Show how to construct a paraboloidal mirror of radius R and focal length f by stress polishing.

(a) Adopt a strategy of polishing the stressed mirror into a segment of a sphere with radius of curvature equal to that of the desired paraboloid at its center, r = 0. By comparing the shape of the desired paraboloid to that of the sphere, show that the required vertical displacement of the stressed mirror during polishing is

where r is the radial coordinate, and we only retain terms of leading order.(b) Hence use Eq. (11.63a) to show that a uniform force per unit area

where D is the flexural rigidity, must be applied to the bottom of the mirror. (Ignore the weight of the mirror.)

(c) Based on the results of part (b), show that if there are N equally spaced levers attached at the rim, the vertical force applied at each of the m must be

and the applied bending torque must be

(d) Show that the radial displacement inside the mirror is

where z is the vertical distance from the neutral surface, halfway through the mirror.(e) Hence evaluate the expansion Θ and the components of the shear tensor ∑, and show that the maximum stress in the mirror is

where h is the mirror thickness. Comment on the limitations of this technique for making a thick, “fast” (i.e., 2R/f large) mirror.

Equation 11.63

n(r) = p.4 64f3 (11.64h)