## D

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.

• 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).

### 1.13.12 Deformation of a Plate in a Gravity Field

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

Fig. 1.56 Saint-Venant's principle of equivalence: example of two quasi-equivalent load configurations applied at the boundary of an axisymmetric mirror

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.

### 1.14 Active Optics 1.14.1 Spherical Polishing

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].

### 1.14.2 Optical Surfaces Free from Ripple Errors

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.

### 1.14.3 Active Optics and Time-Dependence Control

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.

### 1.14.4 Various Aspect of Active Optics

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.

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.

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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

0 0