Abstract

A modified Hartmann test based on the interference produced by a four-hole mask can be used to measure an unknown wavefront. To scan the wavefront, the interference pattern is measured for different positions of the mask. The position of the central fringe of the diamond-shaped interference pattern gives a measure of the local wavefront slopes. Using a set of four-hole apertures located behind an array of lenslets in such a way that each four-hole window is inside one lenslet area, a set of four-hole interference patterns can be obtained in the back focal plane of the lenslets without having to scan the wavefront. The central fringe area of each interference pattern is narrower than the area of the central maximum of the diffraction pattern of the lenslet, increasing the accuracy in the estimate of the lobe position as compared with the Shack–Hartmann wavefront sensor.

© 2010 Optical Society of America

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References

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2009

2007

2006

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc.  371, 323–336 (2006).
[CrossRef]

Z. Jiang, S. Gong, and Y. Dai, “Numerical study of centroid detection accuracy for Shack-Hartmann wavefront sensor,” Opt. Laser Technol.  38, 614–619 (2006).
[CrossRef]

2004

2002

J. Arines and J. Ares, “Minimum variance centroid thresholding,” Opt. Lett.  27, 497–499 (2002).
[CrossRef]

A. Tokovinin, “From differential image motion to seeing,” Publ. Astron. Soc. Pac.  114, 1156–1166 (2002).
[CrossRef]

2001

1999

1994

G. Cao and X. Yu, “Accuracy analysis of a Hartmann-Shack wavefront sensor operated with a faint object,” Opt. Eng.  33, 2331–2335 (1994).
[CrossRef]

1965

C. Lorant, “Variance of a product of independent random variables with small deviations from their means,” Proc. IEEE  53, 1760–1761 (1965).
[CrossRef]

1961

1960

1925

1904

J. Hartmann, “Objektivuntersuchungen,” Z. Instrumentenkd.  24, 1–119 (1904).

Ares, J.

Arines, J.

Bennett, A. H.

Cao, G.

G. Cao and X. Yu, “Accuracy analysis of a Hartmann-Shack wavefront sensor operated with a faint object,” Opt. Eng.  33, 2331–2335 (1994).
[CrossRef]

Chen, H.

H. Chen and C. Rao, “Accuracy analysis on centroid estimation algorithm limited by photon noise for point object,” Opt. Commun.  282, 1526–1530 (2009).
[CrossRef]

Cui, X.

Dai, Y.

Z. Jiang, S. Gong, and Y. Dai, “Numerical study of centroid detection accuracy for Shack-Hartmann wavefront sensor,” Opt. Laser Technol.  38, 614–619 (2006).
[CrossRef]

De, M.

Fusco, T.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc.  371, 323–336 (2006).
[CrossRef]

Gardner, I. C.

Ghozeil, I.

I. Ghozeil, “Hartmann and other screen tests,” in Optical Shop Testing, D.Malacara, ed. (Wiley, 1992), pp. 367–396.

Gong, S.

Z. Jiang, S. Gong, and Y. Dai, “Numerical study of centroid detection accuracy for Shack-Hartmann wavefront sensor,” Opt. Laser Technol.  38, 614–619 (2006).
[CrossRef]

Gupta, K. S.

Gupta, M. K. S.

Hartmann, J.

J. Hartmann, “Objektivuntersuchungen,” Z. Instrumentenkd.  24, 1–119 (1904).

Irwan, R.

Jiang, Z.

Z. Jiang, S. Gong, and Y. Dai, “Numerical study of centroid detection accuracy for Shack-Hartmann wavefront sensor,” Opt. Laser Technol.  38, 614–619 (2006).
[CrossRef]

Lane, R. G.

Lew, X. H. M.

López, D.

Lorant, C.

C. Lorant, “Variance of a product of independent random variables with small deviations from their means,” Proc. IEEE  53, 1760–1761 (1965).
[CrossRef]

Ma, X.

Michau, V.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc.  371, 323–336 (2006).
[CrossRef]

Nicolle, M.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc.  371, 323–336 (2006).
[CrossRef]

Rao, C.

H. Chen and C. Rao, “Accuracy analysis on centroid estimation algorithm limited by photon noise for point object,” Opt. Commun.  282, 1526–1530 (2009).
[CrossRef]

X. Ma, C. Rao, and H. Zheng, “Error analysis of CCD-based point source centroid computation under the background light,” Opt. Express  17, 8525–8541 (2009).
[CrossRef] [PubMed]

Ríos, S.

Rousset, G.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc.  371, 323–336 (2006).
[CrossRef]

Thomas, S.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc.  371, 323–336 (2006).
[CrossRef]

Tokovinin, A.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc.  371, 323–336 (2006).
[CrossRef]

A. Tokovinin, “From differential image motion to seeing,” Publ. Astron. Soc. Pac.  114, 1156–1166 (2002).
[CrossRef]

Vaidya, W.

Yang, C.

Yu, X.

G. Cao and X. Yu, “Accuracy analysis of a Hartmann-Shack wavefront sensor operated with a faint object,” Opt. Eng.  33, 2331–2335 (1994).
[CrossRef]

Zheng, H.

