Abstract

The shape of a focus-modulated point spread function (PSF) is used as a quick visual assessment tool of aberration modes in the PSF. Further analysis in terms of shape moments can permit quantifying the modal coefficients with an accuracy comparable to that of typical wavefront sensors. In this Letter, the error of the moment-based wavefront sensing is analytically described in terms of the pixelation and photon/readout noise. All components highly depend on the (unknown) PSF shape but can be estimated from the measured PSF sampled at a reasonable spatial resolution and photon count. Numerical simulations verified that the models consistently predicted the behavior of the modal estimation error of the moment-based wavefront sensing.

© 2014 Optical Society of America

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References

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C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, Proc. SPIE 8447, 844721 (2012).
[CrossRef]

H. Lee and G. J. Hill, Proc. SPIE 8450, 845019 (2012).

H. Lee, G. J. Hill, S. E. Tuttle, and B. L. Vattiat, Proc. SPIE 8450, 845019 (2012).

2011 (1)

2010 (1)

2009 (1)

2001 (1)

1994 (1)

E. M. Johansson and D. T. Gavel, Proc. SPIE 2200, 372 (1994).
[CrossRef]

1990 (1)

1982 (2)

1976 (1)

Andrei, R.

C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, Proc. SPIE 8447, 844721 (2012).
[CrossRef]

Doelman, N.

C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, Proc. SPIE 8447, 844721 (2012).
[CrossRef]

Dolne, J.

Fienup, J. R.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vitterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge, 1999).

Fraanje, R.

C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, Proc. SPIE 8447, 844721 (2012).
[CrossRef]

Fusco, T.

Gavel, D. T.

E. M. Johansson and D. T. Gavel, Proc. SPIE 2200, 372 (1994).
[CrossRef]

Gonsalves, R.

Hardy, J. W.

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

Hill, G. J.

H. Lee and G. J. Hill, Proc. SPIE 8450, 845019 (2012).

H. Lee, G. J. Hill, S. E. Tuttle, and B. L. Vattiat, Proc. SPIE 8450, 845019 (2012).

Johansson, E. M.

E. M. Johansson and D. T. Gavel, Proc. SPIE 2200, 372 (1994).
[CrossRef]

Keller, C. U.

C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, Proc. SPIE 8447, 844721 (2012).
[CrossRef]

Korkiakoski, V.

C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, Proc. SPIE 8447, 844721 (2012).
[CrossRef]

Lee, H.

H. Lee, G. J. Hill, S. E. Tuttle, and B. L. Vattiat, Proc. SPIE 8450, 845019 (2012).

H. Lee and G. J. Hill, Proc. SPIE 8450, 845019 (2012).

H. Lee, Opt. Lett. 36, 1503 (2011).
[CrossRef]

Meimon, S.

Menicucci, P.

Miccolis, D.

Mugnier, L.

Noll, R. J.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vitterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge, 1999).

Roddier, F.

Schall, H.

Seiden, H.

Suiter, H. R.

H. R. Suiter, Star Testing Astronomical Telescopes, 2nd ed. (Willmann-Bell, 2008).

Teague, M. R.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vitterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge, 1999).

Tuttle, S. E.

H. Lee, G. J. Hill, S. E. Tuttle, and B. L. Vattiat, Proc. SPIE 8450, 845019 (2012).

Vachss, F.

Vattiat, B. L.

H. Lee, G. J. Hill, S. E. Tuttle, and B. L. Vattiat, Proc. SPIE 8450, 845019 (2012).

Verhaegen, M.

C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, Proc. SPIE 8447, 844721 (2012).
[CrossRef]

Vitterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vitterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge, 1999).

Widen, K.

Appl. Opt. (3)

J. Opt. Soc. Am. (2)

Opt. Lett. (3)

Proc. SPIE (4)

E. M. Johansson and D. T. Gavel, Proc. SPIE 2200, 372 (1994).
[CrossRef]

C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, Proc. SPIE 8447, 844721 (2012).
[CrossRef]

H. Lee and G. J. Hill, Proc. SPIE 8450, 845019 (2012).

H. Lee, G. J. Hill, S. E. Tuttle, and B. L. Vattiat, Proc. SPIE 8450, 845019 (2012).

Other (3)

W. H. Press, S. A. Teukolsky, W. T. Vitterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge, 1999).

H. R. Suiter, Star Testing Astronomical Telescopes, 2nd ed. (Willmann-Bell, 2008).

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

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Figures (3)

Fig. 1.
Fig. 1.

(A) Five modulated PSF images and (B)–(D) phase maps. The pixel size in (A) is 5 μm. The color bar unit is wv.

Fig. 2.
Fig. 2.

(A)–(C) Defocused PSF at different resolutions. (D), (E) Error analysis results for (B) and (C). 5 μm pixel in (A).

Fig. 3.
Fig. 3.

(A) Target phase and (B) modulated PSF images used in the moment-based modal sensing error analysis (C)–(H).

Tables (1)

Tables Icon

Table 1. Quantities Computed and Shown in Figs. 2(D) and 2(E)

Equations (12)

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θ=W1Z1+W2Z2+W3Z3++WMZM.
X=fsθ=fθsandY=ftθ=fθt,
μnmg={ΩIXnYmdsdt}/{ΩIdsdt},
μnmd=ΛpxnymdxdyΛpdxdyMnm=k=1Npxpkxknykmk=1Npxpk,
u⃗=MW⃗.
μnm=1Nphk=1NpxΔkpϕnm+p(xϕnm)δxk+p(yϕnm)δyk+O(ϵ)dxdy.
δnmΔ212k=1Npxxϕnmxp+yϕnmypNph/Δ2,
δnm1Nphk=1Npx[l=1q{i=0l1i!j!xiyjϕnm(Δkp0xiyj+(xp)xi+1yj+(yp)xiyj+1dxdy)}],
E[M^nm]=E[kskϕnm,kksk]=E[AB]=E[A]E[1B],Mnm(1+SNR2),
var[M^nm]Mnm2SNR2+kpkϕnm,k2(kpk)2+σ2kϕnm,k2(kpk)2.
c⃗=(ATA)1(ATb⃗)withAij=Fijei,bi=M^nm(Fi)ei.
W⃗=(BTB)1(BTd⃗)withBij=Mijvi,di=uivi,

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