Exit Wave Reconstruction in Microscopy

• Exit wave reconstruction is a process that mathematically recovers the complex-valued wavefield (amplitude and phase) immediately after specimen transmission.
• It employs techniques such as backward propagation, Kramers–Kronig methods, iterative variational algorithms, and deep learning to solve the phase retrieval problem from intensity-only data.
• The method underpins high-resolution imaging applications in TEM, ptychography, and phase-sensitive optical microscopy, enhancing quantitative analysis at atomic scales.

Exit wave reconstruction refers to the mathematical recovery of the complex-valued wavefield—containing both amplitude and phase—immediately after transmission through a specimen, commonly in electron or optical microscopy. The exit wave encodes the full information content imparted by the specimen to a probing field (electrons, photons), and its faithful retrieval underpins high-resolution structural imaging, quantitative phase contrast, and a range of inverse problems in computational and physical optics. The process of exit wave reconstruction is central to quantitative transmission electron microscopy (TEM), holographic and ptychographic imaging, and advanced phase-sensitive optical microscopy.

1. Fundamentals and Governing Equations

The propagation and reconstruction of exit waves is based on the scalar (or vector) wave equation; in the monochromatic, time-harmonic limit, this is the Helmholtz equation: $$\nabla^2 W(\mathbf{r}) + \kappa^2 W(\mathbf{r}) = S(\mathbf{r})$$ where $W(\mathbf{r})$ is the complex wavefield, $\kappa=2\pi/\lambda$ is the wavenumber, and $S(\mathbf{r})$ is the source. The measured data are typically intensity-only (i.e., magnitude-squared of the field at one or more downstream planes), leading to a nonlinear inverse problem for the phase.

A fundamental property underlying many algorithms is propagation symmetry or reciprocity: the Green’s function for the Helmholtz equation,

$h_G(r) = \frac{1}{j \lambda r}e^{j2\pi r/\lambda}$

has a phase-reversal symmetry that forms the basis of "backward propagation" (reverse of free-space evolution), implemented via a sign change in the phase in the Fourier domain (Pallaprolu, 2020).

2. Analytical and Direct Inversion Methods

2.1. Backward Plane-to-Plane Propagation

In the absence of significant multiple scattering or aberrations, the object and image planes can be related through plane-to-plane propagation in the spatial frequency (Fourier) domain. For a field sampled as $W_{\text{image}}(x,y)$ on a grid with pitch $\Delta x, \Delta y$, the exit wave at a distance $-\Delta z$ upstream is computed as:

1. Fourier transform to obtain $S_{\text{image}}(f_x,f_y)$.
2. Multiply each spatial frequency by a propagation phase:

$$H_{\text{back}}(f_x, f_y) = \exp \bigl[ -j 2\pi \Delta z \sqrt{ 1/\lambda^2 - f_x^2 - f_y^2 } \bigr ]$$

with frequency cutoff enforced by $f_x^2 + f_y^2 \le 1/\lambda^2$.

1. Suppress or regularize evanescent components ($f_x^2+f_y^2 > 1/\lambda^2$).
2. Inverse FFT to return to the spatial domain.

This method is exact except for frequencies beyond the “Ewald sphere” and is limited by sampling and finite aperture effects. Aliasing is controlled by ensuring the spatial frequency grid covers the physical bandlimit. Under the paraxial approximation, the operator can be simplified for computational speed (Pallaprolu, 2020).

2.2. Non-iterative Kramers–Kronig Reconstruction

The Kramers–Kronig Single-Aperture Inversion (KKSAI) method reconstructs the complex field from a minimal set of pupil-modulated intensity measurements using analyticity. By shifting a binary amplitude mask in the pupil plane (amplitude modulation), and measuring intensities for at least two configurations with mask-edge passing through the pupil center, the resulting real-space data mimic off-axis holography. The log-transformed intensity is analytic along the aperture-shift direction, enabling phase retrieval by a Hilbert transform (Kramers–Kronig relation):

$$\operatorname{Im} X(r) = -\frac{1}{\pi} \mathcal{P} \int_{-\infty}^{\infty} \frac{\varphi(r')}{r'-r} dr'$$

where $X(r) = \ln[1+\alpha(r)]$ and $\alpha(r)$ encodes the normalized analytic signal. Stitching the Fourier bands from multiple aperture scans recovers the full exit wave spectrum. This approach is non-iterative, parameter-free, and tolerant of strong phase and amplitude objects, requiring only two intensity measurements per field of view (Shen et al., 2020).

