<figure><img src="image_137.png" /><figcaption>**Fig. 18.2.** Deflection parameter versus gap for different period undulators using RADIA (image credits: A. Ghaith).</figcaption></figure>

Progress on Superconducting Undulators

Superconducting technology is routinely applied to the development of high-field-strength magnets for applications such as magnetic resonance imaging (MRI) and particle physics accelerators and detectors. Short-period superconducting undulators (SCUs) are still not a mainstream solution for accelerator-based light sources, with just a few examples being in use today. The reason for this rather slow uptake is, in large part, due to the extremely successful application and ongoing improvements in permanent magnet undulator technology rather than any specific shortcomings in superconducting technologies. Despite the undoubtedly success of permanent magnet systems, there is still a clear margin in performance advantage to be gained through the application of superconducting materials, and it is for this reason that a number of groups around the world have been actively pursuing the detailed development of short-period, high-field SCUs for light source applications over the past ten years or more [458]. This research and development effort has led to the construction of a few SCUs which are now installed and in daily use on storage ring light sources in Germany [459] and the United States [460]. These particular examples have exhibited very good operational performance in terms of reliability, stability, and user experience, and this has increased confidence within the accelerator community that national FEL light source facilities, such as LCLS-II, should carefully assess employing SCUs rather than permanent magnet alternatives in their baseline configurations [461]. The specific advantages of SCUs over PMUs highlighted by the LCLS-II team are the higher magnetic fields at a short period, allowing superior FEL performance or reduced undulator length, the radiation hardness compared to PMUs offering long lifetime and smaller magnet gaps, the anticipated reduction in resistive wakefields due to the cold bore, the much lower vacuum pressure from cryo-pumping, reducing gas scattering, the smaller footprint and simpler magnet field control compared with the massive adjustable-gap PMU, and the easy reorientation for vertical polarization, if desired. The project team estimated that the LCLS-II hard X-ray undulator could be shortened by up to 70 m using an SCU in place of a PMU. Still, LCLS-II undulators are permanent magnet based.
<figure><img src="image_138.png" /><figcaption>**Fig. 18.3.** Typical magnet design for an SCU viewed from the side, with the electron beam direction indicated by the red horizontal arrow. The superconducting windings close to the electron beam are perpendicular to the beam direction and generate the periodic fields indicated by the blue vertical arrows. The windings on the outside of the steel former (shown as grey rectangles) are for the convenience of simplifying the winding configuration.</figcaption></figure>

The magnetic design of planar SCUs is very straightforward, with a similar approach being adopted by all of the research groups working in this area. The typical arrangement is shown in Figure 18.3. Two independent sets of superconducting windings on steel formers are arranged such that the current flows transversely orthogonal to the electron beam and so generates the periodic magnetic field required [462]. The two sets of windings are held apart by a non-magnetic mechanical arrangement, which is not shown in the figure.

The former is made of a good magnetic steel, and the superconducting wire employed is either NbTi or Nb3Sn. The SCU typically operates at between 2 and 4 K. NbTi is more commonly used as it is far easier to work with in terms of winding, insulation, and stability. Nb3Sn has to be heat-treated after winding to create the superconducting alloy and afterwards is rather fragile. Nb3Sn also suffers from instability issues at the field levels required by SCUs (below ~5 T) as it is primarily targeted at far stronger magnetic field applications [463]. Research on the use of special grades of Nb3Sn better suited to SCU applications is ongoing at LBNL [464].

Although the magnetic design of SCUs is straightforward, the engineering challenges are severe, and this is the area which has held back SCUs from widespread adoption so far. The mechanical tolerances are very tight, and these must be maintained as the magnet is cooled down from room temperature to ~4K. The gap separation between the two sets of windings is typically between 5 to 10 mm, with no room for a standard vacuum chamber, and so the SCU is, in effect, another type of in-vacuum undulator. The coils are at ~4 K, and any heat transfer from the electron beam to the coils because of wakefields or synchrotron radiation must be minimised to prevent magnet quenching. This is generally resolved through the insertion of a very thin beam screen between the beam and the windings at an intermediate temperature of 10 to 20 K to absorb any power from the electron beam. A further issue is that the magnetic field quality is not just determined by the steel pole shape and location but also strongly dependent upon the superconducting wire placement. The accurate placement of individual wires, to a tolerance of a few tens of micrometres, is a painstaking process and difficult to maintain over a length of more than a few tens of centimetres. For this reason, and also to achieve the required machining tolerances, the complete SCU magnet is often made up of shorter sections which couple together to form a longer device [465]. Post-assembly magnet shimming, which is a standard technique for permanent magnet undulators, is not easy to implement in an SCU. Many schemes have been proposed, but they add an extra layer of complexity which
teams try to avoid if at all possible [466]. In addition, the proposed schemes often require additional windings or use valuable space within the magnet gap and so also serve to lower the maximum possible peak field that the SCU can achieve. For these reasons, several groups aim to construct SCUs that have excellent field quality on first assembly and have no shimming capability. This is only possible by working to very tight tolerances at all stages of manufacture and assembly but has been shown to be a practical approach.

An example of an SCU installed in a storage ring is the SCU15 in the ANKA synchrotron radiation facility [456]. The SCU15 was installed in the storage ring during the December 2014 shutdown and has been in operation with beam since the beginning of 2015. The SCU15 has a period length of 15 mm and 100 full periods. Since at ANKA, during electron beam injection and energy ramping, a vertical beam stay-clear of 15 mm is needed and at the full energy of 2.5 GeV, a minimum gap of 7 mm is allowed, the beam vacuum chamber is movable from a 7 mm to a 15 mm vacuum gap. The magnetic peak field measured at the maximum coil current of 150 A is 0.73 T. This value is claimed in the paper to be 16% higher than the peak field of 0.62 T achieved with a CPMU using the best available material for the permanent magnets (PrFeB) with the same period length and beam stay-clear of 7 mm. From Figure 18.2, one reaches 0.66 T with a CPMU U18 at 7 mm, reaching 10% increase with superconducting technology for a 7 mm gap. However, it is meaningless to use a CPMU only at a 7 mm gap, as it can reach a much larger field (larger than 1.5 T) at a smaller gaps, for which it is intended to operate.

Another example of a recent SCU installed and successfully operating in a storage ring is SCU1 in the APS. It built upon the experience gained from the 30 cm long SCU0 device, which was their first SCU to be installed [460]. SCU1 has an 18 mm period and a fixed magnet gap of 9.5 mm, achieving a peak field of 0.97 T. For the same gap, the CPMU U18 would give at 9.5 mm a field of 0.53 T, but the design is not optimised. The magnetic length is 1.1 m. The magnet is wound using round NbTi wire with a diameter of only 0.6 mm. Nevertheless, the SCU comfortably operates at a current of 450 A, well within the maximum current achieved of 520 A.

## Advanced and Novel Undulators

Even though unlikely to be used for an LPA-based FEL as a baseline case, more exotic undulators can be considered, such as RF undulators, micro-machined undulators, and plasma undulators.

### Sub-Millimetre Period Undulators

A dramatic reduction of the undulator period is actively investigated using various techniques and concepts, such as the microstructure-driven laser undulator [467], the surface micro-machined undulator [468], the micro-machined magnet undulator [469-471], the RF undulator [472], and the optical undulator [473]. However, because of the magnetic field value and short period, the low value of the deflection parameter limits the emission intensity and the harmonics operation. There could be some issues with wakefields with these extremely low gaps and with magnetic measurements. Because of the low-deflection parameter values and the associated technical challenges, they will not be considered for EuPRAXIA.

### Plasma Undulators

Plasmas represent electromagnetically active media which can produce and sustain very high static fields and give rise to the strong collective phenomena such as channel formation and plasma waves with strong charge separation fields, and for that, they attract high interest for beam applications. The idea to apply plasma fields to operate
and, in particular, to undulate electrons dates back to the 1980s–1990s. In this period, several concepts were proposed involving the oscillations of electrons guided in ion channels [474] or to impose a wiggling motion by coupling an electron beam to plasma waves [475,476]. In these approaches, the plasmas were considered to be created by a high-intensity laser focussed in gas targets. The wavelength corresponding to such gaseous plasmas with densities of $1 \times 10^{15}-1 \times 10^{17} \text{ cm}^{-3}$ corresponds to wiggler periods of $\lambda_w \sim 1 \times 10^{-1}-1 \times 10^2 \text{ µm}$, and the calculated wiggler strength for to-date laser parameters (intensities of $\sim 1 \times 10^{14} \text{ W/cm}^2$) could already reach $a_w \sim 0.1-1$. Such parameters may potentially provide the conditions to generate X-rays using relatively low-energy electron beams with limited collimation quality.

One of the principal challenges identified in these early studies was the low number of oscillation periods which could be produced for realistic parameters, hindering developments of these concepts to the point of FEL applications. Moreover, in the “plasma wave” approach, the laser energy required to create the corresponding plasma wave front has turned rather big, thus requiring high-energy laser systems. Finally, the stability of laser plasmas, which affects directly the reproducibility of interaction conditions, presents another challenge for the experimental tests.

In recent years, the concept of underdense plasma undulators has been revised in more detail. The “plasma wave” scheme has been studied by coupling to the laser-plasma-accelerated electrons with a more advanced numerical approach. In one such study [477], the possibility of keV photon generation along 25 wiggler periods with realistic experimental parameters has been demonstrated. Another approach, derived from the “channel” scheme, consisted of applying to electrons the wakefields of a laser pulse injected off axis into a plasma channel in such a way that the laser centroid oscillates transversely [478]. Potentially, this scheme can be coupled to an LPA and can provide a few tens of $\lambda_w \sim 1-2$ mm wiggler periods with a strength of $a_w \sim 1$. Further numerical studies revealed the potential tunability of such a source in terms of wavelength and polarisation of the produced radiation [479,480].

Recently, an alternative concept of the plasma undulator based on an overdense plasma was proposed [481]. The solid target is composed of nanowires with a sub-micrometre diameter and a few tens of micrometers length, which are arranged in a chessboard fashion on the axis of the LPA, close to its exit. The laser driver of the LPA ionises the wires, producing strong, spatially alternating electrostatic fields on the following electron beam, providing an undulator strength of $a_w \sim 1$. The period of such an undulator does not rely on the plasma density but is fixed only by the target design and can be chosen in the range of $\lambda_w \sim 10-100 \text{ µm}$.

The main interest in plasma undulators is related to their potential to produce strong undulating fields with sub-millimetre periods. In the present state, these schemes remain mainly theoretical concepts, and in the short term, their experimental validation for synchrotron light production is required. For these reasons, plasma undulators are not yet ready to be considered as the candidates for the EuPRAXIA project.

## Transverse Gradient Undulators

Transverse gradient undulators (TGU) are considered to be a promising solution for free-electron lasers (FEL) which aim at utilising electron beams with large energy spread such as beams generated by laser-plasma accelerators (see Fig. 18.4). The TGU enables in principle to restore FEL gain impacted by the energy spread of an electron beam. Some technical issues still remain (e.g. independent tuning of K and gradient) but are in the process of being fixed. Some other points have to be studied in detail, such as the effect on the beam dynamics [482] of additional multipolar terms (not a pure gradient) coming from the finite pole width and the gap variation in the
<figure><img src="image_139.png" /><figcaption>Fig. 18.4. Transverse gradient undulator scheme (image credits: A. Bernhard, KIT).</figcaption></figure>

horizontal plane. As a consequence, TGUs can be proposed for linac-based FELs but hardly for recirculating FELs while the latter point is not clarified. The TGU is considered for LPA-driven FELs and will be further investigated in the context of EuPRAXIA.

The first concept was presented in the latest seventies [483], where it was proposed, on the one hand, to disperse transversally the electrons following their own energy and, on the other hand, to vary transversally the magnetic amplitude of the field of the undulator by generating a gradient. An adequate value of the dispersion and the field gradient minimises the effect of the electron beam energy spread on the FEL resonance condition, leading to a strong increase in the gain. The main interest of the TGU is to keep electrons always energy resonant even at a very large energy spread ($>s10\%$) leading to a shorter saturation length, a higher saturation power, and a narrower bandwidth [64]. However, dispersing the particles increases the beam size in the direction of dispersion (usually the horizontal direction) and reduces the transverse coherence of the FEL radiation. Therefore, a dispersion value of only a few centimetres must not be exceeded and undulators producing transverse gradients of many tens of $m^{-1}$ are necessary to minimise this effect.

Many magnetic designs of TGUs have been proposed, but only two have been built or are under development presently. The first solution, proposed at SINAP, is based on a classical PPM / hybrid undulator whose magnetic system has been transversally tilted [484]. Four fixed-gap TGUs composed of 75 periods of 20 mm have been completely constructed, leading to a K value of 1.15 and a transverse gradient of $50 m^{-1}$. The main drawback is that this solution leads to a compromise between the tilt angle ($<10°$) and the good gradient region. A second TGU based on off-axis operation of a cylindric superconducting undulator is in the commissioning phase at KIT [67]. The undulator is composed of 40 periods of 10.5 mm, with a K value of 1.07 and a transverse gradient of $149.5 m^{-1}$ in a limited region of 5 mm. Both technical solutions do not presently enable the independent tuning of the K value and the transverse gradient, which presently represents a mechanical challenge.

The TGU on-beam impact has also been studied [482]. The presence of the transverse gradient generates a ponderomotive drift [485], which deviates the electrons quadratically from the magnetic axis. The global result is the increase in the transverse beam size if not corrected and, in turn, a drop of the FEL gain. Nonetheless, this effect can be corrected by means of long correction coils inserted in the magnetic structure.
## Conclusion

From the above presented systems, one can pick out from the technologically feasible undulator parameters the most appropriate characteristics for FEL operation in different spectral ranges. Cryogenic undulators seem to be the solution of choice for an FEL project to be realised over the next 10 years. They represent a technological advance over currently used on-air warm permanent magnet undulators used in FEL facilities worldwide. Yet, the underlying technology is, although still cutting-edge, well understood and under control. The combination of short periods with comparatively high on-axis magnetic fields could enable plasma-driven FELs at water-window photon energies and relatively low-energy electron beams of 1 GeV, provided proper electron beam parameters can be reached. In the case where it is more preferable to achieve a high field, superconducting magnets are a good choice to satisfy such a goal; however, these undulators are limited by the minimum magnetic gap reached, which is almost 7 mm so far with a total length shorter than 1.5 m. At 7 mm, cryogenic undulators can obtain a peak field of almost 0.5–0.7 T, but the advantage of such a device is that one is able to reach a magnetic gap of almost 3 mm depending on the magnetic period, and with such a gap, the peak field is enlarged to almost 1.5–1.8 T, with lengths up to 2 m so far. A 12(10) mm CPMU provides a deflection parameter ranging between 1.67 and 0.91 (1.15 and 0.59) from a 3–5 mm gap. A CPMU is more sustainable than an SCU in regards to the operating cost. Since superconducting undulators offer great possibilities and open up the parameter ranges, one could consider also an alternative U12 SCU for EuPRAXIA. Other, more exotic undulator concepts, should, given their early stage of development, currently not be considered for driving a EuPRAXIA FEL facility.

# 19 Beam Transport and Handling Systems

## 19.1 Beam Transport for the Low-Energy Beam

### 19.1.1 Introduction

The low-energy beam transport line (LETL) is relevant for transferring the electron beam from the injector stage (either an RF or plasma injector) to a second plasma-accelerator target (laser-driven or beam-driven) for energy boosting in a staged accelerator setup. With transport from a plasma injector the more challenging scenario, this will be focussed on in the remainder of this section using the example of a plasma injector with the ReMPI technique (Scheme 2, see Chap. 23 for details). The transport lines between the injector and accelerator stages in a setup with an RF injector, on the other hand, are described in Chapter 14.

At low energy (150–250 MeV), space-charge forces are not negligible. A measure of the relative importance of space-charge effects versus emittance pressure is given by the laminarity parameter, defined as the ratio between the space-charge term and the emittance term [486] as follows:

$$ \rho = \frac{\hat{I} \sigma^2}{2I_A \gamma \varepsilon_n^2}, \qquad (19.1) $$

where $I_A = 4\pi\varepsilon_0 \frac{m_e c^3}{e} \approx 17$ kA is the Alfven current, $\hat{I}$ is the peak beam current, $\sigma$ is the RMS beam size, $\varepsilon_n$ is the normalised emittance, and $\gamma$ is the Lorentz factor. When $\rho$ greatly exceeds unity, the beam behaves as a laminar flow – all beam particles move on trajectories that do not cross – and transport and acceleration require a
<figure><img src="image_140.png" /><figcaption>**Fig. 19.1.** Layout of the low-energy beam transfer line (in red: permanent quadrupoles in the capture and matching sections, in green: dipoles, in cyan: BPMs, in gray: ICTs, and in yellow: screens).</figcaption></figure>

careful tuning of focussing and accelerating elements to keep laminarity. A correlated emittance growth is typical in this regime, which can be made reversible if proper beam matching conditions are fulfilled.

The values for the low-energy beam of EuPRAXIA are $\hat{I} \approx 1$ kA, $\gamma \approx 300$, $\sigma \approx 50$ µm, $\epsilon_n \approx 0.5$ µm. After calculation, we find $\rho \approx 2$; the beam dynamics is thus driven by space-charge effects. Nevertheless, minimising momentum spread and beam divergence at the plasma exit will help because a smaller beam divergence implies a smaller average beam size and thus a smaller laminarity parameter.

The beam transfer line will be made up of one capture section consisting of a triplet or a quadruplet of permanent quadrupoles, and one matching section also consisting of a triplet or a quadruplet of permanent quadrupoles. Both capture sections will be separated by a section of at least 1000 mm in length to insert different diagnostics and correctors, like a spectrometer, a dipole corrector, or a beam position monitor (BPM). A sketch of the low-energy transfer line is shown in Figure 19.1.

The electron beam is generated in a laser-plasma injector (LPI) by the ReMPI (Resonant Multi-Pulse Ionisation) technique, then accelerated in the same plasma target to 150 MeV in the quasi-linear regime [170]. The laser beam is split into two pulses, the first of small energy for ionising the gas and the second containing the main part of the energy itself decomposed into a series of several sub-pulses to excite the wakefield without ionising the gas. Start-to-end simulations for this setup have been performed (see Chap. 23), and the final distribution, after further energy boosting in a laser-driven plasma-accelerator stage (LPAS), has been transported to the user areas. We propose in this section to describe the optimised LETL between the LPI and the LPAS. The high-energy beam transport line from the LPAS to the user application will be described in Section 19.2.

### 19.1.2 Compact Beamline with Six Permanent Quadrupoles

The transfer line in this scenario is made of two triplets separated by a drift. The first triplet is used to capture the beam after the LPI, whereas the second triplet is used to match the beam into the LPAS. Matching conditions at the entrance of the LPAS are $\beta_{xy} = 30$ mm and $\alpha_{xy} = 1$ (see Sect. 7.1 for the general equations).

The distance between the triplets and the plasma stages is to be as short as possible to minimise the emittance growth; betatron functions scale as the square of this distance, beam sizes scale linearly with it, and space-charge effects grow as a
<figure><img src="image_141.png" /><figcaption>**Fig. 19.2.** Low-energy transfer line: 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 between the two plasma stages in Scheme 2 (assuming a beam energy of 150 MeV, a total length of 700 mm, and 6 quadrupoles). The calculations were performed with the tracking code TraceWin.</figcaption></figure>

<table><caption>**Table 19.1.** Parameters of the focusing elements in the low-energy transport line between the two plasma stages of Scheme 2, in the case of six quadrupoles and a total length of 700 mm.</caption><thead><tr><th>Quadrupole</th><th>Q1</th><th>Q2</th><th>Q3</th><th>Q4</th><th>Q5</th><th>Q6</th></tr></thead><tbody><tr><th>Length [mm]</th><td></td><td></td><td>45</td><td></td><td></td><td></td></tr><tr><th>Gradient [T/m]</th><td>211</td><td>-189</td><td>86</td><td>-101</td><td>190</td><td>-161</td></tr></tbody></table>

power of the beam size and thus also as a power of this distance. We have assumed a minimum distance between the LPI and the triplet of 50 mm. For start-to-end simulations, the length of the transfer line is 700 mm. As explained above, the drift should host diagnostics like BPMs, screens, or a spectrometer to measure the energy spectrum.

Nevertheless, the distance between both triplets is too short here to insert all diagnostics. The parameters of the quadrupoles are summarised in Table 19.1. The evolution of the beam distribution along the transport line is shown in Figure 19.2, whereas the evolution of the properties of the electron beam slices is shown in Figure 19.3. The final beam properties are summarised in the third column of Table 19.4. If the drift is made longer to host diagnostics (for instance with a total length of 1400 mm) with a parallel beam in between, space-charge effects increase and the emittance growth becomes even larger. The final emittance is then 0.9 µm (compared to 0.66 µm with a total beamline length of 700 mm). Another solution is to over-focus the beam with the first capture section to reach a phase advance between both sections, which cancels the variation of betatron functions with energy.
<figure><img src="image_142.png" /><figcaption>**Fig. 19.3.** Low-energy transfer line: Profile of the current (a), mean energy and energy spread (b), Twiss parameters (c), and normalised emittance (d) along the beam slices at the entrance of the second plasma stage of Scheme 2 (assuming a beam energy of 150 MeV, a total length of 700 mm, and six quadrupoles). The calculations were performed with the tracking code ASTRA.</figcaption></figure>

### 19.1.3 Beamline with Eight Permanent Quadrupoles

This transfer line is also made of two quadruplets separated by a drift. The first quadruplet is used to capture the beam after the LPI, whereas the second quadruplet is used to match the beam into the LPAS. As before, matching conditions at the entrance of the LPAS are $\beta_{xy} = 30$ mm and $\alpha_{xy} = 1$.

