Abstract

In astronomical imaging, the errors in the wave-front slope are a significant cause of aberrations in the detected image. We investigate how the slope can be estimated optimally using an intensity measurement of the propagated wave front. We show that the optimal location for detection of wave-front tilt is the focal plane, and we quantify the error in using defocused images, such as would be obtained from a curvature sensor, for estimating the wave-front tilt. The effect of using broadband light is also quantified.

© 2002 Optical Society of America

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References

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2000 (1)

1999 (2)

1998 (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optimiz. 9, 112–147 (1998).
[CrossRef]

1996 (1)

S. Rios, E. Acosta, S. Bara, “Modal phase estimation from wavefront curvature sensing,” Opt. Commun. 123, 453–456 (1996).
[CrossRef]

1995 (1)

I. Han, “New method for estimating wavefront from curvature signal by curve fitting,” Opt. Eng. 34, 1232–1237 (1995).
[CrossRef]

1993 (1)

1990 (1)

1988 (1)

1983 (1)

1982 (1)

1976 (1)

1966 (1)

Acosta, E.

S. Rios, E. Acosta, S. Bara, “Modal phase estimation from wavefront curvature sensing,” Opt. Commun. 123, 453–456 (1996).
[CrossRef]

Bara, S.

S. Rios, E. Acosta, S. Bara, “Modal phase estimation from wavefront curvature sensing,” Opt. Commun. 123, 453–456 (1996).
[CrossRef]

Becklund, O. A.

C. S. Williams, O. A. Becklund, Introduction to the Optical Transfer Function (Wiley, New York, 1989), pp. 291–299.

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), p. 491.

Cramér, H.

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

Ellerbroek, B. L.

Fried, D. L.

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), pp. 57–100.

Han, I.

I. Han, “New method for estimating wavefront from curvature signal by curve fitting,” Opt. Eng. 34, 1232–1237 (1995).
[CrossRef]

Harding, C. M.

Johnston, R. A.

Lagarias, J. C.

J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optimiz. 9, 112–147 (1998).
[CrossRef]

Lane, R. G.

Noll, R. J.

Reeds, J. A.

J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optimiz. 9, 112–147 (1998).
[CrossRef]

Rios, S.

S. Rios, E. Acosta, S. Bara, “Modal phase estimation from wavefront curvature sensing,” Opt. Commun. 123, 453–456 (1996).
[CrossRef]

Roddier, C.

Roddier, F.

Rousset, G.

G. Rousset, “Wave-front sensors,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge U. Press, Cambridge, UK, 1999), pp. 91–130.

Teague, M. R.

Tyler, D. W.

van Dam, M. A.

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), pp. 66–85, Part 1.

Williams, C. S.

C. S. Williams, O. A. Becklund, Introduction to the Optical Transfer Function (Wiley, New York, 1989), pp. 291–299.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), p. 491.

Wright, M. H.

J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optimiz. 9, 112–147 (1998).
[CrossRef]

Wright, P. E.

J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optimiz. 9, 112–147 (1998).
[CrossRef]

Appl. Opt. (4)

J. Opt. Soc. Am. (4)

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

Opt. Commun. (1)

S. Rios, E. Acosta, S. Bara, “Modal phase estimation from wavefront curvature sensing,” Opt. Commun. 123, 453–456 (1996).
[CrossRef]

Opt. Eng. (1)

I. Han, “New method for estimating wavefront from curvature signal by curve fitting,” Opt. Eng. 34, 1232–1237 (1995).
[CrossRef]

SIAM J. Optimiz. (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optimiz. 9, 112–147 (1998).
[CrossRef]

Other (7)

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), p. 491.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), pp. 57–100.

C. S. Williams, O. A. Becklund, Introduction to the Optical Transfer Function (Wiley, New York, 1989), pp. 291–299.

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

H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), pp. 66–85, Part 1.

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

G. Rousset, “Wave-front sensors,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge U. Press, Cambridge, UK, 1999), pp. 91–130.

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Figures (6)

Fig. 1
Fig. 1

Least-mean-squares slope of a wave front.

Fig. 2
Fig. 2

Geometry to calculate the OTF of a defocused circular aperture.

