Abstract

A new concept of using focus-diverse point spread functions (PSFs) for modal wavefront sensing (WFS) is explored. This concept is based on relatively straightforward image moment analysis of measured PSFs, which differentiates it from other focal-plane WFS techniques. The presented geometric analysis shows that the image moments are nonlinear functions of wave aberration coefficients but notes that focus diversity essentially decouples the coefficients of interest from others, resulting in a set of linear equations whose solution corresponds to modal coefficient estimates. The presented proof-of-concept simulations suggest the potential of the concept in WFS with strongly aberrated high signal-to-noise ratio objects in particular.

© 2011 Optical Society of America

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References

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M. G. Lofdahl, A. L. Duncan, and G. B. Scharmer, Proc. SPIE 3353, 952 (1998).
[CrossRef]

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Dolne, J.

Duncan, A. L.

M. G. Lofdahl, A. L. Duncan, and G. B. Scharmer, Proc. SPIE 3353, 952 (1998).
[CrossRef]

Fineup, J. R.

Fusco, T.

Gonsalves, R.

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J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

Hardy, J. W.

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

Lee, H.

Lofdahl, M. G.

M. G. Lofdahl, A. L. Duncan, and G. B. Scharmer, Proc. SPIE 3353, 952 (1998).
[CrossRef]

Mahajan, V. N.

V. N. Mahajan, Aberration Theory Made Simple (SPIE, 1991).
[CrossRef]

Meimon, S.

Menicucci, P.

Miccolis, D.

Mugnier, L.

Noll, R. J.

Roddier, F.

Schall, H.

Scharmer, G. B.

M. G. Lofdahl, A. L. Duncan, and G. B. Scharmer, Proc. SPIE 3353, 952 (1998).
[CrossRef]

Seiden, H.

Silva, D. E.

Vachss, F.

Wang, J. Y.

Widen, K.

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Figures (3)

Fig. 1
Fig. 1

Polychromatic through-focus PSFs on a 512 2 grid with uniform spectral weight at 11 wavelengths between 514 and 614 nm : top, initial; bottom, after correction; left, 1 mm before focus; middle, at focus; right, + 1 mm after focus. 50 μm squares overlaid for size comparison. Images in square-root scale.

Fig. 2
Fig. 2

Polychromatic through-focus PSFs with 4 × 4 binning: top, initial; bottom, after correction; left, 1 mm before focus; middle, at focus; right, + 1 mm after focus. 50 μm squares overlaid for comparison. Images in square-root scale.

Fig. 3
Fig. 3

Four object images at 1 λ FD (inset). RMS residual error against SNR for 21 objects.

Tables (2)

Tables Icon

Table 1 Slope (in Pixels) and Wave (in λ) Coefficients in Fig. 1

Tables Icon

Table 2 Slope (in Pixels) and Wave (in λ) Coefficients in Fig. 2

Equations (8)

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Φ = a 1 Z 1 + a 2 Z 2 + a 3 Z 3 + + a M Z M ,
X = X 0 + 2 F Φ x , Y = Y 0 + 2 F Φ y ,
2 F Φ / x = a T Γ x Z = ( a x ) T Z 2 F Φ / y = a T Γ y Z = ( a y ) T Z ,
a 1 x = 2 a 2 + 2 2 a 8 , a 1 y = 2 a 3 + 2 2 a 7 , a 2 x = 2 3 a 4 + 6 a 6 , a 2 y = 5 a 5 + 10 a 13 , a 3 x = 5 a 5 + 10 a 13 , a 3 y = 2 3 a 4 6 a 6 , a 4 x = 2 6 a 8 , a 4 y = 2 6 a 7 , a 5 x = 2 3 a 7 + 2 3 a 9 , a 5 y = 2 3 a 8 - 2 3 a 10 , a 6 x = 2 3 a 8 + 2 3 a 10 , a 6 y = 2 3 a 7 + 2 3 a 9 .
μ nm = S 1 Ω I ( X X 0 ) n ( Y Y 0 ) m d Ω ,
μ 20 = i = 2 M ( a i x ) 2 , μ 11 = i = 2 M a i x a i y , μ 02 = i = 2 M ( a i y ) 2 ,
μ 20 a 4 = 4 3 a 2 x , μ 11 a 4 = 4 3 a 3 x , μ 02 a 4 = 4 3 a 3 y .
2 μ 30 a 4 2 = c ( a 4 x + 2 a 6 x ) , 2 μ 12 a 4 2 = d ( 5 a 4 x 3 2 a 6 x ) , 2 μ 03 a 4 2 = c ( a 4 y 2 a 6 y ) , 2 μ 21 a 4 2 = d ( 5 a 4 y + 3 2 a 6 y ) , with a 5 x = 2 a 4 y + a 6 y , a 5 y = 2 a 4 x + a 6 x ,

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