Z2 p4 2C21 p2 8 C22 p2 lnp 4 C23 lnp 4 C24

where the numbers in front of the five coefficients avoid use of fractions in further calculations.

The boundary conditions at the outer circle (C) of the sub-element - where p = 1 and the bending moment Mr contributes to the flexures - is a null radial shearing force, Qr = 0, and a slope perpendicular to the gravity vector g, thus d2Z/dp2 = 0. From (1.182), the radial shearing force writes

dr a2 dp where A2 is a Laplacian operator with respect to p. The continuity conditions at p = b/a are equal sags, equal slopes and equal radial bending moments Mr. From the expression (1.181) of Mr, the latter condition may be readily substituted by equal radial curvatures d2Z/dp2. Hence, the five conditions are the following, dZl dt,2 d2Zl d2Z2

for p = a/b : Zi = Z2, = , -j-y = TT, dp dp dp2 dp2

from which one notes that they all are independent of Poisson is ratio. After solving this equation set, we find b2 b2 b2 Ci,i = -3 + 2-, -2ln-2, C2,i = 1 + 2-», a2 a2 a2

b2 b2 b2 b2 C22 = -1, C2,3 = -2-~ , C2,4 = -2-, + • (8.15)

a2 a2 a2 a2

After substitution into functions Z1, Z2, one beholds that the total sag of the flexure is strongly dependent on the pad radius ratio b/a (Fig. 8.6).

Coefficients C1,1 and C2,i allow determining the maximum sag as well as the maximum slope of the flexure. For instance, from the flexures in Fig. 8.6, we see that the maximum sag is

A Zmax = Z1{b/a} - Z1{0} + Z2{1} Z2{b/a}, b/a e [0,0-549]-

Since we have set Z1{0} = 0 and Z1{b/a} = Z2{b/a}, the second equation in (8.12) entails

After substitution of the coefficients, the dimensionless maximum flexure sag is b2 b2 b Fig. 8.6 Flexure dependence of a mirror sub-element as a function of the pad radius ratio b/a in a constant spaced-pad distribution of step 2a with three-fold symmetry. For b/a = 0, the support pad reduces to a single point. The locus of some inflexion points is shown by the dotted line p = r/a

Fig. 8.6 Flexure dependence of a mirror sub-element as a function of the pad radius ratio b/a in a constant spaced-pad distribution of step 2a with three-fold symmetry. For b/a = 0, the support pad reduces to a single point. The locus of some inflexion points is shown by the dotted line

When 0.549 < b/a < 0.70, the flexure Zi 2 shows a second inflexion point, so that (8.16a) does not apply. We obtain Z2(1) = Zi(0) = 0 for b/a ~ 0.6184, where a ripple arises at p ~ 0.56. The corresponding smallest possible maximum flexure sag, which occurs here, is b

To summarize, from equations (8.16) and further calculations, the dimension-less maximum flexure sag as a function of the pad radius ratio is as follows (Fig. 8.7), b/a 0 0.10 0.20 0.30 0.40 0.50 0.549 0.60 0.6184,

4Zmax 3 2.775 2.324 1.773 1.187 0.613 0.348 0.145 0.0978.

Therefore, the analysis from thin plate theory provides the following general conclusions.

1 ^ If the minimization of the axial flexure of a mirror under gravity must be achieved by use of a minimum pad number Np, of step 2a, then the pad radius ratio must be b/a ~ 0.62 whatever Poisson's ratio.

2 ^ For a narrow support pad (b/a ~ 0) the flexure curve of a mirror sub-element shows that the absolute value of the curvature at p = 0 is 7.75 times as large as that of the radial curvature at p = 1.

The local flexure is much more marked near the supporting pad when it reduces to a point force. Thus, axial support systems that are, for instance, with a pad radius ratio b/a ~ 0.05 require a larger pad number Np than with a larger b/a-value. Current values for various large telescope mirrors that have been built are varying in the range b/a e [0.05, 0.6]. For instance, the 4 m mirror of theMayall Kpno telescope is supported by ring pads that are distributed along two concentric circles; in a radial direction of this mirror, the pad radius ratio is b/a ~ 0.6 (Fig. 8.8). Fig. 8.7 Dimensionless maximum flexure sag A£max as a function of the pad radius ratio b/a. The dotted line corresponds to the region 0.549 < b/a < 0.70 where the mirror sub-element shows a flexure with two inflexion zones Fig. 8.8 Mirror support system of the Kpno 4-m telescope (courtesy Kitt Peak National Observatory)

