Abstract

A Monte Carlo sky coverage model for laser guide star adaptive optics systems was proposed by Clare and Ellerbroek [J. Opt. Soc. Am. A 23, 418 (2006)]. We refine the model to include (i) natural guide star (NGS) statistics using published star count models, (ii) noise on the NGS measurements, (iii) the effect of telescope wind shake, (iv) a model for how the Strehl and hence NGS wavefront sensor measurement noise varies across the field, (v) the focus error due to imperfectly tracking the range to the sodium layer, (vi) the mechanical bandwidths of the tip–tilt (TT) stage and deformable mirror actuators, and (vii) temporal filtering of the NGS measurements to balance errors due to noise and servo lag. From this model, we are able to generate a TT error budget for the Thirty Meter Telescope facility narrow-field infrared adaptive optics system (NFIRAOS) and perform several design trade studies. With the current NFIRAOS design, the median TT error at the galactic pole with median seeing is calculated to be 65  nm or 1.8 mas rms.

© 2006 Optical Society of America

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  1. J. Nelson and G. H. Sanders, "TMT status report," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 745-761 (2006).
  2. I. Hook, G. Dalton, and R. Gilmozzi, "Scientific requirements for a European ELT," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 726-734 (2006).
  3. M. W. Johns, "The Giant Magellan Telescope (GMT)," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 762-776 (2006).
  4. T. E. Andersen, A. Ardeberg, J. Beckers, M. Browne, A. Enmark, P. Knutsson, and M. Owner-Petersen, "From Euro50 toward a European ELT," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 716-725 (2006).
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  7. B. L. Ellerbroek and F. R. Rigaut, "Methods for correcting tilt anisoplanatism in laser-guide-star-based multiconjugate adaptive optics," J. Opt. Soc. Am. A 18, 2539-2547 (2001).
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  11. R. M. Clare and B. L. Ellerbroek, "Sky coverage estimates for adaptive optics systems from computations in Zernike space," J. Opt. Soc. Am. A 23, 418-426 (2006).
    [CrossRef]
  12. G. Herriot, P. Hickson, B. L. Ellerbroek, D. A. Anderson, T. Davidge, D. A. Erickson, I. P. Powell, R. Clare, L. Gilles, C. Boyer, M. Smith, L. Saddlemyer, and J.-P. Véran, "NFIRAOS: TMT narrow field near-infrared facility adaptive optics," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 228-239 (2006).
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    [CrossRef]
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  24. G. Herriot, P. Hickson, B. L. Ellerbroek, J.-P. Véran, C.-Y. She, R. M. Clare, and D. Looze, "Focus errors from tracking sodium layer altitude variations with laser guide star adaptive optics for the Thirty Meter Telescope," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 486-495 (2006).
  25. J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, and A. Ziad, "1998 Gemini site testing campaign: Cerro Pachon and Cerro Tololo," Gemini Document RTP-AO-G0094 (Gemini Observatory, Hilo, Hawaii, 2000).
  26. P. Puxley, "Observing constraints--sky background," Gemini Observatory, October 2002, http://www.gemini.edu/sciops/ObsProcess/obsConstraints/ocSkyBackground.html.

2006 (9)

J. Nelson and G. H. Sanders, "TMT status report," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 745-761 (2006).

I. Hook, G. Dalton, and R. Gilmozzi, "Scientific requirements for a European ELT," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 726-734 (2006).

M. W. Johns, "The Giant Magellan Telescope (GMT)," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 762-776 (2006).

T. E. Andersen, A. Ardeberg, J. Beckers, M. Browne, A. Enmark, P. Knutsson, and M. Owner-Petersen, "From Euro50 toward a European ELT," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 716-725 (2006).

R. M. Clare and B. L. Ellerbroek, "Sky coverage estimates for adaptive optics systems from computations in Zernike space," J. Opt. Soc. Am. A 23, 418-426 (2006).
[CrossRef]

G. Herriot, P. Hickson, B. L. Ellerbroek, D. A. Anderson, T. Davidge, D. A. Erickson, I. P. Powell, R. Clare, L. Gilles, C. Boyer, M. Smith, L. Saddlemyer, and J.-P. Véran, "NFIRAOS: TMT narrow field near-infrared facility adaptive optics," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 228-239 (2006).

L. Gilles, B. L. Ellerbroek, and J.-P. Véran, "Laser guide star multi-conjugate adaptive optics performance of the Thirty Meter Telescope with elongated beacons and matched filtering," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 1025-1032 (2006).

J.-P. Véran and G. Herriot, "Woofer-tweeter tip-tilt control for NFIRAOS for TMT," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 555-562 (2006).

G. Herriot, P. Hickson, B. L. Ellerbroek, J.-P. Véran, C.-Y. She, R. M. Clare, and D. Looze, "Focus errors from tracking sodium layer altitude variations with laser guide star adaptive optics for the Thirty Meter Telescope," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 486-495 (2006).

2005 (3)

D. Kerley, S. Roberts, J. Dunn, N. Stretch, M. Smith, S. Sun, J. Pazder, and J. Fitzsimmons, "Validation and verification of integrated model simulations of a thirty meter telescope," in Optical Modeling and Performance Predictions II, M. A. Kahan, ed., Proc. SPIE 5867, 220-231 (2005).

B. L. Ellerbroek, "Linear systems modeling of adaptive optics in the spatial-frequency domain," J. Opt. Soc. Am. A 22, 310-322 (2005).
[CrossRef]

B. Femenia, "Tip-tilt reconstruction with a single dim natural guide star in multiconjugate adaptive optics with laser guide stars," J. Opt. Soc. Am. A 22, 2719-2729 (2005).
[CrossRef]

2004 (2)

D. B. Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Advancements in Adaptive Optics,Proc. SPIE 5490 (2004).

W. L. Melvin, "A STAP overview," IEEE Aerosp. Electron. Syst. Mag. 19, 19-35 (2004).
[CrossRef]

2002 (1)

2001 (2)

A. Spagna, "Guide star requirements for NGST: deep NIR starcounts and guide star catalogs," Space Telescope Science Institute Document NGST-R-OO13B (Space Telescope Science Institute, Baltimore, Md., 2001).

