Predicting the transverse emittance of space charge dominated beams using the phase advance scan technique and a fully connected neural network

The transverse emittance of a charged particle beam is an important figure of merit for many accelerator applications, such as ultrafast electron diffraction, free electron lasers, and the operation of new compact accelerator concepts in general. One of the easiest to implement methods to determine the transverse emittance is the phase advance scan method using a focusing element and a screen. This method has been shown to work well in the thermal regime. In the space charge dominated laminar flow regime, however, the scheme becomes difficult to apply because of the lack of a closed description of the beam envelope including space charge effects. Furthermore, certain mathematical, as well as beamline design criteria must be met in order to ensure accurate results. In this work, we show that it is possible to analyze phase advance scan data using a fully connected neural network (FCNN), even in setups, which do not meet these criteria. In a simulation study, we evaluate the performance of the FCNN by comparing it to a traditional fit routine based on the beam envelope equation. Subsequently, we use a pretrained FCNN to evaluate measured phase advance scan data, which ultimately yields much better agreement with numerical simulations. To tackle the confirmation bias problem, we employ additional mask-based emittance measurement techniques.

Figure 1

Schematic of a suitable beamline layout for transverse emittance measurements using the phase advance scan technique. The distances are based on what is installed at the ARES linac at DESY, Hamburg.

Figure 2

(a) Schematic of an artificial neuron with $N$ weighted inputs and one output $O$. (b) Exemplary fully connected neural network with one hidden layer.

Figure 3

Diagram showing how the FCNN training, validation, and test data is produced. $N$ is the total number of datasets. $M$ is the number of astra simulations. $M_{\mathrm{in}}$ is the final number of data points to be fed to the FCNN. $M_{\mathrm{out}}$ is the length of the output vector, i.e., the number of predicted beam and simulation parameters. The datasets are split into three categories, where in this study, the common split of $N_{\mathrm{tra}}=0.6·N$, $N_{\mathrm{val}}=0.2·N$ and $N_{\mathrm{tes}}=0.2·N$ is used.

Figure 4

Top: scatter plot of the laminarity parameter $\rho$ at the solenoid according to Eq. (2) vs the bunch charge for all training datasets. $\gamma =6.8$ for this population. The color indicates the laser spot size on the cathode for each dataset. Bottom: distribution of the laminarity parameter.

Figure 5

Top: scatter plot of the fit feasibility parameter according to Eq. (15) vs the bunch charge for all training datasets. $\gamma =6.8$ for this population. The color indicates the laser spot size on the cathode for each dataset. Bottom: distribution of the fit feasibility parameter. Note that the plot shows the values multiplied by 1000 for readability.

Figure 6

Relative error distributions for all verification datasets and label components. The last plot is a radar plot showing the percentage of datasets within a 1% (red), 5% (black), and 10% (gray) relative error interval, respectively. The dashed lines in the distribution plots refer to the same error intervals. $Q$: Bunch charge, ${\sigma }_{t,\mathrm{las}}$: Emission time, ${\sigma }_{x,\mathrm{las}}$: Laser spot size, ${\sigma }_{z,\mathrm{SOL}}$: Bunch length at the solenoid, ${\sigma }_{x^{\prime },\mathrm{SOL}}$: Divergence at the solenoid, ${\sigma }_{x,\mathrm{SOL}}$: Beam size at the solenoid, ${ϵ}_{\mathrm{n},x,\mathrm{SOL}}$: Normalized emittance at the solenoid.

Figure 7

Mean emittance FCNN prediction performance, as well as traditional fit results vs the laminarity parameter (top) and the fit feasibility criterion (bottom). The data are calculated for 100 equally sized laminarity parameter slices. $\gamma =6.8$ for this population. Note that the bottom plot shows the fit criterion values multiplied by 1000 for readability.

Figure 8

Population with less than 5% prediction error on the emittance vs different artificial noise levels. The FCNN layout is the same in both shown cases. Note that in order to enable a direct comparison, the FCNNs used here were trained on the datasets with fixed scan range and constant spacing. This explains the generally better performance compared to Table 3.

Figure 9

Emittance measurements were conducted at the ARES linac at DESY using the phase advance scan technique for different bunch charges. The black dashed line denotes the mean astra simulation result. The shaded areas correspond to the 1, 2, and $3\sigma$ uncertainty of the simulation, based on 100 simulations with normally distributed input parameters.

Figure 10

astra simulation of the evolution of the transverse normalized emittance between gun and grid for a 1-pC bunch with $\gamma =6.8$. The orange dashed line marks the position of the solenoid, which is used to perform the phase advance scan measurements. The black dashed line denotes the mean astra simulation result. The shaded areas correspond to the 1, 2, and $3\sigma$ uncertainty of the simulation, based on 100 simulations with normally distributed input parameters. The blue dashed line shows the evolution of the beam size—the beam reaches its focus slightly before the grid.

Figure 11

Prediction results for other charge dependent beam parameters. The black dashed line denotes the mean astra simulation result. The shaded areas correspond to the 1, 2, and $3\sigma$ uncertainty of the simulation, based on 100 simulations with normally distributed input parameters. The blue dashed line corresponds to an astra simulation using the mean predicted laser spot size and pulse length (see Table 5).