Fig. 3
Fig. 3

CRLB of slope estimate for different levels of turbulence, D=1 m and 1 photon. The curves represent, from top to bottom, D/r0=40, 20, 10, 5, 1, and 0.

Fig. 4
Fig. 4

CRLB of slope estimate for 300 nm<λ<900 nm (top curve) and λ=600 nm (bottom curve) for (a) no turbulence and (b) D/r0=10 at λ=600 nm.

Fig. 5
Fig. 5

Solid curves are the standard deviation of the slope estimate error with use of the ML estimator from one detector plane. The error is plotted in (a) for an infinite number of photons with the top and bottom curves corresponding to a prior distribution of the PSF and the zero LMS slope speckle image, respectively. The circle at 1.5 rad is the error using the curvature-sensing measurements. The error with 100 photons is plotted in (b) with the top and bottom curves corresponding to a prior distribution of the PSF and the zero LMS slope speckle image respectively. The dashed curve is the CRLB for 100 photons with use of the PSF.

Fig. 6
Fig. 6

Displacement of aberrated defocused aperture. The curves, from top to bottom, are for Z2, Z8, Z16, and Z30.

Tables (1)

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Table 1 Effect of Zernike Polynomials on the Tip Term

Equations (30)

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U1(x)=1iλz expi 2πzλexpi πλz x2×-U0(ξ) expi πλz ξ2 exp-i 2πλz ξxdξ.
|U1(x)|2=1λz -U0(ξ)expi πλz ξ2exp-i 2πλz ξxdξ2.
U0(ξ)=A(ξ)exp[iϕ(ξ)],
|U1(x)|2=1λz -U˜0(ξ)expi πλz ξ2×exp-i 2πλz ξ(x-θz)dξ2.
Δx=θz,
OTF(ξ)=-h(x) expi 2πλz ξxdx,
OTF(ξ)=-U0(τ+ξ)U0(τ)dτ.
h(x)=1λz -U0(ξ)expi π(f-z)λfz ξ2×exp-i 2πλz ξxdξ2,
OTFlin,def(ξ)=-P(η)expi π(f-z)λfzη2P(ξ-η)×exp-i π(f-z)λfz(ξ-η)2dη=1-|ξ|Dsincπ(f-z)λfz |ξ|D 1-|ξ|D=(1-u)sincπ(f-z)λfzu(1-u),
P(ξ)=1,ξ{-D/2,D/2}0,otherwise.
δϕ=[(μ+u/2)2+ν2-(μ-u/2)2-ν2] π(f-z)λfz=2μu π(f-z)λfz.
OTFcirc,def(u)
=4π -(1-u)/2(1-u)/2-[1/4-(μ+u/2)2]1/2[1/4-(μ+u/2)2]1/2 expi2μu π(f-z)λfzdνdμ
=8π -(1-u)/2(1-u)/2[1/4-(μ+u/2)2]1/2 cos2μu π(f-z)λfzdμ.
OT̂Fcirc,def(u)=4π[cos-1(u)-u1-u2]J1π(f-z)u(1-u)λfz×λfzπ(f-z)u(1-u).
OTFturb=exp-3.44uDr05/3(1-u1/3),
OTFturb,def=(1-u)sincπ(f-z)λfzu(1-u)×exp-3.44uDr05/3(1-u1/3),
-D/2D/2 π(f-z)λfzu2ϕ(u)du=0.
θˆML=θmaxi=0N-1h(xi|θ).
Var[θˆ(x0)-θ]E ln h(x0|θ)θ2-1,
Var[θˆ(x0)-θ]-E2 ln h(x0|θ)θ2-1.
 ln h(x0|θ)θ=[θˆ(x0)-θ]ψ(θ),
- 1h(x0|θ) h(x0|θ)θ2dx0,
Iz=-λ2π(I·ϕ+I2ϕ),
I1(x)-I2(-x)I1(x)+I2(-x)=λf(f-l)2πl ϕn fxlδc-P2ϕfxl,
zR=2λ(f/D)2=0.12 mm.
si=AiρzΔα=2 fAiD(f-z)zΔα=NfAiπD(f-z)z.
θˆ=s, c/c, c=2NfπDz(f-z)A, c.
ϕ(ξ, η)=k=2dkZk(ξ, η),
Z2n=2 cos(α).

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