An additional constraint is to have a phase advance of roughly 90° between both capture sections. Indeed, initial and final Twiss parameters are quite close, which requires similar capture sections. The main contribution of the trace emittance growth comes from the quadrupoles (and more particularly the variation of Twiss parameters with energy). The pulsation of the betatron wave is twice the phase advance; in other terms, if the phase advance between two points is $\phi$, the phase shift of the perturbation of the Twiss parameters is $2\phi$. Therefore, a perturbation of the Twiss parameters in the first capture section will be cancelled by the second capture section if the phase advance in between is 90° (and thus 180° for the phase advance of the betatron wave). To reach a phase advance of 90°, it is necessary to have $\alpha_{x,y} \approx 0$ and $\beta_{x,y} \approx L_d/2$, where $L_d$ is the length of the drift between both capture sections. A second iteration is then performed to match the beam and to minimise the final emittance with a tracking code. It is worth remarking that this condition is more difficult to fulfil with permanent quadrupoles than with plasma lenses or solenoids because a quadrupole is focusing only in one plane. The cancellation of the betatron wave cannot be perfect because of space-charge effects; this constraint means that only the emittance growth coming from the optics itself can be cancelled and not from collective effects.

We have assumed a minimum distance between the LPI and the triplet of 50 mm. As explained above, the drift should host diagnostics like BPMs, screens, or a spec-
<figure><img src="image_143.png" /><figcaption>**Fig. 19.4.** Low-energy transfer line: 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 between the two plasma stages in Scheme 2 (assuming an electron beam energy of 150 MeV, a total length of 2040 mm, and eight quadrupoles). The calculations were performed with the tracking code TraceWin.</figcaption></figure>

<table><caption>**Table 19.2.** Parameters of the focusing elements in the low-energy transport line between the two plasma stages of Scheme 2 in the case of eight quadrupoles and a total length of 2040 mm.</caption><thead><tr><th>Quadrupole</th><th>Q1</th><th>Q2</th><th>Q3</th><th>Q4</th><th>Q5</th><th>Q6</th><th>75</th><th>Q8</th></tr></thead><tbody><tr><td>Length [mm]</td><td colspan="8">45</td></tr><tr><td>Gradient [T/m]</td><td>96.6</td><td>-182</td><td>162</td><td>-61.6</td><td>-66.7</td><td>-161</td><td>-171</td><td>94</td></tr></tbody></table>

trometer to measure the energy spectrum. Hence, the low-energy beam transfer line has now been lengthened with a total length of 2040 mm. The parameters of the quadrupoles are summarised in Table 19.2. The evolution of the beam distribution along the transport line is shown in Figure 19.4 with the evolution of the properties of the electron beam slices depicted in Figure 19.5. As we can see, the final emittance growth is smaller despite a longer total length. The variation of the betatron functions along the slice is also flatter. The variation of the betatron functions for the slices with $z > 0.5$ µm comes from the initial variation of the Twiss parameters. The final beam characteristics are summarised in the fourth column of Table 19.4.

### 19.1.4 Beamline with Two Active Plasma Lenses

The use of active plasma lenses (APL) has also been considered for the beam capture and final focus (one for each task), as these devices suffer less from chromatic aberrations compared to other focusing systems [38], and are therefore well suited for high-energy-spread beams. Currently, this beamline contains only two APLs. To keep the possibility to separately match horizontal and vertical planes, weak quadrupoles
<figure><img src="image_144.png" /><figcaption>Fig. 19.5. Low-energy transfer line: Profile of the current (a), mean energy and energy spread (b), Twiss parameters (c), and normalised emittance (d) along the beam slices at the entrance of the second plasma stage in Scheme 2 (assuming an electron beam energy of 150 MeV, a total length of 2040 mm, and eight quadrupoles). The calculations were performed with the tracking code ASTRA.</figcaption></figure>

should be inserted in between. Their impact should be low on the final beam emittance.

Nevertheless, at low energy, the needed gradient is reduced and special care must be taken to check that the wakefield generated by the beam itself in the plasma lens is negligible compared to the focusing gradient. We should have the following relation [488]:

$$g_{APL} \gg g_{WF}, \qquad (19.2)$$

$$g_{WF} \approx -\frac{e\mu_0 c}{2} \min \left( n_0, \frac{N k_p^2 \sigma_z}{\pi \sigma_r^2 \left(1 + \frac{k_p^2 \sigma_r^2}{2}\right) \left(1 + \sqrt{8\pi k_p^2 \sigma_z^2}\right)} \right), \qquad (19.3)$$

$$g_{APL} \approx \frac{\mu_0 I}{2\pi R^2}, \qquad (19.4)$$

where $\mu_0$ is the vacuum permeability, $c$ is the speed of light, $g_{WF}$ the maximum wakefield, and $g_{APL}$, $I$, $R$, $n_0$, $k_p$ are, respectively, the gradient, the current, the radius, the plasma density, and the plasma wavenumber of the APL. $\sigma_r$ is the RMS beam size, $\sigma_z$ is the RMS bunch length, and $N$ is the electron population of the bunch.

The APL gradient in the case of a uniform current, a discharge current of 1 A, and a diameter of 1 mm is shown in Figure 19.6. The gradient linearly scales with the discharge current. For the simulations with ASTRA, a more realistic profile has been used, as explained in [487].

To decrease the wakefield in the APL, the plasma lens should be as short as possible and should be located far enough from the LPI exit to increase the beam size. Nevertheless, a trade-off must be found between a small wakefield in the APL
<figure><img src="image_145.png" /><figcaption>**Fig. 19.6.** Magnetic field profile in a 1 mm diameter active plasma lens using the simplified model given in [487] (plotted as solid red curves). One can observe an approximately 35% enhancement in B-field gradient compared to the model with uniform current in the APL. The magnetic field corresponds to a discharge current of 1 A.</figcaption></figure>

(with a large beam size), small chromatic effects (small beam size), and efficient focussing in the APL (small beam size). A first guess is to put the APL at 200 mm from the LPI. At this position, the beam size is about 80 µm, which is roughly 1/6 of the APL radius and thus a good compromise.

The evolution of the beam distribution along the transport line is shown in Figure 19.7. The evolution of the properties of the electron beam slices is shown in Figure 19.8. The parameters of the APLs are summarised in Table 19.3. As we can see, the final emittance growth is smaller thanks to reduced chromatic effects in the APLs. The variation of the betatron functions along the slice is also flat. The final beam characteristics are summarised in the fifth column of Table 19.4.

The order of magnitude of the gradient in the APLs is 300 T/m. If we consider a typical plasma density of $1 \times 10^{17} \text{ cm}^{-3}$, a bunch length of 1 µm, a beam size of 80 µm, and a bunch charge of 33.5 pC, equation (19.3) gives a gradient of 76 T/m for the wakefield, which is 25% of the APL gradient. Some solutions to mitigate the effect of the wakefield are to shorten the APL – but a length of 10 mm is already small and little room is possible – or to put the APL farther – with a larger beam size but increased chromatic effects. The wakefield is thus not negligible and more accurate simulations (with FBPIC [489] for instance) should be performed.

### 19.1.5 Beamline with Two Solenoids

The case of a beamline with two solenoids has also been investigated. The advantage of the solenoid is to be focussing in both planes (same advantage as the APL) and to use a more mature technology. Quadrupoles can also be used to match the beam in both planes after focusing. Nevertheless, at 150 MeV, superconducting solenoids are to be used to keep a compact beamline. The positions of the centre of the solenoids is
<figure><img src="image_146.png" /><figcaption>**Fig. 19.7.** Low-energy transfer line: 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 between the two plasma stages of Scheme 2 (assuming an electron beam energy of 150 MeV, a total length of 2020 mm, and two APLs). The calculations were performed with the tracking code TraceWin.</figcaption></figure>

<figure><img src="image_147.png" /><figcaption>**Fig. 19.8.** Low-energy transfer line: Profile of the current (a), mean energy and energy spread (b), Twiss parameters (c), and normalised emittance (d) along the beam slices at the entrance of the second plasma stage in Scheme 2 (assuming an electron beam energy of 150 MeV, a total length of 2020 mm, and two APLs). The calculations were performed with the tracking code ASTRA.</figcaption></figure>
**Table 19.3.** Parameters of the focusing elements in the low-energy transport line between the two plasma stages of Scheme 2 in the case of two APLs and a total length of 2010 mm.

<table><thead><tr><th>APL</th><th></th><th>APL1</th><th>APL2</th></tr></thead><tbody><tr><td>Length</td><td>[mm]</td><td colspan="2">10</td></tr><tr><td>Current</td><td>[A]</td><td>301</td><td>286</td></tr><tr><td>Gradient</td><td>[T/m]</td><td>316</td><td>301</td></tr></tbody></table>

**Table 19.4.** Comparison of the beam parameters at the end of the LPI and at the entrance of the LPAS for the four different low-energy transfer line configurations investigated. The bunch core contains 33.5 pC of charge.

<table><thead><tr><th></th><th></th><th>LPI exit</th><th colspan="3">LPAS entrance</th></tr><tr><th></th><th></th><th></th><th>6 quads<br>(700 mm)</th><th>8 quads<br>(2040 mm)</th><th>2 APLs<br>(2020 mm)</th><th>2 solenoids<br>(2200 mm)</th></tr></thead><tbody><tr><td>β<sub>x</sub></td><td>[m]</td><td>0.008</td><td>0.029</td><td>0.029</td><td>0.028</td><td>0.030</td></tr><tr><td>β<sub>y</sub></td><td>[m]</td><td>0.008</td><td>0.032</td><td>0.030</td><td>0.027</td><td>0.030</td></tr><tr><td>α<sub>x</sub></td><td>–</td><td>-0.22</td><td>0.816</td><td>0.945</td><td>0.95</td><td>0.965</td></tr><tr><td>α<sub>y</sub></td><td>–</td><td>-0.13</td><td>0.908</td><td>1.041</td><td>1.01</td><td>1.061</td></tr><tr><td>ε<sub>n,x</sub></td><td>[μm]</td><td>0.31</td><td>0.66</td><td>0.59</td><td>0.45</td><td>0.58</td></tr><tr><td>ε<sub>n,y</sub></td><td>[μm]</td><td>0.27</td><td>0.67</td><td>0.41</td><td>0.38</td><td>0.51</td></tr><tr><td>Mean Energy</td><td>[MeV]</td><td>154</td><td>154</td><td>154</td><td>154</td><td>154</td></tr><tr><td>σ<sub>γ/γ</sub></td><td>[10<sup>-3</sup>]</td><td>16.5</td><td>21.6</td><td>29.3</td><td>31.1</td><td>30.0</td></tr><tr><td>I&#773;</td><td>[kA]</td><td>4.2</td><td>3.5</td><td>2.7</td><td>2.5</td><td>2.6</td></tr><tr><td>σ<sub>τ</sub></td><td>[fs]</td><td>3.3</td><td>3.7</td><td>4.5</td><td>4.5</td><td>4.7</td></tr></tbody></table>

at 200 mm and at 2000 mm. The total length of the beamline in this case is 2200 mm. The considered solenoids are 150 mm long, have an inner radius of 15 mm, an outer radius of 40 mm, and a maximum current density of 300 A/mm², which gives a maximum on-axis magnetic field of 8.8 T. The optimisation of the line is similar to the baseline: a waist is generated at the centre of the drift to obtain a phase advance of about 90° between both solenoids. After optimisation, the needed magnetic fields are, respectively, 7.55 T and -7.31 T. The final beam characteristics are summarised in the last column of Table 19.4. The solution with solenoids does not provide better results than the solution with eight quadrupoles. Hence, the solution with eight quadrupoles is preferred, as it enables to differentiate between the horizontal and vertical planes. Nevertheless, it is simpler to tune the beamline with solenoids and thus to reach a phase advance of 90° between them.

### 19.1.6 Conclusion

The beamline between two plasma stages has been optimised and presented. Different schemes have been discussed: a compact beamline with two quadrupole triplets, a longer beamline with two quadrupole quadruplets (to host diagnostics and to get a larger phase advance between both capture sections), a beamline with two active plasma lenses, and a beamline with two solenoids.

The beamline with two APLs gives the best final properties, i.e. minimum emittance growth. However, the wakefield generated by the beam itself in the APL was not taken into account in the simulations. First estimations show it is not negligible and emittance degradation is likely. Further calculations are thus necessary to validate this concept.
<table><caption>**Table 19.5.** Electron beam matching parameters at the entrance of the undulator section in the case of a short-wavelength design (undulator period: $\lambda_u = 20$ mm, module length: $L_u = 2$ m, distance between modules: 360 mm).</caption><thead><tr><th></th><th>⟨β<sub>x,y</sub>⟩ [m]</th><th>β<sub>x</sub> [m]</th><th>β<sub>y</sub> [m]</th><th>α<sub>x</sub></th><th>α<sub>y</sub></th></tr></thead><tbody><tr><td>Scheme 1: LPI-5 GeV</td><td>5</td><td>3.16</td><td>7.31</td><td>-0.697</td><td>1.556</td></tr><tr><td>Scheme 2: LPI-150 MeV + LPAS-5 GeV</td><td>5</td><td>3.16</td><td>7.31</td><td>-0.698</td><td>1.556</td></tr><tr><td>Scheme 3: RFI-500 MeV + LPAS-1 GeV</td><td>5</td><td>3.39</td><td>6.89</td><td>-0.613</td><td>1.174</td></tr><tr><td>Scheme 3: RFI-500 MeV + LPAS-5 GeV</td><td>5</td><td>3.16</td><td>7.31</td><td>-0.698</td><td>1.556</td></tr><tr><td>Scheme 5: RFI-500 MeV + PPAS-1 GeV</td><td>5</td><td>3.40</td><td>6.87</td><td>-0.608</td><td>1.155</td></tr></tbody></table>

Solutions with two quadruplets or two solenoids use mature technology and require less R&D. The final emittance is slightly worse, but stays within specifications. The beamline with solenoids should add some weak quadrupoles to make a fine-tuning between horizontal and vertical planes possible.

Finally, an open problem remains how to remove the laser of the LPI and how to inject the laser for the LPAS. This will be addressed in future studies.

## 19.2 Beam Transport for the High-Energy Beam

### 19.2.1 Introduction

A sketch of the high-energy transfer line (HETL) is shown in Figure 19.9. This is the section transporting the electron beam after its final acceleration stage to an application beamline, here exemplarily assumed to be the entrance of an undulator. Other applications would require slightly different beam parameters, yet the procedure and general beamline layout would be the same.

The transfer line can be divided into three sections: a capture section, a C-chicane, and a matching section.

The capture section is made of permanent quadrupoles and is aimed to capture the electron beam at the plasma exit and focus it. Most of the emittance growth occurs in this section. Careful optimisation must then be performed to minimise this emittance growth. An integrated current transformer (ICT) is inserted behind the capture section to measure the beam current.

The C-chicane is made up of four straight dipoles and is aimed to separate the witness beam from the laser beam (in the case of LWFA), or from the electron drive beam (in the case of PWFA). A collimator will be used for the laser or beam driver removal in these cases. The chicane's other main purpose is to protect the undulators from possible failures at the plasma exit, such as large misalignment errors or energy fluctuations; in these cases the beam will then be dumped into the collimator. Simulations should be performed to validate this concept.

Finally, two doublets are used to match the beam to the undulator entrance. The two magnet pairs are separated by a distance greater than 2 m to enable the insertion of different diagnostics: a BPM to measure the beam position, an X-band transverse deflecting structure (TDS) to measure the time structure [490], and a deflecting dipole to measure the energy spectrum, when needed (see Chap. 21 for details on the diagnostics).

At high electron-beam energy, space-charge forces can be neglected. A python script has been written to match the beam to the undulator and to minimise the emittance growth along the machine. The matching constraints at the entrance of the undulator for the different schemes are given in Table 19.5 for the case of
<figure><img src="image_148.png" /><figcaption>**Fig. 19.9.** Layout of the high-energy beam transfer line (in red: permanent quadrupoles in the capture section, in blue: electromagnet quadrupoles in the matching sections, in green: dipoles (designated as dipole and D1-D4), in cyan: BPMs, in gray: ICTs, and in yellow: screens).</figcaption></figure>

<table><caption>**Table 19.6.** Matching parameters at the entrance of the undulator section in the case of a long-wavelength setup (undulator period: $\lambda_u = 30$ mm, module length: $L_u = 2.1$ m, distance between modules: 360 mm).</caption><thead><tr><th></th><th>⟨β<sub>x,y</sub>⟩ [m]</th><th>β<sub>x</sub> [m]</th><th>β<sub>y</sub> [m]</th><th>α<sub>x</sub></th><th>α<sub>y</sub></th></tr></thead><tbody><tr><td>Scheme 1: LPI-5 GeV</td><td>5</td><td>3.08</td><td>7.44</td><td>-0.668</td><td>1.545</td></tr><tr><td>Scheme 2: LPI-150 MeV + LPAS-5 GeV</td><td>5</td><td>3.07</td><td>7.45</td><td>-0.670</td><td>1.559</td></tr><tr><td>Scheme 3: RFI-500 MeV + LPAS-1 GeV</td><td>4</td><td>2.69</td><td>5.35</td><td>-0.382</td><td>0.627</td></tr><tr><td>Scheme 3: RFI-500 MeV + LPAS-5 GeV</td><td>5</td><td>3.07</td><td>7.45</td><td>-0.670</td><td>1.559</td></tr><tr><td>Scheme 5: RFI-500 MeV + PPAS-1 GeV</td><td>4</td><td>2.75</td><td>5.26</td><td>-0.361</td><td>0.561</td></tr></tbody></table>

a short-wavelength undulator setup and in Table 19.6 for the case of a long-wavelength undulator design. Further details on the undulator parameters are given in Section 24.1. The optimisation is based, in a first instance, on particle swarm optimisation (PSO) [491] to find initial conditions near a global minimum, then, in a second stage, on the conjugate gradient method [492] to speed up the convergence near a minimum; finally, it uses a tracking code, like TraceWin [493], elegant [340] or ASTRA [320]. The variables are the quadrupole gradients and the positions of the different elements. The constraints are the total length of the machine (8 m in our case), minimum and maximum gradients (500 T/m for permanent quadrupoles and 50 T/m for electromagnet quadrupoles), and the minimum distance between elements (30 mm between permanent quadrupoles and 300 mm between electromagnetic quadrupoles of the same doublet to insert BPMs and correctors in between, 2 m between permanent quadrupoles and electromagnetic quadrupoles to insert a C-chicane in between, and 2.5 m between both doublets to insert long diagnostics like the TDS or a spectrometer). Finally, the beam transfer line is optimised with the tracking code TraceWin [493] to match the beam to the undulator entrance and to minimise the emittance growth. This optimisation takes into account the entire beam distribution with no assumptions on the initial conditions.

### 19.2.2 Scheme 1: LPI-5 GeV

The LPI uses the ReMPI injection technique to generate an electron beam inside the plasma and accelerate it to a final energy of 5 GeV. The transfer line is made of one triplet of permanent quadrupoles, used as a capture section, one C-chicane, and four electromagnetic quadrupoles as a matching section. The drive laser will be
<table><caption>**Table 19.7.** Parameters of the focusing elements in the high-energy transport line to the undulator for the case of Scheme 1 (LPI to 5 GeV).</caption><thead><tr><th>Quadrupole</th><th>Q1</th><th>Q2</th><th>Q3</th><th>Q4</th><th>Q5</th><th>Q6</th><th>Q7</th></tr></thead><tbody><tr><th>Length [mm]</th><td colspan="3">120</td><td colspan="4">200</td></tr><tr><th>Gradient [T/m]</th><td>-468</td><td>487</td><td>-266</td><td>-20.0</td><td>29.9</td><td>-12.3</td><td>5.34</td></tr></tbody></table>

<figure><img src="image_149.png" /><figcaption>**Fig. 19.10.** High-energy transfer line – 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 a single LPI to 5 GeV. The calculations were performed with the tracking code TraceWin.</figcaption></figure>

removed at the middle of the C-chicane. Optical functions have been matched at the undulator entrance to the values given in Table 19.5.

The length of the transfer line is 8 m. The evolution of the beam distribution along the transport line is shown in Figure 19.10, whereas the evolution of the slice properties along the longitudinal beam distribution is shown in Figure 19.11. Beam properties are well preserved along the transfer line. The parameters of the quadrupoles are summarised in Table 19.7. The final beam characteristics are listed in Table 19.7.

### 19.2.3 Scheme 2: LPI-150 MeV + LPAS-5 GeV

In this scheme, electrons are first transported from one LPI to the LPAS, as described in Section 19.1. Electrons are then injected into an LPAS to be accelerated up to 5 GeV. As in the previous case, the transfer line is made of one triplet of permanent quadrupoles, used as a capture section, one C-chicane, and four electromagnet quadrupoles used as a matching section. The drive laser is removed at the middle of the C-chicane. Electron beam optical functions have been matched at the undulator entrance to the values given in Table 19.5.

