Abstract

Robustness of the recently proposed “pseudo open-loop control” algorithm against various system errors has been investigated for the representative example of the Gemini-South 8-m telescope multiconjugate adaptive-optics system. The existing model to represent the adaptive-optics system with pseudo open-loop control has been modified to account for misalignments, noise and calibration errors in deformable mirrors, and wave-front sensors. Comparison with the conventional least-squares control model has been done. We show with the aid of both transfer-function pole-placement analysis and Monte Carlo simulations that POLC remains remarkably stable and robust against very large levels of system errors and outperforms in this respect least-squares control. Approximate stability margins as well as performance metrics such as Strehl ratios and rms wave-front residuals averaged over a 1-arc min field of view have been computed for different types and levels of system errors to quantify the expected performance degradation.

© 2005 Optical Society of America

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References

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  1. J. M. Beckers, “Increasing the size of the isoplanatic patch with multi-conjugate adaptive optics,” in Proceedings of the European Southern Observatory Conference and Workshop on Very Large Telescopes and their Instrumentation, M. H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1988), Vol. 30, pp. 693–703.
  2. J. Nelson, “The California extremely large telescope, conceptual design for a thirty-meter telecope,” CELT Rep.34(2002); http://celt.ucolick.org/greenbook/ .
  3. R. Angel, M. Lloyd-Hart, K. Hege, R. Sarlot, C. Peng, “The 20/20 telescope: MCAO imaging at the individual and combined foci,” in Proceedings of the ESO Conference and Workshop Beyond Conventional Adaptive Optics, R. Ragazzoni, N. Hubin, S. Esposito, E. Vernet, eds. (European Southern Observatory, Garching, Germany, 2002), Vol. 58, pp. 17–26.
  4. N. M. Milton, M. Lloyd-Hart, A. Cheng, J. A. Georges, R. Angel, “Design and expected performance of an MCAO system for the Giant Magellan Telescope,” in Astronomical Adaptive Optics Systems and Applications, R. K. Tyson, M. Lloyd-Hart, eds., Proc. SPIE5169, 238–249 (2003).
    [Crossref]
  5. D. C. Johnston, B. M. Welsh, “Analysis of multiconjugate adaptive optics,” J. Opt. Soc. Am. A 11, 394–408 (1994).
    [Crossref]
  6. E. Diolaiti, R. Ragazzoni, M. Tordi, “Stability and optimality of a layer oriented MCAO system,” in Proceedings of the ESO Conference and Workshop Beyond Conventional Adaptive Optics, R. Ragazzoni, N. Hubin, S. Esposito, E. Vernet, eds., (European Southern Observatory, Garching, Germany, 2002), Vol. 58, p. 167.
  7. E. Diolaiti, R. Ragazzoni, M. Tordi, “Closed loop performance of a layer-oriented multi-conjugate adaptive optics system,” Astron. Astrophys. 372, 710–718 (2001).
    [Crossref]
  8. R. Flicker, F. J. Rigaut, B. L. Ellerbroek, “Comparison of multiconjugate adaptive optics configurations and control algorithms for the Gemini-South 8m telescope,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 1032–1043 (2000).
    [Crossref]
  9. B. Le Roux, J. M. Conan, C. Kulcsar, H.-F. Raynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” in Adaptive Optical Systems Technology II, P. L. Wizinowich, ed., Proc. SPIE4839, 878–889 (2002).
    [Crossref]
  10. D. T. Gavel, W. Wiberg, “Towards Strehl-optimizing adaptive optics controllers,” in Adaptive Optical Systems Technology II, P. L. Wizinowich, ed., Proc. SPIE4839, 890–901 (2002).
    [Crossref]
  11. B. L. Ellerbroek, “First-order performance evaluation of adaptive optics systems for atmospheric turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
    [Crossref]
  12. B. L. Ellerbroek, “Efficient computation of minimum variance wave-front reconstructors using sparse matrix techniques,” J. Opt. Soc. Am. A 19, 1803–1816 (2002).
    [Crossref]
  13. L. Gilles, B. L. Ellerbroek, C. R. Vogel, “Preconditioned conjugate gradient wave-front reconstructors for multiconjugate adaptive optics,” Appl. Opt. 42, 5233–5250 (2003).
    [Crossref] [PubMed]
  14. B. L. Ellerbroek, C. R. Vogel, “Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes,” in Propagation and Imaging through the Atmosphere III, M. C. Roggemann, L. R. Bissonnette, eds., Proc. SPIE5169, 206–217 (2003).
  15. B. L. Ellerbroek, “Wavefront reconstruction algorithms and simulation results for multiconjugate adaptive optics on giant telescopes,” in Second Backaskog Workshop on Extremely Large Telescopes, A. L. Ardeberg, T. Andersen, eds., Proc. SPIE5382, 478–489 (2003).
    [Crossref]
  16. L. Gilles, “Closed-loop stability and performance analysis of least-squares and minimum variance control algorithms for multiconjugate adaptive optics,” Appl. Opt., 44, 993–1002 (2005).
    [Crossref] [PubMed]
  17. P. Piatrou, “Phase-to-WFS influence matrix computations cross verification,” presented at the IPAM conference “Estimation and Control Problems in Adaptive Optics,”Los Angeles, Calif., 22–24 January 2004; http://www.ipam.ucla.edu/publications/ao2004/a0_2004_4636.pdf .
  18. “MCAO conceptual design review documents for Gemini-South,” http://www.gemini.edu/sciops/instruments/adaptiveOptics/AOIndex.html .

