Abstract

Explicit analytical expressions are derived for the elastic deformation of a thin or thick mirror of uniform thickness and with a central hole. Thin-plate theory is used to derive the general influence function, caused by uniform and/or discrete loads, for a mirror supported by discrete points. No symmetry considerations of the locations of the points constrain the model. An estimate of the effect of the shear forces is added to the previous pure bending model to take into account the effect of the mirror thickness. Two particular cases of general influence are considered: the actuator influence function and the uniform-load (equivalent to gravity in the case of a thin mirror) influence function for a ring support of k discrete points with k-fold symmetry. The influence of the size of the support pads is studied. A method for optimizing an active mirror cell is presented that couples the minimization of the gravity influence function with the optimization of the combined actuator influence functions to fit low-order aberrations. These low-spatial-frequency aberrations can be of elastic or optical origin. In the latter case they are due, for example, to great residual polishing errors corresponding to the soft polishing specifications relaxed for cost reductions. Results show that the correction range of the active cell can thus be noticeably enlarged, compared with an active cell designed as a passive cell, i.e., by minimizing only the deflection under gravitational loading. In the example treated here of the European Southern Observatory's New Technology Telescope I show that the active correction range can be enlarged by ∼50% in the case of third-order astigmatic correction.

© 1996 Optical Society of America

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  1. A. Couder, “Recherches sur les déformations des grands miroirs employés aux observations astronomiques,” Thesis (Gauthier-Villars et Cie, Paris, 1932).
  2. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Dover, New York, 1944).
  3. S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959), Chap. 9.
  4. J. E. Nelson, J. Lubliner, T. S. Mast, “Telescope mirror supports: plate deflection on point supports,” in Advanced Technology Optical Telescopes I, L. D. Barr, G. Burbridge, eds., Proc. Soc. Photo-Opt. Instrum. Eng.332, 212–228 (1982).
  5. G. Schwesinger, E. D. Knohl, “Comments on a series of articles by L. A. Selke,” Appl. Opt. 11, 200–202 (1972).
  6. L. A. Selke, “Theoretical elastic deformations of solid and cored horizontal circular mirrors having a central hole on a ring support,” Appl. Opt. 10, 939–944 (1971).
  7. G. Schwesinger, “An analytical determination of the flexure of the 3.5 m primary and 1 m mirror of the ESO's New Technology Telescope for passive support and active control,” J. Mod. Opt. 35, 1117–1149 (1988).
  8. G. Schwesinger, “Support configuration and elastic deformation of the 1.5 m prime mirror of the ESO Coudé Auxiliary Telescope (CAT),” ESO Tech. Rep. 9 (European Southern Observatory, Garching-bei-Munchen, Germany, 1979).
  9. D. S. Wan, J. R. P. Angel, R. E. Parks, “Mirror deflection on multiple axial supports,” Appl. Opt. 28, 354–362 (1989).
  10. A. Menikoff, “Actuator influence function of active mirrors,” Appl. Opt. 30, 833–838 (1991).
  11. P. Dierickx, “Optical performance of large ground-based telescopes,” J. Mod. Opt. 39, 569–588 (1992).
  12. A. J. Ostroff, “Evaluation of control laws and actuators locations for control systems applicable to deformable astronomical telescope mirrors,” NASA Tech. Note D-7276 (October. 1973).
  13. F. B. Ray, Y. T. Chung, “Surface analysis of an active controlled telescope primary mirror under static loads,” Appl. Opt. 24, 564–569 (1985).
  14. L. Arnold, “Optimized axial support topologies for thin telescope mirrors,” Opt. Eng. 34, 567–574 (1995).
  15. B. Rule, “Possible flexible mirror and collimation servo-control,” in The Construction of Large Telescopes, D. L. Crawford, eds., IAU Symposium 27 (Academic, London, 1966), pp. 71–75.
  16. R. N. Wilson, F. Franza, L. Noethe, “Active optics I: A system for optimizing the optical quality and reducing the costs of large telescopes,” J. Mod. Opt. 34, 485–509 (1987).
  17. L. Noethe, F. Franza, P. Giordano, R. N. Wilson, “Active optics II: Results of an experiment with a thin 1 m test mirror,” J. Mod. Opt. 35, 1427–1457 (1988).
  18. R. N. Wilson, F. Franza, P. Giordano, L. Noethe, M. Tarenghi, “Active optics III: Final results with the 1 m test mirror and the NTT 3.58 m primary in the workshop,” J. Mod. Opt. 36, 1415–1425 (1989).
  19. R. N. Wilson, F. Franza, L. Noethe, G. Andreoni, “Active optics IV: Setup and performance of the optics of the ESO New Technology Telescope (NTT) in the observatory,” J. Mod. Opt. 38, 219–243 (1991).
  20. D. Enard, B. Delabre, S. D Odorico, F. Merkle, A. Moorwood, G. Raffi, M. Sarazin, M. Schneermann, J. Wampler, R. Wilson, L. Zago, M. Ziebell, “Proposed for the construction of the 16-m Very Large Telescope” (European Southern Observatory, Garching-bei-München, Germany, 1987).
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  29. L. Noethe, European Southern Observatory, Karl Schwarzschild, Strasse 2, D-W8046, Garching-bei-München, Germany (personal communication, 1995).
  30. F. Roddier, “The problematic of adaptive optics design,” in Adaptive Optics for Astronomy, D. M. Alloin, J. M. Mariotti, eds. (Kluwer Academic, Boston, Mass., 1994), Vol. 423, pp. 89–111.
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1995