The length of the transfer line is 8 m. The evolution of the beam distribution along the transport line is shown in Figure 19.12, whereas the evolution of the
<figure><img src="image_150.png" /><figcaption>Fig. 19.11. High-energy transfer line – Scheme 1: Profile of the slice current (a), mean slice energy and slice energy spread (b), slice Twiss parameters (c), and normalised slice emittance (d) along the beam length at the entrance of the undulator for the case of a single LPI to 5 GeV. The calculations were performed with the tracking code ASTRA.</figcaption></figure>

slice properties along the longitudinal beam distribution is depicted in Figure 19.13. Beam properties are well preserved along the transfer line. The parameters of the quadrupoles are summarised in Table 19.9. The final beam characteristics, on the other hand, are presented in Table 19.9.

### 19.2.4 Scheme 3: RFI-500 MeV + LPAS-5 GeV

Electrons are injected from one RFI at 500 MeV to be accelerated up to 5 GeV inside the plasma stage in this scenario. The high-energy transfer line has the same standard setup: made of one triplet of permanent quadrupoles, used as a capture section, one C-chicane, and four electromagnetic quadrupoles used as a matching section. The drive laser is removed at the middle of the C-chicane. Electron beam optical functions have been matched at the undulator entrance to the values given in Table 19.5.

The length of the transfer line is 8 m and the evolution of the beam distribution along the transport line is shown in Figure 19.14. The evolution of the slice properties along the longitudinal beam distribution is shown in Figure 19.15. Beam properties are well preserved along the transfer line. The parameters of the quadrupoles are summarised in Table 19.11, while the final beam characteristics are depicted in Table 19.11.

### 19.2.5 Scheme 4: RFI-240 MeV + LPAS-2.5 GeV + Chicane + LPAS-5 GeV

The current configuration considers splitting the acceleration process into two identical LWFA stages providing a ~2.5 GeV energy gain in each to an externally injected electron beam. This layout is based on the multi-stage dechirping concept presented
**Table 19.8.** High-energy transfer line – Scheme 1: Comparison between the beam parameters at the LPI exit and at the entrance of the undulator. These parameters have been calculated considering only the beam core, i.e. only an 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.

<table><caption>**Table 19.9.** Parameters of the focusing elements in the transport line to the undulator for the case of Scheme 2 (LPI + LPAS to 5 GeV).</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>I&#x0302; [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>

<table><thead><tr><th>Quadrupole</th><th></th><th>Q1</th><th>Q2</th><th>Q3</th><th>Q4</th><th>Q5</th><th>Q6</th><th>Q7</th></tr></thead><tbody><tr><th>Length [mm]</th><td></td><td></td><td>100</td><td></td><td></td><td></td><td>200</td><td></td></tr><tr><th>Gradient [T/m]</th><td></td><td>-347</td><td>471</td><td>-297</td><td>1.18</td><td>22.7</td><td>-39.5</td><td>-25.4</td></tr></tbody></table>

in [494], where the energy chirp given to the electron beam in the first LWFA is inverted by a subsequent magnetic chicane, thus allowing it to be compensated in the second accelerating stage. A simplified representation of the beamline considered for this scheme can be seen in Figure 19.16. The details of the transport line between the two LPAS as well as the transport to the undulator are presented in the following. In the first case, the simulations were performed with the particle tracking code ASTRA [320] for the drifts, quadrupoles, and dipoles, taking into account 3D space-charge effects, while the plasma lenses were simulated with FBPIC [489]. CSRtrack simulations of the chicane were also performed to confirm that CSR effects do not have a significant impact on the beam. The second beamline for the beam transport to the undulator was simulated with the ASTRA code.

## Transport Line Between LPAS

The beam transport between the two plasma acceleration stages is crucial for the performance of this scheme. This beamline has to capture a highly divergent and high-energy-spread beam from the first LPAS, properly transport it through the chicane so that the energy chirp can be effectively inverted, and focus the beam again for injection into the second LPAS. All of this needs to be carried out, while preserving the beam properties, mainly its emittance and current profile.

To achieve this, APLs have been considered for the beam capture and final focus (one for each task), as these devices suffer less from chromatic aberrations compared
<figure><img src="image_151.png" /><figcaption>**Fig. 19.12.** High-energy transfer line – 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 an LPI + LPAS to 5 GeV. The calculations were performed with the tracking code TraceWin.</figcaption></figure>

<figure><img src="image_152.png" /><figcaption>**Fig. 19.13.** High-energy transfer line – Scheme 2: Profile of the slice current (a), mean slice energy and slice energy spread (b), slice Twiss parameters (c), and normalised slice emittance (d) along the beam length at the entrance of the undulator for the case of an LPI + LPAS to 5 GeV. The calculations were performed with the tracking code ASTRA.</figcaption></figure>
**Table 19.10.** High-energy transfer line – Scheme 2: Comparison between the beam parameters at the LPAS exit 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.

<table><thead><tr><th></th><th></th><th>LPAS exit</th><th>Undulator entrance</th></tr></thead><tbody><tr><td>β<sub>x</sub></td><td>[m]</td><td>0.021</td><td>3.16</td></tr><tr><td>β<sub>y</sub></td><td>[m]</td><td>0.023</td><td>7.33</td></tr><tr><td>α<sub>x</sub></td><td>–</td><td>-0.48</td><td>-0.70</td></tr><tr><td>α<sub>y</sub></td><td>–</td><td>-0.54</td><td>1.56</td></tr><tr><td>ε<sub>n,x</sub></td><td>[µm]</td><td>0.76</td><td>0.76</td></tr><tr><td>ε<sub>n,y</sub></td><td>[µm]</td><td>0.90</td><td>0.88</td></tr><tr><td>Mean Energy</td><td>[GeV]</td><td>4.98</td><td>4.98</td></tr><tr><td>σ<sub>γ</sub>/γ</td><td>[10<sup>-3</sup>]</td><td>1.0</td><td>1.0</td></tr><tr><td>I&#770;</td><td>[kA]</td><td>3.2</td><td>3.3</td></tr><tr><td>σ<sub>τ</sub></td><td>[fs]</td><td>0.4</td><td>0.4</td></tr><tr><td>ε<sub>n,x</sub> (slice)</td><td>[µm]</td><td>0.86</td><td>0.85</td></tr><tr><td>ε<sub>n,y</sub> (slice)</td><td>[µm]</td><td>1.00</td><td>1.00</td></tr><tr><td>σ<sub>γ</sub>/γ (slice)</td><td>[10<sup>-3</sup>]</td><td>0.85</td><td>0.76</td></tr></tbody></table>

**Table 19.11.** Parameters of the focusing elements in the transport line to the undulator for the case of Scheme 3 (RFI + LPAS to 5 GeV).

<table><thead><tr><th>Quadrupole</th><th></th><th>Q1</th><th>Q2</th><th>Q3</th><th>Q4</th><th>Q5</th><th>Q6</th><th>Q7</th></tr></thead><tbody><tr><td>Length</td><td>[mm]</td><td colspan="4">120</td><td colspan="3">200</td></tr><tr><td>Gradient</td><td>[T/m]</td><td>-331</td><td>489</td><td>-275</td><td>4.26</td><td>-15.1</td><td>46.1</td><td>-4.63</td></tr></tbody></table>

**Table 19.12.** High-energy transfer line – Scheme 3: Comparison between the beam parameters at the LPAS exit 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.

<table><thead><tr><th></th><th></th><th>LPAS exit</th><th>Undulator entrance</th></tr></thead><tbody><tr><td>β<sub>x</sub></td><td>[m]</td><td>0.034</td><td>3.16</td></tr><tr><td>β<sub>y</sub></td><td>[m]</td><td>0.034</td><td>7.32</td></tr><tr><td>α<sub>x</sub></td><td>–</td><td>-1.79</td><td>-0.70</td></tr><tr><td>α<sub>y</sub></td><td>–</td><td>-1.83</td><td>1.56</td></tr><tr><td>ε<sub>n,x</sub></td><td>[µm]</td><td>0.31</td><td>0.32</td></tr><tr><td>ε<sub>n,y</sub></td><td>[µm]</td><td>0.28</td><td>0.29</td></tr><tr><td>Mean energy</td><td>[GeV]</td><td>5.41</td><td>5.41</td></tr><tr><td>σ<sub>γ</sub>/γ</td><td>[10<sup>-3</sup>]</td><td>1.1</td><td>1.1</td></tr><tr><td>I&#770;</td><td>[kA]</td><td>3.4</td><td>3.3</td></tr><tr><td>σ<sub>τ</sub></td><td>[fs]</td><td>2.9</td><td>2.9</td></tr><tr><td>ε<sub>n,x</sub> (slice)</td><td>[µm]</td><td>0.35</td><td>0.35</td></tr><tr><td>ε<sub>n,y</sub> (slice)</td><td>[µm]</td><td>0.32</td><td>0.32</td></tr><tr><td>σ<sub>γ</sub>/γ (slice)</td><td>[10<sup>-3</sup>]</td><td>0.28</td><td>0.27</td></tr></tbody></table>
<figure><img src="image_153.png" /><figcaption>**Fig. 19.14.** High-energy transfer line – 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 an RFI + LPAS to 5 GeV. The calculations were performed with the tracking code TraceWin.</figcaption></figure>

<figure><img src="image_154.png" /><figcaption>**Fig. 19.15.** High-energy transfer line – Scheme 3: Profile of the slice current (a), mean slice energy and slice energy spread (b), slice Twiss parameters (c), and normalised slice emittance (d) along the beam length at the entrance of the undulator for the case of an RFI + LPAS to 5 GeV. The calculations were performed with the tracking code ASTRA.</figcaption></figure>
<figure><img src="image_155.png" /><figcaption>Fig. 19.16. Illustrative representation of the accelerator beamline of Scheme 4.</figcaption></figure>

<table><caption>**Table 19.13.** Parameters of the focusing elements in the transport line between the two LPAS stages in Scheme 4.</caption><thead><tr><th>Element</th><th></th><th>APL 1</th><th>Q1</th><th>Q2</th><th>Q3</th><th>Q4</th><th>APL 2</th></tr></thead><tbody><tr><td>Length</td><td>[cm]</td><td>1</td><td>20</td><td>20</td><td>20</td><td>20</td><td>1</td></tr><tr><td>Gradient</td><td>[T/m]</td><td>3000</td><td>-35.47</td><td>29.07</td><td>25.59</td><td>-32.46</td><td>3000</td></tr></tbody></table>

to other focusing systems [38] and are therefore well suited for high-energy-spread beams. These two devices are the main components of the transport line and could, in theory, be sufficient to successfully transport the beam. However, since they provide the same focusing in both *x* and *y* planes and the beam parameters (Twiss parameters $\alpha$ and $\beta$) are different in each transverse direction, the presence of quadrupoles is required to properly guide the beam. In addition, including quadrupoles in the beamline provides a higher tunability of the transport line, since now the APLs are only needed to perform the initial capturing and final focusing of the beam. The quadrupoles, on the other hand, can be tuned to transport the beam through the chicane (achieving a beam waist, i.e. $\alpha_x = \alpha_y = 0$, at the chicane centre) and can be adapted for different beamline designs. The parameters of the focussing elements in the beamline, as currently considered in the simulations, can be seen in Table 19.13. The four chicane dipoles are all equal, having a length of 200 mm and a magnetic field of 0.4 T. The evolution of the beam parameters along this transport line is shown in Figure 19.17 and a comparison of the beam current profile before and after the chicane can be seen in Figure 19.18.

## Transport to Undulators

Similarly to the other schemes, this transfer line is made of one triplet of permanent quadrupoles, used as a capture section, one C-chicane, and four electromagnetic quadrupoles used as a matching section. The drive laser is removed in the middle of the C-chicane. Electron beam optical functions have been matched at the undulator entrance to values similar to those given in Table 19.5.

The length of the transfer line is 8 m. The evolution of the beam distribution along the transport line is shown in Figure 19.19 and the evolution of the slice properties along the longitudinal beam distribution is shown in Figure 19.20. Beam properties are well preserved along the transfer line. The parameters of the quadrupoles are summarised in Table 19.15, while the final beam characteristics are depicted in Table 19.15.

### 19.2.6 Scheme 5: RFI-500 MeV + PPAS-1 GeV

Two electron bunches (one driver and one witness) are injected from one RFI at 500 MeV to accelerate the witness up to 1 GeV inside a beam-driven plasma stage
<figure><img src="image_156.png" /><figcaption>**Fig. 19.17.** High-energy transfer line – Scheme 4: Beam beta function (a), normalised emittance (b), transverse size (c), and bunch duration (d) along the transport line between the two LPAS. It should be noted that, although the actual simulations of the beamline were performed with ASTRA and FBPIC, due to the small amount of saved time steps along the transport line in those simulations (insufficient to get a clear picture of the beam evolution), the data shown in this plot come from a simulation performed with the particle-tracking code Wake-T [495]. However, since the effect of space-charge forces and wakefields in the APL on the beam parameters is very limited, the actual beam evolution does not significantly differ from the one shown in this figure.</figcaption></figure>

<figure><img src="image_157.png" /><figcaption>**Fig. 19.18.** Current profile of the beam before and after the chicane in Scheme 4.</figcaption></figure>
<table><caption>**Table 19.14.** Parameters of the focusing elements in the transport line to the undulator for the case of Scheme 4 (RFI + LPAS + LPAS to 5 GeV).</caption><thead><tr><th>Quadrupole</th><th></th><th>Q1</th><th>Q2</th><th>Q3</th><th>Q4</th><th>Q5</th><th>Q6</th><th>Q7</th></tr></thead><tbody><tr><th>Length</th><td>[mm]</td><td colspan="3">120</td><td colspan="4">200</td></tr><tr><th>Gradient</th><td>[T/m]</td><td>-336</td><td>500</td><td>-317</td><td>-4.11</td><td>34.0</td><td>-48.6</td><td>-26.0</td></tr></tbody></table>

<figure><img src="image_158.png" /><figcaption>**Fig. 19.19.** High-energy transfer line – Scheme 4: 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 an RFI + two LPAS to 5 GeV. The calculations were performed with the tracking code TraceWin. Note that small variations in the average beam parameters in this figure compared to Figure 19.20 and Table 19.15 are a consequence of particle filtering; the beam core in this figure has been defined as the particles within 4 sigma of the beam centre.</figcaption></figure>

(PPAS). The transfer line is made of one triplet of permanent quadrupoles, used as a capture section, one C-chicane, and three electromagnetic quadrupoles used as a matching section. The driver is removed at the middle of the C-chicane, where the dispersion is maximum (driver and witness have different energies then, with 500 MeV and 1 GeV, respectively). Electron beam optical functions have been matched at the undulator entrance to the values given in Table 19.5.

The length of the transfer line is 8 m. The evolution of the beam distribution along the transport line is shown in Figure 19.21. The evolution of the slice properties along the longitudinal beam distribution is shown in Figure 19.22. Beam properties change slightly along the transfer line. The parameters of the quadrupoles are summarised in Table 19.16, while the final beam characteristics are depicted in Table 19.17.

### 19.2.7 Conclusion

The high-energy beamlines from plasma stages to user applications (such as the FEL) have been optimised and presented. The optimisation procedure is now well
<figure><img src="image_159.png" /><figcaption>**Fig. 19.20.** High-energy transfer line – Scheme 4: (a) Longitudinal phase space and current profile of the beam core, (b) slice energy spread and normalised slice emittance along the beam, and (c) slice Twiss parameters. This particle distribution corresponds to the beam at the entrance of the undulator for the case of one RFI + two LPAS (5 GeV). The calculations were performed with the tracking code ASTRA.</figcaption></figure>

<table><caption>**Table 19.15.** 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_160.png" /><figcaption>**Fig. 19.21.** High-energy transfer line – 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 an RFI + PPAS to 1 GeV. The calculations were performed with the tracking code TraceWin.</figcaption></figure>

<figure><img src="image_161.png" /><figcaption>**Fig. 19.22.** High-energy transfer line – Scheme 5: Profile of the slice current (a), mean slice energy and slice energy spread (b), slice Twiss parameters (c), and normalised slice emittance (d) along the beam length at the entrance of the undulator for the case of an RFI + PPAS to 1 GeV. The calculations were performed with the tracking code ASTRA.</figcaption></figure>
<table><caption>**Table 19.16.** Parameters of the focusing elements in the high-energy transport line to the undulator for the case of Scheme 5 (RFI + PPAS to 1 GeV).</caption><thead><tr><th>Quadrupole</th><th>Q1</th><th>Q2</th><th>Q3</th><th>Q4</th><th>Q5</th><th>Q6</th></tr></thead><tbody><tr><th>Length [mm]</th><td></td><td>100</td><td></td><td></td><td>200</td><td></td></tr><tr><th>Gradient [T/m]</th><td>-267</td><td>288</td><td>-153</td><td>2.89</td><td>15.6</td><td>-10.1</td></tr></tbody></table>

<table><caption>**Table 19.17.** High-energy transfer line – Scheme 5: Comparison between the beam parameters at the PPAS exit 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.</caption><thead><tr><th></th><th>PPAS exit</th><th>Undulator entrance</th></tr></thead><tbody><tr><td>β<sub>x</sub> [m]</td><td>2.175</td><td>3.39</td></tr><tr><td>β<sub>y</sub> [m]</td><td>4.2664</td><td>6.88</td></tr><tr><td>α<sub>x</sub> –</td><td>0.004</td><td>-0.61</td></tr><tr><td>α<sub>y</sub> –</td><td>1.072</td><td>1.16</td></tr><tr><td>ε<sub>n,x</sub> [µm]</td><td>0.68</td><td>0.91</td></tr><tr><td>ε<sub>n,y</sub> [µm]</td><td>0.59</td><td>0.68</td></tr><tr><td>Mean Energy [GeV]</td><td>1.06</td><td>1.06</td></tr><tr><td>σ<sub>γ</sub>/γ [10<sup>-3</sup>]</td><td>2.6</td><td>2.6</td></tr><tr><td>I&#773; [kA]</td><td>2.06</td><td>1.96</td></tr><tr><td>σ<sub>τ</sub> [fs]</td><td>0.42</td><td>0.56</td></tr><tr><td>ε<sub>n,x</sub> (slice) [µm]</td><td>0.66</td><td>0.62</td></tr><tr><td>ε<sub>n,y</sub> (slice) [µm]</td><td>0.55</td><td>0.48</td></tr><tr><td>σ<sub>γ</sub>/γ (slice) [10<sup>-3</sup>]</td><td>0.33</td><td>0.65</td></tr></tbody></table>

established and transfer lines integrate diagnostics. Results show that, in most of the cases, beam properties can be preserved and the beam is matched to the user application requirements with state-of-the-art elements.

Nevertheless, correction schemes have not been implemented yet and should be addressed in the future. This work will have a direct impact on the diagnostics requiring a validation of the positioning of diagnostics (BPMs for instance). Laser removal is another critical topic and the required space to inject and remove the laser driver should be discussed in the near future.

## 20 Plasma Diagnostics

Diagnostics techniques capable of providing information on plasma density and accelerating field amplitude in the plasma structures envisaged for injectors and accelerators in the EuPRAXIA beamline are described in this chapter.

### 20.1 Introduction

Diagnostic tools suitable for real-time monitoring and control of plasma density are necessary to achieve the long-term stable operation of the accelerator as required for a user facility.
Interferometric diagnostics can be used for the determination of gas or plasma
density in a variety of configurations, giving access to the range of parameters required
for injector and accelerator plasma modules. Their respective sensitivities are com-
pared, and practical considerations are discussed.

Spectroscopic methods and group velocity measurements have the capability to
diagnose plasma density in waveguides and are better suited for the accelerator
plasma.

The spectral analysis of the laser driver after interaction with the plasma in the
quasi-linear regimes provides a measurement of the averaged accelerating field and is
thus a precious tool for monitoring the quality of the accelerator during operation.

Plasma diagnostics are required both in the development stage and in the oper-
ation stage of a facility like EuPRAXIA. They can be classified as diagnostics used
for the following:

- **Alignment, A:** in the preparation phase or start-up after an interruption of operation.

- **Tuning, T:** to monitor plasma parameters while tuning, changing operation mode, or optimising electron beam parameters.

- **Operation, O:** to monitor plasma parameters for measuring reproducibility and stability, and detect any potential deviation during operation.

Controlling and measuring the density of a plasma with a precision better than 10% at the instant of interaction with the short laser or particle pulse is extremely challenging. This report examines several diagnostic techniques that could be useful for either injector or accelerator stages. As plasma density, volume, regime of interaction, and accessibility to detection are different for injectors and accelerators, different techniques may be used. In the quasi-linear regime of the accelerator stage, a measurement of the amplitude of the accelerating field can also be envisaged.

## 20.2 Plasma Diagnostics

Diagnostic tools suitable for a real-time monitoring and control of plasma density are necessary in order to achieve long-term stable operation of the accelerator as required for a user facility. The plasma target envisaged for the EuPRAXIA accelerator stages will be composed of hydrogen gas [220]. Assuming full ionisation, the electron density in the plasma will then be double the neutral particle number density [286], which may be a simpler quantity to measure in some circumstances.

In this section, we consider simple diagnostic methods not requiring sophisticated implementation (e.g. anti-vibration systems) which may be applicable to the plasma preparation phase or resuming after an interruption (A), when changing / optimising parameters (T), and when monitoring parameters for measuring reproducibility / stability (O).

