Abstract

In the context of multiconjugate adaptive optics, the optimum linear estimation of a wave-front phase for a target object using the phases of several surrounding natural guide stars (NGS’s) is studied. A Wiener-filter-type estimator is constructed. The minimum residual wave-front-phase (tomographic) error depends on the turbulence vertical profile, and for typical profiles it is almost insensitive to the presence of strong layers, contrary to current belief. Tomographic error is characterized by a new parameter δK, equivalent profile thickness, which depends on the NGS number K (typically δ5=0.5 km). The angular radius of the NGS configuration must not exceed r0/δK. Exact profile knowledge is not required. When the optimized filters are constructed from some model profile, the loss of the field size is within 10% with respect to exact profile knowledge. Moreover, a method to measure turbulence profile using wave-front-sensor data is outlined. Noise propagation in the restoration algorithm is significant, but not dramatic. Noise increases with increasing size of NGS constellation. Practically, guide stars for tomography should be at least as bright as those for classical adaptive optics.

© 2001 Optical Society of America

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References

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    [CrossRef]
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2000 (3)

R. Ragazzoni, E. Marchetti, G. Valente, “Adaptive-optics corrections available for the whole sky,” Nature 403, 54–56 (2000).
[CrossRef] [PubMed]

M. Le Louarn, N. Hubin, M. Sarazin, A. Tokovinin, “New challenges for adaptive optics: extremely large telescopes,” Mon. Not. R. Astron. Soc. 317, 535 (2000).
[CrossRef]

A. Tokovinin, M. Le Louarn, M. Sarazin, “Isoplanatism in a multiconjugate adaptive optics system,” J. Opt. Soc. Am. A 17, 1819–1827 (2000).
[CrossRef]

1997 (1)

1994 (2)

1990 (1)

M. Tallon, R. Foy, “Adaptive telescope with laser probe: isoplanatism and cone effect,” Astron. Astrophys. 235, 549 (1990).

1982 (1)

1979 (1)

1975 (1)

R. H. Dicke, “Phase-contrast detection of telescope seeing errors and their correction,” Astrophys. J. 198, 605–615 (1975).
[CrossRef]

Avila, R.

Beckers, J. M.

J. M. Beckers, “Increasing the size of the isoplanatic patch with multiconjugate adaptive optics,” in Proceedings of the ESO Conference on Very Large Telescopes and their Instrumentation, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1988), pp. 693–703.

Conan, J.-M.

T. Fusco, J.-M. Conan, V. Michau, L. M. Mugnier, G. Rousset, “Phase estimation for large field of view: application to multiconjugate adaptive optics,” in Propagation through the Atmosphere III, M. C. Roggemann, L. R. Bissonnette, eds., Proc. SPIE3763, 125–133 (1999).
[CrossRef]

Dicke, R. H.

R. H. Dicke, “Phase-contrast detection of telescope seeing errors and their correction,” Astrophys. J. 198, 605–615 (1975).
[CrossRef]

Ellerbroek, B. L.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U., Cambridge, UK, 1992), Chap. 13.3.

Foy, R.

M. Tallon, R. Foy, “Adaptive telescope with laser probe: isoplanatism and cone effect,” Astron. Astrophys. 235, 549 (1990).

Fried, D. L.

Fuchs, A.

A. Fuchs, J. Vernin, “Final report on PARSCA 1992 and 1993 campaigns,” (European Southern Observatory, Garching, Germany, 1996).

Fusco, T.

T. Fusco, J.-M. Conan, V. Michau, L. M. Mugnier, G. Rousset, “Phase estimation for large field of view: application to multiconjugate adaptive optics,” in Propagation through the Atmosphere III, M. C. Roggemann, L. R. Bissonnette, eds., Proc. SPIE3763, 125–133 (1999).
[CrossRef]

Hubin, N.

M. Le Louarn, N. Hubin, M. Sarazin, A. Tokovinin, “New challenges for adaptive optics: extremely large telescopes,” Mon. Not. R. Astron. Soc. 317, 535 (2000).
[CrossRef]

Johnston, D. C.

Lai, O.