L. Arnold, “Optimized axial support topologies for thin telescope mirrors,” Opt. Eng. 34, 567–574 (1995).

1992

P. Dierickx, “Optical performance of large ground-based telescopes,” J. Mod. Opt. 39, 569–588 (1992).

1991

L. Noethe, “Use of minimum-energy modes for modal-active optics corrctions of thin meniscus mirrors”, J. Mod. Opt. 38, 1043–1066 (1991).

R. N. Wilson, F. Franza, L. Noethe, G. Andreoni, “Active optics IV: Setup and performance of the optics of the ESO New Technology Telescope (NTT) in the observatory,” J. Mod. Opt. 38, 219–243 (1991).

A. Menikoff, “Actuator influence function of active mirrors,” Appl. Opt. 30, 833–838 (1991).

1989

D. S. Wan, J. R. P. Angel, R. E. Parks, “Mirror deflection on multiple axial supports,” Appl. Opt. 28, 354–362 (1989).

R. N. Wilson, F. Franza, P. Giordano, L. Noethe, M. Tarenghi, “Active optics III: Final results with the 1 m test mirror and the NTT 3.58 m primary in the workshop,” J. Mod. Opt. 36, 1415–1425 (1989).

1988

G. Schwesinger, “An analytical determination of the flexure of the 3.5 m primary and 1 m mirror of the ESO's New Technology Telescope for passive support and active control,” J. Mod. Opt. 35, 1117–1149 (1988).

L. Noethe, F. Franza, P. Giordano, R. N. Wilson, “Active optics II: Results of an experiment with a thin 1 m test mirror,” J. Mod. Opt. 35, 1427–1457 (1988).

1987

R. N. Wilson, F. Franza, L. Noethe, “Active optics I: A system for optimizing the optical quality and reducing the costs of large telescopes,” J. Mod. Opt. 34, 485–509 (1987).

1985

1984

1981

1978

1976

1974

1972

1971

Andreoni, G.

R. N. Wilson, F. Franza, L. Noethe, G. Andreoni, “Active optics IV: Setup and performance of the optics of the ESO New Technology Telescope (NTT) in the observatory,” J. Mod. Opt. 38, 219–243 (1991).

Angel, J. R. P.

Arnold, L.

L. Arnold, “Optimized axial support topologies for thin telescope mirrors,” Opt. Eng. 34, 567–574 (1995).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

Chung, Y. T.

Couder, A.

A. Couder, “Recherches sur les déformations des grands miroirs employés aux observations astronomiques,” Thesis (Gauthier-Villars et Cie, Paris, 1932).

Delabre, B.

D. Enard, B. Delabre, S. D Odorico, F. Merkle, A. Moorwood, G. Raffi, M. Sarazin, M. Schneermann, J. Wampler, R. Wilson, L. Zago, M. Ziebell, “Proposed for the construction of the 16-m Very Large Telescope” (European Southern Observatory, Garching-bei-München, Germany, 1987).

Dierickx, P.

P. Dierickx, “Optical performance of large ground-based telescopes,” J. Mod. Opt. 39, 569–588 (1992).

Enard, D.

D. Enard, B. Delabre, S. D Odorico, F. Merkle, A. Moorwood, G. Raffi, M. Sarazin, M. Schneermann, J. Wampler, R. Wilson, L. Zago, M. Ziebell, “Proposed for the construction of the 16-m Very Large Telescope” (European Southern Observatory, Garching-bei-München, Germany, 1987).

