abberationen des Menschlichen Auges,” Z. Psychol. Physiol. Sinn. 6, 456–471 (1894).

Barakat, R.

R. Barakat and A. Houston, “The aberrations of non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1–30 (1966).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Sec. 9. 2.

Campbell, F. W.

F. W. Campbell and R. W. Gubisch, “Optical quality of the Human eye,” J. Physiol. 186, 558–578 (1966).

D. G. Green and F. W. Campbell, “Effect of focus on the visual response to a sinusoidally modulated spatial stimulus,” J. Opt. Soc. Am. 55, 1154–1157 (1965).
[CrossRef]

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

Green, D. G.

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

D. G. Green and F. W. Campbell, “Effect of focus on the visual response to a sinusoidally modulated spatial stimulus,” J. Opt. Soc. Am. 55, 1154–1157 (1965).
[CrossRef]

Gubisch, R. W.

F. W. Campbell and R. W. Gubisch, “Optical quality of the Human eye,” J. Physiol. 186, 558–578 (1966).

Hennessy, R. T.

Hopkins, H. H.

H. H. Hopkins, “The application of frequency response techniques in optics,” Proc. Phys. Soc. 79, 889–919 (1962).
[CrossRef]

Houston, A.

R. Barakat and A. Houston, “The aberrations of non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1–30 (1966).
[CrossRef]

Howland, B.

B. Howland, “Electronic aberration synthesizer for measurements of the eye,” J. Opt. Soc. Am. 66, 1121 (1976).

B. Howland and H. C. Howland, “Subjective measurement of high-order aberrations of the eye,” Science 193, 580–582 (1976).
[CrossRef] [PubMed]

B. Howland, “Use of Crossed Cylinder Lens in Photographic Lens Evaluation,” Appl. Opt. 7, 1587–1599 (1968).
[CrossRef] [PubMed]

Howland, H. C.

B. Howland and H. C. Howland, “Subjective measurement of high-order aberrations of the eye,” Science 193, 580–582 (1976).
[CrossRef] [PubMed]

Koomen, M.

M. Koomen, R. Tousey, and R. Scolnik, “The spherical aberration of the eye,” J. Opt. Soc. Am. 55, 370–376 (1949).
[CrossRef]

Krasukopf, J.

Münker, H.

H. Schober, H. Münker, and F. Zolleis, “Die aberration des menschlichen auges und ihre messung,” Opt. Acta 15, 47 (1968).
[CrossRef]

Schober, H.

H. Schober, H. Münker, and F. Zolleis, “Die aberration des menschlichen auges und ihre messung,” Opt. Acta 15, 47 (1968).
[CrossRef]

Scolnik, R.

M. Koomen, R. Tousey, and R. Scolnik, “The spherical aberration of the eye,” J. Opt. Soc. Am. 55, 370–376 (1949).
[CrossRef]

Smirnov, H. S.

H. S. Smirnov, “Measurement of Wave Aberration in the Human Eye,” Biophys. 6, 52–66 (1961).

Smith, W. J.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966).

Tousey, R.

M. Koomen, R. Tousey, and R. Scolnik, “The spherical aberration of the eye,” J. Opt. Soc. Am. 55, 370–376 (1949).
[CrossRef]

Tscherning, M.

M. Tscherning, “Die Monochromatischen abberationen des Menschlichen Auges,” Z. Psychol. Physiol. Sinn. 6, 456–471 (1894).

van Meeteren, A.

A. van Meeteren, “Calculations on the Optical Modulation Transfer function of the Human Eye for White Light,” Opt. Acta 21, 395–412 (1974).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Sec. 9. 2.

Zolleis, F.

