The finite difference methods of computation of X-rays propagation through a system of many lenses

摘要

The propagation of X-ray waves through an optical system consisting of many beryllium X-ray refractive lenses is considered. In order to calculate the propagation of electromagnetic in the optical system, two differential equations are considered. First equation for an electric field of a monochromatic wave and the second equation derived for complex phase of the same electric field. For solving the problems, finite-difference methods are suggested and investigated. It is shown that very small steps of the difference grid are necessary for reliable computation of propagation of X-ray waves through the system of lenses, when the first equation is used. The reason of such a result is that the electric field of the wave after passing through many lenses is a quickly oscillating function of coordinates. It is shown that much larger steps may be utilized if the second equation is used, because the phase of electric field after passing through many lenses is quickly increasing, but not oscillating function. We suggest and recommend using the equation for a phase function instead of the equation for an electric field. The error of simulation obtained for both equations is estimated mathematically and investigated.