Abstract

In the past few years considerable effort has been made to extract basic surface aberrations, such as mean surface power and surface astigmatism, from corneal surface-height data, e.g., as measured by videokeratoscopes. Two applications are corneal surgery and contact lens adjustment. Another primary application is the derivation of the mean power and astigmatism of a wave front, e.g., in interferometers and other wave-front sensors. Both surface and wave-front quantities can be computed by means of Zernike polynominals. One advantage of this computation is that Zernike polynominal expansion provides all the coefficients at once for calculating Seidel aberrations. We present accurate analytical formulas for computing the mean power and astigmatism of both surface and wave front from Zernike coefficients. Therefore, in general, those quantities are not a function of quadratic-order terms alone, as stated in M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 9.2, but include linear-order terms as well.

© 1998 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 9.2.
  2. J. Y. Wang, D. E. Silva, “Wave-front interpretation with Zernike polynominals,” Appl. Opt. 19, 1510–1518 (1980).
    [CrossRef] [PubMed]
  3. For example, Schaum’s Outline, Differential Geometry (McGraw-Hill, New York, 1980).
  4. J. Schwiegerlin, J. E. Greivenkamp, J. M. Miller, “Zernike polynominal representations of videokeratoscope height data,” in Vision Science and Its Applications, Vol. 1 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 41–44.
  5. J. A. Gomez-Pedrero, G. Rueda-Colinas, E. Bernabeu, “Automatic interferometric method for the study of the oblique central refraction in ophthalmic lenses,” Optik (Stuttgart) 104, 171–174 (1997).
  6. G. Löffler, E. Lingelbach, B. Lingelbach, “Cornea: Abbildungseigenschaften und Topometrie, part I,” Deutsch. Optik. Z. 4, 92–96 (1997); “part II,” 104–107 (1997).

1997 (2)

J. A. Gomez-Pedrero, G. Rueda-Colinas, E. Bernabeu, “Automatic interferometric method for the study of the oblique central refraction in ophthalmic lenses,” Optik (Stuttgart) 104, 171–174 (1997).

G. Löffler, E. Lingelbach, B. Lingelbach, “Cornea: Abbildungseigenschaften und Topometrie, part I,” Deutsch. Optik. Z. 4, 92–96 (1997); “part II,” 104–107 (1997).

1980 (1)

Bernabeu, E.

J. A. Gomez-Pedrero, G. Rueda-Colinas, E. Bernabeu, “Automatic interferometric method for the study of the oblique central refraction in ophthalmic lenses,” Optik (Stuttgart) 104, 171–174 (1997).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 9.2.

Gomez-Pedrero, J. A.

J. A. Gomez-Pedrero, G. Rueda-Colinas, E. Bernabeu, “Automatic interferometric method for the study of the oblique central refraction in ophthalmic lenses,” Optik (Stuttgart) 104, 171–174 (1997).

Greivenkamp, J. E.

J. Schwiegerlin, J. E. Greivenkamp, J. M. Miller, “Zernike polynominal representations of videokeratoscope height data,” in Vision Science and Its Applications, Vol. 1 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 41–44.

Lingelbach, B.

G. Löffler, E. Lingelbach, B. Lingelbach, “Cornea: Abbildungseigenschaften und Topometrie, part I,” Deutsch. Optik. Z. 4, 92–96 (1997); “part II,” 104–107 (1997).

Lingelbach, E.

G. Löffler, E. Lingelbach, B. Lingelbach, “Cornea: Abbildungseigenschaften und Topometrie, part I,” Deutsch. Optik. Z. 4, 92–96 (1997); “part II,” 104–107 (1997).

Löffler, G.

G. Löffler, E. Lingelbach, B. Lingelbach, “Cornea: Abbildungseigenschaften und Topometrie, part I,” Deutsch. Optik. Z. 4, 92–96 (1997); “part II,” 104–107 (1997).

Miller, J. M.

J. Schwiegerlin, J. E. Greivenkamp, J. M. Miller, “Zernike polynominal representations of videokeratoscope height data,” in Vision Science and Its Applications, Vol. 1 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 41–44.

Rueda-Colinas, G.

J. A. Gomez-Pedrero, G. Rueda-Colinas, E. Bernabeu, “Automatic interferometric method for the study of the oblique central refraction in ophthalmic lenses,” Optik (Stuttgart) 104, 171–174 (1997).

Schwiegerlin, J.

J. Schwiegerlin, J. E. Greivenkamp, J. M. Miller, “Zernike polynominal representations of videokeratoscope height data,” in Vision Science and Its Applications, Vol. 1 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 41–44.

Silva, D. E.

Wang, J. Y.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 9.2.

Appl. Opt. (1)

Deutsch. Optik. Z. (1)

G. Löffler, E. Lingelbach, B. Lingelbach, “Cornea: Abbildungseigenschaften und Topometrie, part I,” Deutsch. Optik. Z. 4, 92–96 (1997); “part II,” 104–107 (1997).

Optik (Stuttgart) (1)

J. A. Gomez-Pedrero, G. Rueda-Colinas, E. Bernabeu, “Automatic interferometric method for the study of the oblique central refraction in ophthalmic lenses,” Optik (Stuttgart) 104, 171–174 (1997).

Other (3)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 9.2.

For example, Schaum’s Outline, Differential Geometry (McGraw-Hill, New York, 1980).

J. Schwiegerlin, J. E. Greivenkamp, J. M. Miller, “Zernike polynominal representations of videokeratoscope height data,” in Vision Science and Its Applications, Vol. 1 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 41–44.

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Tables (2)

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Table 1 Results for a Spherical Surface with Radius s=50.53 mm, Index of Refraction n=1.502, Mean Power and Astigmatism Calculated in Units of Diopters by Eqs. (8) and (9), and Physical Radius of Expansion r Equal to 1.0 mm

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Table 2 Results for a Spherical Surface with Radius s=50.53 mm, Index of Refraction n=1.502, Mean Power and Astigmatism Calculated in Units of Diopters by Eqs. (9), and Varied Physical Radius of Expansion

Equations (13)

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f(x, y)=i=1NaiZi(x, y),
z=1+a12x/r+a22y/r+a33(-1+2(x/r)2+2(y/r)2)+a46[(x/r)2-(y/r)2]+a526xy/r2,
D=H;A=2H2-K.
H=EN+GL-2FM2(EG-F2),
K=LN-M2EG-F2,
E=1+(dz/dx)2F=(dz/dx)(dz/dy)
G=1+(dz/dy)2,
L=d2z/dx2[(dz/dx)2+(dz/dy)2+1]1/2,
M=d2z/dxdy[(dz/dx)2+(dz/dy)2+1]1/2,
N=d2z/dy2[(dz/dx)2+(dz/dy)2+1]1/2.
D=43[a12(2a3-2a4)+a22(2a3+2a4)-22a1a2a5+a3r2]r4a2+4a22+r23/2;
A=461r2(4a12+4a22+r2)3[4a14(2a32-22a3a4+a42)+4a24(2a32+22a3a4+a42)-16a13a2(2a3-a4)a5+4a22(2a3a4+a42+a52)r2+(a42+a52)r4-8a1a2a5[2a22(2a3+a4)+2a3r2]+4a12[a22(4a3-2a42+4a52)+(-2a3a4+a42+a52)r2]].
D=43 a3r2;A=46 a42+a52r2.

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