H. Schober, H. Münker, and F. Zolleis, “Die aberration des menschlichen auges und ihre messung,” Opt. Acta 15, 47 (1968).
[CrossRef]

Appl. Opt. (1)

Biophys. (1)

H. S. Smirnov, “Measurement of Wave Aberration in the Human Eye,” Biophys. 6, 52–66 (1961).

J. Opt. Soc. Am. (5)

J. Physiol. (2)

F. W. Campbell and R. W. Gubisch, “Optical quality of the Human eye,” J. Physiol. 186, 558–578 (1966).

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

Opt. Acta (3)

A. van Meeteren, “Calculations on the Optical Modulation Transfer function of the Human Eye for White Light,” Opt. Acta 21, 395–412 (1974).
[CrossRef]

H. Schober, H. Münker, and F. Zolleis, “Die aberration des menschlichen auges und ihre messung,” Opt. Acta 15, 47 (1968).
[CrossRef]

R. Barakat and A. Houston, “The aberrations of non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1–30 (1966).
[CrossRef]

Proc. Phys. Soc. (1)

H. H. Hopkins, “The application of frequency response techniques in optics,” Proc. Phys. Soc. 79, 889–919 (1962).
[CrossRef]

Science (1)

B. Howland and H. C. Howland, “Subjective measurement of high-order aberrations of the eye,” Science 193, 580–582 (1976).
[CrossRef] [PubMed]

Z. Psychol. Physiol. Sinn. (1)

M. Tscherning, “Die Monochromatischen abberationen des Menschlichen Auges,” Z. Psychol. Physiol. Sinn. 6, 456–471 (1894).

Other (2)

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Sec. 9. 2.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966).

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Figures (13)

FIG. 1
FIG. 1

Optics of the crossed cylinder aberroscope and eye. The spherocylinder lens RTLB here represents the combined dioptric power of the aberroscope, the eye, and the subject’s prescription. Lines a1 and a2 define the plus and minus axes of the crossed cylinder aberroscope lens and are at 45° to the horizontal and vertical meridians. The two astigmatic foci f1 and f2 are parallel, respectively, to a1 and a2; their separation defines the interval of Sturm. Between the astigmatic foci there lies a plane conjugate to infinity with the aberroscope removed. Points B′, L′, T′, R′ are generated by a pencil of collimated rays parallel to the optical axis passing through corresponding points B, L, T, R. It may be seen that the shadow image defined by this projection of collimated rays has undergone both a 90° rotation and reflection.

FIG. 2
FIG. 2

Optics of viewing aberroscope patterns. White tungsten light from a fiber optic F is shown down a microscope barrel M and brought to a point focus by the microscope objective O. The light is filtered by a Wratten 25 (red) filter mounted on the objective. The subject, whose cornea is at C, views the point source through the crossed cylinder aberroscope A, which is held in a trial frame worn by the subject. No head rest is employed.

FIG. 3
FIG. 3

Aberroscopic patterns for third-order terms, G = H = I = J = 0. 15.

FIG. 4
FIG. 4

Aberroscopic patterns for fourth-order Taylor Terms, K = L = M = N = 0. 05.

FIG. 5
FIG. 5

Definition of coordinate systems. I. Coordinate system centered in subject’s pupil used to describe wave aberration function. View is that of an observer looking at subject’s pupil. Positive z axis points into plane of paper. II. Projection P′ of point P at pupil onto subject’s retina. Due to the rotation of the aberroscope and the inversion of the optics of the eye the retinal coordinate system is rotated and reversed with respect to that of the pupil. III. Subject’s view of the retinal x′, y′ coordinate system and point p. Note that points at the top of the subject’s pupil now appear to his right and vice versa.

FIG. 6
FIG. 6

Extraction of Taylor coefficients from data of Smirnov (Ref. 2). (a) Fitting of 5×5 grid to original data. (b) Contour lines drawn by eye through computed wave aberration from fitted Taylor coefficients: A = 0. 27, B = −0. 01, C = −0. 14, D = −0. 33, E = −0. 22, F = 0. 95, G = 0. 01, H = −0. 03, J = 0. 02, K = 0. 03, L = 0 01, M = −0. 07, N = 0. 01, O = 0. 03. Numbers represent elevations in λ of wave aberration surface.

FIG. 7
FIG. 7

Means, standard deviations, and standard errors of means for the data of the present paper (solid bars, n = 55) and the data of Smirnov (Ref. 2) (open bars, n = 7). Smirnov’s coefficients K and M are significantly different from zero (p ≤ 0. 05, p ≤ 0. 001, one tailed test) and our coefficients K, M, and O are also significantly different from zero (p ≤ 0. 04, 0. 04 and 0. 006 respectively, one tailed test). Use of the one-tailed test is justified because due to the previously reported positive spherical aberration of the eye, all of these coefficients may be expected to be greater than zero as indeed they are. There is also a significant difference in the variance of the J coefficients (p ≤ 0. 01, F Variance ratio test) between the two data sets.