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vettering, B. P. Flannery, Numerical Recipies in C (Cambridge U. Press, 1992).

Franza, F.

R. N. Wilson, F. Franza, L. Noethe, G. Andreoni, “Active optics IV: Setup and performance of the optics of the ESO New Technology Telescope (NTT) in the observatory,” J. Mod. Opt. 38, 219–243 (1991).

R. N. Wilson, F. Franza, P. Giordano, L. Noethe, M. Tarenghi, “Active optics III: Final results with the 1 m test mirror and the NTT 3.58 m primary in the workshop,” J. Mod. Opt. 36, 1415–1425 (1989).

L. Noethe, F. Franza, P. Giordano, R. N. Wilson, “Active optics II: Results of an experiment with a thin 1 m test mirror,” J. Mod. Opt. 35, 1427–1457 (1988).

R. N. Wilson, F. Franza, L. Noethe, “Active optics I: A system for optimizing the optical quality and reducing the costs of large telescopes,” J. Mod. Opt. 34, 485–509 (1987).

Giordano, P.

R. N. Wilson, F. Franza, P. Giordano, L. Noethe, M. Tarenghi, “Active optics III: Final results with the 1 m test mirror and the NTT 3.58 m primary in the workshop,” J. Mod. Opt. 36, 1415–1425 (1989).

L. Noethe, F. Franza, P. Giordano, R. N. Wilson, “Active optics II: Results of an experiment with a thin 1 m test mirror,” J. Mod. Opt. 35, 1427–1457 (1988).

Knohl, E. D.

Love, A. E. H.

A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Dover, New York, 1944).

Lubliner, J.

J. E. Nelson, J. Lubliner, T. S. Mast, “Telescope mirror supports: plate deflection on point supports,” in Advanced Technology Optical Telescopes I, L. D. Barr, G. Burbridge, eds., Proc. Soc. Photo-Opt. Instrum. Eng.332, 212–228 (1982).

Mahajan, V. N.

Markey, J. K.

Mast, T. S.

J. E. Nelson, J. Lubliner, T. S. Mast, “Telescope mirror supports: plate deflection on point supports,” in Advanced Technology Optical Telescopes I, L. D. Barr, G. Burbridge, eds., Proc. Soc. Photo-Opt. Instrum. Eng.332, 212–228 (1982).

Menikoff, A.

Merkle, F.

D. Enard, B. Delabre, S. D Odorico, F. Merkle, A. Moorwood, G. Raffi, M. Sarazin, M. Schneermann, J. Wampler, R. Wilson, L. Zago, M. Ziebell, “Proposed for the construction of the 16-m Very Large Telescope” (European Southern Observatory, Garching-bei-München, Germany, 1987).

Moorwood, A.

D. Enard, B. Delabre, S. D Odorico, F. Merkle, A. Moorwood, G. Raffi, M. Sarazin, M. Schneermann, J. Wampler, R. Wilson, L. Zago, M. Ziebell, “Proposed for the construction of the 16-m Very Large Telescope” (European Southern Observatory, Garching-bei-München, Germany, 1987).

Nelson, J. E.

J. E. Nelson, J. Lubliner, T. S. Mast, “Telescope mirror supports: plate deflection on point supports,” in Advanced Technology Optical Telescopes I, L. D. Barr, G. Burbridge, eds., Proc. Soc. Photo-Opt. Instrum. Eng.332, 212–228 (1982).

Noethe, L.

R. N. Wilson, F. Franza, L. Noethe, G. Andreoni, “Active optics IV: Setup and performance of the optics of the ESO New Technology Telescope (NTT) in the observatory,” J. Mod. Opt. 38, 219–243 (1991).

L. Noethe, “Use of minimum-energy modes for modal-active optics corrctions of thin meniscus mirrors”, J. Mod. Opt. 38, 1043–1066 (1991).

R. N. Wilson, F. Franza, P. Giordano, L. Noethe, M. Tarenghi, “Active optics III: Final results with the 1 m test mirror and the NTT 3.58 m primary in the workshop,” J. Mod. Opt. 36, 1415–1425 (1989).

L. Noethe, F. Franza, P. Giordano, R. N. Wilson, “Active optics II: Results of an experiment with a thin 1 m test mirror,” J. Mod. Opt. 35, 1427–1457 (1988).

