<table><caption>**Table 12.3.** Characteristics of the solenoid focussing magnets per linac section.</caption><thead><tr><th>Parameter</th><th>Unit</th><th>Value</th></tr></thead><tbody><tr><td>Number of coils</td><td></td><td>12</td></tr><tr><td>Inner diameter</td><td>mm</td><td>308</td></tr><tr><td>Outer diameter</td><td>mm</td><td>632</td></tr><tr><td>Coil cross-section (insulated)</td><td>mm<sup>2</sup></td><td>162.82</td></tr><tr><td>Nominal Cu conductor size</td><td>mm<sup>2</sup></td><td>7.5-7.5</td></tr><tr><td>Nominal cooling hole diameter</td><td>mm</td><td>5</td></tr><tr><td>Turns per solenoid</td><td></td><td>200</td></tr><tr><td>Maximum excitation current</td><td>A</td><td>200</td></tr><tr><td>Current density</td><td>A/mm<sup>2</sup></td><td>5.05</td></tr><tr><td>Maximum voltage</td><td>V</td><td>30.15</td></tr><tr><td>Power per coil</td><td>W</td><td>5580</td></tr><tr><td>Hydraulic circuits per coil</td><td></td><td>5</td></tr><tr><td>Water velocity</td><td>m/s</td><td>0.9</td></tr><tr><td>Water flow per magnet</td><td>m<sup>3</sup>/s</td><td>9·10<sup>-5</sup></td></tr><tr><td>Water temperature rise</td><td>°C</td><td>15</td></tr><tr><td>Pressure drop per circuit</td><td>MPa</td><td>0.19</td></tr></tbody></table>

<table><caption>**Table 12.4.** Technical specifications of the S-band accelerating sections.</caption><thead><tr><th>Parameter</th><th>Unit</th><th>Value</th></tr></thead><tbody><tr><td>Structure type</td><td></td><td>Constant gradient, TW</td></tr><tr><td>Working frequency</td><td>GHz</td><td>2.856</td></tr><tr><td>Number of cells</td><td></td><td>86</td></tr><tr><td>Structure length</td><td>m</td><td>3</td></tr><tr><td>Working mode</td><td></td><td>TM<sub>01</sub>-like</td></tr><tr><td>Phase advance between cells</td><td></td><td>2π/3</td></tr><tr><td>Max. average accelerating gradient</td><td>MV/m</td><td>22 (S1)/ 25 (S2)/ 28 (S3)</td></tr><tr><td>Average RF input power (P<sub>IN</sub>)</td><td>MW</td><td>&lt;40 (S1) / &lt;50 (S2) / &lt;60 (S3)</td></tr><tr><td>Shunt Impedance per unit length</td><td>MΩ</td><td>53-60</td></tr><tr><td>Phase velocity</td><td></td><td>c</td></tr><tr><td>Normalised group velocity</td><td>v<sub>g</sub>/c</td><td>0.0202-0.0065</td></tr><tr><td>Filling time (τ<sub>F</sub>)</td><td>ns</td><td>~850</td></tr><tr><td>Structure attenuation constant</td><td>neper</td><td>0.57</td></tr><tr><td>Operating vacuum pressure (typical)</td><td>mbar</td><td>10<sup>-8</sup>-10<sup>-9</sup></td></tr><tr><td>Repetition rate</td><td>Hz</td><td>50</td></tr><tr><td>Average dissipated power</td><td>kW</td><td>~1.3</td></tr></tbody></table>

in the structure (about 1/3) is coupled out from the last RF cell and dissipated on an external load. To meet the severe emittance requirements for the injector, the single-feed couplers (foreseen in the original SLAC-type structures) will be replaced by a dual-feed design [263] to minimise the multipole field effects generated by the asymmetric feeding, which induces transverse kicks along the bunch causing beam emittance degradation.

The maximum achievable accelerating gradient is the most important parameter of such devices. To reach the maximum required nominal energy of about 180 MeV, the average accelerating field in the three SLAC-type sections S1, S2, and S3 needs to be 22, 25, and 28 MV/m, respectively. The beam loading is negligible because of the very small average beam current (energy extracted by the beam is maximum 40 mJ per section; stored energy in each section is about 40 J).

Technical specifications of the S-band accelerating sections are reported in Table 12.4.
<table><caption>**Table 12.5.** Main specifications of the S-band klystrons.</caption><thead><tr><td>Parameter</td><td>Unit</td><td>Value</td></tr></thead><tbody><tr><td>Frequency type</td><td>GHz</td><td>2.856</td></tr><tr><td>RF pulse duration</td><td>µs</td><td>4</td></tr><tr><td>Repetition rate</td><td>pps</td><td>50</td></tr><tr><td>Cathode voltage</td><td>kV</td><td>350–370</td></tr><tr><td>Beam current</td><td>A</td><td>400–420</td></tr><tr><td>HV pulse duration</td><td>µs</td><td>6</td></tr><tr><td>RF peak power</td><td>MW</td><td>60</td></tr></tbody></table>

### 12.2.5 High-Power System Components

Two RF power stations are needed to feed all of the S-band active devices. The first one will power the RF gun and one RF deflecting cavity used for beam longitudinal phase-space diagnostics (see Chap. 21). Both devices are standing wave (SW) cavities, requiring ≈ 1.5µs long RF pulses of ≈ 15MW and ≈ 5MW power, respectively. Because of the pulse length and the moderate power request, this station will not be equipped with a pulse compressor. A total power of ≈ 25MW is sufficient to provide a comfortable safety margin for operation. A circulator will protect the klystron itself against the power reflected by the standing wave cavities. Given the fact that this device employs ferrite materials, it usually operates in a pressurised atmosphere of sulfur-hexafluoride (SF6) to guarantee the required insulation. The waveguide vacuum system will be separated from the circulator SF6 insulating system by means of RF ceramic windows. This type of window will also be used to separate the accelerator and waveguide vacuum systems. A capillary distributed interlock/alarm system, managed by a complex of programmable logic controllers (PLC), will protect the linac and the RF stations in the case of malfunction.

A second RF station will power the three accelerating sections S1, S2, and S3. The total power required is ≈ 150MW in 0.85µs long pulses, which can be obtained by compressing a ≈ 4µs long, 60MW pulse produced by a single klystron. Although the nominal accelerating fields are not extreme, they nevertheless require the use of selected materials, precise machining, a high-quality brazing process, surface treatments and cleaning, ultra-pure water rinsing, careful vacuum, and RF low-power tests. To keep the accelerating structures and the pulse compressor cavities precisely tuned, they need to be kept at a very constant temperature, i.e. $ΔT = ± 0.1°$, by means of regulated cooling water systems.

Three manufacturers – Thales (France), CPI (U.S.), and Toshiba (Japan) – produce 60 MW peak-power S-band klystrons, which meet the requirements of the second station of the EuPRAXIA S-band linac. A set of klystron parameters is given in Table 12.5. Each klystron, equipped with beam-focussing coils, will be supplied by a high-voltage (HV) modulator. As already mentioned the second klystron will be connected to a pulse compressor (SLED), which is used to increase the peak power, feeding the three accelerating structures in parallel. An important specification of the SLED system is the peak power gain, which usually ranges around 7.4 dB, with maximum values of 7.8 dB. The power at the SLED output increases therefore, on average, by a factor of 3, while the pulse length is reduced to 0.85 µs, corresponding to one filling time of the S-band TW structures.

A network of rectangular WR284 copper waveguides distributes the RF power from the klystrons to the gun, SLEDs, and accelerating structures. The waveguides are pumped down to $1 \times 10^{-8}$ mbar with a distributed pumping system and are
<figure><img src="image_88.png" /><figcaption>**Fig. 12.4.** Layout of the EuPRAXIA@SPARC_LAB RF power station as an example for the RF power distribution for the photo-injector.</figcaption></figure>

connected to the accelerating structures through ceramic windows to protect the beamline vacuum.

Variable phase shifters and splitters will be used to allow a fully independent phase and amplitude regulation of the three TW accelerating structures. This option is critical in the optimisation of the machine performance and, in particular, in providing all the capabilities needed for velocity-bunching and chicane-bunching working points. The phase shifter and splitter/attenuators located in the waveguide arms are high-power devices capable of operating in vacuum, and are commercially available. The variable power splitter/attenuators can be integrated in the waveguide network to route in each distribution arm the required power level. Figure 12.4 shows a scheme of the RF power distribution for the photo-injector.

## 12.3 X-band RF Linac

### 12.3.1 Introduction

The EuPRAXIA linac is the core of the accelerator. It is designed to accept the beam coming from the injector and to accelerate it to the final energy with the proper beam characteristics to match the requirements of the various specific applications. In this respect, the linac needs to be flexible enough to cope with different scenarios: (1) injection of a beam comb into a neutral plasma to both excite a plasma wave and exploit it to accelerate the last (or “witness”) bunch of the train (PWFA case) and (2) injection of a single bunch into a plasma wave excited by a laser pulse (LWFA case). The linac RF design is driven by the need of a high accelerating gradient,
i.e. a gradient well above the operational values of the existing FEL sources. This is required to achieve facility compactness, which is one of the main goals of the EuPRAXIA project. Thus, the high-gradient operation motivates the choice of the linac technology. In fact, the use of high RF frequencies is the most suitable and efficient solution for high gradients. C-band technology ($f \approx 6$ GHz) is the chosen baseline of most recent room-temperature FEL source facilities, such as SACLA at Spring 8 (Japan) and SwissFEL at PSI (Switzerland), while X-band ($f \approx 12$ GHz) is the proposed baseline for CLIC, the CERN multi-TeV linear collider project. In principle, both technologies could fulfill EuPRAXIA requirements. The baseline chosen for the EuPRAXIA project, however, is X-band for various reasons: it is more efficient, it has operated at higher gradients, its development is supported by a stronger international effort, and experimental data are more abundant. Moreover, the possibility to establish a partnership with CERN represents an added value to the project. Nevertheless, it has to be mentioned that, at present, some X-band RF components are more expensive than C-band ones, although costs are decreasing as the technology is being adopted by different laboratories, and cavity fabrication and alignment tolerances are becoming more stringent.

### 12.3.2 Linac Technology and Gradient Choice

The total space required for a ~500 MeV linac accelerating section (with ~100 MeV gain in the photoinjector and ~400 MeV gain in the linac itself) is ~10 m, corresponding to an active length of ~8 m, with the remaining space required to accommodate beam diagnostics, magnetic elements, vacuum equipment, and flanges plus the room foreseen for upstream and downstream lattice-matching sections and magnetic chicanes. The minimum gradient required to accomplish the 0.5 GeV scenario is therefore 50 MV/m. However, the linac accelerating sections and the whole RF system need to be designed to ultimately reach a higher gradient to guarantee a comfortable safety margin during operation. The need of generating such high accelerating gradients (i.e. in the range of ≥50 MV/m) has been indeed the strongest motivation addressing the RF technology choice.

In the last 25 years, a huge effort has been invested in the development and consolidation of X-band as a fully reliable RF technology for future linear colliders, initially at SLAC and KEK (NLC/JLC projects) and then at CERN (CLIC project) [264]. The work has led to a better understanding and control of high-gradient phenomena, such as breakdown, as well as to the development and extensive experimental testing of all RF components necessary to build a complete RF accelerating system. This includes RF power sources, pulse compressors, waveguide networks, and vacuum components. To perform long-term tests on various prototypes of the CLIC accelerating structures, three test stand stations, the so-called “X-boxes”, have been put into operation at CERN. Many of the tested structures have exceeded the 100 MV/m threshold, with a breakdown rate compliant with CLIC specifications (≤ 3 × 10⁻⁷ in units of breakdowns per pulse per metre of active length) [265,266].

The development work on C-band technology instead has been more focussed on the performances required by FEL source facilities [267–269] so that less data are available on the ultimate achievable gradients with this approach compared to X-band.

Starting from these considerations, X-band RF technology has been evaluated to be the most appropriate choice for the EuPRAXIA linac. In this regard, INFN has set up a collaboration agreement with CERN for a wide exchange of technical information, expertise, and components. Short-term goals of the collaboration are the refinement of the EuPRAXIA linac RF design and the construction at LNF of an additional X-box to test accelerating structure prototypes.
<figure><img src="image_89.png" /><figcaption>Fig. 12.5. Pulse compression with a SLED-type system.</figcaption></figure>

## 12.4 Accelerating Section Design

The RF system for the EuPRAXIA linac has been designed using the following procedure:

- define the optimal filling time of the structures, considering the necessity to use pulse compressors in the system;

- choose the iris aperture according to beam dynamics considerations (this allows to determine the dimensions of a single cell and the total length of the accelerating structure); and

- define the layout of the complete RF system based on the following:

* optimised TW section parameters;

* RF power source (klystrons) parameters; and

* the various operating scenarios that the linac must support.

The accelerating sections are of the travelling wave (TW) type, and will be powered by means of pulse compression systems. This choice maximises the overall efficiency and decreases costs by reducing the number of power sources required to produce the necessary peak power [270]. Pulse compressors are widely used devices in RF systems of room-temperature linacs. By constructively interfering reflected power pulses of very high-Q cavities with properly phase-modulated forward pulses, it is possible to concentrate a large portion of a klystron RF pulse energy in a small fraction of the original pulse duration, as shown in Figure 12.5.

The peak power in the shortened RF pulse is considerably higher than the original one, and the shorter the pulse, the larger the peak power. On the other hand, the shorter the RF pulse, the smaller the fraction of its energy exploited for acceleration by the TW section. The optimal pulse length of the TW section powered by a compressed pulse can be calculated and depends on the klystron pulse duration and the Q factors of both the accelerating sections and the pulse compressor cavities. For typical values of an X-band system ($f_{RF} \approx 12$GHz, $\tau_{kli} \approx 1.5$µs, $Q_{TW} \approx 6500$, $Q_{SLED} \approx 180000$), the optimal duration of the compressed pulse can be derived from the plots shown in Figure 12.6.

The plots on the left side show the values of the “effective” shunt impedance per unit length $R_s$ normalised to the ordinary shunt impedance per unit length $R$ as a function of the parameter $\tau_s = \alpha \cdot L_s = (\omega_{RF}/2Q) \cdot \tau_F$, with $\alpha$ as the linear attenuation coefficient of the structure, $L_s$ its length, and $\tau_F$ the structure filling
<figure><img src="image_90.png" /><figcaption>Fig. 12.6. Effective shunt impedance as a function of the section attenuation and resulting accelerating field profile.</figcaption></figure>

time. The definitions of R and R_s are the following:

$$R = \frac{E_{acc}^2}{|dP/dz|}; \quad R_s = \frac{\langle E_{acc}^2 \rangle_{avg}}{P_{in}/L_s} \qquad (12.1)$$

The maximum value of the R_s/R ratio is ~3.67, a number larger than 1 since it accounts for the peak power gain factor provided by the pulse compressor. It is reached when τ_s0 ~ 0.68, which corresponds to a filling time value of τ_F ≈ 120 ns. The values obtained with constant impedance and constant gradient sections are similar, so this parameter does not determine the choice between the two options. The dimensions of the basic cell of the TW accelerating section have been parametrised and simulated with the HFSS code to calculate all the characteristic parameters of the structure. The cell model for simulations is shown in Figure 12.7. The aim of the simulations was to scan the structure characteristics for iris radius values a varying in the τ_F ≈ 120 ns range. For each value of a, the cell has been tuned to the nominal frequency (τ_F ≈ 120 ns) by varying the cell radius b, while the other dimensions have been kept fixed. The chosen phase advance of the TW structure is 2π/3, so that the length d of the cell is 1/3 of the RF wavelength.

The main characteristic parameters of the TW structure (such as attenuation constant, group velocity, shunt impedance per unit length, Q factor, and modified Poynting's vector) have been calculated as functions of the iris radius a and are reported in the plots in Figure 12.8. The attenuation constant and group velocity depend critically on the iris aperture, while the shunt impedance per unit length
<table><caption>Fig. 12.7. Basic EuPRAXIA X-band accelerating cell.</caption><thead><tr><th colspan="2">Geometrical parameters</th></tr></thead><tbody><tr><td>a [mm]</td><td>2 &divide; 5</td></tr><tr><td>b [mm]</td><td>9.828 &divide; 10.917</td></tr><tr><td>d [mm]</td><td>8,332 (2&pi;/3 mode)</td></tr><tr><td>r [mm]</td><td>1</td></tr><tr><td>t [mm]</td><td>2.5</td></tr></tbody></table>

<figure><img src="image_91.png" /><figcaption>Fig. 12.8. Characteristics of the TW accelerating section as a function of iris radius.</figcaption></figure>

shows an approximately linear dependence, and the Q-factor excursion is very limited (~5%) in the explored range.