Appl. Opt.

J. Opt. Soc. Am.

Mon. Not. R. Astron. Soc.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc.  371, 323–336 (2006).
[CrossRef]

Opt. Commun.

H. Chen and C. Rao, “Accuracy analysis on centroid estimation algorithm limited by photon noise for point object,” Opt. Commun.  282, 1526–1530 (2009).
[CrossRef]

Opt. Eng.

G. Cao and X. Yu, “Accuracy analysis of a Hartmann-Shack wavefront sensor operated with a faint object,” Opt. Eng.  33, 2331–2335 (1994).
[CrossRef]

Opt. Express

Opt. Laser Technol.

Z. Jiang, S. Gong, and Y. Dai, “Numerical study of centroid detection accuracy for Shack-Hartmann wavefront sensor,” Opt. Laser Technol.  38, 614–619 (2006).
[CrossRef]

Opt. Lett.

Proc. IEEE

C. Lorant, “Variance of a product of independent random variables with small deviations from their means,” Proc. IEEE  53, 1760–1761 (1965).
[CrossRef]

Publ. Astron. Soc. Pac.

A. Tokovinin, “From differential image motion to seeing,” Publ. Astron. Soc. Pac.  114, 1156–1166 (2002).
[CrossRef]

Z. Instrumentenkd.

J. Hartmann, “Objektivuntersuchungen,” Z. Instrumentenkd.  24, 1–119 (1904).

Other

I. Ghozeil, “Hartmann and other screen tests,” in Optical Shop Testing, D.Malacara, ed. (Wiley, 1992), pp. 367–396.

Hamamatsu Photonics, “ieee 1394-based digital camera orca-285 data sheet,” http://sales.hamamatsu.com/assets/pdf/hpspdf/C4742-95-12G04.pdf.

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Figures (5)

Fig. 1
Fig. 1

Four-hole interference geometry.

Fig. 2
Fig. 2

Mask and lenslets array geometry.

Fig. 3
Fig. 3

Intensity distribution at the lenslets back focal plane.

Fig. 4
Fig. 4

Four-hole mask geometry.

Fig. 5
Fig. 5

Intensity distributions in the lenslets back focal plane for (a) the mask with diamond geometry, (b) the mask with square geometry, and (c) lenslets alone.

Tables (3)

Tables Icon

Table 1 Experimental Values of the Central Lobe Sizes and Variance Gain

Tables Icon

Table 2 Variances in the x Direction for Different Approximations

Tables Icon

Table 3 Analysis of the Approximations in the Theoretical Variance

Equations (22)

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

W 1 = W ( x p a , y p ) , W 2 = W ( x p + a , y p ) , W 3 = W ( x p , y p a ) , W 4 = W ( x p , y p + a ) .
I ( x , y ) = 4 + 2 ( cos [ k ( W 2 W 1 ) + x a 2 ] + cos [ k ( W 4 W 3 ) + y a 2 ] + 4 cos [ 1 2 k ( W 3 W 1 + W 4 W 2 ) ] cos { 1 2 [ k ( W 2 W 1 ) + x a 2 ] } × cos { 1 2 [ k ( W 4 W 3 ) + y a 2 ] } ) ,
Δ x = Δ y = λ z a .
x = z 2 a ( W 2 W 1 ) , y = z 2 a ( W 4 W 3 ) ,
Δ x = Δ y = λ f a .
A = 1 2 ( λ f a ) 2 .
A m = ( λ f l d 2 s ) 2 ,
B = ( 2 λ f l ) 2 .
R = A m B = l 2 4 ( l d 2 s ) 2 .
x c = i , j x i j I i j i , j I i j , y c = i , j y i j I i j i , j I i j ,
d x c x c = d U U d V V .
x c x c ¯ x c ¯ = U - U ¯ U ¯ V V ¯ V ¯ ,
( x c x c ¯ ) 2 x c ¯ 2 = ( U U ¯ ) 2 U ¯ 2 + ( V V ¯ ) 2 V ¯ 2 2 ( U U ¯ ) ( V V ¯ ) U ¯ V ¯ .
σ x c 2 x c ¯ 2 = σ U 2 U ¯ 2 + σ V 2 V ¯ 2 2 Cov ( U , V ) U ¯ V ¯ ,
σ U 2 = i , j x i j 2 σ I i j 2 + 2 i j k l x i j x k l Cov ( I i j , I k l ) .
σ U 2 = i , j x i j 2 σ I i j 2 .
σ V 2 = i , j σ I i j 2 + i j k l Cov ( I i j , I k l )
σ V 2 = i , j σ I i j 2 .
Cov ( U , V ) = i , j x i j σ I i j 2 + i j k l x i j Cov ( I i j , I k l ) .
Cov ( U , V ) = i , j x i j σ I i j 2 .
σ x c 2 = x c ¯ 2 ( i , j x i j 2 σ I i j 2 U ¯ 2 + i , j σ I i j 2 V ¯ 2 2 i , j x i j σ I i j 2 U ¯ V ¯ ) .
σ x c 2 = 1 V ¯ ( i j x i j 2 I i j V ¯ U ¯ 2 V ¯ 2 ) .

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