2005 (1)

2003 (1)

2002 (1)

2001 (1)

E. Diolaiti, R. Ragazzoni, M. Tordi, “Closed loop performance of a layer-oriented multi-conjugate adaptive optics system,” Astron. Astrophys. 372, 710–718 (2001).
[Crossref]

1994 (2)

Angel, R.

R. Angel, M. Lloyd-Hart, K. Hege, R. Sarlot, C. Peng, “The 20/20 telescope: MCAO imaging at the individual and combined foci,” in Proceedings of the ESO Conference and Workshop Beyond Conventional Adaptive Optics, R. Ragazzoni, N. Hubin, S. Esposito, E. Vernet, eds. (European Southern Observatory, Garching, Germany, 2002), Vol. 58, pp. 17–26.

N. M. Milton, M. Lloyd-Hart, A. Cheng, J. A. Georges, R. Angel, “Design and expected performance of an MCAO system for the Giant Magellan Telescope,” in Astronomical Adaptive Optics Systems and Applications, R. K. Tyson, M. Lloyd-Hart, eds., Proc. SPIE5169, 238–249 (2003).
[Crossref]

Beckers, J. M.

J. M. Beckers, “Increasing the size of the isoplanatic patch with multi-conjugate adaptive optics,” in Proceedings of the European Southern Observatory Conference and Workshop on Very Large Telescopes and their Instrumentation, M. H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1988), Vol. 30, pp. 693–703.

Cheng, A.

N. M. Milton, M. Lloyd-Hart, A. Cheng, J. A. Georges, R. Angel, “Design and expected performance of an MCAO system for the Giant Magellan Telescope,” in Astronomical Adaptive Optics Systems and Applications, R. K. Tyson, M. Lloyd-Hart, eds., Proc. SPIE5169, 238–249 (2003).
[Crossref]

Conan, J. M.

B. Le Roux, J. M. Conan, C. Kulcsar, H.-F. Raynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” in Adaptive Optical Systems Technology II, P. L. Wizinowich, ed., Proc. SPIE4839, 878–889 (2002).
[Crossref]

Diolaiti, E.

E. Diolaiti, R. Ragazzoni, M. Tordi, “Closed loop performance of a layer-oriented multi-conjugate adaptive optics system,” Astron. Astrophys. 372, 710–718 (2001).
[Crossref]

E. Diolaiti, R. Ragazzoni, M. Tordi, “Stability and optimality of a layer oriented MCAO system,” in Proceedings of the ESO Conference and Workshop Beyond Conventional Adaptive Optics, R. Ragazzoni, N. Hubin, S. Esposito, E. Vernet, eds., (European Southern Observatory, Garching, Germany, 2002), Vol. 58, p. 167.

Ellerbroek, B. L.