R. N. Wilson, F. Franza, L. Noethe, “Active optics I: A system for optimizing the optical quality and reducing the costs of large telescopes,” J. Mod. Opt. 34, 485–509 (1987).

L. Noethe, European Southern Observatory, Karl Schwarzschild, Strasse 2, D-W8046, Garching-bei-München, Germany (personal communication, 1995).

Noll, R. J.

Odorico, S. D

D. Enard, B. Delabre, S. D Odorico, F. Merkle, A. Moorwood, G. Raffi, M. Sarazin, M. Schneermann, J. Wampler, R. Wilson, L. Zago, M. Ziebell, “Proposed for the construction of the 16-m Very Large Telescope” (European Southern Observatory, Garching-bei-München, Germany, 1987).

Ostroff, A. J.

A. J. Ostroff, “Evaluation of control laws and actuators locations for control systems applicable to deformable astronomical telescope mirrors,” NASA Tech. Note D-7276 (October. 1973).

Parks, R. E.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vettering, B. P. Flannery, Numerical Recipies in C (Cambridge U. Press, 1992).

Raffi, G.

D. Enard, B. Delabre, S. D Odorico, F. Merkle, A. Moorwood, G. Raffi, M. Sarazin, M. Schneermann, J. Wampler, R. Wilson, L. Zago, M. Ziebell, “Proposed for the construction of the 16-m Very Large Telescope” (European Southern Observatory, Garching-bei-München, Germany, 1987).

Ray, F. B.

Roddier, F.

F. Roddier, “The problematic of adaptive optics design,” in Adaptive Optics for Astronomy, D. M. Alloin, J. M. Mariotti, eds. (Kluwer Academic, Boston, Mass., 1994), Vol. 423, pp. 89–111.

Rule, B.

B. Rule, “Possible flexible mirror and collimation servo-control,” in The Construction of Large Telescopes, D. L. Crawford, eds., IAU Symposium 27 (Academic, London, 1966), pp. 71–75.

Sarazin, M.

D. Enard, B. Delabre, S. D Odorico, F. Merkle, A. Moorwood, G. Raffi, M. Sarazin, M. Schneermann, J. Wampler, R. Wilson, L. Zago, M. Ziebell, “Proposed for the construction of the 16-m Very Large Telescope” (European Southern Observatory, Garching-bei-München, Germany, 1987).

Schneermann, M.

D. Enard, B. Delabre, S. D Odorico, F. Merkle, A. Moorwood, G. Raffi, M. Sarazin, M. Schneermann, J. Wampler, R. Wilson, L. Zago, M. Ziebell, “Proposed for the construction of the 16-m Very Large Telescope” (European Southern Observatory, Garching-bei-München, Germany, 1987).

Schwesinger, G.

G. Schwesinger, “An analytical determination of the flexure of the 3.5 m primary and 1 m mirror of the ESO's New Technology Telescope for passive support and active control,” J. Mod. Opt. 35, 1117–1149 (1988).

G. Schwesinger, E. D. Knohl, “Comments on a series of articles by L. A. Selke,” Appl. Opt. 11, 200–202 (1972).

G. Schwesinger, “Support configuration and elastic deformation of the 1.5 m prime mirror of the ESO Coudé Auxiliary Telescope (CAT),” ESO Tech. Rep. 9 (European Southern Observatory, Garching-bei-Munchen, Germany, 1979).

Selke, L. A.

Tarenghi, M.

R. N. Wilson, F. Franza, P. Giordano, L. Noethe, M. Tarenghi, “Active optics III: Final results with the 1 m test mirror and the NTT 3.58 m primary in the workshop,” J. Mod. Opt. 36, 1415–1425 (1989).

Tatian, B.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vettering, B. P. Flannery, Numerical Recipies in C (Cambridge U. Press, 1992).

Timoshenko, S.

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959), Chap. 9.

Vettering, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vettering, B. P. Flannery, Numerical Recipies in C (Cambridge U. Press, 1992).

Wampler, J.

D. Enard, B. Delabre, S. D Odorico, F. Merkle, A. Moorwood, G. Raffi, M. Sarazin, M. Schneermann, J. Wampler, R. Wilson, L. Zago, M. Ziebell, “Proposed for the construction of the 16-m Very Large Telescope” (European Southern Observatory, Garching-bei-München, Germany, 1987).

Wan, D. S.

Wang, J. Y.

Wilson, R.

