<figure><img src="image_187.png" /><figcaption>**Fig. 23.11.** Slice analysis with a slice thickness of 0.1 µm. The longitudinal axis corresponds to the slice position with respect to beam centre of mass. a) Current (black), 5D brightness $B_{5D} \equiv 2I/(\pi^2\epsilon_{nx} \times \epsilon_{ny})$ (blue), and 6D brightness $B_{6D} \equiv B_{5D}/(\sigma_E/E/10^3)$ (red). b) Emittances in the *x* (red) and *y* (blue) directions. c) Energy spread using the RMS estimator (red, dashed) or the Mean Absolute Deviation robust indicator (black) (reprinted from [594] with permission, copyright IOP Publishing).</figcaption></figure>

The resulting evolution of the beam envelope, normalised emittance, and other beam parameters along the HETL is shown in Figure 23.12. The properties of the transport line components are listed in Section 19.2.2, while the final electron beam parameters are given in Table 23.6, reproduced from Chapter 19.

## Sensitivity of the Final Beam to Parameter Fluctuations

The final beam quality can be affected in an LWFA stage by several laser and target parameters. Fluctuations of the laser pulse energy are directly linked to variations in both the accelerating field amplitude and phase of the plasma wave, while (even tiny) fluctuations of the background plasma density can affect the resonance condition for the wave excitation. A full evaluation of the final beam parameter stability against the relevant working point parameters has already been performed for a setup related to the 150 MeV injector for Scheme 2 (LPI-150 MeV + LPAS-5 GeV), where a background density of about $1 \times 10^{18} \text{ cm}^{-3}$ was used. In that context, we found that the selected working point is stable, provided that reasonable requirements for laser / target fluctuations are satisfied [591,594]. The multi-pulse nature of the wakefield excitation introduces more constraints on the suitable parameter space, with the pulse-to-pulse delay $T_D$ jitter, for example, being responsible for the loss of the
<figure><img src="image_188.png" /><figcaption>**Fig. 23.12.** High-energy transfer line for Scheme 1: Beam beta function (a), normalised emittance (b), transverse size (c), and bunch duration and energy spread (d) of the core beam along the transport line for the case of one LPI (5 GeV). The calculations were performed with the tracking code TraceWin.</figcaption></figure>

<table><caption>**Table 23.6.** High-energy transfer line for Scheme 1: Comparison between the beam parameters at the end of the LPI and at the entrance of the undulator. These parameters have been calculated considering only the beam core, i.e. only a 11.1 fs long slice in z around the peak current. This bunch core contains 30 pC of charge. The slice parameters and peak current are calculated assuming 0.1 µm long slices.</caption><thead><tr><th></th><th>LPI exit</th><th>Undulator entrance</th></tr></thead><tbody><tr><td>β<sub>x</sub> [m]</td><td>0.030</td><td>3.17</td></tr><tr><td>β<sub>y</sub> [m]</td><td>0.029</td><td>7.28</td></tr><tr><td>α<sub>x</sub> –</td><td>-1.06</td><td>-0.70</td></tr><tr><td>α<sub>y</sub> –</td><td>-1.03</td><td>1.57</td></tr><tr><td>ε<sub>n,x</sub> [µm]</td><td>0.075</td><td>0.095</td></tr><tr><td>ε<sub>n,y</sub> [µm]</td><td>0.050</td><td>0.064</td></tr><tr><td>Mean Energy [GeV]</td><td>4.96</td><td>4.96</td></tr><tr><td>σ<sub>γ</sub>/γ [10<sup>-3</sup>]</td><td>9.0</td><td>9.0</td></tr><tr><td>Î [kA]</td><td>3.8</td><td>3.8</td></tr><tr><td>σ<sub>τ</sub> [fs]</td><td>2.9</td><td>2.9</td></tr><tr><td>ε<sub>n,x</sub> (slice) [µm]</td><td>0.069</td><td>0.068</td></tr><tr><td>ε<sub>n,y</sub> (slice) [µm]</td><td>0.041</td><td>0.041</td></tr><tr><td>σ<sub>γ</sub>/γ (slice) [10<sup>-3</sup>]</td><td>0.8</td><td>0.8</td></tr></tbody></table>

resonance condition (see [594] for further details). Therefore, the time shaper that generates the driver train must possess a very good stability. In our selected configuration, the TEMPI scheme shows a negligible pulse-to-pulse jitter, since the replicas of the stretched pulse (about 1 ps long) are produced by a mechanically stable stack
<figure><img src="image_189.png" /><figcaption>**Fig. 23.13.** Sensitivity of the accelerating field (blue lines) on the resonance condition mismatch. The horizontal axis refers to the number of plasma periods, and in the vertical axis both the accelerating field and pulse amplitude (orange) are shown. While in the optimised case (solid lines) the peak accelerating field reaches the value of $E_{z,ref} = 0.7E_0$, the wakefield excited in a plasma with a background density increased by 5% (dashed lines) has a peak value of $E_z = 0.55E_0$, showing a reduction of more than 20% (reprinted from [594] with permission, copyright IOP Publishing).</figcaption></figure>

of birefringent crystals and polarisers. We also mention that the time jitter between the pulse train and the ionising pulse possesses potential detrimental effects on the beam quality. However, in our proposed setup, only mechanical vibrations can lead to this jitter, limited to a few µm, because the driver and ionisation pulses are amplified replicas of the same master pulse. The phase jitter of the bunch in the plasma wave is therefore very low for a plasma wavelength of $\lambda_p \simeq 75$ µm.

We point out that target density fluctuations represent the most severe source of beam-quality fluctuations. The resonance condition is related to the plasma wavelength, which is dependent on the local background density $n_0 = n_{0,ref} + \delta n_0$. As the efficient resonant excitation of the wave can be expressed as $\delta n_0/n_{0,ref}/2 + \delta T_D/T_{D,ref} \ll 1/N$, where N is the number of pulses in the train (see [594]), very low background density fluctuations on the percent level can be acceptable. This is emphasized in Figure 23.13, where the pulse train and its excited wakefield are shown in two different scenarios: the optimised case with the pair of parameters ($T_{D,ref}$, $n_{0,ref}$) (solid lines) and a case with the “perturbed” pair ($T_{D,ref}$, $n_{0,ref} + \delta n_0$) with $\delta n_0 = 5 \cdot 10^{-2} n_{0,ref}$ (shown with dashed lines). From Figure 23.13, we can infer that a variation of $5 \times 10^{-2}$ of the plasma density, or equivalently a variation of about $2.5 \times 10^{-2}$ of the time delay between all the pulses, will induce a reduction of more than $20 \cdot 10^{-2}$ in the plasma wave amplitude, thus reinforcing the claim that the background plasma density must be controlled at an approximately 1% level.

**Summary of start-to-end simulations for Scheme 1: LPI-5 GeV:**

Start-to-end simulations have been carried out for this configuration. All parameters from the LPI to the HETL let us conclude that this scheme is fully compatible with the EuPRAXIA objectives. Currently, the margin with regard to the required beam parameters is still quite narrow, but further optimisation and thorough tolerance studies are planned in the future.
<figure><img src="image_190.png" /><figcaption>Fig. 23.14. Scheme 2 combining a 150 MeV laser-plasma injector with an LPAS to 5 GeV.</figcaption></figure>

## 23.8 Start-to-End Simulations for Scheme 2: LPI-150 MeV + LPAS-5 GeV

This configuration includes two laser-plasma stages, a laser-plasma injector (LPI) with an output beam energy of 150 MeV, and a laser-plasma acceleration stage (LPAS) with 5 GeV (and also 1 GeV) output energy. These are connected by two transfer lines: a low-energy one (LETL) transporting the witness electron beam from the LPI to the LPAS, and a high-energy one (HETL) driving the beam from the LPAS to the application beamline. Again these include both FEL and other applications, but the FEL case is chosen exemplarily due to its more demanding requirements. As concluded in Section 23.3, the quasi-linear acceleration regime is considered here for the LPAS.

### Optimisation of Particle Injection and Acceleration Within the LPI

The injection scheme considered for the LPI in this case is resonant multi-pulse ionisation injection (ReMPI) [170,591]. It is an injection / acceleration scheme aiming at generating high-quality electron bunches in a flexible fashion in terms of final energy, charge, and duration. Due to beam-loading effects, a configuration with relatively high emittance (about 0.3 mm mrad normalised emittance) and high charge (about 30 pC) is achieved. With a final RMS bunch duration of 3.2 fs, the peak current after the LPI is about 4 kA, while the energy spread has been limited down to 1.7%, well below the 5% limit stated in Table 23.1.

To operate the ReMPI scheme, a single pulse needs to be delivered by a Ti:Sa laser system. This pulse is split into three sub-pulses: a small fraction of laser energy is used to ionise the gas in the plasma target (pre-ionised nitrogen 5+) in order to extract the electrons; another tiny fraction with perpendicular polarisation is used to make the electron beam transversely symmetrical; the remaining main part of the laser is split further into four micro-pulses to resonantly drive the wakefield, as shown in Figure 23.15. The ionising pulse is frequency-tripled by a non-linear crystal and tightly focused behind the wake-driving pulse train. The extracted electrons are quickly trapped by the wake and accelerated up to a final energy of 150 MeV. A round beam is preferred for both the next optics / boosting stages and for minimising beam loading in the current stage; this can be achieved through the application of the perpendicularly polarised laser sub-pulse. No intrinsic timing jitter will be present, as all the laser pulses are fractions of the single initial pulse.

The injection process lasts across a distance of about 250 µm, as a snapshot of the longitudinal field and the electron phase space shows in Figure 23.16 (left). Afterwards, a 31 pC bunch is trapped and longitudinally compressed. After the acceleration phase (about 3 mm from the ionisation pulse focus), the electron bunch is tightly focused (with an RMS size in x, y of about 0.5 µm) and the longitudinal phase-space plot is now as shown in Figure 23.16 (right).

The remarkably low value of 1.6% (RMS) energy spread has been obtained after optimising the density profile according to the non-linear variation of the plasma
<figure><img src="image_191.png" /><figcaption>**Fig. 23.15.** The longitudinal normalised electric field, resonantly excited by a train of laser pulses (red surface).</figcaption></figure>

<figure><img src="image_192.png" /><figcaption>**Fig. 23.16.** The longitudinal phase-space density of the bunch close to the end of the injection process (left) and at the end of the acceleration phase (right). The laser pulses and the electron bunch move from the left to the right. The four driving pulses are represented in black. The amplitudes (a.u.) of the longitudinal electric field on axis, the ionisation pulse, the radial electric field, and the radial force on the relativistic particles are represented by the thin blue, thick blue, red, and green lines, respectively.</figcaption></figure>

wavelength due to relativistic quivering. Moreover, a partial flattening of the accelerating gradient due to beam loading assisted further in reducing the final energy spread. The transverse phase space evolves smoothly as a round beam with a stable size. Unfortunately, beam loading introduces non-linear transverse gradients close to the beam axis, which induces emittance growth up to about 25% in a 2.5 mm propagation distance.

As a final result, the projections of the bunch phase space at the end of the acceleration section of the LPI can be found in Figure 23.17 together with the relevant wakefields.

## Optimisation of the Downramp and Passive Plasma Lens at the LPI Exit

Although this beam has all the features to be properly injected into the next LPAS, its Twiss parameter $\gamma = 5000 \text{ m}^{-1}$ is still too big, i.e. its divergence is too large.
<figure><img src="image_193.png" /><figcaption>**Fig. 23.17.** Electron beam phase space (position vs. normalised momentum p/mc) at the end of the acceleration phase. Red lines represent the longitudinal electric field on axis (top left), the longitudinal electric field at the centre of the beam (top right), and the radial force at the centre of the beam (bottom).</figcaption></figure>

Combined with the 1.6% energy spread, this makes it difficult to transport the beam to the LPAS entrance without significant degradation in beam quality. According to the recommendations in Section 23.6, a density downramp assisted by a passive plasma lens (Fig. 23.18) has thus been introduced to drastically reduce $\gamma$ to 130 m$^{-1}$.

Due to the resonance condition, the mean electric field acting on the bunch varies in a different manner from a single-pulse case, as the density reduces on the ramp. After an optimisation of the density downramp scale, an optimum scale length of 1 mm has been found. At the end of a density downramp of length 1.3 mm, the energy spread has increased up to a value of 1.7% and the Twiss $\gamma$ reaches 1700 m$^{-1}$, which is definitely still too large for a successful beam transport. A flat-density passive plasma lens has been, therefore, inserted just after the downramp so as to gently stabilise the electron beam evolution. The passive plasma lens effect strongly reduces the Twiss parameter by more than a factor of 10 in less than 3 mm of propagation with a tiny increase in normalised emittance.

## Main Features of Particle Injection and Acceleration in the LPI and Downramp

In the following, the laser and plasma parameters used in the simulation study of the LPI are listed together with the achieved output electron beam properties:

- **Laser:** bi-Gaussian, ionising laser: 3rd harmonic, P = 1 TW, E = 0.07 J, $a_0$ = 0.53, $w_0$ = 3.8 µm, $\tau_{FWHM}$ = 45 fs; symmetrisation laser: 3rd harmonic, delay 40 fs, P = 0. TW, E = 0.02 J, $a_0$ = 0.14, $w_0$ = 11 µm, $\tau_{FWHM}$ = 25 fs, drive
<figure><img src="image_194.png" /><figcaption>**Fig. 23.18.** Twiss parameters and chromatic length along the plasma downramp and the passive plasma lens.</figcaption></figure>

laser: delay 160 fs, 4 pulses, each of them has P = 30 TW, E = 0.98 J, a₀ = 1, w₀ = 30 µm, τFWHM = 30 fs (the total need is P=240 TW, E=8 J, because of a 50 % conversion efficiency to 4 pulses)

- **Plasma:** n₀ = 5 × 10¹⁷ cm⁻³, N pre-ionised up to 5+, radially uniform, 3.5 mm length + 1 mm downramp + 3 mm passive plasma lens, where n₀ = 1.4 × 10¹⁶ cm⁻³

- **Output electron beam:** Q = 31 pC, E = 150 MeV, εn,x,y = 0.31, 0.27 mm mrad, β₀ = 0.008 m, γ₀ = 130 m⁻¹, σE/E = 1.7%, τFWHM = 7.9 fs

## Optimisation of Beam Transfer in the LETL

Several different setups for the low-energy transfer line have been investigated and are described in detail in Chapter 19. A clear decision on the optimal design will require further analysis and experimental testing; a decision on a final configuration will thus need to be taken during the technical design phase. For the current setup, the following transfer line design has been used exemplarily in the simulation study.

The objective here is to transfer the output beam from the LPI plasma lens, where β = 0.008 m, γ = 130 m⁻¹, and σ = 2.9 µm, to the LPAS entrance, where β = 0.03 m, γ = 67 m⁻¹ and σ = 10.0 µm, while preserving all other beam qualities. In particular, the beam emittance should practically not be degraded. The optimisation process in this sense is described in Section 23.5. The proposed solution is a 0.7 m long line with six permanent quadrupole magnets: one triplet to catch the divergent beam on the LPI side and another triplet to symmetrically refocus the beam on the LPAS side. These are connected by a drift section in between, where the beam is round and parallel, reserved for diagnostic devices. Note that, instead of six quadrupoles, also two strong solenoids or two active plasma lenses could be used, as discussed in Chapter 19.

The particle distribution at the plasma lens exit looks compact without any tail. The whole distribution can thus be used. The charge and normalised emittance at the LETL exit are Q = 31 pC and εtr,n = 0.61 mm mrad. The evolution of the beam envelope and normalised emittance along the LETL are shown in Figure 23.19. The
<figure><img src="image_195.png" /><figcaption>**Fig. 23.19.** Evolution of the beam size and emittance along the LETL. The six quadrupoles are symbolised by the green boxes.</figcaption></figure>

LETL setup can be seen in this figure as well. The six permanent quadrupoles are each 45 mm long with a 211 T/m maximum gradient.

## Optimisation of Particle Acceleration Within the LPAS

This LPAS was optimised in parallel with the LPI, when the optimised input beam coming from the latter was not yet available. In waiting, a 6D-Gaussian beam with characteristics as required in Table 23.1 at 150 MeV was used for the optimisation. The obtained results are described in the following, and first results with the LPI beam will be given subsequently.

The objective of the LPAS is to boost the witness electron beam to 5 GeV with the required high beam charge and low emittance, low divergence, and low energy spread [582]. The technique of quasi-linear acceleration in a parabolically pre-formed plasma is applied. The laser and plasma parameters are defined following the scaling laws for a resonant wakefield. The plasma depth is defined to match the laser injection. The transversal beam size is defined to minimise emittance growth, and the longitudinal beam size is defined to minimise the energy spread by using the beam-loading effect [595]. The obtained transverse and longitudinal phase spaces after acceleration in the LPAS are shown in Figure 23.20. Simulations have been performed with the 3D PIC code Warp, where the boosted-frame technique is used to speed up the computation [354].

For an acceleration up to only 1 GeV, only a very first study was performed. Starting from the 5 GeV configuration, the first optimisation based uniquely on scal-
<figure><img src="image_196.png" /><figcaption>**Fig. 23.20.** Transverse and longitudinal phase spaces at the plasma exit of Scheme 2 (LPI 150 MeV + LPAS 5 GeV).</figcaption></figure>

ing laws allows one to already obtain a 1 GeV beam with all the parameters meeting the EuPRAXIA beam requirements, except for the slice energy spread. The latter could be improved by tuning more carefully the beam length to take profit of the beam-loading effect, as explained above.

## Optimisation of the Upramp at the LPAS Entrance

At the LPAS entrance, it is mandatory to have a matched input beam in order not to deteriorate the beam emittance upon entering the plasma. If no upramp is considered, i.e. with a sharp edge plasma, this matched beam should obey two constraints:

- its Twiss parameter $\alpha$ must be equal to zero (waist);

- its RMS size must be equal to the inverse of the square root of the wakefield focussing strength at the start of the LPAS, i.e. $\sim 1$ µm.

Such a small beam size at the waist implies a strong focussing in the upstream LETL inducing a significant emittance growth there (see Eq. (23.5)). It is thus mandatory to consider a density upramp to smoothly increase the plasma density from zero to its maximum value in the density plateau to relax the two constraints above. We set ourselves the objective to relax the required beam size by a factor of 10.

Symmetrical at the LPAS exit, exponential profiles with different lengths have also been studied. It is found in [606] that, for each ramp length, there exists an optimal divergence resulting in a minimum emittance growth. First semi-analytical investigations with the ASTRA code [320] show that the longer the ramp, the smaller the emittance growth, but the minimum is more peaked, meaning that it should be more precisely achieved. 3D PIC simulations were conducted with the Warp code [354] in the boosted frame. In order to save computation time, a comparably short ramp of 7 mm characteristic length is studied with different $\alpha_0$ Twiss parameters at the plasma entrance. The evolution of the beam emittance and beam size along the upramp is shown in Figure 23.21. It is interesting to note that, no matter how well the beam is matched, the beam size always converges to its matched value, because of the presence of the very strong plasma wakefield focussing gradient. However, the stronger the beam size oscillates before it reaches its equilibrium value, the more the emittance increases. A minimum emittance growth of 4% is obtained for $\alpha_0 = 2.0$. Those results are obtained with a 6D-Gaussian beam.
<figure><img src="image_197.png" /><figcaption>Fig. 23.21. Beam emittance (a) and RMS size (b) evolution along the upramp of the LPAS.</figcaption></figure>

<figure><img src="image_198.png" /><figcaption>Fig. 23.22. Evolution of the Twiss parameter $\gamma$ and the normalised phase-space emittance $\epsilon_{ph,n}$ along the downramp and the downstream drift for exponential density profiles with characteristic lengths of $L= 3, 5, \text{ and } 7 \text{ mm, compared to the case without ramp. } z = 0 \text{ mm is the downramp entrance.}$</figcaption></figure>

## Optimisation of the Downramp at the LPAS Exit

Although the witness beam at the LPAS exit meets all the requirements dictated for an FEL application, its Twiss parameter $\gamma = 400 \text{ m}^{-1}$ is still too large, i.e. its divergence is too big. Combined with the 1% energy spread, this means that it is extremely difficult to transport the beam to the FEL entrance without significant emittance degradation. According to the recommendations of Section 23.5, a density downramp with different types of profiles and lengths has been considered. It turns out that, independent of the profile shape, it is enough to roughly tune the length of the density downramp to drastically decrease $\gamma$, while deteriorating only marginally the emittance. Figure 23.22 shows the effect of varying ramp lengths on $\gamma$ and the beam emittance. For example, an exponential shape with a 7 mm characteristic length allows one to decrease $\gamma$ from 400 down to $80 \text{ m}^{-1}$ [606]. This beam distribution is taken for the HETL entrance.

## Main Features of Particle Acceleration in the LPAS, Upramp, and Downramp

The following parameter list summarises the laser, plasma and input electron beam parameters assumed in simulations of the LPAS. It also shows the output beam properties at the LPAS exit:
- **Laser**: bi-Gaussian, P = 341 TW, E = 45 J, $a_0 = 2$, $w_0 = 50$ µm, $\tau_{FWHM} = 132$ fs

- **Plasma**: $n_0 = 1 \times 10^{17} \text{cm}^{-3}$, radially parabolic with $\Delta n/n_c = 0.35$, 26 cm long, 7 mm long up- and downramps

- **Input electron beam**: 6D-Gaussian, $Q = 30$ pC, $E = 150$ MeV, $\varepsilon_{n,x,y} = 1$ mm mrad, $\varepsilon_{n,x,y,S} = 1$ mm mrad, $\beta_0 = 0.03$ m, $\gamma_0 = 67 \text{ m}^{-1}$, $\sigma_E/E = 2\%$, $\tau_{FWHM} = 7$ fs

- **Output electron beam**: $Q = 30$ pC, $E = 5$ GeV, $\varepsilon_{n,x,y} = 1.07$ mm mrad, $\varepsilon_{n,x,y,S} = 1.1$ mm mrad, $\beta_0 = 0.13$ m, $\gamma_0 = 79 \text{ m}^{-1}$, $\sigma_E/E = 0.8\%$, $\sigma_{E,S}/E = 0.1\%$, $\tau_{FWHM} = 7$ fs

Note that the first simulation studies of the LPAS have also been carried out with an input beam coming directly from the LPI simulations, as described in Section 23.8, and transferred by the LETL described in Section 23.8. While the setup is not fully optimised yet, small changes in the configuration (e.g. a reduction of the upramp length to 6 mm) already achieved good output beam properties, as shown below:

- **Output electron beam**: $Q = 30$ pC, $E = 5$ GeV, $\varepsilon_{n,x,y} = 1.0$ mm mrad, $\varepsilon_{n,x,y,S} = 0.8$ mm mrad, $\beta_0 = 0.02$ m, $\gamma_0 = 60 \text{ m}^{-1}$, $\sigma_E/E = 0.9\%$, $\sigma_{E,S}/E = 0.1\%$, $\tau_{FWHM} = 7.8$ fs

## Optimisation of Beam Transfer in the HETL

The objective here is to transfer the output beam from the LPAS downramp as described above, i.e. with the Twiss parameters $\alpha = 3.04$ and $\beta = 0.13$ m, towards the FEL entrance, while preserving the beam quality. In particular, the beam emittance should practically not be degraded. The optimisation in this sense is described in detail in Chapter 19. The adopted solution is an 8 m long line with 7 quadrupoles: three as a capture section right behind the plasma target and another four used for matching the beam to the application. They are separated by a C-chicane right after the capture section.