F. Rigaut, J.-P. Véran, O. Lai, “An analytical model for Shack-Hartmann-based adaptive optics systems,” in Adap-tive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 1038–1048 (1998).
[CrossRef]

Le Louarn, M.

A. Tokovinin, M. Le Louarn, M. Sarazin, “Isoplanatism in a multiconjugate adaptive optics system,” J. Opt. Soc. Am. A 17, 1819–1827 (2000).
[CrossRef]

M. Le Louarn, N. Hubin, M. Sarazin, A. Tokovinin, “New challenges for adaptive optics: extremely large telescopes,” Mon. Not. R. Astron. Soc. 317, 535 (2000).
[CrossRef]

Marchetti, E.

R. Ragazzoni, E. Marchetti, G. Valente, “Adaptive-optics corrections available for the whole sky,” Nature 403, 54–56 (2000).
[CrossRef] [PubMed]

Masciadri, E.

Michau, V.

T. Fusco, J.-M. Conan, V. Michau, L. M. Mugnier, G. Rousset, “Phase estimation for large field of view: application to multiconjugate adaptive optics,” in Propagation through the Atmosphere III, M. C. Roggemann, L. R. Bissonnette, eds., Proc. SPIE3763, 125–133 (1999).
[CrossRef]

Mugnier, L. M.

T. Fusco, J.-M. Conan, V. Michau, L. M. Mugnier, G. Rousset, “Phase estimation for large field of view: application to multiconjugate adaptive optics,” in Propagation through the Atmosphere III, M. C. Roggemann, L. R. Bissonnette, eds., Proc. SPIE3763, 125–133 (1999).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U., Cambridge, UK, 1992), Chap. 13.3.

Ragazzoni, R.

R. Ragazzoni, E. Marchetti, G. Valente, “Adaptive-optics corrections available for the whole sky,” Nature 403, 54–56 (2000).
[CrossRef] [PubMed]

Rigaut, F.

F. Rigaut, J.-P. Véran, O. Lai, “An analytical model for Shack-Hartmann-based adaptive optics systems,” in Adap-tive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 1038–1048 (1998).
[CrossRef]

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. XIX, pp. 281–376.

Rousset, G.

T. Fusco, J.-M. Conan, V. Michau, L. M. Mugnier, G. Rousset, “Phase estimation for large field of view: application to multiconjugate adaptive optics,” in Propagation through the Atmosphere III, M. C. Roggemann, L. R. Bissonnette, eds., Proc. SPIE3763, 125–133 (1999).
[CrossRef]

Sarazin, M.

A. Tokovinin, M. Le Louarn, M. Sarazin, “Isoplanatism in a multiconjugate adaptive optics system,” J. Opt. Soc. Am. A 17, 1819–1827 (2000).
[CrossRef]

M. Le Louarn, N. Hubin, M. Sarazin, A. Tokovinin, “New challenges for adaptive optics: extremely large telescopes,” Mon. Not. R. Astron. Soc. 317, 535 (2000).
[CrossRef]

M. Sarazin, in OSA/ESO Topical Meeting on Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching, Germany, 1996), pp. 439–444.

Tallon, M.

M. Tallon, R. Foy, “Adaptive telescope with laser probe: isoplanatism and cone effect,” Astron. Astrophys. 235, 549 (1990).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U., Cambridge, UK, 1992), Chap. 13.3.

Tokovinin, A.

M. Le Louarn, N. Hubin, M. Sarazin, A. Tokovinin, “New challenges for adaptive optics: extremely large telescopes,” Mon. Not. R. Astron. Soc. 317, 535 (2000).
[CrossRef]

A. Tokovinin, M. Le Louarn, M. Sarazin, “Isoplanatism in a multiconjugate adaptive optics system,” J. Opt. Soc. Am. A 17, 1819–1827 (2000).
[CrossRef]

Valente, G.

R. Ragazzoni, E. Marchetti, G. Valente, “Adaptive-optics corrections available for the whole sky,” Nature 403, 54–56 (2000).
[CrossRef] [PubMed]

Valley, G. C.

Véran, J.-P.

