W

na distributed by hydraulic pads along the mirror contour (Fig. 8.10D). The distribution of Nl even lateral forces acting at angles 9n = (2n - 1)n/Nl along arches corresponding to AO = 2n/Nl = constant, determines the radial and tangential components as

On+AO/2

Jon-AO/2

These forces are implicitly generated by pads pushing normally to the edge surface.

Compared to other possible edge force distributions such as, for instance, with two concentrated forces at O = ±45° located each side of the lowest part of the edge, the two latter distributions provide the main advantage of a smooth flexure of the mirror by avoiding large m-values of spacial frequency modes in cos mO. The radial push force distribution, which corresponds to forces generated by a narrow mercury bag, only requires use of normal pushing forces.

8.6.2 Flexure of a Mirror Supported at its Lateral Edge

Elasticity analysis using Fourier expansions were developed early by Schwesinger, [70, 72, 75], who derived mirror surface flexures caused by various lateral support systems. In order to avoid solving the complex three-dimensional problem of the determination of the three displacement components u, v, w in (8.43), it was assumed that the radial and tangential displacements u, v, and axial displacement w - here denoted z for active optics reasons - at the middle surface of the mirror surface can be accurately expressed from Love-Kirchhoff hypotheses (cf. Sect. 1.13.6), i.e. reduce to u\z=0 = 0, v|z=o = 0, w = z(r, 0). (8.46)

In this approximation corresponding to the thin plate theory, the small axial displacements - hereafter called the flexure - can also be assumed as the same for all mirror points of coordinate r, 0, so the problem reduces to the determination of the axial flexure of the middle surface when the mirror axis is horizontal.

Using the radius variable p = r/a normalized with respect to the edge radius, a Fourier expansion of the z-flexure of the middle surface due to the lateral support can be represented by

where cm and Pm are coefficients and polynomials.

If the middle surface of the mirror is without curvature (e.g. a mirror with equi-concave faces), then all Pm = 0, which entails z = uz = 0 from the same above approximation that ur = ut = 0.

In addition to the force distribution applied to the contour, the two parameters entering into the determination of the flexure of the mirror are the curvature 1/ < R > of the middle surface and the thickness distribution t(p). These latter parameters can be characterized by dimensionless quantities a and k as a = a/< R >, t/t0 = 1 + Kp2. (8.48)

Let 1/R be the curvature of the optical surface, the radius of curvature of the middle surface is

<R> = ^t0 + R for a meniscus mirror, and k = 0 for a meniscus mirror.

Assuming hereafter that the mirror is without central hole, the bending and twisting moments Mx, My, Mxy are determined from the components FXkk, Fy,k of the perimeter force distribution and from the a and k parameters. For each mode of order m, the Fourier expansions for FXkk, Fy,k allow expressing the three equilibrium equations of the moments as functions of the stresses axx, oyy, oxy (see equations (1.165) in Sect. 1.13.8), and then provide the flexure.

For instance, considering the radial push force distribution fr <x 1 + cos 9 and a plano-concave mirror for which, from (8.48), k = aa/t0 = a2/2Rt0, Schwesinger  derived the rms flexure (cf. (8.24) in the form

0 0