The particles with energies in the tail of the distribution will also have its Twiss parameters far from those in the inner core. They must be removed in the optimisation of the HETL, which consists of matching those Twiss parameters. The remaining charge is $Q = 25$ pC.

The resulting evolution of the beam envelope, normalised emittance, and other beam parameters along the HETL is shown in Figure 23.23. The properties of the transport line components are listed in Section 19.2.3, while the final electron beam parameters are given in Table 23.7, reproduced from Chapter 19.

### Summary of start-to-end simulations for Scheme 2: LPI-150 MeV + LPAS-5 GeV

Start-to-end simulations have been carried out for this configuration. All parameters from the LPI, LETL, LPAS, and HETL let us conclude that this scheme is fully compatible with the EuPRAXIA objectives. Currently, the margin with regard to the required beam parameters is still quite narrow, but further optimisation as well as thorough tolerance studies are planned in the future.

## 23.9 Start-to-End Simulations for Scheme 3: RFI-500 MeV + LPAS-5 GeV

This configuration includes a radiofrequency injector (RFI), which produces an electron beam of energy 500 MeV to be injected into the laser-plasma acceleration stage
<figure><img src="image_199.png" /><figcaption>**Fig. 23.23.** High-energy transfer line for Scheme 2: Beam beta function (a), normalised emittance (b), transverse size (c), and bunch duration and energy spread (d) of the core beam along the transport line for the case of one LPI + LPAS (5 GeV). The calculations were performed with the tracking code TraceWin.</figcaption></figure>

<table><caption>**Table 23.7.** High-energy transfer line for Scheme 2: Comparison between the beam parameters at the end of the LPAS and at the entrance of the undulator. These parameters have been calculated considering only the beam core, i.e. only a 1.4 fs long slice in z around the peak current. This bunch core contains 5 pC of charge. The slice parameters and peak current are calculated assuming 0.1 µm long slices.</caption><thead><tr><th></th><th>LPAS exit</th><th>Undulator entrance</th></tr></thead><tbody><tr><td>β<sub>x</sub> [m]</td><td>0.021</td><td>3.16</td></tr><tr><td>β<sub>y</sub> [m]</td><td>0.023</td><td>7.33</td></tr><tr><td>α<sub>x</sub> –</td><td>-0.48</td><td>-0.70</td></tr><tr><td>α<sub>y</sub> –</td><td>-0.54</td><td>1.56</td></tr><tr><td>ε<sub>n,x</sub> [µm]</td><td>0.76</td><td>0.76</td></tr><tr><td>ε<sub>n,y</sub> [µm]</td><td>0.90</td><td>0.88</td></tr><tr><td>Mean Energy [GeV]</td><td>4.98</td><td>4.98</td></tr><tr><td>σ<sub>γ</sub>/γ [10<sup>-3</sup>]</td><td>1.0</td><td>1.0</td></tr><tr><td>I&#773; [kA]</td><td>3.2</td><td>3.3</td></tr><tr><td>σ<sub>τ</sub> [fs]</td><td>0.4</td><td>0.4</td></tr><tr><td>ε<sub>n,x</sub> (slice) [µm]</td><td>0.86</td><td>0.85</td></tr><tr><td>ε<sub>n,y</sub> (slice) [µm]</td><td>1.00</td><td>1.00</td></tr><tr><td>σ<sub>γ</sub>/γ (slice) [10<sup>-3</sup>]</td><td>0.85</td><td>0.76</td></tr></tbody></table>

<figure><img src="image_200.png" /><figcaption>**Fig. 23.24.** Scheme 3 combining a 500 MeV RF injector with an LPAS to 5 GeV.</figcaption></figure>
<figure><img src="image_201.png" /><figcaption>**Fig. 23.25.** Evolution of the beam parameters along the RF injector. The graphs show the electron beam transverse normalised emittance ($\epsilon_n$, red line), envelope ($\sigma_t$, blue line), and longitudinal bunch length ($\sigma_z$, green line), as obtained with the TStep code for the case of pure RF compression.</figcaption></figure>

(LPAS). The LPAS, in turn, accelerates the beam to 5 GeV (and also 1 GeV). It is followed by a high-energy transfer line (HETL) driving the beam to the application beamlines. The latter include FEL or other applications, but as the FEL requirements are more demanding, these will be considered primarily here. As concluded in Section 23.3, the quasi-linear acceleration regime is considered for the LPAS.

## Optimisation of the RFI

The main challenge for the RF photo-injector comes from the request to produce ultra-short, high-quality electron beams. The RF injector foreseen for this setup is described in detail in Section 14.3. It generates a witness electron beam of 500 MeV energy at the plasma interface with much less than 1 mm mrad slice emittance and 30 pC charge in a 10 fs FWHM length, which is equivalent to a peak current up to 3 kA. Pure RF compression, applying the velocity-bunching scheme [260], is used to produce in one stage a 3 kA beam at the end of the S-band travelling wave (TW) sections at 100 MeV. The S-band photo-injector [314], operating at 2.856 GHz, consists of a 1.6-cell UCLA/BNL/SLAC-type standing wave (SW) RF gun, including a copper photo-cathode with an emittance-compensating solenoid followed by 3 m long SLAC-type travelling wave (TW) sections operating at 2.856 GHz [262]. The beamline matching foresees a proper set of emittance compensation solenoids and S-band cavity gradients in the velocity bunching scheme [301], according to the invariant envelope criteria [299]. In this configuration, the first and second TW sections can operate far from the crest in the velocity-bunching regime, enabling the RF compression of the beam length, while the third section operates almost on crest to let the electron bunch gain energy and freeze its phase-space properties. Velocity bunching is used in the first two S-band cavities to shorten the beam length from 102 µm (on crest) to ~3 µm (RMS), both cavities working close to the zero-crossing of the field. Emittance minimisation is achieved by setting the gun solenoid to a 3 kG magnetic field and those surrounding the first and second S-band cavities to 0.32 kG and 0.50 kG, respectively. A slightly off-crest operation of the third S-band cavity further reduces the energy spread at the injector exit (see Fig. 23.25).

An additional X-band linac boosts the beam energy up to ~500 MeV, as needed at the plasma entrance with proper matching conditions. The X-band linac mainly consists of two sections, L1 and L2; twelve X-band accelerating sections, 50 cm long,
<figure><img src="image_202.png" /><figcaption>**Fig. 23.26.** Left: Transverse and longitudinal distribution of the electron beam at the capillary entrance. Right: Transverse phase space of the beam at the capillary entrance. $E_0$ is the mean energy, $\sigma_x$ and $\sigma_y$ the RMS transverse size at the plasma capillary entrance, and $n$ is the number of occurrences.</figcaption></figure>

are foreseen for L1, and twenty for L2. According to the RF power system design, the maximum accelerating gradient applied is ~60 MV/m through all of L1 and L2 to reach the required energy and energy spread for the electron beam in the conventional RF operation scheme. An increased power configuration can also be implemented progressively in a machine upgrade plan to provide overhead and flexibility to the operation, and ultimately, to reach higher beam energies with the accelerating gradient raised up to ~80 MV/m.

For this working point, the X-band RF linac provides an electron beam with the following parameters: $Q = 30$ pC, $I = 3$ kA (FWHM), $E = 550$ MeV, and energy spread less than 0.1% (see Fig. 23.26). Before entering the plasma capillary, a focussing triplet of permanent magnet quadrupoles (PMQ) is foreseen at a distance of a few centimetres from the plasma entrance to obtain a typical beta function of $\beta_{x,y} \sim 1-5$ mm. The gradient of the first three PMQs is around $G \sim 300$ T/m with a magnetic length of 5–10 cm; the emittance dilution due to chromatic effects in the quadrupoles is the main concern of the final focussing stage.

## Optimisation of Particle Acceleration Within the LPAS

The goal at the end of the plasma-accelerator stage is to produce a witness electron beam at 5 GeV energy with the required high beam charge and low emittance, low divergence, and low energy spread, as indicated in Table 23.1. The technique of quasi-linear acceleration in a parabolically pre-formed plasma is applied.

The plasma target profile is comprised of equally shaped, exponential input and output ramps, and a constant density plateau. The presence of the ramps helps both in matching the bunch to the plasma at the input and to reduce its divergence at the exit. Their characteristic length of 3.5 mm is chosen to be half of the bunch betatron wavelength at injection. This length is both realistic and has been shown to yield the same results, in terms of beam parameters, as longer ramps [612].

The plasma density is set to $n_0 = 1 \times 10^{17} \text{ cm}^{-3}$ so that the plasma wavelength is much longer than the beam length, to avoid excessive energy spread increase while
<figure><img src="image_203.png" /><figcaption>**Fig. 23.27.** Transverse and longitudinal phase spaces at the exit of the Scheme 3 (RFI 500 MeV + LPAS 5 GeV).</figcaption></figure>

<figure><img src="image_204.png" /><figcaption>**Fig. 23.28.** Evolution of some electron beam parameters during plasma acceleration. Left: transverse size (blue) and emittance (red) evolution. Right: energy spread evolution.</figcaption></figure>

retaining an accelerating gradient around 10 GV/m. The laser parameters are set to $E = 24.4$ J, $w_0 = 70$ µm and $\tau = 112$ fs (FWHM), and $a_0 \sim 1$, to both increase the dephasing length and maximise the laser-to-plasma energy transfer.

The obtained transverse and longitudinal phase spaces are shown in Figure 23.27. Figure 23.28 reports on the optimised beam parameters throughout the acceleration in the plasma, after performing a 20% charge cut in the bunch tail. Figure 23.29 shows the final slice phase-space emittance and energy spread parameters. For an acceleration up to only 1 GeV, equivalent results can be obtained with even higher charges. Results have been reported in [612].

## Main Features of Particle Acceleration in the LPAS

The following laser and plasma parameters have been used in the simulation of the LPAS. Also listed are the input electron beam parameters, as achieved from the RF injector above, and the output parameters of the LPAS simulations:

- **Laser**: bi-Gaussian, $P = 225$ TW, $E = 24.5$ J, $a_0 = 1.15$, $w_0 = 70$ µm, $\tau_{FWHM} = 110$ fs
<figure><img src="image_205.png" /><figcaption>**Fig. 23.29.** Slice characteristics along the beam. Left: slice normalised emittances (red, green). Right: slice energy spread (orange). In both plots, the slice current is plotted in blue.</figcaption></figure>

- **Plasma:** $n_0 = 1 \times 10^{17} \text{ cm}^{-3}$, radially parabolic, 50 cm long, exponential up- and downramp

- **Input electron beam** (from RFI): $Q = 30$ pC, $E = 536.5$ MeV, $\epsilon_{n,x,y} = 0.4$ mm mrad, $\beta_0 = 0.023$ m, $\gamma_0 = 93 \text{ m}^{-1}$, $\sigma_E/E = 0.06\%$, $\tau_{FWHM} = 11$ fs

- **Output electron beam:** $Q = 24$ pC, $E = 5.3$ GeV, $\epsilon_{n,x,y} = 0.39$ mm mrad, $\epsilon_{n,x,y,S} = 0.36$ mm mrad, $\beta_0 = 0.033$ m, $\gamma_0 = 110 \text{ m}^{-1}$, $\sigma_E/E = 0.1\%$, $\sigma_{E,S}/E = 0.04\%$, $\tau_{FWHM} = 11$ fs

## Optimisation of Beam Transfer in the HETL

The objective here is to transfer the output beam from the LPAS downramp as described above, i.e. with the Twiss parameters $\alpha = 1.62$ and $\beta = 0.033$ m, towards the FEL entrance while preserving the beam quality. In particular, the beam emittance should practically not be degraded. According to Chapter 19, the adopted solution is an 8 m long line with 7 quadrupoles: three as a capture section right behind the plasma target and another four used for matching the beam to the application. They are separated by a C-chicane right after the capture section.

The particles with energies in the tail of the distribution will also have its Twiss parameters far from those in the inner core. They must be removed in the optimisation of the HETL, which consists of matching those Twiss parameters. The remaining charge is $Q = 24$ pC.

The resulting evolution of the beam envelope and normalised emittance along the HETL is shown in Figure 23.30. The properties of the transport line components are listed in Section 19.2.4, while the final electron beam parameters are given in Table 23.8, reproduced from Chapter 19.

### Summary of start-to-end simulations for Scheme 3: RFI-500 MeV + LPAS-5 GeV:

Start-to-end simulations have been carried out thoroughly for this configuration. All parameters achieved from the RFI, LPAS, and HETL let us conclude that this scheme is fully compatible with the EuPRAXIA objectives. Currently, the margin with regard to the required beam parameters is still quite narrow, but further optimisation and thorough tolerance studies are planned in the future.
<figure><img src="image_206.png" /><figcaption>**Fig. 23.30.** High-energy transfer line for Scheme 3: Beam beta function (a), normalised emittance (b), transverse size (c), and bunch duration and energy spread (d) of the core beam along the transport line for the case of one RFI + LPAS (5 GeV). The calculations were performed with the tracking code TraceWin.</figcaption></figure>

<table><caption>**Table 23.8.** High-energy transfer line for Scheme 3: Comparison between the beam parameters at the end of the LPAS and at the entrance of the undulator. These parameters have been calculated considering only the beam core, i.e. only a 13 fs long slice in *z* around the peak current. This bunch core contains 22 pC of charge. The slice parameters and peak current are calculated assuming 0.1 µm long slices.</caption><thead><tr><th></th><th>LPAS exit</th><th>Undulator entrance</th></tr></thead><tbody><tr><td>β<sub>x</sub> [m]</td><td>0.034</td><td>3.16</td></tr><tr><td>β<sub>y</sub> [m]</td><td>0.034</td><td>7.32</td></tr><tr><td>α<sub>x</sub> –</td><td>-1.79</td><td>-0.70</td></tr><tr><td>α<sub>y</sub> –</td><td>-1.83</td><td>1.56</td></tr><tr><td>ε<sub>n,x</sub> [0.1 µm]</td><td>0.31</td><td>0.32</td></tr><tr><td>ε<sub>n,y</sub> [0.1 µm]</td><td>0.28</td><td>0.29</td></tr><tr><td>Mean Energy [GeV]</td><td>5.41</td><td>5.41</td></tr><tr><td>σ<sub>γ</sub>/γ [10<sup>-3</sup>]</td><td>1.1</td><td>1.1</td></tr><tr><td>I&#x0304; [kA]</td><td>3.4</td><td>3.3</td></tr><tr><td>σ<sub>τ</sub> [fs]</td><td>2.9</td><td>2.9</td></tr><tr><td>ε<sub>n,x</sub> (slice) [0.1 µm]</td><td>0.35</td><td>0.32</td></tr><tr><td>ε<sub>n,y</sub> (slice) [0.1 µm]</td><td>0.35</td><td>0.32</td></tr><tr><td>σ<sub>γ</sub>/γ (slice) [10<sup>-3</sup>]</td><td>0.28</td><td>0.27</td></tr></tbody></table>
<figure><img src="image_207.png" /><figcaption>Fig. 23.31. Overview of the proposed acceleration scheme (not to scale) and the evolution of the longitudinal phase space of the beam.</figcaption></figure>

## 23.10 Start-to-End Simulations for Scheme 4: RFI-240 MeV + LPAS-2.5 GeV + Chicane + LPAS-5 GeV

Achieving a low beam energy spread below the 0.1% level, as required by the EuPRAXIA design parameters, is one of the major current challenges for plasma-based acceleration. One of the main reasons contributing to this issue is the large slope of the accelerating fields in the plasma wake as a consequence of their high amplitude and small wavelength. This slope imprints a large energy-position correlation along the beam (known as chirp), which leads to a large energy spread. Although several possible solutions to mitigate this chirp have been proposed [33,48,613–616], achieving an energy spread below 0.1% has not yet been demonstrated.

Motivated by this challenge, a new design concept for plasma accelerators has been proposed and developed within the EuPRAXIA framework, which could overcome this issue. The proposed scheme, published in the journal *Physical Review Letters* [34], considers performing the acceleration of an externally injected beam in two identical plasma stages joined by a magnetic chicane, as represented in Figure 23.31. This chicane inverts the beam energy chirp generated in the first plasma stage, which can therefore be naturally compensated in the second. Numerical simulations of a first conceptual implementation of this scheme show that 5.5 GeV electron beams with energy spreads of 0.12% (total) and 0.028% (slice) could be produced while maintaining a sub-micrometre emittance. This energy spread is at least one order of magnitude below the current state of the art and would satisfy the EuPRAXIA requirements. Given these promising initial results, a EuPRAXIA layout based on this chirp compensation method has been designed and is presented in this section.

### Design of a Two-Stage Beamline

The first conceptual implementation of the scheme, presented in [34], showed that a number of potential issues such as space charge or coherent synchrotron radiation (CSR) had a negligible influence on the simulated case, and that the energy chirp could be effectively compensated for in the second stage. This initial setup, with a total length of only 1.5 m, was, however, not meant as a prototype for an experiment, as it did not take into account certain aspects such as space for diagnostics and for the laser in- and outcoupling. In addition, the plasma-to-vacuum transitions (or ramps)
<figure><img src="image_208.png" /><figcaption>Fig. 23.32. Overview of the two-stage accelerator beamline for EuPRAXIA.</figcaption></figure>

<figure><img src="image_209.png" /><figcaption>Fig. 23.33. On-axis density profile of the two plasma stages.</figcaption></figure>

were not considered and the beam transport between accelerating stages relied solely on active plasma lenses. For these reasons, a dedicated conceptual design of this scheme for EuPRAXIA has been developed, which goes a step further towards a realistic implementation by taking all of these issues into account. A simplified view of the accelerator layout can be seen in Figure 23.32. In the following, details of each beamline section are given.

## Plasma-Accelerating Stages

As in the case presented in [34], the two plasma-accelerating stages have a length of 8 cm and operate with an on-axis plasma density at the plateau of $n_{p,0} = 10^{17} \text{ cm}^{-3}$, with a parabolic transverse profile $n_p(r) = n_{p,0} + r^2/\pi r_e w_0^4$ for laser guiding, where is $r$ the radial coordinate, $r_e$ the classical electron radius, and $w_0$ the spot size of the laser driver. The experimental generation of parabolic transverse profiles in this range of parameters has been recently demonstrated by means of laser pre-pulses [44,349]. In addition, the plasma ramps have now been taken into account and optimised for beam matching at the plasma entrance and divergence minimisation at the exit. No transverse guiding profile was assumed in the plasma ramps, while their longitudinal shape was assumed to follow the expression $n_{p,ramp}(z) = n_{p,0}/(1+z/L_r)^2$, which was found to provide good performance for matching and emittance preservation [617]. The total ramp length and the decay parameter $L_r$ were optimised to offer the best possible matching at the injection (minimising emittance growth) and minimise the beam divergence at extraction. The values of these two parameters and the on-axis density profile of the plasma cells can be seen in Figure 23.33. Regarding the laser driver, the same parameters as in [34] have been considered; i.e. a 40 J pulse per stage with a FWHM duration of 50 fs (peak power of 0.75 PW), a spot size of $w_0 = 50 \text{ µm}$, and a peak normalised vector potential of $a_0 = 3$. These parameters are, however, above what is in principle required to achieve a 2.5 GeV energy gain per stage, and preliminary simulations indicate that other designs based on 20 J of laser energy per stage would also be feasible.
<figure><img src="image_210.png" /><figcaption>**Fig. 23.34.** Transverse and longitudinal phase space of the electron beam as delivered by the RF linac at the entrance of the first plasma.</figcaption></figure>

## Space for Laser Removal

Thanks to the plasma downramp in the first stage, the divergence of the beam can be significantly reduced with respect to the original design. This reduces the emittance growth rate after the plasma stage and allows the first plasma lens to be placed farther away. In this case, a distance of 30 cm was chosen (in comparison to only 3 cm in the original design). This provides additional space for placing a plasma mirror and allows the laser pulse to diverge more, reducing its peak intensity from ~ $2 \times 10^{19}$ W cm⁻² to ~ $3 \times 10^{16}$ W cm⁻² after 25 cm. This greatly minimises the magnetic fields generated by the laser on the plasma mirror [618] and thus the risk of emittance increase. In this regard, the use of liquid crystal mirrors [619] instead of tapes would also be strongly beneficial to mitigate emittance growth caused by multiple scattering [620], as they can be much thinner (a few nm vs. a few µm). The same considerations apply for the coupling of the second laser into the next plasma stage.

## Transport Line Between Stages

The transport line, in this case, has been designed to allow for a higher degree of tunability. Instead of relying only on two active plasma lenses, two pairs of electromagnetic quadrupoles have been added. In this way, the role of the plasma lenses is only to control the beam divergence after the plasma stage to suppress further emittance growth and to provide the last strong focus for the injection into the second stage. The beam transport through the chicane is, instead, performed by the quadrupoles. This allows for an independent control of both transverse planes of the beam, which might feature different properties at the plasma exit, as well as the possibility of increasing the length of the transport line to, for example, accommodate diagnostic devices simply by optimising the quadrupole strengths and positions. The full details of the transport line can be found in Section 19.2.5.