The choice of the iris aperture design value is a trade-off between efficiency, pushing towards small values of *a*, and different technical and beam dynamics-related considerations (such as energy spread, emittance growth, beam stay-clear margins, alignment tolerances and so on), pushing towards larger values of *a*. In particular, calculations on beam breakup (BBU) limits caused by the transverse wakefield of the accelerating sections scaled to the EuPRAXIA beam parameters have led to a minimum acceptable value of the iris aperture of *a* = 3.2mm [271,272]. This value, together with the optimal total attenuation and filling time reported in Figure 12.6, is sufficient to complete the TW section design. The main parameters characterising a constant gradient TW X-band section for the EuPRAXIA linac are summarised in Table 12.6.
<table><caption>**Table 12.6.** Characteristics of the EuPRAXIA constant gradient accelerating section.</caption><thead><tr><th>Accelerating section parameter</th><th>Symbol</th><th>Unit</th><th>Value</th></tr></thead><tbody><tr><td>Average iris radius</td><td>⟨a⟩</td><td>mm</td><td>3.2</td></tr><tr><td>Structure length</td><td>L<sub>s</sub></td><td>mm</td><td>500</td></tr><tr><td>Quality factor</td><td>Q</td><td></td><td>6400</td></tr><tr><td>Normalised group velocity</td><td>v<sub>g</sub>/c</td><td>%</td><td>2.5–0.77</td></tr><tr><td>Filling time</td><td>τ<sub>F</sub></td><td>ns</td><td>121</td></tr><tr><td>Number of cells</td><td>N<sub>c</sub></td><td>–</td><td>60</td></tr><tr><td>Average shunt impedance per unit length</td><td>⟨R⟩</td><td>MΩ m<sup>-1</sup></td><td>90</td></tr><tr><td>Effective shunt impedance per unit length</td><td>R<sub>s</sub></td><td>MΩ m<sup>-1</sup></td><td>330</td></tr></tbody></table>

<figure><img src="image_92.png" /><figcaption>**Fig. 12.9.** Schematics of an 8-section RF module powered either by one (left) or two (right) klystrons.</figcaption></figure>

### 12.4.1 X-band Linac Layout

The TW X-band accelerating sections optimised for the EuPRAXIA application are 0.5 m long and show an effective shunt impedance per unit length of 330 MΩ/m, a value that accounts also for the peak power gain provided by the pulse compressor. The fraction of the klystron RF power required by each accelerating section to reach the ≈ 57 MV/m average gradient (i.e. the minimum needed to provide a ≈ 0.5 GeV beam to the magnetic undulator without plasma contribution) is then:

$$P_{in} = \frac{\langle E_{acc}^2 \rangle_{avg}}{Rs/Ls} = 4.9 \text{ MW.} \qquad (12.2)$$

Commercially available X-band klystrons provide up to 50 MW peak power in ≈ 1.5 µs long pulses.

RF losses in the waveguide distribution system are estimated to reduce the available klystron power to the accelerating sections by ≈ 20 % so that a single tube can actually deliver ~40 MW. Thus, up to 8 TW structures can be fed in parallel at the required gradient by a single klystron. For this reason, the basic RF module of the EuPRAXIA X-band linac can be conveniently composed by a group of 8 TW sections assembled on a single girder and powered by a single klystron connected to a pulse compressor system and a waveguide network splitting and transporting the RF power to the input couplers of the sections. Each RF module will provide an active length of 4 m, while its actual physical length will be 5 m to accommodate flanges, vacuum equipment, beam diagnostics stations, and magnets. The 500 MeV
<table><caption>**Table 12.7.** Main parameters of the X-band linac RF system for different scenarios: injection in the plasma (LPWA – PWFA) and ultimate performance.</caption><thead><tr><th></th><th>Parameter</th><th>Unit</th><th colspan="2">Value</th></tr></thead><tbody><tr><td>N<sub>s</sub></td><td>Number of sections</td><td></td><td colspan="2">16 (2 modules x 8 sections)</td></tr><tr><td>P<sub>k</sub></td><td>RF power<br>available per klystron</td><td>MW</td><td colspan="2">50 (at klystron output couplers)<br>40 (at section input couplers)</td></tr><tr><td></td><td></td><td></td><td>PWFA – LWFA</td><td>Ultimate</td></tr><tr><td>⟨E<sub>acc</sub>⟩</td><td>Max. average gradient</td><td>MV/m</td><td>57</td><td>80</td></tr><tr><td>P<sub>RF</sub></td><td>Total RF power required</td><td>MW</td><td>79</td><td>155</td></tr><tr><td>N<sub>k</sub></td><td>Number of klystrons</td><td></td><td>2<br>(reduced power)</td><td>4<br>(full power)</td></tr></tbody></table>

linac will host 2 RF modules in total, corresponding to 16 accelerating sections. This configuration (2 RF modules driven by 2 klystrons) is already sufficient to provide the minimum nominal beam energy for the different operational scenarios; however, its optimisation is still in progress.

The linac energy could be increased by doubling the RF power on one or more modules by simply adding a second klystron. Upgraded modules could run with an increased gradient up to 80 MV/m. Thanks to the modularity of the RF system, the RF power upgrade of the modules can be planned in various steps according to the needs of the facility. This will provide a gradient overhead that could be exploited to increase the operation flexibility and, ultimately, to reach higher beam energies. A sketch of one RF module powered in initial (left) and upgraded (right) configurations is shown in Figure 12.9, while the main parameters of the X-band linac RF system are reported in Table 12.7.

The RF system layout and the waveguide network distribution will remain unchanged downstream of the 2-klystron power combiner. The standard WR-90 rectangular waveguide supporting the TE<sub>10</sub> mode at 12 GHz shows a ≈ 0.1 dB/m attenuation. Thus, the choice is not suitable for long-distance connections, such as the one between the RF power station (klystron gallery) and the RF module (accelerator hall). For this reason, the majority of this length will be covered by a round overmoded WC-50 waveguide, showing a much lower attenuation of ≈ 0.013 dB/m. Two mode converters need to be placed to interface the two waveguide standards. The pulse compressor (SLED) will be placed downstream of the low attenuation connection, just in front of the power splitter network feeding the 8 TW structures.

# 13 Plasma Injector

## 13.1 Introduction

A laser-driven plasma-wakefield accelerator can be used as a laser-plasma injector (LPI) stage by trapping electrons from the plasma source. This would then form the first component of the accelerator in place of a conventional photo-cathode. During the in-plasma injection process, a small fraction of electron trajectories are such that they become trapped in the focussing and accelerating field structure of the plasma wakefield set up by the bulk motion of the plasma. This happens naturally when the plasma oscillation is driven to a large enough amplitude, such as in the ground-breaking “dream beam” papers of 2004 [11,12,273]. Thus, in its simplest form, an LPI can be constructed by focussing a short, relativistically intense laser pulse into a uniform plasma. By ensuring that the laser pulse is focussed to the dimensions of a relativistic plasma wave (≈ $\lambda_p \equiv 2\pi c/\omega_p$), the laser pulse can efficiently drive an electron plasma wave with
<table><caption>**Table 13.1.** Target parameters for electron bunches generated by LPIs.</caption><thead><tr><th>Beam parameter</th><th>Symbol</th><th>unit</th><th>Baseline</th><th>Range (accept-best)</th></tr></thead><tbody><tr><td>Bunch Charge</td><td>Q</td><td>pC</td><td>30</td><td>≥ 30 → 100</td></tr><tr><td>Bunch total energy spread</td><td>ΔE/E</td><td></td><td>5%</td><td>≤ 10% → 1%</td></tr><tr><td>Bunch transverse emittance</td><td>ε<sub>N,⊥</sub></td><td>mm mrad</td><td>1</td><td>≤ 10 → 1</td></tr><tr><td>Bunch length</td><td>τ</td><td>fs</td><td>5</td><td>10 → 3</td></tr><tr><td>Repetition rate</td><td>f</td><td>Hz</td><td>10</td><td>1 → 100</td></tr></tbody></table>

a phase velocity equal to the group velocity of the laser pulse. The normalised vector potential $a_0$ of the laser pulse determines the nature of the plasma response, and for $a_0 \gg 1$, the plasma wave amplitude becomes large, with the motion of the plasma electrons becoming highly relativistic. Indeed, the velocities can become so large that there can be trajectory crossing, and the plasma wave is said to “break” in analogy to water waves breaking at the sea shore. By controlling the laser and plasma parameters, this wave breaking can be harnessed to inject a bunch of charge, which is then accelerated in the longitudinal field of the plasma wave.

To build a high-quality and stable LPI for a potential user facility, control of the injection process has to be maintained to industrial standards. Experimental results over the past 15 years have explored various methods of achieving controlled injection. The main methods considered for a EuPRAXIA LPI are density-downramp injection, ionisation injection, and a new technique called resonant multi-pulse ionisation injection (ReMPI). The first two techniques affect the injection process in complementary ways and can further be incorporated into a single plasma source design, leading potentially to greater control on beam properties. With an emphasis on experimental data, this chapter will focus largely on the techniques of density-downramp and ionisation injection, both demonstrated in test experiments in the past, while the ReMPI technique will be discussed further via simulation studies in Chapter 23.

Two scales of LPI will be implemented in EuPRAXIA: a “low-energy” injector producing electron beams in the range of 150–400 MeV and a 1 GeV injector. Evidently, the low-energy injector is less challenging and can be accomplished with a lower-energy laser driver. However, the GeV injector allows extra versatility for EuPRAXIA, such as enabling a lower transformer ratio or single-stage plasma wakefield acceleration. The GeV injector could also be directly used for some of the secondary applications proposed for the facility. Apart from the energy, the specifications for the electron beam produced by these two LPIs are similar, as summarised in Table 13.1.

## 13.2 Injection Mechanisms

To generate high-quality electron beams, the region over which injection occurs must be small and controlled. The latter implies that a large quantity of charge is placed into a small region of the accelerator, leading to small bunch energy spread and emittance. In addition, the injection ideally happens at a point that is both reproducible and repeatable, leading to high stability in energy and charge. The outcome should be a high-brightness beam with good shot-to-shot stability. LPI beams are most commonly generated by self-injection, whereby the plasma wave amplitude grows so large that wavebreaking can provide the trapped electrons [11,12,273]. This process, however, relies on the non-linear evolution of the laser pulse in the plasma and so can be sensitive to small changes in input parameters [274]. By tailoring the plasma source, injection can be decoupled from the laser pulse evolution, leading to increased stability.
Density-downramp injection can be used to decrease the plasma wave phase velocity over a specific region featuring a negative density gradient [15,275]. As the drive laser propagates through this region, the plasma wavelength increases, and so the back of the plasma wave moves at a reduced velocity. This temporarily decreases the potential required for trapping since a slower moving wave more readily accelerates particles to the velocity required to be trapped. Hence, injection can occur at lower plasma wave amplitudes than pure self-injection, so the electron beam quality can be higher. It also ensures that the injection is localised to the region of the density transition, thereby increasing energy stability [276]. In addition, by fixing the injection point physically in the plasma, the electron beam timing relative to the laser pulse is more precisely defined, allowing for a greater temporal synchronisation of subsequent stages.

Ionisation injection in a laser wakefield accelerator provides further control and increased flexibility by allowing injection only from a high-Z dopant gas species [277–280]. The dopant gas, e.g. nitrogen, is mixed to a concentration of a few percent in a background gas of hydrogen or helium. The outer levels of the dopant atoms are easily ionised at low intensity, and so the electrons are released in the rising edge of the laser pulse and form the plasma wakefield structure along with the electrons from ionisation of the H/He gas. However, the inner electrons of the high-Z dopant are only ionised near the peak of the laser and are therefore born inside the wakefield. They can then be born with a longitudinal velocity greatly different from the background plasma electrons which form the plasma waves. Hence, these electrons can be trapped at wave amplitudes well below the usual trapping threshold [281], and the process can be largely decoupled from the non-linear evolution of the laser pulse. The amount of trapped charge can be adjusted through control of the level of gas impurity, and the location of the injection can be controlled by allowing only a small physical extent to the mix region [282]. This process may also be further controlled by combining it with a negative density gradient to produce the highest level of control and stability.

Either mechanism or a combination of both can achieve the minimum requirements for the LPI stage. Ionisation injection, however, allows a far easier adjustment of electron beam charge and potentially greater stability of output beam parameters. In addition, a plasma source incorporating downramp injection may be easily modified to allow ionisation injection by merely changing the target gas to a mixture of species.

> The LPI stage of EuPRAXIA will employ controlled injection. Both downramp and ionisation injection will be able to produce beams of acceptable quality. In addition, a combination of injection mechanisms could produce beams which can directly achieve the target parameters.

## 13.3 Low-Energy Plasma Injector

The plasma parameter range can be determined from simple considerations. Using scaling laws [185], for an LPI driven by a laser pulse with $a_0 = 2$, the plasma density required to achieve a 150 (400) MeV energy gain is estimated be $n_e \sim 5.0 (1.9) \times 10^{18} \text{ cm}^{-3}$, and the corresponding dephasing length is $L_\varphi \sim 1.5 (6.5) \text{ mm}$. Short pulse Ti:sapphire laser drivers with power as little as 30 TW should be sufficient, though powers in excess of 100 TW provide greater versatility.

A crucial property of the electron bunch exiting the LPI is the emittance of the beam. To mitigate emittance growth at the exit of the injector, the density gradient at the plasma to vacuum transition should have a characteristic length larger than the betatron wavelength $\lambda_B = \sqrt{2\gamma_e\lambda_p}$, where $\gamma_e$ is the electron Lorentz factor and $\lambda_p$ the plasma wavelength [283].
<figure><img src="image_93.png" /><figcaption>**Fig. 13.1.** Optimal plasma profile used in simulations for the LPI using a density downramp to inject the beam before further acceleration. The inset shows a zoom of the profile in the passive plasma lens region.</figcaption></figure>

These values are indicative and will need to be tuned when the electron energy is specified. Published experimental or numerical results relevant to EuPRAXIA plasma properties [28,284,285] as well as specific simulation studies described below and in Chapter 23 are in agreement, giving confidence that these parameters can be achieved.

### 13.3.1 Simulation Results

Complementary to the scalings described above, three specific plasma injector scenarios were simulated using particle-in-cell codes. While injection using the resonant multi-pulse ionisation injection (ReMPI) technique is described in detail as part of the start-to-end simulations in Chapter 23.8, the following shows the predicted performance for an LPI based on density downramp and ionisation injection.

#### Density-Downramp Injection

This chapter presents simulation results for the injector stage where a plasma density transition is used to inject a bunch of electrons in the wakefield. After injection, the beam is accelerated for a few millimetres, reaching an energy of 250 MeV, and then is prepared to be captured at the transfer line after the plasma target. Below, the laser and plasma parameters used in our optimal simulation are described. The simulation was carried out with the PIC code OSIRIS (see Sect. 32.1 for details), and the beam parameters at end of the plasma (i.e., right before the transfer line) are presented.

The plasma profile is similar to the ones obtained in fluid simulations of gas cells [286], and it is shown in Figure 13.1. It consists of an upramp with a Gaussian profile and total length of 1 mm, where the density rises from 0 to $6 \times 10^{18} \text{ cm}^{-3}$. This is followed by a downramp with 150 µm length, which brings the density to $4 \times 10^{18} \text{ cm}^{-3}$, where it reaches a plateau. As the plasma wave expands during the downramp, some electrons at the back of the wakefield are injected in the accelerating structure and form a beam. This beam is accelerated in the plateau for 1.8 mm. We considered a plasma exit with an exponential profile and a total length of 500 µm coupled with a 5 mm long $1 \times 10^{16} \text{ cm}^{-3}$ plasma, which works as a passive plasma lens (see inset of Fig. 13.1). The plasma exit and lens prepare the beam for the transport to the next acceleration stage.