3. Iterative and Variational Algorithms

3.1. Joint Reconstruction and Image Registration

The variational framework for exit wave reconstruction from a focal series formulates the task as a minimization problem: $$E[\Psi, t_1,\dots,t_N] = \frac{1}{N} \sum_{j=1}^N \left \| \Psi \star_{T_{Z_j}} \Psi - \mathcal{F}(g_j \circ \varphi_{t_j}) \right \|_{L^2}^2 + \alpha \|\Psi - \Psi_M \|_{L^2}^2$$ where $\Psi$ is the exit wave, $t_j$ are registration shifts, and $\star_{T_{Z_j}}$ is a weighted cross-correlation (Transmission Cross Coefficient, TCC). Minimization is performed jointly in $(\Psi, \{t_j\})$, typically via nonlinear conjugate gradient methods. Convexity is absent with respect to $\Psi$, requiring good initialization; the Tikhonov term ensures existence of a minimizer. Compared to classical MAL/MIMAP approaches, this simultaneous scheme avoids inconsistencies from alternating updates (Doberstein et al., 2018).

3.2. Ptychographic Algorithms

In Fourier ptychography—especially recently adapted for electrons—the transmission function is illuminated with a succession of plane waves (via beam-tilt), and the resulting intensity patterns are used to reconstruct the exit wave. The phase retrieval proceeds via a modified "Ptychographic Iterative Engine" (PIE):

• For each iteration and tilt, compute simulated image via forward propagation, replace amplitude with square-root measured intensity (preserving phase), propagate back and update the exit-wave estimate.
• Converges in ≈50 iterations for datasets as small as 4–7 tilts.
• Achieves spatial resolution down to 0.63 nm at 4.5×10² e⁻/nm², validated on Cry11Aa protein crystals.
• No additional hardware is required beyond beam-tilt, and pre-processing is harmonized with standard data pipelines (Zhao et al., 12 Feb 2025).

4. Exit Wave Reconstruction via Holography and Phase Plate Imaging

4.1. Phase Plate TEM and Three-Image Inversion

Recording three images with different phase shifts imposed on the undiffracted beam allows for analytic inversion:

$$I_j(\mathbf{r}) = A^2 + B^2 - 2AB \cos[\phi(\mathbf{r}) - \chi_j]$$

for three phase shifts $\chi_j \in \{0,\pm\pi/2\}$. Direct algebraic solution yields both the amplitude and phase of the diffracted component. This method cancels nonlinear TCC terms and incoherent background if the phase plate affects only the central ($\mathbf u = 0$) mode. Phase error decreases as signal-to-noise increases and remains robust to realistic detector noise (Gamm et al., 2010).

4.2. Phase-Shifting Off-Axis Electron Holography

By recording a series of off-axis holograms under known reference phase shifts, the exit wave’s amplitude and phase can be obtained via cosine fitting at each pixel. Advanced drift correction is implemented by constructing a high-SNR reference from vacuum regions, aligning and dividing out this reference, and then cross-correlating in Fourier space to retrieve the true sample drift vector for each frame. This method enables atomic-resolution phase retrieval (down to 1 Å) and very high sensitivity (≈2π/452 rad at 1 Å), validated quantitatively by multislice simulations (Lindner et al., 2023).

5. Machine Learning and Deep Unfolding Approaches

5.1. Data-Driven Reconstruction with Supervised Neural Networks

Convolutional neural networks (CNNs), especially U-Net architectures, have demonstrated high fidelity in reconstructing exit waves from simulated or experimental focal series. For instance, training on thousands of simulated HRTEM images of diverse 2D materials from the Computational 2D Materials Database allows reconstruction with root mean square error below 0.03 (imaginary component) and atomic column positions recovered to better than 10 pm in favorable cases. Extension to experimental data requires dataset-specific retraining and inclusion of aberration, noise, and detector transfer functions. Performance closely tracks the training data priors; generalization to unseen structures or imaging regimes is limited if not represented in the training set (Larsen et al., 2021).