FIG. 8
FIG. 8

The fourth-order Taylor coefficients K and O plotted against each other for the eyes of 55 subjects. Subjects showing classical spherical aberration are represented by points on or near the line K = 0. Points lying on or near the axes at some distance from the origin represent eyes with “cylindrical” aberrations.

FIG. 9
FIG. 9

Apportionment of mean-square wave-front deviations (MSWDs) among Zernike and Taylor terms averaged over 28 eyes as a function of pupil diameter. The eyes used were those of the 25th to the 75th percentile in the MSWD rank order for d = 5 mm. For third-order aberrations Z ¯ n = 100 ( C n F n ) 2 r 6 / ( π W ¯ 2 ), for fourth order: Z ¯ n = 100 ( C n F n ) 2 r 8 / π W ¯ 2).

FIG. 10
FIG. 10

Rank order of mean-square deviations of calculated wave fronts form spherocylindrical surface for 55 eyes with 5, 4, and 3 mm pupil diameters.

FIG. 11
FIG. 11

Cumulative distribution of diffraction limited pupil sizes for the 55 subjects of this study.

FIG. 12
FIG. 12

Calculated monochromatic modulation transfer curves for 5 mm pupil diameter for selected eyes from set of 55 rank ordered eyes. 100% designates best eye. All curves computed by the method of Hopkins (1962). Ordinate is the average of horizontal and vertical meridian best-focus MTFs. Abcissa gives cycles per degree at the retina for standard eye.

FIG. 13
FIG. 13

Frequency distribution for Zernike coefficient, Z11 (spherical aberration). Arrow indicates mean value.

Tables (6)

Tables Icon

TABLE I Mean Zernike coefficients for 55 eyes of this study.

Tables Icon

TABLE II Single Taylor coefficients for freehand drawings and mean values and standard deviations for 10 camera Lucida trials.

Tables Icon

TABLE III Root-mean-square deviations of wave aberration surface at 5 mm pupil as estimated by freehand and camera Lucida drawings.

Tables Icon

TABLE AI An orthonormal polynomial for n = 4 over the interval −1, +1; −1, +1.

Tables Icon

TABLE A II An orthonormal polynomial for n = 5 over the interval −1, +1; −1, +1.

Tables Icon

TABLE B I Zernike polynomial terms and coefficients.

Equations (24)

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W ( x , y ) = A + B x + C y + D x 2 + Exy + F y 2 + G x 3 + H x 2 y + I x y 2 + J y 3 + K x 4 + L x 3 y + M x 2 y 2 + N x y 3 + O y 4 .
Δ x = F ( z / y ) ,
Δ y = F ( z / x ) ,
Δ X = ( z / y ) [ 1 / ( N ) ] ,
Δ Y = ( z / y ) [ 1 / ( N ) ] ,
g ( x , y ) = k 1 L 1 + k 2 L 2 + + k 15 L 15 ,
( 1 / n 2 ) x = 1 + 1 y = 1 + 1 L j ( x , y ) L k ( x , y ) = 0 for j k , = 1 for j = k .
K j = ( 1 / n 2 ) x = 1 + 1 y ¯ = 1 + 1 L j ( x , y ) Z y ( x , y ) .
z / x = B + 2 D x + 3 N y 3
z / y = C + E x + 40 y 3 ,
G = [ k 5 / a 5 + ( k 6 / a 6 ) ] / 3 Q 2 ,
O = ( k 7 / a 7 ) / ( 4 Q 3 ) .
LSA = 24 C 11 r 2 = 0. 0342 r 2
π / 8
π / 8
π / 8
π / 8
π / 5
π / 10
π / 10
π / 10
π / 10
W ( X , Y ) = n = 7 10 C n Z n r 3 + n = 11 15 C n Z n r 4 .
W ¯ 2 = ( 1 / π ) n = 7 10 ( C n F n ) 2 r 6 + ( 1 / π ) n = 11 15 ( C n F n ) 2 r 8 ,