## Start-to-End Simulation Results

Initial simulations to optimise the shape of the plasma ramps and the transport line were performed with Wake-T [495], a particle-tracking code with simplified models for plasma-based accelerators, which was developed within the framework of this work to be able to quickly simulate and optimise this kind of beamlines combining plasma and conventional devices. After the initial optimisations, fully detailed simulations of the beamline have been carried out with a combination of several numerical codes.
<figure><img src="image_211.png" /><figcaption>**Fig. 23.35.** Evolution of the longitudinal phase space of the externally injected electron beam along the beamline. The phase-space distribution is shown at the entrance and exit of each plasma stage. The development of an energy modulation and the micro-bunching are clearly visible. Only the central core of the beam along *z*, which contains ~80% of the charge and is the relevant part for lasing, is shown.</figcaption></figure>

<figure><img src="image_212.png" /><figcaption>**Fig. 23.36.** Comparison of the current profiles of the original and smoothed beams.</figcaption></figure>

The plasma elements, i.e. the accelerating stages and the active plasma lenses, have been simulated with the particle-in-cell code FBPIC [489], while the drifts between them as well as the transport line were simulated with the particle tracker ASTRA [320] in order to account for 3D space-charge effects. In addition, the transport line was also simulated with CSRTRACK [599] to check for possible CSR effects.

The externally injected electron beam, whose phase space can be seen in Figure 23.34, comes from the linac described in Section 14.4. Using this beam as-is led, however, to problems in the current profile and energy distribution of the beam after acceleration. As seen in Figure 23.35, the sharp peaks in the initial current profile of the beam lead to a slight energy modulation at the end of the first plasma stage caused by longitudinal space-charge effects. In the chicane, this energy modulation is translated into a charge-density modulation with sharp peaks or micro-bunches, which, in turn, generate an even larger energy modulation in the second stage. As a result, even though the energy chirp is compensated, the large energy modulation leads to a high average slice energy spread above 0.3% (for 0.1 µm long slices), which does not fulfil the target requirements.

In order to solve this issue, an improved version of the electron beam was created by smoothing its current profile as shown in Figure 23.36. This is something that could, in principle, be achieved by introducing a laser heater before the bunch compressor in the linac, which is a technique used in FEL facilities to prevent the onset of micro-bunching. As a result of this smoothing, the energy modulation is greatly minimised and a final average slice energy spread of ~0.05% (for 0.1 µm long slices) is achieved, fulfilling the EuPRAXIA requirements.

The evolution of the parameters of this improved version of the beam along both plasma stages can be seen in Figure 23.37, while its final phase space is shown in Figure 23.38. The parameters and distributions shown in these figures as well as the following table correspond to the bunch core. This has been defined because there is a small number of particles that are far from the axis and therefore give an unrealistic calculation of beam parameters. In addition, the beam also has a long tail with a small fraction of particles that have a significantly different energy from the rest. This
<figure><img src="image_213.png" /><figcaption>**Fig. 23.37.** Evolution of the parameters of the core part of the beam along both plasma stages. The noisy behaviour of the emittance at the beginning of the first plasma is due to the particle filtering applied to remove halo particles when computing the beam parameters.</figcaption></figure>

<figure><img src="image_214.png" /><figcaption>**Fig. 23.38.** Transverse and longitudinal phase space of the improved (smoothed) beam after the second plasma stage. Only the core of the beam is shown, as defined in the text.</figcaption></figure>

<table><caption>**Table 23.9.** Final parameters of the core of the improved (smoothed) beam at the end of the second plasma.</caption><thead><tr><th>Parameter</th><th>Q<sub>core</sub></th><th>I<sub>peak</sub></th><th>τ<sub>FWHM</sub></th><th>E</th><th>σ<sub>E,rel</sub></th><th>σ<sup>slice</sup><sub>E,rel</sub></th><th>ε<sub>n,x</sub></th><th>ε<sub>n,y</sub></th><th>ε<sup>slice</sup><sub>n,x</sub></th><th>ε<sup>slice</sup><sub>n,y</sub></th></tr><tr><th>Units</th><td>pC</td><td>kA</td><td>fs</td><td>GeV</td><td>%</td><td>%</td><td>μm</td><td>μm</td><td>μm</td><td>μm</td></tr></thead><tbody><tr><th>Value</th><td>23.7</td><td>4.9</td><td>3.1</td><td>6.0</td><td>0.41</td><td>0.05</td><td>1.50 | 0.69</td><td>0.77 | 0.40</td><td></td><td></td></tr></tbody></table>

core, which contains ~ 80% of the charge and is the relevant part of the beam for the FEL process, has been obtained by removing the particles that are more than 5 sigma away from the axis in the transverse phase space and taking into account only a 12 fs long region around the position of the current peak, as seen in the figures.

The high-energy transport line after the last plasma stage is described in detail in Chapter 19. It consists of an 8 m long line with 7 quadrupoles: three as a capture section right behind the plasma target and another four used for matching the beam to the application. They are separated by a C-chicane right after the capture section. Table 23.10 shows the electron beam parameters achieved after the beam transport and ready for use in one of the applications foreseen for EuPRAXIA. The beam distribution at the undulator entrance is shown in Figure 23.39.
<table><caption>**Table 23.10.** High-energy transfer line – Scheme 4: Comparison between the beam parameters at the exit of the second LPAS and at the undulator entrance. These parameters have been calculated considering only the beam core, as defined in the main text. The slice parameters and peak current are calculated assuming 0.1 µm long slices.</caption><thead><tr><th></th><th>2<sup>nd</sup> LWFA exit</th><th>Undulator entrance</th></tr></thead><tbody><tr><td>Q [pC]</td><td>23.7</td><td>23.8</td></tr><tr><td>β<sub>x</sub> [m]</td><td>0.15</td><td>4.07</td></tr><tr><td>β<sub>y</sub> [m]</td><td>0.08</td><td>7.73</td></tr><tr><td>α<sub>x</sub> –</td><td>-3.27</td><td>-0.98</td></tr><tr><td>α<sub>y</sub> –</td><td>-1.91</td><td>1.63</td></tr><tr><td>ε<sub>n,x</sub> [µm]</td><td>1.50</td><td>2.19</td></tr><tr><td>ε<sub>n,y</sub> [µm]</td><td>0.69</td><td>0.90</td></tr><tr><td>Mean energy [GeV]</td><td>6.0</td><td>6.0</td></tr><tr><td>σ<sub>γ</sub>/γ [10<sup>-3</sup>]</td><td>4.1</td><td>3.9</td></tr><tr><td>I<sub>peak</sub> [kA]</td><td>4.9</td><td>5.1</td></tr><tr><td>τ<sub>FWHM</sub> [fs]</td><td>3.1</td><td>3.1</td></tr><tr><td>ε<sub>n,x</sub> (slice) [µm]</td><td>0.77</td><td>0.85</td></tr><tr><td>ε<sub>n,y</sub> (slice) [µm]</td><td>0.40</td><td>0.40</td></tr><tr><td>σ<sub>γ</sub>/γ (slice) [10<sup>-3</sup>]</td><td>0.54</td><td>0.55</td></tr></tbody></table>

<figure><img src="image_215.png" /><figcaption>**Fig. 23.39.** Transverse and longitudinal phase space of the improved (smoothed) beam at the undulator entrance. Only the core of the beam is shown, as defined in the text.</figcaption></figure>

## Conclusion

Start-to-end simulations of the two-stage layout, based on the dechirping concept presented in [34], show that 6.0 GeV beams satisfying the EuPRAXIA requirements and with a slice energy spread of ~ 0.05% could be produced. Achieving this beam quality requires, however, to prevent the onset of micro-bunching in the magnetic chicane, which might require an upgrade of the RF linac to include a laser heater. It is also possible that this issue would not be as strong in reality as the linac simulations might be enhancing it given the limited amount of beam particles that were considered. Further detailed studies of this effect should therefore be carried out.

## Summary of start-to-end simulations for Scheme 4: RFI-240 MeV + LPAS-2.5 GeV + chicane + LPAS-5 GeV:

Start-to-end simulations have been carried out thoroughly for this configuration. All parameters achieved from the RFI, two LPAS stages, LETL and HETL let us conclude that this scheme is fully compatible with the EuPRAXIA objectives. Currently, the margin with regard to the required beam parameters is still quite narrow, but further optimisation and thorough tolerance studies are planned in the future.
<figure><img src="image_216.png" /><figcaption>Fig. 23.40. Scheme 5 combining a 500 MeV RF injector with a PPAS to 1 GeV.</figcaption></figure>

## 23.11 Start-to-End Simulations for Scheme 5: RFI-500 MeV + PPAS-1 GeV

This configuration includes one radiofrequency injector (RFI) injecting a 500 MeV witness beam and a driver beam to the particle-driven plasma-acceleration stage (PPAS), which, in turn, accelerates the witness beam to 1 GeV, and one high-energy transfer line (HETL) driving the beam to the end users. The latter include FEL or other applications, but as the FEL requirements are more demanding, these will be considered primarily here. The weakly non-linear acceleration regime is considered here for the PPAS.

### Optimisation of the RFI

A laser-comb configuration [323,324] has been explored for producing two electron bunches: a 200 pC driver followed by a 30 pC witness bunch. By illuminating the photo-cathode with a train of laser pulses with well-controlled timing, two or more electron bunches can be generated within the same RF accelerating bucket. The witness is created earlier than the driver on the photo-cathode, but then they are reversed in time at the end of the velocity-bunching process, during which the longitudinal phase space is rotated.

This optimisation process consists of setting the parameters of the two electron bunches and the longitudinal distance between them as desired at the next plasma acceleration stage to be at least $\lambda_p/2$, where the plasma wavelength is $\lambda_p = 330 \ \mu m$ for a plasma background density of $n_p = 1 \times 10^{16} \ cm^{-3}$. For this purpose, the fine-tuning of the accelerating cavity RF phases and solenoid magnetic fields has been computed. The results are summarised in Table 23.11. The RFI and the X-band boosting sections are the same as for the LWFA case. Details of these components are reported in Chapters 12 and 14.3.

The parameters of both witness and driver beams at the X-band linac entrance are listed in Table 23.12; it is worth to notice that the witness length is about 3 $\mu m$ FWHM with a normalised transverse emittance of 0.7 mm mrad.

The X-band RF linac has to provide a witness beam for injection in the plasma capillary with $Q= 30 \ pC$, $I= 3 \ kA$ (FWHM), and a final beam energy of $E_{L2exit} \sim 580 \ MeV$, with an energy spread of less than 0.1%.

The driver and witness bunches are characterised by a high charge / low current and a low charge / high current, respectively. Moreover, the initial matching conditions for the injection in the X-band linac are quite different for the two bunches. In this regard, an efficient sharing of the same lattice is achieved by means of a mild transverse focusing that aims to keep the RMS size of the comb beam (formed by both witness and driver) compatible with the beam stay-clear aperture through all the X-band accelerator. The same argument applies also to the focussing stage with the permanent quadrupoles at the entrance of the plasma capillary, where a 1–2 $\mu m$ transverse spot size is required; in this case a residual asymmetry between the horizontal and vertical planes for the witness beam is present (see Figs. 23.41 and 23.42).
<table><caption>**Table 23.11.** Main photo-injector parameters.</caption><thead><tr><th>Parameter</th><th>Unit</th><th>Value</th></tr></thead><tbody><tr><td>Gun electric field amplitude</td><td>MV/m</td><td>120</td></tr><tr><td>Gun electric field operation phase</td><td>deg</td><td>32</td></tr><tr><td>Output gun beam energy</td><td>MeV</td><td>5.6</td></tr><tr><td>Amplitude of electric field in the three TW sections</td><td>MV/m</td><td>20.0/20.0/28.0</td></tr><tr><td>Magnetic field in the emittance compensating solenoid</td><td>kG</td><td>3.0</td></tr><tr><td>Magnetic field in the linac solenoid</td><td>kG</td><td>0.34/0.52</td></tr><tr><td>Total photo-injector length</td><td>m</td><td>12</td></tr></tbody></table>

<table><caption>**Table 23.12.** Driver and witness beam parameters at the end of the photo-injector.</caption><thead><tr><th>Parameter</th><th>Unit</th><th>Witness</th><th>Driver</th></tr></thead><tbody><tr><td>Charge</td><td>pC</td><td>30</td><td>200</td></tr><tr><td>Energy</td><td>MeV</td><td>101.5</td><td>103.2</td></tr><tr><td>RMS energy spread</td><td>%</td><td>0.15</td><td>0.67</td></tr><tr><td>RMS bunch length</td><td>fs</td><td>12</td><td>20</td></tr><tr><td>Peak current (FWHM)</td><td>kA</td><td>6.0</td><td>0.37</td></tr><tr><td>RMS normalised emittance</td><td>mm mrad</td><td>0.69</td><td>1.95</td></tr><tr><td>Repetition rate</td><td>Hz</td><td>10</td><td>10</td></tr></tbody></table>

<figure><img src="image_217.png" /><figcaption>**Fig. 23.41.** Horizontal and vertical phase-space distribution of the PWFA driver (cyan dots) and witness (red dots) beams at the capillary entrance.</figcaption></figure>

## Optimisation of Particle Acceleration Within the PPAS

The objective is to accelerate the bunch to 1 GeV without phase-space dilution, with the required high beam charge and low emittance, low divergence, and low energy spread as indicated in Table 23.1. The so-called weakly non-linear regime is applied,
<figure><img src="image_218.png" /><figcaption>Fig. 23.42. Transverse horizontal and vertical distribution of the PWFA driver (left) and witness (right) beams at the capillary entrance.</figcaption></figure>

characterised by an electric field wave departing from a sinusoidal wave, tending toward a sawtooth profile.

The parameter we used to measure the degree of non-linearity is the reduced charge parameter, which is the beam charge normalised to the electron charge located in a cubic plasma skin depth $Q_{RC} = N_{ik}k_p^3/n_p$ [361,362]. For the foreseen case with a driver carrying about 200 pC of charge, the reduced charge is about $Q_{RC} = 0.8–0.9$. Moreover, recalling that the witness charge is fixed by the downstream applications, simulations identify that beam loading is compensated for a driver–witness distance of 184 µm, which corresponds to $0.55 \lambda_p$ for a $1 \times 10^{16} \text{cm}^{-3}$ number density. With this parameter choice, the accelerating field experienced by the witness beam is around 1.1 GV/m, as reported in Figure 23.43.

The following setup specifically allows to produce a maximum field on the order of 2.5–3.0 GV m⁻¹ at the bubble closure. The maximum peak is achieved at the bubble closure, where the room to allocate the witness will be limited and where the positioning of the witness would significantly change the bubble structure. We recall that the accelerating field together with the plasma wavelength depend upon the plasma number density $n_p$ as $\propto n_p^{1/2}$ and $\propto n_p^{-1/2}$, respectively. The technology related to the capillary plasma discharge is quickly evolving, suggesting that it will be possible to control the density within the capillary with great accuracy and also to compensate for some fluctuations such as the distance between driver and witness. A flat density profile at the required density is achieved with a capillary tube. The capillary tube, confining the ejected gas, permits a high degree of control, to which we can rely on for experimental on-site optimisation.

Much care has been taken to deliver the witness at the plasma entrance according to the transverse and longitudinal matching conditions of a plasma channel. For the case of interest, a witness with a given current for FEL application, requires a closer position to the driver to control at some level the energy spread. The non-bubble-rear witness positioning has some advantages and some disadvantages. The transformer ratio is limited, however, the positioning well inside the bubble allows for a wider room to place the witness, avoiding that any part of the transverse distribution hits the bubble edge with a consequent quality deterioration. In an accelerating length of 40 cm within the plasma, the witness gains about 460 MeV, so as to reach about the 1 GeV desired energy. This value indicates that the accelerating gradient is about 1.1 GV/m and that its transformer ratio is about 3.
<figure><img src="image_219.png" /><figcaption>**Fig. 23.43.** Driver and witness density contour plots with the position of the accelerating electric field overlaid.</figcaption></figure>

<figure><img src="image_220.png" /><figcaption>**Fig. 23.44.** Transverse and longitudinal phase spaces at the plasma exit of Scheme 5 (RFI 500 MeV + PPAS 1 GeV).</figcaption></figure>

The transverse and longitudinal phase spaces obtained at the plasma exit are shown in Figure 23.44. Simulations have been performed with the Architect hybrid code [600,621].

## Main Features of Particle Acceleration in the PPAS

- **Injected bunch 1 (driver):** bi-Gaussian, $Q = 200$ pC, $E = 567$ MeV, $\epsilon_{n,x,y} = 3$ mm mrad, $\sigma_E/E = 0.2\%$, $\tau_{FWHM} = 313$ fs
**Table 23.13.** High-energy transfer line for Scheme 5: Comparison between the beam parameters at the end of the PPAS and at the entrance of the undulator. These parameters have been calculated considering only the beam core, i.e. only electrons in the range ±5 MeV around the mean energy. This bunch core contains 10.5 pC of charge. The slice parameters and peak current are calculated assuming 0.1 µm long slices.

<table><thead><tr><th></th><th></th><th>PPAS exit</th><th>Undulator entrance</th></tr></thead><tbody><tr><td>β<sub>x</sub></td><td>[mm]</td><td>2.175</td><td>3.39</td></tr><tr><td>β<sub>y</sub></td><td>[mm]</td><td>4.2664</td><td>6.88</td></tr><tr><td>α<sub>x</sub></td><td>–</td><td>0.004</td><td>-0.61</td></tr><tr><td>α<sub>y</sub></td><td>–</td><td>1.072</td><td>1.16</td></tr><tr><td>ε<sub>n,x</sub></td><td>[µm]</td><td>0.68</td><td>0.91</td></tr><tr><td>ε<sub>n,y</sub></td><td>[µm]</td><td>0.59</td><td>0.68</td></tr><tr><td>Mean Energy</td><td>[GeV]</td><td>1.06</td><td>1.06</td></tr><tr><td>σ<sub>γ</sub>/γ</td><td>[10<sup>-3</sup>]</td><td>2.6</td><td>2.6</td></tr><tr><td>I&#773;</td><td>[kA]</td><td>2.06</td><td>1.96</td></tr><tr><td>σ<sub>τ</sub></td><td>[fs]</td><td>0.42</td><td>0.56</td></tr><tr><td>ε<sub>n,x</sub> (slice)</td><td>[µm]</td><td>0.66</td><td>0.62</td></tr><tr><td>ε<sub>n,y</sub> (slice)</td><td>[µm]</td><td>0.55</td><td>0.48</td></tr><tr><td>σ<sub>γ</sub>/γ (slice)</td><td>[10<sup>-3</sup>]</td><td>0.33</td><td>0.65</td></tr></tbody></table>

- **Injected bunch 2 (witness):** triangular, Q = 34 pC, E = 575 MeV, ε<sub>n,x,y</sub> = 0.6 mm mrad, σ<sub>E</sub>/E = 0.07%, τ<sub>FWHM</sub> = 12 fs

- **Plasma:** n<sub>0</sub> = 1 × 10<sup>16</sup> cm<sup>-3</sup>, 40 cm long with a 0.5 cm long plasma ramp

- **Output electron beam:** Q = 34 pC, E = 1.03 GeV, ε<sub>n,x,y</sub> = 0.96 mm mrad, ε<sub>n,x,y,S</sub> = 1.2 mm mrad, β<sub>0</sub> = 0.0038 m, γ<sub>0</sub> = 406 m<sup>-1</sup>, σ<sub>E</sub>/E = 1.1%, σ<sub>E,S</sub>/E = 0.036%, τ<sub>FWHM</sub> = 12 fs

## Optimisation of Beam Transfer in the HETL

The objective here is to transfer the output beam from the PPAS downramp as described above, i.e. with the Twiss parameters α = 0.74 and β = 0.0038 m, towards the FEL entrance, while preserving the beam quality. In particular, the beam emittance should practically not be degraded. The optimisation in this sense is described in Chapter 19. The adopted solution is an 8 m long line with 7 quadrupoles: three as a capture section right behind the plasma target and another four used for matching the beam to the application. They are separated by a C-chicane right after the capture section.

The particles with energies in the tail of the distribution will also have their Twiss parameters far from those in the inner core. They must be removed in the optimisation of the HETL, which consists of matching those Twiss parameters. The remaining charge is Q = 28 pC.

The resulting evolution of the beam envelope and normalised emittance along the HETL are shown in Figure 23.45. Table 23.13 shows the electron beam parameters achieved after the beam transport and ready for use in one of the applications foreseen for EuPRAXIA. The properties of the transport line components are listed in Section 19.2.
<figure><img src="image_221.png" /><figcaption>**Fig. 23.45.** High-energy transfer line for Scheme 5: Beam beta function (a), normalised emittance (b), transverse size (c), and bunch duration and energy spread (d) of the core beam along the transport line for the case of one RFI + PPAS (1 GeV). The calculations were performed with the tracking code TraceWin.</figcaption></figure>

**Summary of start-to-end simulations for Scheme 5, RFI-500 MeV + PPAS-1 GeV:**

Start-to-end simulations have been carried out thoroughly for this configuration. All parameters achieved from the RFI, PPAS, and HETL let us conclude that this scheme is fully compatible with the EuPRAXIA objectives. Currently, the margin with regard to the required beam parameters is still quite narrow, but further optimisation and thorough tolerance studies are planned in the future.

In addition, the present start-to-end simulation also considers the injection into a conventional undulator based on permanent magnets to successfully produce SASE FEL radiation at 3 nm, as requested for the FEL user application. This is described further in detail in Section 24.1.

## 23.12 Summary of Start-to-End Simulations

The main conclusions that can be drawn from the start-to-end simulation studies described in this chapter are, for each of the acceleration schemes under investigation, as follows:

**Scheme 1 – LPI-5 GeV:** Start-to-end simulations of Scheme 1 have demonstrated good performance with the final witness electron beam parameters aligning well with the EuPRAXIA requirements. As the scheme has not been tested experimentally and, as a single stage, promises slightly less tunability, it is considered as a development option.

**Scheme 2 – LPI-150 MeV + LPAS-5 GeV:** Full start-to-end simulations of Scheme 2 have not yet been carried out thoroughly yet. Nonetheless, the performance of the LPI, LPAS, and connecting transfer line sections demonstrates
promising results in line with the EuPRAXIA requirements. This option should
therefore be considered as a baseline configuration, under the condition that full
start-to-end simulations can confirm the beamline performance.

