Calculations of HREM image intensity using Monte Carlo integration

Computationally Efficient Handling of Partially Coherent Electron Sources in (S)TEM Image Simulations via Matrix Diagonalization

We introduce a novel method to improve the computational efficiency for (S)TEM image simulation by employing matrix diagonalization of the mixed envelope function (MEF). The MEF is derived by taking the finite size and the energy spread of the effective electron source into account, and is a component of the transmission cross-coefficient that accounts for the correlation between partially coherent waves. Since the MEF is a four-dimensional array and its application in image calculations is time-consuming, we reduce the computation time by using its eigenvectors. By incorporating the aperture function into the matrix diagonalization, only a small number of eigenvectors are required to approximate the original matrix with high accuracy. The diagonalization enables for each eigenvector the calculation of the corresponding image by employing the coherent model. The individual images are weighted by the corresponding eigenvalues and then summed up, resulting in the total partially coherent image.

Comparison of different imaging models handling partial coherence for aberration-corrected HRTEM at 40-80 kV

Partial coherence of the electron waves leaving the specimen is taken into account in the high-resolution transmission electron microscopy (HRTEM) image simulation by mainly three methods - the incoherent summation approach, the transmission cross-coefficient (TCC) model, and the quasi-coherent model. In the incoherent summation approach, coherent images are calculated for each point in the effective source and summed up incoherently. The TCC is the transfer function of the microscope obtained based on the incoherent summation approach. An analytical form of the TCC can be derived by assuming a Gaussian distribution for the source radiation and for the variation of the focal length caused by the energy spread of the effective source. In the quasi-coherent model, the partial coherence effect is simplified by multiplying the wave function at the diffraction plane with the envelope functions. Envelope functions suppress the contributions to the image contrast from waves which do not propagate along the optical axis. The quasi-coherent model is usually sufficient for the image simulation of weak phase objects. This model is more computationally efficient than both the incoherent summation approach and the TCC model. For the C_s-corrected and C_c/C_s-corrected microscopes operating at 80 kV, we have compared images simulated by using the three models with the experimental images. The comparison shows that the quasi-coherent model also provides a sufficient approximation for the image simulation of high-Z materials if chromatic aberration is corrected and the samples comprise only several atomic layers. In the case of only C_s-correction, the incoherent summation approach or the TCC model needs to be employed for modelling the imaging of high-Z samples even though it is more computationally consuming.