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Optical Point-Spread Function via Defocus Phase Retrieval in Scanning Electron Microscopy

This paper presents a defocus phase retrieval method that successfully reconstructs the optical point-spread function of an uncorrected scanning electron microscope by analyzing a focal series of images from gold nanoparticles, thereby enabling experimental aberration quantification at 20 keV.

Original authors: Surya Kamal, Kevin Mogere Nyaburi, Richard K. Hailstone

Published 2026-09-03

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CC BY 4.0

Original authors: Surya Kamal, Kevin Mogere Nyaburi, Richard K. Hailstone

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

For nearly a century, the scanning electron microscope has stood as a cornerstone of modern science, allowing researchers to peer into the microscopic world with a clarity that was once unimaginable. Unlike its cousin, the transmission electron microscope, which requires samples to be sliced paper-thin, the scanning version scans a focused beam of electrons across a surface, revealing details about materials, electronics, and even forensic evidence. However, this powerful tool has a hidden flaw. Just as a camera lens with a scratch or a smudge blurs a photograph, the magnetic lenses inside the microscope suffer from imperfections called aberrations. These flaws distort the electron beam, smearing the image and preventing the machine from reaching its theoretical limits of sharpness. For decades, scientists have tried to fix these lenses by adding complex, expensive hardware to cancel out the distortions, but a simpler question has remained: can we measure exactly how the beam is distorted without needing to fix the machine first?

A team of researchers at the Rochester Institute of Technology has answered this question by developing a new way to see the invisible shape of the electron beam itself. In a standard microscope, the beam hits the sample and creates an image, but the information about the beam's internal structure—specifically its phase, or the timing of its waves—is lost in the process. It is like trying to understand the shape of a shadow without knowing the object casting it. The researchers realized that by taking a series of images of tiny gold nanoparticles while deliberately shifting the focus up and down, they could work backward to reconstruct the missing information. They did not need to install new hardware or correct the lenses; instead, they used a mathematical process to recover the lost phase data from the changing patterns of light and dark in the images.

The experiment involved a standard, uncorrected scanning electron microscope operating at an energy of 20 kiloelectronvolts. The team placed a film containing gold nanoparticles under the beam. They then captured a sequence of images, moving the sample in precise steps of one micrometer along the vertical axis to create a "focal series." In each image, the nanoparticles appeared slightly different depending on how far the beam was from its perfect focus point. By stacking these images together, the researchers first reconstructed the intensity, or brightness, of the electron beam at each step. This intensity acts as a blurring function, determining how much the image of the nanoparticle gets smeared out.

Once they had the intensity maps, the team applied an iterative algorithm to recover the missing phase information. This process is akin to solving a puzzle where the pieces are the images taken at different focus levels. The algorithm starts with a guess about the beam's shape and then repeatedly adjusts it, checking if the guess would produce the observed images when the focus is shifted. With each cycle, the guess becomes more accurate until it perfectly matches the experimental data. The result was a complete, three-dimensional description of the electron wavefunction, including both its intensity and its phase, right at the surface of the sample.

With this full description in hand, the researchers were able to calculate the optical point-spread function of the microscope's lenses. This function is a map that shows exactly how the optics distort a single point of light. In their study, the researchers intentionally introduced a specific type of distortion known as astigmatism, where the beam focuses at different points in different directions, creating an oval shape rather than a circle. The resulting maps clearly showed this distortion, appearing as elongated, contour-like features that matched the theoretical expectations of an astigmatic wave. By comparing their experimental results with computer simulations of a beam passing through a lens with similar distortions, they confirmed that the strange patterns they saw were real physical effects of the aberrations, not just digital artifacts.

The significance of this work lies in its ability to diagnose the health of the microscope's optics without expensive corrections. The researchers demonstrated that they could quantify the aberrations and visualize the point-spread function for multiple datasets, each with different levels of intentional distortion. This approach offers a pragmatic metric for evaluating the quality of the electron optics, moving beyond simple measurements of beam width to a full understanding of the wavefront. While the study focused on uncorrected microscopes, the method provides a foundation for future strategies that could use this wavefront sensing to guide automated corrections or improve image reconstruction. The researchers have made their code and data available to the scientific community, inviting others to use this method to better understand and improve the performance of electron microscopes.

Technical Summary: Optical Point-Spread Function via Defocus Phase Retrieval in Scanning Electron Microscopy

Problem Statement
While Scanning Electron Microscopy (SEM) is a robust imaging tool, its practical resolution is often significantly lower than theoretical limits due to optical aberrations. In conventional, uncorrected SEMs (those without multipole correctors), there is a lack of standardized methods for aberration quantification. Unlike Transmission Electron Microscopy (TEM), where the exit wave is directly imaged, the focused electron probe in an SEM loses phase information as it interacts with the specimen. Consequently, the "probe intensity distribution" is often misidentified as the system's Point-Spread Function (PSF). The authors argue that this is a misnomer; the probe intensity acts as a blurring function, while the true optical PSF ( $PSF_{optics}$ ) describes the coherent wavefront aberrations of the probe-forming optics. Accurately characterizing this $PSF_{optics}$ is critical for understanding system performance and developing pragmatic metrics for optical quality, yet recovering the lost phase information of the electron probe remains a challenging task.