**Scheme 3 – RFI-500 MeV + LPAS-5 GeV:** Start-to-end simulations of Scheme 3 have demonstrated good performance with the final witness electron beam parameters aligning well with the EuPRAXIA requirements. This configuration is considered as a baseline option.

**Scheme 4 – RFI-240 MeV + LPAS-2.5 GeV + chicane + LPAS-5 GeV:**
Start-to-end simulations of Scheme 4 have demonstrated good performance with the final witness electron beam parameters aligning well with the EuPRAXIA requirements. This configuration is considered as a baseline option.

**Scheme 5 – RFI-500 MeV + PPAS-1 GeV:** Start-to-end simulations of Scheme 5 have been carried out thoroughly. The expected performance from this configuration fits well with the EuPRAXIA requirements, thus making it a baseline option for the EuPRAXIA design for acceleration to 1 GeV.

More studies remain to be completed to improve and consolidate the results described in this chapter. With further progress expected, most of the proposed schemes may be, for example, further optimised and tested for their robustness.

It should be noted that all the studies presented here have been performed in close collaboration between the EuPRAXIA work packages WP2 ("Beam Physics and Simulation") and WP5 ("Electron Beam Design and Optimisation"). Intense exchanges and discussion, including the sharing of large particle files, have been necessary between colleagues in different European institutes and between experts in plasma acceleration physics and conventional RF accelerator physics. This collaboration should be extended to the next studies on error effects and tolerance analyses. Considering the important number of parameters subject to errors and the very time-consuming simulations in plasma acceleration, such studies will only be possible in close collaboration between the future clusters on plasma structures, laser technology, and applications.

# 24 Photon User Applications

## 24.1 Free-Electron Laser

### 24.1.1 Introduction

Free-electron lasers use ultra-relativistic free electrons travelling in a periodic magnetic field shaped by an undulator as a gain medium [111,114]. Among the recent trends of free-electron laser (FEL) developments, compact designs are in rapid development for replacing relevant elements with alternative accelerator concepts or novel schemes [622]. Within the EuPRAXIA concept [295], the interest is to consider a plasma-based accelerator to drive an FEL. When considering to implement an FEL based on electron bunches produced by plasma acceleration [180], several new issues have to be considered:

- the electron beam presents an energy spread inherent to the dynamics of plasma acceleration, which is typically one order of magnitude larger than that generally available at conventional linear accelerators driving FELs;

- external injection schemes promise to provide a more reliable control of the beam quality, with emphasis on those effects inducing beam-emittance deterioration [379];

- in order to minimise the energy spread and divergence increase, a high brightness injector is needed and particular care must be taken of the bunch length;
- Twiss parameters have to be chosen such that the $\beta$ parameter is large enough at the plasma exit to allow beam manipulation, but then such that both a decrease in divergence and an increase in size at an almost fixed emittance take place at the undulator entrance.

In the present work, reproduced from [623] with the authors' permission, we analyse the FEL performance of some of the most promising schemes based on laser- and particle-driven plasma acceleration investigated for EuPRAXIA. We also compare the FEL results of the electron beams targeting the same resonant wavelength.

### 24.1.2 Features of the Electron Beams at the Plasma Exit and the Transfer Line

The main EuPRAXIA objective in terms of beam energy lies in providing electron bunches:

- at 1 GeV, perfectly available for FEL applications as a commissioning step;

- at 5 GeV, meeting final FEL user requirements in terms of wavelength and short-pulse duration.

The performance of several schemes has been investigated in this context in terms of beam quality and subsequent efficient light production.

**LPI-150 MeV + LPAS-5 GeV (Scheme 2):** This scheme includes two plasma stages: the laser-plasma injector to produce electrons with beam energy of 150 MeV, and a laser-plasma acceleration stage to accelerate particles to the final energy of 5 GeV. In particular, the beam distribution under study, and denoted hereafter as Scheme 2-5 GeV, is injected with the resonant multi-pulse ionisation technique [170,624] and accelerated through a single stage in the quasi-linear regime [582,595].

**RFI-500 MeV + LPAS-1 / 5 GeV (Scheme 3):** In this scheme [304], a 500 MeV electron beam is injected from an RF section [625] into the plasma acceleration stage, which, in turn, accelerates the electrons up to either a beam distribution with 1 GeV [612] energy, denoted hereafter as Scheme 3-1 GeV, or a beam distribution with 5 GeV energy, denoted hereafter as Scheme 3-5 GeV.

**RFI-500 MeV + PPAS-1 GeV (Scheme 5):** In this scheme, a laser-comb configuration [323,324] produces two electron bunches: a 200 pC driver and a 30 pC witness bunch. Both are injected into the plasma acceleration stage, which, in turn, accelerates the witness bunch up to 1 GeV energy [626]. The resulting beam distribution is denoted hereafter as Scheme 5-1 GeV.

These beam distributions are analysed in terms of the main parameters driving the FEL performance. Values of emittance, energy spread, and peak current are calculated over the width of a single slice in order to have reasonable performance predictions. More in detail, the electron distribution slice with the highest current density is identified and denoted as the “best slice”. Table 24.1 shows the parameter values of the best slice at the plasma exit of the beams under study regarding FEL applications, where $\epsilon_{n,x(y)}$, $\sigma_{x(y)}$, $I_{peak}$, and $\sigma_E/E$ are the slice values of the normalised emittance in $x(y)$, the RMS size in $x(y)$, the peak current, and the RMS energy spread for the specified slice length $l_s$, respectively.

The best slice results are evaluated assuming that the FEL dynamics in one slice are not affected by the electron distribution in another slice if the two slices are significantly separated. However, bunch collective effects could, in principle, affect this picture and should be considered at the stage of a technical design.

The transfer line from the plasma exit stage, where the bunch leaves strong focussing fields to drift into free space, towards the undulator section is designed [606, 627] according to the following considerations:
<table><caption>**Table 24.1.** Best slice values of the relevant parameters at the plasma exit.</caption><thead><tr><th>Name</th><th><i>E</i> [GeV]</th><th><i>I</i><sub>peak</sub> [kA]</th><th>σ<sub>E</sub>/E [%]</th><th>ε<sub>n,x</sub> [µm]</th><th>ε<sub>n,y</sub> [µm]</th><th>σ<sub>x</sub> [µm]</th><th>σ<sub>y</sub> [µm]</th><th>ℓ<sub>s</sub> [µm]</th></tr></thead><tbody><tr><td>Scheme 2-5 GeV</td><td>4.98</td><td>2.93</td><td>0.108</td><td>0.53</td><td>0.59</td><td>0.87</td><td>0.92</td><td>0.11</td></tr><tr><td>Scheme 3-5 GeV</td><td>5.41</td><td>2.85</td><td>0.046</td><td>0.38</td><td>0.32</td><td>1.06</td><td>0.98</td><td>1.3</td></tr><tr><td>Scheme 5-1 GeV</td><td>1.07</td><td>1.95</td><td>0.098</td><td>0.67</td><td>0.59</td><td>0.83</td><td>0.98</td><td>0.9</td></tr><tr><td>Scheme 3-1 GeV</td><td>1.09</td><td>1.88</td><td>0.923</td><td>0.4</td><td>0.41</td><td>2.2</td><td>2.2</td><td>1.2</td></tr></tbody></table>

- the growth of both emittance and energy spread has to be minimised along the line;

- the maximum total length is 8 m and the focussing gradients are chosen to be 700 T/m for the permanent magnet and 100 T/m for the electromagnetic quadrupoles.

Moreover, each beam distribution in Table 24.1 has to be properly matched to the undulator configurations to be discussed in the following section. Further details on the beam transfer lines to the undulator section are also found in Chapter 19.

### 24.1.3 Undulator Line Characteristics

In an FEL, electromagnetic radiation is attained by wiggling the electrons, subject to the periodic magnetic field of an undulator and specified by the undulator period $\lambda_u$ and the deflection strength parameter $K$. These quantities define the resonant FEL wavelength $\lambda_R$ as

$$ \lambda_R = \frac{\lambda_u}{2\gamma^2} (1 + a_u^2), \quad K = \frac{eB_0\lambda_u}{2\pi m_e c}, \qquad (24.1) $$

where $\gamma$ is the beam energy Lorentz factor, $a_u = K/\sqrt{2}$ ($a_u = K$) for planar (helical) undulators, $B_0$ is the peak magnetic field value, and $e$, $m_e$, and $c$ are the electron charge, the electron mass, and the speed of light, respectively.

The main parameter quantifying FEL performance is the Pierce parameter $\rho$ [219, 628–631]. It depends on undulator parameters and on the bunch current density $j$, from which the cooperation length $L_c$ is derived as follows:

$$ \rho \propto \frac{\sqrt[3]{j (\lambda_u K f_b(K))^2}}{\gamma}, \quad L_c = \frac{\lambda_R}{4\pi\rho\sqrt{3}}, \qquad (24.2) $$

where $f_b(K) = J_0(\xi) - J_1(\xi)$ is the planar undulator factor, with the argument

$$ \xi = \frac{K^2}{2} \frac{1}{2 + K^2} $$

described in terms of the Bessel $J_{0,1}(\xi)$ functions.

Each previously discussed beam distribution is analysed and matched to two different undulator configurations, to probe the beam phase-space features with two different cooperation lengths: one at the shortest reasonable $\lambda_R$ and the other such that $L_c \sim O(1\%)$ at $E = 5$ GeV and $L_c \sim O(10\%)$ at $E = 1$ GeV. Table 24.2 shows the features of the chosen undulator configurations.

These values are perfectly feasible [632,633] with the present undulator technology, for both superconducting and cryogenic permanent magnet devices, assuming a
<table><caption>**Table 24.2.** Undulator configurations used for the FEL environment.</caption><thead><tr><th><i>E</i> [GeV]</th><th>λ<sub>R</sub> [nm]</th><th>λ<sub>u</sub> [mm]</th><th>K</th><th><i>B</i><sub>0</sub> [T]</th></tr></thead><tbody><tr><td>5</td><td>0.22</td><td>20</td><td>1.5</td><td>0.81</td></tr><tr><td>5</td><td>1.65</td><td>30</td><td>4.36</td><td>1.56</td></tr><tr><td>1</td><td>5.5</td><td>20</td><td>1.5</td><td>0.81</td></tr><tr><td>1</td><td>41</td><td>30</td><td>4.36</td><td>1.56</td></tr></tbody></table>

<figure><img src="image_222.png" /><figcaption>**Fig. 24.1.** Magnetic unit cell of a system made up of an undulator as well as focussing and defocusing quadrupoles for the $\lambda_u = 3$ cm configuration, associated with the specified electron beams and superimposed with the longitudinal profiles of their Twiss $\beta_x$ (solid red line) and $\beta_y$ (dotted blue line) functions.</figcaption></figure>

minimum undulator gap of 6 mm. This choice mitigates the wakefield-deteriorating effects inside undulators. In-vacuum undulators are expected to provide slightly smaller $B_0$ and K values compared to those of Table 24.2. A more detailed analysis of different possible undulator technologies can also be found in Chapter 18. At both 1 GeV and 5 GeV electron beam energies, the natural focussing of the undulator is rather weak. In order to maintain a small transverse size of the electron beam, the periodic magnetic cell thus has to include alternate gradient quadrupoles between undulator modules.

As an example, Figure 24.1 shows the resulting $\beta_{x,y}$ values as a function of $z$ when matching the electron beams of Table 24.1 to the undulator configuration with $\lambda_u = 3$ cm and $K = 4.36$.

The strategy to match the beams is based on minimising the difference between average Twiss $\beta$ values, $|\langle\beta_x\rangle - \langle\beta_y\rangle|$, while also featuring reasonable magnetic gradients for both the short- and the long-$\lambda_u$ configurations. The undulator period and strength clearly define the Twiss $\alpha$ and $\beta$ parameter values that the electron beams should have at the undulator entrance to be correctly matched.

### 24.1.4 FEL Results

After a proper transport of the beams to the undulator entrance, the best slice parameters have the values described in Table 24.3. As a first step, these values are used to estimate the FEL performance by means of semi-analytical formulae [219,628–631].

Within this picture, the main parameters quantifying the FEL performance are the gain length $L_{G,3d}$ and the saturation power $P_{sat}$, both depending on the Pierce parameter in the following way:

$$L_{G,3d} \approx \chi\left(\rho, \frac{\sigma_E}{E}, \langle\epsilon_n\rangle, \langle\beta\rangle\right) \frac{\lambda_u}{4\pi\rho\sqrt{3}}, \quad P_{sat} \approx \Phi\left(\rho, \frac{\sigma_E}{E}, \langle\epsilon_n\rangle, \langle\beta\rangle\right) \rho P_E, \quad (24.3)$$
<table><caption>**Table 24.3.** Best slice values of the relevant parameters at the undulator entrance and expected cooperation lengths.</caption><thead><tr><th>Name</th><th><i>E</i> [GeV]</th><th><i>I</i><sub>peak</sub> [kA]</th><th>σ<sub>E</sub>/<i>E</i> [%]</th><th>⟨ε<sub>n</sub>⟩ [µm]</th><th>⟨β⟩ [m]</th><th><i>L</i><sub>c</sub><sup>2 cm</sup> [nm]</th><th><i>L</i><sub>c</sub><sup>3 cm</sup> [nm]</th></tr></thead><tbody><tr><td>Scheme 2-5 GeV</td><td>4.96</td><td>2.63</td><td>0.052</td><td>0.58</td><td>5</td><td>20</td><td>61</td></tr><tr><td>Scheme 3-5 GeV</td><td>5.41</td><td>2.74</td><td>0.052</td><td>0.34</td><td>5</td><td>14</td><td>42</td></tr><tr><td>Scheme 3-1 GeV</td><td>1.09</td><td>1.75</td><td>0.103</td><td>0.44</td><td>4</td><td>140</td><td>430</td></tr><tr><td>Scheme 5-1 GeV</td><td>1.07</td><td>1.06</td><td>0.047</td><td>0.55</td><td>4</td><td>320</td><td>990</td></tr></tbody></table>

<table><caption>**Table 24.4.** FEL semi-analytical results for the short-undulator-period configuration based on the best slice parameters of Table 24.3.</caption><thead><tr><th>Name</th><th>λ<sub>R</sub> [nm]</th><th>Pierce ρ [%]</th><th><i>L</i><sub>G,3d</sub> [m]</th><th><i>P</i><sub>sat</sub> [GW]</th></tr></thead><tbody><tr><td>Scheme 3-5 GeV</td><td>0.19</td><td>0.099</td><td>1.7</td><td>4.39</td></tr><tr><td>Scheme 2-5 GeV</td><td>0.3</td><td>0.084</td><td>2.74</td><td>1.75</td></tr><tr><td>Scheme 3-1 GeV</td><td>4.67</td><td>0.236</td><td>0.59</td><td>1.94</td></tr><tr><td>Scheme 5-1 GeV</td><td>4.86</td><td>0.188</td><td>0.63</td><td>1.29</td></tr></tbody></table>

where $\chi$ and $\Phi$ are correction functions accounting for energy spread and emittance contributions, and $P_E$ is the power associated with the electron beam. Then, as a second step, the full longitudinal behaviour of each electron beam distribution is analysed in the self-amplified spontaneous emission (SASE) mode, and the FEL performance is evaluated with the PERSEO simulation code [634], which allows to perform a full time-dependent simulation of the FEL dynamics in a fast way, taking into account the proper energy, current, and energy spread profiles along the bunch. This feature is of particular relevance for the electron bunches under study as the current distribution is not Gaussian and with long asymmetric tails, and as the energy spread distribution exhibits abrupt variations. Furthermore, the time-dependent analysis allows a reliable estimate of the FEL pulse duration and spectrum line width, improved with the aforementioned longitudinal dynamics-deteriorating effects.

Figures 24.2 and 24.3 show the longitudinal slice profiles of energy, current, energy spread, and normalised emittance for the 5 GeV and 1 GeV beam distributions, respectively. In all cases, the normalised emittance is assumed to be constant as the associated variations affect the effective saturation length less dramatically.

## Results with the Short-Undulator-Period Configuration

Table 24.4 shows the FEL results on gain length and saturation power targeting the short resonant wavelength, featuring inhomogeneous broadening effects based only on the best slice parameters.

The normalised emittance value in Scheme 2-5 GeV, which is higher than in Scheme 3-5 GeV, results in a higher gain length and lower saturation power.

The results of the time-dependent simulations obtained with longitudinal dynamics are shown in Table 24.5. The already mentioned argument of a high normalised emittance in Scheme 2-5 GeV is enforced in the time-dependent analysis resulting in a long saturation length and in a small number of photons per pulse. At saturation, Scheme 3-1 GeV and Scheme 5-1 GeV have a similar number of photons per pulse; a lower current profile in Scheme 3-1 GeV is compensated by a smaller normalised emittance than in Scheme 5-1 GeV.

Figures 24.4 and 24.5 show the evolution of the SASE FEL energy per pulse for the beam distributions matched to the short-undulator-period configuration, for
<figure><img src="image_223.png" /></figure>

<figure><img src="image_224.png" /><figcaption>**Fig. 24.2.** Energy, current, energy spread, and normalised emittance slice profiles as a function of the intra-bunch $ζ$ coordinate, for the beam distributions of Scheme 3-5 GeV (top) and Scheme 2-5 GeV (bottom) at the undulator entrance.</figcaption></figure>
<figure><img src="image_225.png" /></figure>

<figure><img src="image_226.png" /><figcaption>**Fig. 24.3.** Energy, current, energy spread, and normalised emittance slice profiles as a function of the intra-bunch $ζ$ coordinate, for the beam distributions of Scheme 3-1 GeV (top) and Scheme 5-1 GeV (bottom) at the undulator entrance.</figcaption></figure>
<figure><img src="image_227.png" /><figcaption>**Fig. 24.4** Growth of the SASE FEL energy per pulse of Scheme 2-5 GeV (top) and Scheme 3-5 GeV (bottom) beam distributions with the short-undulator-period configuration. The fundamental (solid line) and third (dotted line) harmonics are shown.</figcaption></figure>

<table><caption>**Table 24.5.** Results of the time-dependent simulations with longitudinal dynamics, obtained with PERSEO, for the short-undulator-period configuration.</caption><thead><tr><th>Name</th><th>Saturation length [m]</th><th>Linewidth [%]</th><th>Pulse duration [fs]</th><th>Photons per pulse [10<sup>10</sup>]</th><th>Brightness [×10<sup>30</sup> s<sup>-1</sup> (mm mrad)<sup>-2</sup> (0.1%bw)<sup>-1</sup>]</th></tr></thead><tbody><tr><td>Scheme 2-5 GeV</td><td>126</td><td>0.18</td><td>0.4</td><td>0.19</td><td>3.7</td></tr><tr><td>Scheme 3-5 GeV</td><td>38</td><td>0.23</td><td>2.0</td><td>3.2</td><td>40</td></tr><tr><td>Scheme 3-1 GeV</td><td>28</td><td>0.25</td><td>2.4</td><td>2.3</td><td>0.5</td></tr><tr><td>Scheme 5-1 GeV</td><td>16</td><td>0.59</td><td>2.0</td><td>1.3</td><td>0.08</td></tr></tbody></table>

both fundamental and third harmonics. The curves suggest that targeting the shorter resonant wavelength is within reach for 1 GeV beams, while it demands extremely high-quality 5 GeV beam distributions, with very low energy spread and normalised emittance values.

## Results with the Long-Undulator-Period Configuration

Table 24.6 shows the FEL results on gain length and saturation power targeting the long resonant wavelength, featuring inhomogeneous broadening effects based only on the best slice parameters.

The considerations described discussing Table 24.4 above apply also to this case.

The results of the time-dependent simulations obtained with longitudinal dynamics are shown in Table 24.7.

A larger undulator period and strength yield a significantly larger Pierce $\rho$ parameter in each beam distribution. The effective result is a significantly better FEL performance in terms of saturation length, brightness, and photons per pulse, while
<figure><img src="image_228.png" /><figcaption>**Fig. 24.5.** Growth of the SASE FEL energy per pulse of Scheme 5-1 GeV (top) and Scheme 3-1 GeV (bottom) beam distributions for the short-undulator-period configuration. The fundamental (solid line) and third (dotted line) harmonics are shown.</figcaption></figure>

<table><caption>**Table 24.6.** FEL semi-analytical results for the long-undulator-period configuration based on the best slice parameters of Table 24.3.</caption><thead><tr><th>Name</th><th>λ<sub>R</sub> [nm]</th><th>Pierce ρ [%]</th><th>L<sub>G,3d</sub> [m]</th><th>P<sub>sat</sub> [GW]</th></tr></thead><tbody><tr><td>Scheme 3-5 GeV</td><td>1.4</td><td>0.23</td><td>0.83</td><td>17.8</td></tr><tr><td>Scheme 2-5 GeV</td><td>1.67</td><td>0.2</td><td>0.94</td><td>14</td></tr><tr><td>Scheme 3-1 GeV</td><td>34.5</td><td>0.57</td><td>0.36</td><td>4.94</td></tr><tr><td>Scheme 5-1 GeV</td><td>35.9</td><td>0.45</td><td>0.44</td><td>2.51</td></tr></tbody></table>

<table><caption>**Table 24.7.** Results of the time-dependent simulations with longitudinal dynamics, obtained with PERSEO, for the long-undulator-period configuration.</caption><thead><tr><th>Name</th><th>Saturation length [m]</th><th>Linewidth [%]</th><th>Pulse duration [fs]</th><th>Photons per pulse [10<sup>10</sup>]</th><th>Brightness [×10<sup>30</sup> s<sup>-1</sup>(mm mrad)<sup>-2</sup>(0.1%bw)<sup>-1</sup>]</th></tr></thead><tbody><tr><td>Scheme 2-5 GeV</td><td>26</td><td>0.3</td><td>0.71</td><td>4.2</td><td>27.6</td></tr><tr><td>Scheme 3-5 GeV</td><td>20</td><td>0.3</td><td>2.2</td><td>72</td><td>475</td></tr><tr><td>Scheme 3-1 GeV</td><td>16</td><td>0.54</td><td>7.8</td><td>31</td><td>0.86</td></tr><tr><td>Scheme 5-1 GeV</td><td>23</td><td>3.6</td><td>15</td><td>16</td><td>0.02</td></tr></tbody></table>

keeping a good quality spectrum line width and pulse duration, clearly targeting a significantly longer resonant wavelength at a given beam energy. Figures 24.6 and 24.7 show the evolution of the SASE FEL peak power for the beam distributions matched to the long-undulator-period configuration, for both fundamental and third harmonics. All beam distributions conceptually show a fair performance in realising compact SASE FEL facilities driven by a plasma-acceleration stage.
<figure><img src="image_229.png" /><figcaption>**Fig. 24.6.** Growth of the SASE FEL energy per pulse of Scheme 2-5 GeV (top) and Scheme 3-5 GeV (bottom) beam distributions for the long-undulator-period configuration. The fundamental (solid line) and third (dotted line) harmonics are shown.</figcaption></figure>

<figure><img src="image_230.png" /><figcaption>**Fig. 24.7.** Growth of the SASE FEL energy per pulse of Scheme 5-1 GeV (top) and Scheme 3-1 GeV (bottom) beam distributions for the long-undulator-period configuration. The fundamental (solid line) and third (dotted line) harmonics are shown.</figcaption></figure>
<figure><img src="image_231.png" /><figcaption>Fig. 24.8. Cartoon of the trajectory of an electron with maximum Lorentz factor $γ_{z0}$ being accelerated in a non-linear wakefield with an offset of $r_β$, which leads to a strong synchrotron emission within a cone angle of $1/γ_{z0}$.</figcaption></figure>

### 24.1.5 Conclusion

We have analysed some of the most promising electron beam distributions arising from dedicated design studies on injection and – both laser- and particle-driven – plasma acceleration stages, adopting present and near-future solutions for the undulator technology and producing FEL radiation in the SASE configuration. In particular, we have performed time-dependent simulations with full longitudinal dynamics for each beam distribution, targeting two different resonant wavelengths.