B. L. Ellerbroek and F. R. Rigaut, "Methods for correcting tilt anisoplanatism in laser-guide-star-based multiconjugate adaptive optics," J. Opt. Soc. Am. A 18, 2539-2547 (2001).
[CrossRef]

2000 (1)

D. J. Butler, R. I. Davies, H. Fews, R. M. Redfern, N. Ageorges, W. K. Hackenberg, R.-R. Rohloff, S. Rabien, T. Ott, and S. Hippler, "Sodium layer monitoring at Calar Alto by LIDAR," in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE 4007, 358-367 (2000).

1998 (1)

B. L. Ellerbroek and D. W. Tyler, "Adaptive optics sky coverage calculations for the Gemini-North telescope," Publ. Astron. Soc. Pac. 110, 165-185 (1998).
[CrossRef]

1982 (1)

1980 (1)

J. N. Bahcall and R. M. Soneira, "The universe at faint magnitudes. I. Models for the galaxy and the predicted star counts," Astrophys. J. Suppl. 44, 73-110 (1980).
[CrossRef]

1976 (1)

R. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. A 66, 207-211 (1976).
[CrossRef]

1966 (1)

D. L. Fried, "Optical resolution through a randomly inhomogenous medium for very long and very short exposures," J. Opt. Soc. Am. A 56, 1376-1379 (1966).

1953 (1)

H. W. Babcock, "The possibility of compensating astronomical seeing," Publ. Astron. Soc. Pac. 65, 229-236 (1953).
[CrossRef]

Agabi, A.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, and A. Ziad, "1998 Gemini site testing campaign: Cerro Pachon and Cerro Tololo," Gemini Document RTP-AO-G0094 (Gemini Observatory, Hilo, Hawaii, 2000).

Ageorges, N.

D. J. Butler, R. I. Davies, H. Fews, R. M. Redfern, N. Ageorges, W. K. Hackenberg, R.-R. Rohloff, S. Rabien, T. Ott, and S. Hippler, "Sodium layer monitoring at Calar Alto by LIDAR," in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE 4007, 358-367 (2000).

Andersen, T. E.

T. E. Andersen, A. Ardeberg, J. Beckers, M. Browne, A. Enmark, P. Knutsson, and M. Owner-Petersen, "From Euro50 toward a European ELT," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 716-725 (2006).

Anderson, D. A.

G. Herriot, P. Hickson, B. L. Ellerbroek, D. A. Anderson, T. Davidge, D. A. Erickson, I. P. Powell, R. Clare, L. Gilles, C. Boyer, M. Smith, L. Saddlemyer, and J.-P. Véran, "NFIRAOS: TMT narrow field near-infrared facility adaptive optics," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 228-239 (2006).

Ardeberg, A.

T. E. Andersen, A. Ardeberg, J. Beckers, M. Browne, A. Enmark, P. Knutsson, and M. Owner-Petersen, "From Euro50 toward a European ELT," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 716-725 (2006).

Avila, R.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, and A. Ziad, "1998 Gemini site testing campaign: Cerro Pachon and Cerro Tololo," Gemini Document RTP-AO-G0094 (Gemini Observatory, Hilo, Hawaii, 2000).

Azouit, M.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, and A. Ziad, "1998 Gemini site testing campaign: Cerro Pachon and Cerro Tololo," Gemini Document RTP-AO-G0094 (Gemini Observatory, Hilo, Hawaii, 2000).

Babcock, H. W.

H. W. Babcock, "The possibility of compensating astronomical seeing," Publ. Astron. Soc. Pac. 65, 229-236 (1953).
[CrossRef]

Bahcall, J. N.

J. N. Bahcall and R. M. Soneira, "The universe at faint magnitudes. I. Models for the galaxy and the predicted star counts," Astrophys. J. Suppl. 44, 73-110 (1980).
[CrossRef]

Beckers, J.

T. E. Andersen, A. Ardeberg, J. Beckers, M. Browne, A. Enmark, P. Knutsson, and M. Owner-Petersen, "From Euro50 toward a European ELT," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 716-725 (2006).

Boyer, C.

G. Herriot, P. Hickson, B. L. Ellerbroek, D. A. Anderson, T. Davidge, D. A. Erickson, I. P. Powell, R. Clare, L. Gilles, C. Boyer, M. Smith, L. Saddlemyer, and J.-P. Véran, "NFIRAOS: TMT narrow field near-infrared facility adaptive optics," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 228-239 (2006).

Browne, M.

T. E. Andersen, A. Ardeberg, J. Beckers, M. Browne, A. Enmark, P. Knutsson, and M. Owner-Petersen, "From Euro50 toward a European ELT," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 716-725 (2006).

Butler, D. J.

D. J. Butler, R. I. Davies, H. Fews, R. M. Redfern, N. Ageorges, W. K. Hackenberg, R.-R. Rohloff, S. Rabien, T. Ott, and S. Hippler, "Sodium layer monitoring at Calar Alto by LIDAR," in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE 4007, 358-367 (2000).

Calia, D. B.

D. B. Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Advancements in Adaptive Optics,Proc. SPIE 5490 (2004).

Clare, R.

G. Herriot, P. Hickson, B. L. Ellerbroek, D. A. Anderson, T. Davidge, D. A. Erickson, I. P. Powell, R. Clare, L. Gilles, C. Boyer, M. Smith, L. Saddlemyer, and J.-P. Véran, "NFIRAOS: TMT narrow field near-infrared facility adaptive optics," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 228-239 (2006).

Clare, R. M.

