Study of spherical aberrations for the design of ESRF beam lines

The calculation of the spherical aberration, which has been given only in the case of a point source up to now, is extended to the case of source of finite size. It is shown that the real size of the image can be well described by a convolution of the ideal size of the image and the aberrations involved. The expression for an optimum magnification is developed yielding the smallest possible image size for a given optical system in the case of a cylindrical mirror. Calculations are presented for different ESRF sources and the results are compared to simulations performed by the ray-tracing code SHADOW