D. Enard, B. Delabre, S. D Odorico, F. Merkle, A. Moorwood, G. Raffi, M. Sarazin, M. Schneermann, J. Wampler, R. Wilson, L. Zago, M. Ziebell, “Proposed for the construction of the 16-m Very Large Telescope” (European Southern Observatory, Garching-bei-München, Germany, 1987).

Wilson, R. N.

R. N. Wilson, F. Franza, L. Noethe, G. Andreoni, “Active optics IV: Setup and performance of the optics of the ESO New Technology Telescope (NTT) in the observatory,” J. Mod. Opt. 38, 219–243 (1991).

R. N. Wilson, F. Franza, P. Giordano, L. Noethe, M. Tarenghi, “Active optics III: Final results with the 1 m test mirror and the NTT 3.58 m primary in the workshop,” J. Mod. Opt. 36, 1415–1425 (1989).

L. Noethe, F. Franza, P. Giordano, R. N. Wilson, “Active optics II: Results of an experiment with a thin 1 m test mirror,” J. Mod. Opt. 35, 1427–1457 (1988).

R. N. Wilson, F. Franza, L. Noethe, “Active optics I: A system for optimizing the optical quality and reducing the costs of large telescopes,” J. Mod. Opt. 34, 485–509 (1987).

Woinowsky-Krieger, S.

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959), Chap. 9.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

Zago, L.

D. Enard, B. Delabre, S. D Odorico, F. Merkle, A. Moorwood, G. Raffi, M. Sarazin, M. Schneermann, J. Wampler, R. Wilson, L. Zago, M. Ziebell, “Proposed for the construction of the 16-m Very Large Telescope” (European Southern Observatory, Garching-bei-München, Germany, 1987).

Ziebell, M.

D. Enard, B. Delabre, S. D Odorico, F. Merkle, A. Moorwood, G. Raffi, M. Sarazin, M. Schneermann, J. Wampler, R. Wilson, L. Zago, M. Ziebell, “Proposed for the construction of the 16-m Very Large Telescope” (European Southern Observatory, Garching-bei-München, Germany, 1987).

Appl. Opt.

J. Mod. Opt.

L. Noethe, “Use of minimum-energy modes for modal-active optics corrctions of thin meniscus mirrors”, J. Mod. Opt. 38, 1043–1066 (1991).

G. Schwesinger, “An analytical determination of the flexure of the 3.5 m primary and 1 m mirror of the ESO's New Technology Telescope for passive support and active control,” J. Mod. Opt. 35, 1117–1149 (1988).

R. N. Wilson, F. Franza, L. Noethe, “Active optics I: A system for optimizing the optical quality and reducing the costs of large telescopes,” J. Mod. Opt. 34, 485–509 (1987).

L. Noethe, F. Franza, P. Giordano, R. N. Wilson, “Active optics II: Results of an experiment with a thin 1 m test mirror,” J. Mod. Opt. 35, 1427–1457 (1988).

R. N. Wilson, F. Franza, P. Giordano, L. Noethe, M. Tarenghi, “Active optics III: Final results with the 1 m test mirror and the NTT 3.58 m primary in the workshop,” J. Mod. Opt. 36, 1415–1425 (1989).

R. N. Wilson, F. Franza, L. Noethe, G. Andreoni, “Active optics IV: Setup and performance of the optics of the ESO New Technology Telescope (NTT) in the observatory,” J. Mod. Opt. 38, 219–243 (1991).

P. Dierickx, “Optical performance of large ground-based telescopes,” J. Mod. Opt. 39, 569–588 (1992).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

L. Arnold, “Optimized axial support topologies for thin telescope mirrors,” Opt. Eng. 34, 567–574 (1995).

Other

B. Rule, “Possible flexible mirror and collimation servo-control,” in The Construction of Large Telescopes, D. L. Crawford, eds., IAU Symposium 27 (Academic, London, 1966), pp. 71–75.

G. Schwesinger, “Support configuration and elastic deformation of the 1.5 m prime mirror of the ESO Coudé Auxiliary Telescope (CAT),” ESO Tech. Rep. 9 (European Southern Observatory, Garching-bei-Munchen, Germany, 1979).

A. Couder, “Recherches sur les déformations des grands miroirs employés aux observations astronomiques,” Thesis (Gauthier-Villars et Cie, Paris, 1932).

A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Dover, New York, 1944).

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959), Chap. 9.

