Abstract

The conventional way of measuring the average slope of the phase of a wave front is from the centroid of the image formed at the focal plane. We show the limitations of using the centroid and present an optimal estimator along with the derivation of its lower error bound for a diffraction-limited image. The method is extended to slope estimation in the case of a random aberration introduced by atmospheric turbulence. It was found that the variance of the error of the slope estimator can be improved significantly at low turbulence levels by using the minimum mean-square-error estimator instead of the centroid.

© 2000 Optical Society of America

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References

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  1. R. Irwan, R. G. Lane, “Analysis of optimal centroid estimation applied to Shack–Hartmann sensing,” Appl. Opt. 38, 6737–6743 (1999).
    [CrossRef]
  2. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  3. F. Roddier, “The effect of atmospheric turbulence in optical astronomy,” Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), pp. 283–376.
  4. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), pp. 59–65.
  5. M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996), pp. 182–183.
  6. M. R. Teague, “Irradiance moments: their propagation and use for unique retrieval of phase,” J. Opt. Soc. Am. 72, 1199–1209 (1982).
    [CrossRef]
  7. H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part 1 (Wiley, New York1968), pp. 66–85.
  8. S. Gasiorowicz, Quantum Physics (Wiley, New York, 1974), p. 55.
  9. H. Cramér, Mathematical Methods of Statistics (Princeton U. Press, Princeton, N.J., 1946), pp. 498–505.
  10. K. A. Winick, “Cramér–Rao lower bounds on the performance of charge-coupled-device optical position estimators,” J. Opt. Soc. Am. A 3, 1809–1815 (1986).
    [CrossRef]
  11. G. Cao, X. Yu, “Accuracy analysis of a Hartmann–Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2331–2335 (1994).
    [CrossRef]
  12. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  13. C. M. Harding, R. A. Johnston, R. G. Lane, “Fast simulation of a Kolmogorov phase screen,” Appl. Opt. 38, 2161–2170 (1999).
    [CrossRef]

1999

1994

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

1986

1982

1980

1976

Cao, G.

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

Cramér, H.

H. Cramér, Mathematical Methods of Statistics (Princeton U. Press, Princeton, N.J., 1946), pp. 498–505.

Gasiorowicz, S.

S. Gasiorowicz, Quantum Physics (Wiley, New York, 1974), p. 55.

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), pp. 59–65.

Harding, C. M.

Irwan, R.

Johnston, R. A.

Lane, R. G.

Noll, R. J.

Roddier, F.

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

Roggemann, M. C.

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996), pp. 182–183.

Southwell, W. H.

Teague, M. R.

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part 1 (Wiley, New York1968), pp. 66–85.

Welsh, B.

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996), pp. 182–183.

Winick, K. A.

Yu, X.

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

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

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

Other

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

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), pp. 59–65.

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996), pp. 182–183.

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part 1 (Wiley, New York1968), pp. 66–85.

S. Gasiorowicz, Quantum Physics (Wiley, New York, 1974), p. 55.

H. Cramér, Mathematical Methods of Statistics (Princeton U. Press, Princeton, N.J., 1946), pp. 498–505.

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Figures (3)

Fig. 1
Fig. 1

Cramér–Rao bound for the Airy disk with N photons. The points, with 1σ error bars, represent the simulation results.

Fig. 2
Fig. 2

Unnormalized prior distribution, p(a), for W=10.

Fig. 3
Fig. 3

Comparison of the variance of the error (pixels2) of the MMSE and centroid estimators as a function of the level of turbulence on a truncated plane. The error bars are the 1σ uncertainty.

Equations (38)

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I(ξ, η)=1λ2f 2 --A(x)exp[iϕ(x)]×exp-i 2πλf (xξ+yη)dxdy2,
h(ρ)=J1(ρ)2πρ2,
aˆ=amax i=0N-1h(ui|a).
g(u)=12πσ2 exp(-u2/2σ2),
i=0N-1g(ui|a)
=i=0N-1 12πσ2 exp-(ui-au)2-(vi-av)22σ2.
i=0N-1 ln g(ui|a)
=N ln 12πσ2-i=0N-1(ui-au)2+(vi-av)22σ2.
1σ2 i=0N-1(ui-au)=0,
amax h(u0|a)=amax h{[(u0-au)2+(v0-av)2]1/2}.
E[aˆ]=--aˆh(u0|a)du0dv0,
 E[(aˆ-a)2]=--(aˆ-a)2h(u0|a)du0dv0
=--(u0-a)2 J1(u0-a)2π(u0-a)2 du0dv0
=2π0 J1(ρ)2π ρdρ,
Var(a^i-ai)Jii,i1, 2,
J=-E2 ln h(u|a)uu22 ln h(u|a)auav2 ln h(u|a)avau2 ln h(u|a)av2.
2 ln h(u|a)au2=2 ln h(ρ)au2
=ρau2 2 ln h(ρ)ρ2+2ρau2  ln h(ρ)ρ
=(u-au)2ρ2 2 ln h(ρ)ρ2+(v-av)2ρ3  ln h(ρ)ρ.
J11=---(u-au)2ρ2 2 ln h(ρ)ρ2+(v-av)2ρ3  ln h(ρ)ρh(ρ)dudv.
J11+J22=---(u-au)2+(v-av)2ρ2 2 ln h(ρ)ρ2+(u-au)2+(v-av)2ρ3  ln h(ρ)ρ×h(ρ)dudv
=-02π0 2 ln h(ρ)ρ2+1ρ ln h(ρ)ρ×h(ρ)ρdρdθ.
J12=---(u-au)(v-av)×1ρ2  2 ln h(ρ)ρ2-1ρ3  ln h(ρ)ρh(ρ)dudv,
=0
J=1001.
limNVar(a^i-ai)=Jii,i1,2,
s(u)=sin2(u)πu2.
J=-- d2 ln s(u)du2 s(u)du=--2u2-2sin2(u) sin2(u)πu2 du=4/3.
amaxf (u0, a)=amaxh{[(u0-au)2+(v0-av)2]1/2}p(a).
p(a)=12πγ 2 exp(-a2/2γ 2).
JP=-E2 ln p(a)au22 ln p(a)auav2 ln p(a)avau2 ln p(a)av2
=1/γ 2001/γ 2.
J=JD+JP=1/γ 2+N001/γ 2+N,
aˆ=--auavp(a)×i=0N-1h{[(ui-au)2+(vi-av)2]1/2}daudav.
E[aˆ]=-WWug(u-a)du-WWg(u-a)du,
E[aˆ]=-ug(u-a)p(a)du-p(a)da-g(u-a)du=p(a)-ug(u-a)du- p(a)da=p(a)a-p(a)da,
p(a)=-WWug(u-a)dua-WW g(u-a)du.
aˆ=au=-MM av=-MMauavp[(au2+av2)1/2]×i=0N-1h{[(ui-au)2+(vi-av)2]1/2}.

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