Fig. 1. The stages of the Bayesian analysis procedure are summarized. Initially, a distribution of pre-collision electron spectra is predicted by a neural network (for simplicity only one pre-collision spectrum is shown). The pre-collision spectrum is decomposed into a sum of Gaussian sub-bunches that are fed into the inference procedure. The MCMC returns three inference parameters, the laser , longitudinal displacement of the collision from the laser focus, , and the electron beam duration, , which are used to reconstruct the pre-collision phase space of the electron beam and the laser electric field it experiences at the collision. This information is supplied to the forward model (in this case the classical, quantum-continuous or quantum-stochastic model), which predicts the post-collision electron spectrum and photon spectrum for each sub-bunch. The full post-collision electron and photon spectra are obtained by performing a charge-weighted sum over the sub-spectra predicted for each sub-Gaussian. The model predictions, measured post-collision electron and photon spectra and their uncertainties are used to compute the posterior probability, which allows the MCMC algorithm to predict the subsequent region of the posterior to sample. Once the MCMC has converged, model comparison is performed using Bayes factors computed for the different models.

Fig. 2. A collision between an electron beam (red) and a tightly focused, counter-propagating laser (normalized field strength shown in blue) is depicted. The electron beam charge is normally distributed both spatially and temporally, with duration , source size and energy-dependent divergence . The laser intensity, which is proportional to the square of the normalized intensity parameter, , has Gaussian spatial and temporal dependence. The laser waist, , and duration, , are indicated. The collision is longitudinally and transversely offset from the laser focus (yellow cross) by and , respectively.

Fig. 3. (a) The decomposition of a pre-collision electron spectrum predicted by a neural network (cyan) into Gaussian sub-spectra (purple), the sum over which (black) reproduces the original spectrum. (b) The phase-space projection (centre) of a single Gaussian sub-spectrum with the mean, , and standard deviation, , Lorentz factor demarcated by continuous and dashed vertical black lines, respectively. The location, , and width, , of its longitudinal distribution are indicated by continuous and dashed horizontal black lines, respectively. The longitudinal (left) and spectral (bottom) distributions of the Gaussian sub-spectrum (obtained by integrating its phase-space distribution over the spectral and longitudinal axes, respectively) are shown in cyan. (c) Decomposition of the phase-space distribution in (b) into femto-bunches (magenta) with varying numbers of electrons, , evenly spaced mean longitudinal positions, , and 0.85 fs durations, where the latter two properties are indicated for a single femto-bunch by continuous and dashed black horizontal lines. The sum over the femto-bunches yields the spectral (bottom) and longitudinal (left) distributions shown in cyan.

Fig. 4. Overview of the forward models used to predict the post-collision electron and photon spectra. Once the phase-space decomposition has been performed, the mean and standard deviation Lorentz factor and mean longitudinal position of each femto-bunch are fed into five interpolation tables together with the laser . Each interpolation table generates a single output, three of which describe the post-collision electron spectrum location, , scale, , and shape factor, , while the remaining tables output the critical factor, , and photon number, , of the photon spectrum. The interpolation table outputs are used to obtain the post-collision electron and gamma spectra for each femto-bunch, which are then weighted by the number of electrons in the pre-collision femto-bunch and summed, yielding the full post-collision electron and photon spectra, respectively.

Fig. 5. (a) The post-collision electron spectrum obtained from a Monte Carlo simulation for a collision between an electron beam with initial and and a laser with where . The reconstructed electron spectrum obtained using the interpolation tables (magenta) shows good agreement with the simulated post-collision spectrum. (b) The photon spectrum simulated using a Monte Carlo code for the parameters provided in Figure 5 is shown alongside the fit thereto (with Equation (4)) and the photon spectrum constructed using the interpolation tables.

Fig. 6. The mean Lorentz factor of the post-collision electron spectrum predicted by the classical and quantum-stochastic models varies with the deviation of a given collision parameter from its mean value, normalized by the standard deviation. This choice of normalization factor illustrates the probability that a parameter will deviate from its mean value by a given amount.

