Learning a Latent Pulse Shape Interface for Photoinjector Laser Systems

Alexander Klemps; Denis Ilia; Pradeep Kr. Banerjee; Ye Chen; Henrik Tünnermann; Nihat Ay

cs.LGFeb 19, 2026

Learning a Latent Pulse Shape Interface for Photoinjector Laser Systems

Authors: Alexander Klemps, Denis Ilia, Pradeep Kr. Banerjee, Ye Chen, Henrik Tünnermann, Nihat Ay

Organizations: Institute for Data Science Foundations, Hamburg University of Technology, Germany · Deutsches Elektron-Synchrotron (DESY), Hamburg, Germany

Abstract

Controlling the longitudinal laser pulse shape in photoinjectors of Free-Electron Lasers is a powerful lever for optimizing electron beam quality, but systematic exploration of the vast design space is limited by the cost of brute-force pulse propagation simulations. We present a generative modeling framework based on Wasserstein Autoencoders to learn a differentiable latent interface between pulse shaping and downstream beam dynamics. Our empirical findings show that the learned latent space is continuous and interpretable while maintaining high-fidelity reconstructions. Pulse families such as higher-order Gaussians trace coherent trajectories, while standardizing the temporal pulse lengths shows a latent organization correlated with pulse energy. Analysis via principal components and Gaussian Mixture Models reveals a well behaved latent geometry, enabling smooth transitions between distinct pulse types via linear interpolation. The model generalizes from simulated data to real experimental pulse measurements, accurately reconstructing pulses and embedding them consistently into the learned manifold. Overall, the approach reduces reliance on expensive pulse-propagation simulations and facilitates downstream beam dynamics simulation and analysis.

Figures & tables

Figure 1: Layout of the modeled photoinjector laser system. Initially in the front-end created pulses undergo nonlinear transformations during propagation to the photoinjector system.

Parameter	Distribution	Unit
$A(\omega)$	$\mathcal{U}(\{S,P,F,T,G\})$	-
$p_{T}$	$\mathcal{U}(\{1,2,4\})$	-
$p_{G}$	$\mathcal{U}(\{1,2,3,4,5,10\})$	-
$\sigma_{t}$	$\mathcal{U}([2,\,40])$	ps
$\varphi_{2}$	$\mathcal{N}(0,\frac{100}{3\sigma_{t}^{2}})$	$\text{s}^{2}$
$\varphi_{3}$	$\mathcal{N}(0,\frac{100}{3\sigma_{t}^{3}})$	$\text{s}^{3}$

Table 1: Pulse parameters and sample ranges.

Figure 2: Representative simulated pulse pair before and after propagation through the fiber system. The input pulse (blue) and the propagated pulse (red) are shown as normalized intensity profiles after centroid and support alignment. Propagation leads to temporal reshaping due to dispersion and nonlinear effects, highlighting the variety of pulse morphologies within the dataset.

Figure 3: Structure of the convolutional encoder and decoder forming the Wasserstein Autoencoder. Each block consists of one-dimensional convolutions followed by batch normalization and Leaky-ReLU activations. Residual skip connections are included to stabilize training and preserve fine temporal details. Layer output sizes are shown below each block, and kernel sizes are indicated by numbers inside the convolutional layers. The decoder’s final output layer uses a hyperbolic tangent activation to constrain the reconstruction to the normalized signal range.

Figure 4: Comparison between input pulse profiles from the test dataset (blue) and their corresponding reconstructions (orange) obtained with the trained Wasserstein Autoencoder. The reconstructions capture not only the overall pulse envelope but also higher-frequency oscillatory components, demonstrating the model’s ability to preserve fine temporal structure.

Figure 5: Visualization of the learned WAE latent space projected onto the first two principal components. Each point represents an encoded pulse, color-coded by normalized pulse energy. The overlaid trajectories correspond to parameterized pulse families shown on the left, whose latent embeddings evolve smoothly along coherent paths across the manifold.

Figure 6: Latent interpolation between two pulse profiles. Left: source and target pulses with a

W_{2}

distance of

1.080

. Right: sequence of intermediate decoded pulses from latent interpolations via a discrete linear interpolation path

\Gamma_{0}

(blue) and a

W_{2}

-optimized path

\Gamma^{*}

(orange) using

N=10

waypoints. Both yield smooth transitions w.r.t. the

W_{2}

distance, where the optimized interpolation path attains a near-optimal length of

L(\Gamma^{*})=1.084

compared to slightly higher

L(\Gamma_{0})=1.195

Figure 7: Comparison between experimentally measured and preprocessed pulse profiles (colored solid) and corresponding reconstructions (black dotted) produced by the trained Wasserstein Autoencoder. Each row to the right resembles representative samples from recorded sequences of pulses measured at the photoinjector laser system, in which optical filter parameters were varied through a gradient-based optimization to morph the initial profile (triangle/first column) into a desired target shape (square/last column). Pulses in the first two sequences are infrared ones, whereas the third one contains UV pulses. The reconstructions closely match the measured data, demonstrating high accuracy and generalization to real experimental conditions.

Model	Simulated Test Data			Experimental Data
	MSE $\downarrow$	SNR $\uparrow$	COR $\uparrow$	MSE $\downarrow$	SNR $\uparrow$	COR $\uparrow$
$\beta$ -VAE 0.5	6.58e-3	14.74	0.19	8.17e-3	15.29	0.18
$\beta$ -VAE 0.7	8.17e-3	13.89	0.18	1.19e-2	13.77	0.16
$\beta$ -VAE 1.0	9.95e-3	12.86	0.16	1.33e-2	13.03	0.14
WAE	2.40e-4	28.99	0.67	3.70e-4	28.38	0.78

Table 2: Comparison of

\beta

-VAEs for different

\beta

and WAE on simulated test and experimental datasets.

\uparrow

denotes that higher values indicate better performance, while

\downarrow

indicates that lower values are preferable.

Figure 8: Linear correlations between the first 5 latent principal components and

\varepsilon_{xy}

. Oscillations between

2.5\,\mathrm{m}

and

4.5\,\mathrm{m}

arise from an accelerating module within the accelerator lattice, causing fluctuations in transverse momenta.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 9: Left panel: PCA projection of the train and test embeddings colored w.r.t. the fit GMM components and their means. Middle panel: Decoded means of the GMM components. Correspondence to components is indicated by the coloring from the left panel. Right panel: Pairwise normalized 2-Wasserstein distances between GMM components.

Figure 10: Pulse shapes decoded from latent samples drawn via a latent Gaussian mixture model (orange) are interpreted as probability density functions for electron emission time. Inverse transform sampling produces zero-centered emission time histograms (blue) of

200\,000

particles each that faithfully match the decoded pulse profiles.

Figure 11: Left panel: Distribution of reconstruction MSE on the test set for different values of $\beta$ and the WAE. Middle panel: Reconstruction of a representative flattop pulse, illustrating progressive loss of sharp temporal features with increasing $\beta$ , while the WAE preserves the target shape. Right panel: PCA projection of the learned latent spaces with trajectories of selected pulse families, showing degraded geometric organization for $\beta$ -VAEs compared to the smooth and interpretable structure obtained with the WAE.

Representation Learning

Representation learning develops data transformations that expose useful structure for prediction, transfer, generation, and control.

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