Abstract

Images formed by individual Shack–Hartmann wavefront sensor lenslets are displaced proportionally to the average wavefront slope over their aperture. This principle fails when the lenslet illumination is non-uniform. Here we demonstrate that the resulting error is proportional to the linear component of the illumination intensity, the quadratic wavefront component, and the lenslet size. For illustrative purposes, we compare the error due to centered Gaussian illumination decaying by 30% at the pupil edge against the error due to assuming the wavefront at the lenslet center being equal to the wavefront average across each lenslet. When testing up to ninth-order Zernike polynomial wavefronts and simulating nine lenslets across the pupil, the maximum centroid errors due to non-uniform illumination and sampling are 1.4% and 21%, respectively, and 0.5% and 6.7% when considering 25 lenslets across the pupil in the absence of other sources of error.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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C. Robert, J. M. Conan, V. Michau, T. Fusco, and N. Vedrenne, J. Opt. Soc. Am. A 23, 613 (2006).
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B. C. Platt and R. Shack, J. Refract. Surg. 17, S573 (2001).
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M. Mrochen, M. Kaemmerer, and T. Seiler, J. Cataract Refract. Surg. 27, 201 (2001).
[Crossref]

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G. Westheimer, J. Physiol. 192, 309 (1967).
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P. L. Wizinowich, D. Le Mignant, A. H. Bouchez, R. D. Campbell, J. C. Y. Chin, A. R. Contos, M. A. van Dam, S. K. Hartman, E. M. Johansson, R. E. Lafon, H. Lewis, P. J. Stomski, and D. M. Summers, Publ. Astron. Soc. Pac. 118, 297 (2006).
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P. L. Wizinowich, D. Le Mignant, A. H. Bouchez, R. D. Campbell, J. C. Y. Chin, A. R. Contos, M. A. van Dam, S. K. Hartman, E. M. Johansson, R. E. Lafon, H. Lewis, P. J. Stomski, and D. M. Summers, Publ. Astron. Soc. Pac. 118, 297 (2006).
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Carroll, J.

Cense, B.

Chin, J. C. Y.

P. L. Wizinowich, D. Le Mignant, A. H. Bouchez, R. D. Campbell, J. C. Y. Chin, A. R. Contos, M. A. van Dam, S. K. Hartman, E. M. Johansson, R. E. Lafon, H. Lewis, P. J. Stomski, and D. M. Summers, Publ. Astron. Soc. Pac. 118, 297 (2006).
[Crossref]

Conan, J. M.

Contos, A. R.

P. L. Wizinowich, D. Le Mignant, A. H. Bouchez, R. D. Campbell, J. C. Y. Chin, A. R. Contos, M. A. van Dam, S. K. Hartman, E. M. Johansson, R. E. Lafon, H. Lewis, P. J. Stomski, and D. M. Summers, Publ. Astron. Soc. Pac. 118, 297 (2006).
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E. Dalimier, C. Dainty, and J. L. Barbur, J. Mod. Opt. 55, 791 (2008).
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P. L. Wizinowich, D. Le Mignant, A. H. Bouchez, R. D. Campbell, J. C. Y. Chin, A. R. Contos, M. A. van Dam, S. K. Hartman, E. M. Johansson, R. E. Lafon, H. Lewis, P. J. Stomski, and D. M. Summers, Publ. Astron. Soc. Pac. 118, 297 (2006).
[Crossref]

Hebert, T.

Johansson, E. M.

P. L. Wizinowich, D. Le Mignant, A. H. Bouchez, R. D. Campbell, J. C. Y. Chin, A. R. Contos, M. A. van Dam, S. K. Hartman, E. M. Johansson, R. E. Lafon, H. Lewis, P. J. Stomski, and D. M. Summers, Publ. Astron. Soc. Pac. 118, 297 (2006).
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P. L. Wizinowich, D. Le Mignant, A. H. Bouchez, R. D. Campbell, J. C. Y. Chin, A. R. Contos, M. A. van Dam, S. K. Hartman, E. M. Johansson, R. E. Lafon, H. Lewis, P. J. Stomski, and D. M. Summers, Publ. Astron. Soc. Pac. 118, 297 (2006).
[Crossref]

Le Mignant, D.

