Analytical evaluation of the X-ray scattering contribution to imaging degradation in grazing-incidence X-ray telescopes

Abstract

Aims. 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). In order to ensure the imaging quality at all energies, clear requirements have to be established in terms of reflecting surfaces microroughness. Methods. 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. Results. 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. Practical equations are firstly developed for the case of a single-reflection optic with a single-layer reflective coating, and then extended to an optical system with N identical reflections. The results are approximately valid also for multilayer-coated mirrors to be adopted in hard X-rays. These results will be extremely useful in order to establish the surface finishing requirements for the optics of future X-ray telescopes.

Thin glass shell oriented to wide field x-ray telescope

“…In this figure we compare the observed values with the calculated values assuming that the total HEW is the sum (quadratic in magenta or linear in cyan) of a constant term due to the figuring errors and of the scattering. The scattering contribution is calculated starting from the micro-roughness Power Spectral Density (PSD) (Spiga 2007). With this model, the contribution of the scattering is negligible at 0.93 keV.…”

Specification of x-ray mirrors in terms of system performance: a new twist to an old plot

“…A powerful approach to evaluate the effects of surface imperfection with middle and high spatial frequencies (surface finish) to the performance of x-ray optical systems consists of sophisticated x-ray scattering (diffraction) calculations based on the one-or two-dimensional (1-D or 2-D) power spectral density (PSD) distribution of the surface height, allowing for the evaluation of three-dimensional distributions of x-rays scattered by the optics. [20][21][22][23][24][25][26][27][28][29][30] In the early 1990s, Church and Takacs pointed out that the specification of surface figure and finish of x-ray mirrors must be based on their imaging performance, and the results can be expressed in terms of statistical quantities, such as rms roughness and residual slope variation, that are directly accessible from optical metrology. 23,24 The spatial frequency bandwidths of applicability of geometrical and diffraction optics are determined by the radiation source and beamline system parameters rather than by the metrology instruments.…”

“…[33][34][35][36][37] The found, extrapolated, modeled PSD spectrum is used (via inversed Fourier transform) for the simulation of *Address all correspondence to: Valeriy V. Yashchuk, E-mail: VVYashchuk@lbl .gov metrology data for x-ray optics before fabrication and prediction of the performance of the optical system. [27][28][29][30] In the present work, we investigate the applicability of the PSD-based evaluation of beamline performance of prospective x-ray optics for XFELs, for the case when dedicated optical systems deliver the beams over distances of hundreds of meters. The conclusion is that in this case, the PSD-based specification is not sufficient.…”

Roughness scattering in X-ray grazing-incidence telescopes

“…The relation between the half-energy width in grazing-incidence telescopes and the roughness variance was studied in (Spiga 2007) by applying the Debye-Waller expression for the total integrated scattering (Spiller 1994) in 1-dimensional geometry. In the framework of the scalar theory of diffraction (Harvey 1995b;Goodman 1968;Gaskill 1978;Born & Wolf 1980), an exact derivation of the rms width of the angle spread function from the corresponding optical transfer function (Goodman 1968;Gaskill 1978;Born & Wolf 1980;Smith 1963;Harvey et al 1988) in 2-dimensional geometry was reported in (Zocchi 2009), in which the contribution of scattering to the rms width of the angle spread function was shown to depend on only the variance of the roughness slope.…”

“…In the framework of the scalar theory of diffraction (Harvey 1995b;Goodman 1968;Gaskill 1978;Born & Wolf 1980), an exact derivation of the rms width of the angle spread function from the corresponding optical transfer function (Goodman 1968;Gaskill 1978;Born & Wolf 1980;Smith 1963;Harvey et al 1988) in 2-dimensional geometry was reported in (Zocchi 2009), in which the contribution of scattering to the rms width of the angle spread function was shown to depend on only the variance of the roughness slope. However, even in these two papers (Spiga 2007;Zocchi 2009), the cylindrical symmetry of an X-ray telescope was not taken into account.…”