R. M. Clare and B. L. Ellerbroek, "Sky coverage estimates for adaptive optics systems from computations in Zernike space," J. Opt. Soc. Am. A 23, 418-426 (2006).
[CrossRef]

G. Herriot, P. Hickson, B. L. Ellerbroek, J.-P. Véran, C.-Y. She, R. M. Clare, and D. Looze, "Focus errors from tracking sodium layer altitude variations with laser guide star adaptive optics for the Thirty Meter Telescope," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 486-495 (2006).

Conan, R.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, and A. Ziad, "1998 Gemini site testing campaign: Cerro Pachon and Cerro Tololo," Gemini Document RTP-AO-G0094 (Gemini Observatory, Hilo, Hawaii, 2000).

Dalton, G.

I. Hook, G. Dalton, and R. Gilmozzi, "Scientific requirements for a European ELT," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 726-734 (2006).

Davidge, T.

G. Herriot, P. Hickson, B. L. Ellerbroek, D. A. Anderson, T. Davidge, D. A. Erickson, I. P. Powell, R. Clare, L. Gilles, C. Boyer, M. Smith, L. Saddlemyer, and J.-P. Véran, "NFIRAOS: TMT narrow field near-infrared facility adaptive optics," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 228-239 (2006).

Davies, R. I.

D. J. Butler, R. I. Davies, H. Fews, R. M. Redfern, N. Ageorges, W. K. Hackenberg, R.-R. Rohloff, S. Rabien, T. Ott, and S. Hippler, "Sodium layer monitoring at Calar Alto by LIDAR," in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE 4007, 358-367 (2000).

Dunn, J.

D. Kerley, S. Roberts, J. Dunn, N. Stretch, M. Smith, S. Sun, J. Pazder, and J. Fitzsimmons, "Validation and verification of integrated model simulations of a thirty meter telescope," in Optical Modeling and Performance Predictions II, M. A. Kahan, ed., Proc. SPIE 5867, 220-231 (2005).

Ellerbroek, B. L.

L. Gilles, B. L. Ellerbroek, and J.-P. Véran, "Laser guide star multi-conjugate adaptive optics performance of the Thirty Meter Telescope with elongated beacons and matched filtering," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 1025-1032 (2006).

R. M. Clare and B. L. Ellerbroek, "Sky coverage estimates for adaptive optics systems from computations in Zernike space," J. Opt. Soc. Am. A 23, 418-426 (2006).
[CrossRef]

G. Herriot, P. Hickson, B. L. Ellerbroek, D. A. Anderson, T. Davidge, D. A. Erickson, I. P. Powell, R. Clare, L. Gilles, C. Boyer, M. Smith, L. Saddlemyer, and J.-P. Véran, "NFIRAOS: TMT narrow field near-infrared facility adaptive optics," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 228-239 (2006).

G. Herriot, P. Hickson, B. L. Ellerbroek, J.-P. Véran, C.-Y. She, R. M. Clare, and D. Looze, "Focus errors from tracking sodium layer altitude variations with laser guide star adaptive optics for the Thirty Meter Telescope," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 486-495 (2006).

B. L. Ellerbroek, "Linear systems modeling of adaptive optics in the spatial-frequency domain," J. Opt. Soc. Am. A 22, 310-322 (2005).
[CrossRef]

D. B. Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Advancements in Adaptive Optics,Proc. SPIE 5490 (2004).

B. L. Ellerbroek and F. R. Rigaut, "Methods for correcting tilt anisoplanatism in laser-guide-star-based multiconjugate adaptive optics," J. Opt. Soc. Am. A 18, 2539-2547 (2001).
[CrossRef]

B. L. Ellerbroek and D. W. Tyler, "Adaptive optics sky coverage calculations for the Gemini-North telescope," Publ. Astron. Soc. Pac. 110, 165-185 (1998).
[CrossRef]

Enmark, A.

T. E. Andersen, A. Ardeberg, J. Beckers, M. Browne, A. Enmark, P. Knutsson, and M. Owner-Petersen, "From Euro50 toward a European ELT," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 716-725 (2006).

Erickson, D. A.

G. Herriot, P. Hickson, B. L. Ellerbroek, D. A. Anderson, T. Davidge, D. A. Erickson, I. P. Powell, R. Clare, L. Gilles, C. Boyer, M. Smith, L. Saddlemyer, and J.-P. Véran, "NFIRAOS: TMT narrow field near-infrared facility adaptive optics," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 228-239 (2006).

Femenia, B.

Fews, H.

D. J. Butler, R. I. Davies, H. Fews, R. M. Redfern, N. Ageorges, W. K. Hackenberg, R.-R. Rohloff, S. Rabien, T. Ott, and S. Hippler, "Sodium layer monitoring at Calar Alto by LIDAR," in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE 4007, 358-367 (2000).

Fitzsimmons, J.

D. Kerley, S. Roberts, J. Dunn, N. Stretch, M. Smith, S. Sun, J. Pazder, and J. Fitzsimmons, "Validation and verification of integrated model simulations of a thirty meter telescope," in Optical Modeling and Performance Predictions II, M. A. Kahan, ed., Proc. SPIE 5867, 220-231 (2005).

Fried, D. L.

G. A. Tyler and D. L. Fried, "Image-position error associated with a quadrant detector," J. Opt. Soc. Am. 72, 804-808 (1982).
[CrossRef]

D. L. Fried, "Optical resolution through a randomly inhomogenous medium for very long and very short exposures," J. Opt. Soc. Am. A 56, 1376-1379 (1966).

Gilles, L.

L. Gilles, B. L. Ellerbroek, and J.-P. Véran, "Laser guide star multi-conjugate adaptive optics performance of the Thirty Meter Telescope with elongated beacons and matched filtering," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 1025-1032 (2006).

G. Herriot, P. Hickson, B. L. Ellerbroek, D. A. Anderson, T. Davidge, D. A. Erickson, I. P. Powell, R. Clare, L. Gilles, C. Boyer, M. Smith, L. Saddlemyer, and J.-P. Véran, "NFIRAOS: TMT narrow field near-infrared facility adaptive optics," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 228-239 (2006).

