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Applied Optics

Applied Optics


  • Vol. 35, Iss. 7 — Mar. 1, 1996
  • pp: 1095–1106

Uniform-load and actuator influence functions of a thin or thick annular mirror: application to active mirror support optimization

Luc Arnold  »View Author Affiliations

Applied Optics, Vol. 35, Issue 7, pp. 1095-1106 (1996)

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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

Original Manuscript: January 25, 1995
Revised Manuscript: July 31, 1995
Published: March 1, 1996

Luc Arnold, "Uniform-load and actuator influence functions of a thin or thick annular mirror: application to active mirror support optimization," Appl. Opt. 35, 1095-1106 (1996)

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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).
  21. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).
  22. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85 (1981).
  23. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils: errata,” J. Opt. Soc. Am. 71, 1408 (1981).
  24. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils”, J. Opt. Soc. Am. A 1, 685 (1984).
  25. L. Noethe, “Use of minimum-energy modes for modal-active optics corrctions of thin meniscus mirrors”, J. Mod. Opt. 38, 1043–1066 (1991).
  26. J. Y. Wang, J. K. Markey, “Modal compensation of atmospheric turbulence phase distortion,” J. Opt. Soc. Am. 68, 78–87 (1978).
  27. B. Tatian, “Aberration balancing in rotationally symmetric lenses,” J. Opt. Soc. Am. 64, 1083–1091 (1974).
  28. W. H. Press, S. A. Teukolsky, W. T. Vettering, B. P. Flannery, Numerical Recipies in C (Cambridge U. Press, 1992).
  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.
  31. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).

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