Abstract

A Zernike expansion of wind-induced deformations in a segmented mirror is described. The wind model is a frozen turbulent field with a Kolmogorov spectrum for scales smaller than the outer scale and a flat spectrum for scales larger than the outer scale. The approach allows a mode-by-mode comparison of the wave-front error contributions from atmospheric phase distortions, wind-induced deformations, and the mirror control system noise. This is used to design a controller that minimizes the mirror surface errors by application of corrections based on edge sensor measurements and wave-front measurements on a guide star.

© 2002 Optical Society of America

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References

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  1. F. F. Forbes, “The effects of wind loads on large telescopes and their enclosures,” in ESO Conference on Very Large Telescopes and their Instrumentation, ESO Rep. 30, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1988), pp. 831–843.
  2. J. E. Nelson, T. S. Mast, and FaberS. M., eds., Design of the Keck Observatory and Telescope, Keck Observatory Rep. 90 (Keck Observatory, Kamuela, Hawaii, 1985).
  3. G. Chanan, J. E. Nelson, C. Ohara, E. Sirko, “Design issues for the active control system of the California Extremely Large Telescope (CELT),” in Telescope Structures, Enclosures, Controls, Assembly/Integration/Validation and Commissioning, T. Anderson, ed., Proc. SPIE4004, 363–372 (2000).
    [CrossRef]
  4. D. G. MacMartin, T. S. Mast, G. Chanan, J. E. Nelson, “Active control issues for the California Extremely Large Telescope,” presented at the AIAA Guidance, Navigation, and Control Conference, Montreal, 6–9 Aug. 2001 (American Institute of Aeronautics and Astronautics, Reston, Va., 2001), paper AIAA-2001-4035.
  5. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  6. F. Roddier, M. J. Northcott, J. E. Graves, D. L. McKenna, “One-dimensional spectra of turbulence-induced Zernike abberations: time-delay and isoplanacity error in partial adaptive compensation,” J. Opt. Soc. Am. A 10, 957–965 (1993).
    [CrossRef]
  7. F. Roddier, ed., Adaptive Optics in Astronomy (Cambridge University, Cambridge, UK, 1999), Chap. 3.
    [CrossRef]
  8. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
    [CrossRef]
  9. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, Springfield, Va., 1971).
  10. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
    [CrossRef]
  11. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.
  12. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, Oxford, UK, 1998), Chap. 3.
  13. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
    [CrossRef]
  14. M. K. Cho, L. M. Stepp, S. Kim, “Wind buffeting effects on the Gemini 8m primary mirrors,” in Optomechanical Design and Engineering 2001, A. E. Hatheway, ed., Proc. SPIE.4444, 302–314 (2001).
    [CrossRef]
  15. S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959), Chap. 3.
  16. T. S. Mast, J. E. Nelson, “Segmented mirror control system hardware for CELT,” in Optical Design, Materials, Fabrication and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 226–240 (2000).
    [CrossRef]

1993 (1)

1990 (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

1976 (1)

1965 (1)

Chanan, G.

G. Chanan, J. E. Nelson, C. Ohara, E. Sirko, “Design issues for the active control system of the California Extremely Large Telescope (CELT),” in Telescope Structures, Enclosures, Controls, Assembly/Integration/Validation and Commissioning, T. Anderson, ed., Proc. SPIE4004, 363–372 (2000).
[CrossRef]

Cho, M. K.

M. K. Cho, L. M. Stepp, S. Kim, “Wind buffeting effects on the Gemini 8m primary mirrors,” in Optomechanical Design and Engineering 2001, A. E. Hatheway, ed., Proc. SPIE.4444, 302–314 (2001).
[CrossRef]

Forbes, F. F.

F. F. Forbes, “The effects of wind loads on large telescopes and their enclosures,” in ESO Conference on Very Large Telescopes and their Instrumentation, ESO Rep. 30, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1988), pp. 831–843.

Fried, D. L.

Graves, J. E.

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, Oxford, UK, 1998), Chap. 3.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.

Kim, S.

M. K. Cho, L. M. Stepp, S. Kim, “Wind buffeting effects on the Gemini 8m primary mirrors,” in Optomechanical Design and Engineering 2001, A. E. Hatheway, ed., Proc. SPIE.4444, 302–314 (2001).
[CrossRef]

Mast, T. S.

T. S. Mast, J. E. Nelson, “Segmented mirror control system hardware for CELT,” in Optical Design, Materials, Fabrication and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 226–240 (2000).
[CrossRef]

McKenna, D. L.

Nelson, J. E.

G. Chanan, J. E. Nelson, C. Ohara, E. Sirko, “Design issues for the active control system of the California Extremely Large Telescope (CELT),” in Telescope Structures, Enclosures, Controls, Assembly/Integration/Validation and Commissioning, T. Anderson, ed., Proc. SPIE4004, 363–372 (2000).
[CrossRef]

T. S. Mast, J. E. Nelson, “Segmented mirror control system hardware for CELT,” in Optical Design, Materials, Fabrication and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 226–240 (2000).
[CrossRef]

Noll, R. J.

Northcott, M. J.

Ohara, C.