Gilmozzi, R.

I. Hook, G. Dalton, and R. Gilmozzi, "Scientific requirements for a European ELT," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 726-734 (2006).

Hackenberg, W. K.

D. J. Butler, R. I. Davies, H. Fews, R. M. Redfern, N. Ageorges, W. K. Hackenberg, R.-R. Rohloff, S. Rabien, T. Ott, and S. Hippler, "Sodium layer monitoring at Calar Alto by LIDAR," in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE 4007, 358-367 (2000).

Herriot, G.

G. Herriot, P. Hickson, B. L. Ellerbroek, D. A. Anderson, T. Davidge, D. A. Erickson, I. P. Powell, R. Clare, L. Gilles, C. Boyer, M. Smith, L. Saddlemyer, and J.-P. Véran, "NFIRAOS: TMT narrow field near-infrared facility adaptive optics," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 228-239 (2006).

J.-P. Véran and G. Herriot, "Woofer-tweeter tip-tilt control for NFIRAOS for TMT," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 555-562 (2006).

G. Herriot, P. Hickson, B. L. Ellerbroek, J.-P. Véran, C.-Y. She, R. M. Clare, and D. Looze, "Focus errors from tracking sodium layer altitude variations with laser guide star adaptive optics for the Thirty Meter Telescope," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 486-495 (2006).

Hickson, P.

G. Herriot, P. Hickson, B. L. Ellerbroek, J.-P. Véran, C.-Y. She, R. M. Clare, and D. Looze, "Focus errors from tracking sodium layer altitude variations with laser guide star adaptive optics for the Thirty Meter Telescope," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 486-495 (2006).

G. Herriot, P. Hickson, B. L. Ellerbroek, D. A. Anderson, T. Davidge, D. A. Erickson, I. P. Powell, R. Clare, L. Gilles, C. Boyer, M. Smith, L. Saddlemyer, and J.-P. Véran, "NFIRAOS: TMT narrow field near-infrared facility adaptive optics," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 228-239 (2006).

Hippler, S.

D. J. Butler, R. I. Davies, H. Fews, R. M. Redfern, N. Ageorges, W. K. Hackenberg, R.-R. Rohloff, S. Rabien, T. Ott, and S. Hippler, "Sodium layer monitoring at Calar Alto by LIDAR," in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE 4007, 358-367 (2000).

Hook, I.

I. Hook, G. Dalton, and R. Gilmozzi, "Scientific requirements for a European ELT," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 726-734 (2006).

Johns, M. W.

M. W. Johns, "The Giant Magellan Telescope (GMT)," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 762-776 (2006).

Kerley, D.

D. Kerley, S. Roberts, J. Dunn, N. Stretch, M. Smith, S. Sun, J. Pazder, and J. Fitzsimmons, "Validation and verification of integrated model simulations of a thirty meter telescope," in Optical Modeling and Performance Predictions II, M. A. Kahan, ed., Proc. SPIE 5867, 220-231 (2005).

Knutsson, P.

T. E. Andersen, A. Ardeberg, J. Beckers, M. Browne, A. Enmark, P. Knutsson, and M. Owner-Petersen, "From Euro50 toward a European ELT," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 716-725 (2006).

Le Louarn, M.

Looze, D.

G. Herriot, P. Hickson, B. L. Ellerbroek, J.-P. Véran, C.-Y. She, R. M. Clare, and D. Looze, "Focus errors from tracking sodium layer altitude variations with laser guide star adaptive optics for the Thirty Meter Telescope," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 486-495 (2006).

Martin, F.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, and A. Ziad, "1998 Gemini site testing campaign: Cerro Pachon and Cerro Tololo," Gemini Document RTP-AO-G0094 (Gemini Observatory, Hilo, Hawaii, 2000).

Masciadri, E.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, and A. Ziad, "1998 Gemini site testing campaign: Cerro Pachon and Cerro Tololo," Gemini Document RTP-AO-G0094 (Gemini Observatory, Hilo, Hawaii, 2000).

Melvin, W. L.

W. L. Melvin, "A STAP overview," IEEE Aerosp. Electron. Syst. Mag. 19, 19-35 (2004).
[CrossRef]

Nelson, J.

J. Nelson and G. H. Sanders, "TMT status report," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 745-761 (2006).

Noll, R.

R. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. A 66, 207-211 (1976).
[CrossRef]

Ott, T.

D. J. Butler, R. I. Davies, H. Fews, R. M. Redfern, N. Ageorges, W. K. Hackenberg, R.-R. Rohloff, S. Rabien, T. Ott, and S. Hippler, "Sodium layer monitoring at Calar Alto by LIDAR," in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE 4007, 358-367 (2000).

Owner-Petersen, M.

T. E. Andersen, A. Ardeberg, J. Beckers, M. Browne, A. Enmark, P. Knutsson, and M. Owner-Petersen, "From Euro50 toward a European ELT," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 716-725 (2006).

Pazder, J.

D. Kerley, S. Roberts, J. Dunn, N. Stretch, M. Smith, S. Sun, J. Pazder, and J. Fitzsimmons, "Validation and verification of integrated model simulations of a thirty meter telescope," in Optical Modeling and Performance Predictions II, M. A. Kahan, ed., Proc. SPIE 5867, 220-231 (2005).

Powell, I. P.

G. Herriot, P. Hickson, B. L. Ellerbroek, D. A. Anderson, T. Davidge, D. A. Erickson, I. P. Powell, R. Clare, L. Gilles, C. Boyer, M. Smith, L. Saddlemyer, and J.-P. Véran, "NFIRAOS: TMT narrow field near-infrared facility adaptive optics," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 228-239 (2006).

Rabien, S.

