Abstract

New technologies for accurately measuring corneal shape and full eye aberrations are now available. An algorithm that uses these technologies to predict the amount of ablation needed to produce a corneal surface that optimally focuses light is developed. It is found that knowledge of the aberrations is far more important than knowledge of corneal shape. Neglect of corneal shape information introduces an error of less than approximately 0.05 µm in the optimal ablation depth. Neglect of the aberrations is a different story. Small changes in the aberration structure, such as going from the optimal ablation to a spherical ablation, introduce ablation changes of greater than 10 µm. It is argued that there are many occasions when less ablation can lead to improved image quality.

© 1998 Optical Society of America

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References

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  1. G. O. Waring, “Quality of vision and freedom from optical correction after refractive surgery,” J. Refract. Surg. 13, 213–215 (1997); R. L. Lindstrom, “The Barraquer lecture: surgical management of myopia—a clinician’s perspective,” J. Refract. Surg. 13, 287–294 (1997); T. P. Werblin, “20/20—How close must we get?” J. Refract. Surg. 13, 300–301 (1997).
    [PubMed]
  2. The November 1997 issue of Optometry and Vision Science was a special issue on corneal topography containing papers on the state of the art in corneal topography, including several from the corneal topographer manufacturers. R. Mattioli, N. K. Tripoli, “Corneal geometry reconstruction with the Keratron videokeratographer,” Optom. Vision Sci. 74, 881–894 (1997); D. Brenner, “Modeling the cornea with the topographic modeling system videokeratoscope,” Optom. Vision Sci. 74, 895–898 (1997); C. Campbell, “Reconstruction of the corneal shape with the MasterVue Corneal Topography System,” Optom. Vision Sci. 74, 899–905 (1997).
    [CrossRef]
  3. J. Liang, B. Grimm, S. Goelz, J. Bille, “Objective measurement of the wave aberrations of the human eye with the use of a Hartmann–Shack wave-front sensor,” J. Opt. Soc. Am. A 11, 1949–1957 (1994).
    [CrossRef]
  4. J. Liang, D. R. Williams, “Aberrations and retinal image quality of the normal human eye,” J. Opt. Soc. Am. A 14, 2873–2883 (1997).
    [CrossRef]
  5. J. Liang, D. R. Williams, D. T. Miller, “Supernormal vision and high-resolution retinal imaging through adaptive optics,” J. Opt. Soc. Am. A 14, 2884–2892 (1997).
    [CrossRef]
  6. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. A 70, 998–1006 (1980).
    [CrossRef]
  7. J. Liang, “A new method to precisely measure the wave aberrations of the human eye with a Hartmann–Shack wavefront sensor,” Ph.D. dissertation (Universität Heidelberg, Heidelberg, Germany, 1991).
  8. S. A. Klein, R. B. Mandell, “Shape and refractive powers in corneal topography,” Invest. Ophthalmol. Visual Sci. 96, 2096–2109 (1995).
  9. M. S. Smirnov, “Measurement of the wave aberration of the human eye,” Biophys. J. 7, 766–795 (1962).
  10. M. C. Campbell, E. M. Harrison, P. Simonet, “Psychophysical measurement of the blur on the retina due to optical aberrations of the eye,” Vision Res. 30, 1587–1602 (1990).
    [CrossRef] [PubMed]
  11. R. H. Webb, C. M. Penney, K. P. Thompson, “Measurement of ocular local wavefront distortion with a spatially resolved refractometer,” Appl. Opt. 31, 3678–3686 (1992).
    [CrossRef] [PubMed]
  12. M. Tscherning, “Die monochromatischen Abberationen des menschlichen Auges,” Z. Psychol. Physiol. Sinn. 6, 456–471 (1894).
  13. B. Howland, H. C. Howland, “Subjective measurement of high-order aberrations of the eye,” Science 193, 580–582 (1976).
    [CrossRef] [PubMed]
  14. H. C. Howland, B. Howland, “A subjective method for the measurement of monochromatic aberrations of the eye,” J. Opt. Soc. Am. 67, 1508–1518 (1977).
    [CrossRef] [PubMed]
  15. A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, UK, 1995).
  16. R. P. Feynman, QED. The Strange Theory of Light and Matter (Princeton U. Press, Princeton, N.J., 1985).
  17. S. A. Klein, B. A. Barsky, “Method for generating the anterior surface of an aberration-free contact lens for an arbitrary posterior surface,” Optom. Vision Sci. 72, 816–820 (1995).
    [CrossRef]
  18. G. Walsh, W. N. Charman, H. C. Howland, “Objective technique for the determination of monochromatic aberrations of the human eye,” J. Opt. Soc. Am. A 1, 987–992 (1984).
    [CrossRef] [PubMed]
  19. R. A. Applegate, K. A. Gansel, “The importance of pupil size in optical quality measurements following radial keratotomy,” Refract. Corneal Surg. 6, 47–54 (1990).
    [PubMed]
  20. H. C. Howland, R. H. Rand, S. R. Lubkin, “A thin shell model of the cornea and its application to corneal surgery,” Refract. Corneal Surg. 8, 183–186 (1992).
    [PubMed]