8.5.3 Density Criterion for Pad Distribution - Couder's Law

From the latter study of an infinitely large mirror supported by a uniform pad density, we now have all the elements for determining the dimensioned maximum flexure sag Azmax of a mirror sub-element in its complete region r e [0, a}. In addition, we also know the dimensionless maximum sag A £max(b/a) as a function of the pad radius ratio b/a. From the flexure (8.11) of a sub-element, the maximum flexure sag writes

where AZmax is given by the distribution (8.16c) for b/a e [0, 0.62]. From equations (8.9), the ratio q/D is represented by q

After substitution, the maximum flexure sag for the mirror sub-element of circular contour (C) is

1JEt2

In a 3-fold symmetry, the slab filling is with hexagon contour sub-elements. Compared to the radius a of the circle (C), the radial distance of a hexagon corner is 2a/v/3 - 1.154a. On the other hand, the contour line (H) of the hexagon does remain in a plane, so the flexure is somewhat larger than that at circle (C). For continuity and minimum strain energy reasons, the maximum flexure sag AZmax at a corner of the hexagon sub-element - i.e. for all the mirror - can be accurately related to that of the circular sub-element by use of the scaling factor 2/%/3. Hence, the maximum flexure sag for an infinitely large mirror generated by hexagon slabs is

AZm a

= V3 AZmax

8 Et2

• Extended Couder's law: Now for a large mirror of finite diameter d, from (8.8), a2 = nd2/8\f3Np so that a2 can be substituted into (8.20). If we restrain the pad radius ratio such as b/a < 0.549, then equation (8.16a) for AZmax(b/a) applies. Therefore, the substitutions entail the following general result.

^ For a large telescope mirror, of finite diameter d and thickness t, supported by Np axial ring pads of radius b that are distributed in a 3-fold symmetry of step 2a, the maximum flexure sag is

Vgd4

Although in a somewhat different formulation, a first similar result was enounced by Couder which would correspond here to point-pads i.e. b = 0; this is known as the d4/t2 flexure scaling law or Couder's law .3 Equation set (8.21) includes the extension for current values of the pad radius ratio. A somewhat larger extension may be used from the distribution (8.16c).

This result readily allows us to introduce an optics tolerance criterion. After reflection of a wavefront at the mirror, let wptv be the maximum wavefront deflection error introduced by the flexure of the mirror, so we have wptv = 24Zmax. Now if we state that an optics tolerance error is defined by a wavefront ptv criterion such as, say, wptv < A/5, then we obtain the inequality, wptv = 24Zmax < A/5, (8.22)

where we assume that the flexure of the mirror can be potentially corrected by an adaptive optics system. Hence, from (8.21), a wavefront tolerance criterion for the choice of the mirror material (E, v, ¡i), the mirror thickness (t) and its pad support system (b/a,Np), is given by the inequalities n2(1 - v2)

Hgd4 X

Current designs of large telescope mirrors and associated support systems satisfy these inequalities for the shortest wavelength they are optimized for. If one prefers using the pad density per unit surface area np instead of the total number of pads, then, from (8.8), Np = nnpd2/4 must be substituted in (8.23).

Let us consider, for instance, the case of the primary mirror meniscuses of the Eso-Vlt in Zerodur from Schott. The axial support pads are tripods that have a

3 From the thin plate theory, Andre Couder analytically derived the flexure of a constant thickness plane plate supported by a continuous concentric ring. He applied the superposition principle of the flexure for passing from one to several support rings and minimized the resulting flexure sag from a best fit balance of the radius of these rings and of the reacting forces per unit length acting on them.

relatively small radius b with respect to a (Fig. 8.3-Left); thus we can accurately assume that their effect is equivalent to a ring pad of same ratio b/a. The quantities concerned in (8.22) are as follows,

E = 90.2 x 109Pa, v = 0.243, ¡g = 24.82 x 103Pa/m, d = 8.2m, t = 0.175m, b/a = 0.2, Np = Na/3 = 150, where, from (8.16a), 4£max(0.2) = 2.324. After substitution in (8.23), we find for the wavefront deflection wptv = 0.439¡m. Thus, at the wavelength X = 0.5¡m, the wavefront error due to the flexure of the mirror under gravity is wptv = X/5.7.