The laser vacuum parameters are the following: wavelength $\lambda = 800$ nm, pulse duration $\tau_{FWHM} = 30$ fs, and beam waist and normalised vector potential at the
<figure><img src="image_94.png" /><figcaption>**Fig. 13.2.** Beam energy distribution along its length at the end of the density profile of Figure 13.1. The region highlighted in red (between circa 7551.1 µm and 7552.1 µm) is able to fulfill all the requirements for the injector stage. The blue curves along the energy- and z-axes are the integrated profile of the red region.</figcaption></figure>

<table><caption>**Table 13.2.** LPI using a density downramp: beam parameters at the end of the plasma lens.</caption><thead><tr><th></th><th>Units</th><th>Values</th><th></th><th>Units</th><th>Values</th></tr></thead><tbody><tr><td>Energy</td><td>MeV</td><td>255.3</td><td>Energy Spread</td><td>–</td><td>0.78%</td></tr><tr><td>Charge</td><td>pC</td><td>30.66</td><td>Peak Current</td><td>kA</td><td>10.22</td></tr><tr><td>RMS norm. emittance x</td><td>µm</td><td>0.13</td><td>RMS norm. emittance y</td><td>µm</td><td>0.16</td></tr><tr><td>RMS γ Twiss x</td><td>m<sup>-1</sup></td><td>190.6</td><td>RMS γ Twiss y</td><td>m<sup>-1</sup></td><td>181.9</td></tr></tbody></table>

focus of $w_0 = 18$ µm, and $a_0 = 1.8$, respectively. The laser is linearly polarised and focussed at 200 µm into the plasma profile of Figure 13.1. Simulations show that the laser is greatly affected by the plasma; the normalised vector potential gets as high as $a_0 \approx 4$ before the density downramp due to self-focussing.

The beam charge injected at the density transition is 163 pC. While beam loading for the whole beam was not achieved simultaneously, the accelerating field in the central region is nearly constant. If we observe the beam energy distribution along its length at the end of the plasma lens (Fig. 13.2a), the full beam profile (in grey) has regions of high energy spread at the leading edge and tail. In the centre, however, the energy spread is small. The region highlighted in red is able to fulfill all the requirements of the injector stage, as shown in Table 13.2. The integrated profiles for this section are shown in the blue curves in Figure 13.2a, and the transverse beam profile for the selected electrons is shown in Figure 13.2b. One possible way to separate the particles in this region from the ones at the leading edge and tail is the use of magnetic fields, which is possible because the electron energy is higher compared to the centre. The remaining unwanted electrons could be lost at the transfer line or the next plasma stage since they might not be matched to those elements. Details of such a scheme will be developed and tested as part of the technical design phase of EuPRAXIA.

## Ionisation Injection

A parametric study was carried out of the main physical parameters for ionisation injection in order to identify optimised configurations. For this parametric study, the laser is assumed to have a Gaussian form both in space and in time. Non-Gaussian forms will be considered in future studies. Except for this Gaussian form, specific care was taken in choosing configuration parameters already validated in experiments, both for the laser and for the target.
<figure><img src="image_95.png" /><figcaption>Fig. 13.3. Schematic view of the target cell used in the simulation.</figcaption></figure>

<figure><img src="image_96.png" /><figcaption>Fig. 13.4. Longitudinal density profile for different lengths of the exit tube: 500 µm (black solid line), 750 µm (dashed blue line), and 1000 µm (dotted red line) [285].</figcaption></figure>

As described above, in the ionisation injection scheme (IIS), the gas medium in the target is composed of hydrogen with a small amount of high-Z atom (generally nitrogen). If ionised by the laser, the inner-shell electrons of the high-Z atom are more easily trapped in the plasma wakefield. The laser intensity is then adjusted to trap and accelerate only these inner-shell electrons. A simple way for controlling the accelerated charge is through the percentage of the impurity gas. By increasing this percentage, the amount of trapped charge increases. Because the size and duration of the electron bunch should be kept at small values for accelerator applications, the increase in the trapped charge generates an increase in the space-charge fields. As pointed out in [281], when these space-charge fields become significant, their increase yields a reduction of the final electron energy and an enhancement of energy dispersion. The objective of this study is thus to perform a detailed quantitative analysis of this effect within a configuration in accordance with the EuPRAXIA requirements.

## Simulation parameters

The laser configuration is similar to existing multi-TW laser systems. The laser wavelength is 0.8 µm; in the focal plane, its maximum intensity is $5.5 \times 10^{18} \text{ W cm}^{-2}$ with a waist of 16 µm. Its duration is 20 fs FWHM, resulting in a total energy of 0.47 J and a maximum power of 22 TW. The target gas cell is shown in Figure 13.3. It is similar to ELISA (ELectron Injector for compact Staged high energy Accelerator), as described in [280]. It is composed of a central tube with a large radius and a variable length adjusted to a value of 1 mm to reach the required energy range. This cell is connected to the vacuum chamber through two tubes of 600 µm in diameter. It has been verified experimentally that with this diameter, the tubes are not affected by the high-intensity laser. The length of the tube at the entrance is fixed at 500 µm. Different tube lengths can be used at the exit to modify the density gradient. The density profile has been determined by fluid simulations using the OpenFoam
<figure><img src="image_97.png" /><figcaption>**Fig. 13.5.** Laser evolution inside the ionisation injection plasma target. The red area indicates the longitudinal density profile of the gas cell (left axis). The evolution of $a_0$ (right vertical axis) in vacuum (dashed green line) and in plasma is shown for different values of CN2 [285].</figcaption></figure>

software. Examples of the obtained density profiles are shown in Figure 13.4. The target gas is composed of hydrogen with a small percentage CN2 of nitrogen. Simulations have been performed for CN2 = 2, 1, 0.5 and 0.35%, keeping the density of all the outer-shell electrons constant, with a maximum value of $4 \times 10^{18} \text{ cm}^{-3}$ electron density.

The numerical simulation has been performed with the PIC code Warp using the azimuthal Fourier decomposition algorithm in cylindrical geometry with 3 Fourier modes (0, +/−1) (see Sect. 32.1 for further details, also [287]), whereas field ionisation rates are calculated with the ADK model. In our PIC calculations, the numerical grid is a moving window 130 µm in radius, 58.4 µm in length, and with 36 macro-particles per species and per cell. The mesh resolution is chosen to be $\lambda_L/25$ and $\lambda_L/6$ in the longitudinal and transverse directions (with $\lambda_L$ the laser wavelength). A mesh resolution of $\lambda_L/30$ and $\lambda_L/10$ has also been used for the most optimal case for confirmation and for a more precise evaluation of the second-order beam properties, such as the beam emittance. The calculations were performed on 140 cores at the Mésolum computing centre of University Paris-Sud and on 1000 cores of the Curie supercomputer at the TGCC-CEA centre. Using these machines, the required computing time for describing the 5 mm interaction length in the plasma was between 10,000 and 30,000 CPU hours.

## Results and discussion

In Figure 13.5, the evolution of the maximum normalised laser vector potential is reported for the different values of CN2. We observe that this evolution is nearly identical for all values. This indicates that by keeping the same total density of outer-shell electrons, the plasma wave is independent of CN2. It allows a direct
<figure><img src="image_98.png" /><figcaption>Fig. 13.6. Energy distribution of the electrons at the exit of the target. Each plot corresponds to a specific value of CN2. (a) 2%; (b) 1%; (c) 0.5%; (d) 0.35% [285].</figcaption></figure>

identification of the contribution of the space-charge field induced by the trapped electrons. The energy distributions of accelerated electrons at the exit of the injector for different values of CN2 are shown in Figure 13.6. We observe a strong influence of CN2 on the energy distribution. Figure 13.6 also shows that electrons coming from $N^{6+}$ yield the main part of the low-energy pedestal. To obtain a low energy spread, their contribution has to be minimised by avoiding large laser intensities. The main characteristic of the high-energy peak, defined by the energy range of the maximum of the curve $\pm$ the FWHM width, are reported in Table 13.3. As in [281], we observe a decrease in the average energy and an increase in the energy dispersion with CN2. Only the two smallest values of 0.5 and 0.35% have an energy spread within the EuPRAXIA requirements, as described in Chapter 23. Note that these values of CN2 are smaller than the ones reported in several experimental works. In particular, at $CN2 = 1\%$ the energy spread is already much too large. Looking at the other beam characteristics at $CN2 = 0.35-0.5\%$, we observe that they are all in accordance with the EuPRAXIA requirements, except for the normalised emittance $\epsilon_{N,y}$ in the laser polarisation direction, which is larger by a factor of two. In fact in all calculations that we have performed for ionisation injection, a minimum value close to $\epsilon_{N,y} = 2$ mm mrad has been observed (see also [288]). This relatively large value of $\epsilon_{N,y}$ is mainly generated during the trapping process.

## Conclusions

A parametric study has been performed for the LPI with ionisation injection to determine the best conditions to meet the requirements of the EuPRAXIA design.
**Table 13.3.** Main characteristics of the electron bunch for different values of CN2. $\langle E \rangle$: average energy; $Q$: charge; $\delta E$: RMS width of the energy;

$varepsilon_{N,x(y)}$: normalised emittance; $\theta_{x(y)}$: RMS transverse divergence; $\tau_t$: RMS bunch length.

<table><thead><tr><th>C<sub>N2</sub> (%)</th><th>⟨E⟩ (MeV)</th><th>Q (pC)</th><th>δE/⟨E⟩ (%)</th><th>ε<sub>N,x</sub><br>(mm.mrad)</th><th>ε<sub>N,y</sub><br>(mm.mrad)</th><th>θ<sub>x</sub><br>(mrad)</th><th>θ<sub>y</sub><br>(mrad)</th><th>τ<sub>t</sub><br>(fs)</th></tr></thead><tbody><tr><td>2</td><td>93</td><td>108</td><td>29</td><td>2.6</td><td>3.9</td><td>4.5</td><td>5.6</td><td>7.0</td></tr><tr><td>1</td><td>114</td><td>72</td><td>15</td><td>1.9</td><td>2.3</td><td>3.7</td><td>4.0</td><td>3.4</td></tr><tr><td>0.5</td><td>135</td><td>37</td><td>5</td><td>1.1</td><td>1.9</td><td>2.8</td><td>3.1</td><td>3.0</td></tr><tr><td>0.35</td><td>142</td><td>27</td><td>4</td><td>0.8</td><td>1.8</td><td>2.2</td><td>2.9</td><td>2.9</td></tr></tbody></table>

We found that in our configuration, the space-charge field imposes a severe limitation on the beam charge, with a maximum value below 40 pC. For a charge between 20 and 40 pC, the properties of the electron bunch are within the EuPRAXIA requirements except for the normalised emittance in the laser polarisation direction, which is larger by a factor of two. Additional calculations also showed that the case with the lowest charge (here 27 pC) is the most robust when considering variations of the laser-plasma configuration. The next step will be to consider a more realistic description of the laser pulse by using data from experimental diagnostics. The final step will be devoted to a detailed analysis of the sensitivity of the injector performances on the fluctuations of the physical parameters (target and laser) planned as part of the technical design.

### 13.3.2 Plasma Structure

For a 150–250 MeV injector, a short plasma target of the order of 1 mm is considered. Both gas jets and gas cell targets could provide such a medium, and both have their own benefits (more details in Sect. 32.2). Gas jets exhibit an open geometry convenient for transverse probing and in-line density control. They can be used in conjunction with a sharp edge to generate a density transition to trigger electron trapping [284]. However, small-scale flow non-uniformities are thought to lead to a degradation of the injector’s electron beam quality [22]. Also, the typical sharp density gradients obtained from supersonic gas jets [289] are not adequate for mitigating emittance growth at the exit of the injector [283], which would reduce the coupling efficiency of the two stages.

To avoid emittance growth at the exit of the injector, the density gradient at the plasma-to-vacuum transition should have a characteristic length larger than the betatron wavelength $\lambda_{\beta} = \sqrt{2\gamma_e\lambda_p}$, where $\gamma_e$ is the electron Lorentz factor of the accelerated electrons. In the case of 150 MeV electrons accelerated in plasma densities in the range $n_e \simeq [10^{18}; 10^{19}]$ cm$^{-3}$, the betatron wavelength is:

$$ \lambda_{\beta} \approx \begin{cases} 800 \ \mu\text{m} & \text{for } n_e = 10^{18} \ \text{cm}^{-3}, \\ 253 \ \mu\text{m} & \text{for } n_e = 10^{19} \ \text{cm}^{-3}. \end{cases} $$

Gradients with such characteristic lengths, larger than a few hundred $\mu$m, can be achieved in gas cells or capillaries by adjusting the geometry of the transition volume between the inside of the target and the vacuum of the experimental chamber. Cells with various geometries can be used to confine the gas, but the probing of density profiles is more challenging. However, the walls of the gas cell can be in proximity to the laser path and are thus more susceptible to laser ablation of the target in case of misalignment or poor laser beam quality. Clearly, in the context of EuPRAXIA, as
<table><caption>**Table 13.4.** Experimental parameters for the low-energy LPI.</caption><thead><tr><th>Experimental parameter</th><th>Symbol</th><th>unit</th><th>LPI at 150 MeV</th><th>LPI at 400 MeV</th></tr></thead><tbody><tr><td>Plasma electron density</td><td>n<sub>e</sub></td><td>10<sup>18</sup> cm<sup>-3</sup></td><td>3–8</td><td>3–5</td></tr><tr><td>Plasma length</td><td>L<sub>p</sub></td><td>mm</td><td>0.6–1.5</td><td>0.8–1.6</td></tr><tr><td>Exit gradient length</td><td>L<sub>grad</sub></td><td>μm</td><td>300–500</td><td>600–800</td></tr><tr><td>Laser intensity</td><td>I<sub>L</sub></td><td>10<sup>18</sup> Wcm<sup>-2</sup></td><td>4.2–16.8</td><td>15–43</td></tr><tr><td>Normalised vector potential</td><td>a<sub>0</sub></td><td></td><td>1.4–2.8</td><td>2.6–4.5</td></tr></tbody></table>

high reliability is necessary, laser stability and quality should be specified to be good enough for using gas cells.

Both gas cells and gas jets, being quite inexpensive to manufacture, could be implemented in the final design and even share the same target positioning system. However, a different gas injection and extraction system would be needed for each target due to the different backing pressure ranges required. In particular, gas jets typically produce greater demands on the vacuum pumping system, and added to the aforementioned concerns on stability, it is not recommended to use gas jets in the base configuration for the facility. This makes gas cells, fashioned either in capillaries or in a more open geometry, the baseline configuration for the LPI stages. The electron injection scheme considered for this stage – likely relying on density gradient injection, ionisation-induced injection, or a combination of both – should be readily implemented in these different cell configurations.

Table 13.4 summarises the experimental parameter ranges based on scaling laws and published results required to achieve the EuPRAXIA plasma properties for the low-energy injector. It also gives the exit gradient length corresponding to the density range and electron energy.

Both gas cell and capillary targets would be suitable for the low-energy injector. Published results with existing laser systems show that generating these beams is possible with present experimental techniques.

## 13.4 GeV Plasma Injector

Similar to the considerations for the low-energy injector, we can determine experimental parameters to achieve 1 GeV electron beams at the exit of the LPI through scaling laws [185] and published results [19,290]. These well-studied results fix the density and plasma length. The requirements for mitigation of the emittance growth at the exit of the injector are also pertinent for the 1 GeV LPI. However, as the electron energy is higher, the exit gradient section has to be longer.

The length of the 1 GeV injector stage will range between 10 and 40 mm depending on the plasma density lying between ~1 × 10¹⁸ cm⁻³ and ~4 × 10¹⁸ cm⁻³. Laser power between 40 and 250 TW has been reported as necessary to reach these output energies [19,290], though it is likely that powers in excess of 100 TW will be required to be less dependent on non-linear laser evolution [290]. Nevertheless, for the length of target being considered here, it may be preferable to house the injector in some form of laser-guiding structure, such as a capillary discharge or capillary tube. This, though, must be weighed against the potentially shorter lifetimes of these devices as compared to open gas cells. Injection will be achieved in a similar way to the low-energy injector, i.e. ReMPI, downramp, ionisation injection, or a combination of these two schemes. The particular scheme in use will depend on the specific application of the injector and whether one wants to optimise a particular parameter such as charge, energy spread, etc.
<table><caption>**Table 13.5.** Experimental parameters for the 1 GeV LPI.</caption><thead><tr><th>Experimental parameter</th><th>Symbol</th><th>unit</th><th>LPI at 1 GeV</th></tr></thead><tbody><tr><td>Plasma electron density</td><td>n<sub>e</sub></td><td>10<sup>18</sup> cm<sup>-3</sup></td><td>1–6</td></tr><tr><td>Plasma length</td><td>L<sub>acc</sub></td><td>mm</td><td>9–30</td></tr><tr><td>Exit gradient length</td><td>L<sub>grad</sub></td><td>mm</td><td>0.9–2.1</td></tr><tr><td>Laser intensity</td><td>I<sub>L</sub></td><td>10<sup>18</sup> Wcm<sup>-2</sup></td><td>3.1–32.5</td></tr><tr><td>Normalised vector potential</td><td>a<sub>0</sub></td><td></td><td>1.2–3.9</td></tr></tbody></table>

<figure><img src="image_99.png" /><figcaption>**Fig. 13.7.** Schematic of the laser removal and coupling in a multi-stage configuration.</figcaption></figure>

The emittance growth considerations at the exit of the plasma also apply for this stage, and as the betatron wavelength scales as $\lambda_{\beta} \propto \gamma_e^{1/2} n_e^{-1/2}$, the exit gradient length should be longer. For 1 GeV electrons and plasma electron density in the range $n_e \simeq [10^{18}; 4 \times 10^{18}] \text{ cm}^{-3}$, the betatron wavelength is:

$$ \lambda_{\beta} \approx \begin{cases} 2 \text{ mm} & \text{for } n_e = 10^{18} \text{ cm}^{-3} \\ 1 \text{ mm} & \text{for } n_e = 4 \times 10^{18} \text{ cm}^{-3} \end{cases} $$

Table 13.5 shows the experimental parameter range to achieve the EuPRAXIA plasma properties for the 1 GeV LPI. Note that in the range indicated in Table 13.5, the higher values of $a_0$ correspond to the shortest plasma length because of a larger accelerating field. These values may need to be tuned to meet all EuPRAXIA target properties.