At a given beam energy, short resonant wavelengths demand for extremely high-quality beams, with the highest current density and lowest energy spread characteristics extended over many slices of the bunch. In more detail, wavelengths of the order $O(1 \text{ nm})$ are perfectly within reach in such a FEL facility operating in the SASE mode and matching to a rather compact undulator section.

## 24.2 Betatron X-Ray Source

### Betatron Radiation from Wakefield Accelerators

Plasma wakefield accelerators typically exhibit a strong focussing force for forward-going electrons, created by the radial displacement and longitudinal motion of the plasma wave electrons. Electrons with an initial offset from the central axis of the accelerator, or with some initial transverse momentum, will thus oscillate around this axis as they propagate, as shown in Figure 24.8. The resulting wiggling motion leads to the emission of synchrotron radiation, referred to as “betatron radiation”.

Betatron radiation from plasma accelerators has some unique features as compared to other synchrotron radiation sources. In much the same way as the accelerating forces in a plasma accelerator are greater than those in a conventional accelerator, so are the focussing forces also many orders of magnitude greater. They typically reach an order of $E \approx E_0 = mc\omega_p/e \approx 0.1 \cdot [n_e (\text{m}^{-3})] \text{Vm}^{-1}$. For a plasma of density of $10^{18} \text{cm}^{-3}$, this creates fields of magnitude $\approx 10^{11} \text{Vm}^{-1}$ over distances on the scales of tens of micrometres. In these fields, the electrons wiggle with a betatron frequency and wavelength given by the following:

$$ \omega_{\beta} = \omega_{p} / \sqrt{2\gamma_{z0}}; \quad \lambda_{\beta} = \sqrt{2\gamma_{z0}} \lambda_{p}, $$
where $\gamma_{z0}$ is the Lorentz factor due to the longitudinal motion of the electron. For typical densities used to generate GeV-scale electron beams ($n_e \approx 10^{18} \text{cm}^{-3}$), a 1 GeV electron would have $\lambda_\beta \approx 2$ mm. This compares to the undulation period of conventional (magnet-based) undulators, which can be of the order of tens of millimetres. Hence, plasma-based undulators can be much more compact, in the same way that plasma-based accelerators have advantages in scale.

But, of course, the short undulation period and thus correspondingly high beta-tron frequency mean that the radiation produced is also of higher frequency. For low amplitude undulation, the emitted energy is just the Doppler-upshifted photon energy, $E_u \approx 2\gamma_{z0}^2\hbar\omega_\beta$. For this to be valid, the undulator parameter $\alpha_\beta$ (also often denoted as K) given by,

$$\alpha_\beta = \gamma_{z0}r_\beta\omega_\beta/c,$$

should be less than one ($\alpha_\beta < 1$). $\alpha_\beta$ is a measure of the transverse momentum of the oscillation and thus is analogous to the normalised momentum discussed for electrons in a laser field. To have large oscillations, betatron sources commonly use self-injection where the injected electrons are those which were initially part of the plasma-wave motion. Hence, they have energies when arriving on axis comparable to the ponderomotive potential, which generated the plasma wave $E_{\text{pond}} \approx a_0mc^2$. As the electrons are accelerated, the oscillation amplitude $r_\beta$ decreases due to the relativistic increase in the electrons inertia but the energy associated with the transverse motion stays almost unchanged. This means that $r_\beta$ is typically of the order of micrometres at the point where the electrons have their highest energy [635]. This is important because this is when the electrons are also radiating the most, and so the source size is also of the order of 1 µm. This small source size has some advantages, which will be discussed later.

Yet, despite these small excursion sizes caused by the high betatron frequency for the parameters given before (GeV electrons accelerated in $n_e \approx 10^{18} \text{cm}^{-3}$ plasmas), $\alpha_\beta \approx 10$, and the resulting motion is strongly non-linear. As $\alpha_\beta$ becomes greater than 1, the non-linearities induced in the electron motion produce harmonics of the fundamental Doppler-upshifted undulator radiation. The higher $\alpha_\beta$, the higher the harmonic content, and including the effect of the increasing electron energy and the range of $r_\beta$ that would be present (not all electrons are injected with the same momentum), a continuous spectrum is formed similar to the usual synchrotron spectrum [635,636]. The synchrotron spectrum has a peak in the spectral brightness and is usually characterised by a critical energy as follows:

$$E_{\text{cr}} = \frac{3}{2} a_{\beta} \gamma_{z0}^2 \hbar \omega_{\beta},$$

which is defined as the point in the spectrum, where half of the energy is located above and half is below. The critical energy is close to the energy with the peak spectral brightness. Note that the frequency of this radiation scales as $\gamma_{z0}^3$ times the betatron frequency because of the dependence of $\alpha_\beta$ on $\gamma_{z0}$. This means that, even for modest electron energies, the radiation is in the multi-keV range [637]. For example, in experiments performed at the Gemini facility at the Rutherford Appleton Laboratory, acceleration of electrons to ~GeV energies ($\gamma_{z0} \approx 2000$) in plasmas of densities $n_e \approx 10^{18} \text{cm}^{-3}$ led to critical energies in excess of 20 keV [75,638]. Indeed, the photon energies from betatron radiation exceed those of most synchrotron sources except in specialised beamlines.

The (mean) power radiated by an electron experiencing this undulating motion is given by the following equation:

$$\bar{P} = \frac{e^4 \gamma_{z0}^2 \kappa^2 r_\beta^2}{12 \pi \varepsilon_0 m^2 c^3},$$
<table><caption>**Table 24.8.** Important properties of a plasma betatron-radiation source and their implications.</caption><thead><tr><td>Property</td><td>Symbol</td><td>Characteristic Value</td><td>Implication</td></tr></thead><tbody><tr><td>Hard photon energy</td><td>E<sub>crit</sub></td><td>&gt;25 keV</td><td>Deeply penetrating</td></tr><tr><td>Small source size</td><td>r<sub>β</sub></td><td>~μm</td><td>Exhibits high spatial resolution</td></tr><tr><td>Small divergence</td><td>θ</td><td>~10 mRad</td><td>Makes beamline</td></tr><tr><td>Short pulse</td><td>τ</td><td>~10 fs</td><td>Ultra-fast dynamics</td></tr><tr><td>Bright</td><td>N<sub>ph</sub></td><td>&gt;10<sup>9</sup> (photons per shot)</td><td>Single-shot imaging</td></tr></tbody></table>

where $κ$ is the gradient of the focussing field strength. Despite the small oscillation amplitudes, the strong fields give rise to a substantial photon yield, with almost one photon produced per electron. A single shot can produce upwards of $10^8$ photons, more than enough to do single-shot imaging of moderately sized objects (~cm).

The high photon energy is not the only interesting property of these sources. Given the small spatial scales of the accelerator, the region from which the radiation is emitted is also very small, both transversely and longitudinally. The length of the X-ray beam can be controlled by changing the distance over which electrons are injected and accelerated. In addition, high-charge (100 pC) electron beams can be readily produced with a bunch duration that is less than half of the plasma period [639]. Experimental measurements have shown that electron bunches can be a few femtoseconds in duration with peak current of a few kA [550]. The emitted radiation is set by the electron bunch duration and so is a highly attractive source for measurements on the finest temporal scales.

Furthermore, the transverse extent of the radiation is also small, comparable to the amplitude of the undulator oscillation when the electrons reach their highest energy. This is, as noted above, typically of the order of ~ µm. This small source size allows for high-resolution imaging in a compact imaging setup, without the need for expensive and difficult-to-align X-ray optics. Furthermore, the X-rays are “beamed” by the relativistic electrons that emit them, which enables high-magnification imaging without sacrificing imaging brightness greatly. But perhaps the most interesting manifestation of this small source size, is that it allows the beam to exhibit spatial coherence in a compact arrangement. This enables enhanced imaging modalities, specifically phase-contrast imaging, which will be discussed in more detail below.

Table 24.8 lists some of the interesting properties of the plasma betatron-radiation sources and also the implications of these properties.

In the next section, we expand a little on these interesting properties of betatron-radiation sources, and outline a plan for implementing a dedicated betatron beamline within the applications area of the EuPRAXIA facility. Finally, we give some example applications.

## Implementation of a Betatron Source at EuPRAXIA

The first measurements of betatron radiation from a plasma accelerator were made by S. Wang and co-workers on the E-157 plasma-wakefield experiment at SLAC [640]. They measured ≈ 15 keV photons produced by 28.5 GeV electrons. Betatron radiation was soon measured from laser-wakefield-acceleration experiments. Using a laser with many tens of TW power to produce electrons with energies up to 200 MeV, Rousse and co-workers at the LOA in France [573] were able to measure photons with critical energies in the few keV range [641]. The high densities that these LWFA experiments
<figure><img src="image_232.png" /><figcaption>**Fig. 24.9.** Conceptual visualisation of a betatron beamline at the proposed EuPRAXIA laser-driven construction site (based on the CAD model in Sect. 32.9 of the appendix).</figcaption></figure>

work at mean that the scale of the source was reduced from the metre-length E-157 experiment down to sub-centimetre scales.

A step forward in usability of these sources was achieved by using a plasma accelerator in the self-injecting self-guiding mode (“bubble regime”). In these experiments, a significant increase in spectral brightness (by more than three orders of magnitude) was achieved by being able to accelerate large numbers of particles to high energy [637]. More than $10^8$ photons per shot were produced, which was more than sufficient for single-shot imaging [637]. Performing these measurements with greater than 100 TW lasers has allowed the critical energy to be extended to well over 30 keV, which allows for uses in studying dense media [75,638].

Considerations for radiation emission in this highly non-linear regime gives scalings of electron beam energy, betatron critical energy, and photon number, which increase monotonically with laser power. These scalings have now been thoroughly tested in numerical simulations [635] and in experiments. Though petawatt lasers are already operational, there has been little work in the strongly driven regime, which is ideal for betatron applications.

Both plasma-wakefield beams and laser-wakefield drivers can be used as drivers for betatron radiation sources, but, as noted, the significantly higher densities of operation of laser drivers to reach the bubble (“blow-out”) regime is advantageous in terms of photon energies, numbers, pulse lengths, and source sizes. Hence, it is proposed that the LWFA driver that will be usually used to drive a laser-wakefield stage be redirected to a dedicated applications area where a betatron source can be installed. A conceptual visualisation of such a beamline is shown in Figure 24.9.

Using the considerations given in the first section and some of the scalings presented in [575], Table 24.9 gives the expected scalings in photon properties for the PW-class laser that will drive the laser accelerators in EuPRAXIA.

## Applications of Betatron Sources

As we have noted, there are numerous synchrotron facilities, each with beamlines dedicated to enhance their applicability for particular uses. But an application beamline based on a betatron radiation source has a number of useful intrinsic properties that
<table><caption>**Table 24.9.** Scalings for betatron radiation from established scaling laws ([575]). The figures for 10 and 100 TW systems agree well with reported measurements.</caption><thead><tr><th>Laser power</th><th>10 TW</th><th>100 TW</th><th>1 PW</th></tr></thead><tbody><tr><td>Pulse duration τ (fs)</td><td>12</td><td>37</td><td>120</td></tr><tr><td>Spot size w<sub>0</sub> (μm)</td><td>4</td><td>10</td><td>35</td></tr><tr><td>Plasma density n<sub>e</sub> (cm<sup>-3</sup>)</td><td>5 × 10<sup>19</sup></td><td>5 × 10<sup>18</sup></td><td>5 × 10<sup>17</sup></td></tr><tr><td>Plasma length (mm)</td><td>90 × 10<sup>-3</sup></td><td>3</td><td>90</td></tr><tr><td>Peak electron energy γm<sub>e</sub>c<sup>2</sup> (MeV)</td><td>60</td><td>600</td><td>6 × 10<sup>3</sup></td></tr><tr><td>Beam charge (nC)</td><td>0.1</td><td>0.4</td><td>1</td></tr><tr><td>X-ray energy (keV)</td><td>0.6</td><td>8</td><td>110</td></tr><tr><td>Source size (μm)</td><td>1.4</td><td>1.8</td><td>2.4</td></tr><tr><td>Divergence θ (mrad)</td><td>70</td><td>7</td><td>0.7</td></tr><tr><td>Undulator parameter α<sub>β</sub></td><td>7</td><td>9</td><td>12</td></tr><tr><td>Peak X-ray brightness *</td><td>2 × 10<sup>21</sup></td><td>3 × 10<sup>23</sup></td><td>1 × 10<sup>26</sup></td></tr><tr><td>Photon number</td><td>2 × 10<sup>8</sup></td><td>3 × 10<sup>9</sup></td><td>4 × 10<sup>10</sup></td></tr></tbody></table>

can be exploited for particular investigations. Furthermore, the relatively small size and cost of a betatron facility could mean that applications of synchrotron radiation could become more widespread if such facilities were developed. In particular, synchrotron techniques could supplant widespread X-ray techniques that are based on first-generation X-ray sources. Of particular note is the short pulse length, high photon energy, synchronised secondary beams and small source size, which make certain applications relatively easy to implement using a betatron source as compared to other X-ray radiation sources. Though many of these applications have or are being pursued in proof-of-principle experiments, the higher energy, relatively high repetition rate, and industrial robustness of the (laser) drivers would mean a betatron source based on EuPRAXIA would be a major step towards dedicated betatron radiation facilities.

## Synchrotron Applications

The EuPRAXIA betatron source would have sufficient average photon yield to allow various types of experimental stations for spectroscopy, diffraction, and imaging experiments, as are common in existing synchrotron facilities. In particular, it would potentially be the first betatron source to be run at a consistent repetition rate (minimum of 10 Hz) promising time-averaged photon brightness in excess of 10¹³ photons / (mm² mrad² s¹ 0.1%BW), many orders of magnitude greater than available with other common laboratory sources [75].

## Lensless Imaging

The small source size and high photon energy already make betatron radiation sources interesting for standard radiography applications. It allows the high-resolution imaging of dense matter, such as bone specimens or materials of interest in industrial manufacturing with a compact size [75]. The petawatt betatron source at EuPRAXIA offers to take this a step further, with its higher photon energy. This allows imaging of even denser objects, in particular in vivo imaging of humans, which takes photon energies in excess of 100 keV. For such a high photon energy, optics are not readily

<footnote>*Units for peak X-ray brightness are photons / (mm² mrad² s¹ 0.1%BW).</footnote>
available, so coupling the small sources to a beam with reasonable divergence allows high-resolution imaging in a compact size. This can be a considerable advantage over other synchrotron sources, where the beam has to be conditioned to offer similar abilities.

The other advantage of the small source size is that it offers an appreciable degree of spatial coherence. This can be most readily taken advantage of in free-propagation phase-contrast imaging (PCI). Again, the natural properties of the source allow X-ray PCI to be carried out without additional optics such as gratings, which are required for other X-ray sources. Phase-contrast imaging offers increased contrast and the ability to demarcate features in specimens which offer relatively little absorption contrast. This has been demonstrated on small biological [637,642] and industrial samples, especially those of low-Z materials, such as carbon fibres, and those made by additive manufacture. The higher repetition rate of EuPRAXIA offers full 3D tomographic reconstruction capabilities of such objects in a fraction of the time that it takes with many other PCI solutions [643]. The higher energies available make it possible to visualise soft tissue such as cancerous liver or kidney lesions, removing the need for a contrast agent. Though the advantages of PCI for this type of work have been proven, progress has been held back due to the lack of suitable X-ray sources with the necessary spatial coherence properties or due to the requirement for cumbersome imaging techniques. A EuPRAXIA-based synchrotron source using betatron radiation would make a valuable contribution to advancing this field.

## Coherent Imaging

Of course, it is well known that the small wavelength of X-rays allows imaging features of size comparable to the radiation wavelength. In the case of hard X-rays produced by a petawatt betatron source, we can expect to be able to resolve features in the nanometre to angström range. For objectives approaching the wavelength of the radiation, direct imaging is not possible, but instead we rely on diffractive imaging. Even when specimens do not crystallise, computational techniques can be used in a lensless imaging scheme to infer the structure of a specimen from the diffraction pattern so long as the X-ray probe is temporally and spatially coherent. Again, considering the often large scattering angle, it is impractical to use X-ray optics, and so lensless imaging configurations are ideal. Temporal coherence is inversely proportional to spectral width and can be achieved with crystal monochromators, as used in other synchrotron facilities. Though this can greatly reduce the available flux – typical monochromators will reduce the spectral brightness by at least $10^6$ –, this can be compensated somewhat by the higher repetition rate proposed for the EuPRAXIA beamline. Spatial coherence increases proportionally with the distance from the source and with the inverse of the source size. This means conventional X-ray tubes and even synchrotron light sources must be apertured, which becomes difficult for hard X-rays, thus reducing the available photon flux and increasing the required exposure time. Once again, the inherent small size of betatron sources makes them attractive for this application.

## Ultra-Fast Studies

As has been noted, the inherently ultra-fast nature of the betatron source, coupled to the fact that it can be synchronised to numerous other radiation sources, makes it an attractive platform for numerous pump-probe experiments. The secondary radiation source can be other laser beams, but also particle beams that can be generated with lasers. There are two types of dynamic processes, which could be particularly interesting to implement on a betatron beamline.
The first are ultra-fast processes, where an interaction initiated by a secondary sources is then followed over subsequent steps by taking multiple shots with vary- ing delay between the pump and a betatron probe. This has been used in proof- of-principle experiments to study the propagation of shocks in dense media. The betatron beam need not only be used in imaging mode, but can also be used for X-ray Thomson scattering, or even for spectroscopy. Indeed, the broad bandwidth of the betatron source is ideal for ultra-fast absorption spectroscopy.

The other type of interaction, where a high-repetition-rate, short-pulse X-ray source is advantageous, is in the study of longer-scale dynamic effects. Examples for this are the motion in biological systems, e.g. the respiration of insects and the motion of wings, as well as gradual material effects, e.g. the growth of dislocations in materials, annealing and nucleation processes. Here the pulsed nature of the betatron sources is useful in reducing motion blur.

## Gamma-Rays

The availability of petawatt laser energies at EuPRAXIA will extend photon energies to beyond 100 keV, which is not widely available in synchrotron laboratories. As already noted, the greater penetration of these photons can be used in imaging macro- scale objects, such as cracks in airplane wings. They can also be used to identify high-Z materials, such as for use in the discrimination of glass fragments for forensic analysis, which require photon energies in excess of 100 keV. Indeed, the higher end of this photon spectrum can be considered as γ-rays and can be used for nuclear-induced applications. Gamma radiation can be used to identify and study nuclear materials. It can also be used in nuclear resonance fluorescence to study the nuclear structure or identify isotopes, potentially even pumping a gamma laser. Finally gamma-ray radiography is routinely used to scan cargo for security applications.

> The EuPRAXIA laser-plasma beamlines based on industrially robust laser systems offer the opportunity to have a first generation of user experiments based on plasma-accelerator technologies. One of the premier applications for this type of system will be a betatron beamline that can provide high quality X-rays for a number of different types of experiments. Betatron radiation sources could potentially be the first type of plasma-based accelerator device to be used as the basis of an accelerator facility.

## 24.3 Compton Source

### Applications of Gamma Sources

The inverse Compton scattering (ICS) end station at EuPRAXIA will aim to generate unique sources of high-energy (>100 keV) photons, which, in theory, would be tunable up to 300 MeV. The low divergence and small source size allow the study of nuclear physics as well as the interrogation of high-areal-density samples, which may be of interest for security screening at ports.

High-quality, high-energy photon sources are important for nuclear physics as demonstrated by the success of the HiγS facility at TUNL. With photon energies in excess of 100 MeV, it will be possible to produce pions and perhaps even heavier exotic particles from the “clean” photon rather than particle collisions.

The highly penetrating nature of such gamma-rays also allows for the interroga- tion of high-areal-density materials for security purposes.
<figure><img src="image_233.png" /><figcaption>Fig. 24.10. Cartoon of the collision of an electron with Lorentz factor $γ$ with a laser resulting in MeV radiation emission from a small spot and within a cone angle $1/γ$.</figcaption></figure>

## Inverse Compton Scattering

### Introduction

Inverse Compton scattering (ICS) is the process by which light may be upshifted in frequency through collision with a moving electron. The process of X-ray production is straightforward in the photon picture, with the emitted photons experiencing a double relativistic Doppler upshift: once due to the reflection in the electron rest frame, and a second time as the interaction is transformed back into the lab frame. The emitted photon energy, $E_γ$, is dependent on the Lorentz factor of the relativistic electron, $γ$, and the angle of collision, $θ$, where $θ = 0$ in the co-propagating geometry. This gives the following relation:

$$E_γ = \hbar\omega_0 \times [\gamma(1 - \cos\theta)]^2, \qquad (24.4)$$

where $\hbar\omega_0$ is the energy of the incident photon with radial frequency $ω_0$. This results in the generation of high-energy (greater than 100 keV) photons from the collision of an optical laser (≈ eV) with electrons of energy above 100 MeV. Since the radiation may be considered isotropic in the rest frame of the electron, the transformation into the lab frame also results in the radiation being beamed into a small cone. The cone angle may be approximated by $1/γ$, as illustrated by the cartoon in Figure 24.10.

