where Q are unknown constants that are determined from the boundaries. The terms r2 ln r and ln r allow treating cases with a ring force or a central force, and holed plates.

Starting from the stress-relations (1.124b), and similarly as for rectangular plates, we derive the three stress components that differ from zero, namely orr, att and azr. This allows determining the bending and twisting moments, per unit length, as f d2w v dw\ ( d2w 1 dw\ „

Mr = D I ~r~2 +--T > Mt = D v-y +-— , Mrt = 0. (1.181) dr2 r dr dr2 r dr

The shearing forces Qr, Qt and net shearing force Vr, Vt are

The total shearing force acting along the circle of radius r is 2nrQr. This force is in static equilibrium with the force resulting from the total load applied inside this area. For instance, with a uniform load applied all over the surface, 2nrQr + /0r2rcqrdr = 0, so that Qr = -qr/2 in this loading case. Hence, the shearing force Qr can always be determined from the loading configuration.

It is always advantageous to derive the flexure from the shearing force in (1.182). The integration of Poisson's equation is then directly operated from the integro-differential equation d / dm r drKrdr) = - DlQrdr (L183)

by substituting the expression of Qr as a function of the known load.

1.13.11 Circular Plates and Axisymmetric Loading Manifolds

• Sign convention: In all Chapters, the sign convention for the flexure w in the z-direction, sometimes denoted uz or z, is that with a positive flexure - which means towards z positive - when the curvature term w(r2) is positive, this term being considered as the first-order mode of the flexure. Hence the sign of an applied force or load must be conveniently chosen.

- If a force F or a load q is positive, then it acts towards z positive.

- If w(r2) = 0, we apply the convention to the next-order term w(r4).

- A positive moment Mr at outer edge generates a positive curvature.

In the thin plate theory of small deformations one uses this to set the origin of the flexure at the mid-plane vertex of the plate, however the flexure is the same at its external surfaces. We list hereafter the flexure for various loads and boundaries, the associated shearing force Qr = Vr, and the maximum flexure w{a} at the edge. The sign of the load is given for a flexure with positive curvature (term w(r2) > 0) or, if this term is null, for w(r4) > 0.

1. Free Edge and Uniform Bending Moment at Edge: If M > 0 is a constant bending moment at the edge and no other force acts on the plate, then Qr = 0, and

Ma2 r2 Ma2

2. Simply Supported Edge and Uniform Load: If q < 0 is a uniform load applied over all the surface, then Mr{a} = 0, Qr = -qr/2, and w = (r2-r2, wW = -t+vqt. a.i84b)

3. Built-in (or Clamped) Edge and Uniform Load: If q < 0 is a uniform load, then the slope at edge is dw/dr\r=a = 0, Qr = -qr/2, and w = ^ ( rl - 2] r2, w{a} = - qa4 , (1.184c)

4. Simply Supported Edge and Concentrated Force at Center: If F < 0 is a force applied at center, then Mr{a} = 0, Qr = -F/2nr, and

5. Built-in Edge and Concentrated Force at Center17 : If F < 0 is a force applied at center, then dw/dr\a = 0, Qr = —F/2nr, and

Fa2 r2 r2 Fa2

6. Free Edge and Opposite Central Force and Load: If F < 0 is a central force such as F + na2q = 0, then Mr {a} = 0, Qr = -(F/2n)(1/r - r/a2), and

7. Bent and Supported Edge with Uniform Load for r4 Flexure: If q > 0 is a load and by generating an edge moment M =(3 + v) qa2/16 > 0, then Qr = -qr/2, and

qa4 r4 qa4

8. Plate Bent by a Concentric Ring Force: When a ring force acts on a centered circle of radius b and the plate is freely supported or clamped at the edge, the determination of the flexure must be dissociated into an inner zone, r < b, and an outer zone b < r < a. This problem was first solved by Saint-Venant [138] (cf. [158] p. 64).

9. Plate Bent by an Off-Center Force Point: When a force point is applied offcenter of the plate, the determination of the flexure has been solved by Clebsch

17 The flexures of cases 1-5 were first derived by Poisson in his famous and detailed memoir [121] of 1828 where he used the uni-constant theory {E, v = 1/4} and thus created the thin plate theory of elasticity. See also the comments in Love's book [97], p. 489.

[34]. Starting from Poisson's equation in polar coordinates (r,0), one shows that the flexure is represented by Clebsch'spolynomials (cf. Chap. 7).

The volume forces generated by a gravity field act on all the elements of a body. For instance the length of a bar is not the same when placed vertically or horizontally on the ground. The flexure of a solid due to the gravity is sometimes called own weight flexure.

When a flat and horizontal plate is supported on the edge, its flexure under gravity can be easily derived. Denoting y the density of the plate, the sum of the volume forces per unit surface area of the plate over the thickness t is equivalent to the pressure tyg. In the sign convention, the gravity vector g is opposite to the z-axis and a uniform load q is positive towards z positive. Hence, the substitution q ^ t yg, g < 0, (1.185)

straightforwardly provides the flexure of a plate under gravity.