The GeV stage will share some commonality with the low-energy stage but ideally would be driven at higher laser power (>250 TW). Efficiency and reliability in beam pointing can be enhanced by using a guiding plasma target but will result in more regular target replacement.

# 13.5 Implementation in the EuPRAXIA Beamline

## 13.5.1 Staging Requirements

As schematically presented in Figure 13.7, after each plasma stage, the transmitted laser light has to be removed from the axis, and the laser driving the following stage has to be coupled into the plasma. The high intensities limit the proximity of the removal/coupling device to the plasma target, whereas the electron transport prescribes that the electron capture should be as close to the plasma as possible to limit the emittance growth (see [291]). As a mirror placed close to the laser focus would suffer damage from heating and ionisation, plasma mirrors are considered as a solution to remove the laser from the electron axis.
### 13.5.2 Laser Removal

After the LPI interaction, a significant fraction of the laser pulse energy remains and is capable of damaging downstream components. The most compact way to remove the laser pulse is to use a thin mirror to reflect the laser pulse shortly after the interaction. The foil will be damaged upon reflecting and absorbing the laser, and so must be rapidly replenished. A commonly used device for this purpose is a tape drive, which allows a thin foil to be easily translated between shots so that each laser pulse interacts with a fresh surface. A similar device can be used for laser in-coupling for later laser-plasma accelerator stages. However, in the case of out-coupling, since all that is being reflected is spent laser energy, it is not necessary to have a high reflectivity or good reflected wavefront quality as long as the transmitted energy is low. Hence the requirements here are not as demanding.

A metal-coated plasma tape can be used, with the substrate (~10 µm) and metal coating (~20 µm) kept as thin as possible to reduce scattering of the traversing particle beam. It may be necessary to allow some drift space at the end of the plasma source so that the plasma density at the foil surface is low. Otherwise, a wakefield driven by the reflecting laser pulse may interact with the electron bunch and reduce its beam quality. This distance will be determined by the plasma density profile at the gas cell exit, and the tape itself will add a further ~5 mm drift space to the first electron capture optic. Therefore, the total distance before the next downstream element will be 5-10mm depending on the LPI scheme employed. A beam pipe will be required to propagate the laser pulse in vacuum until the laser intensity has dropped sufficiently to be absorbed by a permanent laser dump.

There remains a possibility for the electron beam quality to be degraded because of bremsstrahlung or by interaction with the fields generated by the laser at the beam dump. Further studies will be required to determine the effect of this interaction and if any steps are required to minimise potential adverse effects.

> Laser drivers require the coupling of the optical pulses to the accelerator beamline. For removing spent laser energies, plasma mirrors can be used with little drawback.

### 13.5.3 Stability

In order to achieve reproducibility of the electron energy from shot to shot, the external electron beam issued from the LPI should be injected in the same phase-space volume of the LPAS with every shot. This implies a need to control the position of the injected electron beam relative to the plasma wave, both transversely and longitudinally. This is determined by the pointing stability, and relative timing jitter of the laser driver as well as the pointing stability, transverse emittance, and timing jitter of the generated electron beam. The reproducibility of the plasma wave amplitude is controlled by the reproducibility of laser intensity and plasma electron density.

Table 13.6 summarises the main parameters of interest for the generated electron beam on the left, and the accelerator properties that have a direct influence on their stability on the right.

### 13.5.4 Repetition Rate

To achieve a high-repetition-rate operation of gas targets in a high-vacuum environment, it may be necessary to use differential pumping to reduce the gas load on the main chamber. This has been demonstrated with a 5 Hz gas jet in laser-plasma
<table><caption>**Table 13.6.** Beam parameters and the accelerator properties that impact their stability.</caption><thead><tr><th>Beam parameter</th><th>Injector and accelerator constraints</th></tr></thead><tbody><tr><td>Particle energy</td><td>Accelerating phase stability<br>Drive laser stability<br>Beam loading effect<br>Plasma density and length stability</td></tr><tr><td>Bunch charge</td><td>Laser stability at injection<br>Plasma properties at injection</td></tr><tr><td>Emittance</td><td>Laser and plasma properties at injection<br>Plasma matching sections</td></tr><tr><td>Transverse size</td><td>Laser beam quality<br>Transverse density profile<br>Longitudinal density profile</td></tr><tr><td>Divergence</td><td>Laser pointing stability<br>Input electron pointing stability<br>Plasma transverse gradient stability<br>Laser pulse front tilt stability</td></tr></tbody></table>

experiments at the CLF [292] and LWFA experiments at LUX [66]. The LUX design uses a continuous flow target and a differential pumping system which can achieve $10^{-6}$ mbar vacuum pressures at a distance of 1.5 m from the plasma source. Adopting such an approach would allow for a very-high-repetition-rate operation, limited by recombination and pressure equilibration time, and provides compatibility with accelerated beamline vacuum requirements. However, gas jet targets generally produce high loads on the vacuum system, and smaller targets, such as gas cells and capillaries, can significantly reduce pumping requirements. However, as noted before, this must be weighed with the potentially shorter life-times of such targets. Further tests will be required to determine how to construct a tailored plasma structure in a way which is robust to laser damage. This will be critically important for continuous high-repetition-rate performance with petawatt-class laser systems, such as may be used for the GeV LPI stage.

Numerous factors can affect the repeatability and control of plasma accelerators, and EuPRAXIA should demand state-of-the-art control on all of these parameters.

## 13.6 Conclusion

Though LPI electron sources have only been demonstrated relatively recently as a source of high-energy electrons, they may offer remarkable benefits to the proposed EuPRAXIA beamlines. Their unique nature promises short, relatively high-brightness electron beams in an extremely compact configuration. This can have important beneficial repercussions for the delivery of EuPRAXIA beams, especially to other laser-driven accelerator structures. Furthermore, not only are they suitable for injection into successive plasma accelerator stages, but also, given their potentially high currents, they can be considered as drivers for plasma wakefield stages (see Chap. 26).

However, the novelty of these structures also produces many challenges. Transporting the short and often divergent beams produced in LPIs will be a challenge, especially while maintaining their ultra-short duration. Also, whilst many of the properties required for the EuPRAXIA injectors have been reached in proof-of-principle
experiments, no single experiment has yet simultaneously achieved all of the necessary parameters in one demonstration. Furthermore, most of these experiments have taken place at existing high-power laser facilities where shot-to-shot reproducibility is not emphasised.

Clearly, whilst the methodology is demonstrated and injectors for high-quality beams can be produced in simulation, the direct regular operation of LPIS suitable for EuPRAXIA is a remaining challenge to demonstrate during the technical design phase.

# 14 RF Photo-Injector

## 14.1 Introduction

Multi-GeV acceleration, both laser- and particle-beam-driven, has been already demonstrated in cm-scale plasma structures [11,45,290,293]. Great efforts are currently ongoing in several groups [294–297] worldwide for the acceleration of high-brightness electron beams, which need to be captured from plasma or RF injectors. In all cases, the control of electron injection into the plasma accelerator module is mandatory for efficient acceleration; the beam must satisfy the transverse matching condition at the plasma entrance to prevent envelope oscillations that may cause emittance growth. In particular, the following condition for the Twiss parameters of the witness beam holds in the blow-out regime and assuming negligible beam loading [298]:

$$ \beta_{matching} = \frac{\sqrt{2\gamma}}{k_p} \qquad (14.1) $$

where $k_p = 2\pi/\lambda_p$ is the inverse plasma skin depth, with the plasma wavelength depending on the plasma background density as $\lambda_p(\mu m) \sim 3.3 \cdot 10^{10} n_p^{-1/2}(\text{cm}^{-3})$; $\gamma$ is the Lorentz factor for the electron beam. The matching condition for the beam transverse size is:

$$ \sigma_{matching} = \sqrt{\frac{\beta_{matching}\varepsilon_n}{\gamma}} \qquad (14.2) $$

with the typical numbers involved, e.g. $\gamma=1000$, $n_p=10^{16} \text{cm}^{-3}$, and $\varepsilon_n=1 \text{ mm mrad}$; the beam transverse size is of micrometre scale.

Here, a fully external injection scenario is considered, meaning that bunches (witness for the LWFA case and both driver and witness for the PWFA case) are externally produced by an RF photo-injector and then delivered to the plasma-accelerating section (Schemes 3 to 5, as described in Chap. 8). Studies for the accelerator and start-to-end simulations of the electron beam dynamics from the photo-cathode to the plasma entrance have been performed by means of numerical codes, and they are described in detail in the following sections.

Beam parameters – e.g. charge, emittance, peak current, and energy spread, except for the final energy – must be guaranteed at the plasma entrance.

Concerning the LWFA case (Scheme 3: RFI-500 MeV + LPAS-5 GeV and Scheme 4: RFI-240 MeV + LPAS-2.5 GeV + chicane + LPAS-5 GeV), we have investigated two methods for satisfying the requirements on the peak current, thus on the bunch length, while preserving the beam quality. One method is based on a full RF compression scheme, occurring in the first two S-band traveling wave (TW) sections, and a boost to a final energy of about 500 MeV in the X-band linac. The second method is based on a hybrid compression scheme, combining moderate RF compression in the first S-band TW section and magnetic compression at higher energy through a
<table><caption>**Table 14.1.** Target values for the 1 GeV electron beam parameters to drive a SASE FEL.</caption><thead><tr><th></th><th>Units</th><th>1 GeV case</th></tr></thead><tbody><tr><td>Bunch charge</td><td>pC</td><td>30</td></tr><tr><td>Peak current</td><td>kA</td><td>3</td></tr><tr><td>Repetition rate</td><td>Hz</td><td>10</td></tr><tr><td>RMS normalised emittance</td><td>μm</td><td>1</td></tr><tr><td>RMS energy spread</td><td>%</td><td>1</td></tr><tr><td>RMS bunch length</td><td>fs</td><td>10</td></tr><tr><td>Slice normalised emittance</td><td>μm</td><td>&lt;1</td></tr><tr><td>Slice energy spread</td><td>%</td><td>0.1</td></tr><tr><td>Slice length</td><td>μm</td><td>0.75</td></tr><tr><td>Radiation wavelength</td><td>nm</td><td>4</td></tr><tr><td>Pierce parameter ρ</td><td>1 × 10<sup>-3</sup></td><td>&gt;1</td></tr><tr><td>Undulator period</td><td>cm</td><td>1.5</td></tr><tr><td>Undulator strength K</td><td></td><td>0.872</td></tr></tbody></table>

chicane. The two methods have been optimised for slightly different layouts, but with proper design, only one layout could host both compression schemes.

## 14.2 Laser-Driven External Injection Scheme

In this section, we will discuss two possible RF injector layouts for the successful operation of a laser-driven wakefield accelerator (LWFA), based on the external injection of high-brightness electron bunches (Scheme 3: RFI-500 MeV + LPAS-5 GeV and Scheme 4: RFI-240 MeV + LPAS-2.5 GeV + chicane + LPAS-5 GeV). The requirements at the plasma-accelerating module for the 1 GeV case have been identified in the framework of the EuPRAXIA collaboration and are listed in Table 14.1.

The two layouts differ in the compression scheme adopted for the achievement of kA peak current, while preserving the normalised transverse emittance at the plasma entrance.

## 14.3 Full RF Compression Scheme

At the EuPRAXIA facility, the main challenge for the RF photo-injector comes from the request of producing ultra-short, high-quality electron beams. A case of interest foresees 500 MeV witness beam energy at the plasma interface, with a slice emittance of much less than 1 mm mrad and 30 pC charge in a 10 fs FWHM length, which turns into a peak current of up to 3 kA. Except for the final energy, these parameters are those requested to generate SASE FEL radiation at 3 nm with a 1 GeV electron beam.

A pure RF compression, applying the velocity-bunching scheme, has been considered to produce in one stage a 3 kA beam at the end of the S-band TW sections with 100 MeV.

High-quality electron beams can be achieved in RF photo-injectors by means of RF guns, equipped with laser-driven photo-cathodes, followed by booster sections. An emittance compensation scheme [258] based on a focussing solenoid at the exit of the RF gun can be used in photo-injectors to control emittance growth from space-charge effects. In addition to the invariant envelope theory [299], a proper matching of the
transverse phase space of the electron beam, injected in the downstream accelerating sections (booster), can help to control the transverse emittance oscillations during the acceleration. Under the conditions of an invariant envelope and proper phasing of space-charge oscillations [300], the final emittance is almost compensated down to the thermal emittance value given by cathode emission with an expected emittance scaling like $\epsilon_n \sim \sigma_{cath} \sim \sqrt{Q}$, where $\sigma_{cath}$ is the laser spot size on the photo-cathode, and $Q$ the extracted electron charge. A compression stage can be employed to shorten the beam length so as to achieve the required high peak current. The so-called velocity-bunching method [301] has opened up a new possibility of compressing the beam inside an RF structure and, when integrated in the emittance compensation process [260], can provide the desired bunch current values with the advantage of compactness of the machine and the absence of coherent synchrotron radiation (CSR) effects present in a magnetic compressor [302].

It is interesting to notice that a shortened beam length also permits the containment of the energy spread dilution due to RF curvature degradation; indeed, the energy spread depends on the bunch length, $\sigma_z$, and the accelerating frequency, $f_{RF}$, as $\frac{\Delta\gamma}{\gamma} \sim 2(\pi f_{RF} \sigma_z / c)^2$, where an on-crest operation in full relativistic conditions has been considered. To avoid the energy spread dilution from RF curvature degradation effects, a bunch length of $\sigma_z \le 130 \ \mu m$ must be injected in the X-band linac, ensuring $\frac{\Delta\gamma}{\gamma} \le 0.1\%$.

One proposed layout for the EuPRAXIA photo-injector is based on the experience at the SPARC_LAB test facility [296] in operation at INFN laboratories at Frascati, and devoted to plasma-based experiments [303-305], both to accelerate and focus charge particle beams, and to generate advanced radiation, such as from a multi-color FEL [306-309], $\gamma$-rays through Thomson backscattering [310,311], and both broad- and narrow-band high-peak-power THz radiation [312,313].

The layout of the S-band photo-injector [314], operating at 2.856 GHz, is shown in Figure 14.1; it consists of a 1.6-cell UCLA/BNL/SLAC-type standing-wave RF gun, including a copper photo-cathode with an emittance-compensating solenoid followed by 3 m long SLAC-type travelling-wave sections operating at 2.856 GHz [262]. The first two accelerating sections are embedded by a solenoid; each solenoid is composed of 13 coils, with the first coil and the other twelve coils, in groups of three, independently supplied. The beamline matching foresees a proper set of the emittance compensation solenoids and of the S-band cavity gradients in the velocity-bunching scheme [301], according to the invariant envelope criteria [299]. In this configuration, the first and second TW sections can operate far from the crest in the velocity-bunching regime, enabling the RF compression of the beam length, while the third section operates almost on crest to let the electron bunch gain energy and freeze its phase-space quality.

The main operating parameters of this EuPRAXIA photo-injector for the optimised witness beam in case of LWFA in external injection are summarised in Table 14.2.