Simplifying the expression for the emission energy, by considering only the counter-propagating geometry, we obtain the following simple expression for the maximum radiation energy:

$$E_γ = 4γ^2\hbar\omega_L \qquad (24.5)$$

### ICS from Wakefield Accelerators

Inverse Compton scattering from LWFA electron bunches has been demonstrated in various experiments. Ta Phuoc et al. [644] demonstrated that reflecting the LWFA drive laser from a foil at the end of the accelerator allows guaranteed temporal and spatial overlap of the electron and laser pulse. In this work, the linear scattering generated sources in the tens of keV. Such sources were also generated with a secondary laser [645] by the Umstadter group, who then extended the accessible photon energy to the MeV range by scattering from higher energy electrons [646]. In 2014, Sarri et al. reported the first instance of non-linear ICS, where the scattering laser was sufficiently intense to generate harmonics of the scattering frequency [647].

The increased interest in ICS from LWFA is driven by the uniqueness of the sources which may be generated. Much of the discussion in previous sections about the advantages of LWFA-based photon sources also apply to ICS. For example, the duration of the emission is restricted to the duration of the electron bunch allowing for radiation source durations in the range of few femtoseconds. In addition, since
<table><caption>**Table 24.10.** Unique properties of plasma-accelerator-based ICS radiation sources, and their implications.</caption><thead><tr><td>Property</td><td>Symbol</td><td>Characteristic Value</td><td>Implication</td></tr></thead><tbody><tr><td>High photon energy</td><td>E<sub>γ</sub></td><td>&gt; 100 MeV</td><td>Deeply penetrating</td></tr><tr><td>Small source size</td><td>r<sub>ICS</sub></td><td>~μm</td><td>High spatial resolution</td></tr><tr><td>Low divergence</td><td>θ</td><td>&lt;1 mrad</td><td>Localised irradiation</td></tr><tr><td>Short pulse</td><td>τ</td><td>~ 30 fs</td><td>Ultra-fast dynamics</td></tr></tbody></table>

the scattering comes from the laser focus, the source size can be restricted to order of 1 micrometre.

Table 24.10 lists some of the interesting properties of plasma-based inverse Compton scattering radiation sources, and also the implications of these properties.

### Considerations of ICS Source Quality from a Single Electron

Here, each property of the ICS is addressed in more realistic terms. In equation (24.5), an expression is given to demonstrate the unique photon energies accessible through ICS at EuPRAXIA. However, taking into account the observer angle, $\theta$, the laser strength parameter, $a_0$, and the electron recoil, encapsulated in $\Delta$, a more accurate expression is given by the following [648]:

$$E_{\gamma} \approx \frac{4\gamma^2\hbar\omega_L}{1 + \gamma^2\theta^2 + \frac{a_0^2}{2}} (1 - \Delta) \quad (24.6)$$

### Preservation of Low Bandwidth

From equation (24.6), it is clear that the bandwidth of an ICS source is only small, even in the idealised linear case, when the observation angle is small. To retain spectral selectivity, a hard aperture will be required for the beam.

In addition, if a small bandwidth is required, it will be important to ensure that the intensity of the scattering laser is low such that the spectrum is not broadened by an increase in $a_0$. This low laser intensity regime will also facilitate a small $\Delta$.

### Small Source Size and Short Pulse Duration

An advantage of ICS over other radiation mechanisms lies in the importance of the scattering laser. Since the radiation only emanates from the focal spot, the size and duration of the laser at focus will determine the size and duration of the radiation source.

### Low Divergence

As illustrated in Figure 24.10, the divergence of the emission from an electron is given by $\frac{1}{\gamma}$. Again, this is only preserved for scattering at linear laser intensities ($a_0 < 1$).
## Electron Bunch Considerations

Thus far, arguments for high beam quality have been based on the simplistic picture of a single electron. While it is expected that the beam quality in EuPRAXIA will be high, there are some fundamental beam parameters which must be considered when designing an ICS source. All of the above parameters rely on every electron in the bunch being identical in terms of energy, position, and angle relative to the laser pulse. The deviation from this idealised case may be quantified by the electron beam emittance.

When scattering from a 5 GeV electron beam ($\gamma \sim 10,000$), in order to limit the bandwidth of the source, the acceptance angle of the end station must be limited to <0.1 mrad. If the divergence of the electron beam is greater than this, it will not be possible to limit the bandwidth of any ICS source. This does not erase the applicability of the source, rather imposing limitations on how the source is implemented and requiring flexibility in both electron energy and laser energy / intensity in order to fully access regimes of interest. The study of electron beam quality on ICS sources has been considered [648,649].

## Implementation of an ICS Source at EuPRAXIA

From the previous section, it is clear that a range of photon sources will be feasible from the ICS end station. As with many aspects of facility use, the parameter selection will be a compromise, whereby an increase in flux (through higher $a_0$) or higher photon energy (through higher electron energy) will each come at the price of increased bandwidth. The conclusion from this argument is that flexibility in electron and laser parameters will allow access to a range of unique sources unavailable at other facilities. The specific parameters available at the ICS end station will be sufficiently sensitive to electron beam parameters the precise knowledge of which will be essential.

## Electron Beam Diagnostics

In addition to the plethora of applications outlined thus far, the relationship between emittance and bandwidth, as intimated above, provides an opportunity for a novel beam diagnostic. Measuring the angularly resolved ICS emission from a well-characterised laser focus will provide a complementary transverse emittance measurement of the wakefield-accelerated beam.

# 25 Particle User Applications

## 25.1 High-Energy Physics Test Beams

As complex high-tech devices, modern large particle detectors, such as those used for measurements at the Large Hadron Collider at CERN, typically require extensive R&D, simulation, and testing, before they can start operation. Test beams suitable for such work, however, are only available at large accelerator laboratories. The electron-positron synchrotron DESY II, for example, is one of such test beam facilities, although it is also used as an injector for the PETRA III synchrotron. With a circumference of 293 m, it generates particle beams with energies up to 6.3 GeV, ~5% energy spread, ~1 mrad divergence and around 1000 particles per cm$^2$ in bunch
<figure><img src="image_234.png" /><figcaption>Fig. 25.1. Schematic layout for the electron test beamline, which could be used, for example, for HEP detector timing calibration and testing.</figcaption></figure>

lengths around 30 ps [650]. In view of increasing the availability and reducing the cost of such test beam facilities, plasma accelerators could provide an interesting alternative, pioneering complementary ways to develop and test detectors.

There are indeed a number of differences between electron beams generated from plasma accelerators and RF-based machines. Conventional test beams can consist of a variety of particle species, from protons and neutrons to electrons, positrons, photons, and other more exotic particles. Plasma accelerators, on the other hand, have been shown so far to generally only produce few particle species at the same time in the form of electrons and photons, as well as, in the case of EuPRAXIA (as discussed below), positrons. Additionally, the beam distribution from plasma accelerators is quite unusual for test beams due to its short bunch duration, fs-level synchronisation and compact transverse beam size leading to a large number of particles arriving at the same time and same spot at the detector.

These particular characteristics could, however, be turned into an advantage and allow new types of detector test and timing calibration studies. In the following, three particular use cases are proposed.

**Parallel irradiation of large detectors:** Using a high-flux beam, but with a low flux density, larger areas of a detector could be irradiated simultaneously instead of scanning across the detector area in many small steps. Such an approach would be not only be significantly faster or allow for significantly more data-taking, but also make it possible to calibrate the timing measurement uniformity across the detector. Additionally, the high-flux beam could be used as an excellent tool to stress-test the data acquisition system of the detector under realistic conditions given the high, but tunable number of particles.

**High-flux-density scanning of detector elements:** Using a focussed high-flux beam from a plasma accelerator, high flux density can be reached to study fine effects within individual detector elements. This could be used, for example, to test the saturation and linearity of an active element response, but may also be interesting for studying particle beam effects on the embedded read-out electronics in the detector in an extreme high-density scenario.

**Testing of the uniformity of the timing response:** A third possible use case of plasma-accelerated electron beams is for timing tests of the detector. Thanks to their short bunch duration, a good temporal resolution could be achieved and it could be envisaged to be able to scan through different bunch durations by stretching the electron beam in a controlled way along the transport beamline to the detector measurement.

At EuPRAXIA, several of these testing scenarios could be realised. In the generation of low- and high-energy positron beams at both construction sites, part of the incoming electron beam will remain after propagation through the converter
<figure><img src="image_235.png" /><figcaption>**Fig. 25.2.** Conceptual design of the low-energy positron beamline to be installed at the laser-driven plasma acceleration construction site. The laser (red cone on the left) is focussed on a plasma target to generate an ultra-short (∼fs) and ultra-relativistic electron beam that, after conditioning and potential second acceleration, is sent onto a high-Z converter. A beamline for both the electrons and positrons escaping the solid target is sketched after that.</figcaption></figure>

target. This beam will be of GeV-scale energy, with charge between a few particles and several pC (depending on the converter target thickness), and limited beam quality in terms of energy spread and transverse emittance. It is foreseen to separate this electron beam from the positrons in a dogleg setup, described in more detail in Section 25.2, transport it through a conditioning beamline, and finally direct it into a user area for detector calibration and other test experiments. While a detailed design still needs to be developed during the technical design phase, a schematic beamline layout can be seen in Figure 25.1 showing the key elements of the conditioning line. As Figure 25.5 in the next section shows to some extent, a set of quadrupoles will be used for capturing the electron beam after the converter target and separating it from the positron beam through a dipole section. The latter could also be expanded for momentum selection to fine-tune the energy and energy spread of the electron beam as well as filtering out a specific beam charge. Beam focussing can be tuned downstream through further focussing optics. Finally, a controlled bunch lengthening system would need to be implemented for timing studies, a feasibility study for which remains to be carried out. The combination of these conditioning elements and their potential for tuning the beam properties will thus enable such unique studies, as high-flux-density scans, large-area detector irradiation and uniformity-of-timing tests, as described above. Additionally, this setup could provide a very interesting ground, more generally, for experiments with ultra-short, tunable electron test beams, for example for studying novel diagnostics or beam manipulation elements. It is thus foreseen to implement a test beam user area along the shared positron beamlines at each of the two EuPRAXIA construction sites, thus allowing for more user time and increased flexibility.

Regarding the test beam user areas, several essential requirements can be defined, based on discussions with potential future users during the “EuPRAXIA Workshop
<table><caption>**Table 25.1.** Expected properties of the low-energy positron beam generated at the EuPRAXIA beamline.</caption><thead><tr><th>Positrons/shot<br>(50 keV bandwidth)</th><th>Energy</th><th>Repetition<br>rate</th><th>Beam size<br>at sample</th><th>Beam<br>duration</th><th>Positrons/s</th></tr></thead><tbody><tr><td>10<sup>6</sup></td><td>tunable: 0.5–10 MeV</td><td>100 Hz</td><td>2–5 mm</td><td>90 ps 1 MeV<br>20 ps 2 MeV</td><td>10<sup>8</sup></td></tr></tbody></table>

on Pilot Applications of Electron Plasma Accelerators (PAEPA)” in 2016 and include the following:

- large manipulation space both horizontally and vertically,

- presence of a crane and movable table for large components,

- presence of an accurate timing signal, e.g. through adequate synchronisation with a reference laser,

- presence of several specific diagnostics, including a flux monitor,

- presence of cooling water in the user area.

The current conceptual layout of EuPRAXIA has been developed with these features in mind and a detailed design will be realised together with the technical specification of the conditioning beamline during the next phases of EuPRAXIA.

## 25.2 Secondary Particle Sources

### 25.2.1 Low-Energy Positrons

Using baseline values for the EuPRAXIA plasma injector (charge: 30 pC, duration: 10 fs, energy: 150 MeV – 5 GeV, depending on the number of acceleration stages), one can obtain, after propagating through a mm-scale Pb target, up to 10 pC of positrons with a broad energy spectrum extending up to the maximum electron energy, and a sub-ps duration. The repetition rate of the positron source would be identical to that of the laser beam driving the acceleration of the primary electrons, and is designed to be of the order of 100 Hz. The broadband spectrum of the positrons [148] can be used as an advantage since, with the appropriate beam handling, one can select the energy required, virtually seamlessly from sub-MeV to a few MeV – the spectral region where most of the positrons are present. In Figure 25.2, a conceptual design of the beam handling after the converter, including collimation and energy selection, is shown. After approximately 6 m of beamline, one can achieve a mm-size positron beam with a tuneable energy from 0.5 to 10 MeV and a bandwidth of ±50 keV. The simulated evolution of the transverse size of the positron beam through the beamline (both along the x and y axes) is shown in Figure 25.3.

Once the positrons exit the conditioning beamline, they will enter a dedicated target area with an approximate footprint of 8 × 15 m². The electrons escaping the target will instead be directed along a different beamline and will be finely characterised, on-shot, to provide a live diagnostic of the generated positron beam. Finally, the arrival of the X-ray radiation escaping the solid converter will be used as a definition for time-zero, i.e., the time the positrons are implanted in the sample. The end-user target area will comprise a vacuum vessel where the sample to be studied will be mounted on a 5-D motorised stage, allowing for its fast alignment and scanning, both transversally and in depth. A suite of detectors will be used to perform typical positron-induced material inspections, which include positron annihilation spectroscopy and Doppler-broadening spectroscopy.
<figure><img src="image_236.png" /><figcaption>**Fig. 25.3.** Simulated evolution of the transverse size of the positron beam, in x and y (designated as SIGX and SIGY), through the low-energy beamline shown in Figure 25.2 (after the converter).</figcaption></figure>

Due to the different penetration depths of different positron energies, this allows one to perform the *volumetric* scanning of materials, a feature that is unavailable at the sub-relativistic positron factories currently operational. Moreover, the high positron flux per shot (of the order of $10^6$ positrons in a 50 keV bandwidth) and the considerable repetition rate of 100 Hz result in a considerably high flux of the order of approximately $10^8$ positrons/s.

The intrinsic short duration of the laser-wakefield electron beam, implies that the positron duration at the sample will be predominantly dictated by the temporal spreading during propagation through the beamline. Assuming a 6 m long beamline (see Fig. 25.3), a 2 MeV ± 50 keV (1 MeV ± 50 keV) positron beam will have a duration at the sample of the order of 20 ps (90 ps), which is currently unattainable by more conventional positron factories and, indeed, shorter than the typical timescales of positron annihilation in materials (see Tab. 25.1).

This will not only lead to much higher resolution but can also allow to study real-time dynamic effects in material under stress on picosecond time-scales, a unique feature that would represent a step change in the designing and manufacturing of materials under extreme conditions. A summary of the positron parameters that can be reached in this end-station of EuPRAXIA is given in Table 25.1.

It must be noted here that the number of positrons will scale linearly with the charge of the primary electron beam, making it desirable to have a higher charge in a broader spectrum. Moreover, the number of positrons will scale, empirically, with the square root of the energy of the primary electron beam. Extending the electron beam energy from 150 MeV to 5 GeV, as possible using a multi-staged approach, would thus increase the number of positrons by a factor of approximately 6.

### 25.2.2 High-Energy Positrons

We propose here to use the main electron beam designed for the beam-driven plasma acceleration construction site of EuPRAXIA to drive high-quality and high-energy positron beams that can be used as seed beams for advanced studies of laser wakefield
<figure><img src="image_237.png" /><figcaption>**Fig. 25.4.** Conceptual visualisation of the positron and test electron beamline at the proposed EuPRAXIA laser-driven construction site (based on the CAD model in Section 32.9 of the appendix).</figcaption></figure>
<figure><img src="image_238.png" /><figcaption>**Fig. 25.5.** Conceptual design of the high-energy positron beamline designed for the PWFA construction site.</figcaption></figure>

<figure><img src="image_239.png" /><figcaption>**Fig. 25.6.** Simulated evolution of the transverse size of the beam, in both x and y (designated as SIGX and SIGY here), through the high-energy beamline shown in Figure 25.5. At the end of the beamline, the beam has a radius of the order of 20 µm.</figcaption></figure>

acceleration and the conditioning of positrons. For the laser wakefield case, the baseline parameters of the electron beam are the following (see [162]): energy of 1 GeV, duration of 7 fs, and charge of 30 pC. The possibility of accelerating the electron beam to 5 GeV final energy is also considered.

After the electron acceleration stage, a switchyard will direct the beam into three main areas: an area for the generation of high-quality multi-MeV gamma-ray beams from Inverse Compton Scattering in a laser field, an XFEL line, and a line for positron generation. Once the main electron beam is directed to the positron generation area,
the current design for the positron line is given in Figure 25.5. The electron beam will be steered by a magnetic dipole onto a high-Z solid target (lead) with a thickness of the order of a few millimetres. The quantum electrodynamic cascade inside the solid target will generate a high-flux photon beam and dense populations of electrons and positrons. The positrons escaping the target will present a broad spectrum and energy-dependent duration, source size, and divergence. In a 5% bandwidth around 1 GeV, one expects approximately $10^7$ positrons with a duration comparable to that of the primary electron beam, a source size of 20 µm and a divergence of 10 mrad [148]. The resulting geometrical emittance at 1 GeV at source is then 0.25 mm mrad (normalised emittance of 500 mm mrad). The positron duration at 1 GeV will be comparable to that of the primary electron beam, i.e., sub-10 fs [148]. The simulated evolution of the transverse size of the positron beam at 1 GeV through the beamline illustrated in Figure 25.5 is given in Figure 25.6.

The beamline can be tuned in order to maximise the collection efficiency, which can be as high as 50%. Moreover, tuning of the magnetic components in the beamline allows for a certain level of freedom in tuning the final energy of the positrons, in the range of hundreds of MeV up to 1 GeV. At the end of the beamline, the positron beam will then enter the user area with the following characteristics: an energy of 1 GeV ±5%, a duration of <10 fs, a beam size of 20 µm, and a charge of the order of 1 pC. A versatile area with a footprint of $10 \times 20 \text{ m}^2$ will then be dedicated to experiments in high-energy physics, with a specific focus on providing a test beam for experiments in advanced schemes of positron acceleration. Several plasma columns and plasma cells will be available in vacuum vessels to study laser-driven and particle-driven wakefield acceleration. A dedicated high-power laser system will be installed and synchronised with the positron beam for laser-wakefield studies (800 nm laser system providing 30 J in pulses with a duration of 30 fs and a repetition rate of 10 Hz). It must be noted then that a high-charge electron beam (>10 pC) synchronised with the positron beam will also be collected by the proposed beamline, and thus available in the area for particle-driven wakefield experiments.

# 26 Additional Innovation Paths

## 26.1 Hybrid Laser-Beam-Driven Plasma Wakefield Acceleration

### 26.1.1 Introduction

PWFA has various well-known important advantages when compared to LWFA, such as dephasing-free, dark-current-free operation, and enables unique injection schemes such as the plasma photo-cathode, which promises improved output electron beam qualities such as brightness by a factor of 1,000 to 100,000 better than the state of the art.

PWFA can be realised at linac-driven facilities, such as SLAC FACET(-II), DESY, Daresbury Laboratory, and INFN, where the EuPRAXIA PWFA construction site is foreseen. Linac-driven PWFA is well-established and potentially allows beam acceleration at high repetition rate with stable beams, but also has drawbacks such as the large size of the accelerator due to the linac system, which produces the electron beam used to drive the PWFA stage, and limited synchronisation between electron beam driver and laser systems, which is desirable e.g. for FEL seeding, pump-probe-experiments, and the realisation of plasma photo-cathodes.

In EuPRAXIA's Work Package 14 on "Hybrid Laser-Electron-Beam Driven Acceleration", forward-looking and potentially disruptive approaches were developed, which combine the best of LWFA and PWFA (Scheme 6, as described in Chap. 8).
In addition, hybrid solutions across LWFA and PWFA also strengthen synergies and coherence of the overall EuPRAXIA design. They are a bracket around the key EuPRAXIA approaches and sites for LWFA and PWFA.

> Hybrid plasma accelerators unite the best of LWFA and PWFA and offer a symbiosis between the two main underlying technologies of EuPRAXIA.

Dramatic progress has been seen in applying this novel hybrid philosophy over the lifetime of the EuPRAXIA study (2015–2019) both conceptually but also experimentally. While key concepts [53,601,603,651,652] had been experimentally unproven at the beginning of the project, a major important subset of them could recently be successfully demonstrated experimentally [653,654]. This constitutes a jump in technology readiness level (TRL) from approximately TRL 2 (conceptual) to TRL 6 (experimental technology demonstration) and is a very strong sign of its viability and capability to fulfil the main EuPRAXIA aim of a “Compact European Plasma Accelerator with Superior Beam Quality”.

### 26.1.2 LWFA Beams as PWFA Drivers

The main concept of hybrid LWFA→PWFA staging is to use the electron beam output from LWFA as a driver for PWFA. PWFA requires energetic, dense electron beams with high currents as drivers. This is precisely what LWFAs can inherently deliver: ultra-short, multi-kiloampere current electron bunches. While the sizeable energy spread and emittance of electron bunches from current state-of-the-art LWFAs impose serious challenges to realise key applications, such as free-electron lasers, these properties are excellent prerequisites to realise compact PWFA, as they offer improved resilience to instabilities [655,656]. In this hybrid approach, the electron beam (or beams) produced in an LWFA stage is inserted into a subsequent PWFA stage, which can be less than a millimetre away from the LWFA stage, or further away when an electron beam transport beamline is employed.