• Wavefront variance and rms criterion: The wavefront deflection w caused by the flexure of a mirror under gravity may be minimized by use of a wavefront rms criterion from which the best wavefront deflection is denoted wrms. The variance wjm of the wavefront deflection w is defined by wL = -A/ JA(w+Am)2dA, (8.24)

where the integration is taken over the surface area A of the wavefront, and A00 is the unknown piston term. If a small defocussing is tolerated, then the determination of the variance is obtained from the substitution A00 ^ A00 + A20r2.

Note that the rms values of the slopes (Vw)rms are also relevant for the image degradation. In this case, the variance (Vw)2ms should be calculated by simply minimizing the resulting image size.

• Pad density, mirror thickness, and ptv criterion: Whatever the diameter of a large telescope mirror, the pad density np per unit surface area, the thickness t of the mirror and the ring pad radius ratio b/a are the three fundamental parameters. Once the ratio b/a is fixed - which determines either a ring pad or a point pad -, from (8.8), the substitution of Np = nnpd2/4 into (8.23) provides a tolerance wavefront ptv criterion for the pad density np relative to the mirror thickness t. For a wavefront tolerance, say wptv < X/5, the general result in dimension [L~2] is n2p t2

p - 16^/3 ring pad radius ratio condition 0 < b/a < 0.549.

Given a material (E, v, x), a pad radius ratio (b/a) and an optical tolerance (above X/5 on the wavefront), the latter relations are of more convenient use in practice since they are only dependent on the thickness of the mirror.

• Pad density and mirror thickness geometry: We have seen that the above criterion is derived from the local flexure of a hexagon slab element in a 3-fold symmetry of an infinitely large mirror. On the other hand, large telescope mirrors may have either a constant thickness t (meniscus) or a flat rear side (plano-concave). For these various thickness geometries, the local property of criterion (8.25) must be interpreted as follows.

1 ^ Whatever the thickness distribution t(r) of a large telescope mirror, the tolerance criterion for its flexure under gravity determines the pad support density np per unit surface area from the law "npt = constant."

2 ^ Hence, for a mirror geometry with a flat rear side, if the thickness distribution is represented by t = to(1 + Kr2), where k > 0, then the pad density is np(r) <x [t0(1 + Kr2)]-1.

A general theoretical conclusion is that a concave mirror with flat rear side requires a smaller pad density at the edge than at the center.

8.5.4 Other Axial Flexure Features

The calculation for minimizing the flexure under gravity of mirrors having finite size originated with the case of constant thickness plates simply supported by a concentric and continuous ring. The first theoretical analysis for this problem was investigated by Couder  in a classic study based on the thin plate theory. Couder also carried out interferometric experiments with glass plates allowing him to determine the maximum elastic deflection when the plate is supported by three points in a 3-fold symmetry.4 For two concentric and continuous support rings, the optimizing parameters are the radii of the rings and the reaction distribution per unit length along each ring. In a first stage, separated flexures z(r) are calculated for each support ring of radius ri by use of continuity conditions at the ring and free edge boundary conditions. Then the superposition principle allows summing the flexures zi such that the sum of each reaction R along the corresponding ring is equal to the mirror weight. Variation of parameters r and R allows determining the minimum flexure. For mirrors with flat rear side, further flexure and reaction distribution corrections allow taking into account the thickness variation, thus providing the final flexure. Later investigations introduced the free inner edge boundary conditions for the case of central holed mirrors; for mirror apertures in the 3-4 m class, Schwesinger  optimized a 2-ring support and Lemaitre  a 3-ring support. However for thick mirrors, say with aspect ratios t/d > 1/8, other considerations can also be taken into account as follows.

• Shear stresses and thick plate theory: From Love-Kirchhoff hypotheses of the thin plate theory (cf. Sect. 1.13.6), and in the axisymmetric bending of circular plates, the stress component ozz arising along normal lines to the mid-surface of the plate is not taken into account.