F. Rigaut, J.-P. Véran, O. Lai, “An analytical model for Shack-Hartmann-based adaptive optics systems,” in Adap-tive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 1038–1048 (1998).
[CrossRef]

Vernin, J.

R. Avila, J. Vernin, E. Masciadri, “Whole atmosphere profiling with a generalized SCIDAR,” Appl. Opt. 36, 7898–7905 (1997).
[CrossRef]

A. Fuchs, J. Vernin, “Final report on PARSCA 1992 and 1993 campaigns,” (European Southern Observatory, Garching, Germany, 1996).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U., Cambridge, UK, 1992), Chap. 13.3.

Wandzura, S. M.

Welsh, B. M.

Appl. Opt. (1)

Astron. Astrophys. (1)

M. Tallon, R. Foy, “Adaptive telescope with laser probe: isoplanatism and cone effect,” Astron. Astrophys. 235, 549 (1990).

Astrophys. J. (1)

R. H. Dicke, “Phase-contrast detection of telescope seeing errors and their correction,” Astrophys. J. 198, 605–615 (1975).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Mon. Not. R. Astron. Soc. (1)

M. Le Louarn, N. Hubin, M. Sarazin, A. Tokovinin, “New challenges for adaptive optics: extremely large telescopes,” Mon. Not. R. Astron. Soc. 317, 535 (2000).
[CrossRef]

Nature (1)

R. Ragazzoni, E. Marchetti, G. Valente, “Adaptive-optics corrections available for the whole sky,” Nature 403, 54–56 (2000).
[CrossRef] [PubMed]

Other (7)

J. M. Beckers, “Increasing the size of the isoplanatic patch with multiconjugate adaptive optics,” in Proceedings of the ESO Conference on Very Large Telescopes and their Instrumentation, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1988), pp. 693–703.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. XIX, pp. 281–376.

F. Rigaut, J.-P. Véran, O. Lai, “An analytical model for Shack-Hartmann-based adaptive optics systems,” in Adap-tive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 1038–1048 (1998).
[CrossRef]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U., Cambridge, UK, 1992), Chap. 13.3.

T. Fusco, J.-M. Conan, V. Michau, L. M. Mugnier, G. Rousset, “Phase estimation for large field of view: application to multiconjugate adaptive optics,” in Propagation through the Atmosphere III, M. C. Roggemann, L. R. Bissonnette, eds., Proc. SPIE3763, 125–133 (1999).
[CrossRef]

A. Fuchs, J. Vernin, “Final report on PARSCA 1992 and 1993 campaigns,” (European Southern Observatory, Garching, Germany, 1996).

M. Sarazin, in OSA/ESO Topical Meeting on Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching, Germany, 1996), pp. 439–444.

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Figures (7)

Fig. 1
Fig. 1

Problem layout. The phase of the object wave-front ϕObj is estimated from the phases of the surrounding guide stars ϕi in an optimized way, with use of the a priori information on turbulence profile, noise and turbulence statistics. Variance of the residual phase Res is used as an optimization creterion and as a characteristic of tomographic error.

Fig. 2
Fig. 2

Error transfer function G(f) in the spatial frequency plane (right) for three guide stars and two positions of object: at the FOV center and decentered. The Paranal model profile (see Section 4) was used in the calculations. Only the upper half of the frequency plane is displayed owing to symmetry; coordinate origin is in the middle of the fx axis. Dark shadows are perpendicular to the directions for which the object projects onto one of the guide stars. Bright stripes at ±30° from vertical correspond to the directions for which matrix A would be singular without measurement noise; in these directions the error-transfer function depends on the adopted noise level.

Fig. 3
Fig. 3

For a symmetrical configuration of three guide stars the dependence of residual-phase variance on object position is traced when the object is moved from the FOV center in three directions. Object position along the radius is given in units of guide star distance from center.

Fig. 4
Fig. 4

Cumulative turbulence profiles for Paranal: fraction of total turbulent energy below a given altitude. Two different cases are selected: profile 52, which is similar to the mean profile (dashed and solid curves, respectively) and profile 51 (thin curve), which shows a strong peak at an altitude of 7.5 km, containing approximately 1/3 of the total energy.

