The X-ray Focusing System at the Time-Resolved AMO Instrument.

With the set of terminal bending moments, actuator position, and terminal slopes for each actuator, solve for the set of coefficients k SB 1 sb ~k SB 4 sb . Since no lateral force is imparted on the silicon mirror in between the two end supports (outside the glue pad area and ignoring the effect of GaIn), the bending moment along the long axis of the mirror within the clear optical aperture is a straight line connecting the two terminal bending moments controlled by the actuator. To back calculate actuator positions with a given center line surface shape, one would repeat steps 4 to 5 to obtain the terminal bending moments and terminal slopes, then solve the quadratic equation for actuator position in Eq. (1) with corresponding coefficient sets (k SB 1 sb ~k SB 4 sb ). In essence, each actuator controls the bending moment exerted on their corresponding end of the silicon mirror. [Extracted from the article]