### 20.2.1 Interferometric Density Measurement

In interferometric methods, the density is inferred from measurements of the phase
shift acquired by an optical ray when passing through the target. This is possible as
the refractive index of the sample $\eta(n)$ is dependent on its particle number density
$n$; in particular, the refractivity $\eta(n) - 1$ is proportional to the particle number
density of the atoms / molecules in a gas or free electrons in a plasma. For a plasma,
$\eta_e(\lambda, n_e) - 1 = -e^2\lambda^2n_e/(8\pi^2m_e\varepsilon_0c^2)$, where $e$ is the electron charge, $\lambda$ is the light
wavelength, $m_e$ is the electron mass, $\varepsilon_0$ is the vacuum permittivity, and $c$ is the
speed of light. For neutral atoms / molecules, from the Lorentz-Lorenz equation,
$\eta(\lambda, n) - 1 = (\eta_0(\lambda) - 1) \times n/n_0$, where $\eta_0(\lambda)$ is the refractive index at a particle number density $n_0$ taken as reference value. The additional phase acquired by a light beam when passing through a sample of length $L$ is then $\phi = (2\pi/\lambda) \int_L (\eta(\lambda) - 1) d\ell$. Therefore, with an interferometric measurement, the information acquired is the line-integrated particle density.

Interferometric measurements can be performed over an extended area using an optical imaging system to acquire a 2D interferogram, from which it is possible to retrieve a 2D map of the line-integrated phase shift imparted by the target. Two-dimensional phase retrieval can lengthen the data analysis and introduce some uncertainty in the measurement due to the algorithm used to analyse the interferogram but allows for the reconstruction of complex density profiles which may be most appropriate during the alignment (A) and tuning (T) of the system.

It is also possible to perform a 1D interferometric measurement over a single line of sight, i.e. integrating over the entire light beam using a photodiode or PMT as a light detector, in which case a spatially averaged density will be retrieved. One-dimensional measurements are faster to acquire and process and therefore may be more suitable during operation (O), e.g. to monitor in real time the gas particle number density.

Three classes of interferometric diagnostics are in common usage.

## Two-Arm Interferometer (TAI)

In two-arm interferometers (TAIs), the phase shift is measured relative to the phase of an optical ray not passing through the sample. The two-arm method can be implemented in various configurations depending on the optical design adopted:

- Standard two-arm design, e.g. Mach-Zehnder and Fizeau interferometers, where the light beam is split into two distinct beams following different paths, one of which passes through the target, which then recombine and interfere [352].

- Modified two-arm interferometers, also called two-arm folded interferometers, where the light beam is split into two after passing through the target and interference take place between parts of the beam with and without the sample [496,497]. This method provides more control on the fringe spacing and angle and is often used in LWFA experiments.

- Nomarski interferometer, similar to the two-arm folded interferometer, but the beams are split using a Wollaston polariser or a bi-prism after the target [498,499].

The standard two-arm interferometers suffer from high sensitivity to environmental conditions, e.g. mechanical vibration and temperature changes, while the last two methods are relatively more robust. The typical sensitivity of an ultra-fast-imaging two-arm interferometer can be hundredths of a fringe, i.e. ~100 mrad.

## Wavefront (WF) Sensor

A versatile and robust method is based on wavefront (WF) sensors, e.g. the commercial quadri-wave lateral shearing interferometer (Phasics, Fr) [500]. This instrument measures the differential phase among two adjacent points of the light beam on the wavefront detector. In fact, it is sensitive to the phase gradients along two orthogonal directions, and the actual phase map is retrieved by analysing the data with a proprietary software. Finally, the phase shift is calculated by subtracting the phase map of an undisturbed beam, acquired in a separate measurement without the plasma [501]. It is a single-arm method since interference takes place only at the detector and therefore is a very robust method, allowing for a relatively high phase sensitivity,
<table><caption>**Table 20.1.** Measuring capability of interferometric methods. Phase shift after passing through 1 cm of medium.</caption><thead><tr><td>Method</td><td>Neutral gas at 10<sup>19</sup> cm<sup>-3</sup></td><td>Plasma electrons at 10<sup>17</sup> cm<sup>-3</sup></td></tr></thead><tbody><tr><td>Two-arm, 800 nm</td><td>4 rad</td><td>2.2 rad</td></tr><tr><td>Two-arm, 400 nm</td><td>8.3 rad</td><td>1.1 rad</td></tr><tr><td>Second harmonic, 800 nm</td><td>0.3 rad</td><td>3.4 rad</td></tr></tbody></table>

which is claimed to be <2 nm, i.e. <30 mrad at 400 nm [502]. An RMS sensitivity of 11 mrad has been estimated at 400 nm over 188 acquisitions [503]. It is noted that no single-acquisition value has been reported, so a conservative value of 30 mrad at 400 nm is assumed.

The wavefront sensor measures only the gradient of the density and therefore is not a straight forward analytical relation between the measured quantity and the density. In reference [503], a comparison between wavefront-based sensors and a modified two-arm interferometer resulted in a difference in the absolute value of the measured density of 10–20%. From the WF data sheet, 15 nm accuracy, i.e. 0.24 rad at 400 nm, is claimed, which can explain the deviation with respect to the TAI measurement. Thus, the WF sensor-based instrument would require an accurate calibration before being used for an absolute measurement of density.

## Second-Harmonic Interferometer (SHI)

Another kind of interferometer is the so-called second-harmonic (or dispersion) interferometer (SHI), which has a fully common-path design and measures the phase difference acquired by the fundamental and second-harmonic beams when passing collinearly through the sample [504,505]. The measured phase shift is given by $\Delta\phi_{SHI} = (4\pi/\lambda) \int_L \Delta\eta(\lambda) d\ell$, where $\Delta\eta(\lambda) = \eta(\lambda) - \eta(\lambda/2)$. It is a very robust method against mechanical and thermal vibrations, with a sensitivity $\le 1$ mrad for 1D measurements [506], though there are no data available in the literature regarding sensitivity of the SHI for 2D imaging [507,508]. Here, we assume a conservative sensitivity value of 10 mrad for quadrature detection [509].

Spectral-domain second-harmonic interferometry (SDSHI) has recently been successfully applied to measure the line-integrated longitudinal plasma density. Significantly, the measurement of the group delay in SDSHI at 800 nm results in a sensitivity of 2.84 rad, while a simultaneous measurement of the spectral phase allows one to get a sensitivity of 63 mrad without the need of a fringe-tracking procedure [510].

## Comparison of the Measurement Capability

Given that the refractivity of hydrogen at the fundamental and second-harmonic wavelength of a Ti:sapphire laser are $1.374 \times 10^{-4}$ and $1.426 \times 10^{-4}$, respectively [511], $\Delta\eta(800\text{nm}) = \eta(400\text{nm}) - \eta(800\text{nm}) = 52 \times 10^{-7}$ at standard temperature and pressure.

The measuring capability of the various techniques, measured as phase shift per centimetre of medium, are compared in Table 20.1.

In Figures 20.1 and 20.2, the capabilities of the various interferometric methods are graphically shown for neutral hydrogen and free electrons, respectively; the lower full lines represent the lower detection limit set by the sensitivity, while the upper dashed lines represent the upper limit within a single fringe measurement, i.e. $\Delta\phi < 0$.
<figure><img src="image_162.png" /><figcaption>**Fig. 20.1.** Interferometric measurement capability for neutral molecular hydrogen. The solid lines correspond to the minimum detectable phase while the dashed-lines correspond to the sub-fringe limit, as indicated by the phase value reported in the legend.</figcaption></figure>

π. The black boxes represent the parameters space indicated for free-electron density $n_e$ for EuPRAXIA, 2 × 10¹⁷–1 × 10¹⁹ cm⁻³, with a neutral gas density half of it. Interferometry is performed usually transversally to the main laser and the electron beam. In such a case, the plasma length seen by the interferometer lies between 10 µm, roughly the dimension of the main laser beam spot, and hundreds of µm. In case of a centimetres-long plasma channel/waveguide, e.g. for the accelerator module, longitudinal interferometry may be considered. The length of neutral gas seen by the interferometer can range from a millimetre, e.g. for a pulsed jet [512], to tens of millimetres, as in flowing gas cells [40,513].

From Figure 20.1, it is evident that the SHI and the WF sensors are capable of measuring low-density values even for paths of a few millimetres. The SHI allows a very fast measurement of neutral hydrogen in 1D (acquisition time ~1 µs) and therefore can be implemented as a sensor in a closed-loop gas flow regulation system. This is very important for gas cells as repetitive shots of the high-power main laser beam may eventually modify the cell's orifice, altering the gas flow dynamics and therefore the internal number density for a pre-set backing pressure [286]. On-line regulation of the gas supply system is therefore necessary to achieve a stable and reproducible laser-plasma acceleration process. A 1D measurement with an SHI can achieve a sensitivity <1 mrad, therefore enabling the measurement of density around 10¹⁷ cm⁻³ over a 1 mm length [506]. In comparison, a wavefront sensor is suitable for 2D density mapping for densities above 10¹⁷ cm⁻³ gas lengths above several millimetres.

For one-shot measurements of the free-electron density, Figure 20.2 shows that only the imaging SHI with 10 mrad sensitivity would be suitable to measure the lowest free-electron density in the range of a few 10¹⁷ cm⁻³ for a plasma length <100 µm. However, such an instrument has not yet been tested, and further development is necessary to assess the actual sensitivity achievable with an imaging SHI, though ultra-fast SHI has been presented in the literature [514]. The wavefront sensor is suitable for the measurement of free-electron densities above 10¹⁸ cm⁻³ and a plasma longer than hundreds of µm, and it can readily work with ultra-fast light sources [503].

Concerning plasma channels / waveguides, the measurement of the group delay in SDSHI results in a sub-fringe resolution (2.84 rad), with a simultaneous measurement
<figure><img src="image_163.png" /><figcaption>**Fig. 20.2.** Interferometric measurement capability for free electrons. The solid lines correspond to the minimum detectable phase, while the dashed lines correspond to the sub-fringe limit, as indicated by the phase value reported in the legend, except for the SDSHI, for which both spectral phase and group delay measurements capability are shown by solid and dashed lines, respectively.</figcaption></figure>

<table><caption>**Table 20.2.** Comparison of interferometric methods.</caption><thead><tr><td>Diagnostic</td><td>Advantages</td><td>Disadvantages</td></tr></thead><tbody><tr><td>Two-arm</td><td>Ultra-fast compatible 2D measurement</td><td>Sensitive to environmental conditions Requires multiple beams</td></tr><tr><td>Second-harmonic</td><td>Single beam Compact Easy to implement High phase sensitivity High absolute accuracy</td><td>Ultra-fast and 2D imaging versions to be tested</td></tr><tr><td>Wave-front</td><td>Single beam Compact Easy to implement Ultra fast compatible 2D measurement</td><td>Low absolute phase accuracy Processing frequency &lt; 3 Hz</td></tr></tbody></table>

of the spectral phase to achieve a phase resolution of 63 mrad over multiple fringes without any fringe tracking procedure [510]. These data are also included in Figure 20.2 for better comparison with the other interferometric methodologies.

The advantages and disadvantages of each interferometric method are summarised in Table 20.2.

## Practical Considerations

To retrieve the actual density from an interferometric measurement, some assumptions on the geometry of the sample have to be made. There are two typical cases: cylindrically symmetric samples, e.g. pulsed gas jet from circular nozzles and plasma created by a laser beam, and uniform samples, e.g. gas-filled cells. For uniform and homogeneous samples, the actual density is simply related to the measured phase
shift. For cylindrically symmetric samples, the measured phase shift must be symmetrised and Abel-inverted to extract the radial density profile [497]. The density distribution of anisotropic samples must be measured using tomographic methods [515].

All interferometric methods require a line of sight through the sample. For a freely expanding gas jet, interferometry is easy to implement, while in the case of confined samples – e.g. gas cells, capillaries, and tubes – flat optical side windows are necessary in order to provide the optical path for transverse interferometry. This issue must be considered when designing the geometry of the accelerator stage. Longitudinal interferometry may be adopted [352,510], potentially for aligning (A) / tuning (T), but not during operation (O) because of the geometrical constraints due to the main laser / electron bunch propagation.

### 20.2.2 Spectroscopic Methods

In the case when there is not a straight optical path through the sample, e.g. cylindrical capillaries or tubes, other optical diagnostics to monitor density can be considered. Stark broadening measurements may be envisaged to monitor the free-electron density in this situation [516]. It has to be noted, however, that the data analysis in this case depends (in a mild way) on the electron temperature. Therefore, a direct measure or estimate of the electron temperature may be necessary in parallel with Stark broadening measurements. Spatially resolved emission spectroscopy (along a line) can be implemented using an imaging spectrometer, and measurements at the nanosecond time scale can be achieved using a fast (intensified) camera. For example, the Stark broadening of hydrogen lines (e.g. $H_β$ at 486.1 nm) allows for the determination of the local electron density down to the $10^{16} \text{ cm}^{-3}$ range with a medium-resolution spectrometer.

### 20.2.3 Group Velocity Measurements

For a discharge capillary waveguide producing a radial plasma density distribution, laser group velocity measurements can be used to measure the plasma density [517–519]. This measurement relies on interferometry in the spectral domain between a reference beam travelling in vacuum and a beam probing the plasma. The reference beam and the probe beam are achieved by a 50/50 beam splitter in front of the capillary. The capillary beam is focussed by a lens into the capillary entrance and imaged on a spectrometer. The reference beam is transported in vacuum to a delay stage before being imaged on the same imaging spectrometer. A reference measurement is done with the capillary out and compared to the same measurement, with the capillary guiding the laser to deduce the delay introduced by the lower group velocity ($v_g$) inside the capillary plasma. Particular attention must be put into the laser spot size as laser size fluctuations during its propagation in the capillary discharge can indeed increase the optical path. Simulations of the laser propagation inside the plasma channel must be performed to take this effect into account.

### 20.2.4 Summary

In summary, the methods described above are appropriate for different types of plasma targets:

- For the LPI, flowing gas cells with an electron density of $2 \times 10^{17} \text{ cm}^{-3}$ for a 1 GeV baseline and $3 \times 10^{18} \text{ cm}^{-3}$ for a 150 MeV baseline are considered a promising
target. For the lower density case, the 2D SHI is a potentially useful diagnostic, but further development is necessary to develop the required ultra-fast and 2D capability. At higher density / plasma length, a wavefront sensor is suitable for 2D mapping with ultra-fast capability. Concerning neutral gas density measurements in the range $10^{17} - 10^{19} \text{cm}^{-3}$, 1D SHI provides the best solution for real-time monitoring, while the wave-front sensor is suitable for 2D measurements. TAIs are less suitable during operation because of their limited sensitivity but may be useful during the aligning / tuning of the system.

- Small-diameter plasma channels created with a laser pre-pulse at densities of $10^{17} - 10^{19} \text{cm}^{-3}$ are open structures compatible with transverse interferometry. At lower density, only the ultra-fast 2D SHI may prove useful, while at higher density, a wavefront sensor will work.

- Plasma channels created by a discharge in a capillary or the grazing incidence capillaries do not allow for transverse interferometry; in these cases, Stark broadening measurements and / or longitudinal interferometry may be adopted [352,510,520]. TAIs are less suitable during operation but may be useful during aligning / tuning.

## 20.3 Plasma Wave Diagnostics

The primary diagnostic of electric field and thus plasma wave amplitude in the plasma accelerator is likely to come from the injection and acceleration of fast particles into the structures. However, in the absence of a particle beam, which will be true in preparatory phases (A), optical diagnosis can provide a great amount of information about the relativistic plasma wave. This can be achieved with both longitudinally and transversely directed beams.

### 20.3.1 Longitudinal Probing

#### Spectral Characterisation of the Driver Beam

The plasma wave amplitude can be diagnosed by analysing the spectrum of the transmitted laser radiation after interaction over a long distance in the quasi-linear regime. Laser pulses transmitted through gas jets [521] and gas-filled capillary tubes exhibit broadened spectra [522]. In the range of parameters relevant to the accelerator plasma, spectral modifications of the laser pulse driving the plasma wave, after propagating in the plasma over a large distance, are mainly related to changes in the index of refraction of the plasma during the creation of the plasma wave. The front of the laser pulse creates an increase in electron density as plasma electrons are pushed in front of the intense laser pulse by its ponderomotive push. This leads to a blueshift at the very front of the pulse, sometimes called photon acceleration [523]. However, the main body of the laser pulse sits in a region of decreasing electron density, away from the first maximum. In this region, the laser pulse is redshifted, indicating that the laser photons are transferring energy to the growth of the plasma wave (i.e. “photon deceleration”). This shift of the spectrum toward longer wavelengths is thus a signature of plasma wave excitation.

The average wavelength shift $\Delta\lambda(l)$ over a propagation length $l$ can be expressed as a function of the laser spectrum $S(\lambda, l)$ as follows [524]:

$$ \Delta\lambda(l) = \frac{\int_{0}^{\infty} \lambda S(\lambda, l) d\lambda}{\int_{0}^{\infty} S(\lambda, l) d\lambda} - \lambda_{L} $$

with $\lambda_L = \int_0^\infty \lambda S(\lambda, l = 0) d\lambda / \int_0^\infty S(\lambda, l = 0) d\lambda \approx 2\pi c/\omega_L$ as the laser centre wavelength. For a laser pulse well above the ionisation threshold so that the blueshift
caused by ionisation can be neglected and the wavelength shift is small compared to the laser wavelength, the shift is related to the energy of the plasma wave electric field $E_p$ through

$$ \frac{\Delta\lambda(l)}{\lambda_L} \approx \frac{1}{16\pi E_{out}} \int_V E_p^2 dV $$

with $E_{out}$ as the transmitted laser energy and $V$ as the interaction volume. For a Gaussian laser pulse propagating in a monomode dielectric capillary of radius $r_c$, the wavelength shift can be expressed analytically as follows:

$$ \Delta\lambda(l) \approx \left[ 0.178 + 1.378 \frac{c^2}{(\omega_p r_c)^2} \right] \left( \frac{\omega_p}{\omega_L} \right)^3 a_L^2 D(\Omega)l $$

with $D(\Omega) = \Omega \exp(-\Omega^2/4)$ and $\Omega = \omega_p \tau_L / \sqrt{2 \ln 2}$, $\omega_p$ as the plasma angular frequency and $\tau_L$ as the laser pulse duration; $c$ is the velocity of light in vacuum and $a_L$ the peak normalised amplitude of the laser field.

The measurement of the total energy and spectral composition of the transmitted laser pulse can therefore be used to diagnose the amplitude of the plasma wave [525]. This diagnostic can be used to adjust plasma and laser parameters for optimum acceleration.

## Photon Acceleration Probe

This process can be extended by the use of secondary laser beams to probe the plasma density. A long laser pulse can be sent behind the short pulse drive, which generates the wakefield, so that it copropagates with the wakefield. Frequency shifts along the length of the probe beam caused by the varying refractive index can be compared to the simulation to allow the plasma wave shape to be unfolded [526]. Alternatively, full spectral-phase measurements can be performed with an instrument such as a FROG (frequency-resolved optical gating) to allow the variation in phase along the plasma wave to be determined [527]. This can then be directly related to the density of the plasma wave causing the phase shifts.

Problems with this technique, which are common with those for diagnosing the transmitted beam, are the longitudinal displacement of photons due to group velocity dispersion, photon trapping, and instabilities such as Raman scattering. These become particularly an issue over long interaction lengths. Sending the probe beam at an angle to limit the interaction length has also been considered to overcome some of these problems.

## Frequency Fomain Interferometry

These measurements can be extended by directly measuring the phase shifts rather than spectral shifts using frequency-domain interferometric methods. The 1D structure of the wave can be found by stepping an ultra-short probe through the plasma wave so that the plasma density can be interrogated as a function of longitudinal displacement [528,529]. An extension of this technique is the overlapping of a long probe pulse with one that has not been through the plasma to produce a complete spatio-temporally resolved image in three-dimensions of the plasma wave in a technique called frequency-domain holography (FDH) [530]. FDH is particularly useful for diagnosing linear plasma waves over limited distance, but again, when spectral and phase shifts become too large because of long interaction lengths or strong density gradients in the plasma wave, then the same limitations as other longitudinal plasma wave density measurements exist.
## 20.3.2 Transverse Probing

Recently, it has been demonstrated that the same transverse probing techniques used for imaging plasma channels can be used for measuring the density profiles of relativistic plasma waves as well [531–533]. An ultra-short probe can essentially freeze the motion of the plasma wave. Since the plasma wave can be as short as 30 fs in temporal duration at a plasma density of $1 \times 10^{19} \text{ cm}^{-3}$, then this demands that the probe beam should be much shorter. Typically, probe beams of duration <10 fs are employed using the latest pulse-compression techniques. Additionally, polarisation measurements can allow sensitive measurements of the magnetic field within the plasma wave [531,532], which can be produced as a combination of the self-generated fields of the plasma wave as well as the magnetic field of an accelerating bunch. Indeed, this has allowed an injected bunch inside a plasma wave to be imaged for the first time [532], and is a useful diagnostic of the injection process [533]. The short pulse means that interferometry is relatively difficult in this configuration, since the short pulse length necessitates a large bandwidth. The short pulse length limits the spatial extent over which interference occurs, and the broad bandwidth can reduce fringe visibility. As a result, these first experiments have relied on transverse probe deflection (shadowgraphy) for imaging the plasma wave.

