Abstract

Phase estimates in adaptive-optics systems are computed by use of wavefront sensors, such as Shack–Hartmann or curvature sensors. In either case, the standard error of the phase estimates is proportional to the standard error of the measurements; but the error-propagation factors are different. We calculate the ratio of these factors for curvature and Shack–Hartmann sensors in dependence on the number of sensors, n, on a circular aperture. If the sensor spacing is kept constant and the pupil is enlarged, the ratio increases as n0.4. When more sensing elements are accommodated on the same aperture, it increases even faster, namely, proportional to n0.8. With large numbers of sensing elements, this increase can limit the applicability of curvature sensors.

© 2011 Optical Society of America

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References

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  1. B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wave-front sensing,” J. Refractive Surg. 17, 573–577(2001).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2010 (1)

2009 (1)

P. Lena, “Adaptive optics: a breakthrough in astronomy,” Exp. Astron. 26, 35–48 (2009).
[CrossRef]

2008 (2)

C. Torti, S. Gruppetta, and L. Diaz-Santana, “Wavefront curvature sensing for the human eye,” J. Mod. Opt. 55, 691–702(2008).
[CrossRef]

M. Watanabe, S. Oya, Y. Hayano, H. Takami, M. Hattori, Y. Minowa, Y. Saito, M. Ito, N. Murakami, M. Iye, O. Guyon, S. Colley, M. Eldred, T. Golota, and M. Dinkins, “Implementation of 188-element curvature-based wavefront sensor and calibration source unit for the Subaru LGSAO system,” Proc. SPIE 7015, 701564 (2008).
[CrossRef]

2006 (1)

2002 (1)

L. C. Roberts and C. R. Neyman, “Characterization of the AEOS adaptive optics system,” Publ. Astron. Soc. Pac. 114, 1260–1266(2002).
[CrossRef]

2001 (1)

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wave-front sensing,” J. Refractive Surg. 17, 573–577(2001).

1997 (1)

1989 (1)

N. Roddier, “Curvature sensing for adaptive optics: a computer simulation,” Master’s thesis (University of Arizona, 1989).

1988 (1)

1981 (1)

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics (North-Holland, 1981), Vol.  19, pp. 281–376.
[CrossRef]

1977 (2)

Arjona, M.

Colley, S.

M. Watanabe, S. Oya, Y. Hayano, H. Takami, M. Hattori, Y. Minowa, Y. Saito, M. Ito, N. Murakami, M. Iye, O. Guyon, S. Colley, M. Eldred, T. Golota, and M. Dinkins, “Implementation of 188-element curvature-based wavefront sensor and calibration source unit for the Subaru LGSAO system,” Proc. SPIE 7015, 701564 (2008).
[CrossRef]

Diaz-Douton, F.

Diaz-Santana, L.

C. Torti, S. Gruppetta, and L. Diaz-Santana, “Wavefront curvature sensing for the human eye,” J. Mod. Opt. 55, 691–702(2008).
[CrossRef]

Dinkins, M.

M. Watanabe, S. Oya, Y. Hayano, H. Takami, M. Hattori, Y. Minowa, Y. Saito, M. Ito, N. Murakami, M. Iye, O. Guyon, S. Colley, M. Eldred, T. Golota, and M. Dinkins, “Implementation of 188-element curvature-based wavefront sensor and calibration source unit for the Subaru LGSAO system,” Proc. SPIE 7015, 701564 (2008).
[CrossRef]

Eldred, M.

M. Watanabe, S. Oya, Y. Hayano, H. Takami, M. Hattori, Y. Minowa, Y. Saito, M. Ito, N. Murakami, M. Iye, O. Guyon, S. Colley, M. Eldred, T. Golota, and M. Dinkins, “Implementation of 188-element curvature-based wavefront sensor and calibration source unit for the Subaru LGSAO system,” Proc. SPIE 7015, 701564 (2008).
[CrossRef]

Fried, D.

Golota, T.

M. Watanabe, S. Oya, Y. Hayano, H. Takami, M. Hattori, Y. Minowa, Y. Saito, M. Ito, N. Murakami, M. Iye, O. Guyon, S. Colley, M. Eldred, T. Golota, and M. Dinkins, “Implementation of 188-element curvature-based wavefront sensor and calibration source unit for the Subaru LGSAO system,” Proc. SPIE 7015, 701564 (2008).
[CrossRef]

Gruppetta, S.