A photo-cathode laser pulse with a Gaussian longitudinal profile of RMS length of $\sigma_z = 105 \ \mu m$ and a transverse uniform distribution with a spot size $\sigma_r = 175 \ \mu m$ has been chosen, as reported in Figure 14.2. The velocity-bunching regime is applied to the first two S-band structures to shorten the beam length from 102 $\ \mu m$ (on crest) to $\sim 3 \ \mu m$ (RMS), both cavities working close to the zero crossing of the field. The emittance minimisation is obtained, setting the gun solenoid at $\sim 3 \ kG$ and those surrounding the first and second S-band cavities at 0.32 kG and 0.50 kG, respectively. A slightly off-crest operation of the third S-band cavity further reduces the energy spread at the injector exit. In this configuration, the design electron beam parameters at the photo-injector exit are listed in Table 14.3, as obtained from beam dynamics
<figure><img src="image_100.png" /><figcaption>Fig. 14.1. Layout of the SPARC-like RF photo-injector: a 1.6 cell UCLA/BNL-type SW RF gun, equipped with a copper photo-cathode and an emittance compensation solenoid, followed by three S-band TW SLAC type sections; another two compensation solenoids surround the first and the second structures for the operation in the velocity-bunching scheme.</figcaption></figure>

<table><caption>Table 14.2. Main photo-injector parameters.</caption><thead><tr><th>Parameter</th><th>Unit</th><th>Value</th></tr></thead><tbody><tr><td>Gun electric field amplitude</td><td>MV/m</td><td>120</td></tr><tr><td>Gun electric field operation phase</td><td>deg</td><td>30</td></tr><tr><td>Output gun beam energy</td><td>MeV</td><td>5.6</td></tr><tr><td>Amplitude of electric field in the three TW sections</td><td>MV/m</td><td>20.0/20.0/28.0</td></tr><tr><td>Magnetic field in the emittance compensating solenoid</td><td>kG</td><td>3.05</td></tr><tr><td>Magnetic field in the linac solenoids</td><td>kG</td><td>0.32/0.50</td></tr><tr><td>Total photo-injector length</td><td>m</td><td>12</td></tr></tbody></table>

<figure><img src="image_101.png" /><figcaption>Fig. 14.2. Charge distribution at the cathode surface produced by the photo-cathode laser pulse, as obtained with 2D TStep simulations.</figcaption></figure>

simulations using 30k macro-particles with the code TStep [315]. The simulation results are shown in Figure 14.3 for the optimised witness-like beam; the left plot shows the evolution of the transverse normalised emittance (red line), spot size (blue line), and longitudinal bunch length (green line), while the right plot illustrates the energy (blue line) and energy spread (green line) from the cathode down to the photo-injector exit, as obtained with the TStep code.

The longitudinal and transverse phase spaces at the photo-injector exit are reported in Figure 14.4. The longitudinal profile of the beam exiting the photo-injector is typical of the velocity-bunching process showing a spike of current at the head of the bunch and a long tail, with a FWHM bunch length of 3 µm.

The 100 MeV beam exiting the photo-injector is then matched to the X-band linac to boost the energy up to about 500 MeV. The X-band linac mainly consists of two sections located before and after the magnetic chicane, named as L1 and L2, respectively. Twelve X-band accelerating sections, 50 cm long, are foreseen for L1 and twenty for L2. According to the RF power system design, the maximum accelerating gradient applied might be up to ~80 MV/m. For the study of this working point, a more conservative accelerating gradient, ~60 MV/m, has been considered. Between
<table><caption>**Table 14.3.** Witness beam parameters at the end of the photo-injector in the case of pure RF compression.</caption><thead><tr><td>Parameter</td><td>Unit</td><td>Value</td></tr></thead><tbody><tr><td>Charge</td><td>pC</td><td>30</td></tr><tr><td>Energy</td><td>MeV</td><td>100</td></tr><tr><td>RMS energy spread</td><td>%</td><td>0.27</td></tr><tr><td>RMS bunch length</td><td>fs</td><td>10</td></tr><tr><td>Peak current (FWHM)</td><td>kA</td><td>4.0</td></tr><tr><td>RMS normalised emittance</td><td>mm mrad</td><td>0.44</td></tr><tr><td>Repetition rate</td><td>Hz</td><td>10</td></tr></tbody></table>

<figure><img src="image_102.png" /><figcaption>**Fig. 14.3.** Evolution along the injector of the electron beam transverse normalised emittance ($\epsilon_T$, red line), envelope ($\sigma_T$, blue line), and longitudinal bunch length ($\sigma_z$, green line), as obtained with the TStep code in the case of pure RF compression. The right-hand plot shows the evolution of mean energy ($\langle E \rangle$, blue line) and energy spread ($\Delta\gamma/\gamma$, green line).</figcaption></figure>

L1 and L2, a 10 m long magnetic chicane is foreseen for phase-space manipulation and / or longitudinal compression of the bunch in case hybrid compression schemes want to be addressed; at the same time, when the chicane dipoles are switched off, the straight beamline accommodates the middle energy diagnostic station for beam parameter measurements. Note that this setup is somewhat different from the minimum linac configuration described in Chapter 12. Compared to the latter, a longer X-band section was chosen in this particular design to allow more flexibility in selecting between different compression schemes and beam energies (through variations in the accelerating gradient) without underestimating the potential linac length in the final design.

The following simulations have been performed with the chicane between the two linac sections off. The two linac sections L1 and L2 have been optimised to provide the required beam acceptance, from the photo-injector and after the magnetic chicane, for the considered working points described in this chapter; the best focussing strength for the lattice has been found with a betatron phase advance per cell of 15° and 28° for L1 and L2, respectively. In Table 14.4, the L1 and L2 main parameter list is reported.

For this working point, the X-band RF linac is meant to provide at the plasma module an electron beam of $Q = 30$ pC, $I = 3$ kA (FWHM), with a transverse spot size of 4 µm; the electron bunch is fully compressed in the photo-injector by means of the velocity-bunching operation scheme. An accelerating gradient of $E_{acc} \sim 20-36m^{-1}$
<figure><img src="image_103.png" /><figcaption>Fig. 14.4. Upper plots: transverse and longitudinal phase spaces. Lower plots: transverse (x and y) distribution, energy, and current profile. The results are outputs from the TStep code at the photo-injector exit.</figcaption></figure>

<table><caption>Table 14.4. L1 and L2 linac parameter list.</caption><thead><tr><th rowspan="2">Beam Parameter</th><th rowspan="2">Unit</th><th>L1</th><th>L2</th></tr><tr><th>LWFA</th><th>PWFA</th></tr></thead><tbody><tr><td>Initial energy</td><td>GeV</td><td>0.098</td><td>0.212</td></tr><tr><td>Final energy</td><td>GeV</td><td>0.212</td><td>0.550</td></tr><tr><td>Active linac length</td><td>m</td><td>6.0</td><td>10.0</td></tr><tr><td>Accelerating gradient</td><td>MV/m</td><td>20.0</td><td>36.0</td></tr><tr><td>RF phase (0 crest)</td><td>deg</td><td>-20.0</td><td>-19.5</td></tr><tr><td>Initial energy spread</td><td>%</td><td>0.27</td><td>0.15</td></tr><tr><td>Final energy spread</td><td>%</td><td>0.15</td><td>0.07</td></tr><tr><td>Final bunch length</td><td>mm</td><td>0.005</td><td>0.005</td></tr></tbody></table>

is applied in the L1 and L2 linac sections, respectively (see Tab. 14.4), and the final electron beam energy is $E_{L2exit} \sim 550$ MeV, with an energy spread of less than 0.1% (Fig. 14.5).

Before entering the plasma capillary, a focussing triplet of permanent magnet quadrupoles (PMQ) is foreseen at a distance of a few centimetres from the plasma entrance to obtain a typical $\beta$-function $\beta_{x,y} \sim 1-5$ mm. The gradient of the first three PMQs is around $G \sim 300$ T/m with a magnetic length of 5–10 cm; the emittance dilution due to chromatic effects in the quadrupoles is the main concern of the final focussing stage. A longitudinal position adjustment setup to tune the strength of the final focus array and latest generation of tunable permanent quadrupoles are under study to increase as much as possible the tunability of the magnet arrangement and widen the energy acceptance of the transfer line.

This working point for the RF photo-injector has been optimised and validated by means of its integration into a start-to-end simulation, which included the plasma-accelerating module and the matching sections to provide the complete evolution of the bunch transverse and longitudinal properties from photo-cathode to the undulator entrance; the results are reported in Figures 14.6 and 14.7, respectively.
<figure><img src="image_104.png" /><figcaption>Fig. 14.5. Longitudinal phase space of the LWFA-accelerated electron beam at the L2 linac exit.</figcaption></figure>

<figure><img src="image_105.png" /><figcaption>Fig. 14.6. Start-to-end simulation results for the witness bunch from the photo-cathode to the undulator entrance: evolution of the electron beam transverse normalised emittance ($\epsilon_{nx}$ red line, $\epsilon_{ny}$ red dotted line) and spot sizes ($\sigma_x$ blue line, $\sigma_y$ blue dotted line).</figcaption></figure>

<figure><img src="image_106.png" /><figcaption>Fig. 14.7. Start-to-end simulation results for the witness bunch from the photo-cathode to the undulator entrance: evolution of the energy ($E$ red line) and energy spread ($\Delta E/E$ red dotted line) as well as longitudinal bunch length ($\sigma_z$ blue line).</figcaption></figure>
<figure><img src="image_107.png" /><figcaption>Fig. 14.8. Injection layout for the hybrid compression scheme.</figcaption></figure>

## 14.4 Hybrid Compression Scheme

An alternative to full RF compression is the integration of a hybrid compression scheme, consisting of RF based at low energy and magnetic based at higher energy, to finalise the longitudinal compression. In this regard, the layout of the RF photo-injector is based on the ARES linac [316] at the SINBAD facility [317], hosted by DESY Hamburg.

The layout of the proposed RF injector is shown in Figure 14.8. The electron bunch generated by the 1.5-cell S-band photo-cathode RF gun is velocity bunched and accelerated by five 4.2 m long S-band traveling wave structures. The first two structures are surrounded by solenoids in order to provide better emittance compensation and focussing. Afterwards, the bunch is matched into the magnetic chicane compressor with an $R_{56}$ of about 9 mm for further compression. To match the electron beam at the plasma module, the final focus section consists of a triplet of permanent magnet quadrupoles (PMQ) 3 cm in length and with gradients of 300 T/m, 600 T/m, and 600 T/m, respectively.

The electron bunch is compressed using a two-stage hybrid compression scheme, i.e. the beam is compressed by means of velocity bunching [260] in the first S-band structure and magnetic compression successively. The initial bunch charge at the photo-cathode is 50 pC; to meet the beam requirements at the plasma module (see Tab. 14.1) and, in particular, a bunch charge of 30 pC, a slit collimator is located in the middle of the chicane with a full width of 0.6 mm. Situated at a position of maximum horizontal dispersion inside the chicane, the slit collimator is used to control both charge and bunch length by allowing only the central slice of the dispersed bunch to pass through [318,319]. The two-stage compression scheme, with a slit collimator integrated in the magnetic chicane, also helps to reduce the non-linearity of the longitudinal phase space in case strong bunch compression is required (i.e. ~ fs bunch duration). Compared to the pure magnetic compression scheme, the hybrid compression scheme has a much lower charge loss at the slit collimator. In addition, since a much smaller amount of charge is required to be extracted from the photo-cathode, the beam emittance will be smaller. Finally, this scheme helps to reduce both the energy and peak current jitter after bunch compression [318]. The energy jitter affects the longitudinal location of the focal point, while the peak current jitter affects the beam loading effect as well as the FEL output after the LWFA stage.

Preliminary studies have been performed to define the minimum injection energy at the plasma module based on the requirements listed in Table 14.1, in particular to guarantee at the plasma entrance the 3 kA peak current, <1 mm mrad slice normalised emittance, and ≪1% energy spread needed to drive a SASE FEL. In addition, to meet the plasma matching condition of equation (14.2), the transverse beam size must be in the μm range. Afterwards, start-to-end simulations have been performed by using the combination of two codes. The beam dynamics from the photo-cathode to the linac exit was simulated using ASTRA [320] with a two-dimensional
<figure><img src="image_108.png" /><figcaption>**Fig. 14.9.** Evolution of the transverse beam size and the normalised emittance (here defined as $\epsilon_{x,y}$) from the photo-cathode to the linac exit.</figcaption></figure>

<figure><img src="image_109.png" /><figcaption>**Fig. 14.10.** Beam optics from the linac exit to the injection point.</figcaption></figure>

cylindrical-symmetric space-charge algorithm. Afterwards, it was simulated up to the plasma by using IMPACT-T [321] with a 3D space-charge algorithm and 1D CSR model. The photo-cathode laser pulse was assumed to have a Gaussian longitudinal shape with an RMS duration of 3 ps and a uniform transverse laser intensity distribution ($\sigma_x = \sigma_y \sim 0.17$ mm) taken at the photo-cathode. The maximum gradient of the 1.5-cell S-band RF gun was assumed to be 110 MV/m. By operating the five structures with the same gradient of 25.5 MV/m and different off-crest phases (–487.0, –55.0, –41.0, –41.0, and –41.0) in degrees, the final beam energy is about 240 MeV. The evolution of the transverse beam sizes and emittances from the photo-cathode to the linac exit are shown in Figure 14.9.

The beam optics from the linac exit to the injection point is shown in Figure 14.10. The beam parameters at the linac exit and the injection point are summarised in Table 14.5, and the transverse and longitudinal phase spaces are shown in
<table><caption>Table 14.5. Beam parameters at the linac exit and the injection point.</caption><thead><tr><th></th><th>Linac exit</th><th>Injection point</th></tr></thead><tbody><tr><td>Energy (MeV)</td><td>242.0</td><td>240.8</td></tr><tr><td>Bunch charge (pC)</td><td>50.0</td><td>29.8</td></tr><tr><td>RMS bunch length (fs)</td><td>160.0</td><td>7.5</td></tr><tr><td>Peak current (kA)</td><td>0.13</td><td>4.0</td></tr><tr><td>Projected emittance ε<sub>n,x</sub>/ε<sub>n,y</sub> (μm)</td><td>0.30/0.30</td><td>0.81/0.46</td></tr><tr><td>Slice emittance ε<sub>n,x</sub>/ε<sub>n,y</sub> (μm)</td><td>0.28/0.28</td><td>0.59/0.34</td></tr><tr><td>β<sub>x</sub>/β<sub>y</sub> (mm)</td><td>/</td><td>3.1/3.0</td></tr><tr><td>RMS energy spread (%)</td><td>0.50</td><td>0.27</td></tr><tr><td>Slice RMS energy spread (%)</td><td>0.05</td><td>0.23</td></tr></tbody></table>

<figure><img src="image_110.png" /><figcaption>Fig. 14.11. Transverse and longitudinal phase spaces of the beam at the linac exit (upper row) and the injection point (lower row). The normalised emittance is defined as ε<sub>x,y</sub> here.</figcaption></figure>

Figure 14.11. The FWHM bunch length at the injection point is only 5 fs. However, there is merely one 0.75 µm long slice with a peak current higher than 3 kA. The influence of the space-charge effects after bunch compression is found to be significant. From the chicane exit to the injection point, the RMS energy spread of the bunch increases from 0.18% to 0.27%, and the vertical slice emittance increases from 0.27 µm to 0.34 µm. However, it is interesting to find out that the horizontal slice emittance decreases slightly in this region. The decrease in the horizontal slice emittance could be caused by the cancellation of the correlations built by the strong CSR effect in the chicane. It should be noted that the chromatic aberration at the PMQ triplet also contributes considerably to the vertical emittance growth [322].

## 14.5 Particle-Driven External Injection Scheme

For the beam-driven external injection case (Scheme 5: RFI-500 MeV + PPAS-1 GeV), a comb-like configuration for the electron beam, consisting of a 200 pC driver followed by a 30 pC witness bunch, has been explored with the aim to optimise the witness parameters and to set the longitudinal distance between the two bunches at the desired value.

