Abstract

Turbulence correction in a large field of view by use of an adaptive optics imaging system with several deformable mirrors (DM’s) conjugated to various heights is considered. The residual phase variance is computed for an optimized linear algorithm in which a correction of each turbulent layer is achieved by applying a combination of suitably smoothed and scaled input phase screens to all DM’s. Finite turbulence outer scale and finite spatial resolution of the DM’s are taken into account. A general expression for the isoplanatic angle θM of a system with M mirrors is derived in the limiting case of infinitely large apertures and Kolmogorov turbulence. Like Fried’s isoplanatic angle θ0,θM is a function only of the turbulence vertical profile, is scalable with wavelength, and is independent of the telescope diameter. Use of angle θM permits the gain in the field of view due to the increased number of DM’s to be quantified and their optimal conjugate heights to be found. Calculations with real turbulence profiles show that with three DM’s a gain of 7–10× is possible, giving the typical and best isoplanatic field-of-view radii of 16 and 30 arcseconds, respectively, at λ=0.5 μm. It is shown that in the actual systems the isoplanatic field will be somewhat larger than θM owing to the combined effects of finite aperture diameter, finite outer scale, and optimized wave-front spatial filtering. However, this additional gain is not dramatic; it is less than 1.5× for large-aperture telescopes.

© 2000 Optical Society of America

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References

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1998 (2)

M. Chun, “The useful field of view of an adaptive optics system,” Publ. Astron. Soc. Pac. 110, 317–329 (1998).
[CrossRef]

F. Martin, A. Tokovinin, A. Ziad, R. Conan, J. Borgnino, R. Avila, A. Agabi, M. Sarazin, “First statistical data on the wavefront outer scale at La Silla Observatory,” Astron. Astrophys. 336, L49–L52 (1998).

1994 (4)

1990 (1)

M. Tallon, R. Foy, “Adaptive telescope with laser probe—isoplanatism and cone effect,” Astron. Astrophys. 235, 549–557 (1990).

1989 (1)

F. Chassat, “Calcul du domaine d’isoplanétisme d’un systéme d’optique adaptative fonctionnant à travers la turbulence atmosphérique,” J. Opt. (Paris) 20, 13–23 (1989).
[CrossRef]

1982 (1)

1979 (1)

Agabi, A.

F. Martin, A. Tokovinin, A. Ziad, R. Conan, J. Borgnino, R. Avila, A. Agabi, M. Sarazin, “First statistical data on the wavefront outer scale at La Silla Observatory,” Astron. Astrophys. 336, L49–L52 (1998).

Avila, R.

F. Martin, A. Tokovinin, A. Ziad, R. Conan, J. Borgnino, R. Avila, A. Agabi, M. Sarazin, “First statistical data on the wavefront outer scale at La Silla Observatory,” Astron. Astrophys. 336, L49–L52 (1998).

Borgnino, J.

F. Martin, A. Tokovinin, A. Ziad, R. Conan, J. Borgnino, R. Avila, A. Agabi, M. Sarazin, “First statistical data on the wavefront outer scale at La Silla Observatory,” Astron. Astrophys. 336, L49–L52 (1998).

Chassat, F.

F. Chassat, “Calcul du domaine d’isoplanétisme d’un systéme d’optique adaptative fonctionnant à travers la turbulence atmosphérique,” J. Opt. (Paris) 20, 13–23 (1989).
[CrossRef]

Chun, M.

M. Chun, “The useful field of view of an adaptive optics system,” Publ. Astron. Soc. Pac. 110, 317–329 (1998).
[CrossRef]

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]

Conan, R.

F. Martin, A. Tokovinin, A. Ziad, R. Conan, J. Borgnino, R. Avila, A. Agabi, M. Sarazin, “First statistical data on the wavefront outer scale at La Silla Observatory,” Astron. Astrophys. 336, L49–L52 (1998).

Ellerbroek, B.

Foy, R.

M. Tallon, R. Foy, “Adaptive telescope with laser probe—isoplanatism and cone effect,” Astron. Astrophys. 235, 549–557 (1990).

Fried, D.

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]

Hu, P. H.

Johnston, D. C.

Ma, S.

Martin, F.

F. Martin, A. Tokovinin, A. Ziad, R. Conan, J. Borgnino, R. Avila, A. Agabi, M. Sarazin, “First statistical data on the wavefront outer scale at La Silla Observatory,” Astron. Astrophys. 336, L49–L52 (1998).