C. Torti, S. Gruppetta, and L. Diaz-Santana, “Wavefront curvature sensing for the human eye,” J. Mod. Opt. 55, 691–702(2008).
[CrossRef]

Guyon, O.

M. Watanabe, S. Oya, Y. Hayano, H. Takami, M. Hattori, Y. Minowa, Y. Saito, M. Ito, N. Murakami, M. Iye, O. Guyon, S. Colley, M. Eldred, T. Golota, and M. Dinkins, “Implementation of 188-element curvature-based wavefront sensor and calibration source unit for the Subaru LGSAO system,” Proc. SPIE 7015, 701564 (2008).
[CrossRef]

Hattori, M.

M. Watanabe, S. Oya, Y. Hayano, H. Takami, M. Hattori, Y. Minowa, Y. Saito, M. Ito, N. Murakami, M. Iye, O. Guyon, S. Colley, M. Eldred, T. Golota, and M. Dinkins, “Implementation of 188-element curvature-based wavefront sensor and calibration source unit for the Subaru LGSAO system,” Proc. SPIE 7015, 701564 (2008).
[CrossRef]

Hayano, Y.

M. Watanabe, S. Oya, Y. Hayano, H. Takami, M. Hattori, Y. Minowa, Y. Saito, M. Ito, N. Murakami, M. Iye, O. Guyon, S. Colley, M. Eldred, T. Golota, and M. Dinkins, “Implementation of 188-element curvature-based wavefront sensor and calibration source unit for the Subaru LGSAO system,” Proc. SPIE 7015, 701564 (2008).
[CrossRef]

Hudgin, R.

Ito, M.

M. Watanabe, S. Oya, Y. Hayano, H. Takami, M. Hattori, Y. Minowa, Y. Saito, M. Ito, N. Murakami, M. Iye, O. Guyon, S. Colley, M. Eldred, T. Golota, and M. Dinkins, “Implementation of 188-element curvature-based wavefront sensor and calibration source unit for the Subaru LGSAO system,” Proc. SPIE 7015, 701564 (2008).
[CrossRef]

Iye, M.

M. Watanabe, S. Oya, Y. Hayano, H. Takami, M. Hattori, Y. Minowa, Y. Saito, M. Ito, N. Murakami, M. Iye, O. Guyon, S. Colley, M. Eldred, T. Golota, and M. Dinkins, “Implementation of 188-element curvature-based wavefront sensor and calibration source unit for the Subaru LGSAO system,” Proc. SPIE 7015, 701564 (2008).
[CrossRef]

Kellerer, A.

Lena, P.

P. Lena, “Adaptive optics: a breakthrough in astronomy,” Exp. Astron. 26, 35–48 (2009).
[CrossRef]

Liang, J.

Luque, S. O.

Miller, D. T.

Minowa, Y.

M. Watanabe, S. Oya, Y. Hayano, H. Takami, M. Hattori, Y. Minowa, Y. Saito, M. Ito, N. Murakami, M. Iye, O. Guyon, S. Colley, M. Eldred, T. Golota, and M. Dinkins, “Implementation of 188-element curvature-based wavefront sensor and calibration source unit for the Subaru LGSAO system,” Proc. SPIE 7015, 701564 (2008).
[CrossRef]

Murakami, N.

M. Watanabe, S. Oya, Y. Hayano, H. Takami, M. Hattori, Y. Minowa, Y. Saito, M. Ito, N. Murakami, M. Iye, O. Guyon, S. Colley, M. Eldred, T. Golota, and M. Dinkins, “Implementation of 188-element curvature-based wavefront sensor and calibration source unit for the Subaru LGSAO system,” Proc. SPIE 7015, 701564 (2008).
[CrossRef]

Neyman, C. R.

L. C. Roberts and C. R. Neyman, “Characterization of the AEOS adaptive optics system,” Publ. Astron. Soc. Pac. 114, 1260–1266(2002).
[CrossRef]

Oya, S.

M. Watanabe, S. Oya, Y. Hayano, H. Takami, M. Hattori, Y. Minowa, Y. Saito, M. Ito, N. Murakami, M. Iye, O. Guyon, S. Colley, M. Eldred, T. Golota, and M. Dinkins, “Implementation of 188-element curvature-based wavefront sensor and calibration source unit for the Subaru LGSAO system,” Proc. SPIE 7015, 701564 (2008).
[CrossRef]

Platt, B. C.

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wave-front sensing,” J. Refractive Surg. 17, 573–577(2001).

Pujol, J.

Roberts, L. C.