L. Gilles, B. L. Ellerbroek, C. R. Vogel, “Preconditioned conjugate gradient wave-front reconstructors for multiconjugate adaptive optics,” Appl. Opt. 42, 5233–5250 (2003).
[Crossref] [PubMed]

B. L. Ellerbroek, “Efficient computation of minimum variance wave-front reconstructors using sparse matrix techniques,” J. Opt. Soc. Am. A 19, 1803–1816 (2002).
[Crossref]

B. L. Ellerbroek, “First-order performance evaluation of adaptive optics systems for atmospheric turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
[Crossref]

B. L. Ellerbroek, C. R. Vogel, “Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes,” in Propagation and Imaging through the Atmosphere III, M. C. Roggemann, L. R. Bissonnette, eds., Proc. SPIE5169, 206–217 (2003).

B. L. Ellerbroek, “Wavefront reconstruction algorithms and simulation results for multiconjugate adaptive optics on giant telescopes,” in Second Backaskog Workshop on Extremely Large Telescopes, A. L. Ardeberg, T. Andersen, eds., Proc. SPIE5382, 478–489 (2003).
[Crossref]

R. Flicker, F. J. Rigaut, B. L. Ellerbroek, “Comparison of multiconjugate adaptive optics configurations and control algorithms for the Gemini-South 8m telescope,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 1032–1043 (2000).
[Crossref]

Flicker, R.

R. Flicker, F. J. Rigaut, B. L. Ellerbroek, “Comparison of multiconjugate adaptive optics configurations and control algorithms for the Gemini-South 8m telescope,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 1032–1043 (2000).
[Crossref]

Fusco, T.

B. Le Roux, J. M. Conan, C. Kulcsar, H.-F. Raynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” in Adaptive Optical Systems Technology II, P. L. Wizinowich, ed., Proc. SPIE4839, 878–889 (2002).
[Crossref]

Gavel, D. T.

D. T. Gavel, W. Wiberg, “Towards Strehl-optimizing adaptive optics controllers,” in Adaptive Optical Systems Technology II, P. L. Wizinowich, ed., Proc. SPIE4839, 890–901 (2002).
[Crossref]

Georges, J. A.

N. M. Milton, M. Lloyd-Hart, A. Cheng, J. A. Georges, R. Angel, “Design and expected performance of an MCAO system for the Giant Magellan Telescope,” in Astronomical Adaptive Optics Systems and Applications, R. K. Tyson, M. Lloyd-Hart, eds., Proc. SPIE5169, 238–249 (2003).
[Crossref]

Gilles, L.

Hege, K.

R. Angel, M. Lloyd-Hart, K. Hege, R. Sarlot, C. Peng, “The 20/20 telescope: MCAO imaging at the individual and combined foci,” in Proceedings of the ESO Conference and Workshop Beyond Conventional Adaptive Optics, R. Ragazzoni, N. Hubin, S. Esposito, E. Vernet, eds. (European Southern Observatory, Garching, Germany, 2002), Vol. 58, pp. 17–26.

Johnston, D. C.

Kulcsar, C.

B. Le Roux, J. M. Conan, C. Kulcsar, H.-F. Raynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” in Adaptive Optical Systems Technology II, P. L. Wizinowich, ed., Proc. SPIE4839, 878–889 (2002).
[Crossref]

Le Roux, B.

B. Le Roux, J. M. Conan, C. Kulcsar, H.-F. Raynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” in Adaptive Optical Systems Technology II, P. L. Wizinowich, ed., Proc. SPIE4839, 878–889 (2002).
[Crossref]

Lloyd-Hart, M.

R. Angel, M. Lloyd-Hart, K. Hege, R. Sarlot, C. Peng, “The 20/20 telescope: MCAO imaging at the individual and combined foci,” in Proceedings of the ESO Conference and Workshop Beyond Conventional Adaptive Optics, R. Ragazzoni, N. Hubin, S. Esposito, E. Vernet, eds. (European Southern Observatory, Garching, Germany, 2002), Vol. 58, pp. 17–26.

N. M. Milton, M. Lloyd-Hart, A. Cheng, J. A. Georges, R. Angel, “Design and expected performance of an MCAO system for the Giant Magellan Telescope,” in Astronomical Adaptive Optics Systems and Applications, R. K. Tyson, M. Lloyd-Hart, eds., Proc. SPIE5169, 238–249 (2003).
[Crossref]

Milton, N. M.