1997 (4)

G. O. Waring, “Quality of vision and freedom from optical correction after refractive surgery,” J. Refract. Surg. 13, 213–215 (1997); R. L. Lindstrom, “The Barraquer lecture: surgical management of myopia—a clinician’s perspective,” J. Refract. Surg. 13, 287–294 (1997); T. P. Werblin, “20/20—How close must we get?” J. Refract. Surg. 13, 300–301 (1997).
[PubMed]

The November 1997 issue of Optometry and Vision Science was a special issue on corneal topography containing papers on the state of the art in corneal topography, including several from the corneal topographer manufacturers. R. Mattioli, N. K. Tripoli, “Corneal geometry reconstruction with the Keratron videokeratographer,” Optom. Vision Sci. 74, 881–894 (1997); D. Brenner, “Modeling the cornea with the topographic modeling system videokeratoscope,” Optom. Vision Sci. 74, 895–898 (1997); C. Campbell, “Reconstruction of the corneal shape with the MasterVue Corneal Topography System,” Optom. Vision Sci. 74, 899–905 (1997).
[CrossRef]

J. Liang, D. R. Williams, “Aberrations and retinal image quality of the normal human eye,” J. Opt. Soc. Am. A 14, 2873–2883 (1997).
[CrossRef]

J. Liang, D. R. Williams, D. T. Miller, “Supernormal vision and high-resolution retinal imaging through adaptive optics,” J. Opt. Soc. Am. A 14, 2884–2892 (1997).
[CrossRef]

1995 (2)

S. A. Klein, R. B. Mandell, “Shape and refractive powers in corneal topography,” Invest. Ophthalmol. Visual Sci. 96, 2096–2109 (1995).

S. A. Klein, B. A. Barsky, “Method for generating the anterior surface of an aberration-free contact lens for an arbitrary posterior surface,” Optom. Vision Sci. 72, 816–820 (1995).
[CrossRef]

1994 (1)

1992 (2)

H. C. Howland, R. H. Rand, S. R. Lubkin, “A thin shell model of the cornea and its application to corneal surgery,” Refract. Corneal Surg. 8, 183–186 (1992).
[PubMed]

R. H. Webb, C. M. Penney, K. P. Thompson, “Measurement of ocular local wavefront distortion with a spatially resolved refractometer,” Appl. Opt. 31, 3678–3686 (1992).
[CrossRef] [PubMed]

1990 (2)

R. A. Applegate, K. A. Gansel, “The importance of pupil size in optical quality measurements following radial keratotomy,” Refract. Corneal Surg. 6, 47–54 (1990).
[PubMed]

