Joint ptychographic tomography

Electron tomography has revolutionized materials characterization by enabling three-dimensional imaging at the nanoscale. Traditional approaches treat electrons like classical particles, assuming they travel in straight lines through the sample similar to X-rays in medical CT scans. While this simplification has enabled remarkable achievements, it cannot capture the wave-optical interactions crucial for atomic-resolution imaging.

Limitations of classical tomography

Classical electron tomography faces two fundamental limitations:

  1. Loss of Phase Information: By measuring only the intensity \(I\) of transmitted electrons, it discards the phase information \(\phi\). This phase carries crucial details about local electromagnetic fields and is especially important for imaging light elements.

  2. Independent Processing: Each tilt angle \(\theta\) is processed independently, failing to leverage physical relationships between different views of the same sample. This approach misses valuable correlations in the data.

Joint ptychographic tomography overcomes these limitations by embracing the wave nature of electrons while simultaneously considering data from all tilt angles. This approach recovers both the complex transmission per tilt (amplitude + phase) and a globally consistent 3-D potential.

Physics model

Electron Probe Model

The electron probe in a transmission electron microscope is shaped by the microscope’s optical system at each tilt angle \(\theta\). This probe includes various aberrations and can be described mathematically as:

\[P_\theta(\mathbf{r}) = \mathcal{F}^{-1}\{A(q)e^{i\chi_\theta(q)}\}\]

The aperture function \(A(q)\) defines the probe-forming angles, where \(A = 1\) for \(q \leq q_{\text{max}}\) and 0 elsewhere. The phase term \(\chi_\theta(q)\) captures all aberrations, including both shared base aberrations and per-tilt variations:

\[\chi_\theta(q) = \pi\lambda\Delta f_\theta |q|^2 + \frac{\pi}{2}C_s\lambda^3|q|^4 + \pi\lambda A_1|q|^2\cos(2(\phi - \phi_{A_1})) + \text{(coma, higher terms)}\]

Here, \(|q|\) represents the spatial frequency magnitude and \(\phi\) is the azimuthal angle. These aberrations evolve continuously as the sample is tilted, providing essential physical constraints for reconstruction.

Intensity and transmission function

The interaction between the electron probe and sample is described by the transmission function \(T_\theta(\mathbf{r})\):

\[T_\theta(\mathbf{r}) = A_\theta(\mathbf{r})e^{i\sigma\Phi_\theta(\mathbf{r})}\]

where \(A_\theta(\mathbf{r})\) represents the absorption amplitude and \(\Phi_\theta(\mathbf{r})\) is the phase shift. The phase shift is determined by integrating through the rotated potential:

\[\Phi_\theta(\mathbf{r}) = \int V(R_\theta[x, y, z])dz\]

Here, \(z\) represents the beam direction after applying the rotation \(R_\theta\). This makes \(\Phi_\theta(\mathbf{r})\) a line integral through the rotated potential, capturing how the sample modifies both the amplitude and phase of the electron wave at each tilt angle.

The measured intensity at each probe position \(\mathbf{r}_{\theta,j}\) is then modeled as:

\[I_{\theta,j}^{model}(q) = |\mathcal{F}\{P_\theta(\mathbf{r} - \mathbf{r}_{\theta,j})T_\theta(\mathbf{r})\}|^2\]

This formulation creates an inverse-physics problem: we must determine the 3D potential \(V\) that best explains our measurements, considering both data fidelity and physical constraints.

\[\mathcal{L}(V, P, \{R_\theta\}) = \sum_{\theta,j} \|I_{\theta,j}^{\text{meas}} - I_{\theta,j}^{\text{model}}(V, P, R_\theta)\|_2^2 + \lambda R(V)\]

Here:

  • \(V\) is the 3D electrostatic potential we’re reconstructing

    • Higher values indicate likely atomic positions

    • Shape and magnitude help identify atomic species

    • Includes both nuclear and electronic contributions

  • \(P\) represents the probes at all tilt angles

  • \(R_\theta\) is the rotation operator for tilt angle \(\theta\). Transforms coordinates from lab frame to tilted sample frame

  • \(R(V)\) adds prior knowledge about the potential

    • Common choices include nonnegativity or TV priors

    • Can enforce sparsity (atoms are localized)