N. M. Milton, M. Lloyd-Hart, A. Cheng, J. A. Georges, R. Angel, “Design and expected performance of an MCAO system for the Giant Magellan Telescope,” in Astronomical Adaptive Optics Systems and Applications, R. K. Tyson, M. Lloyd-Hart, eds., Proc. SPIE5169, 238–249 (2003).
[Crossref]

Mugnier, L. M.

B. Le Roux, J. M. Conan, C. Kulcsar, H.-F. Raynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” in Adaptive Optical Systems Technology II, P. L. Wizinowich, ed., Proc. SPIE4839, 878–889 (2002).
[Crossref]

Nelson, J.

J. Nelson, “The California extremely large telescope, conceptual design for a thirty-meter telecope,” CELT Rep.34(2002); http://celt.ucolick.org/greenbook/ .

Peng, C.

R. Angel, M. Lloyd-Hart, K. Hege, R. Sarlot, C. Peng, “The 20/20 telescope: MCAO imaging at the individual and combined foci,” in Proceedings of the ESO Conference and Workshop Beyond Conventional Adaptive Optics, R. Ragazzoni, N. Hubin, S. Esposito, E. Vernet, eds. (European Southern Observatory, Garching, Germany, 2002), Vol. 58, pp. 17–26.

Piatrou, P.

P. Piatrou, “Phase-to-WFS influence matrix computations cross verification,” presented at the IPAM conference “Estimation and Control Problems in Adaptive Optics,”Los Angeles, Calif., 22–24 January 2004; http://www.ipam.ucla.edu/publications/ao2004/a0_2004_4636.pdf .

Ragazzoni, R.

E. Diolaiti, R. Ragazzoni, M. Tordi, “Closed loop performance of a layer-oriented multi-conjugate adaptive optics system,” Astron. Astrophys. 372, 710–718 (2001).
[Crossref]

E. Diolaiti, R. Ragazzoni, M. Tordi, “Stability and optimality of a layer oriented MCAO system,” in Proceedings of the ESO Conference and Workshop Beyond Conventional Adaptive Optics, R. Ragazzoni, N. Hubin, S. Esposito, E. Vernet, eds., (European Southern Observatory, Garching, Germany, 2002), Vol. 58, p. 167.

Raynaud, H.-F.

B. Le Roux, J. M. Conan, C. Kulcsar, H.-F. Raynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” in Adaptive Optical Systems Technology II, P. L. Wizinowich, ed., Proc. SPIE4839, 878–889 (2002).
[Crossref]

Rigaut, F. J.

R. Flicker, F. J. Rigaut, B. L. Ellerbroek, “Comparison of multiconjugate adaptive optics configurations and control algorithms for the Gemini-South 8m telescope,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 1032–1043 (2000).
[Crossref]

Sarlot, R.

R. Angel, M. Lloyd-Hart, K. Hege, R. Sarlot, C. Peng, “The 20/20 telescope: MCAO imaging at the individual and combined foci,” in Proceedings of the ESO Conference and Workshop Beyond Conventional Adaptive Optics, R. Ragazzoni, N. Hubin, S. Esposito, E. Vernet, eds. (European Southern Observatory, Garching, Germany, 2002), Vol. 58, pp. 17–26.

Tordi, M.

E. Diolaiti, R. Ragazzoni, M. Tordi, “Closed loop performance of a layer-oriented multi-conjugate adaptive optics system,” Astron. Astrophys. 372, 710–718 (2001).
[Crossref]

E. Diolaiti, R. Ragazzoni, M. Tordi, “Stability and optimality of a layer oriented MCAO system,” in Proceedings of the ESO Conference and Workshop Beyond Conventional Adaptive Optics, R. Ragazzoni, N. Hubin, S. Esposito, E. Vernet, eds., (European Southern Observatory, Garching, Germany, 2002), Vol. 58, p. 167.

Vogel, C. R.

L. Gilles, B. L. Ellerbroek, C. R. Vogel, “Preconditioned conjugate gradient wave-front reconstructors for multiconjugate adaptive optics,” Appl. Opt. 42, 5233–5250 (2003).
[Crossref] [PubMed]

B. L. Ellerbroek, C. R. Vogel, “Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes,” in Propagation and Imaging through the Atmosphere III, M. C. Roggemann, L. R. Bissonnette, eds., Proc. SPIE5169, 206–217 (2003).