M. C. Campbell, E. M. Harrison, P. Simonet, “Psychophysical measurement of the blur on the retina due to optical aberrations of the eye,” Vision Res. 30, 1587–1602 (1990).
[CrossRef] [PubMed]

1984 (1)

1980 (1)

W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. A 70, 998–1006 (1980).
[CrossRef]

1977 (1)

1976 (1)

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

1962 (1)

M. S. Smirnov, “Measurement of the wave aberration of the human eye,” Biophys. J. 7, 766–795 (1962).

1894 (1)

M. Tscherning, “Die monochromatischen Abberationen des menschlichen Auges,” Z. Psychol. Physiol. Sinn. 6, 456–471 (1894).

Applegate, R. A.

R. A. Applegate, K. A. Gansel, “The importance of pupil size in optical quality measurements following radial keratotomy,” Refract. Corneal Surg. 6, 47–54 (1990).
[PubMed]

Barsky, B. A.

S. A. Klein, B. A. Barsky, “Method for generating the anterior surface of an aberration-free contact lens for an arbitrary posterior surface,” Optom. Vision Sci. 72, 816–820 (1995).
[CrossRef]

Bille, J.

Campbell, M. C.

M. C. Campbell, E. M. Harrison, P. Simonet, “Psychophysical measurement of the blur on the retina due to optical aberrations of the eye,” Vision Res. 30, 1587–1602 (1990).
[CrossRef] [PubMed]

Charman, W. N.

Feynman, R. P.

R. P. Feynman, QED. The Strange Theory of Light and Matter (Princeton U. Press, Princeton, N.J., 1985).

Gansel, K. A.

R. A. Applegate, K. A. Gansel, “The importance of pupil size in optical quality measurements following radial keratotomy,” Refract. Corneal Surg. 6, 47–54 (1990).
[PubMed]

Goelz, S.

Grimm, B.

Harrison, E. M.

M. C. Campbell, E. M. Harrison, P. Simonet, “Psychophysical measurement of the blur on the retina due to optical aberrations of the eye,” Vision Res. 30, 1587–1602 (1990).
[CrossRef] [PubMed]

Howland, B.

Howland, H. C.

H. C. Howland, R. H. Rand, S. R. Lubkin, “A thin shell model of the cornea and its application to corneal surgery,” Refract. Corneal Surg. 8, 183–186 (1992).
[PubMed]

G. Walsh, W. N. Charman, H. C. Howland, “Objective technique for the determination of monochromatic aberrations of the human eye,” J. Opt. Soc. Am. A 1, 987–992 (1984).
[CrossRef] [PubMed]

H. C. Howland, B. Howland, “A subjective method for the measurement of monochromatic aberrations of the eye,” J. Opt. Soc. Am. 67, 1508–1518 (1977).
[CrossRef] [PubMed]

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

Klein, S. A.

S. A. Klein, B. A. Barsky, “Method for generating the anterior surface of an aberration-free contact lens for an arbitrary posterior surface,” Optom. Vision Sci. 72, 816–820 (1995).
[CrossRef]

S. A. Klein, R. B. Mandell, “Shape and refractive powers in corneal topography,” Invest. Ophthalmol. Visual Sci. 96, 2096–2109 (1995).

Liang, J.

Lubkin, S. R.

H. C. Howland, R. H. Rand, S. R. Lubkin, “A thin shell model of the cornea and its application to corneal surgery,” Refract. Corneal Surg. 8, 183–186 (1992).
[PubMed]

Mandell, R. B.

S. A. Klein, R. B. Mandell, “Shape and refractive powers in corneal topography,” Invest. Ophthalmol. Visual Sci. 96, 2096–2109 (1995).

Mattioli, R.

