Tilt-based Aberration Estimation in Transmission Electron Microscopy

TL;DR

This work addresses fast, drift-aware aberration estimation in transmission electron microscopy by exploiting tilt-induced image shifts to obtain information about aberration coefficients. It integrates a Kalman-filter framework with a dynamic tilt-aberration model, enabling simultaneous estimation of multiple aberrations while accounting for temporal drift and specimen motion. The tilt sequence is designed offline via an A-optimality objective by receding-horizon gradient optimization, with an EM step to tailor measurement noise to the specimen. Experimental validation on a real TEM demonstrates that optimized tilt patterns substantially reduce alignment time (to under a minute) while maintaining estimation accuracy, enabling faster high-resolution imaging.

Abstract

Transmission electron microscopes (TEMs) enable atomic-scale imaging but suffer from aberrations caused by lens imperfections and environmental conditions, reducing image quality. These aberrations can be compensated by adjusting electromagnetic lenses, but this requires accurate estimates of the aberration coefficients, which can drift over time. This paper introduces a method for the estimation of aberrations in TEM by leveraging the relationship between an induced electron beam tilt and the resulting image shift. The method uses a Kalman filter (KF) to estimate the aberration coefficients from a sequence of image shifts, while accounting for the drift of the aberrations over time. The applied tilt sequence is optimized by minimizing the trace of the predicted error covariance in the KF, which corresponds to the A-optimality criterion in experimental design. We show that this optimization can be performed offline, as the cost criterion is independent of the actual measurements. The resulting non-convex optimization problem is solved using a gradient-based, receding-horizon approach with multi-starts. Additionally, we develop an approach to estimate specimen-dependent noise properties using expectation maximization (EM), which are then used to tailor the tilt pattern optimization to the specific specimen being imaged. The proposed method is validated on a real TEM set-up with several optimized tilt patterns. The results show that optimized patterns significantly outperform naive approaches and that the aberration and drift model accurately captures the underlying physical phenomena. In total, the alignment time is reduced from typically several minutes to less than a minute compared to the state-of-the-art.

Figures (7)

• Figure 1: Phase plates of the single wave aberrations in TEM. The columns and rows correspond to the foldness and order of the aberration, respectively. Each phase plate displays the phase shift of the wavefront as a function of the spatial frequency $g \in \mathbb{C}$ in gray scale. Contour lines are plotted in white for positive phase shifts and in black for negative phase shifts. This figure has been adapted from JuriBarthel2003.
• Figure 2: Example of a user-defined tilt constraint $\tilde{\theta}_k$ as a function of time $k$. The red lines indicate the boundaries of the feasible set. Note that the tilt values are normalized.
• Figure 3: Schematic representation of receding horizon optimization. Starting at time $k$, we solve an optimization problem with a reduced horizon $H$ to find the optimal tilts $\left\{\theta^\star(i)\right\}_{i=k}^{k+H-1}$. We then store the first tilt $\theta_k = \theta^\star(k)$, move up one time step in the planning domain and solve the $k+1$-th optimization problem with the updated state $P(k+1\mid k)$. We can use the previous solution $\left\{\theta^\star(i)\right\}_{i=k}^{k+H-1}$ as a warm start for the next optimization problem. This process is repeated until we reach the end of the tilt sequence.
• Figure 4: Comparison of different normalized tilt patterns. The patterns are: 1) Lissajous curve with frequency ratio 3:2, 2) completely randomized feasible pattern, 3) optimized pattern with $H=1$, and 4) optimized pattern with $H=10$.
• Figure 5: Weighted posterior covariance trace ($\text{tr}(WP_{k\mid k})$) at each time index for selection of tilt patterns. In this figure, the estimation error of every normalized state $x_k$ is weighted equally using weight matrix $W=\frac{1}{d}I_{d}$.