Abstract

A wave-front sensing scheme based on placing a lenslet array at the focal plane of the telescope with each lenslet reimaging the aperture is analyzed. This wave-front sensing arrangement is the dual of the Shack–Hartmann sensor, with the wave front partitioned in the focal plane rather than in the aperture plane. This arrangement can be viewed as the generalization of the pyramid sensor and allows direct comparisons of this sensor with the Shack–Hartmann sensor. We show that, as with the Shack–Hartmann sensor, when subdividing in the focal plane, the quality of the wave-front estimate is a trade-off between the quality of the slope measurements over each region in the aperture and the resolution to which the slope measurements are obtained. Open-loop simulation results demonstrate that the performance of the lenslet array at the focal plane is equivalent to that of the Shack–Hartmann sensor when no modulation is applied to the lenslet array. However, when the array is modulated in a manner akin to that of the pyramid sensor, subdivision at the focal plane provides advantages when compared with the Shack–Hartmann sensor.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. Roddier, “The effect of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), pp. 283–376.
  2. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, New York, 1998), pp. 135–175.
  3. J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1598–1608 (1990).
    [CrossRef]
  4. M. A. van Dam, R. G. Lane, “Wave-front slope sensing,” J. Opt. Soc. Am. A 17, 1319–1324 (2000).
    [CrossRef]
  5. R. G. Lane, M. Tallon, “Wavefront reconstruction using a Shack–Hartmann sensor,” Appl. Opt. 31, 6902–6908 (1992).
    [CrossRef] [PubMed]
  6. R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
    [CrossRef]
  7. S. Esposito, A. Riccardi, “Pyramid wavefront sensor behavior in partial correction adaptive optic systems,” Astron. Astrophys. 369, L9–L12 (2001).
    [CrossRef]
  8. R. Ragazzoni, J. Farinato, “Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics,” Astron. Astrophys. 350, L23–L26 (1999).
  9. I. Iglesias, R. Ragazzoni, Y. Julien, P. Artal, “Extended source pyramid wavefront sensor for the human eye,” Opt. Express 10, 419–428 (2002), http://www.opticsexpress.org
    [CrossRef] [PubMed]
  10. R. Ragazzoni, E. Diolaiti, E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
    [CrossRef]
  11. F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225 (1988).
    [CrossRef] [PubMed]
  12. G. A. Tyler, D. L. Fried, “Image-position error associated with a quadrant detector,” J. Opt. Soc. Am. 72, 804–808 (1982).
    [CrossRef]
  13. B. L. Ellerbroek, D. W. Tyler, “Adaptive optics sky coverage calculations for the Gemini-North telescope,” Publ. Astron. Soc. Pac. 110, 165–185 (1998).
    [CrossRef]
  14. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), pp. 63–75.
  15. R. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  16. N. F. Law, R. G. Lane, “Wavefront estimation at low light levels,” Opt. Commun. 126, 19–24 (1996).
    [CrossRef]
  17. C. M. Harding, R. A. Johnston, R. G. Lane, “Fast simulation of a Kolmogorov phase screen,” Appl. Opt. 38, 2161–2170 (1999).
    [CrossRef]
  18. D. L. Fried, “Optical resolution through a randomly inhomogenous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1376–1379 (1966).
    [CrossRef]
  19. B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive-optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
    [CrossRef]

2002 (2)

2001 (1)

S. Esposito, A. Riccardi, “Pyramid wavefront sensor behavior in partial correction adaptive optic systems,” Astron. Astrophys. 369, L9–L12 (2001).
[CrossRef]

2000 (1)

1999 (2)

R. Ragazzoni, J. Farinato, “Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics,” Astron. Astrophys. 350, L23–L26 (1999).

C. M. Harding, R. A. Johnston, R. G. Lane, “Fast simulation of a Kolmogorov phase screen,” Appl. Opt. 38, 2161–2170 (1999).
[CrossRef]

1998 (1)

B. L. Ellerbroek, D. W. Tyler, “Adaptive optics sky coverage calculations for the Gemini-North telescope,” Publ. Astron. Soc. Pac. 110, 165–185 (1998).
[CrossRef]

1996 (2)

N. F. Law, R. G. Lane, “Wavefront estimation at low light levels,” Opt. Commun. 126, 19–24 (1996).
[CrossRef]

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
[CrossRef]

1992 (1)

1990 (1)

1989 (1)

1988 (1)

1982 (1)

1976 (1)

1966 (1)

D. L. Fried, “Optical resolution through a randomly inhomogenous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1376–1379 (1966).
[CrossRef]

Artal, P.