G. Chanan, J. E. Nelson, C. Ohara, E. Sirko, “Design issues for the active control system of the California Extremely Large Telescope (CELT),” in Telescope Structures, Enclosures, Controls, Assembly/Integration/Validation and Commissioning, T. Anderson, ed., Proc. SPIE4004, 363–372 (2000).
[CrossRef]

Roddier, F.

F. Roddier, M. J. Northcott, J. E. Graves, D. L. McKenna, “One-dimensional spectra of turbulence-induced Zernike abberations: time-delay and isoplanacity error in partial adaptive compensation,” J. Opt. Soc. Am. A 10, 957–965 (1993).
[CrossRef]

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
[CrossRef]

Roddier, N.

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Sirko, E.

G. Chanan, J. E. Nelson, C. Ohara, E. Sirko, “Design issues for the active control system of the California Extremely Large Telescope (CELT),” in Telescope Structures, Enclosures, Controls, Assembly/Integration/Validation and Commissioning, T. Anderson, ed., Proc. SPIE4004, 363–372 (2000).
[CrossRef]

Stepp, L. M.

M. K. Cho, L. M. Stepp, S. Kim, “Wind buffeting effects on the Gemini 8m primary mirrors,” in Optomechanical Design and Engineering 2001, A. E. Hatheway, ed., Proc. SPIE.4444, 302–314 (2001).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, Springfield, Va., 1971).

Timoshenko, S.

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959), Chap. 3.

Woinowsky-Krieger, S.

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959), Chap. 3.

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Other (12)

M. K. Cho, L. M. Stepp, S. Kim, “Wind buffeting effects on the Gemini 8m primary mirrors,” in Optomechanical Design and Engineering 2001, A. E. Hatheway, ed., Proc. SPIE.4444, 302–314 (2001).
[CrossRef]

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959), Chap. 3.

T. S. Mast, J. E. Nelson, “Segmented mirror control system hardware for CELT,” in Optical Design, Materials, Fabrication and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 226–240 (2000).
[CrossRef]

F. F. Forbes, “The effects of wind loads on large telescopes and their enclosures,” in ESO Conference on Very Large Telescopes and their Instrumentation, ESO Rep. 30, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1988), pp. 831–843.

J. E. Nelson, T. S. Mast, and FaberS. M., eds., Design of the Keck Observatory and Telescope, Keck Observatory Rep. 90 (Keck Observatory, Kamuela, Hawaii, 1985).

G. Chanan, J. E. Nelson, C. Ohara, E. Sirko, “Design issues for the active control system of the California Extremely Large Telescope (CELT),” in Telescope Structures, Enclosures, Controls, Assembly/Integration/Validation and Commissioning, T. Anderson, ed., Proc. SPIE4004, 363–372 (2000).
[CrossRef]

D. G. MacMartin, T. S. Mast, G. Chanan, J. E. Nelson, “Active control issues for the California Extremely Large Telescope,” presented at the AIAA Guidance, Navigation, and Control Conference, Montreal, 6–9 Aug. 2001 (American Institute of Aeronautics and Astronautics, Reston, Va., 2001), paper AIAA-2001-4035.

F. Roddier, ed., Adaptive Optics in Astronomy (Cambridge University, Cambridge, UK, 1999), Chap. 3.
[CrossRef]

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
[CrossRef]

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, Springfield, Va., 1971).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, Oxford, UK, 1998), Chap. 3.

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Figures (6)

Fig. 1
Fig. 1

Spatial decorrelation of the wave front for small offsets between the target and the guide star and L 0 = 10R [Eq. (13)]. Γ n (ρ) is the correlation coefficient between the wave-front variances in two directions offset by ρ = hθ/R cos γ, where θ is the angle between the two directions, h is the height of the turbulent atmospheric layer, R is the mirror radius, and γ is the zenith angle.

Fig. 2
Fig. 2

Wave-front variance on the guide star on time scales longer than τint for turbulence with L 0 = 10R [Eq. (18)].

Fig. 3
Fig. 3

Wave-front variance that is due to wind-induced deformations on time scales shorter than τint, normalized to the total wave-front variance in the range 0.1u/ Rf ≤ 20u/ R and 1 ≤ n ≤ 50 for a normal-incidence wind model with outer scale L m = R [Eqs. (25) and (28)]. For R = 15 m and u = 1 m s-1, the frequency range is 0.0067 ≤ f ≤ 1.33 Hz, and the outer scale is 15 m, which corresponds to a frequency of 0.067 Hz.

Fig. 4
Fig. 4

Same as Fig. 3, but for a frozen screen wind model [Eqs. (33) and (28)].