For instance, in a thick cantilever beam loaded at its free end, Saint Venant showed that a normal plane to the beam section becomes an S-shaped surface. This means that the stresses axx acting through the thickness of the beam are not linear but vary accordingly to an odd cubic law from one face to the other. Around 1910 or

4 At that time it was unclear whether or not a glass material may show a slight viscosity. In a dedicated interferometric experiment, where a glass plate was bent by weights for more than a year, Couder  showed that a variation of the fringe pattern could not be detected.

somewhat earlier, taking into account the shear deformations in circular and elliptical plates, Augustus Love elaborated the shear theory usually referred to as thick plate theory.5

When a uniform load is applied to a face of a plain constant thickness plate, Love  derived the bending and shear flexure of the middle surface for a circular plate simply supported at the edge, and for circular or elliptic plates built-in at the edge. In these cases, he also derived the complete equation set expressing the radial and axial displacements for surfaces located at distance z € [-t/2, t/2] from the middle surface. Woinowsky-Krieger  showed that in the simple case of a plain built-in plate with an aspect ratio t/d = 1/10, the shear flexure contributes ~17% of the total flexure. Comparisons of the flexures between thin and thick plate theories are also given in Timoshenko and Woinowsky-Krieger . As shown hereafter, this effect is less pronounced for a simply supported plate.

For instance, if we denote z the bending flexure of a constant thickness plate simply supported at its edge r = a, we have seen in Sect. 1.13.10 [case 2, (1.184b)], that Poisson's equation V2V2z = q/D of the thin plate theory leads to the flexure qa4 f r2 3 + v\r2

where q = ¡igt is negative. Referring to Love  or Woinowsky-Krieger in , the shear flexure of the middle surface of a thick plate simply supported at the edge is zs = -^^ £ (8.27a)

but, from comparison with the finite element analysis result, it is more accurate to adopt the expression qa4 1 - v t2r2

From (8.27c), a more exact expression for the flexure including the shear component is qa4 ( r2 3 + v 1 - v t2 \ r2

5 Analytical investigations of the shear deformations of beams were originated by Saint Venant  in the case of a cantilever beam. He also pointed out that the shear deformations can be considered as additive components to those of the bending deformations. Further development of the shear theory was briefly generalized to Michell , in 1900, who obtained the equilibrium equations including the shear stress components. This led to the elaboration of the thick plate theory by Love .

so the maximum flexure sag, obtained for r = a, is

The two quantities inside the parentheses allow evaluation of the effect of the shear component relative to the bending component of the flexure. At the edge, where this effect is maximum, this can be deduced from the ratio

For instance, with a Poisson's ratio v = 1/5 and aspect ratios t/d = t/2a = 1/12,1/8,1/6, we obtain for the shear effect at the middle surface zs/z(a) = 1.7%, 3.8%, 6.8% respectively (Fig. 8.9).

The above flexure component zs of the shear deformation only applies to the middle surface of the plate. The contribution of the shear deformation in the vertical direction of the plate is not a constant function over the thickness. The shear strains entail that the volume elements where the supporting forces act are much more deformed than those of the middle surface whilst the elements near the opposite outer surface of the plate are less deformed. In the local regions where the support acts, the shear flexure shows a slope variation somewhat similar to that of a footprint.

The shear effects are implicitly taken into account in three-dimensional finite element analysis.

• Meniscus mirrors and shallow shell theory: An important advance in elasticity theory was realized by Eric Reissner [64, 65] who introduced the so-called shallow shell theory. In this theory, the tensions and compressions arising in the two tangential directions of the middle surface of a meniscus plate are taken into account. Fig. 8.9 Normalized flexures Z = [Z(a) — Z(r)\/(qa4/64D) of the middle surface of constant thickness plates simply and continuously supported at the edge r = a. Aspect ratios t/2a = 1/12, 1/8, 1/6, 1/4. The flexures represented by full lines include the shear component in t2r2/a4. The dotted line is from the thin plate theory

Fig. 8.9 Normalized flexures Z = [Z(a) — Z(r)\/(qa4/64D) of the middle surface of constant thickness plates simply and continuously supported at the edge r = a. Aspect ratios t/2a = 1/12, 1/8, 1/6, 1/4. The flexures represented by full lines include the shear component in t2r2/a4. The dotted line is from the thin plate theory

Including this shell effect, Reissner derived a complete equation set allowing the determination of the bending and shear deformations. For astronomical mirror menis-cuses faster than, say, f/2.5, and in the axisymmetric case, this equation set allows introducing continuity conditions at the ring support radius; however solving such problems requires the complex use of four Kelvin functions (cf. Sect. 8.6.3).

From the shallow shell theory, Selke  derived some flexures - although not optimized - of a plain mirror meniscus with 1-ring and 2-ring supports. For an aspect ratio t/d = 1/12 and a single ring support, he showed that the shear component of the flexure does not exceed ~6%. Schwesinger  commented on the non-linear stress distribution over the thickness of a plate.

