Abstract

A modified Hartmann test is proposed for measuring corneal topography. The plane screen with holes used in the typical Hartmann test is replaced with a curved object surface. This object surface yields a plane image for a spherical mirror surface. We show that the object surface is an oval of revolution that can be modeled by an ellipsoid. The plane image will be formed by a square array of circular spots, all with the same diameter. To obtain the square array in the image, we calculated the spatial distribution of the spots on the object surface.

© 2001 Optical Society of America

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References

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  1. A. Gullstrand, in Helmholtz’s Treatise on Physiological Optics, J. P. C. Southall, ed. (Optical Society of America, Washington, D.C., 1924), Vol. 1, 305–358.
  2. H. A. Knoll, R. Stimson, C. L. Weeks, “New photokeratoscope utilizing a hemispherical object surface,” J. Opt. Soc. Am. 47, 221–222 (1957).
    [CrossRef] [PubMed]
  3. H. M. Dekking, “Zyr Photographie der Hornhautoberfläche,” Albrecht von Graefes Arch. Ophthalmol. 124, 708–730 (1930).
    [CrossRef]
  4. A. H. Knoll, “Corneal contours in the general population as revelated by the photokerastoscope,” Am. J. Optom. 38, 389–397 (1961).
    [CrossRef]
  5. Keratoscope, Optikon.
  6. A. E. Conrady, Applied Optics and Optical Design (Dover, New York, 1985).
  7. D. Malacara, Z. Malacara, Handbook of Lens Design (Marcel Dekker, New York, 1994).
  8. S. Wittenberg, W. M. Ludlam, “Planar reflected imagery in photokeratoscopy,” J. Opt. Soc. Am. 60, 981–985 (1970).
    [CrossRef] [PubMed]
  9. S. A. Klein, “Corneal topography reconstruction algorithm that avoids the skew ray ambiguity and the skew ray error,” Optom. Vis. Sci. 74, 931–944 (1997).
    [CrossRef] [PubMed]
  10. R. H. Rand, H. C. Howland, R. A. Applegate, “Mathematical model of a Placido disk keratometer and its implications for recovery of corneal topography,” Optom. Vis. Sci. 74, 926–930 (1997).
    [CrossRef] [PubMed]
  11. M. A. Halstead, B. A. Barsky, S. A. Klein, R. B. Mandell, “A spline surface algorithm for reconstruction of the corneal topography from a video-keratographic reflection pattern,” Optom. Vis. Sci. 72, 821–827 (1995).
    [CrossRef] [PubMed]
  12. W. E. Humphrey, “Method and apparatus for analysis of corneal shape,” U.S. patent4,420,228 (13Dec.1983).
  13. D. Malacara, Optical Shop Testing (Wiley, New York, 1992).
  14. E. Kreyszig, Differential Geometry (Dover, New York, 1991).
  15. D. Malacara, Z. Malacara, “Testing and centering of the lenses by means of a Hartmann test with four holes,” Opt. Eng. 31, 1551–1555 (1992).
    [CrossRef]

1997 (2)

S. A. Klein, “Corneal topography reconstruction algorithm that avoids the skew ray ambiguity and the skew ray error,” Optom. Vis. Sci. 74, 931–944 (1997).
[CrossRef] [PubMed]

R. H. Rand, H. C. Howland, R. A. Applegate, “Mathematical model of a Placido disk keratometer and its implications for recovery of corneal topography,” Optom. Vis. Sci. 74, 926–930 (1997).
[CrossRef] [PubMed]

1995 (1)

M. A. Halstead, B. A. Barsky, S. A. Klein, R. B. Mandell, “A spline surface algorithm for reconstruction of the corneal topography from a video-keratographic reflection pattern,” Optom. Vis. Sci. 72, 821–827 (1995).
[CrossRef] [PubMed]

1992 (1)

D. Malacara, Z. Malacara, “Testing and centering of the lenses by means of a Hartmann test with four holes,” Opt. Eng. 31, 1551–1555 (1992).
[CrossRef]

1970 (1)

1961 (1)

A. H. Knoll, “Corneal contours in the general population as revelated by the photokerastoscope,” Am. J. Optom. 38, 389–397 (1961).
[CrossRef]

1957 (1)

1930 (1)

H. M. Dekking, “Zyr Photographie der Hornhautoberfläche,” Albrecht von Graefes Arch. Ophthalmol. 124, 708–730 (1930).
[CrossRef]

Applegate, R. A.

R. H. Rand, H. C. Howland, R. A. Applegate, “Mathematical model of a Placido disk keratometer and its implications for recovery of corneal topography,” Optom. Vis. Sci. 74, 926–930 (1997).
[CrossRef] [PubMed]

Barsky, B. A.

M. A. Halstead, B. A. Barsky, S. A. Klein, R. B. Mandell, “A spline surface algorithm for reconstruction of the corneal topography from a video-keratographic reflection pattern,” Optom. Vis. Sci. 72, 821–827 (1995).
[CrossRef] [PubMed]

Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design (Dover, New York, 1985).

Dekking, H. M.

