Analysis of noise-induced errors in vector-field electron tomography

Vector-field electron tomography (VFET) reconstructs electromagnetic vector fields of magnetic nanomaterials using transmission electron microscopy. The theory behind this reconstruction process is well established, but the practical implications of experimental errors and how they affect the accuracy of the reconstructed vector fields is not well understood, hindering progress in the use of these techniques for routine magnetic characterization of nanomaterials. Here, we present an analysis of the propagation of stochastic errors through a VFET algorithm. A method for determining the contribution of image noise to errors in a reconstructed vector potential is derived. Simulations are performed to test the validity of this method when applied to shot noise, which shows good agreement with theory.

Figure 1

Schematic of the experimental setup. The exit phase of the electron wave is modified by the potentials of the sample, and is measured by recording out-of-focus micrographs and applying a phase retrieval algorithm. The focal plane image (indicated by a dashed border) is approximated from the underfocus (left) and overfocus (right) micrographs. The exit phase is measured for multiple angles about two orthogonal axes ($\alpha$ and $\theta$), enabling reconstruction of the vector potential associated with the sample.

Figure 2

Acquisition of tilt series $T^{\theta }$ and $T^{\alpha }$. The curved arrows indicate the direction that the object $O$ is rotated in each tilt series, and the double arrows indicate the propagation direction of the electron beam (after Yu et al. [3]).

Figure 3

Shape of sample mask used for our simulations (gray) and surface point cloud obtained using scalar tomography [42] applied separately to two tilt series [red (dark gray) and blue (light gray)]. For an animated version of this figure, see Supplemental Material [43].

Figure 4

Example of through focal derivatives computed from micrographs with $\Delta f=-10\mu \mathrm{m}$ and ${\sigma }_{\mathrm{in}}=1\%$ using (a) simulated micrographs and (b) experimental micrographs.

Figure 5

Slice of the $x$ component of the exact vector potential at $z=0$ (top), and the same slice taken from reconstructions using various values of $\Delta f$ and ${\sigma }_{\mathrm{in}}$.

Figure 6

Contour plots of $E_{\mathrm{noise}}^{\mathrm{tot}}$ as a function of ${\sigma }_{\mathrm{in}}$ and $\Delta f$ for (a) reconstructions using simulated data, obtaining the phase via Eq. (5), and (b) analytical results obtained using Eqs. (34) and (43). The numbers given on the contour lines are total rms errors expressed as a percentage.

Figure 7

(a) Analytically derived values ${\overline{E}}_{\mathrm{noise}}^{\mathrm{tot}}$ [red (dark gray)] obtained using Eqs. (34) and (43), along with the results of simulations $E_{\mathrm{noise}}^{\mathrm{tot}}$ using Eq. (5) for the phase retrieval [blue (black)] and Eq. (6) [green (light gray)]. (b) Normalized residual ${\chi }^{\mathrm{tot}}$ plotted against ${\sigma }_{\mathrm{in}}/\Delta f$ for each method (blue circles and green crosses, respectively).

Figure 8

Analytically derived values ${\overline{E}}_{\mathrm{noise}}^{\mathrm{dir}}$ [red (dark gray)], along with the results of simulations $E_{\mathrm{noise}}^{\mathrm{dir}}$ using Eq. (5) for the phase retrieval [blue (black)], and Eq. (6) [green (light gray)]; (a) using the approximation given by Eq. (52), and (b) using the higher-order approximation given by Eq. (54). (c), (d) Normalized residual ${\chi }^{\mathrm{dir}}$ plotted against ${\sigma }_{\mathrm{in}}/\Delta f$ for each approximation, respectively, showing both the results obtained using Eqs. (5) (blue circles) and (6) (green crosses).

Figure 9

(a) Analytically derived values ${\overline{E}}_{\mathrm{noise}}^{\mathrm{abs}}$ obtained using Eq. (56) [red (dark gray)], along with the results of simulations $E_{\mathrm{noise}}^{\mathrm{abs}}$ using Eq. (5) for the phase retrieval [blue (black)], and Eq. (6) [green (light gray)]. (b) Normalized residual ${\chi }^{\mathrm{abs}}$ plotted against ${\sigma }_{\mathrm{in}}/\Delta f$ for each method (blue circles and green crosses, respectively).

Figure 10

Surface plots showing reconstruction errors (a) $E^{\mathrm{tot}}$, (b) $E^{\mathrm{dir}}$, and (c) $E^{\mathrm{mag}}$. Each subfigure shows both the results obtained using Eq. (5) for the phase retrieval, in blue (black), and Eq. (6), in green (gray).