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Depending on the mirror length and on the maximum value of the grazing angle, arctan (b/£), we shall consider hereafter two alternatives for the aspheriza-tion by stress figuring and restrain ourself to a third-order expansion of the mirror shape.

• Stress figuring with straight generatrix segments: Let us assume that the stress figuring provides both the mirror curvature 1/R and the biquadratic term in (10.51). Hence, the inner surface of the mirror is cylindrical when at rest. During the surfacing, a positive load q is applied to its outer surface from the edge x = —L/2 to the other at L/2 and the surfacing tool is constituted of straight segments that are generatrices of the cylinder. For glass or vitroceram materials, we have seen that the easiest case is when the boundary conditions are satisfied with free edges, so the loading system - for instance by air pressure - must avoid any radial force or bending moment at the cylinder ends.

In the determination of the flexure w with these boundaries, we must set the sag origin at the edges and also introduce a constant a2 to avoid any pole singularity in the thickness distribution. This latter constant provides the same free retraction all along the cylinder. From (10.51) and with these conditions, the flexure w and its dimensionless associate W may be represented by

W s CEw = a2 + 2R (i2 —x2) + 8R2 (i4 — x4), (l°.-52a)

and where C is a constant.

From the inverse proportional law TW = constant, the dimensionless thickness is

8bR2

where the constant a2 is given a value so the expression into the brackets appropriately differs from zero. Introducing the distance 21 between the conjugate foci, from (10.50a), the thickness t/a writes

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