The difficulty with transverse probing is that the relatively short interaction distances mean that the sensitivity of this technique is less than that of longitudinal techniques. Indeed, until now, it has only been operated at high density ($>1 \times 10^{19} \text{ cm}^{-3}$) and for non-linear plasma waves. However, the use of longer wavelength probes, which increases sensitivity, and the use of advanced techniques, such as SHI or wavefront sensors, would allow these techniques to be extended to lower plasma density stages.

## 20.3.3 Summary

In summary, a number of optical techniques have been developed to diagnose the full 3D structure of an accelerating relativistic plasma wave. Generally, these techniques make use of the variation in refractive index associated with the plasma density modulations of the plasma wave. Probing longitudinal to the plasma wave can allow an integrated effect, allowing for a particularly high sensitivity. This will be particularly important for the long low-density plasmas, which are envisaged for the high-energy stages for EuPRAXIA. These density modulations can then be diagnosed using a combination of spectral and interferometric techniques. In turn, this will allow both a longitudinal and a transverse characterisation of the accelerating fields excited in the wake of the laser pulses over the long distances necessary for the efficient acceleration of electrons to high energies.

## 20.4 Conclusion

A set of diagnostics for monitoring plasma density and plasma wave amplitude with EuPRAXIA parameters can be designed from existing techniques. The final choice of diagnostics will depend on the choice of target: (i) gas cell or gas jet for the injector; and (ii) type of waveguide for the accelerator. A detailed implementation will depend on the compatibility with other components such as coupling and removal devices for the high-power laser beam, electron diagnostics or coupling, and transport components.
# 21 Electron Beam Diagnostics

## 21.1 Introduction

A plasma accelerator sets new and interesting challenges for every device, including diagnostic tools. While our investigation of electron beam diagnostics found reasonable solutions with state-of-the-art devices, we also focus on ongoing R&D to develop new diagnostics tailored to plasma accelerators.

The EuPRAXIA machine will produce electron beams with high brightness, accelerated by one or several plasma-accelerator stages. The inherent instability of plasma acceleration, observed in experiments in the past, lends itself to single-shot measurements. Additionally, diagnostics must measure beam transverse dimensions of the order of µm (RMS) and bunch lengths of the order of a few fs (FWHM), ideally with single-shot devices. This is not trivial and exceeds, in certain cases, the state of the art.

Different schemes can be foreseen to inject electron beams into the plasma booster stage, some including an RF accelerator. However, even the performance of this RF-based machine will be far from conventional, as the high accelerating gradient, the need to operate at very high frequencies (in the X-band range), requests to control the beam trajectory position / transverse size at the µm level and bunch lengths at the fs level set a very high standard for the diagnostics, even if multi-shot measurements were foreseen.

Finally, in a compact machine, all the components must be compact as well to avoid wasting space and increasing the overall machine length. This is reflected in a complete redesign and re-engineering of many existing devices, as described in this chapter.

## 21.2 Charge and Trajectory Measurements

One of the first questions in the commissioning of a new accelerator is the number of particles in the machine, i.e. the beam intensity. This fundamental property, which can be obtained by measuring the beam current (charge), is important for the beam transmission and for avoiding beam losses. For laser wakefield accelerators and particle wakefield accelerators, charges of a few pC have to be measured. Many technologies exist to measure such values.

Integrated current transformers (ICTs, [534]) measure the magnetic field of the beam using magnetic induction to effectively detect intensity variations. The measurement is linear, perfectly adapted for low charges and in line with the single-shot requirement. Additionally, charge can be determined in absolute terms. However, many studies reported that ICTs can overestimate the beam charge compared to a measurement based on imaging plates and scintillating screens [535,536]. The source of this discrepancy is mainly attributed to the electromagnetic pulse (EMP) from the laser-plasma interaction. However, a more recent investigation [537] presents a good correlation between an ICT and a Lanex screen. Authors explained this agreement with their particular focus on separating the EMP from the laser-plasma interaction, direct particle/radiation hits on the ICT, and low-energy electrons. A new generation of ICTs called Turbo-ICT, developed by Bergoz, can be a good choice to detect low charges (few pC) in a single bunch mode.

Faraday cups (FC) are another charge diagnostic. They use an absorber block (usually made of copper) to absorb the beam. The block is connected to the ground through an ammeter to measure the total charge deposited by the beam. Unlike ICTs, they intercept the beam and thus can only be installed at the end of the beamline. Especially in the commissioning phase, a Faraday cup will be very useful.
Beam alignment and intensity measurements are essential operations for parti-
cle accelerators and can be carried out with a beam position monitor (BPM) as a
non-destructive element. For LWFA, because of beam shape fluctuations and beam
instabilities, those devices must allow an extended detection area and be as insensitive
as possible to the bunch profile. Several LWFA and PWFA accelerator experiments
have used stripline BPMs [538]. These can offer a good resolution of tens of µm with
a charge of a few pC. However, those devices are not very compact to be inserted
into a plasma-accelerator beamline, where an increase by every centimetre leads to a
growth of the machine length and a decrease in the effective accelerating gradient.

For LWFA and PWFA machines, a resolution of a few µm with a few pC is
mandatory, and cavity BPMs show a better resolution than their stripline and button
counterparts. For this reason, the choice of cavity BPM seems to be an interesting
option for the beam position measurement. In the EuPRAXIA project, a re-entrant
cavity BPM [539] can be a good choice in particular for the beam orbit measurement
because of its mechanical simplicity and excellent resolution.

21.3 Transverse Size Measurement

Transverse diagnostics can perform two types of measurements: emittance and envelope. The envelope is very important to properly match the beam along the machine comparing the measured dimensions with the simulated ones. Usually, scintillator screens, like YAG:Ce or optical transition radiation (OTR) monitors, are in use for such a task. The LWFA community is also using Kodak Lanex fast and Gd2O2S:Tb screens [537]. YAG screens in particular are a must, when the beam charge is below a few tens of pC due to their better photon yield. OTR is a prompt surface effect proportional to the number of charges, while scintillator light is emitted isotropically, and the process involves the material bulk. In the latter case, there is a limit on the charge amount beyond which the emission is not linear anymore.

We plan to use scintillator screens, where the resolution and the saturation are not
an issue. OTR screens will be employed to produce radiation useful for longitudinal
diagnostics and wherever high-resolution conditions demand a different choice of
screen other than scintillator-based ones.

## 21.4 Longitudinal Diagnostics

A bunch of electrons can only be successfully accelerated without a large energy
spread by a plasma if the bunch length is much shorter than the plasma wavelength.
Additionally, particular longitudinal shaping can be used in order to increase the
transformer ratio in a beam-driven plasma accelerator. There are several techniques
that can be fruitful to thus measure the bunch length and profile. Some of them are
single shot, while others are not intercepting. Usually, a certain redundancy is needed
in every machine; hence, it makes sense to use several different techniques.

Transverse Deflection Structures

Transverse deflection structures (TDS), often called RF deflectors, are RF cavities
providing a time-dependent transverse force. The longitudinal distribution of an elec-
tron beam injected into such a structure is mapped onto the transverse dimension
thanks to the transversely deflecting field. The transverse distribution of the beam
can be measured using a screen placed downstream of the cavity. This device is
**Table 21.1.** Electron bunch parameters at different diagnostic points (DP1 to DP3) along the beamline, as defined in the main text. When the properties in x-y are different, the average value is given.

<table><thead><tr><th>Parameter</th><th>DP1</th><th>DP2</th><th>DP3</th></tr></thead><tbody><tr><td>Charge [pC]</td><td>30</td><td>30</td><td>30</td></tr><tr><td>Normalised RMS emittance [µm]</td><td>0.44</td><td>0.46</td><td>0.5</td></tr><tr><td>RMS bunch length [fs]</td><td>10</td><td>7.5</td><td>16</td></tr><tr><td>Beam energy [MeV]</td><td>100</td><td>240</td><td>550</td></tr><tr><td>σ<sub>x,y</sub> [mm]</td><td>0.125</td><td>0.100</td><td>0.06</td></tr><tr><td>β<sub>x,y</sub> [m]</td><td>7</td><td>10</td><td>8</td></tr></tbody></table>

used for longitudinal diagnostics, such as bunch length measurements, longitudinal charge profile measurements, longitudinal phase-space measurements (in combination with a dipole spectrometer), slice emittance, and slice transverse phase-space reconstruction measurements (in combination with quadrupole scan techniques or a multi-screen lattice). The slice parameters that can be retrieved are relative to the plane perpendicular to the streaking direction of the RF field.

In Table 21.1, we report illustrative beam parameters at different energies obtained for some of the working points analysed in the design of the RF injector:

- DP1 (Diagnostic Point 1) is taken at the exit of the SPARC-like photo-injector (Sect. 14.3).

- DP2 (Diagnostic Point 2) is taken at the exit of the ARES-like photo-injector, considered for the hybrid compression scheme (Sect. 14.4).

- DP3 (Diagnostic Point 3) is taken at the exit of the X-band accelerator designed after the SPARC-like photo-injector (Sect. 12.3).

Those working points are not exhaustive but give an overview of typical electron beam parameters that we can expect at different locations of the accelerator considering the different RF designs that are discussed within EuPRAXIA. More challenging transverse parameters are expected at the plasma input, where a spot size on the order of a few µm RMS is required.

X-band TDS cavities are reliable tools that are currently being operated in many facilities worldwide [540,541] and have been proven to be capable of achieving sub-fs temporal resolution. In FEL facilities, RF deflectors can be placed after the undulators and thus allow for the reconstruction of the X-ray pulse profile [542,543]. Moreover, measurements of the longitudinal phase space of the electron bunch driver and witness in plasma wakefield accelerators have been planned in projects, such as FLASHForward [544] and SINBAD-ARES [317] at DESY. Such a use of this diagnostic could also be of potential interest for EuPRAXIA.

More recently, a new project [545] has been born thanks to the collaboration of three institutes – namely, DESY, CERN, and PSI – for the development of a novel type of TDS called PolariX TDS, having the potential of changing the streaking direction of the field [546]. The prototype of this novel cavity design is currently being characterised and tested [547] and, if successful, would open up opportunities for a more complete characterisation of the phase space of the beam. One possible application of such a technology would, for example, be the multi-shot reconstruction of the 3D charge density distribution of an electron bunch, which would be of extreme interest as a diagnostic tool for visualising the properties of the electron beam before the injection into the plasma [548,549]. This technology fits very well with a plasma-accelerator design, and we plan to use such a device in our project.
## COTR-Based Methods

The ability to produce ultra-short electron bunches in plasma acceleration opens yet another possibility of using coherent transition radiation (COTR) spectroscopy techniques in the infrared or even optical wavelength range, where well-established spectrum characterisation methods exist. Such schemes have already been demonstrated, even as single-shot diagnostics, in [550,551], where the CTR spectrum was directly measured using a far-infrared spectrometer based on a dispersive prism, or in [552], where a cascaded grating setup was used.

Another method based on coherent radiation uses Smith-Purcell radiation. This type of radiation occurs when a relativistic beam of charged particles passes near a metallic diffraction grating [553]. The phenomenon is described in detail in [554]. When the bunch is short enough, the radiation emitted will be coherent, and like in similar phenomena, it will encode the electron bunch length in the radiation pulse. Several experimental demonstrations have been performed [555–558]. Beams of a few hundred femtoseconds in duration have been characterised, and it is believed that shorter beams could be measured if suitable test facilities were found.

## 21.5 Time-of-Arrival Monitors

Measuring the beam arrival time is of great importance, especially in a facility that wants to drive an FEL, because it can supply crucial information for the beam-based feedback system. The best result in this field (with a resolution of a few fs) is achieved with beam-arrival monitors (BAMs) developed at DESY for XFEL; we refer mainly to this system reported in [559]. The BAM system is composed of three parts: the RF unit, the electro-optical unit (EOM), and the data acquisition (DAQ) system. In a BAM, the electromagnetic field induced by an electron bunch is captured by four broadband pickups. The electro-optical unit then combines the signals from the RF unit and a reference signal provided by an external source to perform the measurement. The result is finally stored in the DAQ system.

Another possible diagnostic tool for measuring the time of arrival is electro-optical sampling (EOS), a technique widely used to monitor electron bunch length [560–562]. It relies on the modification of the refractive index of a crystal, placed very close to the beamline, caused by the strong electric field co-propagating with the electron beam. The polarisation of a probe laser is rotated by the non-linearity induced in this crystal. Orthogonal polarisers are then placed before and after the crystal, so the probe laser signal is recorded only when there is a polarisation rotation, i.e. when there is an interaction with the beam. A decoding scheme transforms the measured polarisation rotation into the bunch profile.

By using as a probe laser the same laser source that arrives on the photo-cathode and produces the electron emission or an optically linked oscillator, there is a natural self-synchronisation between the laser and the electrons. In this way, the fluctuations in time, that are always measured in the readout are linked only to the beam jitter, and they are a direct measurement of the beam arrival time. A resolution of a few fs has already been demonstrated [563], highlighting the great feasibility of this monitor in measuring the time of arrival in different machine configurations and helping in optimising jitters and synchronisation.

## 21.6 Beam Loss Monitors

Beam loss monitors are very important in plasma acceleration. In the commissioning phase, for instance, the beam can easily be lost, and better knowledge of its fate
is of paramount importance to properly tune the plasma parameters. In brief, the
principle of optical beam loss monitors (oBLMs) depends upon on-line measurements,
taken along the whole beamline by optical fibres laid on or near the beam pipe. The
detection of fast charged particles lost from the beam is by Cherenkov light emitted
when an electron traverses a medium at speeds greater than that of light in the
medium. Photons emitted within the critical angle will be totally internally reflected
down the fibre and so are able to reach the end, where they are detected by a silicon
photo-multiplier (SiPM) device.

## 21.7 Transverse Emittance Monitors

In a linac, there are several methods to measure the beam emittance, the main one
being multiple screen measurements with a quadrupole scan [564]. This method is the
most commonly used worldwide. Unfortunately, it is a multi-shot method, whereas
we are more interested in single-shot measurements in the context of EuPRAXIA.
While in the operational phase of the machine, a multi-shot technique may be feasible,
during the commissioning of a plasma accelerator, single-shot techniques are needed.
However, so far, there is a lack of this kind of technique for emittance measurements.
To fill this gap, the EuPRAXIA collaboration is developing and investigating pos-
sible new instrumentation methods, brief descriptions of which are presented in the
following.

### High-Energy Pepper Pot

The low-energy limitation of the pepper-pot technique comes from the penetration
depth of high-energy particles in matter. In recent years, it has been proposed to use
pepper pots at several hundreds of MeV in energy [565]. Several experiments have
confirmed the validity of this approach at energies up to 3 GeV [566], and higher
energies could be considered with a suitable test beam. Of course, an accurate analysis
of the angular acceptance of the device must be carried out depending on the beam
energy and transverse beam parameters. It should be noted that thick masks can cut
the phase space, resulting in an underestimation of the beam emittance.

### Multiscreen OTR Measurement

The beam transverse emittance can also be measured by inserting quasi-non-destructive screens at several locations. These screens can measure the beam envelope at different positions in a single shot, from which the transverse emittance can be deducted. The method has been demonstrated experimentally [567] at 3 GeV, and there is no upper limitation on the energy that could be reached. On the other hand, for lower energies, multiple scattering will be a limiting factor. Also, in order to have enough betatron phase advance for the measurement, a large distance between screens is needed.

### Multiscreen OTR Measurement with Active Plasma Lens

The simultaneous measurement of beam size on different screens in the same shot, described in the section above, is very attractive. However, as with all the other diagnostics in a plasma accelerator, it must be very compact, occupying the smallest possible space in the beamline. Conventional quadrupoles have focal lengths on the order of metres and this results in screens placed at roughly this distance. Plasma
lenses [38,305] have regained significant interest in recent years because they are able to focus in both planes at the same time with a strength at least one order of magnitude better than permanent quadrupoles. A recent study [568] demonstrated the possibility to use a plasma lens without appreciable emittance dilution. Additionally, the focal length of these devices can be tuned with the current and can be on the order of a few centimetres. This fact paves the way to replace conventional quadrupoles with plasma lenses in multi-screen measurements.

The main advantage of this setup is the extreme compactness. The screens can be very thin (about 1 µm), and they can be used in the same shot, realising a single-shot emittance measurement. However, further investigations are needed to better validate the working principle of plasma lenses, to study their chromatic aberration – which will, in any case, be less important than in a conventional quadrupole – and side effects, such as beam misalignment, plasma density fluctuations, and so on.

## Permanent Quad Scan

In a conventional quadscan, the current of the quadrupoles is varied. In a multi-screen setup, on the other hand, the position of the measurement is varied by using different screens. If we consider a beam with high energy spread (more than 1%) and we use permanent quadrupoles with a dipole, we can focus the beam in different transverse positions after the dipole, corresponding to different energies and different parts of the beam. With the beam dimension depending on the quads’ focal length and this focal length depending on the beam energy, we find that particles at different energies are focussed with different spot sizes. In this way, we do not need to change the focal length of the quads or the position of the measurement, but the inherent energy spread already produces different beam sizes at the same measurement position.

This method was first introduced a few years ago [569] and later successfully implemented by several groups [570,571]. We consider using this method in the commissioning phase of the plasma injector because in this scenario, with an energy spread likely on the order of 1% or worse, this system can offer a single-shot emittance measurement. As the technique is already implemented by several other groups, we can also benefit from their experience. However, with the goal of the whole project being the production of a high-quality electron beam with reduced energy spread with respect to these mentioned values, we do not plan to reserve room in the final accelerator for this scheme.

## Measurements Using Cherenkov Radiation

Recently, a breakthrough paper has been published [572] about the use of Cherenkov radiation as a non-intercepting monitor for beam size measurements. A high-purity fused silica slab is placed along the beamline at a mm-range distance to the electron beam. The electromagnetic field, co-propagating with the electron bunch, interacts with the crystal and produces Cherenkov radiation. By observing the crystal from above, it is possible to see a bright line, the thickness of which corresponds to the electron beam size.

The ultimate resolution of this system is still unknown because so far, there exists no analytic expression for the spread point function yet. A beam size of about 70 µm has been measured and positively cross-checked with other methods. One of the crucial points to further investigate is a possible dependence between resolution and beam energy.
Measurements Using Betatron Radiation

Measurements inside the plasma cell can be performed by making use of the beta-
tron radiation generated during the acceleration process (see Sect. 24.2 for more
details on betatron radiation). Diagnostics based on betatron radiation [573] have
been developed in recent years in several laboratories, relying on the measurement of
the radiation spectrum (see for instance [574]) or on the diffraction of the radiation
from a knife edge [575].

However, these systems were able to measure just the beam profile and divergence,
neglecting the correlation term. Only recently, a new algorithm has been developed
to retrieve the correlation term [576]. Using the simultaneous measurement of the
electron and radiation energy spectrum together with the plasma density, it is possible
to reconstruct the whole phase space.

To collect the betatron radiation, we have to, sooner or later, separate the radia-
tion from the electron beam with a dipole. Unfortunately, the bending of the beam
produces synchrotron radiation, and its spectrum can overlap with the betatron pho-
tons. Usually, the betatron radiation is much stronger than the synchrotron radiation
though, and there is an additional degree of freedom from the bending angle. The
possibility to increase the magnetic field also allows to move the peak of the syn-
chrotron radiation to a higher frequency, resulting in a better separation from the
betatron radiation. However, an open problem remains with the separation between
the betatron radiation coming from the witness beam and that from the driver in a
beam-driven plasma acceleration scheme. In this case, the driver contains much more
charge with respect to the witness, and so only a clear energy separation of the two
spectra can solve the problem. In the case of laser-driven plasma acceleration, this
problem obviously disappears.

21.8 Accelerator Diagnostics Layout Before the Plasma-Accelerator Stage
(Plasma Booster)

Laser Plasma Injector (LPI, relevant for Scheme 2: LPI-150 MeV + LPAS-5 GeV)

Before the installation of the plasma booster, temporary diagnostics will be placed
after the LPI, as seen in Figure 21.1.

A driver exclusion and removal zone is foreseen to almost completely remove
the laser and to protect the following instrumentation. The length and the schemes
employed in this removal section are outside the scope of this chapter and will be
defined as part of the technical design.

The beamline is schematically shown in Figure 21.1. A triplet of permanent quads
will follow the exclusion region. They can be used both to capture the beam for the
following diagnostics and to allow for a single-shot emittance measurement following
the scheme in Section 21.7 by using the downstream dipole and the screen on the
dogleg.

After the triplet, a view screen will be placed, including the possibility to host an
EOS station. EOS will be used as a time-of-arrival monitor, while the view screen
allows the measurement of the beam envelope. A couple of cavity BPMs are needed
to properly measure the beam trajectory, while a Faraday cup at the end of the line
will give us a measurement of the total charge. Two electromagnetic quadrupoles will
be used to properly adjust the optics, providing also the possibility of a multi-shot
measurement with a quadrupole scan. A pepper-pot station will be implemented as
well to have the possibility to perform single-shot high-energy emittance measure-
ments. The bunch length can be evaluated in the range down to 50 fs by means of
<figure><img src="image_164.png" /><figcaption>Fig. 21.1. Temporary beamline for the electron diagnostics after the LPI.</figcaption></figure>

EOS, while for shorter bunches, a longitudinal single-shot coherent radiation-based measurement setup or a Smith-Purcell device can be located in the chamber placed 2.5 m behind the beginning of the beamline. The overall length of this line is not a critical point because it will be removed later. However, the need to have several different diagnostics to cover a large range of beam parameters as well as the requirement to easily move the setup downstream later prevents us from building a longer beamline.

