Abstract

An expression is derived for the spatial power spectrum of wave-front errors after correction with a segmented mirror. This includes estimates of the spectral contributions of segment piston and tilt corrections and spatial aliasing by a regular array of segments. The approach allows rapid computation of wave-front error spectra in systems with highly segmented mirrors.

© 2003 Optical Society of America

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References

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  1. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, Oxford, UK, 1998), p. 196.
  2. F. Rigaut, J. Veran, O. Lai, “An analytical model for Shack-Hartmann-based adaptive optics systems,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1038–1048 (1998).
    [CrossRef]
  3. J. M. Beckers, “Adaptive optics for astronomy,” Annu. Rev. Astron. Astrophys. 31, 13–62 (1993).
    [CrossRef]
  4. D. G. Sandler, S. Stahl, J. R. P. Angel, M. Lloyd-Hart, D. McCarthy, “Adaptive optics for diffraction-limited infrared imaging with 8-m telescopes,” J. Opt. Soc. Am. A 11, 925–945 (1994).
    [CrossRef]
  5. J. E. Nelson, T. S. Mast, S. M. Faber, eds., “The design of the Keck Observatory and Telescope,” Keck Observatory Rep. 90 (Keck Observatory, Kamuela, Hawaii, 1985).
  6. P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969), p. 103.
  7. R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1986), p. 117.
  8. G. Chanan, M. Troy, “Strehl ratio and modulation transfer function for segmented mirror telescopes as functions of segment phase error,” Appl. Opt. 38, 6642–6647 (1999).
    [CrossRef]
  9. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, U.S. Department of Commerce, Springfield, Va., 1971), Chap. 1B.

1999 (1)

1994 (1)

1993 (1)

J. M. Beckers, “Adaptive optics for astronomy,” Annu. Rev. Astron. Astrophys. 31, 13–62 (1993).
[CrossRef]

Angel, J. R. P.

Beckers, J. M.

J. M. Beckers, “Adaptive optics for astronomy,” Annu. Rev. Astron. Astrophys. 31, 13–62 (1993).
[CrossRef]

Bevington, P. R.

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969), p. 103.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1986), p. 117.

Chanan, G.

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, Oxford, UK, 1998), p. 196.

Lai, O.

F. Rigaut, J. Veran, O. Lai, “An analytical model for Shack-Hartmann-based adaptive optics systems,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1038–1048 (1998).
[CrossRef]

Lloyd-Hart, M.

McCarthy, D.

Rigaut, F.

F. Rigaut, J. Veran, O. Lai, “An analytical model for Shack-Hartmann-based adaptive optics systems,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1038–1048 (1998).
[CrossRef]

Sandler, D. G.

Stahl, S.

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, U.S. Department of Commerce, Springfield, Va., 1971), Chap. 1B.

Troy, M.

Veran, J.

F. Rigaut, J. Veran, O. Lai, “An analytical model for Shack-Hartmann-based adaptive optics systems,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1038–1048 (1998).
[CrossRef]

Annu. Rev. Astron. Astrophys. (1)

J. M. Beckers, “Adaptive optics for astronomy,” Annu. Rev. Astron. Astrophys. 31, 13–62 (1993).
[CrossRef]

Appl. Opt. (1)

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

Other (6)

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, Oxford, UK, 1998), p. 196.

F. Rigaut, J. Veran, O. Lai, “An analytical model for Shack-Hartmann-based adaptive optics systems,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1038–1048 (1998).
[CrossRef]

J. E. Nelson, T. S. Mast, S. M. Faber, eds., “The design of the Keck Observatory and Telescope,” Keck Observatory Rep. 90 (Keck Observatory, Kamuela, Hawaii, 1985).

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969), p. 103.

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1986), p. 117.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, U.S. Department of Commerce, Springfield, Va., 1971), Chap. 1B.

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Figures (4)

Fig. 1
Fig. 1

Mirror surface profile calculation for wave-front error correction in one dimension, with segment tilts from a least-squares fit to the wave front. The operations run from top to bottom; × and * represent multiplication and convolution, and ↑ is an impulse symbol. Applying the measurement and sampling functions yields the wave-front piston and tilt at each segment, and the correction functions generate the corresponding mirror surface. The residual wave-front error is w(x) + δ M (x) + θ M (x).

Fig. 2
Fig. 2

Normalized wave-front error spectra for an array of square segments, after correction for (a) segment pistons, (b) tilts calculated by least-squares wave-front fitting, (c) pistons and least-squares tilts together, and (d) tilts measured by a Shack-Hartmann sensor. In each plot, the bold solid curve is a simulation of 27 × 27 segments, with 23 × 23 wave-front samples per segment. The thin solid curve (which is mostly hidden by the bold curve) is the spectrum predicted by Eqs. (15) and (16), with measurement and correction functions from Table 2 and ξ j,k = (j/ d, k/ d). The dashed curve is the 1-D spectrum predicted by Eq. (14), and the dotted curve in (a) is the normalized uncorrected wave-front error spectrum.

Fig. 3
Fig. 3

Normalized wave-front error spectra for an array of hexagonal segments. The bold solid curve is a simulation of piston and least-squares tilt correction for 27 × 27 segments, with 56 wave-front samples per segment. The thin solid curve is the spectrum predicted by Eqs. (15) and (16), with piston functions p(u) = -P(u) = 2[J 1(s)]/s, where s = πd| u| 1.22 cos(π/6)], tilt functions from Table 2, and ξ j,k = ({j - [(-1) k - 1]/4}/d cos(π/6), k/ d ). The dashed curve is the 1-D spectrum predicted by Eq. (14), and the dotted curve is the normalized uncorrected wave-front error spectrum. Predicted spectra include an estimate of the aliasing that is due to the gridding of the wave front in the simulation.

Fig. 4
Fig. 4

Same as Fig. 3, but for an uncorrected wave-front error spectrum Ψ(|u| ≤ 1/L 0) = 1 and Ψ(|u| > 1/L 0) = (|u|L 0)-11/3, with L 0 = 5d. This represents atmospheric phase fluctuations for Kolmogorov turbulence9 with an outer scale of five segment diameters.

Tables (2)

Tables Icon

Table 1 Measurement and Correction Functions in One Dimension

Tables Icon

Table 2 Measurement and Correction Functions for Square Segments

Equations (16)

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δLSx=1dx-d/2x+d/2 wsds=1dwx * Πx/d,
ΔLSu=Wusincπdu,
θLSx=x-d/2x+d/2s-xwsds-d/2d/2 s2ds=12d3x-d/2x+d/2s-xwsds=-12d3wx * xΠx/d,
ΘLSu=Wu6idsinπduπdu2-cosπduπdu.
θSHx=1dwx * Πx/d.
ΘSHu=i2πuWusincπdu=Wu2idsinπdu.
δSHx=0x θSHsds,
ΔSHu=ΘSHu,
ΔSHu=ΘSHui2πu=Wusincπdu,
δMx=-δxIIIx/d * Πx/d,
ΔMu=-Δu * IIIudsincπdu;
θMx=-θxIIIx/d * xΠx/d,
ΘMu=id2Θu * IIIud×sinπduπdu2-cosπduπdu.
Φu=|Wu|2|1+puPu+tuTu|2+j0Wu+jd2pu+jdPu+tu+jdTu2,
Φu=|Wu|2|1+puPu+txuTxu +tyuTyu|2 +j,kξ0 |Wu+ξj,k|2|pu+ξj,kPu +txu+ξj,kTxu+tyu+ξj,kTyu|2,
Ψ|u|=12π|W0|2-ππ Φ|u|, ϕdϕ,

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