For instance, this substitution in (1.184b), (1.184c) or (1.184f) gives the flexure of a simply supported plate at the edge, of a built-in plate or of a plate suspended from its vertex, respectively.

1.13.13 Saint-Venant's Principle

The small deformation theory of circular plates allows us to simply express the component w(r) of the displacement vector u, v, w of the middle surface and any other point departing from this surface is displaced by the same amount w(r) at the same radius r. The large deformation theory of thin circular plates (cf. Chap. 2) considers that the radial strain err becomes a function of both u and w. The thick plate theory of small deformations takes into account the shear strains that lead to cross-sections over the plate thickness which are no longer orthogonal to the middle surface and become S-shaped. With these improvements of the basic thin plate theory, the difficulties involved for finding the mathematical expressions of the displacements do not generally allow us to obtain explicitly represented solutions.

In the general case of a solid or plate where both thickness and flexure are not small, the complexity is such as it is out of purpose to search for the functions satisfying a partial derivative equation set, this even if the boundary conditions are particularly simple.

In other respects from the practical point of view, it is experimentally illusory to consider that for all loading cases we could exactly apply given surface force distributions F over a given area 8A. Although a uniform load can be accurately distributed by a pressure difference whatever the flexure is, in most cases local deformations only arise due to the application of concentrated forces as generally

happens at the boundaries of the solid. Of course, these local deformations can be determined, for instance by using Hertz's contact formulas [74] (cf. Landau and Lifshitz [92] p. 42) or Dirac's function, but when the main purpose is to derive the whole displacements of the solid, it is clear that these local deformations do not substantially affect them.

These remarks led Saint-Venant to enounce a useful principle which introduced some flexibility in the practical application of the boundary conditions.18 We recall that a set of forces define a torsor which, at any given point, is globally represented by a resultant force and a resultant moment. An excellent statement of Saint-Venant's principle of equivalence has been given by Germain and Muller [69] as follows:

^ If one substitutes a first distribution of given surface forces F, acting on a part SAb of a boundary area, by a second one acting on the neighborhood and determining the same torsor whilst the other boundary conditions on the complementary parts of Ab relatively to A remain unchanged, then, in all regions of A sufficiently distant from AB the stress and strain components are practically unchanged.

The application of Saint-Venant's principle allows determining several quasi-equivalent loading configurations at the contour of a solid (Fig. 1.56).

In active optics methods, the application of Saint-Venant's principle allows finding external force configurations at the boundaries which minimize the local deformations of the optical surface near the clear aperture contours. We will often use it in the next chapters such as, for instance, with the monomode and multimode deformable mirrors in Chap. 7.

1.13.14 Computational Modeling and Finite Element Analysis

Computational modeling - sometimes called the "third branch of science" for bridging analytical theory and experimentation - is the ultimate method to accurately

18 Saint-Venant first enounced the equivalence principle in Sur la Torsion des Prismes [139] pp. 298-299.

solve any sort of equilibrium or time-dependent problems. Finite element analysis allows determining the elastic deformations of a solid in static equilibrium.

Developed for more than three decades, evolute software for finite element analysis are now plainly efficient to solve complex three-dimensional elasticity problems. Finite element analysis can be briefly summarized as follows. For each finite volume element, the three-dimensional equations of elasticity allow writing the continuity conditions from three equilibrium equations that use the six stresses aik associated to this element. The loads acting at the boundary of concerned volume elements determine the stresses of all elements. Navier's stress-strain relations (see (1.123b) in Sect. 1.13.4) allow us to derive the strains £ik for all finite elements, thus providing the component u(r,d, z), v(r, 0, z), w(r,0, z) (1.186)

of the displacement vector for each element (cf. (1.121b) in Sect. 1.13.3). Iteration algorithms allow repeating the solving process until no variation occurs in the displacement vectors, which thus corresponds to the static equilibrium. A convenient accuracy is reached when increasing the number of finite elements entails quasi-equivalent displacements.

The sphere is the natural shape obtained in the surfacing with abrasive grains of two rigid blanks of the same size that are brought into contact in a relative movement with three degrees of freedom. These movements are three rotations which reduce for a plane surface to a rotation and two translations. By progressively decreasing the size of the abrasive grains, this process provides extremely accurate spheres as was known by mankind in the "polished stone age" for the elaboration of hatchets and low reflection mirrors.

For astronomical optics, the finishing process generally uses square segments in a soft material like pitch - originally, a hardened pine resin - which are thermally sealed on the tool substrate. The spherical polishing within a diffraction limited criteria is naturally achieved by a rigid tool of the same diameter as the optical surface.

Let d be the diameter of the tool or optical surface. In a cylindrical frame z, r, 0, some appropriate rules are as follows: (i) the duration of a z-rotation 2n between the two surfaces must be at least 7 times greater than that of a full loop relative displacement in a z, r plane, (ii) the full displacement in the z, r plane must be Ar ~ d/3, (iii) the lateral off centering i of the two surfaces in contact must be varied such as i/d e [0, 1/7].