D. J. Butler, R. I. Davies, H. Fews, R. M. Redfern, N. Ageorges, W. K. Hackenberg, R.-R. Rohloff, S. Rabien, T. Ott, and S. Hippler, "Sodium layer monitoring at Calar Alto by LIDAR," in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE 4007, 358-367 (2000).

Ragazzoni, R.

D. B. Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Advancements in Adaptive Optics,Proc. SPIE 5490 (2004).

Redfern, R. M.

D. J. Butler, R. I. Davies, H. Fews, R. M. Redfern, N. Ageorges, W. K. Hackenberg, R.-R. Rohloff, S. Rabien, T. Ott, and S. Hippler, "Sodium layer monitoring at Calar Alto by LIDAR," in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE 4007, 358-367 (2000).

Rigaut, F. R.

Roberts, S.

D. Kerley, S. Roberts, J. Dunn, N. Stretch, M. Smith, S. Sun, J. Pazder, and J. Fitzsimmons, "Validation and verification of integrated model simulations of a thirty meter telescope," in Optical Modeling and Performance Predictions II, M. A. Kahan, ed., Proc. SPIE 5867, 220-231 (2005).

Rohloff, R.-R.

D. J. Butler, R. I. Davies, H. Fews, R. M. Redfern, N. Ageorges, W. K. Hackenberg, R.-R. Rohloff, S. Rabien, T. Ott, and S. Hippler, "Sodium layer monitoring at Calar Alto by LIDAR," in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE 4007, 358-367 (2000).

Saddlemyer, L.

G. Herriot, P. Hickson, B. L. Ellerbroek, D. A. Anderson, T. Davidge, D. A. Erickson, I. P. Powell, R. Clare, L. Gilles, C. Boyer, M. Smith, L. Saddlemyer, and J.-P. Véran, "NFIRAOS: TMT narrow field near-infrared facility adaptive optics," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 228-239 (2006).

Sanchez, L.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, and A. Ziad, "1998 Gemini site testing campaign: Cerro Pachon and Cerro Tololo," Gemini Document RTP-AO-G0094 (Gemini Observatory, Hilo, Hawaii, 2000).

Sanders, G. H.

J. Nelson and G. H. Sanders, "TMT status report," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 745-761 (2006).

She, C.-Y.

G. Herriot, P. Hickson, B. L. Ellerbroek, J.-P. Véran, C.-Y. She, R. M. Clare, and D. Looze, "Focus errors from tracking sodium layer altitude variations with laser guide star adaptive optics for the Thirty Meter Telescope," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 486-495 (2006).

Smith, M.

G. Herriot, P. Hickson, B. L. Ellerbroek, D. A. Anderson, T. Davidge, D. A. Erickson, I. P. Powell, R. Clare, L. Gilles, C. Boyer, M. Smith, L. Saddlemyer, and J.-P. Véran, "NFIRAOS: TMT narrow field near-infrared facility adaptive optics," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 228-239 (2006).

D. Kerley, S. Roberts, J. Dunn, N. Stretch, M. Smith, S. Sun, J. Pazder, and J. Fitzsimmons, "Validation and verification of integrated model simulations of a thirty meter telescope," in Optical Modeling and Performance Predictions II, M. A. Kahan, ed., Proc. SPIE 5867, 220-231 (2005).

Soneira, R. M.

J. N. Bahcall and R. M. Soneira, "The universe at faint magnitudes. I. Models for the galaxy and the predicted star counts," Astrophys. J. Suppl. 44, 73-110 (1980).
[CrossRef]

Spagna, A.

A. Spagna, "Guide star requirements for NGST: deep NIR starcounts and guide star catalogs," Space Telescope Science Institute Document NGST-R-OO13B (Space Telescope Science Institute, Baltimore, Md., 2001).

Stretch, N.

D. Kerley, S. Roberts, J. Dunn, N. Stretch, M. Smith, S. Sun, J. Pazder, and J. Fitzsimmons, "Validation and verification of integrated model simulations of a thirty meter telescope," in Optical Modeling and Performance Predictions II, M. A. Kahan, ed., Proc. SPIE 5867, 220-231 (2005).

Sun, S.

D. Kerley, S. Roberts, J. Dunn, N. Stretch, M. Smith, S. Sun, J. Pazder, and J. Fitzsimmons, "Validation and verification of integrated model simulations of a thirty meter telescope," in Optical Modeling and Performance Predictions II, M. A. Kahan, ed., Proc. SPIE 5867, 220-231 (2005).

Tallon, M.

Tyler, D. W.

B. L. Ellerbroek and D. W. Tyler, "Adaptive optics sky coverage calculations for the Gemini-North telescope," Publ. Astron. Soc. Pac. 110, 165-185 (1998).
[CrossRef]

Tyler, G. A.

Véran, J.-P.

J.-P. Véran and G. Herriot, "Woofer-tweeter tip-tilt control for NFIRAOS for TMT," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 555-562 (2006).

G. Herriot, P. Hickson, B. L. Ellerbroek, D. A. Anderson, T. Davidge, D. A. Erickson, I. P. Powell, R. Clare, L. Gilles, C. Boyer, M. Smith, L. Saddlemyer, and J.-P. Véran, "NFIRAOS: TMT narrow field near-infrared facility adaptive optics," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 228-239 (2006).

L. Gilles, B. L. Ellerbroek, and J.-P. Véran, "Laser guide star multi-conjugate adaptive optics performance of the Thirty Meter Telescope with elongated beacons and matched filtering," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 1025-1032 (2006).

G. Herriot, P. Hickson, B. L. Ellerbroek, J.-P. Véran, C.-Y. She, R. M. Clare, and D. Looze, "Focus errors from tracking sodium layer altitude variations with laser guide star adaptive optics for the Thirty Meter Telescope," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 486-495 (2006).

Vernin, J.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, and A. Ziad, "1998 Gemini site testing campaign: Cerro Pachon and Cerro Tololo," Gemini Document RTP-AO-G0094 (Gemini Observatory, Hilo, Hawaii, 2000).

