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.

Figure 1

(Color online) Schematic view of the experimental setup and two-dimensional display-type analyzer. Angular distribution without distortion is directly projected onto the screen. An example of the observed image is shown in the figure.

Figure 2

Measured Cu(001) PEH. The kinetic energy of the photoelectron is 818 eV, and the initial state is the $\text{Cu}3p$ state. The lines in the figure indicate the Kikuchi band.

Figure 3

Measured Au(001) IXFH. The photon energy is 12 keV.

Figure 4

(Color online) (a) Schematic view of the geometry of the scattering pattern function. (b) Scattering pattern function of an electron caused by the Cu atom. The kinetic energy is 818 eV. (c) Scattering pattern function of an x ray caused by the Au atom. The photon energy is 12 keV.

Figure 5

Atomic images reconstructed from an experimental single-energy PEH of Cu(001). Each image is a cross section of the volume image at $z=0$, 0.18, and 0.36 nm. The darkness is proportional to the function $\left|\mathbf{a}\right|g\left(\mathbf{a}\right)$. The circles show the expected position of the atoms.

Figure 6

Atomic images reconstructed from an experimental single-energy XFH of Au. Each image is a cross section of the volume image at $z=0$, $\pm 0.20$, and $\pm 0.41\text{nm}$. The darkness is proportional to the function $\left|\mathbf{a}\right|g\left(\mathbf{a}\right)$. The circles show the expected position of the atoms.

Figure 7

Atomic images reconstructed from an experimental single-energy PEH of Cu(001) using the scattering pattern function without the factor $\left|\mathbf{a}\right|$. Each image shows a cross section of the volume image at $z=0$, 0.18, and 0.36 nm. The darkness is proportional to the function $g\left(\mathbf{a}\right)$.

Figure 8

Schematic view of mixture of voxels by translational symmetry.

Figure 9

Atomic image reconstructed from the experimental single-energy PEH of Cu(001) using the SPEA-MEM with the translational symmetry operation.

Figure 10

Reconstructed atomic image from the experimental single-energy IXFH of Au by using the SPEA-MEM with the translational symmetry operation.