Characterizing Photoelectron Momentum Distributions: Comparing Spherical Harmonic Reconstruction Methods

Abstract

This Master’s thesis explores the reconstruction of photoelectron angular distributions using spherical harmonic decomposition with a view of characterize the polarization state of the ionizing light field. Photoelectron angular distributions describe the measured angular distribution of emitted electrons, which follows from the probability of detecting an electron at different angles. These distributions contain information about the polarization properties of the ionizing field.

Three different reconstruction approaches are investigated. Firstly, a direct fitting method is explored, where a coherent sum of spherical harmonics is directly fitted to the angular distribution. Secondly, a projective method is investigated, where the distribution is first projected onto an orthogonal basis of spherical harmonics before being fitted. Finally, a machine learning approach based on a convolutional neural network with a physics-informed decoder is considered. The performance of each method is examined by investigating the reconstruction performance for several angular distributions corresponding to different polarization states, decreasing signal intensities, and varying noise levels. Both additive white Gaussian noise and Poisson noise are used in order to evaluate the methods under experimental-like conditions.

The results show that all three methods are able to accurately reconstruct the parameters describing the polarization state for both high-intensity and moderate-noise distributions. The direct fitting method gives the highest accuracy for varying signal intensity, while the projective method is most robust against Poisson noise. The machine learning method achieves comparable reconstruction performance to the two other methods, and provides more consistent predictions for different distributions, whereas the numerical optimization-based methods show a stronger dependence on the specific polarization state. A limitation observed for all methods is the difficulty in accurately reconstructing polarization states where one of the spherical harmonic coefficients used to describe the angular distribution approach zero.

Overall, the findings give insight into the potentials and limitations of the different reconstruction methods and their performance under different conditions. Possible improvements of the current methods, as well as future applications to more complex systems, are also discussed.

Popular Abstract

Light is an important tool when we want to study the dynamics of electrons in atoms and other small quantum systems. In particular, very short light pulses are useful because they are short enough to capture processes that happen on the timescale of electron motion. In attosecond science, attosecond pulses are used, which are one quintillionth of a second long.

One important property of light is its polarization, which describes the direction in which the electric field of the light oscillates. You might be familiar with the concept from 3D cinema. The two lenses in a pair of 3D glasses filter different polarizations so that each eye receives a differently polarized image, which when combined in our brains will make the image appear three dimensional. But polarization is not only useful in 3D cinema, it can also be used in scientific research. When a photon with sufficient energy is absorbed by the system, an electron can be emitted. These photoelectrons can be detected, and the probability of detecting an electron at different angles forms a photoelectron angular distribution. The shape of this distribution depends on the properties of the ionizing light, meaning that different polarization states produce different patterns. This Master’s thesis investigates how these photoelectron angular distributions can be used to determine the polarization of the light that caused the electron emission.

To describe these patterns, spherical harmonics can be used, which are mathematical functions that can be combined in different ways to describe spherical shapes and patterns. For different polarizations of the light, there will be different coefficients connected to the spherical harmonics. In other words, these coefficients contain information about the light. Therefore, if these coefficients can be retrieved from a given angular distribution, they can be used to determine the polarization of the light that produced the distribution. This coefficient reconstruction is the main focus of this thesis.

Three different reconstruction approaches are investigated in this thesis. The first method directly com- pares a reconstructed angular distribution with the input distribution, and adjusts the coefficients in the reconstructed distribution until the best agreement has been found. For the second method, the angular distribution is first projected onto a new basis of spherical harmonics before being compared to the input distribution. Instead of comparing the full angular distribution pixel by pixel, this gives us a smaller set of coefficients that describe the main features of the distribution, which reduces the number of parameters that are needed to be optimized. The third and final method uses machine learning, more specifically a convolutional neural network. The network is trained to recognize the relationship between the angular distributions and their corresponding coefficients.
To evaluate the methods, a large variety of simulated angular distributions describing different polar- ization states is used. The methods are also tested for varying signal intensities and noise levels, to investigate how well they could perform under conditions similar to those in a real experiment. The types of noise considered are Gaussian noise and Poisson noise. Gaussian noise can be seen as back- ground noise from false particle detections, while Poisson noise describes the statistical fluctuations that arise from having a limited number of detected electrons.
The results show that all three methods can accurately reconstruct the spherical harmonic coefficients for high intensities and moderate noise, especially for relatively high levels of Gaussian noise. The direct method generally maintains the best performance as the signal intensity is reduced, whereas the projective method is most robust, especially with respect to Poisson noise. The machine learning method achieves comparable performance to the other two methods, and gives more consistent results for different polarization states. The two other methods appear to be more dependent on the specific polarization state being reconstructed. They perform much better for some polarization states than for others.

Overall, the results demonstrate that photoelectron angular distributions can be used to retrieve informa- tion about the polarization of the light used for electron emission. The comparison of the three methods highlights their different strengths and limitations, and provides a basis for further development of cur- rent reconstruction methods. Although only relatively simple systems are investigated in this work, the results provide insights which could be useful for further investigation of more complex processes and the use of experimental data. It could especially provide guidance on what signal quality is needed when collecting experimental data.

• author: Randsalu, Nora
• supervisor: Mathieu Gisselbrecht; Emilia My Kjærsdam Telléus
• organization: Synchrotron Radiation Research; Department of Physics
• course: PHYM01 20261
• year: 2026
• type: H2 - Master's Degree (Two Years)
• subject: Physics and Astronomy
• language: English
• id: 9251761
• date added to LUP: 2026-09-25 08:37:56
• date last changed: 2026-09-25 08:37:56