The thin shallow shell theory was also used by Arnold  to derive the flexure of a meniscus mirror from a discrete support pad distribution operating with active optics correction modes. This required expressing the load function q(r, 9) in a Fourier series form and solving both the stress function F and normal flexure z from the following equations (cf. (6.6) and (6.8) in Chap. 6),

where 1/ <R > is the mean curvature of the mirror meniscus. Several explicit forms of the load function q(r, 9) are given hereafter.

• Variable thickness mirrors: For large mirrors which are not meniscuses, such as mirrors with a flat rear side, Poisson's equation does not apply since the flexural rigidity D = constant. A classic process for determining the flexure z under gravity is to distribute the mirror thickness variation into N constant thickness segments. For instance, the thin plate theory leads to solving the equation set

V2V2zn(r, 9) = q/Dn, r e K rn+x], V n e [1, N], (8.33)

where the flexural rigidities Dn are constants. Four continuity conditions at each segment junction and two boundary conditions at the center and at the edge must be solved.

Wan et al.  make the assumption that the shear component of the flexure can be related to that of the bending flexure by use of zS ^ 12V2z. Although giving a demonstration, this assumption is not clearly proved; from (8.26) and one of the equations in (8.27), it appears to be not verified for axisymmetric flexures. These authors derived the Clebsch polynomials expressing the total flexure z + zS of a large plano-concave mirror supported by discrete point forces by use of the two-dimensional Dirac function (see hereafter).

• Infinitely small support pad areas: Schwesinger  proposed a method to extend the flexure analysis of a mirror under gravity for k discrete point pads distributed along a concentric ring in a k-fold symmetry. This method was subsequently investigated by Nelson et al.  and Arnold [3, 4] by use of Dirac functions.

Let W and a be the total weight (here negative because of the negative acceleration) and outer radius of the mirror. A two-dimensional Dirac 5 function must satisfy

which allows expressing the load q per unit area as

W -k, fi q(r, 9) = — + X f 8(r - bj) 8(9 - 9j), (8.35)

na j=0bj where the support system has k-fold symmetry and each support point j is characterized by its force fj located on a circle of radius r = bj at the azimuth angle 9j = 9o + 2jn/k. Replacing the Dirac function 8(9 - 9j) by its complex Fourier series leads to

q(r, 9) = —2 + X 2b8(r - bj) X exp [im(9 - 9j)]. (8.36)

The condition of the static equilibrium for the forces and moments are kk

From the thin plate theory, the flexure is expressed by z = X Rnm(r) cos [km(9 - eo)], (8.38)

where the Clebsch solutions Rnm(r) are polynomial forms expressed by the set (7.7) in Sect. 7.2. However, the Dirac functions restrain the problem to infinitely small support points, which is a quite pessimistic option.

• Finite support pad areas: Monolithic mirrors of presently large telescopes are often designed in a meniscus shape with aspect ratios t/d < 1/20. Given the small value of such ratios, the shear flexure from the support pad is fully negligible. Although the f-ratios of these primary mirrors often reach values such as f/1.8 or f/1.6 that would require use of Reissner's shallow shell theory, the low level of stress involved in their flexure under gravity renders it unnecessary. It is also reasonable to assume that the maximum flexure sag Azmax for generating any active optics correction modes znm ^ rn cos m9 does not give rise to significant stresses in the "in plane" middle surface of the meniscus. Hence, for most of large telescope mirror menis-cuses, the basic thin plate theory can be assumed as accurate enough for the determination of the mirror thickness and its passive and active optics support system.

In establishing a support pad density criterion for an infinitely large mirror (Sect. 8.5.3), we have seen from the thin plate theory that when a mirror is supported by ring pads, the pad radius ratio is a useful free parameter [cf. criterion (8.25)].

Also using the thin plate theory, another approach by Arnold  consists of determining the flexure under gravity and the influence function from plain actuated pads having a finite acting area. Compared to infinitely small pads represented by Dirac functions, pads with finite surface area spread the load under the mirror and smooth the optical surface. For mathematical convenience, and without changing significantly the final result, the pads considered by Arnold are almost squares of dimension dp and analytically defined by the product of two top-hat n functions as n

where bj and Oj = O0 + 2 jn/k are the polar coordinates of the k pads in the mirror frame r,9, z, and n[x] = 1 for x < 0.5, n[x] =0 for x > 0.5.

For a mirror of weight W with a hole of radius r = c, the general expression of the load q can be written as q(r, 9)

0 0