Useful information on grinding abrasives, polishing oxides, optical cements, polishing pitches, and cinematics of surfacing machines can be found in [5, 154].

It has been sometimes said that the most important tool for the aspherization of a surface is an efficient optical testing. This affirmation implicitly admits the principle of the conventional method by zonal retouches with conveniently small polishing tools to obtain an optical surface with the required peak-to-valley (ptv) or root mean square (rms) tolerance. However, the small size of these polishing tools makes it difficult to avoid generating extremely local footprints on the optical surface whose number increases when the tool size decreases. This effect of slope discontinuities, known as ripple errors or extremely high spatial frequency errors, provides a scattering light which may become difficult to measure even with strict diffraction limited tolerance criteria on encircled energy (cf. Sect. 1.11.2). The stress lap polishing is an alternative with controlled flexible tools to partly avoid the ripple errors, but is not fully satisfactory however.

Active optics methods directly applied by elastic deformation of the optical surface are of particular interest because the surface can be figured as a sphere by full-aperture grinding and polishing tools which, therefore, naturally provide the advantages of continuity, smoothness, and accuracy. Compared to the conventional method of generating aspherics, active optics allows avoiding the zonal defects of slope discontinuities due to inherent local polishing tools. Then, optical surfaces generated from active optics are free from "ripple errors" and "high spatial frequency errors."

Active optics methods allow us to generate an aspherical surface from spherical polishing, but also allow us to generate shape variations of the surface.

Optical surfaces can be obtained from "active optics" in the three following cases: (i) after spherical stress polishing when in an elastically relaxed state, (ii) during in situ stressing after a spherical polishing, or, (iii) by a combination of the latter two cases. The flexure may reach a 10 mm range or more, without time dependence.

Some optical systems require an "in situ active optics" control, such as a telescope mirror, a variable curvature mirror for field compensation in two-arm interferometers, etc. Generally, these systems use a low frequency bandpass control.

In contrast, "adaptive optics" is a high frequency bandpass control essentially concerned with wavefront corrections of the atmospheric seeing, thus cannot compensate for more than a few wavelengths range, i.e. 1 or 1.5 ym in the visible.

Active optics methods provide both axisymmetrical surfaces and non-axisymmetrical surfaces. Current research and developments in active optics may be subdivided into the following aspects:

• Large amplitude aspherization by stress figuring and/or by in situ stressing.

• In situ correction of shape and drift due to the optics orientation in field gravity.

• Available variable asphericity for various focii selected by mirror interchanging.

• Field compensation by variable curvature mirrors in telescope interferometers.

• Optics and diffraction gratings by replication techniques from active submasters.

• Diffractive corrections by photosensitive recording with active compensators.

• Mirror concepts for optics modal corrections with adaptive optics systems.

In 1965, the International Astronomical Union held a symposium in Tucson on the construction of large telescopes. On this occasion, the second aspect above for improving telescope imaging quality was discussed. As found in the proceedings, it was stated that "the primary mirror would be actively aligned with the relay optics" and it was also observed that: 19

[...] Active optics is a sophistication that astronomers haven't worried about to date, but I'm afraid when we consider very large optical systems the tolerances of alignments of the optical elements will require us to consider actively controlling collimation as well as the flgufe.

Aden B. Meinel [107]

This same year, active optics allowed the complete aspherization of a telecope corrector plate using the stress figuring method suggested in 1930 by Bernard Schmidt:

[...] The method easily yields zone-free plates. [...] Although the theory is elementary and the process is not difficult, the method appears to have been long neglected.

Edgar Everhart [54]

Various theoretical aspects and application fields of active optics are described throughout the next chapters. Active optics methods constitute the main subject of the book.

References

1. E. Abbe, in Schultze's Archiv für Mikroskopische Anatomie, IX, 413-468 (1873) 22, 55

2. G.B. Airy, Trans. Camb. Phil. Soc., 5, 283 (1835) 21,68

3. C.W. Allen, Allen's Atrophysical Qüantites, Fourth issue by A.N. Cox, Springer, 263 (2000) 88

4. J. Allington-Smith, R. Content, R. Haynes, I. Lewis, in Optical Telescopes of Today and Tomorrow, Spie Proc., 2871, 1284-1294 (1996) 82

19 Thanks to Marc Ferrari for this reference.

5. Amateur Telescope Making, A.G. Ingalls ed., Scientific American Inc. publ., Book one (1953) and two (1954) 128

6. J.M. Arnaudies, P. Delezoide, Nombres(2,3)-constructibles, Advances in Mathematics, 158

(2001), Constructions geometriques par intersection de coniques, Apmep bull. 446, 367-382 (2003), and www.apmep.asso.fr/BV446Som.html 1,4, 8

7. G. Avila, G. Rupprecht, J. Beckers, in Optical Telescopes of Today and Tomorrow, Spie Proc., 2871,1135 (1996) 53,88,89