Fig. 5
Fig. 5

Residual phase variance for a wavelength of 500 nm as a function of NGS configuration radius for (a) three NGS’s and (b) five NGS’s. Curves are labeled by NGS magnitudes. Dashed–dotted horizontal curves mark the rms residual levels of 0.5 and 1 rad2, corresponding to the Strehl ratio degradation of 0.37 and 0.61, respectively. Dashed curves show the 5/3 scaling law [Eq. (26)]. The object is at the FOV center. A Paranal model profile with r0=0.15 m is used.

Fig. 6
Fig. 6

Maximum Strehl ratio achievable at 500 nm as limited by combination of tomographic and photon noise errors with three NGS’s, as a function of their magnitude. A model Paranal profile is used for calculation. The radius of the NGS configuration is indicated near the curves. The dashed curve shows for comparison an on-axis performance of a classic single-NGS AO system.

Fig. 7
Fig. 7

Distribution of Strehl ratio at 500 nm over the FOV for a random constellation of five NGS’s of magnitudes (from left to right) 7, 6, 11, 8, and 10 (NGS positions are marked by pluses). The faintest star is at the center. Paranal profile 51 was used. Contours with numbers indicate Strehl ratio levels.

Tables (2)

Tables Icon

Table 1 Equivalent Thickness of a Uniform Layer for Different Numbers of Guide Stars

Tables Icon

Table 2 Equivalent Thickness of Atmosphere δK

Equations (33)

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

ϕ(r, ϕ)=n=1Nϕn(r-θhn).
ϕ˜(f, θ)=n=1Nϕ˜n(f)exp(-2πifθhn),
Mx(f)=2πifx sinc(dfx)sinc(dfy).
Sk(f)=M(f)ϕ˜(f, θk)+ν˜k(f),
νk2(f)=(2π/λ)2α2N-1 sinc2(dfx)sinc2(dfy),
ψ˜(f, θ)=k=1Kgk(f, θ)Sk(f).
˜(f, θ)=ϕ˜-ψ˜=n=1NP(f, θ, hn)ϕ˜n(f)-k=1Kgk(f, θ)ν˜k(f),
P(f, θ, h)=exp(-2πifθh)1-k=1Kgk(f, θ)M(f)×exp[2πif(θ-θk)h].
Wϕ(f)=W0(f)Cn2(hn)dh,
W0(f)=9.69×10-3(2π/λ)2|f|-11/3.
W(f)=W0(f)|P(f, θ, h)|2Cn2(h)dh+k=1K|gk(f, θ)|2νk2(f),
|P(f, θ, h)|2=1-2 Rek=1KgkMexp[2πif(θ-θk)h]+k=1Kk=1Kgkgk*|M|2 exp[2πif(θk-θk)h].
C˜(ξ)=Cn2(h)exp(2πiξh)dh.
W=W0c01-2 Rek=1Kgkck+k=1Kk=1Kgkgk*akk+k=1K|gk|2νk,
c0=C˜(0),
ck=MC˜[f(θ-θk)]/c0,
akk=|M|2C˜[f(θk-θk)]/c0,
νk=νk2/(W0c0).
c0=6.882.91λ2π2r0-5/3.
g=[(A+νI)-1c]*,
ν(f)=103.2α2N-1c0-1|f|11/3 sinc2(dfx)sinc2(dfy).
G(f, θ)=1-2 Re(cTg)+gTAg*
W(f)=c0W0(f)[G+gT(νI)g*].
G(f, θ)1-cTA-1c*.
tom2(θ)=c0W0(f)GK(f, θ)df.
tom2(θ)=c0Ө5/3W0(f)GK(f, θ)df.
tom2(θ)=Ө5/3r0-5/3δK5/3eK(θ),
δK5/3eK(θ)=0.0229|f|-11/3GK(f, θ)df.
tom2(θ)=eK(θ)(Ө/γK)5/3
γK=r0/δK.
Npixel=λλ01/54Dr0(λ0)δKλ0.
Wkk(f)=SkSk*=|M|2n=1Nϕ˜n(f)ϕ˜n*(f)×exp[-2πif(θk-θk)hn],
Wkk(f)=|M|2W0(f)C˜[f(θk-θk)].

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