Reconstruction of x-ray free-electron laser pulse duration and energy chirp from spectral intensity fluctuations

Accurate measurements of the x-ray pulse duration produced by x-ray free-electron lasers (XFELs) typically rely on longitudinal electron beam phase space diagnostics, e.g., an x-band transverse deflecting cavity (XTCAV). An alternative group of methods is based on measurements of radiation statistics and spectral correlations. In this paper, we present a statistical analysis of FEL radiation which extends the previous technique by Lutman et al. [Phys. Rev. ST Accel. Beams 15, 030705 (2012)], including the electron beam energy chirp in the FEL gain model. In doing so, we show that with measurements of the average spectrum and the spectral intensity correlations, one can reconstruct the x-ray pulse length, e-beam chirp, and corresponding spectrometer resolution. We validate our approach with 1D and 3D simulations before moving to an experimental demonstration in which our method finds results similar to those obtained from XTCAV measurements.

Figure 1

The steps of the reconstruction method are outlined. (a) shows a set of simulated spectra. These are used to calculate $G_2(\omega ,\delta \omega )$ [Eq. (3)] and the weight function [Eq. (10)] $W(\omega )$ (b). The titles of (a) and (b) act as labels for their color bars. Averaging $G_2(\omega ,\delta \omega )$ weighted by $W(\omega )$ yields $G_2(\delta \omega )$ [Eq. (11)] (d) and an average of the spectra yields $⟨\tilde{S}(\omega )⟩$ (c). Fitting Gaussian functions to each of them allows us to extract widths and amplitudes that can be used to solve for ${\sigma }_t$ (e), $h$ (f), and ${\sigma }_m$ (g) as functions of ${\sigma }_{\omega }$ [Eqs. (7) and (9)]. The black lines indicate these reconstructed values. The blue horizontal lines indicate the true values of each parameter, whereas the green dot indicates the predicted value at the guessed SASE bandwidth.

Figure 2

Results for a scan of the e-beam chirp in 1D simulations. The left and right show the x-ray pulse length and e-beam chirp, respectively, as obtained from the simulation (blue), from our reconstruction model (orange), and from the model of [15] (green).

Figure 3

Results for a scan of the e-beam length in 1D simulations. The left and right show the x-ray pulse length and e-beam chirp, respectively, as obtained from the simulation (blue), from our reconstruction model (orange), and from the model of [15] (green).

Figure 4

Results from 1D simulations scanning the spectrometer resolution for a fixed $7.5\text{−}\mathrm{MeV}/\mathrm{\mu }\mathrm{m}$ e-beam chirp. (a) shows the average radiation power along the undulator marking $z=20\mathrm{m}$ (blue), $z=40\mathrm{m}$ (orange), and $z=50\mathrm{m}$ (green). (b), (c), and (d) show the reconstructed pulse length, e-beam chirp, and spectrometer resolution as a function of imposed spectrometer resolution at those three locations.

Figure 5

A typical single-shot spectrum from 1D simulations at $z=40\mathrm{m}$ showing the impact of 0-, 0.5-, and 1.0-eV measurement resolution.

Figure 6

Results from 3D simulations for three different values of the e-beam chirp. (a) shows the average radiation power for 0, 15, and $30\mathrm{MeV}/\mathrm{\mu }\mathrm{m}$ chirp. (b) and (c) show the reconstructed pulse length and chirp (dots) alongside the true values obtained from the simulations (lines) in the exponential gain and early saturation regimes. (a) also shows the average far field transverse intensity profile as well as the transverse intensity correlation function $\mathrm{\Gamma }(x,y)$ at $z=57.5\mathrm{m}$. The image axes run between $\pm 100\mathrm{\mu }\mathrm{m}$ in both directions.

Figure 7

Representative XTCAV images are shown for 4.7 (a) and 14.7 (b) $\mathrm{MeV}/\mathrm{\mu }\mathrm{m}$ chirps. In each case, the beam has undergone lasing that is evident from the energy loss around 0 fs. Blue lines give estimates of the linear chirp in the lasing regions.

Figure 8

The average spectra (a) and spectral intensity correlation functions (b) are plotted for the two chirp configurations described in the text, including Gaussian fits to each. From these, we extract pulse length (c) and chirp (d) for the two cases as a function of the SASE bandwidth.