Welsh, B. M.

Wiberg, W.

D. T. Gavel, W. Wiberg, “Towards Strehl-optimizing adaptive optics controllers,” in Adaptive Optical Systems Technology II, P. L. Wizinowich, ed., Proc. SPIE4839, 890–901 (2002).
[Crossref]

Appl. Opt. (2)

Astron. Astrophys. (1)

E. Diolaiti, R. Ragazzoni, M. Tordi, “Closed loop performance of a layer-oriented multi-conjugate adaptive optics system,” Astron. Astrophys. 372, 710–718 (2001).
[Crossref]

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

Other (12)

E. Diolaiti, R. Ragazzoni, M. Tordi, “Stability and optimality of a layer oriented MCAO system,” in Proceedings of the ESO Conference and Workshop Beyond Conventional Adaptive Optics, R. Ragazzoni, N. Hubin, S. Esposito, E. Vernet, eds., (European Southern Observatory, Garching, Germany, 2002), Vol. 58, p. 167.

P. Piatrou, “Phase-to-WFS influence matrix computations cross verification,” presented at the IPAM conference “Estimation and Control Problems in Adaptive Optics,”Los Angeles, Calif., 22–24 January 2004; http://www.ipam.ucla.edu/publications/ao2004/a0_2004_4636.pdf .

“MCAO conceptual design review documents for Gemini-South,” http://www.gemini.edu/sciops/instruments/adaptiveOptics/AOIndex.html .

B. L. Ellerbroek, C. R. Vogel, “Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes,” in Propagation and Imaging through the Atmosphere III, M. C. Roggemann, L. R. Bissonnette, eds., Proc. SPIE5169, 206–217 (2003).

B. L. Ellerbroek, “Wavefront reconstruction algorithms and simulation results for multiconjugate adaptive optics on giant telescopes,” in Second Backaskog Workshop on Extremely Large Telescopes, A. L. Ardeberg, T. Andersen, eds., Proc. SPIE5382, 478–489 (2003).
[Crossref]

R. Flicker, F. J. Rigaut, B. L. Ellerbroek, “Comparison of multiconjugate adaptive optics configurations and control algorithms for the Gemini-South 8m telescope,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 1032–1043 (2000).
[Crossref]

B. Le Roux, J. M. Conan, C. Kulcsar, H.-F. Raynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” in Adaptive Optical Systems Technology II, P. L. Wizinowich, ed., Proc. SPIE4839, 878–889 (2002).
[Crossref]

D. T. Gavel, W. Wiberg, “Towards Strehl-optimizing adaptive optics controllers,” in Adaptive Optical Systems Technology II, P. L. Wizinowich, ed., Proc. SPIE4839, 890–901 (2002).
[Crossref]

J. M. Beckers, “Increasing the size of the isoplanatic patch with multi-conjugate adaptive optics,” in Proceedings of the European Southern Observatory Conference and Workshop on Very Large Telescopes and their Instrumentation, M. H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1988), Vol. 30, pp. 693–703.

J. Nelson, “The California extremely large telescope, conceptual design for a thirty-meter telecope,” CELT Rep.34(2002); http://celt.ucolick.org/greenbook/ .

R. Angel, M. Lloyd-Hart, K. Hege, R. Sarlot, C. Peng, “The 20/20 telescope: MCAO imaging at the individual and combined foci,” in Proceedings of the ESO Conference and Workshop Beyond Conventional Adaptive Optics, R. Ragazzoni, N. Hubin, S. Esposito, E. Vernet, eds. (European Southern Observatory, Garching, Germany, 2002), Vol. 58, pp. 17–26.

N. M. Milton, M. Lloyd-Hart, A. Cheng, J. A. Georges, R. Angel, “Design and expected performance of an MCAO system for the Giant Magellan Telescope,” in Astronomical Adaptive Optics Systems and Applications, R. K. Tyson, M. Lloyd-Hart, eds., Proc. SPIE5169, 238–249 (2003).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1

z-domain block diagrams of (a) nonideal MCAO system driven by POLC and (b) the pseudo open-loop turbulence estimator T(z).