Diolaiti, E.

R. Ragazzoni, E. Diolaiti, E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
[CrossRef]

Ellerbroek, B. L.

B. L. Ellerbroek, D. W. Tyler, “Adaptive optics sky coverage calculations for the Gemini-North telescope,” Publ. Astron. Soc. Pac. 110, 165–185 (1998).
[CrossRef]

Esposito, S.

S. Esposito, A. Riccardi, “Pyramid wavefront sensor behavior in partial correction adaptive optic systems,” Astron. Astrophys. 369, L9–L12 (2001).
[CrossRef]

Farinato, J.

R. Ragazzoni, J. Farinato, “Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics,” Astron. Astrophys. 350, L23–L26 (1999).

Fontanella, J. C.

Fried, D. L.

G. A. Tyler, D. L. Fried, “Image-position error associated with a quadrant detector,” J. Opt. Soc. Am. 72, 804–808 (1982).
[CrossRef]

D. L. Fried, “Optical resolution through a randomly inhomogenous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1376–1379 (1966).
[CrossRef]

Gardner, C. S.

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), pp. 63–75.

Harding, C. M.

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, New York, 1998), pp. 135–175.

Iglesias, I.

Johnston, R. A.

Julien, Y.

Lane, R. G.

Law, N. F.

N. F. Law, R. G. Lane, “Wavefront estimation at low light levels,” Opt. Commun. 126, 19–24 (1996).
[CrossRef]

Noll, R.

Primot, J.

Ragazzoni, R.

I. Iglesias, R. Ragazzoni, Y. Julien, P. Artal, “Extended source pyramid wavefront sensor for the human eye,” Opt. Express 10, 419–428 (2002), http://www.opticsexpress.org
[CrossRef] [PubMed]

R. Ragazzoni, E. Diolaiti, E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
[CrossRef]

R. Ragazzoni, J. Farinato, “Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics,” Astron. Astrophys. 350, L23–L26 (1999).

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
[CrossRef]

Riccardi, A.

S. Esposito, A. Riccardi, “Pyramid wavefront sensor behavior in partial correction adaptive optic systems,” Astron. Astrophys. 369, L9–L12 (2001).
[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 effect of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), pp. 283–376.

Rousset, G.

Tallon, M.

Tyler, D. W.

B. L. Ellerbroek, D. W. Tyler, “Adaptive optics sky coverage calculations for the Gemini-North telescope,” Publ. Astron. Soc. Pac. 110, 165–185 (1998).
[CrossRef]

Tyler, G. A.

van Dam, M. A.

Vernet, E.

R. Ragazzoni, E. Diolaiti, E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
[CrossRef]

Welsh, B. M.

Appl. Opt. (3)

Astron. Astrophys. (2)

S. Esposito, A. Riccardi, “Pyramid wavefront sensor behavior in partial correction adaptive optic systems,” Astron. Astrophys. 369, L9–L12 (2001).
[CrossRef]

R. Ragazzoni, J. Farinato, “Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics,” Astron. Astrophys. 350, L23–L26 (1999).

J. Mod. Opt. (1)

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

Opt. Commun. (2)

N. F. Law, R. G. Lane, “Wavefront estimation at low light levels,” Opt. Commun. 126, 19–24 (1996).
[CrossRef]

R. Ragazzoni, E. Diolaiti, E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
[CrossRef]

Opt. Express (1)

Publ. Astron. Soc. Pac. (1)

B. L. Ellerbroek, D. W. Tyler, “Adaptive optics sky coverage calculations for the Gemini-North telescope,” Publ. Astron. Soc. Pac. 110, 165–185 (1998).
[CrossRef]

Other (3)

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), pp. 63–75.

F. Roddier, “The effect of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), pp. 283–376.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, New York, 1998), pp. 135–175.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Propagation of a region of a wave front of width Δx over a distance z allows the estimation of the displacement a with an uncertainty Δa.

Fig. 2
Fig. 2

Comparison of (a) the Shack–Hartmann wave-front sensor and (b) the pyramid sensor implemented as a lenslet array.

Fig. 3
Fig. 3

Effect of a local tilt at a point in the aperture on the corresponding point in the aperture images.

