Method for Retrieval of the Three-Dimensional Object Potential by Inversion of Dynamical Electron Scattering

Dynamical scattering of fast electrons can be inverted by recasting the multislice algorithm as an artificial neural network, enabling the iterative retrieval of the three-dimensional object potential. This allows a nonheuristic treatment of the modulation transfer function of the CCD, partial spatial and temporal coherence, and inelastic scattering through an absorptive potential. Furthermore, prior knowledge about the atomic potential shape and the sparseness and positivity of the object can be used. The method is demonstrated on simulated bright field images recorded at 40 kV.

Figure 1

The MS algorithm written as an ANN. The incoming wave, ${\psi }^1$, is multiplied with the transmission function $t^1$ of the first slice, then ${\psi }^1{t}^1$ is fed into the identity function (id.). The next layer of nodes and edges realizes a real space convolution with the propagator $p$, resulting in ${\psi }^2$. This is repeated until the exit wave ${\psi }^{N+1}$ is reached. The next layer encodes a real space convolution with the lens function if in imaging mode, or a Fourier transform if in diffraction mode, resulting in ${\psi }^{N+2}$. The last layers convert ${\psi }^{N+2}$ to an intensity $I$ and calculate the error metric $E$ from the measurements $J$.

Figure 2

Five typical measurements, the tilt around the horizontal axis of these images is 0° and around the vertical axis, from left to right, $-10°$, $-5°$, 0°, 5°, and 10°.

Figure 3

Comparison between a dynamical and a kinematical simulation. The particle is imaged at zero tilt.

Figure 4

Upper two rows: Slices with the projected potential of the Au particle; the gray scale is logarithmic. Middle two rows: Reconstructed projected potential, for comparison, the solution has been convolved with $V_0$ after intensities below the CF threshold have been set to zero. Lower two rows: Reconstructed absorptive potential.