Tham khảo tài liệu 'modern developments in x-ray and neutron optics episode 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 144 G. Schneider et al. Fig. . Part of the mathematical functions describing the transmission grating consisting of material A upper and material B lower with the permittivities SA and SB. The local zone plate period A is given by A L S X axis where the mathematical functions p x z and q x z denote Fourier series which are used to describe the spatial distribution of the permittivity of the grating. This is directly related to the material distribution of the grating which is taken into account in the wave equation by a periodically changing permittivity x z represented by a Fourier expansion with the grating vector G .z . . . 2 L x z A l S oo z L Esincl hn L S h 1 2 3. The average permittivity of the grating is introduced as - _ . L 2 l 2 2 L B A - B L S nB nA - nB L S and the difference A between the permittivities of the materials A and B in terms of the refractive indices is A A - b nA - nB which follows from Maxwell s relation n2. This allows one the calculation of the function x z in terms of the complex indices of refraction usually used in X-ray physics nA 1 - ỖA jpA and nB 1 - ỖB - jpB. The X-ray optical parameters Ỗ and 3 can be determined directly from the atomic scattering factor 1 jf2 . They are tabulated for all relevant elements in the energy range interesting for X-ray microscopy 15 16 17 . Introducing 8 Volume Effects in Zone Plates 145 the periodically changing permittivity e x z in leads to the scalar wave equation describing the modulated region V2 E x z k0 Ae 2 L L S hnL S L S cos hG r E x z 0 GO X 2 sinc h which is mathematically a linear second-order differential equation with periodic coefficients Mathieu differential equation . It may be concluded from Floquet s theory that this differential equation has a solution for the electrical field E x z of the form oo oo E x z Em x z Eo 22 Am z exp j Pm r mo mo with Pm r Pm x x Pm z z. This solution of the wave equation can be interpreted as an infinite sum of .
