Comparison of the vectorial diffraction theory and Fraunhofer approximation method on diffractive images of Fresnel zone plates

Both the vectorial diffraction theory and the Fraunhofer approximation method are used to calculate diffractive images. In those cases, 13.5 nm light is normally incident on Fresnel zone plates (FPZs) of which the focuses are all 0.3 mm. By investigating the calculations using the vectorial diffraction theory, the full width at half maximum (FWHM) intensities of the first-order diffraction along x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x$$\end{document}-axis (ϕ=0∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi =0^{\circ }$$\end{document}) are 17.2 and 8.7 nm for 2,000-zone and 10,000-zone FZPs, respectively. The comparison between two methods shows that FWHM errors along ϕ=0∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi =0^{\circ }$$\end{document} are 4.2 and 22.3 %, and along ϕ=45∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi =45^{\circ }$$\end{document} are 3.3 and 33.1 % for 2,000-zone and 10,000-zone FZPs, respectively. Furthermore, maximum intensity errors are 12.3 % for the 2,000-zone FZP and 33.5 % for the 10,000-zone FZP in our study cases.