Ziad, A.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, and A. Ziad, "1998 Gemini site testing campaign: Cerro Pachon and Cerro Tololo," Gemini Document RTP-AO-G0094 (Gemini Observatory, Hilo, Hawaii, 2000).

Astrophys. J. Suppl. (1)

J. N. Bahcall and R. M. Soneira, "The universe at faint magnitudes. I. Models for the galaxy and the predicted star counts," Astrophys. J. Suppl. 44, 73-110 (1980).
[CrossRef]

IEEE Aerosp. Electron. Syst. Mag. (1)

W. L. Melvin, "A STAP overview," IEEE Aerosp. Electron. Syst. Mag. 19, 19-35 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Proc. SPIE (4)

G. Herriot, P. Hickson, B. L. Ellerbroek, D. A. Anderson, T. Davidge, D. A. Erickson, I. P. Powell, R. Clare, L. Gilles, C. Boyer, M. Smith, L. Saddlemyer, and J.-P. Véran, "NFIRAOS: TMT narrow field near-infrared facility adaptive optics," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 228-239 (2006).

D. B. Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Advancements in Adaptive Optics,Proc. SPIE 5490 (2004).

J.-P. Véran and G. Herriot, "Woofer-tweeter tip-tilt control for NFIRAOS for TMT," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 555-562 (2006).

G. Herriot, P. Hickson, B. L. Ellerbroek, J.-P. Véran, C.-Y. She, R. M. Clare, and D. Looze, "Focus errors from tracking sodium layer altitude variations with laser guide star adaptive optics for the Thirty Meter Telescope," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 486-495 (2006).

Publ. Astron. Soc. Pac. (2)

H. W. Babcock, "The possibility of compensating astronomical seeing," Publ. Astron. Soc. Pac. 65, 229-236 (1953).
[CrossRef]

B. L. Ellerbroek and D. W. Tyler, "Adaptive optics sky coverage calculations for the Gemini-North telescope," Publ. Astron. Soc. Pac. 110, 165-185 (1998).
[CrossRef]

Other (10)

L. Gilles, B. L. Ellerbroek, and J.-P. Véran, "Laser guide star multi-conjugate adaptive optics performance of the Thirty Meter Telescope with elongated beacons and matched filtering," in Advances in Adaptive Optics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 1025-1032 (2006).

D. J. Butler, R. I. Davies, H. Fews, R. M. Redfern, N. Ageorges, W. K. Hackenberg, R.-R. Rohloff, S. Rabien, T. Ott, and S. Hippler, "Sodium layer monitoring at Calar Alto by LIDAR," in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE 4007, 358-367 (2000).

J. Nelson and G. H. Sanders, "TMT status report," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 745-761 (2006).

I. Hook, G. Dalton, and R. Gilmozzi, "Scientific requirements for a European ELT," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 726-734 (2006).

M. W. Johns, "The Giant Magellan Telescope (GMT)," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 762-776 (2006).

T. E. Andersen, A. Ardeberg, J. Beckers, M. Browne, A. Enmark, P. Knutsson, and M. Owner-Petersen, "From Euro50 toward a European ELT," in Ground-Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE 6267, 716-725 (2006).

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, and A. Ziad, "1998 Gemini site testing campaign: Cerro Pachon and Cerro Tololo," Gemini Document RTP-AO-G0094 (Gemini Observatory, Hilo, Hawaii, 2000).

P. Puxley, "Observing constraints--sky background," Gemini Observatory, October 2002, http://www.gemini.edu/sciops/ObsProcess/obsConstraints/ocSkyBackground.html.

A. Spagna, "Guide star requirements for NGST: deep NIR starcounts and guide star catalogs," Space Telescope Science Institute Document NGST-R-OO13B (Space Telescope Science Institute, Baltimore, Md., 2001).

D. Kerley, S. Roberts, J. Dunn, N. Stretch, M. Smith, S. Sun, J. Pazder, and J. Fitzsimmons, "Validation and verification of integrated model simulations of a thirty meter telescope," in Optical Modeling and Performance Predictions II, M. A. Kahan, ed., Proc. SPIE 5867, 220-231 (2005).

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Figures (7)

Fig. 1
Fig. 1

AO control system model used in this paper. Inside the dotted box is the control algorithm. See the text for the matrix definitions.

Fig. 2
Fig. 2

Strehl ratio in the J band as a function of the position of the NGS in the field. The squares represent the Strehls generated using LAOS, and the curve is a best-fit cubic approximation.

Fig. 3
Fig. 3

(Color online) Residual telescope wind-shake tilt jitter after the correction as a function of the sampling frequency of the NGS WFS.

Fig. 4
Fig. 4

(Color online) Estimated focus wavefront error due to variations in the mean altitude of the sodium layer as a function of the sampling frequency of the NGS WFS.

Fig. 5
Fig. 5

FOV for the NFIRAOS showing the sodium LGS (○), the 2 arc min diameter patrol field for the NGS (dotted circle), a random constellation of four NGSs (+), and the science evaluation points (×) for the 30 arc sec square science field.

Fig. 6
Fig. 6

Median TT error (nm) over 500 NGS constellations versus the patrol field diameter (arc sec) for two TT NGSs and one TTFA NGS in the J band.

Fig. 7
Fig. 7

CDF [ Pr ( σ < ε )  versus   ε ( nm ) ] from 500 different NGS constellations in the J band with ten electrons of read noise for the NFIRAOS for one TTFA NGS WFS and two TT NGS WFSs (solid curve), a single TTFA NGS WFS (dashed curve), and a single TT NGS WFS used in conjunction with a Rayleigh LGS (dotted curve).

