In the weak phase object approximation the wave function in the objective lens’ back focal plane is a convolution of the illuminating wave, ψin, with the projected potential, V, multiplied with the coherent transfer function, H, i.e. ( ψin * ( 1 + σV ) ) H, with σ the interaction constant and * the convolution operator. Therefore, frequencies from above the cutoff-frequency in H’s objective aperture enter the image formation in aliased form, as is illustrated in Fig. 1a. In this abstract, fully dynamical multislice simulations are carried out to investigate if these higher frequencies can be disentangled and superresolution can be achieved.
Simulations of  Si are set up with FDES , the supercell is 5.12 nm by 5.12 nm wide and is 5 nm thick on the left side and 2.5 nm on the right-hand side. The in-plane sampling distance and the slice thicknesses were set to 0.01 nm. Si in the  direction displays dumbbells that are 0.075 nm apart, corresponding to a spatial frequency of 13.3 nm-1 or 33.4 mrad at an acceleration voltage of 200 kV. The imaging is done with illumination that is structured to have random-but-known phases and that is bandwidth limited to 20 mrad, see Fig. 1b and Fig. 2a. The objective lens behind the sample is set to Scherzer conditions with an objective aperture of 20 mrad, a defocus of -18.8 nm and a spherical aberration of 94.0 µm, thus yielding a point resolution of 0.125 nm that is clearly insufficient for resolving the dumbbells. Eight images are recorded with the simulation shifted by 0.27 nm between consecutive recordings; the first image is shown in Fig. 2b, the others look similar.
A three-dimensional reconstruction was set up in IDES [2,3] with three slices 2.5 nm apart and a pixel size of 0.01 nm; amplitude and phase of the impinging wave and the settings of the objective lens were assumed known. Multiple scattering was accounted for by propagating the wave between slices with the Fresnel propagator. A Polak-Ribière conjugate gradient search was used to minimize an error function that was chosen as the sum of absolute differences between measurement and corresponding data simulated from the current object estimate. Contrary to [2,3] no sparse regularization has been applied. In Fig. 3a the middle slice of the reconstruction is shown, displaying a clear separation of the Si-dumbbells. Furthermore, the lower slice displayed in Fig. 3b shows that depth sensitivity with a resolution of at least 2.5 nm is achieved as the right-hand side, which by construction does not contain any atoms in its lower half, indeed does not display any. 
 W. Van den Broek, et al. “FDES, a GPU-based multislice algorithm with increased efficiency of the computation of the projected potential.” Ultramicroscopy 158 (2015), pp. 89–97.
 W. Van den Broek and C.T. Koch. “Method for retrieval of the three-dimensional object potential by inversion of dynamical electron scattering.” Phys. Rev. Lett. 109 (2012), p. 245502.
 W. Van den Broek and C.T. Koch. “General framework for quantitative three-dimensional reconstruction from arbitrary detection geometries in TEM.” Phys. Rev. B 87 (2013), p. 184108.
 The Carl Zeiss Foundation is gratefully acknowledged by all authors. C.T. Koch also acknowledges the DFG (KO 2911/7-1).
To cite this abstract:Wouter Van den Broek, Christoph T. Koch; Superresolution and depth sensitivity in HRTEM through structured illumination. The 16th European Microscopy Congress, Lyon, France. https://emc-proceedings.com/abstract/superresolution-and-depth-sensitivity-in-hrtem-through-structured-illumination/. Accessed: September 23, 2019
EMC Abstracts - https://emc-proceedings.com/abstract/superresolution-and-depth-sensitivity-in-hrtem-through-structured-illumination/