Fig. 4
Fig. 4

(a) Linear operation occurs in the Shack–Hartmann quad cell when the image intersects all four pixels. Alternatively, for the 2 × 2 lenslet array at the focal plane, linear operation occurs when the spot in the focal plane intersects all four lenslets. (b) Saturation occurs in the Shack–Hartmann quad cell when the image does not intersect all four pixels. For the 2 × 2 lenslet array at the focal plane, saturation occurs when the spot does not intersect all four lenslets.

Fig. 5
Fig. 5

Normalized global tilt characteristics for a lenslet array in the focal plane with (a) 2 × 2 array of lenslets such that the spot size is less than the lenslet width, (b) 4 × 4 array of lenslets such that the spot size is less than the lenslet width, (c) 64 × 64 array of lenslets such that the spot size is equal to the lenslet width.

Fig. 6
Fig. 6

Normalized global tilt characteristics for a 4 × 4 lenslet array in the focal plane where the modulation width of the array is (a) zero, (b) half the width of the lenslet, (c) equal to the width of the lenslet, (d) twice the width of the lenslet.

Fig. 7
Fig. 7

Comparison of the MSE (rad2) for different sized lenslet arrays at the aperture plane (Shack–Hartmann) and the focal plane with and without modulation, in turbulence of severity: (a) D/r0=8 and (b) D/r0=12.

Fig. 8
Fig. 8

Optical layout of a lenslet of width d and focal length x2 at the focal plane of an aperture of width D and focal length x1.

Equations (26)

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

σϕ=σθ2πΔxλ,
σθ=θb/SNR,
σϕ2π/SNR,
SNR=Np(Np+4Nb+4σe2)1/2,
σϕ2π/d,
a(u, v)=exp[j2πf/λ]exp[jπ(α2+β2)/λf]jλf×--A(x, y)×exp-j 2πλf (αx+βy)dxdy,
a(u, v)=F[A(x, y)]{u, v},
F[F(x, y)]=f(u, v)=--F(x, y)exp[-j2π(xu+yv)]×dxdy,
F-1[f(u, v)]=F(x, y)=--f(u, v)exp[j2π(xu+yv)]dudv.
I(ξ, η)|F-1(H(u, v)×F{P(ξ, η)exp[jϕ(ξ, η)]})|2.
I(ξ, η)|F-1{F[F-1[H(u, v)]  F-1(F{P(ξ, η)×exp[jϕ(ξ, η)]})]}|2.
I(ξ, η)|h(ξ, η)  P(ξ, η)exp[jϕ(ξ, η)]|2.
H(u, v)=1u-Δu2uu+Δu2,  v-Δv2vv+Δv20otherwise,
h(ξ, η)=Δu sinc(ξΔu)exp(j2πξu)Δv sinc(ηΔv)×exp(j2πηv),
I(ξ, η)|Δu sinc(ξΔu)exp(j2πξu)Δv sinc(ηΔv)×exp(j2πηv)P(ξ, η)exp[jϕ(ξ, η)]|2.
I(ξ, η)u-w/2u+w/2|Δu sinc(ξΔu)×exp(j2πξu)Δv sinc(ηΔv)×exp(j2πηv)  P(ξ, η)exp[jϕ(ξ, η)]|2du.
hm,n(ξ, η)=F-1[Hm,n(u, v)]=(n-1/2)d-δv(n+1/2)d-δv(m-1/2)d-δu(m+1/2)d-δuexp[j2π(uξ+vη)]dudv,
hm,n(ξ, η)=d2 sinc(ξd)exp[j2πξ(md-δu)]sinc(ηd)exp[j2πη(nd-δv)].
Im,n(ξ, η)=|d2 sinc(ξd)exp[j2πξ(md-δu)]sinc(ηd)exp[j2πη(nd-δv)] P(ξ, η)exp[jϕ(ξ, η)]|2.
ϕ(ξ, η)ξm=-M/2+1M/2n=-N/2+1N/2(md-δu)Im,n(ξ, η)m=-M/2+1M/2n=-N/2+1N/2Im,n(ξ, η),
ϕ(ξ, η)ηm=-M/2+1M/2n=-N/2+1N/2(nd-δv)Im,n(ξ, η)m=-M/2+1M/2n=-N/2+1N/2Im,n(ξ, η).
c=(Kzz-1+ΘTKnn-1Θ)-1ΘTKnn-1s,
d=2λx1D.
d/D=x2/x1.
d=2λx2d=2λF2,
d=2λF2=2λx2d,

Metrics