9. W.W. Rouse Ball, Short Account of the History of Mathematics, Sterling Publ., London,

10. A. Baranne, F. Launay, Cassegrain: un celebre inconnu de l'astronomie instrumentale, J. Opt., 28, 158-172(1997) 17

11. A. Baranne, Un nouveau montage spectrographique, Comptes Rendus Acad. Sc., 260, 2383 (1965) 79,81

12. A. Baranne, White pupil story, in Very Large telescope and their Instrumentation, Eso Proc., Garching, Vol. II, 1195-1206 (1998) 79

13. D. Bernoulli, 26th letter to Euler (Oct. 1742), in Correspondance Mathématique et Physique by Fuss, t. 2, St Petersburg (1843). Also cf. Love (loc. cit.) 93

14. B. Di Biagio, E. Le Coarer, G.R. Lemaitre, in Instrumentation in Astronomy VII, Spie Proc., 1235, 422-427 (1990) 81

15. R.G. Bingham, in Very large Telescopes and their Instrumentation, Eso Proc., Garching, 2, 1157 (1988) 88

16. J. Bland-Hawthorn, in Tridimensional Optical Spectroscopic Methods in Astrophysics, I au Proc., G. Comte & M. Marcelin eds., Asp Conf. Ser., 71, 72-84 (1995) 82

17. M. Born, E. Wolf, Principles of Optics, Cambridge Univ. Press (1999) 7,21,22,27, 30, 44, 45,68,69, 70, 71

18. J. Boussinesq, J. Math., Paris, ser. 2, 16, 125-274 (1871), and ser. 3, 5, 329-344 (1879) 120

19. A. Bouwers, Achievements in Optics, Elsevier edt., New York (1948)

21. I.S. Bowen, A.H. Vaughan, The optical design of the 40-in. telescope and the Irenee DuPont telescope at Las Campanas, Appl. Opt., 12, 1430-1434 (1973) 87

22. G. Bruhat, Thermodynamique, Masson edt., issue 8, p. 74, (1968) 40

23. C.R. Burch, On aspheric anastigmatic systems, Proc. Phys. Soc., 55, 433-444 (1943) 24

24. J. Caplan, private communication, and Marseille Observatory Inventory No. IM13000003 in www.oamp.fr/patrimoine/museevirtuel-lunettes.html 8

25. J. Caplan, in Tridimensional Optical Spectroscopic Methods in Astrophysics, Iau Proc., G. Comte & M. Marcelin eds., Asp Conf. Ser., 71, 85-88 (1995) 82

26. C. Caratheodory, Geometrische Optik, Springer, Berlin (1937) 31

27. L. Cassegrain, Journal des Scavans, J.-B. Denis edt., 7, 71-74, April 25 issue (1672) 16

28. E.F.F. Chladni, Die Akustik, Leipzig (1802) 119

29. H. Chretien, Calcul des Combinaisons Optiques, 5th ed., Masson edt., Paris, 422 (1980) 12, 23,45,46, 55

30. H. Chretien, Le telescope de Newton et le telescope aplanetique, Rev. d'Optique, 1, 13-22 and 51-64 (1922) 22,58

31. J.A. Church, Refractor designs: Clairaut's forgotten legacy, Sky & Telescopes, 66(3), 259-261 (1983) 11,88

32. A.C. Clairaut, Memoires sur les moyens de perfectionner les lunettes d'approche, Mem. Acad. Roy. Sc., 380-437 (1756), 524-550 (1757) and 378-437 (1762). Due to the Seven Years' War, the publications of these memoirs were somewhat delayed; they appear in the 1761, 1762 and 1764 Memoirs of the ARS. 9,36

33. R. Clausius, Poggendorf Ann., 121, 1-44 (1864) 55

34. A.R.F. Clebsch, Theorie der Elasticitat fester Korper, Teubneredt., Leipzig (1862) 91,95,96,98, 119, 126

35. E. Le Coarer, P. Amram et al., Astron. Astrophys., 257, 289 (1992) 82

36. A.E. Conrady, Applied Optics and Optical Design, Oxford (1929), reissued by Dover Publ., New York (1957) 46

37. R. Content, in Optical Telescopes of Today and Tomorrow, Spie Proc., 2871, 1295-1307 (1997) 83,84

38. A. Cornu, Methode optique pour l'etude des deformations elastiques, Comptes Rendus Acad. Sc. Paris, vol. 69, 333-337 (1869) 122

39. C. Coulomb, Histoire de l'Académie for 1784, 229-269 (1787) 100

40. G. Courtes, Comptes Rendus Acad. Sc., 234, 506 (1952) 81

42. G. Courtes, An integral field spectrograph (IFS) for large telescopes, Proc. Iau Conf., C.M. Humphries ed., Reidel Publ. Co., 123-128 (1982) 82

43. G. Courtes, in Tridimensional Optical Spectroscopic Methods in Astrophysics, Iau Proc., G. Comte & M. Marcelin eds., Asp Conf. Ser., 71, 1-11 (1995) 82

44. A. Danjon, A. Couder, Lunettes et Telescopes (1933), (reissued: Blanchard edt., Paris, 1979) 5, 14

45. R. Descartes, La Geometrie LivreII and La Dioptrique in Discours de la Methode, Adam & Tannery edt., 389-441 (1637), reissue Vrin edt., Paris (1996) 7, 8, 52