L. C. Roberts and C. R. Neyman, “Characterization of the AEOS adaptive optics system,” Publ. Astron. Soc. Pac. 114, 1260–1266(2002).
[CrossRef]

Roddier, F.

F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225 (1988).
[CrossRef] [PubMed]

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics (North-Holland, 1981), Vol.  19, pp. 281–376.
[CrossRef]

Roddier, N.

N. Roddier, “Curvature sensing for adaptive optics: a computer simulation,” Master’s thesis (University of Arizona, 1989).

Saito, Y.

M. Watanabe, S. Oya, Y. Hayano, H. Takami, M. Hattori, Y. Minowa, Y. Saito, M. Ito, N. Murakami, M. Iye, O. Guyon, S. Colley, M. Eldred, T. Golota, and M. Dinkins, “Implementation of 188-element curvature-based wavefront sensor and calibration source unit for the Subaru LGSAO system,” Proc. SPIE 7015, 701564 (2008).
[CrossRef]

Shack, R.

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wave-front sensing,” J. Refractive Surg. 17, 573–577(2001).

Takami, H.

M. Watanabe, S. Oya, Y. Hayano, H. Takami, M. Hattori, Y. Minowa, Y. Saito, M. Ito, N. Murakami, M. Iye, O. Guyon, S. Colley, M. Eldred, T. Golota, and M. Dinkins, “Implementation of 188-element curvature-based wavefront sensor and calibration source unit for the Subaru LGSAO system,” Proc. SPIE 7015, 701564 (2008).
[CrossRef]

Torti, C.

C. Torti, S. Gruppetta, and L. Diaz-Santana, “Wavefront curvature sensing for the human eye,” J. Mod. Opt. 55, 691–702(2008).
[CrossRef]

Watanabe, M.

M. Watanabe, S. Oya, Y. Hayano, H. Takami, M. Hattori, Y. Minowa, Y. Saito, M. Ito, N. Murakami, M. Iye, O. Guyon, S. Colley, M. Eldred, T. Golota, and M. Dinkins, “Implementation of 188-element curvature-based wavefront sensor and calibration source unit for the Subaru LGSAO system,” Proc. SPIE 7015, 701564 (2008).
[CrossRef]

Williams, D. R.

Appl. Opt. (1)

Exp. Astron. (1)

P. Lena, “Adaptive optics: a breakthrough in astronomy,” Exp. Astron. 26, 35–48 (2009).
[CrossRef]

J. Mod. Opt. (1)

C. Torti, S. Gruppetta, and L. Diaz-Santana, “Wavefront curvature sensing for the human eye,” J. Mod. Opt. 55, 691–702(2008).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

J. Refractive Surg. (1)

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wave-front sensing,” J. Refractive Surg. 17, 573–577(2001).

Opt. Lett. (1)

Proc. SPIE (1)

M. Watanabe, S. Oya, Y. Hayano, H. Takami, M. Hattori, Y. Minowa, Y. Saito, M. Ito, N. Murakami, M. Iye, O. Guyon, S. Colley, M. Eldred, T. Golota, and M. Dinkins, “Implementation of 188-element curvature-based wavefront sensor and calibration source unit for the Subaru LGSAO system,” Proc. SPIE 7015, 701564 (2008).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

L. C. Roberts and C. R. Neyman, “Characterization of the AEOS adaptive optics system,” Publ. Astron. Soc. Pac. 114, 1260–1266(2002).
[CrossRef]

Other (2)

N. Roddier, “Curvature sensing for adaptive optics: a computer simulation,” Master’s thesis (University of Arizona, 1989).

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics (North-Holland, 1981), Vol.  19, pp. 281–376.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Analysis of the error propagation in SH and RC sensors is separated into two steps: signals to mean curvatures and mean curvatures to phase estimates.

Fig. 2
Fig. 2

Upper curve of symbols, e 2 , RC . Lower curve of symbols, e 2 , SH . The results are given for the number n of sensors that occur within circular regions, with the grid center positioned among four sensors.

Fig. 3
Fig. 3

Ratio r 2 = e 2 , RC / e 2 , SH . The results are given for the number n of sensors that occur within circular regions, with the grid center positioned among four sensors.

Fig. 4
Fig. 4

Error-propagation factors e 2 , RC (upper pair of curves) and e 2 , SH (lower pair of curves). In each pair of curves, the upper curve gives the total e 2 factor, while the lower curve gives the reduced value that remains when tip and tilt are removed from the estimates. The results without tip and tilt compare to the ones in Fig. 2. The results are obtained for an orthogonal grid of n sensors within a circular aperture. The grid center is positioned among four sensors.