The November 1997 issue of Optometry and Vision Science was a special issue on corneal topography containing papers on the state of the art in corneal topography, including several from the corneal topographer manufacturers. R. Mattioli, N. K. Tripoli, “Corneal geometry reconstruction with the Keratron videokeratographer,” Optom. Vision Sci. 74, 881–894 (1997); D. Brenner, “Modeling the cornea with the topographic modeling system videokeratoscope,” Optom. Vision Sci. 74, 895–898 (1997); C. Campbell, “Reconstruction of the corneal shape with the MasterVue Corneal Topography System,” Optom. Vision Sci. 74, 899–905 (1997).
[CrossRef]

Miller, D. T.

Penney, C. M.

Rand, R. H.

H. C. Howland, R. H. Rand, S. R. Lubkin, “A thin shell model of the cornea and its application to corneal surgery,” Refract. Corneal Surg. 8, 183–186 (1992).
[PubMed]

Simonet, P.

M. C. Campbell, E. M. Harrison, P. Simonet, “Psychophysical measurement of the blur on the retina due to optical aberrations of the eye,” Vision Res. 30, 1587–1602 (1990).
[CrossRef] [PubMed]

Smirnov, M. S.

M. S. Smirnov, “Measurement of the wave aberration of the human eye,” Biophys. J. 7, 766–795 (1962).

Southwell, W. H.

W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. A 70, 998–1006 (1980).
[CrossRef]

Thompson, K. P.

Tripoli, N. K.

The November 1997 issue of Optometry and Vision Science was a special issue on corneal topography containing papers on the state of the art in corneal topography, including several from the corneal topographer manufacturers. R. Mattioli, N. K. Tripoli, “Corneal geometry reconstruction with the Keratron videokeratographer,” Optom. Vision Sci. 74, 881–894 (1997); D. Brenner, “Modeling the cornea with the topographic modeling system videokeratoscope,” Optom. Vision Sci. 74, 895–898 (1997); C. Campbell, “Reconstruction of the corneal shape with the MasterVue Corneal Topography System,” Optom. Vision Sci. 74, 899–905 (1997).
[CrossRef]

Tscherning, M.

M. Tscherning, “Die monochromatischen Abberationen des menschlichen Auges,” Z. Psychol. Physiol. Sinn. 6, 456–471 (1894).

Walsh, G.

Walther, A.

A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, UK, 1995).

Waring, G. O.

G. O. Waring, “Quality of vision and freedom from optical correction after refractive surgery,” J. Refract. Surg. 13, 213–215 (1997); R. L. Lindstrom, “The Barraquer lecture: surgical management of myopia—a clinician’s perspective,” J. Refract. Surg. 13, 287–294 (1997); T. P. Werblin, “20/20—How close must we get?” J. Refract. Surg. 13, 300–301 (1997).
[PubMed]

Webb, R. H.

Williams, D. R.

Appl. Opt. (1)

Biophys. J. (1)

M. S. Smirnov, “Measurement of the wave aberration of the human eye,” Biophys. J. 7, 766–795 (1962).

Invest. Ophthalmol. Visual Sci. (1)

S. A. Klein, R. B. Mandell, “Shape and refractive powers in corneal topography,” Invest. Ophthalmol. Visual Sci. 96, 2096–2109 (1995).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

J. Refract. Surg. (1)

G. O. Waring, “Quality of vision and freedom from optical correction after refractive surgery,” J. Refract. Surg. 13, 213–215 (1997); R. L. Lindstrom, “The Barraquer lecture: surgical management of myopia—a clinician’s perspective,” J. Refract. Surg. 13, 287–294 (1997); T. P. Werblin, “20/20—How close must we get?” J. Refract. Surg. 13, 300–301 (1997).
[PubMed]

Optom. Vision Sci. (2)

The November 1997 issue of Optometry and Vision Science was a special issue on corneal topography containing papers on the state of the art in corneal topography, including several from the corneal topographer manufacturers. R. Mattioli, N. K. Tripoli, “Corneal geometry reconstruction with the Keratron videokeratographer,” Optom. Vision Sci. 74, 881–894 (1997); D. Brenner, “Modeling the cornea with the topographic modeling system videokeratoscope,” Optom. Vision Sci. 74, 895–898 (1997); C. Campbell, “Reconstruction of the corneal shape with the MasterVue Corneal Topography System,” Optom. Vision Sci. 74, 899–905 (1997).
[CrossRef]