47. K. Dohlen, A. Origne, D. Pouliquen, B.M. Swinyard, in UV, Optical andIR Space Telescopes and Instrumentation, Spie Proc., 4013, 119-128 (2000) 82

48. H. Draper, Smithsonian Contributions to Knowledge, 14(article 4), (1864), (reissued 1904) 21

49. J. Dyson, Unit magnification system without Seidel aberrations, J. Opt. Soc. Am., 49, 713 (1946) 63

50. H.W. Epps, J.P.R. Angel, E. Anderson, in Very Large Telescopes, their Instrumentation and Programs, Iau Proc., 79, 519 (1984) 86, 88

51. L. Euler, Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes in the Additamentum of De Curvis Elasticis, Lausanne (1744) 93

52. L. Euler, Sur la Force des Colonnes, Memoires Acad. Sc. Berlin, t. XIII, 252-281 (1758) 99

53. L. Euler, Sur la Force des Colonnes, Acta Acad. Petropolitanae, Pars prior edt., 121-193 (1778) 99

54. E. Everhart, Making corrector plates by Schmidt's vacuum method, Appl. Opt., 5(5), 713-715 (1966) 130

55. M. Faulde, R.N. Wilson, Astron. Astrophys., 26, 11 (1973) (1934) 86

56. C. Fehrenbach, Principes fondamentaux de classification stellarie, Annales d'Astrophysique, 10, 257 (1947) 79

57. C. Fehrenbach, R. Burnage, Comptes Rend. Acad. Sc. Paris, 281-B, 481-483 (1975) 79

58. C. Fehrenbach, Des Hommes, des Telescopes, des Etoiles, Cnrs edt., Isbn 2-222-04459-6 (1991) and 2nd issue with complements, Vuibert edt., Isbn 978-2711740383 (2007). Note: This book gives an historical account on the French development of astronomical telescopes and instrumentation during the first half of the 20th century. 79

59. L. Foucault, Essai d'un nouveau telescope en verre argente, C.R. Acad. Sc., 49, 85-87 (1859)

60. L. Foucault, C.R. Acad. Sc., 44, 339-342 (1857) 20

61. M. Francon, M. Cagnet, J.-C. Thrierr, Institut d'Optique de Paris, in Atlas of Optical Phenomena, Springer-Verlag edt. (1962) 72

62. A. Fresnel, Ann. Chim. et Phys, 1(2), 239 (1816) (cf. also Mem. Acad. Sc. Paris, Vol.5, 338-475 (1821-22)) 66

63. G. Galilei, Discorsi e Dimostrazioni Matematiche Intorno a due Nuove Scienze, Leiden, Elsevier edt. (1638) 91,92

64. S.C.B. Gascoigne, The Observatory, 85, 79 (1965) 86

65. S.C.B. Gascoigne, Recent advances in astronomical optics (p. 1419-1429), Appl. Opt., 12(4), 1419(1973) 86

66. K.F. Gauss, Dioptrische Untersuchungen, Göttingen, Memories from 1838 to 1841 (1841) 32

67. Y.P. Georgelin, G. Comte et al., in Tridimensional Optical Spectroscopic Methods in Astrophysics, Iau Proc., G. Comte & M. Marcelin eds., Asp Conf. Ser., 71, 300-307 (1995) 82

68. S. Germain, Recherches sur la Theorie des Surfaces Elastiques, Mme. V. Courcier edt., Paris (1821) 119

69. P. Germain, P. Muller, Introduction à la Mécanique des Milieux Continus, Masson edt., Paris, 2nd issue, 140 (1995) 127

70. Y. Le Grand, in Optique Physiologique, Revue d' Optique edt., Paris, 3rd issue, Vol. I, 68, 74 and 103 (1965) 76

71. SirW.R. Hamilton, Report Brit. Assoc., 3,360(1833) 21,45

72. H. Harting, Zur Theorie der Zweitheiligen Verkitteten Fermrohrobjective, Z. Instrum., 18, 357-380(1898) 12

73. C. Henry, P. Tannery, Oeuvres de Fermat - 5 Vol., Gauthier-Villars edt., Paris. Vol. 2, 354 (1891) 29

74. H. Hertz, Uber die Berührung fester elastischer Körper (1881), English translation in H. Hertz Miscellaneous Papers, Macmillan edt., New York, 146-183 (1896) 127

75. M. Herzberger, Modern Geometrical Optics, Interscience Publ., New York (1958) 45

76. P. Hickson, E.H. Richardson, A curvature-compensated corrector for drift-scan observations, Publ. Astron. Soc. Pac., 110, 1081-1086 (1998) 65

77. R. Hooke, De Potentia, or Of Spring Explaining the Power of Springing Bodies, London (1678) 92

78. E. Hugot, Performance evaluation of toroid mirrors generated by elasticity, Phase B study on Planet-Finder/Vlt, report Obs. Astron. Marseille Provence (2006) 51