Tables (5)

Tables Icon

Table 1 Six-Layer Turbulence Profile Typical of Cerro Pachon (Ref. [25]25) a

Tables Icon

Table 2 System and Atmospheric Parameters

Tables Icon

Table 3 Median TT Errors (nm) for the NFIRAOS for the Three NGS WFS Options with SH Sensors Using Quad-Cell Detectors

Tables Icon

Table 4 Median TT Errors (nm) for the NFIRAOS Broken Down into Wind Shake, Noise, Tilt Anisoplanatism, and Servo Lag for the Three, WFS Options in the J Band a

Tables Icon

Table 5 Median TT Errors (nm) for the NFIRAOS with Three TT NGS WFSs for the Quad-Cell and Matched-Filter Approaches

Equations (106)

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

ϕ ( i ; r , θ ) = n = 0 N m = 0 n a n m ( i ) Z n m ( r , θ ) ,
b = T a ,
a = [ a ( 1 ) a ( i ) a ( N l ) ] T .
b L = T L a ,
b n = T n a + n ,
b s = T s a ,
b m = T m P m a ^ ,
P m = ( T m T W T m ) 1 T m T W T s .
a ^ L = E L b L = E L T L a ,
a res = a a ^ L ,
a ^ res = a ^ a ^ L .
b n = H ol ( s ) 1 + H ol ( s ) ( T n a + n ) = H cl ( s ) ( T n a + n ) = T n [ H cl ( s ) a ] + H cl ( s ) n = T n a + n ,
δ b n = b n b ^ n = b n E n b L ,
δ b n = T n a + n E n T L a .
a ^ res = E res δ b n ,
a ^ L = a b L T b L b L T b L = E L T L a ,
E L = a a T T L T ( T L a a T T L T ) ,
b ^ n = b n b L T b L b L T b L = E n T L a ,
E n = T n a a T T L T ( T L a a T T L T ) .
( a ^ L b ^ n ) = ( a b n ) b L T b L b L T b L .
( a r e s δ b n ) ( a r e s δ b n ) T = ( a a ^ L b n b ^ n ) ( a a ^ L b n b ^ n ) T = ( a b n ) ( a b n ) T ( a b n ) b L T b L b L T b L ( a b n ) T = [ a a T a b L T b L b L T b L a T b n a T b n b L T b L b L T b L a T a b n T a b L T b L b L T b L b n T b n b n T b n b L T b L b L T b L b n T ] .
b n b n T = T n a a T T n T + C N ,
b n a T = T n a a T .
b n b L T = T n a a T T L T .
C ϕ = [ C ϕ ( 1 ) 0 C ϕ ( i ) 0 C ϕ ( N l ) ] ,
C ϕ ( i ) = a ( i ) a ( i ) T = γ ( i ) C ,
C ϕ ( i ) = a ( i ) a ( i ) T = a ( i ) a ( i ) T .
C ϕ ( i ) = a ( i ) a ( i ) T .
( a a ^ L b n b ^ n ) ( a a ^ L b n b ^ n ) T = [ C ϕ E L T L C ϕ C ϕ T n T C ϕ T L T E n T T n C ϕ E n T L C ϕ T n C ϕ T n T E n T L C ϕ T n T + C N ] [ A B T B D + C N ] .
a ^ res = a res δ b n T δ b n δ b n T 1 δ b n = E res δ b n ,
E res = B T ( D + C N ) 1 .
( a res a ^ res ) ( a res a ^ res ) T = A E res B B T E res T +  E res ( D + C N ) E res T
= A B T ( D + C N ) 1 B .
a a ^ = a res a ^ res ,
( a a ^ ) ( a a ^ ) T = A B T ( D + C N ) 1 B .
Σ N = ( E res n ) ( E res n ) T = ( E res C N E res T ) .
B a = T n C ϕ E n T L C ϕ ,
D a = T n C ϕ T n T E n T L C ϕ T n T .
Σ a = ( a a ^ a ) ( a a ^ a ) T = A E res B a B a T E res T + E res D a E res T .
Σ l = ( a a ^ ) ( a a ^ ) T Σ N Σ a = δ A E res δ B δ B T E res T + E res δ D E res T ,
δ A = 0,
δ B = T n ( C ϕ C ϕ ) ,
δ D = T n ( C ϕ C ϕ ) T n T E n T L ( C ϕ C ϕ ) T n T ,
σ MOAO 2 = Tr [ T s T W T s ( a a ^ ) ( a a ^ ) T ] ,
σ MOAO 2 = Tr { T s T W T s [ A B T ( D + C N ) 1 B ] } .
σ MCAO 2 = σ est 2 + σ f i t 2 .
σ f i t 2 = Tr [ ( T s T m P m ) T W ( T s T m P m ) C ϕ ] ,
σ est 2 = Tr [ ( T m P m ) T W ( T m P m ) ( a a ^ ) ( a a ^ ) T ] .
σ est 2 = Tr { ( T m P m ) T W ( T m P m ) [ A B T ( D + C N ) 1 B ] } .
σ ( m ) = σ 2 ( rad 2 ) 2 π / λ e ,
s x ( k ) = G x b ( k ) + θ x ( k ) ,
s y ( k ) = G y b ( k ) + θ y ( k ) ,
σ θ 2 ( k ) = | θ x ( k ) | 2 = | θ y ( k ) | 2 .
