The design principles and special advantages of monolithic adaptive metal mirrors have been described in an earlier paper 1 • The present paper provides analysis for understanding the response of such mirrors to the unintended torques they receive from the flexural hinges which connect them to the bending system. The analysis includes mirrors both with and without water-cooling channels. The torsional rigidities of the usual types of flexural hinge and of the most common mirrors are calculated thus allowing the hinge-induced distortions of any mirror surface to be estimated. Two strategies for reducing such errors are proposed. One involves the design of sufficiently flexible hinges and the other the elimination of the hinge rotations (and therefore their torques) altogether by means of a new design principle. Some analysis of the latter scheme is provided including a prescription for choosing suitable design parameters.
KEYWORDSGrazing-incidence mirrors, adaptive optics, flexural hinges, x-ray focusing, high-power mirrors 1.0 INTRODUCTION Monolithic adaptive mirrors for grazing incidence applications can be cut from a single block of metal using a wire, electric discharge machining (EDM) system. The cut shape is designed to act as a single flexural mechanism comprising both the mirror and its bending device. This type of all-metal component is particularly well-suited for grazing-incidence x-ray mirrors for synchrotron radiation beam lines and is compatible with intensive water-cooling arrangements. In an earlier paper 1 we have described the principles of monolithic design and its advantages for mirror systems as well as the theory that allows one to achieve prescribed mirror shapes by means of a controlled variation of the mirror thickness with position. The ideas in reference 1 have already been applied successfully to a number of mirrors manufactured commercially by Rockwell Power Systems 2 and Photon Sciences Intemational 3 . The above paper also provides a number of references describing earlier work on adaptive x-ray mirrors. The closest antecedent to the monolithic schemes considered here is the mirror described by Ice et al 4 which is a water-cooled metal mirror bent into a circular shape by a single, point load. In the present paper we continue the treatment of the theory of adaptive monolithic mirrors paying special attention to the effects of the three hinges which together provide the force system which deforms the mirror.In reference 1 we discussed the design shown in Fig. 1 which would, for example, provide a circular reflecting surface if the mirror thickness were made proportional to the cube root of the distance from the end. The mirror is bent by applying a point load in the center and the circular shape will be good over the region where the cube root law is obeyed provided that the assumptions of beam theory are satisfied 5 and that the three narrow webs of material linking the mirror to the bending mechanism behave as perfect (torque-free) hinges. In the earlier paper we also discu...