J. E. Nelson, J. Lubliner, T. S. Mast, “Telescope mirror supports: plate deflection on point supports,” in Advanced Technology Optical Telescopes I, L. D. Barr, G. Burbridge, eds., Proc. Soc. Photo-Opt. Instrum. Eng.332, 212–228 (1982).

A. J. Ostroff, “Evaluation of control laws and actuators locations for control systems applicable to deformable astronomical telescope mirrors,” NASA Tech. Note D-7276 (October. 1973).

W. H. Press, S. A. Teukolsky, W. T. Vettering, B. P. Flannery, Numerical Recipies in C (Cambridge U. Press, 1992).

L. Noethe, European Southern Observatory, Karl Schwarzschild, Strasse 2, D-W8046, Garching-bei-München, Germany (personal communication, 1995).

F. Roddier, “The problematic of adaptive optics design,” in Adaptive Optics for Astronomy, D. M. Alloin, J. M. Mariotti, eds. (Kluwer Academic, Boston, Mass., 1994), Vol. 423, pp. 89–111.

D. Enard, B. Delabre, S. D Odorico, F. Merkle, A. Moorwood, G. Raffi, M. Sarazin, M. Schneermann, J. Wampler, R. Wilson, L. Zago, M. Ziebell, “Proposed for the construction of the 16-m Very Large Telescope” (European Southern Observatory, Garching-bei-München, Germany, 1987).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

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Figures (7)

Fig. 1
Fig. 1

Top view of a mirror with a central hole. In this example, four discrete axial forces are applied to the mirror. Three of them are arranged with threefold symmetry.

Fig. 2
Fig. 2

Estimated extra deflection caused by shear forces, as computed for the ESO NTT 3.6-m primary mirror (Table 1) on a hypothetical six-point support with sixfold symmetry. This simple support, instead of the real 78-point one, was chosen for clarity.

Fig. 3
Fig. 3

Radial shear deflections, equally angularly spaced, for azimuth on the mirror going from a radius with a supporting point to a radius between two supporting points. The artifact along the supporting radius (1.2 m) is visible. The same six-point support as in Fig. 2 is shown.

Fig. 4
Fig. 4

Deformation of the ESO NTT primary mirror along a radius with four supports. The dashed curve represents the deformation for supports of neglected size. The surface error is 81 nm ptv and 8.8 nm rms. The solid curve represents the deformation for roughly square supports of 50 mm × 50 mm. The surface error is reduced to 56 nm ptv but is still 8.3 nm rms. In the two cases the calculation takes into account an estimation of the noticeable effect of shear forces because of the 241-mm thickness.

Fig. 5
Fig. 5

NTT primary-mirror active support: computed decrease d of the residual wave-front error obtained with optimized active supports with respect to the residual wave-front error obtained by the current ESO NTT active support. The topologies are optimized only over the clear aperture, from the internal stop to the outer edge. The improvement in the wave front is calculated with respect to the performances measured over the same clear aperture for the current support.

Fig. 6
Fig. 6

NTT primary-mirror active support: expected multiplicative factor MF applied to the correction range from the current ESO NTT active support for the case of astigmatic correction.

Fig. 7
Fig. 7

Three-dimensional views of the mirror deflection: A, deflection calculated for the current support design, representing the pure ULIF for an aberration-free mirror. B, deflection calculated for the same support as A, representing the mirror after the active supports have compensated for pure third-order astigmatism, 5λρ2 cos(2θ) = 2800ρ2 cos(2θ) nm, on the glass. It can be seen that high-order bending modes of azimuthal symmetry, m = 2, are excited by the actuators. C, deflection calculated for an optimized active support design, representing the pure ULIF for an aberration-free mirror. The mirror support was calculated so that a large amount of pure third-order astigmatism, 5λρ2 cos(2θ) = 2800ρ2 cos(2θ) nm, on the glass could be corrected. D, deflection calculated for the same optimized support as for C, representing the mirror after the active supports have compensated the pure third-order astigmatism integrated in the active support optimization. High-order bending modes are now practically absent. Note that the effects of the small bumps on the inner supports are not visible because they are in the shadow of the aperture stop. Peak-to-valley and rms errors for each figure are in Table 2.