Fig. 7. Similar to Figure 6, where the standard deviation of the post-collision electron spectrum is shown along the y-axis.

Fig. 8. The effect of electron beam divergence and source size on the relative transverse sizes of the electron beam and colliding laser is shown as a function of longitudinal displacement from the electron beam source and the laser focus, respectively.

Fig. 9. The location of the post-collision electron Lorentz factor, , as a function of electron beam source size and longitudinal displacement of the collision from the laser focus for the classical and quantum-stochastic models.

Fig. 10. The scale of the electron spectrum, , predicted by the classical and quantum-stochastic models of radiation reaction as the electron beam source size and the longitudinal displacement of the collision from the laser focus are varied.

Fig. 11. The location, , of the post-collision electron Lorentz factor distribution predicted by the classical and quantum-stochastic models of radiation reaction is shown with varying longitudinal and transverse displacement of the collision from the laser focus.

Fig. 12. The scale of the post-collision electron Lorentz factor distribution, , predicted by the classical and quantum-stochastic models of radiation reaction for varying transverse and longitudinal alignment between the electron beam and the colliding laser.

Fig. 13. The percentage difference between the simulated and inferred values for the average, , and standard deviation, , of the post-collision electron Lorentz factor distribution, and the average energy of the photon distribution, , are shown as the longitudinal and transverse offset of the collision from the laser focus and the electron beam source size are varied. The total error is given by the root mean squared deviation of the inferred , and from the simulated values.

Fig. 14. The quantum-stochastic model of radiation reaction was used to simulate a collision between a focusing, Gaussian laser pulse with , and (the remaining laser and electron beam parameters are provided in Tables 2 and 3, respectively) and the pre-collision electron spectrum. Simulated data and classical, quantum-continuous and quantum-stochastic inferences are shown in red, green, blue and magenta, respectively. This colour scheme will be used consistently for the remaining figures in this section. (a) The simulated post-collision electron spectrum, predicted pre-collision electron spectra (orange), and its median (black), alongside the inferred post-collision electron spectra. (b) The simulated and inferred responses of the photon spectrometer as a function of photon propagation depth.

Fig. 15. Inference parameters obtained for the first test case, where the quantum-stochastic model was used to simulate the collision. The collision parameters inferred by the classical (green), quantum-continuous (blue) and quantum-stochastic (magenta) models are compared to the simulation input parameters (red star). (a) , the average effective collision that the electron beam interacts with during the collision. The collision distribution of stems from the finite size of the electron beam, the spatio-temporal dependence of laser intensity and their overlap. Hence, is a function of all three inference parameters. (b) The mean and standard deviation of the collision distribution of due to the broadband electron spectrum and the range of the electron beam experiences during the collision.

Fig. 16. The classical model of radiation reaction was used to simulate a collision between a focusing, Gaussian laser pulse with , , and an electron beam, which were offset transversely by 1.05 μm (the remaining laser and electron beam parameters are provided in Tables 2 and 3, respectively). (a) The simulated post-collision electron spectrum, predicted pre-collision electron spectra (orange), and its median (black), alongside the inferred post-collision electron spectra. (b) The simulated and inferred responses of the photon spectrometer as a function of photon propagation depth.

Fig. 17. Similar to Figure 15, where the input and inferred parameters pertain to the transversely offset classical test case.

Fig. 18. The quantum-stochastic model of radiation reaction was used to simulate a collision between a focusing, Gaussian laser pulse with , , and transverse offset of 2.1 μm (the remaining laser and electron beam parameters are provided in Tables 2 and 3, respectively) and the pre-collision electron spectrum. Simulated data and classical, quantum-continuous and quantum-stochastic inferences are shown in red, green, blue and magenta, respectively. (a) The simulated post-collision electron spectrum, predicted pre-collision electron spectra (orange), and its median (black), alongside the inferred post-collision electron spectra. (b) The simulated and inferred responses of the photon spectrometer as a function of propagation depth.

Fig. 19. Similar to Figure 15, where the input and inferred parameters pertain to the transversely offset stochastic test case.