5.2. One-Shot Deep Learning Phase Recovery

Single-image "one-shot" CNN or GAN-based models have been trained on amplitude images simulated by multislice propagation. Such models can reconstruct the full complex wavefunction in ≈25 ms for a 224×224 image without microscope modification or multiple exposures. Mean absolute errors range from ≈0.12 (narrow class) to 0.61 (broad class) for unseen transforms; random baselines are ≈0.75. Domain adaptation is essential, as distribution mismatch between training and test data results in reconstruction ambiguity (Ede et al., 2020).

5.3. Deep Unfolding and Generalization Theory

Deep unrolling of algorithmic solvers (proximal gradient algorithms, PGAs) for variational exit wave inversion leads to interpretable, trainable networks, with each layer corresponding to a PGA iteration and a shared learned dictionary. Generalization bounds have been established: the generalization error scales as $O\left(\sqrt{L}\right)$ in the number of unrolled layers $L$, provided parameter perturbations are controlled. Empirical studies demonstrate exponential sensitivity to parameter drift as depth increases, recommending moderation in depth unless sample sizes are very large. Regularization and smoothing help control the exponential growth of layerwise perturbations (Atwi et al., 9 Nov 2025).

6. Performance Metrics, Limitations, and Practical Considerations

Methodology	Measurement Requirement	Achievable Resolution / Metrics
Plane-to-plane backprop	1 complex field (intensity+phase)	Fourier-limited, analytic, ~λ
Kramers–Kronig (KKSAI)	≥2 intensity images, pupil scans	Closed form; robust to high phase
PIE/eFP	4–7 intensity tilts, simple setup	0.63 nm @ 4.5×10² e⁻/nm², PSNR ~3.6-4.9 dB
Phase plate TEM	3 phase shifted images	PSNR ≥20 dB above 500 e⁻/px
Drift-corrected PS-EH	Hologram series, phase shifts	1 Å limit, phase sensitivity 2π/452 rad
U-Net ML	≥2 (often 3) defocus images	RMSE ~0.03 (simple); generalization limited
One-shot DL	1 intensity image	MAE ~0.12 (restricted); ~0.61 (broad)

• All approaches require careful matching of modeling assumptions to experimental parameters, including spatial frequency coverage, sampling, and inclusion of noise, aberrations, coherence and detector effects.
• Analytical methods are non-iterative but often require multiple intensity measurements or complex-valued inputs.
• Iterative or variational methods may converge to local minima and are sensitive to initialization.
• ML methods' performance is contingent on successful domain adaptation.
• Loss of evanescent (non-propagating) components is intrinsic to free-space propagation-based inversion schemes.

7. Applications and Future Directions

Exit wave reconstruction is essential in atomic-resolution imaging, quantifying electrostatic potentials, mapping electric/magnetic fields, and in electron ptychography for both radiation-hard and beam-sensitive materials. Recent developments have enabled phase-sensitive imaging at sub-nanometer and Angstrom limits, routine operation on standard TEMs (with only beam tilt or phase plate), and computationally efficient, high-throughput pipelines compatible with in-situ and low-dose applications. Future advances will likely focus on:

• Improved robustness to experimental uncertainties (e.g., aberration correction, sample drift).
• Domain-adaptive and semi-supervised ML methods to address experimental-simulation gaps.
• Joint inversion across modalities (e.g., integrating spectroscopic signals).
• Exploiting algorithm unrolling with provable generalization for interpretable, high-accuracy solutions (Atwi et al., 9 Nov 2025).
• Expanding non-iterative approaches to more general specimen classes and imaging geometries.

The field continues to integrate rigorous mathematics, algorithmic innovation, and experimental validation to approach the physical limits of information extraction from intensity-only measurements.