79. D.M. Hunter, in Methods of Experimental Physics, L. Marton edt., Academic, New York, Vol. 12, Part 4, 193 (1974) 83

80. C. Huygens, Traite de la Lumiere, (completed in 1678), Leiden (1690) 65

81. P. Jacquinot, The luminosity of spectrometers with prisms, gratings or Fabry-Perot etalons, J. Opt. Soc. Am., 44, 761-765 (1954) 39, 82

82. E. Jahnke, F. Emde, Tables of Functions, Dover Publ., 4th issue, 149 (1945) 68

83. W. Kelvin (Lord, Thomsom), G.P. Tait, Treatise of Natural Philosophy, vol.1, part 2, 188 (1883) 120

84. A. Kerber, Ueber di chromatische Korrektur von doppelobjektiven, Central-Zeitung für Optik und Mechanik, t. 8, p. 145 (1887). Chretien also refers to Kerber's paper in Central Ztg. f. Opt. u. Mech., 1.10, p. 147 (1889) 12

85. H.C. King, The History of the Telescope, C. Griffin Co. edt., London (1955) 5, 9, 11, 15, 16,17,18, 19, 36

86. R. Kingslake, Lens Design Fundamentals, Academic Press, New York (1978) 46

87. G. Kirchhoff, Berl. Ber., 641 (1882), Ann. d Physik, 18(2), 663 (1883) 66

88. G.R. Kirchhoff, Uber das Gleichewicht und die Bewegung iener elastischen Scheibe, Journ. Crelle, 40, 51 (1850) 120

89. G.R. Kirchhoff, Vorlesungen über Mathematische Physik, Mechanik, 450 (1877) 120

90. J.L. Lagrange, Miscellanea Taurinensia, vol. 5 (1773) 100

91. G. Lame, Leçons sur la Theorie de l'Elasticité des Corps Solides, Ecole Polytechnique, Paris (1852) 101,102, 103

92. L.D. Landau, E.M. Lifshitz, Theory of Elasticity in Course of Theoretical Physics - Vol. 7, USSR Acad. of Sc., Butterworth & Heinemann eds, 3rd edition (1986) 91, 93,104, 118,127

93. G.R. Lemaitre, Reflective Schmidt anastigmat telescopes and pseudo-flat made by elasticity, J. Opt. Soc. Am., 66, No. 12, 1334-1340 (1976) 61

94. G.R. Lemaitre, private communication to P. Connes (1978) and G. Monnet (1978) 83, 84

95. G.R. Lemaitre, Equal curvature and equal constraint cantilevers: Extension of Euler (sic) and Clebsch formulas, Meccanica, 32, 459-503 (1997). In fact Euler did not do any research on this subject and only Clebsch must be credited with the first advances on this problem. 91, 96,97,98

96. P. Lena, D. Rouan, F. Lebrun, F. Mignard, D. Pelat, in L'Observational en Astrophysique, edt. Edp Science/Cnrs (2008). 75

97. A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, Dover Publ., 4th edition (1944), reissue (1980) 91,92,93,120,125

98. D. Lynden-Bell, Exact optics: a unification of optical telescopes, Mont. Not. R. Astron. Soc., 334, 787-796 (2002) 59

99. J.-P. Maillard, in Tridimensional Optical Spectroscopic Methods in Astrophysics, Iau Proc., G. Comte & M. Marcelin eds., Asp Conf. Ser., 71, 316-327 (1995) 82

100. D. Maksutow, New catadioptric menicus systems, Journ. Opt. Soc. Am., 34, 270 (1944)

101. D. Malacara, Optical Shop Testing, John Wiley & Sons edt., New York, 2nd edition (1992) 26

102. E. Malus, Optique Dioptrique, Journ. Ecole Polytechn., 7, 1-44, 84-129 (1808) 31

104. E. Mariotte, Traite du Mouvement des Eaux, Paris, 1886 92

105. J.C. Maxwell, Cambridge and Dublin Math. J., 8, 188 (1854) 30

106. A.B. Meinel, Astrophys. J., 118, 335-344 (1953) 86

107. A.B. Meinel, in The Construction of Large Telescopes, Proc. of Iau Symposium No. 27, D.L. Crawford ed., Section: optical design, 31 (1966) 130

108. M. Mersenne, L'Harmonie Universelle, Paris (1636) 14,62

109. G. Monnet, in Tridimensional Optical Spectroscopic Methods in Astrophysics, Iau Proc., G. Comte & M. Marcelin eds., Asp Conf. Ser., 71, 12-17 (1995) 81

110. O.F. Mossotti, Nuova theoria degli instrumenti ottici, Anal. Univ. Toscana, Pisa, 4, 38-165 (1853) 12

111. O.F. Mossotti, Nuova theoria degli instrumenti ottici, Anal. Univ. Toscana Pisa, 5, 5-95 (1858)

112. O.F. Mossotti, Nuova Theoria Strumenti Ottici, Casa Nistri edt., Pisa, 171-191 (1859) 12

113. C.L. Navier, Sur les Lois de l'Equilibre et du Mouvement des Corps Solides Elastiques, Mem. Acad. Sc. Paris, Vol. 7, 375-393 (1827). (The memoir was read in 1821. In this memoir, Navier refers to Mécanique Analytique, vol. 1, which seems to have been published by him in 1815 probably as a revised version of Lagrange's book) 106