The comb-like operation foresees the generation of two or more bunches within the same RF accelerating bucket through the so-called laser-comb technique
<table><caption>Table 14.6. L1 and L2 linac parameter list for the PWFA external injection scheme.</caption><thead><tr><th>Beam Parameter</th><th>Unit</th><th>L1</th><th>L2</th></tr></thead><tbody><tr><td>Initial energy</td><td>GeV</td><td>0.102</td><td>0.222</td></tr><tr><td>Final energy</td><td>GeV</td><td>0.222</td><td>0.582</td></tr><tr><td>Active linac length</td><td>m</td><td>6.0</td><td>10.0</td></tr><tr><td>Accelerating gradient</td><td>MV/m</td><td>20.0</td><td>36.0</td></tr><tr><td>RF phase (0 crest)</td><td>deg</td><td>0.0</td><td>0.0</td></tr><tr><td>Initial energy spread</td><td>%</td><td>0.15</td><td>0.11</td></tr><tr><td>Final energy spread</td><td>%</td><td>0.11</td><td>0.07</td></tr><tr><td>Final bunch length</td><td>mm</td><td>0.006</td><td>0.007</td></tr></tbody></table>

[323,324], consisting of a train of laser time-spaced pulses that illuminate the photo-cathode. The witness arrives earlier than the driver on the photo-cathode, and then they are reversed in time at the end of the velocity-bunching process, during which the longitudinal phase space is rotated. Experimental results have been obtained at SPARC_LAB, where the laser-comb technique is routinely used to produce trains of multiple electron bunches [325] for narrow-band THz generation [326], two-color FEL experiments [327,328], and resonant particle-driven plasma acceleration [303].

Computational studies have been dedicated to provide at the plasma module two bunches, i.e. driver and witness, separated by at least 0.55 ps, which corresponds to $\lambda_p/2$ with the plasma wavelength $\lambda_p = 330 \mu m$ for a plasma background density $n_p = 10^{16} \text{cm}^{-3}$. Both driver and witness bunches must be compressed down to $\sim 50$ fs and $\sim 10$ fs (FWHM), respectively; the witness bunch length must be much less than the plasma wavelength to minimise the energy spread growth. In addition, one more request is on the minimisation of the emittance growth, which unavoidably occurs because of the witness and driver overlapping during the velocity-bunching regime.

A linac setup equal to the one described in Chapter 14.3 is assumed in this scenario. The same accelerating gradient of $E_{acc} \sim 20-36 \text{ MeV m}^{-1}$ is applied in the L1 and L2 linac sections, respectively; the final electron beam energy is $E_{L2exit} \sim 580 \text{ MeV}$, with an energy spread of less than 0.1%. The linac parameters are listed in Table 14.6, and the longitudinal phase space at the end of the linac is depicted in Figure 14.12.

The photo-cathode laser has been shaped to provide at the cathode a witness and a driver bunch separated by 4 ps. A Gaussian longitudinal distribution with $\sigma_z = 120 \mu m$ (RMS) and a uniform transverse distribution of radius $r = 0.35$ mm have been assumed for the witness pulse at the cathode. Figure 14.13 depicts the transverse and longitudinal distributions of the photo-cathode laser at the cathode surface. The photo-cathode laser has been shaped via the laser-comb technique, also in intensity to provide at the cathode a 30 pC witness beam (red) separated by 4 ps from the 200 pC driver bunch (blue).

On the other hand, the driver spot size on the cathode has been chosen, looking at the witness quality, as the witness emittance and longitudinal profile are dependent on it. The optimal value for the driver transverse dimension at the cathode surface, in terms of emittance and duration of the witness beam at the injector exit, is $\sigma_{rD} = 350 \mu m$.

Besides an appropriate shaping and relative spacing of the laser comb pulses at the cathode surface, a proper set of active and passive accelerator elements allows us to obtain the required comb beam at the photo-injector exit. The choice of the accelerator setup starts from the optimised witness working point illustrated in Section 14.3, with additional fine-tuning of accelerating cavity RF phases and solenoid magnetic
<figure><img src="image_111.png" /><figcaption>Fig. 14.12. Longitudinal phase space of the PWFA-accelerated electron beam at the L2 linac exit.</figcaption></figure>

<figure><img src="image_112.png" /><figcaption>Fig. 14.13. Transverse and longitudinal distributions of the photo-cathode laser at the cathode surface. The photo-cathode laser has been shaped via the laser-comb technique, also in intensity to provide at the cathode a 30 pC witness beam (red) separated by 4 ps from the 200 pC driver bunch (blue).</figcaption></figure>

<table><caption>Table 14.7. Main photo-injector parameters.</caption><thead><tr><td>Parameter</td><td>Unit</td><td>Value</td></tr></thead><tbody><tr><td>Gun electric field amplitude</td><td>MV/m</td><td>120</td></tr><tr><td>Gun electric field operation phase</td><td>deg</td><td>32</td></tr><tr><td>Output gun beam energy</td><td>MeV</td><td>5.6</td></tr><tr><td>Amplitude of electric field in the three TW sections</td><td>MV/m</td><td>20.0/20.0/28.0</td></tr><tr><td>Magnetic field in the emittance compensating solenoid</td><td>kG</td><td>3.0</td></tr><tr><td>Magnetic field in the linac solenoids</td><td>kG</td><td>0.34/0.52</td></tr><tr><td>Total photo-injector length</td><td>m</td><td>12</td></tr></tbody></table>

fields. The main operating parameters of the EuPRAXIA photo-injector for the optimised comb beam in case of PWFA are summarised in Table 14.7.

The best compromise in terms of final spacing and witness profile has been obtained with a laser-comb operation, with two laser pulses spaced by $\Delta t = 4.8$ ps on the cathode. In this configuration, adopting the setup described in Section 14.3, the beam crossing occurs in the second TW accelerating cavity, and a fine-tuning of the RF phases suffices to provide 0.55 ps spaced beams, corresponding to $\lambda_p/2$, and the desired witness and driver longitudinal lengths, i.e. 3 µm (FWHM) and in the range of 30–50 µm (RMS), respectively.
<figure><img src="image_113.png" /><figcaption>**Fig. 14.14.** Upper plots: Transverse phase space (x and y). Lower plots: transverse distribution and longitudinal phase space for a comb-like beam at the photo-injector exit. The blue and red dots are related to the driver and witness, respectively.</figcaption></figure>

<figure><img src="image_114.png" /><figcaption>**Fig. 14.15.** Driver (left plots) and witness (right plots) beam energy and current profiles (red and blue lines, respectively) after velocity bunching.</figcaption></figure>

Both witness and driver bunches have been simulated with 30 k and 200 k macro-particles, corresponding to 30 pC and 200 pC, respectively. In the described configuration, the driver arrives 0.58 ps earlier than the witness at the X-band booster. The parameters of both witness and driver beams at the X-band linac entrance are listed in Table 14.8; it is worth noticing that the witness length is about 3 µm (FWHM) with a normalised transverse emittance of ~0.7 mm mrad. Figure 14.14 reports longitudinal and transverse phase spaces at the photo-injector exit for both witness and driver beams as obtained with TStep. Figure 14.15 shows the energy and current profiles for both driver and witness bunches as naturally produced by the velocity-bunching regime, i.e. a spike-like distribution with the charge gathered on the head of the bunch. Even though this longitudinal particle distribution is suitable for the witness beam to take profit of the beam loading, it is not the optimum for the driver beam. Indeed, to increase the transformer ratio, the opposite charge distribution, i.e. low charge on the head and maximum charge on the tail, is mandatory. In this regard, further manipulation of the longitudinal phase space is required for the driver.
<table><caption>Table 14.8. Driver and witness beam parameters at the end of the photo-injector.</caption><thead><tr><td>Parameter</td><td>Unit</td><td>Witness</td><td>Driver</td></tr></thead><tbody><tr><td>Charge</td><td>pC</td><td>30</td><td>200</td></tr><tr><td>Energy</td><td>MeV</td><td>101.5</td><td>103.2</td></tr><tr><td>RMS energy spread</td><td>%</td><td>0.15</td><td>0.67</td></tr><tr><td>RMS bunch length</td><td>fs</td><td>12</td><td>20</td></tr><tr><td>Peak current (FWHM)</td><td>kA</td><td>6.0</td><td>0.37</td></tr><tr><td>RMS normalised emittance</td><td>mm mrad</td><td>0.69</td><td>1.95</td></tr><tr><td>Repetition rate</td><td>Hz</td><td>10</td><td>10</td></tr></tbody></table>

<figure><img src="image_115.png" /><figcaption>Fig. 14.16. Horizontal and vertical phase-space distribution of the PWFA driver (black dot) and witness (red dot) beams at the L1 linac entrance.</figcaption></figure>

The X-band RF linac has to provide an electron beam for injection in the plasma capillary with $Q = 30$ pC, $I = 3$ kA (FWHM), and 1–2 µm of transverse spot size; the comb-like electron beam undergoes deep over-compression in the photo-injector by means of the velocity-bunching scheme.

The driver and witness bunches are characterised by a high charge / low current and a low charge / high current, respectively. Moreover, the initial matching conditions for the injection in the X-band linac are quite different for the two bunches, as shown by their transverse phase space at the linac entrance (i.e. injector exit) and depicted in Figure 14.16. In this regard, an efficient sharing of the same lattice is achieved by means of a mild transverse focussing that aims to keep the RMS size of the comb beam compatible with the beam stay-clear aperture throughout the X-band accelerator, as seen in Figure 14.17.

The same argument applies also to the focussing stage, with the permanent quadrupoles at the entrance of the plasma capillary, where a residual asymmetry between the horizontal and vertical planes for the witness beams is present, as shown in Figures 14.18 and 14.19, unavoidable up to now if not at the expense of a much greater dilution of the transverse emittance of the driver bunch; optimisation and improvement of the lattice are due in progress to minimise the issue.

This working point for the RF photo-injector in the laser comb configuration has been optimised and validated by means of its integration into a start-to-end simulation, which included the plasma-accelerating module and the matching sections to provide the complete evolution of both driver and witness bunches. The transverse properties from the photo-cathode to the undulator entrance for both bunches are reported in Figures 14.20 and 14.21, respectively.

The longitudinal properties from the photo-cathode to the undulator entrance for both bunches are reported in Figures 14.22 and 14.23, respectively.
<figure><img src="image_116.png" /><figcaption>Fig. 14.17. Transverse RMS size of the electron beam (driver plus witness) along the linac.</figcaption></figure>

<figure><img src="image_117.png" /><figcaption>Fig. 14.18. Horizontal and vertical phase-space distribution of the PWFA driver (cyan dot) and witness (red dot) beams at the capillary entrance.</figcaption></figure>

<figure><img src="image_118.png" /><figcaption>Fig. 14.19. Transverse horizontal and vertical distribution of the PWFA driver (left) and witness (right) beams at the capillary entrance.</figcaption></figure>
<figure><img src="image_119.png" /><figcaption>Fig. 14.20. Start-to-end simulation results for the driver bunch from the photo-cathode to the undulator entrance: evolution of transverse normalised emittance ($\epsilon_{nx}$ red line, $\epsilon_{ny}$ red dotted line) and spot size ($\sigma_x$ blue line, $\sigma_y$ blue dotted line).</figcaption></figure>

<figure><img src="image_120.png" /><figcaption>Fig. 14.21. Start-to-end simulation results for the witness beam from the photo-cathode to the undulator entrance: evolution of transverse normalised emittance ($\epsilon_{nx}$ red line, $\epsilon_{ny}$ red dotted line) and spot size ($\sigma_x$ blue line, $\sigma_y$ blue dotted line).</figcaption></figure>

## 14.6 Future Plans and Conclusions

Further studies are ongoing to optimise the current distribution in particular for particle-driven plasma acceleration, and to better control the matching of both driver and witness beams along the machine up to the plasma entrance. Additional studies will be dedicated to the definition of a unique solution for the photo-injector layout to integrate both compression schemes and guarantee optimal performances for both external injection acceleration schemes. The start-to-end simulations shown here have taken profit from the collaboration and synergy with other work packages, mainly WP2 and WP9.
<figure><img src="image_121.png" /><figcaption>**Fig. 14.22.** Start-to-end simulation results for the driver bunch from the photo-cathode to the undulator entrance: evolution of the energy (*E* red line) and energy spread ($\Delta E/E$ red dotted line) as well as longitudinal bunch length ($\sigma_z$ blue line).</figcaption></figure>

<figure><img src="image_122.png" /><figcaption>**Fig. 14.23.** Start-to-end simulation results for the witness beam from the photo-cathode to the undulator entrance: evolution of the energy (*E* red line) and energy spread ($\Delta E/E$ red dotted line) as well as longitudinal bunch length ($\sigma_z$ blue line).</figcaption></figure>

# 15 Sub-Femtosecond Synchronisation of Electrons to the Laser

One of the proposed methods for improving beam quality in EuPRAXIA is to use an external injection scheme, where the electron beam is provided by an external RF linac rather than produced within the plasma (Schemes 3 and 4, as described in Chap. 8). This allows good control of the beam properties so that the emittance and energy spread growth during acceleration can be minimised by properly shaping [329] and matching [35,330] the beam into the plasma.

However, operating a plasma accelerator with external injection brings in new technical challenges. In particular, it requires an extremely low timing jitter between the drive laser and witness beam. Given the small size of the accelerating plasma buckets (typically on the order of 10–100 µm) and the steep field gradients in them, an accuracy on the femtosecond level is required, smaller than what is available today in state-of-the-art setups. In this context, a new timing jitter compensation scheme
<figure><img src="image_123.png" /><figcaption>Fig. 15.1. Schematic view of the synchronising stage.</figcaption></figure>

to control synchronisation between the RF-generated electron bunch and the laser-
driven plasma wakefield was developed by A. Ferran Pousa et al. [331]. The scheme is
based on the same underlying idea already explored in previously proposed concepts
[332–334], but with an already pre-bunched and ultrashort beam. In the following,
the results from this first paper on the synchronisation mechanism are reproduced
with the authors’ permission.

**15.1 A Scheme for Timing Jitter Compensation**

We propose a new concept for external injection that would allow the achievement of unprecedented sub-femtosecond timing jitter between the laser and the witness beam. The scheme, as seen in Figure 15.1, relies on adding an intermediate synchronising stage before the main plasma acceleration module.

In this concept, the laser pulse is initially split into two, with one of them contain-
ing only a small fraction of the energy (e.g. ~5%). These two pulses are intrinsically
synchronised and will serve different purposes: the weak one will be used in the syn-
chronising stage, while the strong pulse will be used as a driver in the main plasma
acceleration module.

As seen in Figure 15.1, the main idea is to use the weak pulse to drive the first plasma stage, in which the electron beam will get a small energy change proportional to its arrival time. In the speed-of-light frame, $\xi = z - ct$, beams with an offset $\Delta\xi_i = \xi_i - \xi_0$ with respect to the zero crossing, $\xi_0$, of the accelerating field, $E_z$, will gain or lose energy depending on whether they arrived too late ($\Delta\xi_i < 0$) or too soon ($\Delta\xi_i > 0$), therefore correlating arrival time and beam energy.

Due to this correlation, a magnetic chicane will then be able to correct the ini-
tial offsets thanks to the differences in path length for each beam energy, virtually
removing the timing jitter between the electron beam and the main laser pulse for
injection into the second plasma stage.

Additionally, the scheme contains two quadrupole sections to match the beam
into and out of the plasma modules.

The laser parameters should be such that the weak pulse is able to excite a linear
wake in the first plasma stage, so powers on the order of 1–10 TW should suffice. For
the strong one, the necessary power will depend on the desired final energy of the
beam after acceleration.

The design energy of the chicane, $E_0$, should be selected as the energy, after the
first plasma, of a beam with $\Delta\xi = 0$. In case of no beam loading, this means that
$E_0 = E_i$, where $E_i$ is the initial energy of the beams coming from the linac.
## 15.2 Linear Model

In the first plasma stage, beams will get an energy deviation $\delta = \frac{E-E_0}{E_0}$ depending on their longitudinal position. For small offsets, this can be approximated as

$$ \delta = -\frac{eE'_{z}L_{p}}{E_{0}} (\xi_{i} - \xi_{0}), \qquad (15.1) $$

where $e$ is the electron charge, $E'_z = \partial_\xi E_z$ is the slope of the accelerating field around $\xi_0$, and $L_p$ is the length of the plasma stage.

To the first order, the final position of the beam after the chicane will be given by

$$ \xi_f = \xi_i - R_{56}\delta, \qquad (15.2) $$

where the minus sign comes from the fact that in the co-moving coordinate $\xi$, as commonly used in plasma acceleration, the bunch head is on the right.