G x ( i , j ) = λ e 2 π sa i Z j ( x , y ) x  d x d y sa i 1 d x d y ,
G y ( i , j ) = λ e 2 π sa i Z j ( x , y ) y  d x d y sa i 1 d x d y ,
G x = 2 λ e π D [ 1 0 ] ,
G y = 2 λ e π D [ 0 1 ] ,
G x = 2 λ e π D [ 1 0 8 3 π 4 6 3 π 4 6 3 π 1 0 8 3 π 4 6 3 π 4 6 3 π 1 0 8 3 π 4 6 3 π 4 6 3 π 1 0 8 3 π 4 6 3 π 4 6 3 π ] ,
G y = 2 λ e π D [ 0 1 8 3 π 4 6 3 π 4 6 3 π 0 1 8 3 π 4 6 3 π 4 6 3 π 0 1 8 3 π 4 6 3 π 4 6 3 π 0 1 8 3 π 4 6 3 π 4 6 3 π ] .
b ^ k = { [ G x T G y T ] Θ N 1 ( G x G y ) } 1 [ G x T G y T ] Θ N 1 ( s x s y ) ,
Σ b 2 ( k ) = σ θ 2 ( k ) { [ G x T G y T ] ( G x G y ) } 1 ,
{ [ G x T G y T ] ( G x G y ) } 1 = ( π D 2 λ e ) 2 [ 1 0 0 1 ] ,
{ [ G x T G y T ] ( G x G y ) } 1 = ( π D 4 λ e ) 2 [ 1 0 0 0 0 0 1 0 0 0 0 0 3 π 2 128 0 0 0 0 0 3 π 2 64 0 0 0 0 0 3 π 2 64 ] .
C N = ψ ( f s ) [ Σ b 2 ( 1 ) 0 Σ b 2 ( k ) 0 Σ b 2 ( N NGS ) ] ,
σ θ = θ B SNR ,
SNR = N p N p + 4 N b + 4 σ e 2 ,
θ B = λ 4 r 0 0 exp ( 3.44 κ 5 / 3 ) d κ ,
θ B = 3 π λ N sa 16 D ,
N p = z 10 m / 2.5 χ t s A T N sa ,
N b = z b χ t s A T w 2 N sa ,
I ( r ) = N p I 0 ( r + θ ) .
I = N p 1 { { I 0 } exp ( j 2 π κ θ ) } ,
I θ x | θ = 0 = N p 1 { j 2 π κ x { I 0 } } ,
I θ y | θ = 0 = N p 1 { j 2 π κ y { I 0 } } .
Q = [ Q x Q y ] = [ I θ x I θ y ] .
θ ^ i = ( Q T C N 1 Q ) 1 Q T C N 1 ( I i I 0 ) ,
C N = I + N b + σ e 2 .
Σ b 2 ( 0 ) = ( Q T C N 1 Q ) 1 .
Σ b 2 ( α ) = T α Σ b 2 ( 0 ) T α T ,
T θ = [ cos α sin α sin α cos α ] .
b n = T n a + n ,
H n ( s ) = H cl ( s ) H WFS ( s ) ,
H WFS ( s ) = f s s [ 1 exp ( s f s ) ] .
n ( t ) = n ( t ) h n ( t ) .
N ( f ) = N ( f ) H n ( f ) ,
n ( t ) = N ( f ) exp ( j 2 π f t ) H n ( f ) d f ,
| n ( t ) | 2 = | N ( f ) | 2 | H n ( f ) | 2 d f .
| n ( t ) | 2 = | N ( f ) | 2 f s / 2 f s / 2 | H n ( f ) | 2 d f .
t s = 1 f s .
| N ( f ) | 2 = t s Σ b 2 = Σ b 2 f s ,
| n ( t ) | 2 = Σ b 2 f s f s / 2 f s / 2 | H n ( f ) | 2 d f = Σ b 2 ψ ( f s ) ,
ψ ( f s ) = 1 f s f s / 2 f s / 2 | H n ( f ) | 2 d f .
a n m a n * m * T = ( 0.046 / π ) ( R / r 0 ) 5 / 3 [ ( n + 1 ) ( n * + 1 ) ] 1 / 2 × ( 1 ) ( n + n * 2 n ) / 2 δ m m * I n n * ,
I n n * = 0 κ 1 J n + 1 ( 2 π κ ) J n * + 1 ( 2 π κ ) [ κ 2 + ( R L 0 ) 2 ] 11 / 6 d κ ,
H cl ( s ) = H ol ( s ) 1 + H ol ( s ) .
ϕ ^ ( i ; κ ) ϕ ^ * ( i ; κ ) = | ϕ ^ ( i ; κ ) | 2 h ( κ ; i ) ,
h ( κ ; i ) = 0 2 π H cl [ j v ( i ) κ cos ( θ ) ] d θ ,
| ϕ ^ ( i ; κ ) | 2 = | ϕ ^ ( i ; κ ) | 2 h ( κ ; i ) ,
h ( κ ; i ) = 0 2 π | H cl [ j v ( i ) κ cos ( θ ) ] | 2 d θ .
a n m ( i ) a n * m * ( i ) T = ( 0.046 / π ) ( R / r 0 ) 5 / 3 × [ ( n + 1 ) ( n * + 1 ) ] 1 / 2 × ( 1 ) ( n + n * 2 n ) / 2 δ m m * I n n * ( i ) ,
I n n * ( i ) = 0 π / 2 sec 2 ( β ) tan ( β ) J n + 1 [ tan ( β ) ] J n * + 1 [ tan ( β ) ] [ ( tan ( β ) 2 π ) 2 + ( R L 0 ) 2 ] 11 / 6 × h [ tan ( β ) v ( i ) 2 π R ; i ] d β ,
a n m ( i ) a n * m * ( i ) T = ( 0.046 / π ) ( R / r 0 ) 5 / 3 × [ ( n + 1 ) ( n * + 1 ) ] 1 / 2 × ( 1 ) ( n + n * 2 n ) / 2 δ m m * I n n * ( i ) ,
I n n * ( i ) = 0 π / 2 sec 2 ( β ) tan ( β ) J n + 1 [ tan ( β ) ] J n * + 1 [ tan ( β ) ] [ ( tan ( β ) 2 π ) 2 + ( R L 0 ) 2 ] 11 / 6 × h [ tan ( β ) v ( i ) 2 π R ; i ] d β .
σ 2 ( f s ) = σ atm 2 ( f s ) + σ res 2 ( f s ) + σ Na 2 ( f s ) .
f s * = arg min f s { σ atm 2 ( f s ) + σ res 2 ( f s ) + σ Na 2 ( f s ) } .
σ ( mas ) = 4 × 1000 × 180 × 60 × 60 D π σ ( m ) .

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