114. C.L. Navier, Memoire sur la Flexion des Plans Elastiques, lithographic edition, Paris, 38 pages (1820) and issued in Bulletin de la Societe Philomatique, Paris (1823). This paper was presented to the French Academy in 1820. The original manuscript is in the library of the Ecole des Ponts et Chaussees. 123

115. I. Newton, Phil. Trans., 7,4006-4007 (1672) 16

116. K. Nienhuis, Thesis, University of Groningen (1948) 72, 73

117. R.J. Noll, Zernike polynomials and atmospheric turbulence, J. Opt. Soc. Am., 66, 207-211 (1976) 50

118. A. Offner, New concept in projection mask aligners, Opt. Eng., 14, 131 (1975) 63

119. M. Paul, Systemes correcteurs pour réflecteurs astronomiques, Rev. d'Optique 14(5), 169-202 (1935)

120. J.-C. Pecker, E. Schatzman, Astrophysique Generale, Masson edt., Paris, 121-122 (1959) 39

121. S.D. Poisson, Memoire sur l'Equilibre et le Mouvement des Corps Solides, Mem. Acad. Sc. Paris, vol. 8 (1829) 119,123,125

122. L.W. Ramsey, T.A. Sebring, C. Sneden, in Advanced Technology Telescopes V, Spie Proc., 2199,31 (1994) 87

123. Lord Rayleigh, Phil. Mag., 8(5), 403 (1879) 22,72,74

124. J.W. Rayleigh (Lord, Strutt), Proc. London Mathematical Society, No. 86, 20 (1873) 119

125. J.W. Rayleigh (Lord, Strutt), The Theory of Sound, London, vol. 1 (1877), vol. 2 (1878) 119

126. E.H. Richardson, The spectrographs of the Dominion Astrophysical Observatory, J. Royal Astron. Soc. Canada, 62, 313 (1968) 82

127. E.H. Richardson, Canadian J. Phys., 57-9, 1365-1369 (1979) 80

128. E.H. Richardson, Proc. Spie Conf. on Instrumentation in Astronomy IV, 331, 253 (1982) 80

129. E.H. Richardson, J.M. Fletcher, W.A. Grundman, in Very Large Telescopes, their Instrumentation and Programs, Iau Proc., 79, 469 (1984) 82

130. E.H. Richardson, C.F.H. Harmer, W.A. Grundmann, Mnras, 206, 47-54 (1984) 87

131. E.H. Richardson, in Encyclopedia ofAstronomy and Astrophysics (2003) 76

132. R. Riekher, Fernrohre und ihre Meister, Verlag Tecknik edt., Berlin (1957), (reissued 1990) 5,13

133. W.Ritz, Gessamelte Werke (1911) 119

134. R.J. Roark, W.C. Young, Formulas for Stress and Strain, McGraw-Hill Book Co., 5th issue (1975) 95

137. N.J. Rumsey, A compact three-reflection astronomical camera, in Optical Instruments and Techniques, Ico8 Meeting, London, Home Dickson edt., Oriel Press Newcastle, 514-520(1969) 26

138. A. Saint-Venant (Barre de), Flamant Theorie de l'Elasticite des Corps Solides de Clebsch, Dunod edt., Paris, 858-859 (1881). (This is a French translation of Clebsch's book including many important annotations and complements. This book is often referred to as "Clebsch Annoted Version") 91, 98, 125

139. A. Saint-Venant (Barre de), La Torsion des Prismes, Memoires des Savants Etrangers, Acad. Sc., Paris, vol. 14 (1855) 127

140. Schmidt, B., Mitteilungen der Hamburger Sternwarte, R. Schorr edt., 10 (1930) 60

141. D.J. Schroeder, Astronomical Optics, Academic Press edt. (1987) 85

142. Schwarzschild, K., Untersuchungen zur geometrischen Optik, I, II, III, Göttinger Abh, Neue Folge, Band IV, No. 1 (1905) 22,60

143. T.A. Sebring, J.A. Booth, J.M. Good, V.L. Krabbendam, F.B. Ray, in Advanced Technology Telescopes V, Spie Proc., 2199, 565 (1994) 87

144. L. von Seidel, Astronomische Nachrichten, 43, Nos 1027 p289, 1028 p305 and 1029 p321 (1856) 21,45

145. M. Serrurier, Structural structure of 200-inch telescope for Mount Palomar Observatory, Civil Engineering, 8, 524 (1938) 26