H. M. Dekking, “Zyr Photographie der Hornhautoberfläche,” Albrecht von Graefes Arch. Ophthalmol. 124, 708–730 (1930).
[CrossRef]

Gullstrand, A.

A. Gullstrand, in Helmholtz’s Treatise on Physiological Optics, J. P. C. Southall, ed. (Optical Society of America, Washington, D.C., 1924), Vol. 1, 305–358.

Halstead, M. A.

M. A. Halstead, B. A. Barsky, S. A. Klein, R. B. Mandell, “A spline surface algorithm for reconstruction of the corneal topography from a video-keratographic reflection pattern,” Optom. Vis. Sci. 72, 821–827 (1995).
[CrossRef] [PubMed]

Howland, H. C.

R. H. Rand, H. C. Howland, R. A. Applegate, “Mathematical model of a Placido disk keratometer and its implications for recovery of corneal topography,” Optom. Vis. Sci. 74, 926–930 (1997).
[CrossRef] [PubMed]

Humphrey, W. E.

W. E. Humphrey, “Method and apparatus for analysis of corneal shape,” U.S. patent4,420,228 (13Dec.1983).

Klein, S. A.

S. A. Klein, “Corneal topography reconstruction algorithm that avoids the skew ray ambiguity and the skew ray error,” Optom. Vis. Sci. 74, 931–944 (1997).
[CrossRef] [PubMed]

M. A. Halstead, B. A. Barsky, S. A. Klein, R. B. Mandell, “A spline surface algorithm for reconstruction of the corneal topography from a video-keratographic reflection pattern,” Optom. Vis. Sci. 72, 821–827 (1995).
[CrossRef] [PubMed]

Knoll, A. H.

A. H. Knoll, “Corneal contours in the general population as revelated by the photokerastoscope,” Am. J. Optom. 38, 389–397 (1961).
[CrossRef]

Knoll, H. A.

Kreyszig, E.

E. Kreyszig, Differential Geometry (Dover, New York, 1991).

Ludlam, W. M.

Malacara, D.

D. Malacara, Z. Malacara, “Testing and centering of the lenses by means of a Hartmann test with four holes,” Opt. Eng. 31, 1551–1555 (1992).
[CrossRef]

D. Malacara, Optical Shop Testing (Wiley, New York, 1992).

D. Malacara, Z. Malacara, Handbook of Lens Design (Marcel Dekker, New York, 1994).

Malacara, Z.

D. Malacara, Z. Malacara, “Testing and centering of the lenses by means of a Hartmann test with four holes,” Opt. Eng. 31, 1551–1555 (1992).
[CrossRef]

D. Malacara, Z. Malacara, Handbook of Lens Design (Marcel Dekker, New York, 1994).

Mandell, R. B.

M. A. Halstead, B. A. Barsky, S. A. Klein, R. B. Mandell, “A spline surface algorithm for reconstruction of the corneal topography from a video-keratographic reflection pattern,” Optom. Vis. Sci. 72, 821–827 (1995).
[CrossRef] [PubMed]

Rand, R. H.

R. H. Rand, H. C. Howland, R. A. Applegate, “Mathematical model of a Placido disk keratometer and its implications for recovery of corneal topography,” Optom. Vis. Sci. 74, 926–930 (1997).
[CrossRef] [PubMed]

Stimson, R.

Weeks, C. L.

Wittenberg, S.

Albrecht von Graefes Arch. Ophthalmol. (1)

H. M. Dekking, “Zyr Photographie der Hornhautoberfläche,” Albrecht von Graefes Arch. Ophthalmol. 124, 708–730 (1930).
[CrossRef]

Am. J. Optom. (1)

A. H. Knoll, “Corneal contours in the general population as revelated by the photokerastoscope,” Am. J. Optom. 38, 389–397 (1961).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Eng. (1)

D. Malacara, Z. Malacara, “Testing and centering of the lenses by means of a Hartmann test with four holes,” Opt. Eng. 31, 1551–1555 (1992).
[CrossRef]

Optom. Vis. Sci. (3)

S. A. Klein, “Corneal topography reconstruction algorithm that avoids the skew ray ambiguity and the skew ray error,” Optom. Vis. Sci. 74, 931–944 (1997).
[CrossRef] [PubMed]

R. H. Rand, H. C. Howland, R. A. Applegate, “Mathematical model of a Placido disk keratometer and its implications for recovery of corneal topography,” Optom. Vis. Sci. 74, 926–930 (1997).
[CrossRef] [PubMed]

M. A. Halstead, B. A. Barsky, S. A. Klein, R. B. Mandell, “A spline surface algorithm for reconstruction of the corneal topography from a video-keratographic reflection pattern,” Optom. Vis. Sci. 72, 821–827 (1995).
[CrossRef] [PubMed]

Other (7)

W. E. Humphrey, “Method and apparatus for analysis of corneal shape,” U.S. patent4,420,228 (13Dec.1983).

D. Malacara, Optical Shop Testing (Wiley, New York, 1992).

E. Kreyszig, Differential Geometry (Dover, New York, 1991).

Keratoscope, Optikon.