S. A. Klein, B. A. Barsky, “Method for generating the anterior surface of an aberration-free contact lens for an arbitrary posterior surface,” Optom. Vision Sci. 72, 816–820 (1995).
[CrossRef]

Refract. Corneal Surg. (2)

R. A. Applegate, K. A. Gansel, “The importance of pupil size in optical quality measurements following radial keratotomy,” Refract. Corneal Surg. 6, 47–54 (1990).
[PubMed]

H. C. Howland, R. H. Rand, S. R. Lubkin, “A thin shell model of the cornea and its application to corneal surgery,” Refract. Corneal Surg. 8, 183–186 (1992).
[PubMed]

Science (1)

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

Vision Res. (1)

M. C. Campbell, E. M. Harrison, P. Simonet, “Psychophysical measurement of the blur on the retina due to optical aberrations of the eye,” Vision Res. 30, 1587–1602 (1990).
[CrossRef] [PubMed]

Z. Psychol. Physiol. Sinn. (1)

M. Tscherning, “Die monochromatischen Abberationen des menschlichen Auges,” Z. Psychol. Physiol. Sinn. 6, 456–471 (1894).

Other (3)

A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, UK, 1995).

R. P. Feynman, QED. The Strange Theory of Light and Matter (Princeton U. Press, Princeton, N.J., 1985).

J. Liang, “A new method to precisely measure the wave aberrations of the human eye with a Hartmann–Shack wavefront sensor,” Ph.D. dissertation (Universität Heidelberg, Heidelberg, Germany, 1991).

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Figures (3)

Fig. 1
Fig. 1

Geometry of the HS sensors in relation to measuring aberrations of the eye. Light is focused on the retina from a light source that is not shown. The light not absorbed by the retina then travels to the left, being refracted by the lens and cornea of the eye. After emerging from the eye, it gets focused by one of the HS lenslets onto the CCD array. For simplicity of discussion, the rays are considered to travel in reverse. Thus Eqs. (1) and (2) of the present algorithm are based on rays traveling from the HS sensor toward the cornea in the direction VˆHS. After the rays are refracted by the cornea, their direction becomes , as indicated in (a). Inset: Blown-up version of one of the lenslets, with two adjacent rays shown striking the HS plane at points a and b. The triangle in the inset is used in the derivation of Eq. (9). (b) shows a portion of (a). Dark-shaded region, the region of the cornea that is to be ablated for a myopic correction. Light-shaded region, the eye after the ablation. The distance from the HS sensor to the cornea, DHS-c, is calculated by an iterative method in step 2 of the algorithm. The distance from the preablated to the postablated cornea, Dc-A, is calculated in step 7. In actual practice the HS array is placed conjugate to the eye’s entrance pupil (roughly 3 mm behind the cornea), by use of relay lenses, rather than in front of the cornea as shown in this diagram. (c) Similar to (b), except that it is for correcting a hyperopic eye. In (c) the reference plane for measuring the distance to the postablated eye is coincident with the HS plane. In (b) the reference plane is to the left of the HS plane, resulting in a deeper ablation near the axis than is the case with hyperopic correction.

Fig. 2
Fig. 2

Maximum difference in ablation owing to the exact algorithm versus the algorithm that ignores corneal shape by replacing the cornea with a flat plane. The corneal shapes tested were axially symmetric ellipsoids whose apical radii ranged from R=6.5 to R=10 mm, shown on abscissa, and whose p value (oblateness) was either 0 (paraboloid) or 1 (sphere). The plots show that a maximum error of 0.25 µm would be made by this approximation. The horizontal line that intersects the p=1 curve at R=8 mm provides an estimate of the error incurred by replacement of the flat plane approximation with an approximate shape (an 8-mm sphere in this case). When a reasonable corneal shape is used together with the exact algorithm, the ablation error is reduced to less than 0.05 µm, which is smaller than the intrinsic errors associated with estimating the aberrations.

