Abstract

An efficient way of estimating orthonormal aberration coefficients on variable noncircular pupils is proposed. The method is based on the fact that all necessary pieces of information for constructing orthonormal polynomials (via the Gram–Schmidt process) can be numerically obtained during a routine least-squares fit of Zernike polynomials to wavefront data. This allows the method to use the usual Zernike polynomial fitting with an additional procedure that swiftly estimates the desired orthonormal aberration coefficients without having to use the functional forms of orthonormal polynomials. It is also shown that the method naturally accounts for the pixelation effect of pupil geometries, intrinsic to recording wavefront data on imaging sensors (e.g., CCDs), making the coefficient estimate optimal over a given pixelated pupil geometry. With these features, the method can be ideal for real-time wavefront analysis over dynamically changing pupils, such as in the Hobby–Eberly Telescope (HET), which is otherwise inefficient with analytic methods used in past studies.

© 2010 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 7th ed.(Oxford, 1999).
  2. F. Zernike, Mon. Not. R. Astron. Soc. 94, 377 (1934).
  3. N. Roddier, Opt. Eng. 29, 1174 (1990).
    [CrossRef]
  4. H. Lee, G. B. Dalton, I. A. Tosh, S.-W. Kim, Opt. Express 15, 3127 (2007).
    [CrossRef] [PubMed]
  5. W. B. King, Appl. Opt. 7, 197 (1968).
    [CrossRef] [PubMed]
  6. Hubble Space Telescope, http://hubblesite.org/.
  7. Keck Telescopes, http://www.keckobservatory.org/.
  8. Hobby–Eberly Telescope, http://www.as.utexas.edu/.
  9. South African Large Telescope, http://www.salt.ac.za/.
  10. C. F. Dunkl, SIAM J. Appl. Math. 47, 343 (1987).
    [CrossRef]
  11. R. Upton and B. Ellerbroek, Opt. Lett. 29, 2840 (2004).
    [CrossRef]
  12. G. Dai and V. Mahajan, Opt. Lett. 32, 74 (2007).
    [CrossRef]
  13. V. N. Mahajan and G. Dai, J. Opt. Soc. Am. A 24, 2994 (2007).
    [CrossRef]
  14. G. B. Arfken, H. J. Weber, and F. Harris, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).
  15. Maxima version 5.18.1 (a computer algebra system), http://maxima.sourceforge.net/.
  16. H. Lee, M. Hart, and G. J. Hill are preparing a manuscript to be called “Optimal estimation of wavefront slope aberrations on variable non-circular pupils.”

2007 (3)

2004 (1)

2001 (1)

G. B. Arfken, H. J. Weber, and F. Harris, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

1999 (1)

M. Born and E. Wolf, Principles of Optics, 7th ed.(Oxford, 1999).

1990 (1)

N. Roddier, Opt. Eng. 29, 1174 (1990).
[CrossRef]

1987 (1)

C. F. Dunkl, SIAM J. Appl. Math. 47, 343 (1987).
[CrossRef]

1968 (1)

1934 (1)

F. Zernike, Mon. Not. R. Astron. Soc. 94, 377 (1934).

Arfken, G. B.

G. B. Arfken, H. J. Weber, and F. Harris, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed.(Oxford, 1999).

Dai, G.

Dalton, G. B.

Dunkl, C. F.

C. F. Dunkl, SIAM J. Appl. Math. 47, 343 (1987).
[CrossRef]

Ellerbroek, B.

Harris, F.

G. B. Arfken, H. J. Weber, and F. Harris, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

Hart, M.

H. Lee, M. Hart, and G. J. Hill are preparing a manuscript to be called “Optimal estimation of wavefront slope aberrations on variable non-circular pupils.”

Hill, G. J.

H. Lee, M. Hart, and G. J. Hill are preparing a manuscript to be called “Optimal estimation of wavefront slope aberrations on variable non-circular pupils.”

Kim, S.-W.

King, W. B.

Lee, H.

H. Lee, G. B. Dalton, I. A. Tosh, S.-W. Kim, Opt. Express 15, 3127 (2007).
[CrossRef] [PubMed]

H. Lee, M. Hart, and G. J. Hill are preparing a manuscript to be called “Optimal estimation of wavefront slope aberrations on variable non-circular pupils.”

Mahajan, V.

Mahajan, V. N.

Roddier, N.

N. Roddier, Opt. Eng. 29, 1174 (1990).
[CrossRef]

Tosh, I. A.

Upton, R.

Weber, H. J.

G. B. Arfken, H. J. Weber, and F. Harris, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed.(Oxford, 1999).

Zernike, F.

F. Zernike, Mon. Not. R. Astron. Soc. 94, 377 (1934).

Appl. Opt. (1)

J. Opt. Soc. Am. A (1)

Mon. Not. R. Astron. Soc. (1)

F. Zernike, Mon. Not. R. Astron. Soc. 94, 377 (1934).

Opt. Eng. (1)

N. Roddier, Opt. Eng. 29, 1174 (1990).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

SIAM J. Appl. Math. (1)

C. F. Dunkl, SIAM J. Appl. Math. 47, 343 (1987).
[CrossRef]

Other (8)

M. Born and E. Wolf, Principles of Optics, 7th ed.(Oxford, 1999).

G. B. Arfken, H. J. Weber, and F. Harris, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

Maxima version 5.18.1 (a computer algebra system), http://maxima.sourceforge.net/.

H. Lee, M. Hart, and G. J. Hill are preparing a manuscript to be called “Optimal estimation of wavefront slope aberrations on variable non-circular pupils.”

Hubble Space Telescope, http://hubblesite.org/.

Keck Telescopes, http://www.keckobservatory.org/.

Hobby–Eberly Telescope, http://www.as.utexas.edu/.

South African Large Telescope, http://www.salt.ac.za/.

Supplementary Material (1)

» Media 1: AVI (701 KB)     

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Figures (2)

Fig. 1
Fig. 1

Two sample wavefronts on two pupils with a unit circle (dashed line): left, pupil A; right, pupil B (Media 1).

Fig. 2
Fig. 2

Variance estimates by different methods.

Tables (2)

Tables Icon

Table 1 β ^ i by Method X and Y on Pupil A

Tables Icon

Table 2 D i j by Analytic Integration and D ^ i j by Proposed Method on Pupil A

Equations (9)

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

W = α 1 Z 1 + α 2 Z 2 + + α M Z M .
j = 1 M α j F i j = j = 1 M α j E Z i Z j A d A = E Z i W A d A .
α i , e = j = 1 M G i j E Z i W A d A ,
V j = Z j k = 1 j - 1 C j k U k , U j = V j A / E V j V j d A ,
E V i V j A d A = F i j + m = 1 i 1 { n = 1 j 1 C i m δ m n C n j } n = 1 i - 1 C i n C n j m = 1 j 1 C i m C m j ,
C i i = F i i k = 1 i 1 C i k 2 , C i j = F i j C j j k = 1 j 1 C i k C j k C j j .
W i = 1 M α i , e Z i = i = 1 M j = 1 i α i , e C i j U j = j = 1 M β j U j , leading to β j = i = 1 M α i , e C i j .
W = Z α α ^ e = ( Z T Z ) 1 Z T W α ,
β ^ j = i = 1 M α ^ i , e C ^ i j β j ,

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