Variable coherence model for free-electron laser pulses

TL;DR

The paper introduces the variable coherence model (VCM) to simulate SASE-FEL pulses with a tunable coherence width, bridging minimally coherent partial-coherence behavior and fully coherent Fourier-limited pulses. The method builds on and extends Pfeifer's partial coherence framework by parameterizing phase variation across the pulse bandwidth through a Lévy process, enabling continuous control of sub-pulse structure while keeping the central frequency and bandwidth fixed. Across three FEL parameter regimes, the authors perform extensive statistics of sub-pulse intensities and counts in both time and frequency domains and examine cross-domain correlations, showing that increasing coherence width narrows distributions and reduces sub-pulse numbers toward a single Fourier-limited pulse. They also demonstrate that the coherence width materially affects nonlinear absorption spectra, requiring many pulses for convergence at low coherence and approaching the fully coherent limit as width grows. The VCM thus provides a practical tool for pulse shaping and for interpreting nonlinear spectroscopic results under realistic, stochastic FEL pulse conditions.

Abstract

We introduce the variable coherence model (VCM) for simulating free-electron laser (FEL) pulses generated through self-amplified spontaneous emission. Building on the established partial coherence model of [T. Pfeifer et. al, Opt. Lett. 35, 3441 (2010)], we demonstrate that the implementation of a variable coherence width allows for continuous control over the pulses' characteristic noise, while keeping the average pulse parameters such as the bandwidth fixed. We demonstrate this through systematic statistical analyses of the intensity and number of sub-pulses in VCM pulses, in both time and frequency. In particular, we analyze how the sub-pulse statistics are affected by the coherence width parameter. We perform our analyses across three distinct regimes of FEL parameters and demonstrate how the VCM can generate pulses that range from maximally random to fully coherent. Finally, we illustrate the effect of the VCM variable coherence width on an absorption simulation.

Figures (15)

• Figure 1: Examples of VCM pulses of equation \ref{['eq:VCM_model']} generated using three different coherence widths $W$, as indicated in the title of each panel. Each pulse is generated using the FLASH parameter set in table \ref{['tab:FEL_table']}.
• Figure 2: Examples of typical VCM pulses with zero coherence width for each parameter set in table \ref{['tab:FEL_table']}: long soft X-ray (LCLS-I), short soft X-ray (LCLS-II), and XUV (FLASH). In each panel, the red markers mark the sub-pulse peaks selected by our sub-pulse detection method -- see section \ref{['sec:Methods:Sub-pulse_detection']}.
• Figure 3: Sub-pulse intensity probability distributions as a function of the coherence width for the FLASH parameters of table \ref{['tab:FEL_table']} in the (a) frequency and (b) time domains. In each panel, the inset zooms in on the lower coherence widths where the probability transitions from low to high average value distributions. The dotted and solid curves mark the mean and mode of each distribution at a fixed coherence width, respectively. The vertical red line marks the bandwidth of the FLASH parameter set (0.55 eV).
• Figure 4: Mean sub-pulse intensities (dotted) and sub-pulse modes (solid) as functions of the relative coherence width, defined as the ratio between the coherence width and average pulse bandwidth, for the LCLS-I (blue), LCLS-II (orange), and FLASH (purple) FEL parameter sets of table \ref{['tab:FEL_table']} in the (a) frequency and (b) time domains. For each data set, the upper and lower bounds of the shaded regions represent the 90$^{th}$ and 10$^{th}$ percentiles of the sub-pulse intensity distributions, respectively. In panel (a), we scaled the FLASH results for visual clarity.
• Figure 5: (a-b) Average number of sub-pulses (curves) plotted against the relative coherence width for the LCLS-I (blue), LCLS-II (orange), and FLASH (purple) FEL parameter sets of table \ref{['tab:FEL_table']} in the (a) frequency and (b) time domains. For each curve, the shaded regions represent $\pm$1 standard deviation around mean value. (c-f) Histograms of the sub-pulse number distributions for three exemplary values of the coherence width for the (c,d) LCLS-I (blue) and (e,f) FLASH (purple) parameter sets in the (c,e) frequency and (d,f) time domains.