Tables (2)

Tables Icon

Table 1 ESO NTT Primary Mirror and Support a

Tables Icon

Table 2 NTT Primary Support: Topologies for an Optimized Active Support Designa

Equations (69)

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D 2 ( 2 w ) = q
q ( r , θ ) = q = P π ( a 2 c 2 ) + j = 1 k f j b j δ ( r b j ) δ ( θ θ j )
q ( r , θ ) = q = P π ( a 2 c 2 ) + j = 1 k f j 2 π b j × δ ( r b j ) m = exp [ i m ( θ θ j ) ] ,
j = 1 k f j P = 0 ,
j = 1 k f j b j exp ( i θ j ) = 0 ,
M r ( a ) = M r ( c ) = 0 ,
Q r ( a ) 1 r M r θ ( a ) θ = Q r ( c ) 1 r M r θ ( c ) θ = 0 ,
w ( r , θ ) = m = w m ( r ) exp ( i m θ ) .
( 2 r 2 + 1 r r ) 2 w 0 = P π ( a 2 c 2 ) D + j = 1 k f j 2 π b j D δ ( r b j ) .
j = 1 k f j P = 0 ,
( 2 r 2 + 1 r r ) 2 w 0 = j = 1 k f j π D [ 1 a 2 c 2 + δ ( r b j ) 2 b j ] .
( 2 r 2 + 1 r r m 2 r 2 ) 2 w m = j = 1 k f j 2 π b j D exp ( i m θ j ) δ ( r b j ) .
lim r b j + w m = lim r b j w m ,
lim r b j + w m r = lim r b j w m r ,
lim r b j + 2 w m r 2 = lim r b j 2 w m r 2 ,
lim r b j + 3 w m r 3 lim r b j 3 w m r 3 = f j 2 π b j D exp ( i m θ j ) .
m = 0 ,      w 0 ( r ) = A 0 + B 0 r 2 + C 0 In ( r ) + D 0 r 2 In ( r ) ,
m = 1 ,    w 1 ( r ) = A 1 r + B 1 r 3 + C 1 / r + D 1 r  In ( r ) ,
m > 1 ,    w m ( r ) = A m r m + B m r m + 2 + C m r m + D m r m + 2 ,
Δ j A = lim r b j + A lim r b j A .
Δ j B 0 = q j 8 π D ( ln  b j + 1 ) .
w m 1 ( r , θ ) = Re [ 2 × w m 1 ( r ) × exp imθ ] ,
w s = 3 2 h D G 2 w .
m = 0 ,    2 w 0 ( r ) = 4 { B 0 + D 0 [ ln ( r ) + 1 ] } ,
m = 1 ,      2 w 1 ( r ) = 8 B 1 r + 2 D 1 / r ,
m > 1 , 2 w m ( r ) = 4 ( m + 1 ) B m r m + 4 ( 1 m ) D m r m .
( 2 r 2 + 1 r r ) 2 w 0 = j = 1 k f j 2 π b j D δ ( r b j )
B 0 = α 0 c 2 c 2 a 2 ,
C 0 = 2 α 0 ( 1 + v 1 v ) c 2 a 2 c 2 a 2 ,
α 0 = j = 1 k Δ j B 0 1 2 c 2 ( 1 v 1 + v ) j = 1 k Δ j C 0 ,
B 1 = α 1 c 4 c 4 a 4 ,
C 1 = α 1 ( 3 + v 1 v ) c 4 a 4 c 4 a 4 ,
α 1 = j = 1 k Δ j B 1 + 1 c 4 ( 1 v 3 + v ) j = 1 k Δ j C 1 .
q = P π ( a 2 c 2 ) + P k δ ( r b ) b j = 1 k δ ( θ 2 j π k θ 0 ) ,
j = 1 k δ ( θ 2 j π k θ 0 )
j = 1 k δ ( θ 2 j π k θ 0 ) = m = c m exp ( i k m θ )
c m = 1 2 π j = 1 k 0 2 π δ ( θ 2 j π k θ 0 ) exp ( i k m θ ) = k 2 π exp ( i k m θ 0 ) .
q = P π ( a 2 c 2 ) + P δ ( r b ) 2 π b + P δ ( r b ) π b × m = 1 cos  [ k m ( θ θ 0 ) ] ,