Methodology
The authors propose a non-interferometric, wave-optical approach to recover the complete electron probe wavefunction (both intensity and phase) and subsequently derive the $PSF_{optics}$ in an uncorrected SEM. The methodology proceeds in three main stages:

Probe Intensity Reconstruction:
- Data Acquisition: A focal series of Back-Scattered Electron (BSE) images is captured of gold (Au) nanoparticles on a carbon film using a TESCAN MIRA3 SEM at 20 keV. The specimen is moved in controlled defocus steps ( $\Delta z = 1 \, \mu m$ ) to create a through-focal series.
- Signal Processing: Individual nanoparticles are segmented and stacked to improve the Signal-to-Noise Ratio (SNR). An ideal object structure is generated via Monte Carlo simulation (CASINO).
- Deconvolution: Using a Wiener-filter approach, the authors deconvolve the real image from the ideal object structure to reconstruct the probe intensity ( $|\psi_{sp}|^2$ ) at each defocus plane. This generates a focal series of probe intensities.
Iterative Phase Retrieval:
- The reconstructed focal series of probe intensities serves as the "ground truth" constraints for an iterative phase retrieval algorithm based on defocus diversity.
- The algorithm initializes the probe wavefunction at the in-focus plane with a constant phase. It propagates the wavefunction to adjacent planes using the Angular Spectrum Method (ASM).
- At each plane, the magnitude of the propagated wavefunction is constrained by the experimentally reconstructed probe intensity, while the phase is retained. A relaxation parameter ( $\gamma$ ) is used to update the wavefunction.
- This loop continues until the normalized sum-squared error between the predicted and ground-truth magnitudes at the in-focus plane converges, yielding the recovered phase map for the entire focal series.
$PSF_{optics}$ Calculation:
- With the complete probe wavefunction (intensity and recovered phase) known at the specimen plane, the authors invert the convolution relationship defined in the theory section.
- By dividing the Fourier spectrum of the complete probe wavefunction by the Fourier spectrum of the geometrical demagnified virtual source, they calculate the coherent point-spread function of the optics ( $PSF_{optics}$ ) via inverse Fourier transform.

Key Results

Phase Recovery: The method successfully recovered the phase information of the electron probe for three distinct datasets, each with different levels of controlled astigmatism introduced via stigmators.
Visualization of Aberrations: The reconstructed $PSF_{optics}$ clearly visualized the effects of aberrations. Specifically, the contour features in the $PSF_{optics}$ rotated as the focal plane was traversed, indicating the presence of the Gouy phase shift associated with astigmatic beams.
Qualitative Validation: The experimental $PSF_{optics}$ contours were qualitatively compared to a simulation of a coherent wavefront passing through a 2-fold astigmatic mask. The simulation reproduced the contour-like features observed experimentally, supporting the interpretation that these structures arise from coherent wavefront aberrations rather than numerical artifacts.
Dataset Specifics: The study utilized three datasets with varying astigmatism settings (e.g., Data Set 1: 0.4% x-stigmator, 0.8% y-stigmator). The phase retrieval converged with low error metrics (e.g., $\xi \approx 1.82 \times 10^{-4}$ for Data Set 1).

Significance and Claims
The paper claims to represent an important step toward experimental aberration quantification in uncorrected SEMs without the need for expensive multipole hardware. The primary contributions are:

Methodological Shift: Moving beyond Gaussian beam kernel approximations and hardware-specific feedback loops (used in previous automatic correction methods) to a full wave-optical characterization of the electron probe.
Diagnostic Metric: Establishing the $PSF_{optics}$ as a pragmatic metric for evaluating the true quality of SEM optics, distinct from simple probe width measurements.
Feasibility: Demonstrating that it is possible to recover the lost phase information and reconstruct the complete optical PSF using standard SEM equipment and non-interferometric phase retrieval techniques.

The authors maintain a modest scope, noting that while they have demonstrated the reconstruction of the $PSF_{optics}$ , a rigorous quantitative estimation of specific aberration coefficients and a full simulation of the optical system using those coefficients are left for future work. The current work serves as a proof-of-concept for wavefront sensing in SEMs to enable future aberration correction strategies.

Technical Summary: Optical Point-Spread Function via Defocus Phase Retrieval in Scanning Electron Microscopy

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