A. E. Conrady, Applied Optics and Optical Design (Dover, New York, 1985).

D. Malacara, Z. Malacara, Handbook of Lens Design (Marcel Dekker, New York, 1994).

A. Gullstrand, in Helmholtz’s Treatise on Physiological Optics, J. P. C. Southall, ed. (Optical Society of America, Washington, D.C., 1924), Vol. 1, 305–358.

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Figures (11)

Fig. 1
Fig. 1

In the paraxial approximation a virtual object of height h separated from the vertex of the mirror of radius R by l yielding a real image of height h′ separated from the vertex of the mirror by -l′. The system stop is on the image plane. For a small field the sagittal longitudinal astigmatism can be calculated by the third-order aberration theory.

Fig. 2
Fig. 2

Petzval and astigmatic curves for the mirror of Fig. 1 on a meridional plane. Because the stop is at the Gaussian image plane the longitudinal astigmatism does not depend on the location of the stop.

Fig. 3
Fig. 3

Geometry used to apply the Coddington equations for a spherical mirror: P, point on the object surface; Q, point on the mirror; I, virtual image point; -L, object distance; and L′, image distance that can be either the sagittal or the tangential image distance. The stop distance l V from the mirror is equal to the vertex distance from the mirror of the object surface. The axial conjugate point is at a distance l V ′ = R - z I .

Fig. 4
Fig. 4

Ovals ΣS, ΣT, and ΣM for obtaining plane sagittal, tangential, and mean image surfaces for a spherical mirror and for a wide field, respectively. Oval ΣP is an ellipsoidal surface that yields a Petzval plane surface.

Fig. 5
Fig. 5

Radius of curvature of a meridional curve of object surfaces ΣS and ΣT (heavy curves) when the radius of the mirror is R = 7.8 mm and the stop is |l V | = 180 mm away from the mirror. Angle α is measured from the center of an ellipse that fits each oval (thin curves). The closest ellipses are calculated by using the minimum and the maximum radius of curvature of the ovals. These radii of curvature should be equal to the radii of curvature at the vertices (α = 0 and α = π/2) of the closest ellipse.

Fig. 6
Fig. 6

Square array on the image plane for applying the Hartmann test. The location of any point of the array can be described by a position vector, r I = h I (cos σ, sin σ), where h I = (x I 2 + y I 2)1/2 is the distance between the spot and the center of coordinates; σ is the direction.

Fig. 7
Fig. 7

Coordinates (z M , y M ) of oval ΣM for a meridional curve as a function of image height h I (R = 7.8 mm and |l V | = 180 mm). Most of the object points for testing the mirror are on the half of the object surface nearer the mirror.

Fig. 8
Fig. 8

Spatial distribution of the object points for a quadrant of oval ΣM. This distribution enables one to observe a square array of spots in a virtual plane image for a reference spherical mirror.

Fig. 9
Fig. 9

Sagittal S and tangential T astigmatic surfaces for oval ΣM. The plane mean image surface M bisects S and T. In this plane image surface we obtain the circles of least confusion for object points.

Fig. 10
Fig. 10

Meridional magnification for the object points on oval ΣM as a function of image height.

Fig. 11
Fig. 11

Geometry of object spots on oval ΣM as spots shaped like ellipses. Δs a and Δs b are the lengths of the major and the minor axes of these ellipses. Each point in Fig. 8 must be replaced by its corresponding ellipse, so that a square array of circles of 50-µm diameter in the image plane is obtained.

Tables (1)

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Table 1 Parameters of the Fitting Ellipses for Ovals ΣS, ΣT, and ΣM

Equations (32)

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AstLs=yn/nn-ni+ui¯22nu2,
2R=1l+1l,
AstLs=-h24R.
Ptz=h2R.
S=Ptz+AstLs=3h24R,
T=Ptz+3AstLs=h24R.
M=S+T2=h22R.
RPtz=R2,
RS=2R3,
RM=R,
RT=2R.
1LS+1LS=2 cos ϕR,
1LT+1LT=2R cos ϕ,
L=z-zIcos θ,
LS=Rz-zI2z-zIcos ϕ-R cos θ,
rS=z-LS cos2ϕ-θ, y-LS sin2ϕ-θ.
LT=Rz-zIcos ϕ2z-zI-R cos ϕ cos θ
rT=z-LT cos2ϕ-θ, y-LT sin2ϕ-θ.
LM=z-zIcos θ=LS+LT2.
LM28LM cos ϕ-2Rcos2 ϕ+1+LM2R2 cos ϕ-4LMRcos2 ϕ+1+2LMR2 cos ϕ=0.
rM=z-LM cos2ϕ-θ, y-LM sin2ϕ-θ,
rP=z0p+aP cosα, bP sinα,
aP=2zI2R4zI2-R2,
bP=zIR4zI2-R21/2,
z0P=zIR24zI2-R2.
2aS=2bS2/RS,
2bS=2RS2RmaxΣS1/3,
rSF=z0S+aS cosα, bS sinα,
Dlc=AstL2F/#,
DI=mΔsa * Dlc * Ddif,
Δsb=mΔsayMhI.
F/#=AstL/2.441/2,

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