Eventually, a second stage for plasma acceleration will be placed behind the plasma injector. In the space between the LPI and the LPAS and after the laser mirror, a dedicated and permanent beamline will be placed. This line must contain not only the diagnostics setup but also the permanent capture optics for the electron beam. A conceptual layout is thus shown in Figure 21.2.

The triplet of permanent quads shown in Figure 21.2 can be one of the two already foreseen for the capture optics. However, the distance between two quads must, in this case, be increased to at least 1 m, which should not be a critical issue as the gradient of the permanent quadrupole design can be adjusted accordingly. Even further in the future, when active plasma lenses [38,305] will be routinely used in operational machines and even higher gradients will be available, longer spacing between magnetic elements should not be a critical issue anymore. It should be noted that in our layout, the driver removal zone is considered prohibited for the diagnostics. Behind this region, a couple of cavity BPMs will track the orbits, while an ICT will measure the charge bunch by bunch. A permanent but removable dipole can be inserted to drive the beam into a spectrometer arm to measure its energy and energy spread. Furthermore, view screens can be used to image the beam and check the envelope for proper beam matching in the section following the diagnostics.
<figure><img src="image_165.png" /><figcaption>Fig. 21.2. Layout of the permanent beamline for electron diagnostics between the LPI and LPAS.</figcaption></figure>

In one of the dedicated positions, a bunch length measurement setup by means of COTR will also be placed. A beam loss monitor will be located around the beamline to measure beam losses and help in aligning the beam.

### RF Injector (relevant for Schemes 3 to 5)

The RF injector, even if it will be built using emerging technologies like X-band structures, is a conventional machine from the point of view of diagnostics. However, there are two main issues: orbit control and 6D phase-space control at the plasma injection point, where the bunch size will be on the µm level and the bunch length in the few fs range. We consider two possible approaches to the RF injector using the designs developed in two projects of the collaboration partners INFN and DESY (EuPRAXIA@SPARC_LAB and SINBAD-ARES, respectively).

In EuPRAXIA@SPARC_LAB, the linac is realised with an S-band injector, followed by a X-band linac, equivalent to the proposed design in Chapter 12. The conceptual layout of the injector diagnostics is shown in Figure 21.5.

Stripline or button BPMs are inserted before and after every S-band structure, while an ICT will measure the bunch charge. The layout of part of the linac is shown in Figure 21.6.

The rest of the linac, about a factor of 2 longer, will be a repetition of this structure. After a section with a TDS and a dipole for single-shot longitudinal phase-space
<figure><img src="image_166.png" /><figcaption>Fig. 21.3. Layout of the diagnostics in the electron gun section of SINBAD-ARES.</figcaption></figure>

<figure><img src="image_167.png" /><figcaption>Fig. 21.4. Layout of the diagnostics in the linac section of ARES.</figcaption></figure>

measurements, there are cavity BPMs (CBPMs) placed after every two accelerating structures and view screens after every four. High-resolution OTR monitors will be implemented before the plasma channel. Critical measurements, like emittance or bunch length, will be carried out with diagnostics placed after the plasma, while turning the plasma off. They will thus be discussed in the next chapter.

The gun area in the ARES-like linac (equivalent to the described design in Sect. 14.4) is shown in Figure 21.3.

The diagnostics setup in this region is standard with BPMs, view screens, a pepper pot for emittance measurements, a spectrometer, a BAM monitor, and a Faraday cup, with the latter also useful as a beam dump.

In the linac section, a 1.36 m long empty space is available between every two accelerating modules. A view screen and a BPM can be placed in these positions. Moreover, one toroid for charge measurement can be placed additionally between the second and third RF cavities of the linac.

The linac is followed by a matching and collimation section to prepare the beam for entering the plasma-accelerator stage. Between these two sections, a spectrometer
<figure><img src="image_168.png" /><figcaption>Fig. 21.5. Layout of the diagnostics for the RF injector at EuPRAXIA@SPARC_LAB.</figcaption></figure>

is placed in the following configuration: a dipole is followed by one BPM, two view screens, and a toroid for charge measurement. The presence of a TDS in the matching section can further allow the measurement of the full longitudinal phase space in single-shot mode. In the matching section, a toroid, two BPMs, a beam arrival time monitor, and a view screen will be placed. Additionally, it is foreseen to include a view screen and a BPM in the middle part of the magnetic bunch compressor. After the bunch compressor, if allowed by space constraints, we foresee one additional BPM and / or a view screen as well as a beam arrival time cavity.

## 21.9 Accelerator Diagnostics Layout After the Plasma-Accelerator Stage
(LPAS / PPAS, relevant for Schemes 1 to 6)

After the plasma booster and the driver removal, a diagnostics section is needed to fully characterise the 6D beam phase space and to properly match the beam to the undulator or other applications. As we pointed out previously, there exist so far some examples of single-shot longitudinal phase-space measurements that can be considered almost state of the art. However, only few experiments were already carried out on the topic of single-shot transverse emittance measurements. Hence, in this beamline, we consider the use of not only novel techniques under development but also well-established, multi-shot approaches to the emittance measurement. We will use only betatron radiation to characterise the emittance at the plasma exit, while we plan to measure the emittance and the Twiss parameters of the beam before its entrance into the undulator. This choice arises from the need to capture the beam emerging from the plasma as soon as possible, thus avoiding emittance dilution [577].

One of the fundamental requirements of the diagnostics layout must be the compatibility with the capture optics. A couple of permanent quadrupoles will be followed by several electromagnetic quadrupoles. To avoid wasting space and to keep the machine as compact as possible, we tailored our diagnostics following this layout, as can be seen in the diagnostics sketch in Figure 21.8.

A beam loss monitor will be placed around the system along the full length of this section to measure any beam losses. An ICT will check the charge of the bunch
<figure><img src="image_169.png" /><figcaption>Fig. 21.6. Layout of the diagnostics for a section of the linac at EuPRAXIA@SPARC_LAB.</figcaption></figure>
<figure><img src="image_170.png" /><figcaption>Fig. 21.7. Layout of the diagnostics in the beam-matching section of SINBAD-ARES.</figcaption></figure>
<figure><img src="image_171.png" /><figcaption>Fig. 21.8. Layout of the matching optics and diagnostics setup behind the plasma booster.</figcaption></figure>

emerging from the plasma interaction. Along the beamline, there are also view screens and cavity BPMs foreseen. Additionally, an X-band TDS can be placed here during the commissioning and later, during operation, moved behind the undulator chain. This setup is followed by a spectrometer to measure the energy, energy spread, and longitudinal phase space. A separate station to measure the longitudinal profile also in parasitic mode will be placed after the dipole in the main line. To collect and analyse betatron radiation to measure the transverse phase space at the plasma exit, we need to place a chamber with an X-ray camera and a proper set of filters and attenuators into the beamline. This chamber should be transversally very wide so that the camera can be moved onto the beamline only for this kind of phase-space measurement and only with the dipole in operation. Until the new methods will be ready to be implemented, it is planned to measure the emittance using a multi-shot method with a quadrupole scan (where one of the available quads in the line can be employed).

## 21.10 Conclusion

We have made an extensive survey of the tools available to measure the beams to be produced at EuPRAXIA. We identified diagnostics to measure the bunch charge, the bunch position, the transverse envelope, the transverse emittance, and the longitudinal profile. We also identified means of measuring the bunch arrival time and beam losses along the accelerator. Following this survey, we concluded that the existing diagnostics are sufficient for basic beam measurements. However we also found that for some advanced measurements, especially single shot (e.g. transverse emittance and longitudinal profile), significant additional R&D is still needed.

We also investigated how these diagnostics could be used at EuPRAXIA, and we produced conceptual designs of the main beamlines. This required iterative work with the beam transport group, and in most cases, we succeeded in finding a suitable layout. In some cases, it was found that the design of individual diagnostics should be made more compact, a task to be completed as part of the technical design.

Overall, at this conceptual stage, we find that from a diagnostics point of view, we are ready to build EuPRAXIA, although additional research on some diagnostics in the future will allow better performances.
<table><caption>Table 22.1. Parameters of the electron beam hitting the beam dump.</caption><thead><tr><th></th><th></th><th>EuPRAXIA</th><th>Beam Dump<br>Beam Dump</th><th>Beam Dump<br>Beam Dump</th></tr></thead><tbody><tr><td>Nominal electron energy [GeV]</td><td>E<sub>0</sub></td><td>1–5</td><td>5.8</td><td>3.4</td></tr><tr><td>Nominal charge per bunch [pC]</td><td>N<sub>e</sub></td><td>30</td><td>200</td><td>200</td></tr><tr><td>Repetition rate [Hz]</td><td></td><td>100</td><td>100</td><td>100</td></tr><tr><td>Maximum no. of bunches per RF pulse</td><td></td><td>1</td><td>1</td><td>1</td></tr><tr><td>Average beam power [W]</td><td>P<sub>ave</sub></td><td>3–15</td><td>116</td><td>68</td></tr></tbody></table>

## 22 Electron Beam Dump

### 22.1 Introduction

The electron beam dump is one of the most important components for any particle accelerator facility to absorb or dispose beam kinetic energy in a safe way. Expertise from similar existing machines can be used when looking at a conventional design of the electron beam dump [158] with the basic concept and functions of a beam dump summarised as follows [578]:

- to disperse the beam by magnets in front of the beam entry, i.e. decreasing the power density by means of beam defocussing / spreading;

- to stop the beam part by part in a spatially separated block, i.e. reducing the power deposition in small regions of the beam dump, for example by designing the absorber with a decreasing transverse cross-section diameter to deposit the beam energy in subsequent steps; and

- to minimise the induction of radioactivity, i.e. eliminating the use of water to stop the beam. The electron beam parameters to be taken into account are reported in Table 22.1, together with values related to the SwissFEL beam dumps. With similar requirements, these can be taken as examples, although overestimated, for the EuPRAXIA case, thus showing clearly that conventional solutions for the beam dump design exist. While first estimates and considerations are discussed below, the development of a specific design of beam dumps for EuPRAXIA will be carried out as part of the technical machine design.

Besides the conventional systems, an innovative proposal based on a plasma beam dump is also reported here. This will be considered as a complementary option to conventional solutions and could possibly, depending on further prototyping and test experiments, be implemented as a future upgrade of the facility.

### 22.2 The SwissFEL Beam Dumps, Aramis and Athos

At the end of each undulator line, a permanent beam dump is foreseen by means of which the beam is deflected for a few degrees towards the ground and stopped in a copper block. To shield against produced secondary radiation, the core block is surrounded by layers of iron and heavy concrete. The setup scheme adapted from the SwissFEL CDR [158] is shown in Figure 22.1.

According to the SwissFEL operation modes, the full electron beam is dumped in two beam dumps. During the beam tuning, a smaller fraction of the beam is stopped in the beam stopper to protect the undulator magnets. There is also one stopper for each beamline. For a conservative estimate of the residual activation, the beam energy has been set to 7 GeV and the beam power to 560 W for the beam dump and 14 W for the beam stopper (Aramis line case). Like at SLS, a 30 % downtime of the
<figure><img src="image_172.png" /><figcaption>**Fig. 22.1.** SwissFEL Aramis beam stopper in front of the first undulator (left). A dipole magnet is inserted in the beamline to deflect the electrons on a copper target surrounded by shielding materials (right) (image credits: [158]).</figcaption></figure>

facility has been assumed, which is used as the downtime for the beam dump. The calculation of the residual dose rates has been carried out with FLUKA based on the design of the beam dump and stopper. The layout used for the beam dump consists of a cylindrical carbon core surrounded by iron. The outer layer consists of normal concrete. At the very front, a small layer of PE suppresses low-energetic neutrons to protect electronic devices for diagnostics. From the calculations, the dose rates after 1 day on the outer sides of the beam dump are of the order of a few µSv h⁻¹ and therefore acceptable for a controlled zone. The beam stopper has a copper core of 20 cm thickness followed by 10 cm of tungsten, which can be moved in and out of the beam. It is surrounded by steel and finally normal concrete of about half a metre on each side. With this design, the residual dose rates should be low after 1 h of beam-off.

As a corollary, the nuclide inventory also needs to be calculated for every material block in view of the disposal after the operation shutdown to estimate the time to wait before dismantling the beam dump and the beam stopper. The residual activation and resulting dose rates should also be considered for the safety requirements during deconstruction.

## 22.3 The Plasma Beam Dump

Since a conventional beam dump can affect the overall compactness of a machine, such as EuPRAXIA, aiming for a small footprint, an alternative design is proposed with the implementation of a plasma target to absorb the kinetic energy of the EuPRAXIA electron beams. Such a setup would improve the overall compactness of the facility and generate less radioactivity. In this section, some simulation results of a plasma beam dump based on EuPRAXIA beam parameters are briefly described.

The development of laser-plasma-based compact high-quality electron accelerators have already attracted tremendous interest worldwide. The initial idea was proposed by Tajima and Dawson 40 years ago [4]. The basic principle behind this is to inject a beam of electrons behind the driver pulse to achieve an extremely high acceleration gradient, usually three orders of magnitude higher than the field in conventional accelerators. Nowadays, an electron beam of several GeV energy can be routinely generated in laser wakefield accelerators within centimetre-long plasma targets by using petawatt laser drivers [9,290].

On the other hand, the use of plasma wakefields for the compact deceleration of relativistic beams has not been fully explored yet. In 2010, Wu et al. proposed
the collective deceleration of electron beams in plasma for the first time [579]. The idea is to utilise the large decelerating wakefields, with amplitudes as high as those of the accelerating fields, to absorb the beam energy as fast as possible. This would allow beam deceleration to be achieved in a short distance if compared to equivalent conventional beam dumps. Moreover, it could help to mitigate the requirements of conventional beam dumps, which often suffer from complicated designs and large sizes (and costs), when the beam power is high. Moreover, the use of a low-density plasma greatly reduces radio-activation hazards if compared to conventional beam dumps, in which energetic particles interact with dense media, such as metals, graphite, or water, causing nuclear reactions and the production of secondary particles.

### 22.3.1 Simulation Study of a Plasma Beam Dump

Generally speaking, there are two types of plasma beam dumps: the so-called passive plasma beam dump (PPBD) and the active plasma beam dump (APBD) [580]. For the PPBD, a relativistic particle bunch propagates in a uniform plasma and excites its own wakefield. As a consequence, the head of the bunch will experience no decelerating field due to the finite response time of the plasma, while particles closer to the bunch tail will experience a decelerating field. After some time, the fraction of the bunch experiencing the maximum decelerating field will become non-relativistic, and it will fall behind the rest of the bunch until it reaches an accelerating phase of the wakefield. This causes beam re-acceleration, which leads to the saturation of the beam net energy loss [581]. On the other hand, in the APBD, this drawback is eliminated in the following way. A laser pulse is employed to excite a wakefield in the plasma prior to beam propagation in such a way that the combination of both laser-driven and beam-driven wakefields flattens the decelerating field along the bunch. This enables a uniform energy extraction, thus preventing the formation of re-acceleration peaks. Although the energy extraction is more efficient in the APBD, the need for a laser pulse and the precise synchronisation required between this laser and the beam causes this scheme to be far more complex to be experimentally implemented than the PPBD.

### 22.3.2 Passive Beam Dump for EuPRAXIA

To simplify the design, we propose to use a passive beam dump system for the EuPRAXIA facility. We aim to absorb most of the energy from most of the particles in the bunch by tailoring the plasma density profile. Typical EuPRAXIA beam parameters were used in our studies, as follows [582]: beam energy of 1 GeV, bunch charge of 30 pC, transverse beam size of 1.4 µm, longitudinal bunch length of 2 µm, beam energy spread of 1%, and angular divergence of $1 \times 10^{-5}$. This corresponds to a beam density of $3 \times 10^{18} \text{ cm}^{-3}$. The 2D particle-in-cell code EPOCH is used to perform simulations of beam-plasma interaction [583]. As a first step, we choose the plasma density of $4.4 \times 10^{17} \text{ cm}^{-3}$ so that the wakefield excited is in the quasi-linear to non-linear regime. The results show that the particles lose their energy very fast. After about 8 cm in plasma, the particles at the tail of the bunch lose most of their energy and reach energy loss saturation. As a result, the bunch length increases during the energy dump. If the bunch continues to propagate further in the plasma, the particles at the tail of the bunch will reach the acceleration phase of the wakefield and start to absorb energy again.

Figure 22.2 shows the beam longitudinal phase space after propagating 8.5 cm in plasma. It can be seen clearly that the beam energy at the head of the bunch does not
<figure><img src="image_173.png" /><figcaption>Fig. 22.2. Beam longitudinal phase space after 8.5 cm in plasma.</figcaption></figure>

change, while the particles at the tail start to gain energy. To eliminate the particles gaining energy from the wakefield, the plasma density is tuned just before energy loss saturation occurs. In doing so, the low-energy particles at the bunch tail will move to the defocussing phase of the wakefield, being ejected by the transverse wakefield. Figure 22.3 shows a typical plasma density profile in which the density starts to increase quadratically at about 7.5 cm and reaches a value 10 times higher than the initial density of $n_0 = 4.4 \times 10^{17} \text{ cm}^{-3}$. As a comparison, the beam longitudinal phase space after 21 cm in the plasma is shown in Figure 22.4. It is found that the re-acceleration peak (as shown in Fig. 22.2) is removed and particles continue to lose their energy in the plasma.

Figure 22.5 provides the beam energy as a function of propagation distance in the plasma for the plasma density profile shown in Figure 22.3. The results demonstrate that the total beam energy is reduced to 12% of its initial energy. Almost 80% of the beam’s initial energy is deposited in the plasma, and only approximately 10% of the initial beam energy is transversely ejected. This plasma density tailoring scheme will guarantee a relatively low beam energy deposited in the plasma vessel, ensuring a safe operation of the plasma beam dump.

## 22.4 Conclusions

The study shown here demonstrates the viability of the PPBD for the EuPRAXIA 1 GeV beam. It shows that a 21 cm plasma with a density ramp can absorb most of the total beam energy (≈80%) through the passive beam dump scheme. The PPBD
<figure><img src="image_174.png" /><figcaption>Fig. 22.3. Density profile used to eliminate the re-acceleration of particles in the bunch tail.</figcaption></figure>

<figure><img src="image_175.png" /><figcaption>Fig. 22.4. Beam longitudinal phase space after 21 cm in plasma. The plasma density exhibits quadratic growth starting from $z = 7.5$ cm and reaches a value 10 times higher at $z = 21.0$ cm, compared to initial plasma density.</figcaption></figure>

set up can make the overall facility compact and less radioactive. On the other hand, we have not discussed the APBD scheme here given its complexity. However, if an active beam dump could be used for the EuPRAXIA machine, where a laser pulse can drive the plasma wakefield for beam deceleration, in principle, almost 100% of the beam energy could be absorbed inside the plasma dump. In addition, recent experiments performed at Rutherford Appleton Laboratory (RAL) on multiple laser pulses driving a plasma wakefield have shown the possibility of energy recovery by the trailing laser pulse picking up energy from the plasma [584]. All these possibilities
<figure><img src="image_176.png" /><figcaption>**Fig. 22.5.** Energy as a function of propagation distance in plasma, for the plasma density profile shown in Figure 22.3.</figcaption></figure>

could pave the way for a future very compact and green beam dump facility and will
be investigated further as part of the technical design.

23 Start-to-End Simulations of the Accelerator

**23.1 General Aspects**

The continuous progress in plasma-based acceleration would suggest that the era of
plasma-based accelerators is approaching. However, from plasma acceleration as a
physics experiment towards accelerators as part of a facility delivering beams to user
communities, important challenges remain to be solved. The advanced components
of such a plasma accelerator, in particular the plasma and laser systems, need to be
designed and constructed in a way to ensure reproducibility and reliability beyond the
start of the art. On the user end of the accelerator, the machine must be capable of
providing high beam energy, high charge, and exceptional beam quality for a variety of
applications and user experiments. Table 23.1 summarises the main requirements that
were defined for the EuPRAXIA design based on the needs of foreseen applications
and expected technological developments. They are provided for three different points
along the machine: the electron beam at the exit of the injector, which can be either
(1) a laser-plasma injector or (2) a radiofrequency (RF) injector; and (3) at the
accelerator exit just before the application beamlines. We can see that these beam
characteristics, suitable for a highly demanding application such as a free-electron
laser, are particularly challenging as they require simultaneously a high final beam
energy of 5 GeV (with a commissioning step at 1 GeV), a high charge of 30 pC in a
**Table 23.1.** Main requirements for the electron beam at the exit of the injector, which can be either a laser-plasma (LP) injector or a radiofrequency (RF) injector, and at the accelerator exit, which means at the user applications. E, Q, $\tau$(FWHM), $\sigma_E$/E, $\sigma_{E,s}$/E, $\varepsilon_n$, and $\varepsilon_{n,s}$ stand, respectively, for beam energy, charge, length (full width at half maximum), energy spread, slice energy spread, normalised phase-space emittance, and slice normalised phase-space emittance. Note that slice values are driven by application requirements and are therefore only defined at the accelerator exit.