P. L. Wizinowich, D. Le Mignant, A. H. Bouchez, R. D. Campbell, J. C. Y. Chin, A. R. Contos, M. A. van Dam, S. K. Hartman, E. M. Johansson, R. E. Lafon, H. Lewis, P. J. Stomski, and D. M. Summers, Publ. Astron. Soc. Pac. 118, 297 (2006).
[Crossref]

Lewis, H.

P. L. Wizinowich, D. Le Mignant, A. H. Bouchez, R. D. Campbell, J. C. Y. Chin, A. R. Contos, M. A. van Dam, S. K. Hartman, E. M. Johansson, R. E. Lafon, H. Lewis, P. J. Stomski, and D. M. Summers, Publ. Astron. Soc. Pac. 118, 297 (2006).
[Crossref]

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Llorente, L.

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M. Mrochen, M. Kaemmerer, and T. Seiler, J. Cataract Refract. Surg. 27, 201 (2001).
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B. C. Platt and R. Shack, J. Refract. Surg. 17, S573 (2001).
[Crossref]

R. V. Shack and B. C. Platt, in Spring Meeting of the Optical Society of America (Optical Society of America, 1971).

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B. Vohnsen, A. Carmichael, N. Sharmin, S. Qaysi, and D. Valente, J. Vis. 17(12):18 (2017).
[Crossref]

Queener, H.

Rao, C.

Rao, X.

Robert, C.

Romero-Borja, F.

Roorda, A.

Seiler, T.

M. Mrochen, M. Kaemmerer, and T. Seiler, J. Cataract Refract. Surg. 27, 201 (2001).
[Crossref]

Shack, R.

B. C. Platt and R. Shack, J. Refract. Surg. 17, S573 (2001).
[Crossref]

Shack, R. V.

R. V. Shack and B. C. Platt, in Spring Meeting of the Optical Society of America (Optical Society of America, 1971).

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B. Vohnsen, A. Carmichael, N. Sharmin, S. Qaysi, and D. Valente, J. Vis. 17(12):18 (2017).
[Crossref]

Stiles Walter, S.

S. Stiles Walter, B. H. Crawford, and H. Parsons John, Proc. R. Soc. London B 112, 428 (1933).
[Crossref]

Stomski, P. J.

P. L. Wizinowich, D. Le Mignant, A. H. Bouchez, R. D. Campbell, J. C. Y. Chin, A. R. Contos, M. A. van Dam, S. K. Hartman, E. M. Johansson, R. E. Lafon, H. Lewis, P. J. Stomski, and D. M. Summers, Publ. Astron. Soc. Pac. 118, 297 (2006).
[Crossref]

Sulai, Y.

Summers, D. M.

P. L. Wizinowich, D. Le Mignant, A. H. Bouchez, R. D. Campbell, J. C. Y. Chin, A. R. Contos, M. A. van Dam, S. K. Hartman, E. M. Johansson, R. E. Lafon, H. Lewis, P. J. Stomski, and D. M. Summers, Publ. Astron. Soc. Pac. 118, 297 (2006).
[Crossref]

Thyagarajan, K.

A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge University, 1998).

Tian, Y.

Valente, D.

B. Vohnsen, A. Carmichael, N. Sharmin, S. Qaysi, and D. Valente, J. Vis. 17(12):18 (2017).
[Crossref]

van Dam, M. A.

P. L. Wizinowich, D. Le Mignant, A. H. Bouchez, R. D. Campbell, J. C. Y. Chin, A. R. Contos, M. A. van Dam, S. K. Hartman, E. M. Johansson, R. E. Lafon, H. Lewis, P. J. Stomski, and D. M. Summers, Publ. Astron. Soc. Pac. 118, 297 (2006).
[Crossref]

Vedrenne, N.

Vohnsen, B.

B. Vohnsen, A. Carmichael, N. Sharmin, S. Qaysi, and D. Valente, J. Vis. 17(12):18 (2017).
[Crossref]

Wang, Q.

Westheimer, G.

G. Westheimer, J. Physiol. 192, 309 (1967).
[Crossref]

Williams, D. R.

Wilson, B. J.

Wizinowich, P. L.

