Curved reflecting mirrors are widely used as x-ray optical elements for both laboratory and synchrotron radiation sources. In general, the mirror parameters are optimized by numerical simulation. We discuss an analytical approach that is useful for deriving the mirror parameters, including eccentricity, length, angular acceptance, and magnification. We have examined in particular an elliptical surface from which we learned that, given the distance between the foci of the ellipse, the magnification, and the critical angle of total external reflection, it is possible to find analytically the optimal eccentricity that maximizes the angular acceptance and the optimal mirror length. We found that the last-named parameter, in a first approximation, depends only on the distance between the foci of the ellipse and on the magnification factor. We present as well a comparison of optimal parameters obtained with analytical calculation and with ray-tracing simulation that yielded good agreement.
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