FINITE-DIFFERENCE SOLUTION OF PARABOLIC EQUATION AND NUMERICAL SIMULATION FOR X-RAY FOCUSING

Abstract

In this paper we apply the finite-difference method to solve a parabolic equation of a general problem of short wave diffraction in a conducting medium. It is based on the implicit Runge-Kutta method of the second order combined with the iterative procedure. We also present new numerical simulations for X-rays focusing using the mentioned approach. We consider CRLs with a parabolic profile with a radius of curvature up to 0.2 mm. The main goal of this work is to elaborate an X-ray calculator for a PC which would present new possibilities compared to conventional ones. The correspondent code is written in FORTRAN to obtain the focal distance and diffraction spot profiles. Simulations for two cases were performed, the first one with 33 Al lenses for X-ray energy 15 keV, the results showed that we needed to consider more than 50000 points in each direction which forced us to consider a one-dimensional simulation only. For the second case we performed a simulation for several lenses, up to 15 Al lenses to perform the 2-d simulation. We have good agreement with the experimental data for the focal distance, and for the intensity at the focal plane while, for the spot size, we have smaller FWHM for the Gaussian beam at the detector than in the experimental data. We believe that the FWHM we have is smaller as our lenses are ideal without any defects.