146. P.J. Smith, http://www.users.bigpond.com/pgifl/lndex.html 77

148. J.-M.E. Stephan, C.R. Acad. Sc., 78, 1008-1012 (1874) 21

149. C. Sterken, J. Manfroid, Astronomical Photometry, R. Boyd edt., Kluwer Acad. Publ. Dordrecht (1992) 39

150. J. Strong, Procedure in Experimental Physics, Prentice-Hall edt., Englewood Cliffs, N.J., 24th issue (1966) 25

151. D.-q. Su, Astron. Astrophys., 156, 381 (1986) 88

152. A. Szulc, Improved solution forthe cemented doublet, Appl. Opt., 35(19), 3548-3558 (1996) 10,13

153. E.W. Taylor, The inverting eyepiece and its evolution, J. Sci. Instrum., 22(3), 43 (1945) 77

154. J. Texereau, How to Make a Telescope, Willmann-Bell Inc., 2nd issue (1998) 128

155. S.P. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill edt. (1959) 91,113,115,120,123

156. S.P. Timoshenko, Theory of Elastic Stability, McGraw-Hill edt. (1961)

157. S.P. Timoshenko, Elements of Strength of Materials, Wadsworth Publ., 5th issue, Sect. 55 (1968) 107

158. S.P. Timoshenko, Theory of Elasticity, McGraw-Hill edt. (1970) 101,122, 125

159. S.P. Timoshenko, History of Strength of Materials, McGraw-Hill edt. (1983)

160. W. Tobin, in Leon Foucault, Cambridge Univ. Press (2002) 20

161. I. Todhunter, K. Pearson, A History of the Theory of Elasticity, Dover Publ., Vol. I and Vol. II, reissue (1960) 91,98

162. G.J. Toomer, DIOCLES - On Burning Mirrors, Sources in the History of Mathematics and the Physical Sciences 1, Springer-Verlag, New York (1976) 3, 7

164. T. Walraven, J.H. Walraven, in Auxiliary Instrumentation for Large Telescopes, Eso Proc., Garching, 175 (1972) 83

165. Y.-n. Wang, D.-q. Su, Astron. Astrophys., 232, 589 (1990) 88

166. W.T. Welford, A note on the skew invariant in optical systems, Optica Acta, 15, 621 (1968) 55

167. W.T. Welford, Aberrations of Optical Systems, Adam Hilger edt., 4th edition (2002) 29, 39, 45,46, 55, 57

168. W.B. Wetherell, in Applied Optics and Optical Engineering, R.R. Shannon & J.C. Wyant eds., Academic Press, London, Vol. VII, Chap. 6, 214 (1980) 70,74

169. R.V. Willstrop, D. Lynden-Bell, Exact optics - II: Exploration of designs on- and off-axis, Mont. Not. R. Astron. Soc., 342, 33-49 (2003) 59

170. R.N. Wilson, Reflecting Telescope Optics I, Springer edt. (1996) 5, 13,14,21, 24, 26, 36,37,39, 45, 46, 57

171. R.N. Wilson, Karl Schwarzschild and Telescope Optics, Karl Scharzschild Lecture to the German Astronomical Society, Bochum, 1993, published in Review of Modern Astronomy, 7, 1 (1994) 32

173. G.G. Wynne, Astrophys. Jour., 152(3), 675 (1968) 87

174. C.G. Wynne, The Observatory, 104, 140 (1984) 88

175. C.G. Wynne and S.P. Worswick, Mont. Not. R. Astron., 620, 657 (1986) 88

177. T. Young, A Course of Lectures on Natural Philosophy and the Mechanical Arts, London, Lecture XIII (1807) 92

178. F. Zernike, Diffraction theory of the knife-edge test and its improved form, the phase-contrast method, Mont. Not. R. Astron. Soc., 94 377-384 (1934) 24,44

180. F. Zernike, B.R.A. Nijboer, Contribution to La Theorie des Images Optiques, Revue d'Optique, Paris (1949) 72

Chapter 2

Dioptrics and Elasticity - Variable Curvature Mirrors (VCMs)

The elastic deformation modes corresponding to the first-order modes of the optical matrix characterizing the wavefront shape are the curvature (Cv 1) and tilt (Tilt 1). These are the two fundamental modes involved in Gaussian optics. Because a tilt is easily obtained by a global rotation of a rigid substrate, this chapter only reduces to mirrors generating a Cv 1 mode. Such variable curvature mirrors (VCM) are also sometimes called zoom mirrors.

Let us denote z(r) - instead of w(r), because z is usual for representing an optical surface - the optical figure achieved by flexure of a circular plate which is flat at rest. In the thin plate theory, a curvature mode Cv 1 is represented by where R is the radius of curvature of the bent optical surface.

Two classes of substrate thicknesses provide the curvature mode as investigated hereafter: Constant thickness distribution (CTD) and Variable thickness distribution (VTD).

2.1 Thin Circular Plates and Small Deformation Theory 2.1.1 Plates of Constant Thickness Distribution - CTD

Let us consider a possibly holed plane circular plate with a constant thickness t and rigidity D = Et3/[12(1 — v2)] = constant, where E and v are the Young modulus and Poisson's ratio, respectively. If an external pair of concentric circle forces or a bending moment are applied to the perimeter region without surface load (q = 0), then bilaplacian Poisson's equation representing the flexure z of the plate reduces to

whose general solution z = B20 + C20 lnr + D20r2 + E20r2lnr

Was this article helpful?

## Post a comment