Fig. 2
Fig. 2

Representative Gemini-South 8-m telescope MCAO system guide stars arrangement and scientific field of view.

Fig. 3
Fig. 3

Wave-front variance time histories for different levels of translational misregistration in DMs (upper panel) and WFSs (lower panel). DMs’ shifts are done in the 60 deg, 150 deg, and 300 deg directions. WFSs’ shifts are done in the 0 deg, 45 deg, 135 deg, 225 deg, and 315 deg directions for the LGS WFSs and in the 0 deg, 90 deg, 180 deg, and 270 deg directions for the tip-tilt WFSs. Norms of shifts are quoted as a fraction of LGS WFSs’ subaperature size (0.5 m). WFS shift directions make a 45° angle with each other.

Fig. 4
Fig. 4

Wave-front variance time histories for different levels of rotational misregistration in DMs (upper panel) and WFSs (lower panel). Rotations are taken to be clockwise on DMs 1 and 3 and anticlockwise on DM 2. All WFSs are rotated clockwise.

Fig. 5
Fig. 5

Wave-front variance time histories for different levels of mismagnification in DMs (upper panel) and WFSs (lower panel). Mismagnifications are set equal for each DM or WFS.

Fig. 6
Fig. 6

Wave-front variance time histories for different levels of tilt errors in DMs (upper panel) and WFSs (lower panel). Tilt directions on each DM are in the 60 deg, 150 deg, and 300 deg directions. Tilt directions on each WFS are in the 0 deg, 45 deg, 135 deg, 225 deg, and 315 deg directions for the LGS WFSs and in the 0 deg, 90 deg, 180 deg, and 270 deg directions for the tip-tilt WFSs.

Fig. 7
Fig. 7

Wave-front variance time histories for different levels of additive noise in the DM actuator command vector. White Gaussian noise is assumed. The noise level is given as a fraction of the largest actuator-command vector component at each time step.

Tables (5)

Tables Icon

Table 1 Six-Layer Turbulence Model Used in the Simulation of the Gemini-South 8-m Telescope MCAO System

Tables Icon

Table 2 Gemini-South 8-m Telescope MCAO System Deformable-Mirror Parameters

Tables Icon

Table 3 Approximate Stability Margins for Various Misalignments in DMs for LSC (First Number) and for POLC (Second Number)

Tables Icon

Table 4 Approximate Stability Margins for Misalignments in WFSs

Tables Icon

Table 5 Average Strehl Ratios Obtained for Various Types and Degrees of System Errorsa

Equations (23)

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

s o = G x x + η s ,
= H x x - H a a ,
a = R s o ,
σ 2 = K T W = W 2 = H x x - H a R s o W 2 ,
R ^ = arg min R σ 2
R ^ = F E ,
x ^ = E s o = ( G x T C η η - 1 G x + C x x - 1 ) - 1 G x T G η η - 1 s o ;
a = F x ^ = ( H a T W H a + α I ) - 1 H a T W H x x ^ ,
x ^ = arg min x { s o - G x x C η η - 1 2 + x C x x - 1 2 } ,
F x ^ = arg min a { H x x ^ - H a a W 2 + α a 2 } .
e ^ x = E s = arg min e x { s - G x e x C η η - 1 2 + e x C e e - 1 2 } .
s ^ 0 = s + G a a ,
u ^ = arg min u { s ^ 0 - G x u C η η - 1 2 + u C x x - 1 2 } ,
e ^ x = u ^ - x ^ , u ^ = E s ^ 0 = E ( s + G a a ) .
s = G x x + δ s + η s - G a ( a + δ a + η a ) ,
= H x x - H a ( a + δ a + η a ) .
T ( z ) = g ( z ) [ I + g ( z ) ( I - E G a F ) ] - 1 E ,
Δ s = H CL ( z ) x ( z )
H CL = G a F Q ( z ) E G x ,
Q ( z ) = g ( z ) [ I + g ( z ) M ] - 1 ,
M = I + E ( G a - G a ) F .
g i ( z ) = g ( z ) 1 + λ i g ( z ) ,
g ( z ) = δ z 2 - α z - β ,

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