<table><thead><tr><th>Parameter</th><th>LP Injector Exit</th><th>RF Injector Exit</th><th>Accelerator Exit</th></tr></thead><tbody><tr><td>E</td><td>150 MeV</td><td>250–500 MeV</td><td>5 GeV (1 GeV)</td></tr><tr><td>Q</td><td>30 pC</td><td>30 pC</td><td>30 pC</td></tr><tr><td>τ(FWHM)</td><td>10 fs</td><td>10 fs</td><td>10 fs</td></tr><tr><td>σ<sub>E</sub>/E</td><td>5%</td><td>0.2%</td><td>1%</td></tr><tr><td>σ<sub>E,s</sub>/E</td><td>N/A</td><td>N/A</td><td>0.1%</td></tr><tr><td>ε<sub>n</sub></td><td>1 mm mrad</td><td>1 mm mrad</td><td>1 mm mrad</td></tr><tr><td>ε<sub>n,s</sub></td><td>N/A</td><td>N/A</td><td>1 mm mrad</td></tr></tbody></table>

<figure><img src="image_177.png" /><figcaption>**Fig. 23.1.** The accelerator schemes under investigation. Schemes 1 to 4 are based on laser-driven wakefield acceleration where the laser beam is represented in red. Scheme 5 is based on beam-driven wakefield acceleration. Scheme 6 is a hybrid configuration mixing laser-driven and beam-driven acceleration. RFI stands for radiofrequency injector, LPI for laser-driven plasma injector, PPI for beam-driven plasma injector, LPAS for laser-driven plasma acceleration stage, and PPAS for particle-driven plasma acceleration stage. The energy at the different stages is also given [171].</figcaption></figure>

short length of 10 fs (i.e. producing a high peak current of 3 kA), a low emittance of $\leq 1$ mm mrad, a low energy spread of $\leq 1\%$, and an even lower slice energy spread of $0.1\%$.

In order to meet those challenging requirements, it was decided to explore broadly various plasma injection and acceleration schemes in order to down-select subsequently. This necessary but long procedure is only feasible thanks to the many contributors of the EuPRAXIA collaboration.

The main schemes under investigation are sketched in Figure 23.1, where the final beam energy is also indicated at each stage exit. Schemes 1 to 4 refer to LWFA, laser-driven wakefield acceleration, where the electron beam can be internally injected (Scheme 1) or else externally generated by a laser-plasma injector (Scheme 2) or an RF
**Table 23.2.** Naming convention of the different accelerator schemes investigated as part of the EuPRAXIA conceptual design study. In each case, the energy value after a stage's name indicates the electron beam energy at the end of this stage. Different stages are connected by transport lines which are detailed in Chapter 19. The following abbreviations are used: LPI = laser-plasma injector, RFI = radiofrequency injector, LPAS = laser-driven plasma-accelerator stage, and PPAS = beam-driven plasma-accelerator stage.

<table><thead><tr><td>Scheme no.</td><td>Short descriptive name</td></tr></thead><tbody><tr><td>1</td><td>LPI-5 GeV</td></tr><tr><td>2</td><td>LPI-150 MeV + LPAS-5 GeV</td></tr><tr><td>3</td><td>RFI-500 MeV + LPAS-5 GeV</td></tr><tr><td>4</td><td>RFI-240 MeV + LPAS-2.5 GeV + chicane + LPAS-5 GeV</td></tr><tr><td>5</td><td>RFI-500 MeV + PPAS-1 GeV</td></tr><tr><td>6</td><td>LPAS-3 GeV + PPAS-5 GeV</td></tr></tbody></table>

injector (Schemes 3 and 4). Acceleration to the final beam energy can be achieved in a single laser-plasma accelerator stage (LPAS) (Schemes 2 and 3) or split into two symmetric parts separated by a magnetic chicane for energy dechirping (Scheme 4). The laser considered in all cases is a Ti:Sa laser system operating at 0.8 µm wavelength, described in detail in Chapter 10. Scheme 5 refers to PWFA, particle-driven wakefield acceleration, where this chapter only examines a final beam energy of 1 GeV using a 500 MeV RF injector for external injection; a scenario for an acceleration to 5 GeV is presented in Section 17. Finally, Scheme 6 describes a hybrid configuration where the first LWFA plasma stage generates a 3.5 GeV particle beam to drive the wakefield in the second PWFA plasma stage, accelerating witness electrons to 5 GeV. Table 23.2 gives a short summary of the naming conventions for each scheme.

Further configurations with multiple plasma-accelerator stages were also considered initially. First simulations of the LWFA schemes showed, however, that acceleration to 5 GeV is quite accessible in one plasma-accelerator stage. It is therefore generally not useful to consider schemes with two acceleration stages that will need one more transfer line between the two stages (with the exception of Scheme 4, as described in Sect. 23.10).

Four LWFA schemes, one PWFA scheme, and one hybrid scheme have been studied in detail and are presented here.

In the following, various injection and acceleration techniques, studied for the different schemes of Figure 23.1, are presented. Based on the beam requirements in Table 23.1, the most promising configurations, i.e. combinations of a given scheme and given injection / acceleration techniques, are selected for the next studies. Consequently, theoretical considerations for an optimised electron beam extraction / transport are derived and applied to these selected configurations. Finally, start-to-end simulation results for the most promising machine setups are described, involving injector stages, acceleration stages, and beam transport lines.

## 23.2 Study of Injection Techniques

Two different *RF injectors* have been optimised so as to provide electron beams with 240 or 500 MeV energy and beam parameters meeting the requirements indicated in Table 23.1:

(a) An S-band linac with a hybrid compression scheme combining RF and magnetic compressors is studied. It shows that an electron bunch of 30 pC, 7.5 fs RMS
length, and ~0.5 mm mrad emittance, as required, can be obtained at the plasma injection point. This is achieved by accelerating the beam up to the energy of 240 MeV where space-charge forces are less harmful [585].

(b) Another strategy with pure RF compression (using velocity bunching) [260] has been considered to produce in one stage a 100 MeV, 3 kA beam at the end of a set of S-band travelling wave sections [262]. An additional X-band linac with an accelerating gradient of ≥60 MV/m further boosts the beam energy to ~500 MeV, while proper matching conditions at the plasma entrance are achieved with a triplet of permanent quadrupoles. For the beam-driven plasma acceleration case, a laser-comb configuration [262,323] has been applied 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.

A detailed description of both RF injector designs is found in Chapter 14.

For the laser-plasma injector (LPI), on the other hand, the following five different injection techniques have been simulated and optimised:

(a) Wave-breaking injection and acceleration in the non-linear regime. This is a self-injection technique that arises when the speed of the electrons in the plasma is higher than that of the wakefield [586]. Simulations have been performed with the PIC code SMILEI [587].

(b) Shock-front injection and acceleration in the bubble regime. The principle is to focus the laser beam on a plasma density plateau, which is preceded by a steep upramp followed by a downramp of around 100 µm length [588]. Simulations have been performed with the PIC code CALDER-Circ [589].

(c) Ionisation injection and acceleration in the quasi-linear regime. This study is based on an experimental setup composed of a 5 mm long gas cell equipped at both ends with variable length tubes of smaller radius allowing to adjust the plasma ramps [285,590]. Simulations have been performed with the 3D PIC code WARP [354].

(d) Downramp injection and acceleration in the blowout regime. The principle is similar to that of the shock-front technique in (b), but a smoother and longer density step is used, with more parameters to adjust to improve the beam quality at the injection. The acceleration afterwards occurs in a ~2 mm long plasma density plateau. Simulations have been performed with the 3D PIC code OSIRIS [336].

(e) Resonant multi-pulse ionisation injection (ReMPI). This technique relies on the localised ionisation of gas as in (c) above, but the laser beam is, in this case, split into three components: the main one (decomposed in four sub-pulses) to drive the plasma wakefield, a second small component to ionise the gas, and a third tiny component with perpendicular polarisation to symmetrise the beam [170,584,591]. Simulations have been performed with the 3D PIC ALADYN [592] and QFluid [593] codes, the latter having been benchmarked with the FBPIC code [489].

More detailed descriptions of the simulation codes used are provided in the appendix, in Section 32.1.
## 23.3 Study of Acceleration Stages

The acceleration stages of the four LWFA schemes have been studied with the following different acceleration techniques leading to various and revealing results.

(a) The LPI in Scheme 1 has been studied with the ReMPI technique detailed above. However, to accelerate an electron beam with the required parameters up to 5 GeV, a more sophisticated configuration is considered [594]: the drive laser is decomposed into eight sub-pulses, the fourth harmonic of the ionisation laser is used, and the plasma is split into two components, a gas jet for ionising the doped gas and trapping the ionised electrons followed by a 25 cm long capillary for acceleration.

(b) The LPAS in Scheme 2 has been studied in the quasi-linear acceleration regime with external injection. In the first step, the injected beam considered has a bi-Gaussian density profile and parameters as required in Table 23.1 [582,595]. Simulations have been performed with the 3D PIC code WARP [354] in the boosted frame. In the second step, the considered input beam directly comes from simulations of the 150 MeV LPI applying the ReMPI technique (described in the previous section) and transferred by a dedicated transport line. Simulations have been performed with the quasi-3D code FBPIC [489].

(c) The LPAS in Scheme 3 has also been studied in the quasi-linear acceleration regime with external injection. Two input beams have been considered, coming from the two RF injectors at 240 and 540 MeV described in the previous section. Simulations have been performed, respectively, with the quasi-3D code FBPIC [489], and hybrid fluid-PIC tools – the QFluid code [596] and the LAPLAC code [597,598]. In the latter codes, the plasma is assumed to behave like a cylindrically symmetric fluid, while the electron beam is treated using a full 3D PIC model.

(d) The setup of two LPAS separated by a magnetic chicane in Scheme 4 has been studied in the blowout acceleration regime with external injection. Two input beams have been considered: the first one is an ideal Gaussian beam and the second one is a beam coming from the RF injector at 240 MeV, described in the previous section. Simulations are performed with the FBPIC code for the plasma stages as well as the ASTRA [320] and CSRtrack [599] codes for the magnetic parts.

The acceleration stage of the PWFA Scheme 5 has been studied in the weakly non-linear regime, characterised by a wakefield departing from a sinusoidal wave tending toward a saw tooth profile. The objective is to accelerate the bunch from 540 MeV to 1 GeV without phase-space dilution. Simulations have been performed with the Architect code [600], where the electron bunch is treated in a 3D PIC model and the plasma background in a cylindrical fluid model.

The hybrid Scheme 6 is based on two plasma stages. The first LPAS generates a high-charge electron beam which is used as a driver for the second PPAS. The injection process in the second stage is studied with three different injection techniques:

(a) Plasma torch injection (PTI) is a specific type of density downramp injection. Instead of shaping the plasma through gas jet configurations or a knife edge, a low-power laser pulse is used to ionise helium in a hydrogen / helium mixture. This process generates a steep density downramp, which can be used for the injection [601].

(b) Wakefield ionisation injection (WII) is a specific type of ionisation injection [602]. The accelerated beam from the first stage drives a strong wakefield in the second stage, ready for accelerating witness electrons in the bubble regime. The same wakefield presents a location where its amplitude combined with an optimised dopant gas concentration allows the injection of electrons with very high beam
<table><caption>**Table 23.3.** Laser and electron input beam parameters of the LPAS. $P_L$, $E_L$, $a_{0L}$, and $\tau_{FWHM}$ stand for, respectively, the power, energy, strength, and duration at full width at half maximum of the laser pulse(s). E, $\varepsilon_{x,y}$, $\sigma_E/E$, and $\tau_{FWHM}$ stand for, respectively, the energy, emittance, RMS energy spread, and duration of the input electron beam.</caption><thead><tr><th rowspan="2">LPAS</th><th colspan="4">Laser beam</th><th colspan="3">Electron input beam</th></tr><tr><th>P<sub>L</sub></th><th>E<sub>L</sub></th><th>a<sub>0L</sub></th><th>τ<sub>FWHM</sub></th><th>E</th><th>ε<sub>x,y</sub></th><th>σ<sub>E</sub>/E</th><th>τ<sub>FWHM</sub></th></tr></thead><tbody><tr><td>Scheme 2</td><td>341 TW</td><td>45 J</td><td>2.0</td><td>132 fs</td><td>150 MeV</td><td>1.0 μm</td><td>2 %</td><td>7 fs</td></tr><tr><td>Scheme 3</td><td>225 TW</td><td>25 J</td><td>1.15</td><td>110 fs</td><td>537 MeV</td><td>0.4 μm</td><td>0.06 %</td><td>11 fs</td></tr><tr><td>Scheme 4<br>(for laser: per LPAS)</td><td>750 TW</td><td>40 J</td><td>3.0</td><td>50 fs</td><td>241 MeV</td><td>0.8, 0.5 μm</td><td>0.3 %</td><td>6.5 fs</td></tr></tbody></table>

quality [53]. Simulations have been carried out with the 3D PIC code OSIRIS [336].

(c) Trojan horse injection (THI) is a specific type of plasma photo-cathode injection [603]. A PPAS containing a hydrogen / helium gas mixture is considered, fed by an upstream LPAS and a 10 GW laser beam [603]. The latter ionises the hydrogen gas to generate the witness beam. The LPAS, itself fed by two laser beams, delivers (i) a driver beam that ionises the helium gas by means of its self-field and simultaneously generates a blowout acceleration regime, and (ii) a large-size escort beam at the location where the witness beam already reaches a high enough relativistic energy [48]. Simulations have been performed with the 3D PIC code VSim [604].

## 23.4 Selection of the Configurations to Be Submitted to Start-to-End Simulations

From the extensive results obtained by the broad exploration described above on various injection and acceleration schemes, we will proceed to the down-selection of those configurations producing results closest to the EuPRAXIA requirements.

With the laser-plasma injector (LPI) using wave-breaking injection, a very big charge of 1 nC can be self-injected, then accelerated to 204 MeV after only 1.35 mm propagation. This results, however, in a normalised emittance of 7 mm mrad and 15% energy spread, both well outside the requirements of Table 23.1. Using shock-front injection, a witness beam of 80 pC charge can be created and accelerated to more than 100 MeV with an emittance of 1.5 mm mrad, but an energy spread of around 15%. Again, this latter parameter is well above the required value. The three other injection techniques give results much closer to the requirements. The beam parameters at the LPI exit obtained with those techniques are presented in Figure 23.2, where the parameter requirements are shown as orange bars. While ionisation injection still produces beams with emittance slightly above the required value in the laser polarisation direction, both downramp injection and ReMPI meet all the requirements.

The results obtained by the different acceleration techniques up to 5 GeV are gathered and compared to the parameter requirements (marked as orange bars) in Figure 23.3 (results from the PWFA scheme are not shown here). We can see that the two hybrid schemes lead to very good beam quality but low beam charge around 10 pC. In contrast, the five LWFA schemes, both with external and internal injection, all exhibit results closer to the requirements: four of them, in particular, meet almost all requirements, although only with a small margin. Not shown here are the LWFA Scheme 1 results using wave-breaking and shock-front injection techniques; these were found to accelerate a big charge up to 1 GeV, but with insufficient beam quality.
<figure><img src="image_178.png" /><figcaption>**Fig. 23.2.** Beam parameters obtained at the LPI exit, compared to the requirements, for three different injection techniques.</figcaption></figure>

It is also important to note that, in many of the LWFA schemes, the quasi-linear acceleration regime is applied within the LPAS, although simulations are performed with three different simulation codes and both plasma and laser parameters vary to some extent. The fact that, even with these differences and input electron beams at varying energies, the witness beams at the LPAS exit demonstrate similar properties is a very positive indicator towards the robustness of the simulated results. Not only do the simulation codes appear to be consistent with each other, but also the acceleration process in the quasi-linear LPAS seems to be largely unaffected by moderate variations in the input parameters. We will aim to quantify these indications further in future studies on the robustness of the machine to errors and parameter fluctuations.

The PWFA Scheme 5, accelerating an externally injected witness beam to 1 GeV (from a 500 MeV RFI), demonstrates, although not shown in the figure, results quite close to the requirements. The LWFA Scheme 3 reaching 5 GeV with a 500 MeV RFI has also been optimised for reaching a final beam energy of 1 GeV. Here, the beam parameters meet all output requirements as well. The other acceleration schemes have not yet been thoroughly optimised for 1 GeV, but first results show that, a priori, equivalent beam quality could be reached as for the 5 GeV cases.

In summary, the following configurations have been selected for start-to-end simulations:
<figure><img src="image_179.png" /><figcaption>Fig. 23.3. Beam parameters obtained at the 5 GeV LPAS exit, compared to the requirements, for seven different acceleration techniques and beam injection mechanisms. “qlr”, “bor”, “thi”, and “wii” stand for quasi-linear regime, blowout regime, Trojan horse injection, and wakefield ionisation injection, respectively.</figcaption></figure>

- Scheme 1: LPI with ReMPI injection to 5 GeV

- Scheme 2: LPI with ReMPI injection to 150 MeV + LPAS with quasi-linear acceleration to 5 GeV

- Scheme 3: RFI with RF bunching to 540 MeV + LPAS with quasi-linear acceleration to 5 GeV

- Scheme 4: RFI with RF and magnetic bunching to 240 MeV + LPAS in two sections separated by a magnetic chicane, with blowout acceleration to 5 GeV

- Scheme 5: RFI with the COMB technique to 500 MeV + PPAS with weakly non-linear acceleration to 1 GeV

## 23.5 Preserving Beam Emittance When Matching In and Out of Plasma Stages

As seen above, the EuPRAXIA requirements can be met considering well-optimised injection and acceleration stages. An important concern for full start-to-end simulations, however, is the preservation of these beam properties when transporting the beam from one plasma stage to another or towards the application beamlines (see Sect. 7.1.3). It is indeed well known that the transverse beam emittance can grow very strongly, by a factor up to 10 or more, when the beam abruptly leaves a plasma wakefield area with very strong focussing to enter into free space [605]. Although this is a very well-known phenomenon and many theoretical studies [35,194] have
been dedicated to study mitigation through smooth adiabatic plasma density transitions [36], a consistent description and strategy to fully understand the emittance growth and its mitigation for the concrete cases studied in EuPRAXIA should be developed. It is, for example, important to be clear which emittance, trace- or phase-space emittance, increases the most, and whether this occurs pre-dominantly in the free drifts or the focusing elements of the transfer lines. Additionally, the adiabatic transitions were studied only without beam loading effects, which are not applicable to EuPRAXIA's high-charge beams. For these reasons, a thorough study has been performed to understand emittance growth and to mitigate it in the most general case [606].

Based on the definitions in Section 7.1, let us recall that the trace-space emittance $\epsilon_{tr}$ and the normalised trace-space emittance $\epsilon_{tr,n}$ are defined as:

$$ 
\begin{aligned}
\epsilon_{tr} &= \sqrt{\langle x^2 \rangle \langle x'^2 \rangle - \langle xx' \rangle^2}, \\
\epsilon_{tr,n} &= \beta_r \gamma_r \epsilon_{tr},
\end{aligned} 
\qquad (23.1) 
$$

where $x$ and $x'$ are the particle position and momentum angle, and $\beta_r$ and $\gamma_r$ are the relativistic coefficients (not to be confused with the Twiss parameters $\alpha, \beta, \gamma$).

Equivalently, the phase-space emittance and the normalised phase-space emittance are defined as:

$$ 
\begin{aligned}
\epsilon_{ph} &= \sqrt{\langle x^2 \rangle \langle p_x^2 \rangle - \langle xp_x \rangle^2}, \\
\epsilon_{ph,n} &= \frac{\epsilon_{ph}}{m_0 c},
\end{aligned} 
\qquad (23.2) 
$$

where $p_x$ is the particle momentum, $m_0$ the electron rest mass, and $c$ is the speed of light.

Based on these relations, the following three conclusions can be derived:

1. When the Twiss parameter $\alpha$ equals zero, i.e. at a beam waist where the beam changes from divergent to convergent and vice versa, the two normalised emittance values are equal:

$$ \epsilon_{ph,n} = \epsilon_{tr,n} \quad \text{when} \quad \alpha = 0. \qquad (23.3) $$

2. Through a free drift of length $l$, the trace-space emittance remains constant, whereas the phase-space emittance varies (subscript 0 corresponds to the drift entrance and no subscript to the drift exit):

$$ 
\begin{aligned}
\epsilon_{tr,n}^2 - \epsilon_{tr0,n}^2 &= 0, \\
\epsilon_{ph,n}^2 - \epsilon_{ph0,n}^2 &= \epsilon_{tr0,n}^2 \left(\frac{\sigma_p}{p_0}\right)^2 \gamma_0 l (\gamma_0 l - 2\alpha_0).
\end{aligned} 
\qquad (23.4) 
$$

3. Inversely, through a thin lens of integrated normalised gradient $k$, the phase-space emittance remains constant, whereas the trace-space emittance varies (subscript 0 corresponds to the lens entrance and no subscript to the lens exit):

$$ 
\begin{aligned}
\frac{\epsilon_{tr,n}^2 - \epsilon_{tr0,n}^2}{\epsilon_{tr0,n}^2} &= \beta_0^2 k^2 \left(\frac{\sigma_p}{p_0}\right)^2, \\
\epsilon_{ph,n}^2 - \epsilon_{ph0,n}^2 &= 0.
\end{aligned} 
\qquad (23.5) 
$$

Equation (23.4) points out that the variation of phase-space emittance in a drift is higher when the initial trace-space emittance, the energy spread, or the Twiss $\gamma_0$ is larger. Those parameters are known to be particularly large in plasma accelerators. We thus propose to take advantage of this equation by using a plasma downramp for
minimising $\gamma_0$ at the drift entrance in order to minimise the phase-space emittance growth. $\epsilon_{tr0}^2$ and $\sigma_p^2$, on the other hand, must be minimised upstream in the wakefield acceleration process, and, of course, it is helpful to reduce the drift length $l$ as much as possible.