Fig. 5
Fig. 5

The ratio, r 2 , of the error-propagation factors e 2 , RC and e 2 , SH represented in Fig. 4. The upper curve of symbols relates to the entire standard error of the phase estimates. The lower curve of symbols relates to the standard error without the contribution of tip and tilt. It is in reasonable agreement with the approximate results represented in Fig. 3.

Equations (31)

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x i , k = F i + 1 , k F i , k , y i , k = F i , k + 1 F i , k .
c i , k = ( x i , k x i 1 , k + y i , k y i , k 1 ) / 4 = ( F i 1 , k + F i + 1 , k + F i , k 1 + F i , k + 1 ) / 4 F i , k .
F i , k = l , m x f i , k ( l , m ) x l , m + l , m y f i , k ( l , m ) y l , m .
F i , k = l , m f i , k ( l , m ) c l , m .
F i , k ( F i 1 , k + F i + 1 , k + F i , k 1 + F i , k + 1 + x i , k x i 1 , k + y i , k y i , k 1 ) / 4 = ( F i 1 , k + F i + 1 , k + F i , k 1 + F i , k + 1 ) / 4 c i , k .
c = U · F .
F = R · c .
F = R x · x + R y · y ,
σ F , SH = σ z · e SH , σ F , RC = σ v · e RC .
e 1 , SH = σ c , SH / σ z , e 1 , RC = σ c , RC / σ v .
e 2 , SH = σ F , SH / σ c , SH , e 2 , RC = σ F , RC / σ c , RC .
r = e RC / e SH = e 1 , RC / e 1 , SH · e 2 , RC / e 2 , SH = r 1 · r 2 .
s = 2 π λ d p f l z ,
e 1 , SH = σ c , SH / σ s = π d p / λ f l .
e 1 , SH n 11 / 12 .
c = 2 π λ d 2 l f 2 v .
e 1 , RC = σ c , RC / σ v = 2 π λ d 2 l f 2 .
e 1 , RC n 1 / 2 .
r 1 = e 1 , RC / e 1 , SH n 5 / 12 .
F i , k = ϵ / 4 · ( f i , k ( l , m ) f i , k ( l + 1 , m ) ) .
SH 2 = ϵ 2 / 16 · [ ( f ( l , m ) f ( l + 1 , m ) ) 2 ( f ( l , m ) f ( l + 1 , m ) ) 2 ] = ϵ 2 / 16 · [ f ( l , m ) 2 f ( l , m ) 2 + f ( l + 1 , m ) 2 f ( l + 1 , m ) 2 2 ( f ( l , m ) · f ( l + 1 , m ) f ( l , m ) · f ( l + 1 , m ) ) ] ,
SH 2 = [ var ( f ( l , m ) ) + var ( f ( l + 1 , m ) ) 2 cov ( f ( l , m ) , f ( l + 1 , m ) ) ] / 4.
σ F , SH 2 = e 2 , SH 2 = l , m [ var ( f ( l , m ) ) cov ( f ( l , m ) , f ( l + 1 , m ) ] .
RC 2 = ϵ 2 2 / 16 · [ f ( l , m ) 2 f ( l , m ) 2 ] + ϵ 1 2 / 16 · [ f ( l + 1 , m ) 2 f ( l + 1 , m ) 2 ] ϵ 1 ϵ 2 / 8 · [ f ( l , m ) · f ( l + 1 , m ) f ( l , m ) · f ( l + 1 , m ) ] .
σ F , RC 2 = e 2 , RC 2 = l , m var ( f ( l , m ) ) .
d i 2 + k 2 = d · u .
f i , k = 2 / π ln ( u ) = ln ( i 2 + k 2 ) / π , f 0 , 0 = f ( 0 ) = 1
f i , k ( l , m ) = ln ( ( i l ) 2 + ( k m ) 2 ) / π , f l , m ( l , m ) = 1.
var ( f ( l , m ) ) = f i , k ( l , m ) 2 f i , k ( l , m ) 2 cov ( f ( l , m ) , f ( l + 1 , m ) ) = f i , k ( l , m ) · f i , k ( l + 1 , m ) f i , k ( l , m ) · f i , k ( l + 1 , m ) .
e 2 , RC = n 0.5 R i , k 2 .
e 2 , SH = n 0.5 ( R i , k 2 x + R i , k 2 x ) / 4 = n 0.5 R i , k 2 x / 2.

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