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]

Mills, S. P.

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]

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.

Roggemann, M. C.

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

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.

F. Martin, A. Tokovinin, A. Ziad, R. Conan, J. Borgnino, R. Avila, A. Agabi, M. Sarazin, “First statistical data on the wavefront outer scale at La Silla Observatory,” Astron. Astrophys. 336, L49–L52 (1998).

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

Stone, J.

Tallon, M.

M. Tallon, R. Foy, “Adaptive telescope with laser probe—isoplanatism and cone effect,” Astron. Astrophys. 235, 549–557 (1990).

Tokovinin, A.

F. Martin, A. Tokovinin, A. Ziad, R. Conan, J. Borgnino, R. Avila, A. Agabi, M. Sarazin, “First statistical data on the wavefront outer scale at La Silla Observatory,” Astron. Astrophys. 336, L49–L52 (1998).

Valley, G. C.

Vernin, J.

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

Wandzura, S. M.

Welsh, B.

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

Welsh, B. M.

Yan, J.

J. Yan, R. Zhou, X. Yu, “Problems with multiconjugate correction,” Opt. Eng. 33, 2942–2944 (1994).
[CrossRef]

Yu, X.

J. Yan, R. Zhou, X. Yu, “Problems with multiconjugate correction,” Opt. Eng. 33, 2942–2944 (1994).
[CrossRef]

Zhou, R.

J. Yan, R. Zhou, X. Yu, “Problems with multiconjugate correction,” Opt. Eng. 33, 2942–2944 (1994).
[CrossRef]

Ziad, A.

F. Martin, A. Tokovinin, A. Ziad, R. Conan, J. Borgnino, R. Avila, A. Agabi, M. Sarazin, “First statistical data on the wavefront outer scale at La Silla Observatory,” Astron. Astrophys. 336, L49–L52 (1998).

Astron. Astrophys. (2)

M. Tallon, R. Foy, “Adaptive telescope with laser probe—isoplanatism and cone effect,” Astron. Astrophys. 235, 549–557 (1990).

F. Martin, A. Tokovinin, A. Ziad, R. Conan, J. Borgnino, R. Avila, A. Agabi, M. Sarazin, “First statistical data on the wavefront outer scale at La Silla Observatory,” Astron. Astrophys. 336, L49–L52 (1998).

J. Opt. (Paris) (1)

F. Chassat, “Calcul du domaine d’isoplanétisme d’un systéme d’optique adaptative fonctionnant à travers la turbulence atmosphérique,” J. Opt. (Paris) 20, 13–23 (1989).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Eng. (1)

J. Yan, R. Zhou, X. Yu, “Problems with multiconjugate correction,” Opt. Eng. 33, 2942–2944 (1994).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

M. Chun, “The useful field of view of an adaptive optics system,” Publ. Astron. Soc. Pac. 110, 317–329 (1998).
[CrossRef]

Other (5)

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

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.

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.

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]

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Figures (6)

Fig. 1
Fig. 1

Weighting functions F0 (dotted curve), F1 (dashed curve) and F2 (solid curve) plotted against altitude. Conjugation altitudes are 5 km for F1, 1 and 8 km for F2.

Fig. 2
Fig. 2

Paranal-01 (the best) and Paranal-10 (typical) turbulence profiles. Heights are in kilometers above the observatory; Cn2 is in units of 10-17 m-2/3. For clarity, the profiles have been convolved with a 500-m standard deviation Gaussian. The arrows show the optimum heights for a three-DM MCAO configuration.

Fig. 3
Fig. 3

Isoplanatic angle (in arc sec) as a function of the number of deformable mirrors. Solid curve, median θM obtained from the Cerro Paranal data; dashed curve, linear fit with a slope of 5.8. This slope depends on a turbulence profile.

Fig. 4
Fig. 4

Residual phase variance as a function of off-axis angle for the 8-m telescope, the two-DM MCAO system, and Hufnagel–Valley turbulence profile. Wavelength, 0.5 μm. Solid curves, spatial filtering optimized for 1, 4, and 8; dashed curve, limiting curve (θ/θ2)5/3.

Fig. 5
Fig. 5

Same as in Fig. 4 but for a 4-m telescope working at λ=2.2 μm. The spatial filtering (solid curves) is optimized for 6, 24, and 60 FOV size.