Equation (23.5) exhibits the well-known chromaticity effect. Due to the energy spread, there is a jump in the trace-space emittance when crossing a focussing element. To limit this increase, the focussing gradient $k^2$ should be as low as possible (smoothest focussing), and the beam beta function at the element's entrance, $\beta_0^2$, should be as small as possible. Notice that the latter condition can be met by minimising the Twiss parameter $\gamma$ in the drift section preceding the focussing lens.

Regarding these two results, minimising $\gamma$ at the plasma exit must be the key strategy; it is doubly beneficial, for minimising the phase-space emittance growth when going through a free drift, and for minimising the trace-space emittance growth when crossing a thin lens. Seeing that $\gamma$ is very large within the accelerating plasma and is necessarily constant in a consequent free drift, the only location to minimise this parameter is right at the plasma exit.

The above studies show precisely the relation between the two beam emittances, which emittance grows in which context, and all the parameters governing these growths. Equations (23.4) and (23.5) enable one, furthermore, to distribute different roles to each component of a multi-stage accelerator in the emittance preservation task.

The three key tasks are the following:

1. Minimising the emittance and energy spread during acceleration, therefore this should be the exclusive role of the acceleration section of the plasma-accelerator stage.

2. Minimising the Twiss parameter $\gamma_0$ at the transfer line entrance, i.e. at the plasma exit, therefore this should be the exclusive role of the density downramp of the plasma-accelerator stage, with the reservation that the latter would not itself induce significant emittance growth.

3. Minimising the total length and the integrated focussing strength in the transfer line, therefore this should be the exclusive role of the focussing elements in the transfer line.

It is important here to consider the exclusive role of each stage. Once the best is achieved at each of those three components as recommended, the emittance growth is ensured to be minimised. The advantage is that optimisation can be achieved separately at each stage without affecting the next stage. On the contrary, if optimisation is not carried out correctly at a given stage, the effects cannot be compensated elsewhere. These minimisation procedures will be pursued all along the following start-to-end simulations.

## 23.6 Optimising and Designing Beam Transfer Lines

At least one type of transfer line is needed to drive the electron beam at high energy from the last acceleration stage to the application beamlines. In the case of external injection, a second type is needed, which drives the beam at low energy from the injector (LPI or RFI) to the acceleration stage (LPAS or PPAS). A general strategy for optimising and designing these two transfer lines is discussed in the following (see also [606]). Detailed calculations and design considerations for both transfer lines, on the other hand, are found in Chapter 19.
## The Number of Constraints and Quadrupoles

In both cases of transfer lines, the inputs are the beam parameters at the entrance of the transport section, and the constraints are the beam parameters at the section's exit, required either by the LPAS or the application beamline. In addition, as derived in Section 23.5, the transfer line should be as short as possible and should provide the smoothest focussing possible in order to minimise the emittance growth.

Gathering all these points, three constraints in each transverse direction must be considered at the end of the transfer line: the RMS beam size, divergence, and emittance. This leads to six constraints in total and, therefore, the transfer line should be designed with at least six quadrupoles.

For the low-energy transfer line, linking two plasma stages that have beam sizes and divergences on the same order of magnitude, a good solution is to use two triplets with antisymmetric polarities. As the divergence is rather high, short and strong magnets are needed. Two triplets, including six permanent magnets, should be used. We will see in the following that a section of $\ge 0.7$ m length is the minimum length for the EuPRAXIA schemes, with a place reserved for diagnostic devices in between the triplets where the beam envelopes are parallel.

For the high-energy transfer line, as the required beam parameters on the application side (in particular for the beam propagation into the undulator section) are very different from those on the plasma side – the beam size is much bigger for the former, whereas the beam divergence is much bigger for the latter – a doublet of permanent magnets can be used to capture the beam, followed by a quadruplet of electromagnets. This offers the needed flexibility to shape the beam according to various user demands. We will see in the following that, for the different EuPRAXIA cases, a minimum section length of $\ge 4$ m is adequate.

With six constraints and six degrees of freedom, the solution found for the quadrupole gradients should be unique once their lengths, positions, and polarities are fixed and when considering only their linear effects, as is the case here. However, in order to ensure that there are no better configurations with other quadrupole lengths and positions, or even with more quadrupoles, we have explored different possibilities.

## Exploration of Solutions with More Quadrupoles

When using more than six quadrupoles, we are in a situation where the number of free parameters is larger than the number of constraints. There is thus an infinite set of solutions. For example, two close quadrupoles with various strengths can produce the same effect, as long as the difference between their strengths is the same. Searching for the best solution in this context means searching the minimum of a function that is on a hyper-surface in a multidimensional space, where it is essential to find out the lowest minimum while avoiding the local minima. The particle swarm optimisation (PSO) algorithm [491] is well suited for this kind of problem as it employs an “army of swarms” to explore the topology of the studied space before converging toward the lowest minimum.

A code based on the PSO algorithm has thus been written in Python to optimise the transfer lines. When the program is near converging, a gradient conjugate method is used to speed up the convergence. Although requiring massive simulations, this optimisation code is not very time-consuming because it relies uniquely on analytical transfer matrices (see [606]) so that no particle tracking has to be performed. Only once the solution is found, the latter is checked with the tracking codes TraceWin [493] and ASTRA [320], where space-charge effects are taken into account.
<figure><img src="image_180.png" /><figcaption>**Fig. 23.4.** Scheme 1 using a single-stage injector / accelerator generating an FEL-quality 5 GeV beam.</figcaption></figure>

<figure><img src="image_181.png" /><figcaption>**Fig. 23.5.** The ReMPI scheme: The incoming laser pulse passes through a beam splitter; a portion of it is time-shaped as a train of eight pulses (transparent surface), while a smaller portion in the fourth harmonic is tightly focused in the rear of the train (purple surface) and is used to extract electrons from the Argon dopant. The driver train resonantly excites a high-amplitude plasma wave (black line), which traps and accelerates the electrons (reprinted from [594] with permission, copyright IOP Publishing).</figcaption></figure>

> This extensive procedure demonstrates that the solution using six quadrupoles for achieving six constraints (beam size, divergence, and emittance in x and y) is adequate for a compact beamline aiming at transferring the beam from one stage to another. More complex beam transport solutions, adding extra space for diagnostics or using more unconventional components, can also be considered. They are described in some of the following start-to-end simulation cases as well as, in detail, in Chapter 19.

## 23.7 Start-to-End Simulations for Scheme 1: LPI-5 GeV

In the single-stage injection-and-acceleration scheme, the electron beam is trapped and further accelerated inside the plasma, without propagation in vacuum after the injection occurs and, therefore, without the need of beam transport optics. This approach has the disadvantage of being less flexible than one based on two stages, where both the injection and acceleration parts can be optimised independently. The single-stage scheme, however, offers the possibility to avoid potential beam quality degradation occurring at the plasma exit for low-energy electron beams and during the beam transport line, which has a sizeable impact, particularly on the beam emittance.

In the following, we will show that a single-stage setup in the ReMPI configuration [170] is flexible enough to generate, after a 25 cm long acceleration distance in a guiding capillary, a 5 GeV beam with a charge of 30 pC and FEL-compliant quality [594] as shown in Table 23.4.

The selected working configuration (see [594] for further details) is based upon a 1 PW Ti:Sa laser system, whose pulses are shaped as a train of eight pulses in the fundamental harmonics, each delivering about 6.5 J in 55 fs. Moreover, a tightly focussed
<table><caption>**Table 23.4.** Requested beam quality (see Tab. 23.1) and quality parameters obtained by means of the simulations reported in this section. The relative energy spread, $\delta E_{rms}/E$ and $\delta E_{rms|slice}/E$, and normalised emittance, $\epsilon_n$ and $\epsilon_n|slice$, are also shown as projected and slice parameters.</caption><thead><tr><th>Parameter</th><th>δE<sub>rms</sub>/E</th><th>ε<sub>n</sub></th><th>δE<sub>rms|slice</sub>/E</th><th>ε<sub>n</sub>|slice</th><th>Charge</th><th>Peak Current</th></tr></thead><tbody><tr><td>Requested</td><td>&lt; 10<sup>-2</sup></td><td>&lt; 1 µm rad</td><td>≤ 10<sup>-3</sup></td><td>≈ 1 µm rad</td><td>≥ 30 pC</td><td>&gt; 1 kA</td></tr><tr><td>Obtained</td><td>0.9 · 10<sup>-2</sup></td><td>0.085 µm rad</td><td>0.3 · 10<sup>-3</sup> (min)</td><td>0.085 µm rad</td><td>30 pC</td><td>2.5 kA</td></tr></tbody></table>

<table><caption>**Table 23.5.** Parameters of the driver train and the ionisation pulse. The total delivered energy (E in J), the FWHM duration of the pulse(s) (T in fs), the minimum waist ($w_0$ in µm), the pulse-to-pulse delay (in fs), and the normalised pulse amplitude $a_0$ are shown.</caption><thead><tr><th>Pulse parameters</th><th>E</th><th>T</th><th>w<sub>0</sub></th><th>Delay</th><th>a<sub>0</sub></th></tr></thead><tbody><tr><td>Driver pulse</td><td>50 J</td><td>55 fs</td><td>90 µm</td><td>250 fs</td><td>0.64</td></tr><tr><td>Ionisation pulse</td><td>0.06 J</td><td>45 fs</td><td>5.9 µm</td><td>85 fs</td><td>0.25</td></tr></tbody></table>

45 fs long “ionisation” pulse in the fourth harmonic, obtained by an amplified portion of the same master pulse of the train, is propagated just behind the driver. Considerations for the realisation of such a laser setup are described in Section 10.2.8. The experimental arrangement of the time shaper could make use either of multiplexing techniques for the amplified pulse [584,607–609], or of pulse shaping at an early stage of the amplification chain, as in the TEMPI scheme proposed by L. Labate et al. and numerically tested with start-to-end simulations [228]. The TEMPI scheme results in a more compact and stable setup with respect to the methods handling the full-power version, yet it should operate with an energy conversion efficiency marginally below 100%. Finally, we point out that, as the driving train and the ionisation pulse are amplified replicas of the same master pulse, no standard synchronisation jitter issues are expected, except for femtosecond-level fluctuations due to µm-size mechanical vibrations. Table 23.5 collects the main laser driver and ionisation pulse parameters.

The plasma target consists of two contiguous sections. In the first section (an Ar 50% + He 50% filled gas cell), K-shell electrons are extracted by the ionising pulse field and trapped in the plasma wave excited by the driver train. In the adjacent He-filled capillary, the pulse train remains focussed and excites the plasma wave for about 25 cm [9]. The background plasma density is set to $n_e = 2.1 \times 10^{17} \text{cm}^{-3}$ in the whole gas cell and along the capillary axis, thus sustaining a plasma wave linear wavelength of about 75 µm.

Simulations beyond 25 cm of propagation in the plasma have been performed with the quasi-static, hybrid fluid / PIC code QFluid [593]. QFluid simulations assume a 2D cylindrical symmetry of the fields, while particles of the bunch move in a full 3D space. QFluid is equipped with a mesh-refining technique, which is activated in the longitudinal portion of the cylinder where the bunch is placed. The bunch is sampled with $N_b \approx 10^6$ macro-particles and the simulation cylinder has a radius and length of 320 µm and 690 µm, respectively. The resolution of the whole (coarse) cylinder is $dz_{coarse} = 0.47 \text{ µm}$ (longitudinal) and $dr_{coarse} = 0.93 \text{ µm}$ (radial), while the refined cylinder spacings are $dz_{fine} = 0.0125 \text{ µm}$ and $dr_{fine} = 0.1 \text{ µm}$.

We remark also that the laser pulses' complex envelope space / time evolution [610] has been performed maintaining the second order derivative in the time evolution, thus ensuring the most accurate description of the pulse evolution.
<figure><img src="image_182.png" /><figcaption>Fig. 23.6. Driver train evolution. a) Peak intensity and overall pulse energy evolution. b) Initial (upper) and final (lower) maps of the pulse envelope. Pulses move from the left to the right (reprinted from [594] with permission, copyright IOP Publishing).</figcaption></figure>

## Evolution of the Driving Pulse Train

The evolution of the driver train is highly non-trivial, due to the propagation of the pulses in a plasma perturbed by the preceding parts of the train. This can lead to either focussing or a defocussing of different portions of the pulses, especially in the rear side of the train where the plasma wave amplitude reaches its maximum [594].

In the simulation, the evolution of the pulses at the train tail is stabilised through an optimisation procedure. During the 20 cm of propagation in the capillary, the peak intensity of the train, nonetheless, experiences large fluctuations (see Fig. 23.6a), and severe pump depletion of about 70% of the laser energy occurs, mostly due to the erosion of pulses in the tail (see Fig. 23.6b). The lead pulse, on the other hand, becomes strongly self-focussed, yet only experiences a depletion of around 10% (not shown). Overall, a large-amplitude plasma wave is excited by the train for almost all of its propagation distance, as will be apparent in the next section.

## Extraction of the Electrons by the Ionising Pulse

In the first target section (gas cell containing Argon), it is supposed that the nanosecond and picosecond laser pre-pulses, as well as the first few cycles of the first driver pulse, are able to ionise the Argon atoms up to the 8th level. The large jump of the ionisation energy occurring in the passage to the K-shell (i.e. to the 9th electron, $U_{i,9} \simeq 422$ eV) realises the conditions for the controlled extraction of the K-shell electrons with a large-amplitude electric field. This is accomplished by tightly focussing the fourth-harmonic "ionisation" pulse just behind the train, tuning its electric field in the vicinity of the $Ar^{8+} \rightarrow Ar^{9+}$ transition threshold. In the following, a pulse converted to the fourth harmonic (see [594] for details) will be considered, along with the resulting normalised amplitude $a_{0,ion} = 0.25$ and a minimum waist of $w_{0,ion} = 5.82$ µm. As the electric field amplitude of the laser is inversely proportional to its wavelength, this design choice allows a sufficient field amplitude despite the low $a_0$ value.

As the electron leaves the atom, it starts quivering in the laser field. After the ionising pulse overtakes it, a residual transverse momentum along the polarisation axis remains, thus constituting a source of bunch emittance. Simulations and analytical results (see [611]) show that the normalised emittance achievable by using a linearly polarised pulse can be as low as

$$ \epsilon_n \approx 2^{-\frac{1}{2}} w_{0,ion} \cdot a_{0,ion} \cdot \Delta^2 = 2^{-\frac{1}{2}} w_{0,ion} \cdot a_{0,ion}^2 / a_c \quad (23.6) $$
<figure><img src="image_183.png" /><figcaption>**Fig. 23.7.** On-axis snapshot at the early stage of bunch trapping. A line-out of the driving train (red line) and ionising pulse (purple line) normalised amplitudes, as well as a line-out of the longitudinal normalised electric field $E_z/E_0$ are shown. The Lorentz factor $\gamma$ associated with the wakefield is $\gamma_{ph} \simeq 90$. The longitudinal phase space of the bunch (black dots) is $(z + ct[\mu m], u_z/\gamma_{ph})$, where $u_z = -p_z/mc$. Particles with $u_z/\gamma_{ph} \ge 1$ are trapped by the wave (reprinted from [594] with permission, copyright IOP Publishing).</figcaption></figure>

where $\Delta \equiv \sqrt{a_{0,ion}/a_c}$ and $a_c \simeq 0.108 \cdot \lambda_{ion}(U_i/U_H)^{3/2}$ ($U_H \simeq 13.6$ eV; see Eqs. (26) in [611] and (4) in [170]). In our configuration, equation (23.6) predicts a minimum achievable emittance of $\varepsilon_{n,min} \simeq 0.05$ µm rad. However, other mechanisms than initial quivering can be responsible for an emittance increase. Among them, we mention a residual non-linearity of the transverse ponderomotive force and non-linear dependence of the radial force on the transverse coordinates. In our QFluid simulations, an emittance of $\varepsilon_{nx} = 0.08$ µm rad at the end of the gas-cell section has been obtained, which should be compared with the minimum value of 0.05 µm rad deduced by equation (23.6).

## Beam Trapping and Energy Boosting Up to 5 GeV

Electrons leaving the $Ar^{9+}$-ion with a negligible speed start to quiver in the ionising pulse polarisation plane and slip back into the plasma wave while they are accelerated by the longitudinal electric field. Particle trapping occurs if electrons reach the plasma wave phase speed $v_{ph}$ before they slip into the de-accelerating region of the wave. In our configuration, the newborn electrons reach the plasma wave speed close to the longitudinal electric field peak, which means that trapping occurs close to the “strong” trapping threshold [170] regions. In Figure 23.7, the bunch longitudinal phase space in the middle of the gas cell containing Argon is shown. A portion of the bunch (black dots) is already trapped, while most of the particles still have a longitudinal momentum $u_z = -p_z/mc$ well below the trapping threshold $\gamma_{ph} = (1 - \beta_{ph}^2)^{-1/2} \simeq 90$. In Figure 23.7, the driving pulse train (red line) and ionisation pulse (purple line) are also shown. After the passage of the last train pulse, a moderately non-linear wave (blue line) with amplitude $E_z/E_0 \simeq 0.7$ is excited (here $E_0 = mc\omega_p/e$ is the non-relativistic wave-breaking limit).

Close to the end of the gas cell, the fully trapped beam experiences both longitudinal and transverse phase-space rotations. As the bunch is fully trapped by the wave,
<figure><img src="image_184.png" /><figcaption>**Fig. 23.8.** The phase space ((z + ct)[µm], x[µm], E[MeV]) of the bunch at the end of the 25 cm long capillary. The longitudinal electric field map is also shown (reprinted from [594] with permission, copyright IOP Publishing).</figcaption></figure>

a longitudinal phase-space rotation occurs, resulting in a quasi-monochromatic beam adiabatically squeezed down to a (quasi-round) beam of about 0.8 µm in diameter.

Just after the phase-space rotations, the driving train and the bunch enter into the helium-filled capillary “booster”, which guides the laser for more than 20 cm. Inside the capillary, an average longitudinal electric field of ≈ 46 GV/m is obtained, corresponding to a mean normalised field of $E_z/E_0$ ≈ 0.46. At the capillary exit, the driving train energy is depleted by 70% and the bunch energy is about 5 GeV, with a tiny increase in the normalised emittance up to $\epsilon_{nx} = 0.086$ µm rad and $\epsilon_{ny} = 0.081$ µm rad. The longitudinal electric field map (in the bucket containing the bunch), as well as the bunch longitudinal phase-space at the end of the capillary are shown in Figure 23.8.

## Final Beam Quality

An FEL-driving beam should be analysed from both the projected and slice perspectives. While the former provides relevant information about quality degradation during the beam transport [379,606] up to the undulators, the latter shows if some slices of the bunch are suitable for lasing [219].

In Figure 23.9, several snapshots of the six-dimensional final phase space are shown. Cuts in the x-y and x-u_x planes show that a (quasi)-round and matched beam was obtained (see [591] for details). The z-u_z cut highlights a high-energy tail with a fractional charge of 8% (see Fig. 23.10). This tail, however, can be easily removed through simple fine-tuning of the transfer line energy acceptance. Finally, the transported beam (with about 92% of the total charge) complies with all the requirements of the “Requested” raw in Table 23.4, comprising a projected energy spread of $\sigma_E/E = 9 \cdot 10^{-3}$.

Figure 23.11a shows the slice current profile as well as the 5D brightness ($B_{5D} \equiv 2I/(\pi^2\epsilon_{nx} \times \epsilon_{ny})$) and 6D brightness ($B_{6D} \equiv B_{5D}/(\sigma_E/E/10^3)$) of the beam. We mention here that our definition of brightness includes a $1/\pi^2$ factor, so the comparison of peak brightness values found in the literature must be performed with care. Moreover, the current distribution shows a gently varying profile with a peak value of 3.5 kA. In Figure 23.11b, the normalised emittances along the x (ionisation) and y (driver) polarisation axes are reported. At the peak current, i.e. at a longitudinal position of about 1 µm from the bunch centre of mass, emittances of 0.066 µm rad and 0.041 µm rad along the x- and y-axes are reported. Remarkably, the slice energy
<figure><img src="image_185.png" /><figcaption>**Fig. 23.9.** Phase-space cuts of the final beam. Here, $u_{x,y} = p_{x,y}/mc$ and $u_z = -p_z/mc$. The bunch moves from the left to the right (reprinted from [594] with permission, copyright IOP Publishing).</figcaption></figure>

<figure><img src="image_186.png" /><figcaption>**Fig. 23.10.** Longitudinal phase-space plot (($z + ct$)[µm], ($E - \langle E \rangle$)/($\langle E \rangle$)) and current profile of the final beam. By using a standard beam optics for selecting a maximum energy below $1.3 \times \langle E \rangle$, about 92% of the bunch charge is transported to the final undulator stage for lasing (reprinted from [594] with permission, copyright IOP Publishing).</figcaption></figure>

spread reported in Figure 23.11c shows that more than 80% of the bunch charge lies in slices with an energy spread below the upper limit of $10^{-3}$, required for FEL lasing (see [594] for further details).

## Optimisation of Beam Transfer in the HETL

The final objective for the proposed beamline for Scheme 1 is to transfer the output beam from the plasma target towards the application beamlines, e.g. 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.