Fig. 5
Fig. 5

Contributions to the wave-front variance for 1 ≤ n ≤ 50 for the parameters in Table 2 and a normal-incidence wind model [Eq. (25)]. 〈σCS 2(n)〉 [approximation (34)] is the wave-front variance contributed by noise in the mirror control system; 〈σwind 2int, K(n), n]〉 [Eqs. (29) and (30)] is the wave-front variance that is due to wind-induced deformations on time scales shorter than τint; 〈σatm 2int, θ, n)〉 [Eqs. (19) and (20)] is the wave-front variance that is added to the target as a result of the mirror being corrected on time scales longer than τint by measurements on the guide star; and 〈σWFS 2(n)〉 [approximation (35)] is the wave-front sensor noise for a 14th magnitude guide star in 0.1 s. 〈στ 2int = 0, n)〉 [Eq. (18)] is the total atmospheric phase distortion in the direction of the guide star, i.e., the seeing limit. 〈σatm 2int = 0, θ = 10 arc sec, n)〉 is the total atmospheric phase distortion on the target, after the atmosphere is corrected in the direction of a guide star 10 arc sec away. This is the seeing after an adaptive correction on the guide star. The line at Σ 〈σ2 〉 = 1 represents a simple model of the diffraction limit, in which a wave-front variance of 1 rad2 is uniformly spread across the first 50 radial degrees. Because there are n + 1 azimuthal modes in the nth radial degree, this model corresponds to each azimuthal mode contributing a wave-front variance roughly proportional to 1/n. The line labeled 20 nm represents a similar model, but with 20-nm total wave-front error. The filled circles indicate the wave-front error contributions after the mirror is corrected by wave-front and edge sensor measurements.

Fig. 6
Fig. 6

Same as Fig. 5, but for a frozen screen wind model [Eq. (33)].

Tables (2)

Tables Icon

Table 1 Mirror Control System Noise Model Parameters for the CELT and Keck Telescopes

Tables Icon

Table 2 Model Parameters for Figs. 5 and 6

Equations (35)

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φr-vR t=jajvR tZjr,
ajvR t= WrZjrφr-vR tdr,
Φjk=ΦkFTWrZjr2=ΦkQjk2,
Φk=0.023Rr05/3k-11/3
Qjk=n+1Jn+12πkπk×-1n-m/22 cos mϕ even j-1n-m/22 sin mϕ odd j-1n/2m=0
Bjρ=ajrajr+ρ = FTΦjk,
Bjρ=0.023Rr05/3n+1π2--k-17/3Jn+12πk2×expi2πρkx2 cos2 mϕ even j2 sin2 mϕ odd j1 m=0dkxdky,
Φk=0.023Rr05/31k11/3+R/L011/3,
Bjρ=0.023Rr05/3n+1π2--×Jn+12πk2 expi2πρkxk11/3+R/L011/3k2×2 cos2 mϕ even j2 sin2 mϕ odd j1 m=0dkxdky.
σθ2ρ, n=jBj0-Bjρ,
σθ2ρ, n=1-Γnρj Bj0,
Γnρ=j Bjρj Bj0
1-Γnρ=2π2ρ2×--kx2Jn+12πk2k11/3+R/L011/3k2dkxdky--Jn+12πk2k11/3+R/L011/3k2dkxdky.
Cjτ=ajtajt+τ=ajrajr+vτR=FTΦjk.
Cjτ=-- Φjkx, ky expi2π vR τkxdkxdky=-- ΦjfRv, ky expi2πfτRvdfdky,
FjfRv=FTCjτvR=- ΦjfRv, kydky;
FjfRv=0.023Rr05/3n+1π2-Jn+12πfR/v2+ky21/22fR/v2+ky211/6+R/L011/3fR/v2+ky2×2 cos2 mϕ even j2 sin2 mϕ odd j1 m=0dky.
στ2τint, n=2 j0R/vτint FjfRvdfRv,
στ2τint, n=2 j0 TfRvFjfRvdfRv.
σatm2τint, ρ, n=στ2τint, n 1-Γnρστ2τint, nρ2n5/3for ρ  1.
Vk  1k0 Du2rsin2πkrdr  1k0 r1/3 sin2πkrdr  k-7/3,
Ωk  1k Vk  k-13/3
Θjkx, ky, kz=Akx2+ky2+kz213/6+R/Lm13/3×|Qjkx, ky|2,
ΨjfRu=-- Θjkx, ky, fRudkxdky,
ψnfRu=--n+1π2AJn+12πkx2+ky21/22kx2+ky2+fR/u213/6+R/Lm13/3kx2+ky2dkxdky=0n+1π2AJn+12πξ2ξ2+fR/u213/6+R/Lm13/3ξ2 2πξdξ.
σw2=2 2πλAsegK 2ρairuσu2,
σw2=2 nfmR/uψnfRudfRu,
σwind2τint, n=2 R/uτint ΨnfRudfRu.
σwind2τint, n=2 01-TfRuΨnfRudfRu.
σwind2τint, Kn, n=KKn2σwind2τint, n.
KnK  n2N21-K1K+K1Kfor N  1,
ΨjfRu=-n+1π2×AJn+12πfR/u2+ky21/22fR/u2+ky213/6+R/Lm13/3fR/u2+ky2×2 cos2 mϕ even j2 sin2 mϕ odd j1 m=0dky,
ΨnfRu=-n+1π2AJn+12πfR/u2+ky21/22fR/u2+ky213/6+R/Lm13/3fR/u2+ky2dky.
σCS2n  n+12 2πλ δeRd1n22.
σWFS2n  π2pr022n+1N+1N+2,

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