Then, in order to correct the initial offsets, $\xi_f = \xi_0$ should be imposed. This, together with equation (15.1), yields

$$ R_{56} = -\frac{E_0}{eE'_{z}L_p}. \qquad (15.3) $$

For a plasma in the linear regime ($a_0 \ll 1$) [180,335] and a circularly polarised Gaussian pulse, $a^2 = a_0^2 \exp(-\xi^2/2\sigma_z^2)$, the slope of the field $E_z$ around its zero crossing is

$$ E'_{z} = \frac{mc^2}{e} \sqrt{\frac{\pi}{2}} a_0^2 k_p^3 \sigma_z e^{-\frac{k_p^2 \sigma_z^2}{2}}, \qquad (15.4) $$

where $m$ is the electron mass, $c$ is the speed of light, $a_0$ is the peak amplitude of $a$, the laser's normalised vector potential, $k_p$ is the plasma wavenumber, and $\sigma_z$ is the laser RMS length. In the case of linear polarisation, $a_0^2$ should be replaced by $a_0^2/2$. By substituting this into equation (15.3), one finds the general expression

$$ R_{56} = -\sqrt{\frac{2}{\pi}} \frac{E_0}{mc^2 a_0^2 k_p^3 \sigma_z L_p} e^{\frac{k_p^2 \sigma_z^2}{2}}. \qquad (15.5) $$

Now if the laser pulse length satisfies the resonant condition ($\sigma_z k_p = 1$), one can find a simplified equation which, written in a convenient way, reads,

$$ R_{56} [\text{mm}] = -7.27 \times 10^{13} \frac{E_0 [\text{MeV}]}{a_0^2 n_p [\text{cm}^{-3}] L_p [\text{mm}]} \qquad (15.6) $$

In reality, depending on the beam energy, it might be necessary to take into account the path length effect given the deviation from the speed of light not being negligible. This would decrease the necessary $R_{56}$ at the chicane, but a precise value can only be given once the full geometry of the beamline up to the second plasma has been decided. Also, Eqs. (15.5) and (15.6) do not take into account beam loading, so if this effect becomes significant, $R_{56}$ might have to be determined from the chirp of $\delta$ as obtained from simulations.

Another important remark is that the quality of the jitter correction will depend on the stability of the parameters in equation (15.5) as well as the pointing jitter of the electron beam after the first plasma. For the case presented below, assuming the pointing jitter does not exceed the natural beam divergence, the resulting maximum timing jitter would be around 0.2 fs per metre of beamline downstream of the plasma, still on the sub-fs range.

Moreover, $\delta$ should always be much bigger than the intrinsic energy spread of the beam; otherwise, its longitudinal phase space might be perturbed in the chicane.
<figure><img src="image_124.png" /><figcaption>**Fig. 15.2.** On-axis longitudinal electric field after 1.2 mm of propagation in the plasma, with the electron beam placed at the zero crossing (case with $\Delta\xi = 0$).</figcaption></figure>

## 15.3 Numerical Simulations

In order to test this method, a series of start-to-end simulations going from the injection of the beam into the first plasma until the end of the chicane have been performed.

The simulation of the plasma stage was performed in 3D with the fully relativistic particle-in-cell code OSIRIS [336] using the Ponderomotive Guiding Center (PGC) algorithm [337]. WinAGILE [338] and MAD-X [339] were used for the optimisation of the quadrupole section, and the tracking of the electron beam from the plasma exit until the end of the chicane was done with ELEGANT [340].

Effects derived from coherent synchrotron radiation (CSR) at the dipoles have been taken into account, but a full treatment including space charge after exiting the plasma has not yet been possible. However, because of the low charge density of the beam after the plasma and its relativistic energy, strong space-charge effects are not expected.

### Main Parameters

The electron beam after the linac is Gaussian with a charge of 0.1 pC in 1 fs, an energy of 100 MeV with 0.1% relative energy spread, a normalised emittance $\epsilon_{nx,y}$ of 0.3 µm, and initial offsets due to timing jitter between -20 and 20 fs. These parameters were chosen to avoid significantly perturbing the plasma wakefield with the beam (see Fig. 15.2) and are within the range of those expected to be produced at the SINBAD facility at DESY [322,341,342]. The first plasma stage has a length of 2 mm with a density of $1 \times 10^{17} \text{ cm}^{-3}$, and the laser parameters are based on those considered for the EuPRAXIA conceptual design (see Chap. 10), using only around 3.5% of its energy for the synchronising stage. This means an 800 nm linearly polarised pulse of 3.5 J with an $a_0$ of 0.6, a waist $w_0$ of 54 µm, and a length of 93 fs (FWHM in intensity), having a peak power of 35 TW. With these parameters, the matched beam size for emittance preservation [35] is $\sigma_x = 1.3 \text{ µm}$ for the focussing fields around $\xi_0$.

### Quadrupole Section

The set of magnets is placed 4 cm after the plasma and consists of 4 permanent quadrupoles with a width of 1 cm separated by drifts of 4, 6, and 4 cm. Their strengths were left as free parameters for optimisation in order to achieve, as seen in Figure 15.3, a beam waist at the centre of the chicane and a beta function, $\beta_{x,y} = \gamma_r \sigma_x^2 / \epsilon_{nx,y}$, as constant as possible, where $\gamma_r$ is the Lorentz factor of the beam.
<figure><img src="image_125.png" /><figcaption>Fig. 15.3. Beta function and emittance evolution for beams with different offsets. The solid line corresponds to $\Delta\xi_i = 0$, while the dashed and dotted lines represent offsets of 20 and -20 fs, respectively. z=0 represents the end of the plasma stage.</figcaption></figure>

The resulting geometric strengths of the quadrupoles read, from first to last, -878, 1406, -1497, and 823 in units of m⁻², which, for this beam energy, translate into field gradients of up to 500 T/m, requiring tight apertures.

## Chicane Setup

For this case, equation (15.5) predicts an $R_{56} = -0.222$ mm. However, for higher accuracy, the slope of $\delta$ was directly measured from the simulation results (see Fig. 15.4), obtaining an $R_{56} = -0.267$ mm. This could be due to the laser $a_0$ being 0.6, slightly high for the linear theory to hold, since $a_0 \ll 1$ is assumed.

The bending angle $\theta$ of the dipoles was determined from

$$R_{56} = -2\theta^2 \left( L_d + \frac{2}{3}L_m \right) \qquad (15.7)$$

by assuming a magnet length $L_m = 10$ cm and a drift length $L_d = 2.5$ cm, obtaining $\theta = 2.19^{\circ}$.

## Simulation Results

As seen in Figure 15.4, the plasma stage correctly imprints a linear kick to the beam energy depending on its initial offset, allowing the chicane to compensate the jitter.
<figure><img src="image_126.png" /><figcaption>Fig. 15.4. Jitter correction thanks to plasma stage and chicane. The vertical lines delimit a 1 fs jitter.</figcaption></figure>

All beam offsets between ±10 fs have been reduced to sub-fs level, showcasing the potential of this method.

In terms of quality, as seen in Figure 15.3, the beam experiences an increase in emittance of around 10–20% due to chromatic effects at the quadrupoles. Additionally, given the strong focussing, the beam develops a slightly curved shape in the transverse plane, increasing its RMS length to about 0.5 fs.

Also, as seen in Figure 15.3, beams with $\Delta\xi_i \neq 0$ will present variations in the evolution of the beta function because of their different energies, so some way of compensating this chromatic effect before the second plasma stage will be required.

## 15.4 Conclusion

A new scheme for arrival time jitter minimisation between laser pulse and witness beam for LWFA with external injection is presented. Simulation results show that jitters of 10 fs can be reduced to the sub-femtosecond level with minimal loss of beam quality.

Further studies are required to determine the stability and tolerances of the setup, as well as the influence of space-charge effects and finding ways of matching the beam into the second plasma stage, taking into account the variations in beta function seen in Figure 15.3.

Also, although only ultra-short beams have been tested, it is possible that for longer beams ($\sigma_z \sim 10$ fs), this scheme could be used both for synchronisation and compression.

# 16 Laser-Driven Plasma Accelerator

## 16.1 Introduction

The plasma accelerator stages form the heart of the EuPRAXIA facility. As will be discussed, these structures can readily support accelerating fields exceeding 5 GV/m. The direct consequence of this is the ability to attain particles with multi-GeV gain in a fraction of the distance that would be required by more conventional techniques.

Figure 16.1 demonstrates the location of the plasma acceleration structures within the proposed accelerator developments, with those of long length (which here means >cm length) circled in red. The structures will be driven either by intense laser or electron beams. For beam-driven plasma acceleration experiments, the accelerating structures, though similar in concept to the laser-driven structures, have some
<figure><img src="image_127.png" /><figcaption>**Fig. 16.1.** Schematic of the proposed developments at the two EuPRAXIA construction sites, with the plasma accelerator stages circled. Those circled in darker red are where longer acceleration stages (>cm length) will be employed.</figcaption></figure>

key differences. Their characteristics are generally dependent on their driver, and so they are more naturally discussed in Chapter 17, dedicated to particle-beam-driven wakefield acceleration in EuPRAXIA.

This chapter describes the general requirements to implement plasma-accelerating structures driven by laser beams. Therefore, these discussions primarily apply to the developments planned at the construction site for laser-driven plasma acceleration, where there will be an availability of co-located particle and laser beams.

There are also two types of plasma components to consider for laser wakefield accelerators. Components of the first type are those operated for a *laser-plasma injector* (LPI), which is typically designed to have relatively low output energy (<1 GeV). These LPIs are of less than a centimetre in length and are thus close to the natural diffraction (Rayleigh) length of the laser used to drive the accelerating fields inside them. As such, the use of advanced techniques for the guiding of high-power laser beams is usually considered unnecessary. The electrons accelerated in these devices originate from the plasma, which hosts the accelerating structure. Numerous schemes have been devised and tested to facilitate this injection of electrons into the accelerating structure, and they have been investigated in detail both numerically and in experiments. The strengths and weaknesses of the different approaches are discussed further in Chapter 13, as well as developments to optimise these devices for implementation as an injector for EuPRAXIA.

The second function, as which plasma accelerators will be implemented, is as a *laser-plasma accelerator stage* (LPAS).

In EuPRAXIA, the laser-plasma accelerator stage(s) (LPAS) will take sub-GeV beams and accelerate them to 1 to 5 GeV. Given the need to couple electron and laser beams, these stages form a beamline that must be designed to holistically incorporate electron and laser optics. Also, the need to operate these structures for distances which will exceed tens of centimetres means that some form of laser beam guiding is required to maintain large-amplitude plasma wave growth over the whole length of the accelerator stage.

We will begin by providing the parameters of interest for development of these plasma-accelerator stages since little experimental work in this area exists at the present for laser-driven structures. The parameters are determined predominantly from the limits of well-tested scaling laws and, where appropriate, through confirmation via simulations. These parameters have to be refined when including stability considerations and tolerances, which is a task begun in simulations performed in the frame of Work Package 2 on "Physics and Simulation".
## 16.2 Parameters of Plasma Accelerator Sections

### 16.2.1 Density

Within the laser-driven components of the EuPRAXIA facility, plasma components, as highlighted in Figure 16.1, with the following energy gain characteristics need to be considered:

i. $\Delta W = 1$ GeV injector and accelerator primarily for application beamlines;

ii. $\Delta W = 0.85 - 4.85$ GeV for a test beam;

iii. $\Delta W = 0.85 - 4.85$ GeV for FEL applications.

The LPAS will thus have to accelerate the electron beam to an energy of $\Delta W = 0.85 - 4.85$ GeV for a test beam or FEL applications. In general, these stages will be some form of laser wakefield accelerator [4]. To limit non-linear effects, the accelerator stages will usually be operated in the linear ($a_0 < 1$) or quasi-linear ($a_0 \simeq 1$) regime, where the energy gain is given by the following:

$$W_{\max} = 2a_0^2 \gamma_{ph}^2 mc^2 \approx a_0^2 (n_{cr}/n_e) \text{ MeV}, \quad (16.1)$$

where $\gamma_{ph}$ is the Lorentz factor associated with the plasma wave. Operating in the linear regime ensures higher reproducibility and predictable energy gain. The energy gain is mainly determined by the laser intensity ($a_0$) and the ratio of critical density to plasma density ($n_{cr}/n_e$), the first because it determines the amplitude of the plasma wave, and the latter because it determines its velocity.

An energy gain up to 4 GeV is required; setting $a_0 \simeq 1$ ensures the maximum acceleration whilst remaining in the linear regime, for $n_e \approx 2.5 \times 10^{-4} n_{cr} \approx 4 \times 10^{17} \text{ cm}^{-3}$. These plasma densities are readily obtained by laser ionisation of near-atmospheric-density gas targets and thus determine the type of targets required for the LPA stages.

### 16.2.2 Interaction Length

The length over which this acceleration is achieved is optimally set, in an LPAS, by dephasing. This is the length over which the accelerated electrons, travelling at close to the speed of light $c$, outrun the plasma wave, which travels at close to the group velocity of the driving laser pulse within the plasma. In the linear regime, the dephasing length is approximately given by

$$L_{dp} \approx \gamma_{ph}^2 \lambda_p \approx (n_{cr}/n_e)^{3/2} \lambda_0 \quad (16.2)$$

for a plasma wave of wavelength $\lambda_p$. For the 4 GeV gain required, $(n_{cr}/n_e) \approx 4000$, and so this increases to $L_{dp} \approx 25$ cm.

This can be compared to the distance over which a high-intensity laser pulse naturally diffracts, the Rayleigh range:

$$z_R \approx \pi w_0^2 / \lambda_0. \quad (16.3)$$

Typically, one focusses the laser pulse to a spot size $w_0 \approx \lambda_p = (2\pi c/\omega_p) \approx 30(60) \text{ µm}$ for 1(4) GeV acceleration. This gives $z_R \approx 3(12)$ mm, which means that the interaction length $L_{dp} \approx 10(20) z_R$, motivating the need for some form of guiding of the laser pulse to maintain a high-intensity interaction over the necessary length.

We note that 4 GeV energy gain in a length of 25 cm corresponds to an average acceleration of 16 GeV/m, demonstrating the remarkable accelerating potential of laser-plasma accelerators.
The EuPRAXIA LPA stages will typically operate at densities down to $n_e \approx 1 \times 10^{17} \text{ cm}^{-3}$ over lengths up to $\sim 1 \text{ m}$. Hence, they will be formed in gaseous targets with the requirement of some form of laser pulse guiding at high intensity.

# 16.3 Intense Laser Beam Confinement

## 16.3.1 Requirements

As noted above, the laser drivers for the EuPRAXIA LPA stages will need to be focussed with spot sizes in the range of 10–100 µm for distances up to one metre in length, which corresponds to many tens of Rayleigh ranges. Thus, some form of radial confinement of the laser pulse is required to maintain the high-intensity interaction or guide the laser pulse over these lengths.

The guiding of short laser pulses in an underdense plasma requires modification of the plasma refractive index, such that it has a maximum on axis, i.e. $\partial\eta/\partial r < 0$. This causes the laser wavefronts to be retarded on axis, countering the natural tendency of the focussed laser pulse to diffract. In the weakly non-linear regime ($n_e \ll n_{cr}; a_0 < 1$), the refractive index of the plasma is given by

$$ \eta \approx 1 - \frac{\omega_p^2}{2\omega_0^2} \left( 1 - \frac{a^2}{4} + \frac{\Delta n_p}{n_e} + \frac{\delta n}{n_e} \right) \quad (16.4) $$

for a linearly polarised laser. Equation (16.4) shows there are three methods for producing the necessary refractive index modification: an increase in intensity (a), a self-generated decrease in axial density ($\delta n$), or an initially applied density modification ($\Delta n_p$). The relativistic (intensity) dependent and ponderomotive (self-generated density depression) effects come naturally in laser wakefield acceleration. Since they depend only on the applied laser beam, they are termed collectively as self-guiding. However, relying on self-guiding by itself makes the accelerator highly dependent on the quality and reproducibility of the laser driver. Hence, it is preferable to enforce some kind of guiding structure (which is captured by the last term in equation (16.4)), such as plasma channels and capillary tubes. In the low-intensity (linear) regime, this increases the efficiency of guiding too. This is because the density increase in the front of the plasma wave has a defocussing effect on the laser pulse, which is detrimental to guiding short pulses [343,344]. These externally applied guiding structures are discussed in more detail below.

## 16.3.2 Possible Waveguides

Several types of configurations are candidates for the LPA stages of EuPRAXIA and are described in detail in Chapter 32.2. The main features and most promising options are highlighted in this section.

The transverse refractive index variation required to guide a laser pulse can be created prior to the arrival of the LPA-driving laser pulse by suitable modification of the density profile. A column of plasma, in which $n_e(r)$ increases smoothly with radial distance from the axis, will have a refractive index which decreases away from the axis. As such, each transverse slice of the plasma column acts as a positive lens, focussing the light as it propagates. This *plasma channel* is therefore an example of a gradient refractive index (GRIN) waveguide.

