  The focusing performance of X-ray optics (conveniently expressed in terms of
HEW, Half Energy Width) strongly depend on both mirrors deformations and photon
scattering caused by the microroughness of reflecting surfaces. In particular,
the contribution of X-ray Scattering (XRS) to the HEW of the optic is usually
an increasing function H(E) of the photon energy E. Therefore, in future hard
X-ray imaging telescopes of the future (SIMBOL-X, NeXT, Constellation-X, XEUS),
the X-ray scattering could be the dominant problem since they will operate also
in the hard X-ray band (i.e. beyond 10 keV). [...]
  Several methods were proposed in the past years to estimate the scattering
contribution to the HEW, dealing with the surface microroughness expressed in
terms of its Power Spectral Density (PSD), on the basis of the well-established
theory of X-ray scattering from rough surfaces. We faced that problem on the
basis on the same theory, but we tried a new approach: the direct, analytical
translation of a given surface roughness PSD into a H(E) trend, and - vice
versa - the direct translation of a H(E) requirement into a surface PSD. This
PSD represents the maximum tolerable microroughness level in order to meet the
H(E) requirement in the energy band of a given X-ray telescope.
  We have thereby found a new, analytical and widely applicable formalism to
compute the XRS contribution to the HEW from the surface PSD, provided that the
PSD had been measured in a wide range of spatial frequencies. The inverse
problem was also solved, allowing the immediate evaluation of the mirror
surface PSD from a measured function H(E). The same formalism allows
establishing the maximum allowed PSD of the mirror in order to fulfill a given
H(E) requirement. [...]
