The part f (x) = j2 - X2 shall be considered as the requested flexure function representing the optical shape to generate, whilst the additional constant a2 -which represents a constant retraction (or extension) along the x-axis - is a necessary elasticity condition to avoid a large or infinite thickness at any region of the cylinder. An appropriate setting of the free parameters a2 and C2 provides a mirror thickness geometry that can be readily fabricated (Fig. 10.10).

Parabolic and cone flexure: A flexure of complete quadratic form can be easily generated. Conserving the additional constant a2 for rejecting singular poles of the thickness outside the cylinder, we obtain the flexure

where 7 is a constant determining the "cone term" of the flexure. From (10.38), the dimensionless thickness distribution is

Fig. 10.10 Thickness distribution T = C2/(a2 + ji2 — x2) of three cylinders that generates a x2 flexure mode by uniform load only. Length parameters j = 0.45, 0.90, 1.35. Mirror aspect ratios L/a = 1.092 j
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