Abstract

In eye aberrometry it is often necessary to transform the aberration coefficients in order to express them in a scaled, rotated, and/or displaced pupil. This is usually done by applying to the original coefficients vector a set of matrices accounting for each elementary transformation. We describe an equivalent algebraic approach that allows us to perform this conversion in a single step and in a straightforward way. This approach can be applied to any particular definition, normalization, and ordering of the Zernike polynomials, and can handle a wide range of pupil transformations, including, but not restricted to, anisotropic scalings. It may also be used to transform the aberration coefficients between different polynomial basis sets.

© 2006 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1998), pp. 464-466, 767-772.
  2. L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, "Standards for reporting the optical aberrations of eyes," in Vision Science and Its Applications 2000, V.Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.
  3. J. Schwiegerling, "Scaling Zernike expansion coefficients to different pupil sizes," J. Opt. Soc. Am. A 19, 1937-1945 (2002).
    [CrossRef]
  4. C. E. Campbell, "Matrix method to find a new set of Zernike coefficients from an original set when the aperture radius is changed," J. Opt. Soc. Am. A 20, 209-217 (2003).
    [CrossRef]
  5. A. Guirao, D. R. Williams, and I. G. Cox, "Effect of rotation and translation on the expected benefit of an ideal method to correct the eye's higher-order aberrations," J. Opt. Soc. Am. A 18, 1003-1015 (2001).
    [CrossRef]
  6. E. Acosta and S. Bará, "Variable aberration generators using rotated Zernike plates," J. Opt. Soc. Am. A 22, 1993-1996 (2005).
    [CrossRef]
  7. R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-211 (1976).
    [CrossRef]
  8. American National Standards Institute, Methods for Reporting Optical Aberrations of Eyes, ANSI Z80.28-2004 (Optical Laboratories Association, 2004), Annex B, pp. 19-28.

2005 (1)

2004 (1)

American National Standards Institute, Methods for Reporting Optical Aberrations of Eyes, ANSI Z80.28-2004 (Optical Laboratories Association, 2004), Annex B, pp. 19-28.

2003 (1)

2002 (1)

2001 (1)

2000 (1)

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, "Standards for reporting the optical aberrations of eyes," in Vision Science and Its Applications 2000, V.Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.

1998 (1)

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1998), pp. 464-466, 767-772.

1976 (1)

Acosta, E.

Applegate, R. A.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, "Standards for reporting the optical aberrations of eyes," in Vision Science and Its Applications 2000, V.Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.

Bará, S.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1998), pp. 464-466, 767-772.

Campbell, C. E.

Cox, I. G.

Guirao, A.

Noll, R. J.

Schwiegerling, J.

Schwiegerling, J. T.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, "Standards for reporting the optical aberrations of eyes," in Vision Science and Its Applications 2000, V.Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.

Thibos, L. N.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, "Standards for reporting the optical aberrations of eyes," in Vision Science and Its Applications 2000, V.Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.

Webb, R.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, "Standards for reporting the optical aberrations of eyes," in Vision Science and Its Applications 2000, V.Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.

Williams, D. R.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1998), pp. 464-466, 767-772.

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Other (3)

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1998), pp. 464-466, 767-772.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, "Standards for reporting the optical aberrations of eyes," in Vision Science and Its Applications 2000, V.Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.

American National Standards Institute, Methods for Reporting Optical Aberrations of Eyes, ANSI Z80.28-2004 (Optical Laboratories Association, 2004), Annex B, pp. 19-28.

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Figures (7)

Fig. 1
Fig. 1

Definition of the reference frames.

Fig. 2
Fig. 2

Gray-scale plot of the elements of the transformation matrix T up to radial order 5 for a pure change of scale ( β = 1.2 ) .

Fig. 3
Fig. 3

Gray-scale plot of the elements of the transformation matrix T up to radial order 5 for a pure rotation ( α = + 25 ° ) .

Fig. 4
Fig. 4

Gray-scale plot of the elements of the transformation matrix T up to radial order 5 for a lateral displacement of δ = ( 0.15 , + 0.2 ) .

Fig. 5
Fig. 5

Gray-scale plot of the elements of the transformation matrix T up to radial order 5 for a simultaneous change of scale ( β = 1.2 ) , rotation ( α = + 25 ° ) , and lateral displacement δ = ( 0.15 , + 0.2 ) .

Fig. 6
Fig. 6

(a) Aberration function W with Zernike coefficients a, represented as a wrapped profile in the X Y reference frame; the region outside the unit radius circle G R is shown with a reduced contrast. The gray scale spans an optical path length of 633 nm . The white circle represents a new reference pupil G R , located at δ = ( 0.15 , + 0.2 ) , with R R = 1.2 and whose axes X Y (not shown) are rotated + 25 deg counterclockwise from X Y . (b) The function W represented in the reference frame of G R , after the coefficient transformation a = T a has been performed. (c) The same function after the piston has been corrected to set W to zero at the center of the new pupil.

Fig. 7
Fig. 7

Circles: Zernike aberration coefficients a of the function W, described in the X , Y frame ( G R reference pupil); rhombs: the new coefficients a corresponding to the representation of W in the transformed pupil G R (see Fig. 6); asterisks: Zernike aberration coefficients after setting W to zero at the center of G R . Note that only the piston differs from the previous set. After performing the inverse transformation on a , the original a are recovered (see text). Vertical axis: values of the Zernike coefficients in micrometers. Horizontal axis: mode index. The values of the coefficients corresponding to modes 21–35 (those at the right-hand side from the vertical line) have been multiplied by 10 to ease visualization.

Equations (10)

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r = M ( α ) ( r d ) ,
ρ = L ( ρ ) = β M ( α ) ( ρ δ ) ,
W ( ρ ( P ) ) = j = 0 M a j Z j ( ρ ) ,
W ( ρ ( P ) ) = W ( ρ ( P ) ) = j = 0 M a j Z j ( ρ ) ,
a = T a .
j = 0 M a j Z j ( ρ ) = j = 0 M a j Z j ( ρ ) .
Z a = Z a .
a = ( Z T Z ) 1 Z T Z a ,
T = ( Z T Z ) 1 Z T Z .
ρ = L ( ρ ) = β M ( α ) ( ρ δ ) ,

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