Fig. 6
Fig. 6

Residual phase variance as a function of off-axis angle for the 8-m telescope, the two-DM MCAO system, and the Hufnagel-Valley turbulence profile. Wavelength, 2.2 μm. Solid curves, turbulence outer scale L0=26 m; dashed curves, infinite outer scale. The curves are for proportional control and for the spatial filtering optimized for a 40 field.

Tables (1)

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Table 1 Isoplanatic Angles for λ=500 nm

Equations (45)

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(x, θ)=ϕ(x-θh)-m=1Mψm(x-θHm).
˜( f )=ϕ˜(f)exp[-2πi( f θh)]-m=1Mψ˜m( f )exp[-2πi( fθHm)].
ψ˜m( f )=gm( f )r( f )ϕ˜( f ).
p2=2π0fWϕ( f )|G¯( f )|2p( f )df.
p( f )=1-J1(2πRf )πRf2.
|G¯( f )|2=1-2bTg+gTAg,
bm=r( f )J0(2πfxm),
amm=r2( f )J0[2πf(xm-xm)],xm=θ(Hm-h).
Wϕ( f )=0.38λ-2( f 2+L0-2)-11/6Cn2(h)dh.
p2=0hmaxCn2(h)F(h)dh,
F(h)=2.40λ-20f( f 2+L0-2)-11/6p( f )|G¯( f )|2df.
2=2.905(2π/λ)2|θ|5/3Cn2(h)FM(h)dh,
FM(h)=b^Tc-0.5cTAˆc.
b^m=|h-Hm|5/3,a^mm=|Hm-Hm|5/3.
g=A-1b.
α=1TA^-11,β=1TA^-1bˆ,γ=b^TA^-1bˆ.
δ=(1-β)/α,
c=A^-1(bˆ+δ1),
Fm(h)=0.5[γ-(1-β)2/α].
A=1J0[2πf(x1-x2)]J0[2πf(x1-x2)]1,
b=J0(2πfx1)J0(2πfx2).
g1( f )=J0(2πfx1)-J0[2πf(x1-x2)]J0(2πfx2)1-J02[2πf(x1-x2)].
g1( f )=g2( f )=J0(πfθH)1+J0(2πfθH),
c1=0.5|H2-H1|-5/3(|h-H2|5/3-|h-H1|5/3+|H2-H1|5/3).
2=(|θ |/θM)5/3,
θM-5/3=2.905(2π/λ)2(sec z)8/30hmaxCn2(h)FM(h)dh,
F2(h)=0.5[|h-H1|5/3+|h-H2|5/3-0.5|H2-H1|5/3-0.5|H2-H1|-5/3(|h-H1|5/3-|h-H2|5/3)2].
˜( f )=G( f )ϕ˜( f ),
G( f )=exp[-2πi( fθh)]×1-m=1Mgm( f )r( f )exp[-2πi(fxm)].
|G( f )|2=1-r( f )m=1Mgm( f )×[exp(-2πifxm)+exp(2πifxm)]+ r2( f )m=1 Mm=1Mgm( f )gm( f )×exp[2πif (xm-xm)].
|G¯( f )|2=1-2r( f )m=1Mgm( f )J0(2πfxm)+r2( f )×m=1 Mm=1Mgm( f )gm( f )J0[2πf(xm-xm)].
0=P(x)(x)dx.
p2=P(x)[(x)-0]2dx=2-202+02.
2=W(f )df.
p2=W(f )p(f )df,
p(f )=1-J1(2πRf)πRf2.
p2=2π0fWϕ(f )|G¯(f )|2p(f )df,
Γϕ(x)=ϕ( y)ϕ( y+x)=Wϕ(f )exp[-2πi(fx)]df.
2=σϕ2-2m=1McmΓϕ(xm)+m=1Mm=1McmcmΓϕ(xm-xm),
2=σϕ21-m=1Mcm2+m=1McmDϕ(xm)-m=1 Mm=1McmcmDϕ(xm-xm).
m=1Mcm=1.
Dϕ(x)=2.905(2π/λ)2|x|5/3Cn2(h)dh.
2=2.905(2π/λ)2|θ |5/3Cn2(h)FM(h)dh,
FM(h)=b^Tc-0.5cTAˆc,
b^m=|h-Hm|5/3,a^mm=|Hm-Hm|5/3.

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