Reconstruction algorithm for atomic-resolution holography using translational symmetry

The most widely used methods for reconstructing three-dimensional atomic arrangements from a photoelectron hologram and an x-ray fluorescence hologram are based on an integral kernel, for example, the Fourier transformation. These methods require many holograms that are recorded using multiple energies since the Fourier transformation requires an infinite integral interval. Therefore, it is difficult to reconstruct an atomic arrangement from a single-energy hologram. In order to accomplish to reconstruct the three-dimensional atomic arrangement from a single-energy hologram, we have proposed a scattering pattern extraction algorithm using the maximum-entropy method (SPEA-MEM) for photoelectron holography. In this paper, we also describe the application of this algorithm to x-ray fluorescence holography. We have succeeded in reconstructing 58 Au atoms from a single-energy x-ray fluorescence hologram that we have measured. However, artifacts have been observed in the reconstructed image. This is due to the long coherent length and the mean-free path of the x rays. Hence, we have incorporated crystal translational symmetry into SPEA-MEM to solve this problem. We have applied this algorithm to an x-ray fluorescence hologram of Au and a photoelectron hologram of Cu that we have measured. We have succeeded in reconstructing a very clear atomic arrangement with an accuracy of 0.01 nm in three-dimensional real space for both holograms.