Figure 26.1 shows the schematic underlying concept. First, a short high-current and high-density electron beam is produced via LWFA. This capability is an inherent advantage of LWFA: the short duration of high-power laser pulses makes it possible to work at a wide range of plasma densities (10¹⁷ – 10¹⁹ cm⁻³) to drive a resonant plasma wave. In particular at high densities, the laser pulse group velocity is significantly reduced with respect to the speed of light, which together with the high accelerating and transverse plasma wakefields, favours self-injection and the rapid acceleration of high-current-density electrons. An alternative strategy lies in operating at lower plasma densities to prevent self-injection and exploit the implementation of more controlled injection techniques for tunability [18,279,657–660]. In particular, ionisation-based injection techniques [279,661] allow for a direct control of the amount of injected charge, which can be tuned to find nearly optimal beam-loading conditions for a reduced correlated energy spread along the witness beam [329]. A relatively recent LWFA experiment at the Helmholtz Centre Dresden-Rossendorf (HZDR) employing a ~100 TW laser system demonstrated that electron beams with around 300 pC charge can be produced at a reduced energy spread [28] by adjusting the loaded charge. With a duration of ~10 fs, these LWFA beams deliver tens of kiloamperes of current at around 300 MeV mean energy. Multi-kA level electron beams from LWFA can also be obtained by other LWFA installations, e.g. in the UK at the Scottish Centre for the Application of Plasma-based Accelerators and the Central Laser Facility, in Germany at LMU Munich and in France at LOA Paris. A high current density is the critical requirement for PWFA drivers, as it allows the PWFA system to operate in the highly non-linear “blowout” regime [367]. This regime
<figure><img src="image_240.png" /><figcaption>**Fig. 26.1.** Three-dimensional electron density maps, extracted from particle-in-cell simulations with OSIRIS [336], for LWFA (left) and PWFA (right). The schematic shows the underlying principle of the hybrid approach. In LWFA, laser pulses in plasma propagate at a velocity significantly smaller than $c$, and their oscillatory field structure excite the plasma wave via the ponderomotive force. In PWFA, the drive beam moves with $\sim c$ and its space-charge fields excite the plasma wave, such that the driving field values involved in the process are orders of magnitude lower than in LWFA. This enables unique injection methods based on field ionisation for electron bunch production with ultra-high quality [601,603,652].</figcaption></figure>

offers key advantages for the production of high-quality beams from PWFA, such as the possibility for advanced injection techniques [54,601,603,652,662] and a particularly steady wakefield formation for beam-loading control [48,53]. While in LWFA, the currently achievable levels of energy spread and normalised emittance constrain the application of LWFA beams, e.g. for X-ray lasing in an FEL setup, this is not a critical aspect for using LWFA beams as PWFA drivers. In fact, a significant initial energy spread and emittance of the electron beam driver have advantageous implications for PWFA as they offer improved resilience to well-known instabilities [655,656]. These features suggest that LWFA-generated electron bunches are extremely promising candidates to realise PWFA and explore its physics [653,663].

LWFA-generated electron beams are natural near-ideal drivers for PWFA.

### 26.1.3 LWFA→PWFA Transition

One challenge of hybrid LWFA→PWFA staging is the transition between LWFA and PWFA. LWFA-generated electron beams typically have relatively large divergence ($>1$ mrad), which is due to the strong focussing fields inside the plasma-wakefield cavity that compress the electron beam to a small transverse size on the µm-scale. For this reason, the LWFA beam will quickly diverge after its release in vacuum, thus decreasing quadratically its charge density as the spot size grows linearly with the propagation distance. As mentioned above, maintaining a high beam current density is crucial for driving a strong PWFA, and therefore, the distance to the PWFA stage shall be minimised. Although a gradual transition from plasma to vacuum at
the end of the LWFA stage permits to significantly reduce the divergence of the
beam [664], a capture and transport line between the LWFA and PWFA stages may
be desirable. Such a beamline interface will provide control over the focal position of
the beam in the PWFA stage, while at the same time alleviating the requirements
for out-coupling the spent drive laser and in-coupling assisting lasers for injection
in the PWFA stage. As an alternative to capturing the electron beam from LWFA
with permanent magnet quadrupoles, passive [665] and active [666] plasma lenses
are under development. These tasks are synergistic with LWFA approaches, which
attempt to directly use the electron beam output for applications.

A direct transition from LWFA to PWFA could also be achieved during the plasma
upramp in a PWFA stage placed close by, providing capturing and re-focussing of the
LWFA electron beam [663,665]. The diverging remnant laser pulse from the LWFA
stage can either be used to pre-ionise the PWFA stage immediately prior to arrival
of the electron beam or it can be blocked by a thin foil or tape [653,663,667,668].
The space-charge field of the electron beam itself can then be sufficient to self-
ionise the PWFA stage in an elegant solution, yet there are drawbacks, such as
an increased head erosion and a strong sensitivity to shot-to-shot jitter of the LWFA
output. Substantially more control can be provided by a separate pre-ionisation laser
pulse for the PWFA stage, as it is done in conventional linac-driven PWFA, such as
[46,654,669]. Several options exist for the implementation of the pre-ionisation laser,
for example using axicons, axilenses, or simply Gaussian focussing in a collinear geom-
etry [46,653,654,669]. More comprehensive discussions of complementary advantages
and combinations of LWFA and PWFA can be found, for example, in [602,670,671].

26.1.4 Experimental Status

A transition from LWFA to PWFA was first observed experimentally within a single
plasma stage, naturally occurring when the field strength of the accelerated elec-
tron beam became stronger than the depleting laser. However, only indirect signs
of PWFA were obtained in this kind of single-stage experiments, based on either
electron energy measurements [672], pulse duration measurements [673], or X-ray
emission diagnostics [674]. An experimental milestone towards a well-defined and
controllable LWFA→PWFA transition is the physical separation of the LWFA and
PWFA stages with two different gas reservoirs. This has been demonstrated, for
example, in [665], where a passive plasma lensing effect driven by the LWFA beam
in the second gas reservoir was observed; in [667], where self-ionisation and electron
beam deceleration in the second gas reservoir has been demonstrated; and in [653],
where the unambiguous observation of a plasma wave driven by laser-accelerated
electrons has been achieved using few-cycle shadowgraphy.

Proof-of-Concept Experiment at HZDR

The actual feasibility of the hybrid LWFA→PWFA concept is being tested
through a proof-of-concept experiment implemented at HZDR [602,663], using the
DRACO [675] laser system for the LWFA stage and the thereby produced electron
beam as driver of a subsequent PWFA stage. As a proof of concept, the first goal
of the experiment is to demonstrate the injection and acceleration of a new witness
beam in the PWFA stage driven by the LWFA beam. Ultimately, the resulting PWFA
beam is ought to feature a substantially higher energy and brightness than the initial
LWFA beam [602].

The experimental setup consists of two consecutive supersonic gas jets, one for the
generation of a high-current electron beam in an LWFA stage driven by the DRACO
<figure><img src="image_241.png" /></figure>

**Fig. 26.2.** Schematic of the double-jet plasma target for the LPWFA proof-of-concept experiment at HZDR. In the first gas jet, an LWFA stage is driven by the DRACO laser for the generation of a high-current electron beam. In the second gas jet, the LWFA-produced electron beam drives a PWFA for the production of a new electron beam with largely improved energy and brightness [602].

laser, and a second jet for the injection of a new electron beam in a PWFA stage, driven by the previously produced electron beam (see Fig. 26.2). A thin foil made of steel is placed at the entrance of the second jet, aiming to reflect the main laser from the second stage, while letting the electron beam go through with a tolerable impact on its transverse size and emittance. A counter-propagating low-intensity laser can be used in order to fully pre-ionise the hydrogen to facilitate the beam refocussing and enhance the blowout formation in the second stage.

Promising results have been obtained during the experimental campaigns of 2018, where conclusive evidence for the witness beam acceleration in a PWFA driven by LWFA beams has been achieved, in both self-ionised and pre-ionised PWFA regimes [T. Kurz, T. Heinemann et al., to be submitted]. This result represents a breakthrough milestone in the development and for future implementations of hybrid LWFA→PWFA. The goal is to use those as beam brightness and energy transformers, delivering high-quality beams at multi-GeV energies while maintaining a compact setup [53,602,603,651,663,670,671]. Such electron beams would meet the quality requirements for driving compact free-electron lasers powered by plasma beams, one of the major goals of EuPRAXIA.

## 26.2 LWFA→PWFA Staging as a Beam Energy and Brightness Transformer

Beyond the original prospects for mono-energetic energy doubling in a hybrid LWFA→PWFA setup [651], novel concepts for PWFA powered by LWFA beams aim for largely improved beam quality [53,601–603]. These concepts rely on the advantages that PWFA offers with respect to LWFA for the injection and acceleration of high-quality beams. PWFA drivers propagate through plasma at nearly the speed of
light, thus permitting to lock the phase position of the witness beam with respect to the wakefields. Besides, the strong focussing field generated by the driver allows for a diffraction-free propagation, which results in a remarkably steady wakefield generation. Both features combined make it possible to find persistent beam-loading conditions that suppress the build-up of correlated energy spread in the witness beam [329]. In addition, the relatively low electric fields (orders of magnitude lower than drive lasers, see Fig. 26.1) associated with the space charge of the drive beam can avoid the ionisation of high-ionisation-threshold species (e.g. helium), which can be then used as dopants for advanced injection techniques based on selective ionisation. The underdense plasma photo-cathode [603], also known as Trojan Horse injection, uses a low-intensity laser to rapidly induce ionisation over a small phase region of the wakefield, enabling the production of witness beams with ultra-low emittances, about two orders of magnitude below the state of the art. The wakefield-induced ionisation (WII) injection method [53,652] uses only the wakefields to induce ionisation and trapping of high-quality electron beams intrinsically synchronised with the driver, thus alleviating the pointing jitter requirements at high plasma densities, where normalised emittances as low as 0.1 µm can be obtained. The optical plasma torch injection method [601] relies on a laser-ionised plasma density bump to induce the injection, even for lower-current drive beam scenarios. These injection techniques uniquely apply to PWFA and promise to deliver superior beam quality with respect to LWFA. As Trojan Horse injection, WII injection, and plasma torch methods rely on ionisation, they can be tuned to find optimal conditions for beam loading [48,53] resulting in beams with high current and low energy spread, which together with an intrinsic low emittance yield ultra-high brightness beams.

Hybrid plasma accelerators can improve the brightness of the electron beam output by a factor of up to 10,000 compared to the state of the art, and hence can fulfil the main EuPRAXIA aim of “Superior Beam Quality”.

Specific results considering different injection techniques for high-quality witness beam production in a conceptual design for a hybrid LWFA→PWFA staging are shown in Section 26.2.1.

## 26.2.1 Injector for a Hybrid Plasma Accelerator

### Introduction

Beam-driven plasma wakefield accelerators have unique features for the production of high-quality and high-energy electron beams. On the one hand, PWFAs allow for the generation of stable and dephasing-free plasma wakefields, which can be sustained over longer distances, therefore providing higher energy gains and improved control over accelerated witness beams and their characteristics, such as their energy spread. On the other hand, operating PWFAs in the blowout regime enables novel injection techniques specifically designed to deliver ultra-low emittance and ultra-high brightness witness beams [54,601,603,652,654,662] with much higher quality than the drivers.

The hybrid plasma accelerator is a combination of a laser-driven plasma-wakefield accelerator with a PWFA in a staged setup, which uses the output beam of the LWFA to drive a PWFA, rather than an external beam from an RF linac. Initially proposed as an energy booster for LWFA beams [651], the concept was further developed in the following years towards a beam brightness transformer [53,602,603,670,671], aiming to reach the demanding beam quality requirements of accelerator-driven light sources
<figure><img src="image_242.png" /><figcaption>**Fig. 26.3.** The 6D brightness reach of the three related “plasma-cathode” schemes described here exceeds the state of the art by many orders of magnitude.</figcaption></figure>

without sacrificing the small spatial footprint and the relatively low cost offered by LWFAs.

One of the steps towards the realisation of such energy, emittance and brightness transformers is the implementation of advanced injection techniques for PWFAs delivering superior-quality witness beams. It should be noted that these concepts do also work for linac-driven PWFA and hence are solutions, which are accessible across LWFA and PWFA, and hence globally for the EuPRAXIA approach. In this chapter, we discuss three options for a hybrid plasma accelerator to deliver electron beams with superior quality. Various parameters determine the quality of electron beams, and the priorities may change from application to application, however, a key performance parameter is the brightness of the electron beam. Brightness is a composite parameter of key beam parameters involving the current, $I$, the normalised emittance, $\epsilon_n$, and the relative energy spread, $\bar{\sigma}_{\gamma} = \sigma_{\gamma}/\bar{\gamma}$, with $\bar{\gamma}$ the average energy and $\sigma_{\gamma}$ the energy spread (RMS). The 5D-brightness is defined as $B_{5D} = 2I/\epsilon_{n,x}\epsilon_{n,y}$ and the 6D-brightness as $B_{6D} = B_{5D}/\bar{\sigma}_{\gamma}$, which also includes the relative energy spread. A high current, low energy spread, and in particular a low normalised emittance are therefore key targets of these injector schemes. While short, kA-scale electron beams are the natural outcome of plasma accelerators, these “plasma-cathode” schemes stand out by being able to produce beams with an emittance (in both planes) orders of magnitude less than conventional methods. This is key to achieving peak brightness levels orders of magnitude better than the state of the art, and thus superior-quality electron beams, as shown in Figure 26.3.

The brightness enhancement of the PWFA output electron beam with respect to the input driver electron beam from LWFA or a linac is about five orders of magnitude.

## Plasma-Torch Injection

The use of a sharp density downramp for a localised injection had been first suggested for PWFA [15, 54, 676–678], but while today it is a standard method of injection in LWFA, until recently it never had been realised experimentally for PWFA. Supported
<figure><img src="image_243.png" /><figcaption>**Fig. 26.4.** 3D PIC-simulations reproducing the plasma-torch downramp injection experiment during the E-210: Trojan Horse programme at SLAC FACET ([654] and D. Ullmann, to be published).</figcaption></figure>

by EuPRAXIA, as part of the “E-210: Trojan Horse” experiment at SLAC FACET, an advanced all-optical method of downramp injection has been developed and demonstrated [654], which does not require a *gaseous* density downramp. Instead, another fundamental advantage of PWFA has been exploited, namely the fact that the electric fields associated with even ultra-intense electron beam drivers have a ceiling at the scale of tens of GV/m, the same scale of the plasma wakefields they produce. This means that the electron driver beams themselves do not tunnel-ionise media or ionisation levels with high ionisation thresholds such as helium. For example, in a hydrogen / helium mixture, hydrogen with a comparably low ionisation threshold can be used to support the PWFA process, leaving the helium fraction in a neutral state. An additional low-power laser pulse can then be used to locally ionise helium and thereby generate a plasma density spike, with a sharp plasma density downramp. This avoids the necessity and complexity of hydrodynamic gas inhomogeneities, e.g. by shock-front production with a knife or a localised gas jet, which requires hydrodynamic timescales and interferes also with the hydrogen density, thus impacting on the PWFA process itself. We call the all-optical method of downramp injection plasma-torch injection [601,679].

At SLAC FACET in E-210, we demonstrated the concept experimentally by sending an ionising few-mJ-class, ~100 fs scale Ti:Sapphire laser pulse perpendicularly across the beam-driven hydrogen plasma wave. At intensities of the order of only few $10^{15}$ W/cm$^2$, this non-relativistic laser pulse then ionises a helium plasma torch with tunable width and shape along its path on top of the hydrogen background plasma. The downramp at the end of this plasma spike then leads to the tunable injection of hydrogen and helium electrons. Figure 26.4 shows how this process works, based on 3D PIC-simulations with VSim.

Further details on how to achieve and tune the process is found in references [601,679] (theory), and [654] and [D. Ullmann, to be published] (experiments). The parameter space experimentally available by gas-density downramps is a subset of
what can be realised by plasma-torch downramps. Theoretical studies, which investigate PWFA downramp injection such as [15,54], indicate that very high quality electron bunches are possible. Beyond that, a wide, as of yet unexplored injector parameter space arises from the superior tunability of the plasma torch, e.g. as regards the peak density and steepness of gradients, and shapes. For example, since the helium (or another high ionisation threshold (HIT) gas) density is independently tunable from the hydrogen density, local peak plasma densities up to the critical density are possible. On the other hand, due to the direct dependency of the plasma profile on the generating laser pulse intensity profile, extremely sharp downramps and / or extremely thin or asymmetric plasma density profiles are possible [D. Ullmann, to be published]. The demands put on the driver beam current are comparably low, as the trapping threshold for density-downramp injection is around 1 kA, and the extreme tunability, e.g. as regards downramp steepness enabled by the all-optical plasma torch density transition, may allow to push this entry level further. Electron beams produced by LWFA, for example, naturally easily exceed currents of 1 kA scale and can therefore be used as plasma-torch-capable PWFA drivers. The plasma-torch injection method combines high feasibility and very high tunability with very high output beam quality.

Plasma-torch injection is a proven technique with low driver beam requirements, high tunability, and excellent output electron beam quality reach and high charge levels.

## Wakefield-Induced Ionisation Injection

The wakefield-induced ionisation (WII) injection method [53,652] exploits PWFAs operating at a high transformer ratio to induce ionisation and trapping of high-quality electron bunches by the accelerating wakefields of the plasma wake. By definition, in a high transformer ratio PWFA, the electric fields at the rear of the first plasma blowout are significantly higher than at the front. Thus, this configuration permits to trigger ionisation from the wakefields only while avoiding spurious contributions from the fields associated with the drive beam. In this way, the initial phase-space population of the injected electron bunch can be drastically constrained, resulting in low-emittance beams. The electrons forming the witness beam originate from field-induced ionisation of an atomic species with appropriate ionisation threshold, which is doped into the background plasma in a short axial region of the plasma target. Therefore, by tuning the concentration of the dopant, the total injected charge can be controlled to find near-optimal beam-loading conditions for a largely reduced time-correlated energy spread. As demonstrated in [329] and confirmed by simulations considering the WII injection method [53], the required peak currents are on the order of tens of kiloamperes, thus resulting in extremely high-brightness beams with a sub-percent energy spread.

The WII injection method is especially well suited for hybrid LWFA→PWFA staging, where drive beams above 10 kA peak current and around 10 fs duration can be produced in a dedicated LWFA stage [28]. The short duration of LWFA-produced beams make them suitable to operate as PWFA drivers at high plasma densities, close to $10^{19} \text{ cm}^{-3}$, where witness beams with normalised emittance as low as 0.1 µm can be produced by means of WII injection. Another fundamental advantage of hybrid LWFA→PWFA staging with WII injection relies on its relative simplicity: since the injection is triggered by the wakefields themselves, there is no need for precise spatiotemporal alignment of external components, such as lasers, and thus the WII injection method becomes insensitive to the pointing jitter of the electron beam emerging from the LWFA stage.
<figure><img src="image_244.png" /><figcaption>**Fig. 26.5.** Schematic of a hybrid LWFA→PWFA staged setup with wakefield-induced ionisation injection [602]. The setup consists of two quasi-identical plasma acceleration stages coupled to each other. In each stage, the injection of a witness beam is induced by field-ionisation from the dopant species contained in the gas jet. Immediately after the gas jet, a longer plasma cell with no dopant is used to further accelerate the generated witness beam.</figcaption></figure>

A conceptual design study for hybrid LWFA→PWFA staging with WII injection, based on particle-in-cell (PIC) simulations and well-known scaling rules have been recently published [602]. There, we describe the schematics of a possible design consisting of two quasi-identical plasma acceleration modules coupled to each other with a minimal distance in between (Fig. 26.5). The first plasma stage is meant to be driven by a high-power laser (LWFA), and optimised for the production of a highly relativistic (order GeV energy) and high-current ~10 kA electron beam via ionisation injection [279,661]. The LWFA electron beam is then used as driver of the subsequent plasma stage (PWFA), where a new high-quality witness beam is produced via WII injection and then accelerated to high energies. Both plasma stages need to be sufficiently close to each other, such that the LWFA beam can be refocused into the second plasma by means of its self-driven plasma wakefields. A thin steel foil can be positioned at the entrance of the second plasma stage to reflect the spent laser, while letting the electron beam pass through and drive the PWFA stage. In addition, a counter-propagating low-intensity laser can be used to preionise the PWFA stage to facilitate the beam focusing, enhance the wakefield excitation and further constrain the dopant section.

PIC simulations considering a PWFA scenario as the one sketched above (Fig. 26.5) show that, by operating the PWFA stage with a short and high-current electron beam at plasma densities ~ $8 \times 10^{18}$ cm⁻³, it is possible to generate ultra-short electron beams (sub-femtosecond duration), which double the initial energy of the LWFA beam in just few cm of acceleration, for the parameters considered. Moreover, the normalised emittance of the newly generated PWFA beam can be on the order of hundred nanometers, at the same time that its current can reach values of tens of kiloamps, as required for optimum beam loading.

The WII injection method is especially well-suited for hybrid PWFA at high plasma densities, providing ultra-short and high 6D-brightness beams from a straightforward setup.

Figures 26.6 a and b show electron density snapshots of the simulation for the PWFA stage during the process of injection and acceleration, respectively. The parameters
<figure><img src="image_245.png" /><figcaption>**Fig. 26.6.** 3D OSIRIS simulation [336] for a PWFA stage with wakefield-induced ionisation injection [53,652], driven by a high-current LWFA-produced beam. (a): electron density of the drive beam, the background plasma and the helium dopant during the injection process. (b): same quantities after injection and acceleration. (c) (top): evolution of the average longitudinal momentum of the witness beam (dark grey), the total relative energy spread (red) and the average sliced relative energy spread (dashed red). (c) (bottom): evolution of the projected normalised emittance of the beam in the *x* (dark grey) and *y* (light grey) planes. (b) (top): longitudinal phase space of the witness beam after 12 mm of propagation. (b) (bottom): sliced values of the current (blue), relative energy spread (red) and normalised emittance in the *x* (dark grey) and *y* (light grey) planes.</figcaption></figure>

of the witness beam after 12 mm of propagation are shown in Figure 26.6d, featuring a peak current of 15 kA, 11 pC charge, 0.8 fs duration (fwhm), ~5 GeV average energy and ~160 nm normalised emittance in both transverse planes. The total relative energy spread of the witness beam is ~3% in total, but only ~0.5% within the fwhm of the energy peak. The average sliced relative energy spread is around 0.2%. Figure 26.6c shows the evolution of the witness beam parameters as a function of the propagation distance, showcasing an essentially constant energy gain rate and steady beam-loading conditions, derived from the unique wakefield stability of PWFA in the blowout regime. The brightness of the witness beam produced in the PWFA stage is about five orders of magnitude larger than that of the drive beam generated in the LWFA stage.