P. L. Wizinowich, D. Le Mignant, A. H. Bouchez, R. D. Campbell, J. C. Y. Chin, A. R. Contos, M. A. van Dam, S. K. Hartman, E. M. Johansson, R. E. Lafon, H. Lewis, P. J. Stomski, and D. M. Summers, Publ. Astron. Soc. Pac. 118, 297 (2006).
[Crossref]

Yang, J.

Appl. Opt. (1)

Biomed. Opt. Express (1)

J. Cataract Refract. Surg. (1)

M. Mrochen, M. Kaemmerer, and T. Seiler, J. Cataract Refract. Surg. 27, 201 (2001).
[Crossref]

J. Mod. Opt. (1)

E. Dalimier, C. Dainty, and J. L. Barbur, J. Mod. Opt. 55, 791 (2008).
[Crossref]

J. Opt. Soc. Am. (1)

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

J. Physiol. (1)

G. Westheimer, J. Physiol. 192, 309 (1967).
[Crossref]

J. Refract. Surg. (1)

B. C. Platt and R. Shack, J. Refract. Surg. 17, S573 (2001).
[Crossref]

J. Vis. (1)

B. Vohnsen, A. Carmichael, N. Sharmin, S. Qaysi, and D. Valente, J. Vis. 17(12):18 (2017).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Phil. Trans. R. Soc. A (1)

M. J. Booth, Phil. Trans. R. Soc. A 365, 2829 (2007).
[Crossref]

Proc. R. Soc. London B (1)

S. Stiles Walter, B. H. Crawford, and H. Parsons John, Proc. R. Soc. London B 112, 428 (1933).
[Crossref]

Publ. Astron. Soc. Pac. (1)

P. L. Wizinowich, D. Le Mignant, A. H. Bouchez, R. D. Campbell, J. C. Y. Chin, A. R. Contos, M. A. van Dam, S. K. Hartman, E. M. Johansson, R. E. Lafon, H. Lewis, P. J. Stomski, and D. M. Summers, Publ. Astron. Soc. Pac. 118, 297 (2006).
[Crossref]

Other (4)

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998), Vol. 16.

R. V. Shack and B. C. Platt, in Spring Meeting of the Optical Society of America (Optical Society of America, 1971).

B. Dörband, H. Müller, and H. Gross, Handbook of Optical Systems, Volume 5: Metrology of Optical Components and Systems (Wiley, 2012), Vol. 5.

A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge University, 1998).

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Figures (2)

Fig. 1.
Fig. 1. Slopes of Zernike polynomials ( Z 0 2 to Z 2 6 ) over the unit circle (first two columns), normalized to their peak values. The following columns show the theoretical centroid errors that would be seen in a SHWS due to the finite lenslet size (sampling) and non-uniform lenslet intensity profile, when using either nine or 25 lenslets across the diameter of a circular pupil. Some polynomials have been excluded due to symmetry.
Fig. 2.
Fig. 2. Slopes of Zernike polynomials ( Z 4 6 to Z 1 9 ) over the unit circle (first two columns), normalized to their peak values. The following columns show the theoretical centroid errors that would be seen in a SHWS due to the finite lenslet size (sampling) and non-uniform lenslet intensity profile, when using either nine or 25 lenslets across the diameter of a circular pupil. Some polynomials have been excluded due to symmetry.

Equations (6)

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ϱ x = I ( x , y ) W ( x , y ) x d x d y I ( x , y ) d x d y ,
ϱ x 1 A W ( x , y ) x d x d y ,
ϱ x 1 D D / 2 D / 2 ( I 0 + I 1 x + I 2 x 2 ) ( W 0 + W 1 x + W 2 x 2 ) x d x D / 2 D / 2 ( I 0 + I 1 x + I 2 x 2 ) d x .
ϱ x 1 D I 0 W 1 + ( 2 I 1 W 2 + I 2 W 1 ) D 2 12 I 0 + I 2 D 2 12 .
ε x 1 D = ϱ x 1 D ( I 0 , I 1 , I 2 , W 0 , W 1 , W 2 ) ϱ x 1 D ( I 0 , 0 , 0 , W 0 , W 1 , W 2 ) ,
ε x 1 D ( 2 I 1 12 I 0 + I 2 D 2 ) W 2 D 2 .

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