Focusing performance of hard X-ray single Kinoform lens

Nowadays, X-ray nanoprobe plays an important role in many research fields, ranging from materials science to geophysics and environmental science, to biophysics and protein crystallography. Refractive lenses, mirrors, and Laue lenses, can all focus X-rays into a spot with a size of less than 50 nm. To design a refractive lens at fixed wavelengths, absorption in the lens material can be significantly reduced by removing 2 pi phase-shifting regions. This permits short focal length devices to be fabricated with small radii of curvatures at the lens apex. This feature allows one to obtain a high efficiency X-ray focusing. The reduced absorption loss also enables optics with a larger aperture, and hence improving the resolution for focusing. Since the single Kinoform lens can focus hard X-ray into a spot on a nanoscale efficiently, it has very important application prospect in X-ray nano-microscopy and nano-spectroscopy. We present a theoretical analysis of optical properties of the single Kinoform lens. Using Fermat's principle of least time, an exact solution of the single Kinoform lens figure is derived. The X-ray diffraction theory is reviewed. The complex amplitude transmittance function of the X-ray single Kinoform lens is derived. According to Fourier optics and optical diffraction theory, we set up the physical model of X-ray single Kinoform lens focusing. Employing this physical model, we study how the focusing performance of hard X-ray single Kinoform lens is influenced by the material, the photon energy, the number of steps and the vertex radius of curvature. We find that diamond single Kinoform lens can achieve a smaller focusing beam size with higher intensity gain than Al and Si single Kinoform lens. The single Kinoform lens designed at a certain photon energy can also focus other photon energies with different lateral beam sizes, axial beam sizes, intensity gains and focusing distances. The numbers of steps of a single Kinoform lens can be lessened with the thickness of step increasing, while the single Kinoform lens keeps good focusing performance. To improve the focusing performance further, reducing the vertex radius of curvature is proposed. Following these rules, a single Kinoform lens is optimally designed to focus 30 keV hard X-ray down to a lateral size of 14 nm (full-width at half-maximum, FWHM) and an axial size of 62 mu m (FWHM) with an intensity gain of four orders of magnitude and transmittance of 30%.