w = w 0 + m = 1 w m   cos [ k m ( θ θ 0 ) ]
w m = A m r n + B m r n + 2 + C m r n + D m n + 2 .
A m A m = α = b n n ( n 1 ) Q ,
B m B m = β= b n 2 n ( n + 1 ) Q ,
C m C m = γ= b n n ( n + 1 ) Q ,
D m D m = δ= b n 2 n ( n 1 ) Q ,
Q = P b 2 8 π D
A m = λ ( 3 + v ) ( v 1 ) n ( n + 1 ) ( 1 ρ 0 2 n ) μκ ( 1 ρ 0 2 n + 2 ) a 2 n a 2 n [ ( 3 + v ) 2 ( 1 ρ 0 2 n ) 2 + μ ( ρ 0 n 1 ρ 0 n + 1 ) 2 ] ,
B m = ( v 1 ) [ λ ( 3 + v ) n ( 1 ρ 0 2 n 2 ) a 2 κ ( n 1 ) ( v 1 ) ( 1 ρ 0 2 n ) ] a 2 n [ ( 3 + v ) 2 ( 1 ρ o 2 n ) 2 μ ( ρ 0 n 1 ρ 0 n + 1 ) 2 ] ,
C m = ( n 1 ) ( v 1 ) 3 + v c 2 n A m + μ n ( 3 + v ) ( v 1 ) c 2 n + 2 B m ,
D m = n 1 v 3 + v c 2 n 2 A m + ( 1 v ) ( n + 1 ) 3 + v c 2 n B m ,
κ= 3 + v 1 v δ n a 2 n 2 α ( n + 1 ) α 2 n β,
λ= ( n 1 ) ( 1 v ) 3 + v a 2 n α + μ ( 3 + v ) ( 1 v ) n a 2 n + 2 β+γ,
μ= n 2 ( 1 v ) 2 + 8 ( 1 + v ) .
A m × d a = α n c 2 n [ μ ( ρ 0 2 1 ) ( 3 + v ) 2 ( ρ 0 2 n 1 ) ] + βμ ( n + 1 ) c 2 n + 2 ( 1 ρ 0 2 ) γ ( 3 + v ) ( 1 v ) n ( n + 1 ) ( 1 ρ 0 2 n ) δμ 3 + v 1 v ( 1 ρ 0 2 n + 2 ) a 2 ,
d a = n a 2 n [ ( 3 + v ) 2 ( 1 ρ 0 2 n ) 2 + μ ( ρ 0 n 1 ρ 0 n + 1 ) 2 ] ,
B m × d b = α n ( 1 v ) 2 ( n 1 ) c 2 n 2 ( 1 ρ 0 2 ) + β c 2 n [ μ ( 1 ρ 0 2 ) + ( 3 + v ) 2 ( ρ 0 2 n 1 ) ] γ ( 3 + v ) ( 1 v ) n ( 1 ρ 0 2 n 2 ) a 2 δ ( 1 v ) ( 3 + v ) ( n 1 ) ( 1 ρ 0 2 n ) ,
d b = d a n .
[ ( r b j ) d p ] × [ b ( θ θ j ) d p ] ,
[ x ] = 1 , x < 0.5 ,      [ x ] = 0 ,    x 0.5.
q = P π ( a 2 c 2 ) + j = 1 k f j d p 2 [ ( r b j ) d p ] [ b ( θ θ j ) d p ]
q = P π ( a 2 c 2 ) + j = 1 k f j 2 π b j d p [ ( r b j ) d p ] × m = sinc ( m d p 2 π b j ) exp [ i m ( θ θ j ) ]
w r m s 2 = pupil ( w + p 0 ) 2 d S = pupil [ i = 1 I η i w i + p 0 ] 2 d S
w r m s 2 = pupil [ i = 1 I η i w i + p 0 + p 2 ρ 2 ] 2 d S ,
w r m s 2 = pupil [ i = 1 I η i w i + p 0 + p 2 ρ 2 + n , m ( j 1 J I F j w m , j a n m Z n m + p 0 n m + p 2 n m ρ 2 ) ] 2 d S .
f j s = cos [ 2 π m ( s 1 ) S + θ j ] ,
w r m s 2 = pupil [ ( i = 1 I η i w i ) 2 + ( p 0 + p 2 ρ 2 ) 2 + n , m ( j 1 J I F j w m , j a n m Z n m ) 2 ] d S ,
w r m s 2 = pupil ( i = 1 I η i w i + p 0 + p 2 ρ 2 ) 2 d S + n , m pupil ( j 1 J I F j w m , j a n m Z n m + p 0 n m + p 2 n m ρ 2 ) 2 d S .
Z 2 2 = ρ 2   cos  ( 2 θ+θ 22 ) .
M F = 1 1 d ,
δ F = 16 ( F 2 a ) 2 p 2 ~ 200 μm.

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