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Since a is the radius of the mid-thickness surface of the cylinder at its center (X = 0), the corresponding optical radius of the mirror is b = a -1 (0)/2. By an appropriate set up of the free parameters T(0) and a2, the thickness geometry of the mirror can be readily fabricated either in a linear metal alloy, i.e. satisfying Hooke's law, or in glass, or in vitroceram (Fig. 10.12-Left).

Stress figuring with circle generatrix segments: Assume now that the stress figuring provides the aspherization by minimizing the quantity of material to remove, thus using circle generatrix segments. For mean graze angles up to 3°, then l/b > 19. Considering the osculating circle defined from the x2 coefficient in (10.51), the ratio of the x4-term of the osculating circle over that of the ellipse is b/R. From (10.50b), this ratio is ~ b2/l2 ~ 1/361 which is small. The x4 coefficient of the circle figuring segment will be close to that of the above osculating circle, and then is similarly small.

Fig. 10.12 Thickness distribution T(x) for the active optics stigmatism of a tubular image transport M = -1. Cylinder length L = 2a. Grazing angle arctan(b/l) = 2°. (Left) Stress figuring by uniform load only and straight generatrix segments. (Right) Stress figuring by uniform load only and circle generatrix segments

Fig. 10.12 Thickness distribution T(x) for the active optics stigmatism of a tubular image transport M = -1. Cylinder length L = 2a. Grazing angle arctan(b/l) = 2°. (Left) Stress figuring by uniform load only and straight generatrix segments. (Right) Stress figuring by uniform load only and circle generatrix segments

Thus, we shall assume hereafter that the x4-term of the optics is totally generated by the flexure. Minimizing the volume to remove with the circle tool, we obtain for the dimensionless flexure5

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