## Trojan Horse: Plasma Photo-cathode Injection

Complementary to the thinking behind the hybrid LWFA→PWFA as an accelerator [651], the plasma photo-cathode also hybridises LWFA and PWFA elements and thus combines the best of both worlds to engender an injector [603]. It exploits electron beams – either from LWFA or from a linac – as superior plasma wakefield drivers, and the capability of focused laser pulses to produce localised plasma. The electron beam driver excites a plasma wakefield in low ionisation threshold (LIT) plasma such as hydrogen, and a spatiotemporally synchronised and aligned, focused laser
<figure><img src="image_246.png" /><figcaption>**Fig. 26.7.** The plasma photo-cathode mechanism in collinear geometry in the co-moving frame (left) and laboratory frame (right). The blue profile (solid line) is the longitudinal electric field, and the released electrons are shown in green as a small ellipse in the centre of the left-hand illustration.</figcaption></figure>

pulse – the plasma photo-cathode laser pulse – is ionising a high ionisation threshold (HIT) medium such as helium *inside* the LIT plasma wakefield. The plasma photo-cathode laser pulse can propagate in arbitrary geometry with respect to the driver beam [680], and can be applied over a wide range of plasma densities such as from $10^{15}$ – $10^{19}$ cm$^{-3}$ range and the position, shape and amount of plasma as well as the position of release within the plasma blowout can be tuned by tuning the laser pulse energy and other parameters. Sub-mJ laser pulse energies are sufficient to achieve the required peak intensities of $10^{15}$ W/cm$^2$ for Ti:Sapphire laser pulses to ionise helium, for example. The liberated helium electrons are then captured by the plasma wakefield and produce an electron beam of ultralow emittance. The main reason for this is that the residual transverse momentum of these helium plasma electrons is negligible. Figure 26.7 visualises the plasma photo-cathode in co-moving and lab frame for collinear geometry.

The normalised emittance reach of this method goes down to the tens of nmrad range [603] or even single digit nmrad range in both planes by optically confining the laser pulse release region e.g. via SSTF [681,682] or with colliding laser pulses [662], but also in simple Gaussian laser geometry, in particular when working at larger plasma wavelengths. The obtainable emittances are up to three orders of magnitude better than with state-of-the-art plasma accelerators, or radiofrequency-based linac, with typical output electron beam normalised emittances of the order of $\mu$mrad. Due to the quadratic scaling, this corresponds to a 5D brightness improvement of up to six orders of magnitude. The initial plasma photo-cathode idea and considerations regarding e.g. the negligible residual momentum due to the vanishing ponderomotive potential of the release laser pulse [603] were in the following confirmed and e.g. scalings were further developed by various groups [683–685] and have spawned various variations of the scheme [662,681,682,686,687].

Great progress has also been made experimentally. The “E-210: Trojan Horse” collaboration at SLAC FACET has successfully demonstrated feasibility of the plasma photo-cathode process and first core results are now published [654]. These experiments have been carried out under sub-optimal boundary conditions as regards maximum available blowout size, and geometry. That the experiment nevertheless succeeded, is therefore extremely encouraging. By increasing the plasma blowout size and by improving spatiotemporal alignment and synchronization of electron driver beam and laser pulse, the stability and quality of the output electron beam can be systematically improved. With regard to stability, a key advantage of this injection method is that the injection is largely decoupled from the plasma wakefield strength
<figure><img src="image_247.png" /><figcaption>**Fig. 26.8.** Plasma photo-cathode beam brightness transformer based on the Trojan Horse mechanism as a transformative source for applications.</figcaption></figure>

and that injection happens around the zero-crossing of longitudinal and transverse wakefields (see Fig. 26.7). Therefore, even in case of significant spatiotemporal jitter of the plasma photo-cathode laser, or driver electron beam jitter in terms of charge or current profile, the generated and accelerated output electron beam will be relatively unaffected by these jitters. For example, operation in a 500 µm-scale blowout with state-of-the art input beam jitter suggests an output beam stability at the 1%-level around emittance levels of around 10 nm rad, few pC charge and 1-fs-scale electron beam duration.

Plasma photo-cathode injection is a proven mechanism which promises superior electron beam quality, stability and tunability.

It is applicable as a beam brightness transformer both with linac-generated electron beam drivers as well as with LWFA-generated electron beam drivers, where elegantly the inherent synchronization between electron beam and plasma photo-cathode can be exploited. The extreme beam quality, and stability, is a unique path towards key EuPRAXIA applications. In particular, various light sources profit immensely from dramatically higher electron beam brightness, and high energy physics profits from the much lower emittance. These considerations are visualised in Figure 26.8.

Generating the witness output beam within a single plasma stage e.g. via the plasma photo-cathode mechanism has a further advantage versus external injection or the use of multiple stages, namely that challenges with injection into a plasma stage are completely avoided and that only extraction from the single plasma stage has to be regarded e.g. as regards emittance preservation. This thinking has been extended to minimization of energy spread and optimization of 6D brightness [48]. This is based on tailored beam loading. Such beam loading can be achieved either with a single plasma photo-cathode process as charge values e.g. in the few hundred pC range are accessible just by increasing the HIT density and/or the plasma photo-cathode laser pulse intensity. The obtainable emittance and brightness values of such high-charge bunch generation are still excellent, the obtainable emittance values are increased somewhat due to non-negligible space-charge effects. We therefore conceived and developed a decoupled dechirping process which nevertheless takes place in a single plasma stage [48]. The underlying principle is the use of “escort” bunches, which are added on top of the ultralow emittance witness beam to provide the necessary beam loading. These allow to remove the energy chirp completely, and scaling laws suggest
that at larger plasma blowouts, residual energy spread can be reduced [48,685] and that energy spreads down to the sub-0.01% already at an energy of 2-3 GeV can be obtained. This corresponds to ultrahigh 6D brightness [48], again exceeding state-of-the-art by orders of magnitude.

## 26.3 Plasma Injection Mechanisms

To achieve high beam quality stably from LWFA, control over both electron injection into the accelerating buckets and acceleration processes are essential. Nowadays, electron injection schemes via field ionisation and plasma density gradients are widely applied in experiments due to their simplicity and robustness. However, the resulting electron beam quality is generally not high enough as compared to those beams produced by conventional accelerators. In the following, we discuss the possibility to achieve better control of electron injection by combining different schemes, which may enable one to control the electron bunch duration, charge, and energy spread in order to satisfy the requirements for a LWFA-based user facility.

### 26.3.1 Optimisation of Ionisation Injection

Even though ionisation injection is stable and easy to implement in experiment, it often leads to broad beam energy spread mainly for two causes. First, electron injection can occur continuously as long as the injection thresholds are satisfied. Second, the trapped electrons are often released via ionisation at different wakefield phases, thereby undergoing very different accelerating fields. In the following, we discuss briefly three schemes to reduce the energy spread of accelerated electrons, i.e., self-truncated ionisation injection, two-colour laser ionisation injection, and magnetic-field-controlled ionisation injection.

#### Self-Truncated Ionisation Injection

In order to further reduce the ionisation injection length beyond the mechanical limit, a simple single-stage scheme called self-truncated ionisation injection (STII) was proposed, which predicts a small energy spread of the trapped electron beams well below 10% [661]. This scheme utilises the self-focussing of a laser beam in a plasma to truncate the ionisation injection process, so that the injection length can be reduced to just a few hundred micrometres. Usually, people choose the laser spot size satisfying the so called matching condition $k_p w_0 = 2a_0^{1/2}$ [185], where $k_p$ is the wavenumber of the plasma, $w_0$ is the laser waist radius, and $a_0$ is the amplitude of the normalised laser vector potential. If $w_0$ is chosen to be larger than the above matching condition requirement, due to the development of laser self-focussing, the injection is suppressed at the distance of a few hundred micrometres. The reason is explained as follows. The wake strength can be expressed as the pseudo-potential $\psi$. To trap an electron, the pseudo-potential drop from the releasing point of the inner-shell electrons to the bottom of psi should satisfy $\Delta\psi>1$. For the unmatched case, this condition is satisfied at the beginning. But it is violated when the self-focussing occurs at a few hundred micrometers. Numerical simulations suggest that electron beams with energy spread within 5% can be produced with this scheme. The STII scheme has recently been experimentally verified by several groups with 100 TW scale laser systems, where electrons are trapped and accelerated stably to above 1 GeV with relatively smaller energy spread [28,660].
Ionisation Injection with Two-Colour Lasers

To further reduce the energy spread to the sub-percent level, one needs to further reduce the ionisation injection length. It is found that the use of a dual-colour laser can limit the ionisation injection in a few separated regions [688]. The injection length of each of the regions is within 100 µm, thus the energy spread can be reduced to the sub-percent level. Let us consider a dual-colour laser with its electric field in the form of $E = E_{10} [\sin(\omega t) + \sin(\frac{3\omega t}{3})]$ (the first two terms of the Taylor series of a square wave). Because of the dispersion of the laser in plasma, the different frequency components of the laser propagate at different phase velocities. The wave transforms to $E = E_{10} [\cos(\omega t) + \cos(\frac{3\omega t}{3})]$ after propagating a distance of $\Delta z = \frac{3}{8}\lambda_p \frac{\omega}{\omega_p}$, where $\lambda_p$ is the plasma wavelength and $\omega_p$ is the plasma frequency. With the change of the wave form, the amplitude increases by a factor of $\sqrt{2}$. According to the Ammosov-Delone-Krainov (ADK) model, the ionisation rate is exponentially dependent on the electric field. Thus, the $\sqrt{2}$-factor can largely influence the ionisation probability. Numerical simulations suggest that this scheme may lead to high-quality electron beams with an energy spread less than 1% [689]. An experimental demonstration of this scheme still remains to be carried out.

Magnetic-Field-Assisted Ionisation Injection

The above studies make use of self-consistent laser evolutions to truncate the ion-
isation injection distance, generating high-quality electron beams of small energy
spread. It is found that an external transverse magnetic field (ETMF) of a few tens
of Tesla can be applied to further control the ionisation injection in LWFA [23]. The
ETMF changes the trapping threshold, and the efficient electron injection occurs only
if the self-generated magnetic field in the wakefield exceeds a certain critical thresh-
old. For example, the ionisation injection is totally suppressed by a large ETMF at
early stages when $B_y^{ext}=50$ T. At a later time, when the self-generated magnetic field
in the wakefield exceeds the ETMF, the injection occurs at a higher rate, leading to
a larger beam charge. The non-linear injection process also shapes the longitudinal
charge profile of injected electron beams, as shown in Figure 26.10. When using a
stronger ETMF of $B_y^{ext}=50$ T, for example, a trapezoidal-like charge profile can be
generated. Such a beam profile is known to cause a nearly uniform beam-loading
wakefield $E_z$ [329], which can reduce the energy spread due to longitudinal phase
mixing and accelerates the electron beam to a higher peak energy and a lower energy
spread. When using $B_y^{ext}=50$ T, it is shown that the relative energy spread can grad-
ually decrease from 6.2% ($z = 4$ mm) to 4.3% ($z = 5$ mm), while the peak energy
increases from ~224 MeV to ~290 MeV. This is attributed to the uniform beam-
loading wakefield. So far this scheme is only demonstrated by 3D PIC simulation.

26.3.2 Generation of Attosecond Electron Bunches by Density-Gradient-Controlled Injection

Generally, shorter electron bunches help to achieve higher beam brightness, and hold
the promise of generating shorter radiation pulses [690], which are essential for ultra-
fast science and applications. While it is common to produce fs-duration beams from
LWFA [29], it remains a challenge to go to the sub-fs or attosecond regime. In order
to have such extremely short bunches, it is expected to apply fine control over the
wakefield acceleration, especially during the injection phase. The key requirement
<figure><img src="image_248.png" /><figcaption>**Fig. 26.9.** Bubble structures and injected electron bunches in a density-profile-tailored plasma with $n_0 = 0.0002n_{cr}$, the density bump size 100 µm and four different values of the external magnetic fields and height of the density bump $\alpha$ over the background plasma [693] (copyright IOP Publishing, reprinted with permission).</figcaption></figure>

<figure><img src="image_249.png" /><figcaption>**Fig. 26.10.** (a)–(c) Phase space of injected electrons (colour contour) and the accelerating field $E_z$ (black curve) for the external field $B_y^{ext} = 0, 20, 50$ T. Insets compare the charge profiles of electron beams from the simulations (red solid curve) with the optimised trapezoidal-shaped profiles (red dash curve). The figure is taken from [23].</figcaption></figure>
is to make the electron injection occur in a ultra-short period and in a very small area, meanwhile with a high rate so that there is a considerable number of electrons trapped. Following are several possible schemes based upon numerical simulations, which are yet to be demonstrated experimentally.

## Attosecond Electron Bunches Injected at a Density Transition from an Upramp to a Plateau

A density tailoring scheme was proposed that makes use of a highly non-linear plasma wave [31,662], which forms extremely short (well within attoseconds duration) high electron density peaks. Usually only a small fraction of electrons in the density peaks can be injected when plasma wave breaking occurs. To achieve injection of a whole density peak almost simultaneously, the plasma density is tailored to include an upramp followed by a plateau. With such a density profile, the phase velocity of the plasma wave is changed abruptly along the wave-propagation direction from superluminal in the upramp region to subluminal at the beginning of the density plateau. This can lead to the sudden injection of the first density peak behind the driving laser pulse, generating dense attosecond electron sheets. The above mechanism is essentially of a 1D nature, and in high dimensions it works well for a relatively broad laser spot size (larger than a plasma wavelength). Such dense electron sheets (~ the critical plasma density, ~100 attoseconds in duration) may be used as relativistic flying mirrors for generating attosecond X-ray pulses [691] or may self-consistently generate giant half-cycle attosecond light pulses [692].

## Attosecond Electron Bunch Generation Using a Modulated Plasma Density Profile

In addition to injection control with a density upramp, density downramps have been widely used to control electron injection. If the density downramp is steep enough, it is found that ultra-short bunches with attosecond duration can be produced [30]. Generally, the velocity of the back of the bubble can be manipulated through the plasma density, because the plasma bubble length scales as $n_e^{-1/2}$, where $n_e$ is the electron density. As the density is increased, the bubble contracts, and the rear of the bubble speeds up, suppressing injection. In contrast, if the density decreases, then the back of the bubble slows down, and some electrons can overtake the rear of the bubble and be injected. Thus, by varying the plasma density, self-injection can be controlled. It is found that attosecond electron bunches can be generated by introducing a density bump in a homogeneous plasma. Simulations suggest that injected charge scales approximately linearly with bunch length. Under some optimised cases, it is possible to reduce the electron bunch duration to even less than 100 attoseconds.

## Attosecond Electron Bunches via Injection Suppression with a Magnetic Field

The wakefield structure in the LWFA can be modified by a static external magnetic field. This provides an alternative approach to control the electron injection. Because of the suppression of the electron transverse motion, electron injection can be enhanced by use of a static longitudinal magnetic field of a few tens of tesla [693]. In particular, it is found that attosecond electron bunches can be produced by combining a density-profile-tailored plasma with a longitudinal magnetic field. The static longitudinal magnetic field modifies the transverse structure of the bubble, while the density gradient changes its longitudinal structure. The magnetic field induces a
radial density hole in the bubble rear, which expands as the bubble evolves along the density downramp. Electron injection is triggered by the decreasing phase velocity of the bubble along the density downramp, and then suppressed by the expanding hole at the bubble rear. In this way, the position of electron injection can be controlled, leading to injection of isolated sub-femtosecond electron bunches. Compared to the scheme of density modulation mentioned above [30], the longitudinal magnetic field enables the realisation of sub-femtosecond electron bunches with high charge and with readily accessible parameters both for density profiles and magnetic field strength. Figure 26.9 shows examples of sub-femtosecond electron bunch generation by use of a reasonable density modulation and a longitudinal magnetic field.

## 26.4 Compact Radiation Generation

Based upon energetic electrons from LWFA, X-rays and gamma-rays can be produced via different schemes, such as betatron radiation, non-linear Thomson scattering, undulator radiation, and bremsstrahlung radiation [690]. Among them, the betatron radiation from LWFA in plasma has been demonstrated as a unique source of X-ray emission typically up to 100 keV. It makes use of the wakefield structure, so that the electron acceleration and radiation are combined in a compact way. However, the brilliance and spectrum control needs to be further improved for practical applications. In addition, even though XFELs based upon LWFA-produced electrons have been proposed, the required electron beam quality is yet to be demonstrated in experiment. In this case, coherent X-ray emission based upon LWFA may be achieved in alternative ways rather than based upon the conventional XFEL concept. In the following, some unique potential opportunities with LWFA for X-ray radiation will be discussed.

### 26.4.1 Betatron Radiation Control in a Plasma Channel

Plasma channels have been used normally for guiding laser propagation for electron acceleration via LWFA, where the laser pulses are typically incident along the channel axis. When the laser pulse is off-axially injected or injected with a certain angle against the channel axis, its centroid oscillates transversely in the plasma channel. This results in a wiggler motion of the whole accelerating structure and the self-trapped electrons produced behind the laser pulse, leading to strong synchrotron-like radiation [694]. The produced photon energy or radiation spectrum depend upon the electron energy and betatron oscillation amplitude, where the latter can be controlled by the transverse offset of the laser pulse or the incident angle about the channel axis. Moreover, usually the X-rays from betatron radiation show random polarisation. When the plasma channel is used, the X-ray polarisation can be fixed and tuned from linear to circular polarisation. This is realised by controlling the electron trajectories. If the laser pulse is initially injected off-axially into the plasma channel or at some injection angle in a plane across the plasma channel axis, the laser centroid oscillation and electron betatron oscillation all will remain in this plane. The resulting X-rays will be linearly polarised in this plane. When the laser pulse axis is non-collinear within the plane of the channel axis, the laser centroid will perform a spiral trajectory around the channel axis as shown in Figure 26.11. The resulting X-rays will be elliptically polarised [480]. Therefore, by use of plasma channels, one can realise the control of betatron radiation spectra and polarisation, which is attractive for applications and make up the shortage of large synchrotron radiation facilities.
<figure><img src="image_250.png" /><figcaption>**Fig. 26.11.** Sketch of the radiation source from a helical plasma undulator based on LWFA in a plasma channel and the resulting far-field X-ray radiation distribution [480].</figcaption></figure>

## 26.4.2 Schemes for Enhanced Betatron Radiation

Betatron X-ray emission from LWFA usually utilises low-Z gases in the self-injection regime. In this regime, even though the electron energy can be controlled with the laser energy and acceleration distance, the betatron oscillation amplitude cannot be easily controlled. To enhance the betatron radiation yields and photon energy, a few schemes have been proposed to enhance the betatron oscillation amplitudes. For example, one may use two laser pulses, one to drive LWFA and another propagating behind to drive direct laser acceleration (DLA) [695]. The DLA or laser-assisted betatron resonance can considerably enhance betatron oscillation amplitudes and thus the X-ray yields and photon energy [71]. It is found that ionisation injection and the use of cluster targets can lead to enhanced betatron oscillation amplitudes, as demonstrated experimentally [646,696]. In these two cases, DLA is also involved at the later stage of electron acceleration, which increases the betatron oscillation amplitudes in addition to enhancing the electron energy.

## 26.4.3 Coherent Bright XUV and X-Ray Radiation from LWFA

Laser-driven coherent XUV and X-ray radiation is produced so far via high-harmonics generation from laser-atom interactions at moderate laser intensity [697] or from relativistic laser-solid interactions [698]. However, their conversion efficiencies are usually low and fall off quickly with the harmonic order for both cases. Currently, coherent X-ray sources mainly rely on XFELs based upon conventional accelerator technologies. Nevertheless, there are some schemes proposed for coherent XUV and X-ray radiation generation based upon non-linear laser-wakefield excitation and LWFA-based ultra-short electron bunches. It was suggested that the high electron density spikes associated with the laser wakefields can be used as relativistic flying mirrors (RFM) to achieve laser pulse compression, frequency up-shift, and focussing in a counter-propagating geometry [699,700]. To realise coherent X-ray radiation, the flying mirror
<figure><img src="image_251.png" /><figcaption>**Fig. 26.12.** Results of 2D PIC simulations of the X-ray generation from LWFA using flying mirrors [691] (reprinted with permission of AIP Publishing). (a) Spatial profile of the electric field of the scattered pulse and (b) the corresponding spectrum in space kx-y. (c) Schematic of the backscattering off the flying mirror.</figcaption></figure>

(or the electron density spikes) must be dense enough and electrons in the mirror must have a uniform velocity or very small energy spread. This is technically very challenging.

On the other hand, it has been demonstrated numerically by PIC simulation that a relativistic dense electron sheet can be produced from LWFA with a tailored density profile composed of a density upramp and a density plateau [662]. With this density electron sheet, coherent Thomson scattering can be realised as demonstrated by integrated simulations including both the electron sheet formation and Thomson scattering. Simulations indicate that coherent X-ray emission at a keV photon energy can be realised with this scheme [691]. The required experimental conditions for the generation of dense electron beams from LWFA are within current technical capabilities. Typically multi-100 TW lasers will provide sufficiently high power for the required laser intensity and relatively broad focal spots larger than a plasma wavelength are preferable here. In addition, with the formation of dense electron sheets with a relatively large radius, they will be focused quickly by the transverse field in the laser wakefield. During this process, radially polarised coherent sychrotron radiation with ~100 attosecond duration in the XUV regime (~30 eV) will be produced with an energy conversion efficiency as high as 10⁻³ from the driving laser [692]. The emission has a conical angle typically less than 10°. As gas targets are used for this source, it has the potential to be produced with a high repetition rate.

## 26.5 Dielectric Technology for Diagnostics & Beam Control

While the study of particle acceleration in dielectrics is still limited to low energy gains, alternative use cases of dielectrics are being studied worldwide. The following lists three examples:

- Passive energy dechirper: The beam is sent through a dielectric structure. The induced wakefields act back on the electron beam and, if adjusted correctly, can compensate the correlated energy spread of the beam. This may be of great interest to the beams produced in laser-plasma accelerators due to their large correlated energy spread. On the other hand, in order to work best, these concepts require the charge and beam distribution not to vary from shot to shot, which is a currently not the case in LWFA. Therefore, its application is currently mainly studied in conventional accelerators, e.g. in [701].