Fig. 3
Fig. 3

Effect of oblateness (p value) on the height and refractive power of an ellipsoidal single-surface schematic eye. In both panels the abscissa is the oblateness of the axially symmetric ellipsoid, the solid line is for an ellipsoid whose apical refractive power is 45 D, and the dashed line is for an ellipsoid whose apical power has been adjusted [see Eq. (31)] so that the refractive power r=1 mm is 45 D, where r specifies the distance of the corneal point from the axis. The r=1 mm point is chosen because it is close to the average refractive power across the pupil. The two lines intersect at p=p0=1-e2=1-1/n2=0.441, which is the oblateness for an ellipsoid with zero spherical aberration. For this value of p, all corneal points have the same refractive power for incoming axial rays. (a) shows the height (z distance) from the vertex to the ellipsoid at r=3 mm, as given by Eq. (27). The quantity of interest for a myopic correction is the difference in height from p=p0, the optimal postablation value, to a larger oblateness, such as p=1, for a sphere. For a hyperopic correction the relevant height difference would vary from p=p0 to lower values of p. (b) shows the refractive power at r=1 mm. This plot is useful for checking that the change in oblateness does not cause a large change in the refractive power across the optic zone of the cornea.

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

xc=xHS+(zc-zHS)SHS-x,
yc=yHS+(zc-zHS)SHS-y,
SHS-x=VHS-x/VHS-z,
SHS-y=VHS-y/VHS-z.
n2Vˆ=n1VˆHS+γHS(xc, yc)nˆ(xc, yc),
n2 sin(θ2)=n1 sin(θ1).
γHS=n2 cos(θ2)-n1 cos(θ1),
L(xHS, yHS)=λ/(n12π)ϕ(xHS, yHS),
L(xHS, yHS)=j,kajkxHSjyHSk.
ΔL=VˆHS*ΔrHS
=VHS-xΔxHS+VHS-yΔyHS,
Vx(xHS, yHS)=L/xHS=j,kjajkxHSj-1yHSk,
Vy(xHS, yHS)=L/yHS=j,kkajkxHSjyHSk-1.
OPLHS=L(xHS, yHS)+DHS-c+nDc-A,
DHS-c=(zc+d)/cos(θHS).
cos(θHS)=VHS-z,
OPLA=E+zc+Dc-A cos(θc),
cos(θc)=Vz,
L+(zc+d)/VHS-z+nDc-A=E+zc+Dc-AVz,
Dc-A=[E+zc-L-(zc+d)/VHS-z]/(n-Vz).
E=-min[zc-L-(zc+d)/VHS-z],
E=-max[zc-L-(zc+d)/VHS-z],
xA=xc+Dc-AVx,
yA=yc+Dc-AVy,
zA=zc+Dc-AVz.
D(x, y)=[E-d/VHS-z-L(x, y)]/(n-1).
D(x, y)={d[max(1/VHS-z)-1/VHS-z]+max(L)-L(x, y)}/(n-1),
d[max(1/VHS-z)-1/VHS-z]/(n-1)
d[1-cos(θ)]/(n-1)dθ2/2(n-1)-3(0.003×5)2/2(n-1)-1.0µm.
depth=Pr2/2(n-1)=5×32/(2×0.376)60 µm,
SHS-x=xHS/(zHS-u),
SHS-y=yHS/(zHS-u),
p=1-e2.
z=r2/[R+(R2-pr2)1/2]
z=[R-(R2-pr2)1/2]/p,
heightdifference:32/[R+(R2-p32)1/2].
P=n/f,
f=z+r/tan(θ),
R=7.5+0.0666(p-p0),
p0=1-e2=1-1/n2=0.441,

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