    • May include known atomic spacings or symmetries

    • Helps suppress noise and artifacts

    • Gradients flow through rotations, projection sum, complex exponential, and FFT

  • \(\lambda\) balances data fitting vs. prior constraints

Electrostatic potential reconstruction

Our goal is to reconstruct the 3D electrostatic potential \(V(x,y,z)\) throughout the sample volume. At each voxel position (x,y,z), \(V(x,y,z)\) tells us the strength of the electrostatic potential at that point in space. Update the potential at each voxel following gradient descent:

\[V_{n+1}(x,y,z) = V_n(x,y,z) - \alpha \frac{\partial \mathcal{L}}{\partial V}\]

where:

  • \(V_n(x,y,z)\) is the current estimate of the potential at voxel (x,y,z)

  • \(V_{n+1}(x,y,z)\) is the updated potential at that voxel

  • \(\alpha\) is the learning rate

  • \(\frac{\partial \mathcal{L}}{\partial V}\) is the gradient of the loss

Through these iterative updates, we reconstruct the complete electrostatic potential field \(V(x,y,z)\) of our sample.

Principles of Operation

Joint ptycho-tomography advances beyond classical approaches by treating electrons as waves rather than particles. This wave-optical treatment captures quantum mechanical interactions that traditional straight-line trajectory models miss.

Understanding Electron Wave Interactions

The electron wave undergoes two sequential transformations during the experiment:

  1. Probe Formation (\(P(r)\)): The microscope’s electromagnetic lenses shape the initial electron wave into a focused probe. This probe has both amplitude and phase components, controlled by the aberration function \(\chi_\theta(q)\) of our microscope.

  2. Sample Interaction (\(t(r)\)): The probe then interacts with the sample through its transmission function, which modifies both the amplitude and phase:

    \(t(r) = A(r)e^{i\sigma \int V(r)dz}\)

    where:

    • \(A(r)\): Amplitude modification (absorption)

    • \(\sigma \int V(r)dz\): Phase shift from potential

    • \(V(r)\): Sample’s electrostatic potential

    • \(\sigma\): TEM interaction constant (\(\approx 6.5 \times 10^6\) rad V:math:^{-1} m:math:^{-1} at 300 kV)≠≠±∆ß´

Physical Constraints in Joint Reconstruction

Joint ptycho-tomography employs a holistic reconstruction approach by optimizing a single consistent 3D model that explains all measurements simultaneously. This approach incorporates important physical constraints:

  1. Consistent Optics: While the base optical system remains unchanged during tilting, small variations in effective defocus and astigmatism can occur due to: - Eucentric height offsets - Sample height variations - Minor tilt axis misalignments

  2. Smooth Aberration Evolution: Any changes in optical aberrations must follow physically realistic patterns. For example, astigmatism at 30 degrees tilt is mathematically related to astigmatism at 0 degrees through the fixed lens geometry.

These physical constraints guide the reconstruction toward realistic solutions by enforcing consistency across all tilt angles.

Understanding Transmission Functions

The relationship between a sample and its transmission function often causes confusion. Here’s a clear breakdown:

Sample vs. Transmission Function:

  • The sample is a physical object with a specific structure and composition

  • The transmission function \(t(r)\) describes how this sample affects electron waves

  • \(t(r)\) is a mathematical operator, not a direct image of the sample

Wave-Sample Interaction:

  1. The sample’s structure creates an electrostatic potential \(V(r)\)

  2. This potential produces a transmission function \(t(r)\)

  3. The transmission function modifies incoming waves through: - Amplitude changes (absorption) - Phase shifts (from potential)

Note: The term “object” sometimes used in literature can be misleading. We don’t directly image the sample’s structure; instead, we measure how it transforms our electron probe. From these measurements, we reconstruct the underlying potential \(V(r)\) that caused these transformations.

Comparing Reconstruction Approaches

The power of joint optimization lies in its ability to maintain consistency across all measurements while preserving quantum mechanical effects. Let’s compare the two approaches:

Classical Direct Tomography

  1. Sequential Processing: - Processes each tilt angle \(\theta\) independently - Reconstructs 3D structure one projection at a time - Ignores relationships between different views

  2. Limitations: - Loses valuable correlations between tilt angles - Misses quantum mechanical effects - May produce inconsistent reconstructions

Joint Ptycho-tomography

  1. Unified Approach: - Maintains a single 3D potential \(V(r)\) across all tilts - Updates structure using all tilt data simultaneously - Preserves quantum mechanical interactions

  2. Physical Consistency: - Ensures smooth evolution of probe aberrations \(\chi_\theta(q)\) - Optimizes both sample structure and probe properties - Operates within the elastic, single-scattering regime

This holistic optimization ensures a physically consistent reconstruction that captures both the structure and the quantum mechanical nature of electron-sample interactions.