An alternative configuration, known as a *hollow plasma channel*, is one, in which the plasma column comprises a uniform density core surrounded by a coaxial cylinder
of higher density. Light propagating at sufficiently small angles to the axis of this structure will experience total internal reflection at the core-cladding boundary and hence be guided. This is an example of a step-index waveguide and is analogous to a standard silica optical fibre.

The most widely used type of plasma channel waveguide is the *capillary discharge*. A gas filled capillary discharge readily spans the densities of interest with the required matched spot size and over the lengths suitable for the LPA stages of EuPRAXIA. Although the parameters of interest have not yet been achieved simultaneously, it can be assumed that these parameters will be available. Furthermore, with recent advances in reproducibility, these targets should be able to operate many days at EuPRAXIA repetition rates without the need for replacement.

Plasma channels can also be formed by the *hydrodynamic expansion* of a hot plasma column into a cold gas [345]. More recently optical-field-ionised (OFI), hydrodynamically generated plasma channels have been investigated [346–349]. Here, the initial plasma is formed by the direct field ionisation of a short, more intense laser pulse ($I > 1 \times 10^{15} \text{ W cm}^{-2}$). This allows access to fully ionised media such as hydrogen and helium plasmas. The channel is formed by plasma expansion due to the above-threshold ionisation heating of the ionised electrons. As well as allowing access to a wider range of target material, the plasma can be formed in common targets such as gas jets and gas cells, offering ease of operation. Additionally, OFI-generated channels can be relatively easily diagnosed since they offer unhindered transverse access.

Another device for guiding a high-intensity laser is a *capillary tube*. In these devices, a laser beam can be guided by reflections at the inner wall of the capillary, and so the laser guiding can be achieved either in vacuum or at low plasma density [350]. This does not rely on laser power or plasma density and provides the opportunity to explore a large domain of laser-plasma parameters. An important aspect associated with the use of a capillary tube is the material damage threshold at the inner wall where reflection occurs. For a glass capillary, the maximum intensity on axis which can be guided by a capillary of 50 µm radius in the fundamental $EH_{11}$ mode without wall ionisation is $\sim 10^{19} \text{ Wcm}^{-2}$, which is at least 10 times higher than the one required for LPAS.

For LPA stages, some form of laser confinement is required. Gas-filled capillary discharges and gas-filled capillaries are the two leading candidates. Their operation has been demonstrated over the density range and lengths required for the EuPRAXIA LPAS.

Hydrodynamically generated waveguides are an interesting alternative for the testing and optimisation of the LPA stages for EuPRAXIA due to their improved diagnostic access.

### 16.3.3 Example of Waveguide Implementation

Figure 16.2 shows examples of the two types of waveguides, a capillary discharge waveguide and a dielectric capillary tube that can be envisaged to create plasmas for accelerator stages. The capillary discharge waveguide scheme relies on a pulsed high-voltage discharge to ionise the gas and on plasma relaxation to create a plasma density channel to ensure laser guiding, as shown in Figure 16.2a.

In dielectric capillary tubes, wall reflections ensure laser guiding. The plasma creation relies on the ionisation of gas by the drive laser. Figure 16.2b illustrates the gas distribution in the stationary regime achieved inside a glass capillary tube. The gas density is constant along the axis between the two slots, shown by the red area in Figure 16.2b.
(a)

<figure><img src="image_128.png" /></figure>

(b)

<figure><img src="image_129.png" /></figure>

**Fig. 16.2.** (a) Example of hydrogen-filled capillary discharge waveguide from [351] (copy- right (2002) by the American Physical Society); (b) Simulation of gas distribution in a dielectric capillary tube from [352] (reprinted with permission of AIP Publishing).

Some form of density variation can be accommodated in both types of targets by having unbalanced fill pressures at either end. Additionally, specialised regions of high density or different gas species can also be easily accommodated in both designs by having extra inlets. These sections can be used for dechirping the beam, providing additional energy gain, reducing emittance growth from the plasma exit, and for internal injection into the LPAS from the plasma. The latter subject is covered in detail in the section on laser-plasma injectors (LPIs).

The targets for the EuPRAXIA LPAS will be either laser-ionised or discharge plasmas. Uniform density gas capillaries, either discharge or laser-ionised, will be the baseline configuration for the EuPRAXIA LPA stages.

Many different advanced options exist integrating density ramps and localised areas of high density or special gases. The EuPRAXIA facility will allow many of these schemes to be tested under directly comparable conditions.

## 16.4 Staged Accelerator Implementation

Despite the many possibilities, the baseline recommendation for the LPA stages for the EuPRAXIA facilities should be based on gas-filled dielectric capillaries or gas- filled capillary discharges as these are proven technologies for guiding the laser pulse over the 10–100 cm lengths required here. Actual experimental demonstrations of acceleration to the scale required in EuPRAXIA are not yet available, but the feasibility of both approaches has been demonstrated in numerical simulations. Useful scaling and physics considerations for the design of plasma-accelerating stages in the quasi-linear regime can also be found in [83]. The table below details possible operating parameters for both of the external guiding schemes found for EuPRAXIA, as determined from simulations.

The first option, based on WAKE-EP simulations described in [353], uses dielec- tric capillary tubes to guide the laser beam and excite a wakefield to accelerate a 50 MeV externally injected electron bunch up to 5 GeV. The second option is based on simulations with WARP, described in [354], to accelerate electrons to 10 GeV over 0.75 m from an injected bunch at 100 MeV. These parameter sets are a starting point and are being refined using specific simulations with EuPRAXIA parameters performed by WP2.

For a 1 GeV stage, it would be straightforward to downscale the parameters given above, ideally scaling up the required density to ensure extraction at close to the dephasing length.
<table><caption>**Table 16.1.** Example of parameter sets to achieve an energy gain of the order of 5 GeV in a laser-driven plasma-accelerator stage.</caption><thead><tr><th>Type of guiding</th><th>Dielectric capillary<br>based on [353]</th><th>Plasma channel<br>based on [354]</th></tr></thead><tbody><tr><td>n<sub>e</sub> (cm<sup>-3</sup>)</td><td>1 &times; 10<sup>17</sup></td><td>1 &times; 10<sup>17</sup></td></tr><tr><td>Plasma characteristic length (m)</td><td>~ 1</td><td>~ 0.4</td></tr><tr><td>Guiding</td><td>Capillary walls</td><td>Density distribution</td></tr><tr><td>Gas</td><td>H<sub>2</sub></td><td>H<sub>2</sub></td></tr><tr><td>I<sub>L</sub> (W/cm<sup>2</sup>)</td><td>4.3 &times; 10<sup>18</sup></td><td>4.3 &times; 10<sup>18</sup></td></tr><tr><td>a<sub>0</sub></td><td>1.42</td><td>1.41</td></tr><tr><td>&tau;<sub>L</sub> (FWHM fs)</td><td>40</td><td>47</td></tr><tr><td>w<sub>0</sub> (FWHM &mu;m)</td><td>118</td><td>74</td></tr><tr><td>E<sub>L</sub> (J)</td><td>29</td><td>27</td></tr></tbody></table>

<figure><img src="image_130.png" /><figcaption>**Fig. 16.3.** Schematic of the laser removal and coupling in a multi-stage configuration.</figcaption></figure>

A schematic of the functional implementation of the accelerator, including the injector stage, is presented in Figure 16.3.

The figure shows some of the key challenges common to all staged wakefield acceleration proposals:

- Transport of the electron beam to the accelerator stage

- Transport of the driver (high-intensity laser) to the accelerator stage

- Diagnosis of the interaction

- Separation of the accelerated beam from the spent driver

Some other important considerations include the following:

- The whole interaction should take place in vacuum to prevent degradation of the laser driver and injected particle beam.

- The target and beams should be positioned, ideally, to sub-micron precision, and spatial movement of both should be below this level.

- Diagnostics should be non-invasive where possible but also easily translated if not, for example in the case of an electron spectrometer.

As schematically presented in Figure 16.3, at each plasma stage, the laser driving the accelerator stage has to be coupled to the plasma, and the transmitted laser light has to be removed from the axis after the interaction. The high intensities limit the proximity of the removal / coupling device to the plasma target. However, the electron transport prescribes that the electron capture should be as close to the plasma as possible to limit the emittance growth. There are two main options for the coupling of the laser to the accelerator.

A perforated mirror can be used to couple the second laser beam to the accelerator plasma stage. A hole in the centre of the mirror would free the electron path, and the outer part would reflect the laser beam on to the electron axis. With typical parameters considered for the EuPRAXIA facilities, the operating distance of this coupling mirror would be at least of the order of metres. This can greatly increase the
overall length of the accelerator, consequently reducing the average acceleration. Also, care must be taken that the mirror perforation does not contribute to the unwanted degradation of the driver profile on the target, which will be a serious consideration for some of the targets that may be used. Nevertheless, a perforated mirror is the technically proven solution for laser-beam coupling, and the EuPRAXIA facilities should provide sufficient space to accommodate this solution.

As a mirror placed close to the laser focus would suffer damage from heating and ionisation, plasma mirrors are considered as a solution to remove the laser from the electron axis. These are mirror surfaces which become almost instantaneously overdense plasmas and hence become highly reflecting on irradiation with a high-intensity laser. Such plasma mirrors have long been in use as pulse contrast enhancers [355] but are conventionally constructed of AR-coated fused silica blocks which must be rastered between laser shots in order to expose a fresh optical-quality surface to the laser. This motion limits the repetition rate at which the plasma mirror can operate, and the number of pulses which can be reflected before the mirror is replaced.

Many different schemes for plasma mirrors exist and are being considered for implementation at EuPRAXIA. However, the greatly reduced operating distance of the plasma mirrors is very important. Not only does it maintain the compactness of the facility, but also by making the distance to transport optics for the accelerated beams shorter, it greatly reduces emittance growth in the drift sections. This may prove very important for achieving the high brightness projected for the EuPRAXIA facility.

Another important consideration for a wakefield acceleration facility is the stability of the device. Staged acceleration places exacting requirements on the positioning and timing of laser and particle beams. Spatial positioning to micrometre precision and a temporal overlap down to 10 fs is anticipated. This places high pointing and wavefront constraints on the laser system ($<\mu rad$). Also, though beams may be derived from the same initial source, the system should be devised to avoid or at least compensate long-term drifts. In addition, spatio-temporal properties of the laser, such as pulse front tilts, must be monitored and controlled to maintain high beam quality and pointing stability.

For the implementation of the staged approach to plasma acceleration that forms the basis of the design of the EuPRAXIA facilities, the coupling of driver and particle beams is of paramount importance. We note that for reasons of reliability and field-proven performance, conventional solutions, such as high-damage-threshold perforated optics for laser beam transport and quadrupole transport lines for particle beam focussing should be accommodated in the EuPRAXIA LPA beamlines. This will necessitate allocating sufficient distances to manoeuvre these components in the base designs.

However, we note that in keeping with the ethos of the EuPRAXIA facilities, the application of plasma technology, e.g. plasma lenses for laser transport and plasma lenses for particle beam transport, will allow dramatic improvements in compactness and potentially performance. Hence, the ability to assimilate these technologies should be made in the design phase.

## 16.5 Conclusion

The plasma accelerator stage would be operated in the quasi-linear regime to prevent the additional internal injection of electrons which would be detrimental to bunch quality. As a consequence, the plasma electron density for this stage would be of the order of $10^{17} \text{ cm}^{-3}$. The laser normalised vector potential would be around $a_0 \simeq 1.4$, which corresponds to an intensity of $I_L \simeq 4.3 \times 10^{18} \text{ Wcm}^{-2}$. The low density and
intensity considered imply that the acceleration length should be of the order of 0.5–1 m.

To sustain accelerating gradients over such a long distance, the laser beam has to be transversely confined to prevent diffraction. A self-guiding scheme relies on the non-linear self-focussing effect and requires laser power in excess of the critical power $P_{cr} = 17.4 (n_{cr}/n_e)$ GW. At the density considered for the accelerating stage, $n_e = 10^{17} \text{ cm}^{-3}$, the critical power is around $P_{cr} \sim 100$ TW, which is below the laser power considered for this stage, $\sim 0.5-1$ PW (see Chap. 10). However, as self-guiding relies on non-linear effects, it is sensitive to fluctuations of laser power and spatial profile. In the context of building an accelerator to provide electron beams of industrial quality, this can add undesired instability. Hence, for a robust accelerator, one should operate in the linear or near (quasi-)linear regime of $a_0 \simeq 1$, with some form of external guiding.

An external confining structure, such as a plasma density channel or dielectric capillary tube, would be beneficial, as it can help tame fluctuations caused by non-linearities, such as pointing instabilities. In an extreme case, the laser power can be operated below the self-focussing threshold so that the guiding and acceleration are only determined by the properties of the guiding channel. In that case, good shot-to-shot reproducibility of the guiding structure is imperative to maintain reproducible acceleration.

One issue for applying external guiding structures is their susceptibility to damage due to a poor laser structure and pointing instabilities. For the case of a capillary tube, the angular variation between the laser axis and the capillary tube should be less than 5 mrad, and the displacement of the laser spot centre to the capillary centre should be less than 10% of $R_{cap}$. This imposes stringent requirements on the beam quality of the high-power lasers that will drive the LPA stages of EuPRAXIA.

The coupling of laser and particle beams to the accelerator stage is another area of important consideration for the EuPRAXIA developments. Though laser beams can be directed by mirrors with perforation along the axis to allow the ingress of the particle beams, necessarily, EuPRAXIA should be designed with the possibility of using some form of plasma mirror due to the dramatic reduction in size that this affords. Continued research in the area of plasma mirror development is necessary. Similarly, particle beam coupling can be designed based on conventional fixed magnet optics, quadrupoles, etc., but given the space and challenging beam parameter constraints of the LPA beams, it is unlikely that these optics can provide an adequate solution for compact beam transport. In this case, the advent of plasma lenses based on capillary discharges offers equally important potential savings in space and complexity. Considering the rapid developments on these systems, it is now considered likely that they will form an important component of the EuPRAXIA developments.

# 17 Beam-Driven Plasma Accelerator

For the beam-driven case, acceleration to 1 GeV can be achieved in a single-stage plasma accelerator that is coupled with an RF linac operating at 500 MeV (Scheme 5, as described in Chap. 8). This stage is a few centimetres long, has a plasma density of approximately $10^{16} \text{ cm}^{-3}$, to produce electric fields of 1–2 GV/m, and with a plasma wavelength of $\lambda_p \sim 330 \text{ µm}$, which allows for the acceleration of electron bunch trains with the use of the laser-comb technique [323].

For the 5 GeV case, a transformer ratio of 5–6 is essential, and a single plasma stage would have to be a few metres long. The high transformer ratio is, in this scenario, accomplished with the use of a shaped train of bunches as the driver. This bunch structure guarantees a higher transformer ratio [192] that can be used to
accelerate the trailing bunch beyond the energy-doubling limit for symmetric bunches (see Sect. 7.1). In the design study, we assume an ideal train of bunches generated by the RF linac to an energy of 1.2 GeV, which is then boosted in a plasma stage with a plasma density of approximately $2.5 \times 10^{16} \text{ cm}^{-3}$, to produce electric fields of 1–2 GV/m and with a plasma wavelength of ~210 µm.

# 17.1 Start-to-End Simulation for the 1 GeV Case

A sequence of several codes has been used to study beam-driven plasma acceleration through start-to-end simulations without any phase-space manipulation or remapping. The TStep code is used to simulate the generation and transport of both driver and witness bunches in the photo-cathode section. The ELEGANT code is used to track particles up to the plasma entrance (see Chap. 14); the particle phase space is then imported into the Architect code for the evolution in the plasma section. A summary of these results is presented in Chapter 23. In the following, the study of the beam-driven plasma accelerator stage (PPAS) is discussed.

## 17.1.1 Plasma Simulation Parameters and Code

The state-of-the-art code Architect is used for the plasma stage [356,357]. Given the necessity to run long simulations, a classical particle-in-cell approach was computationally too expensive. Architect is a reduced model code relying on a fluid background (see Sect. 32.1 for more details), which has been tested and verified against a classical particle-in-cell code for the parameter range of interest [357]. Simulations described in this section for the 1 GeV case have been performed with a longitudinal resolution of 1 µm and a transverse resolution of 0.4 µm in a mesh that allows resolving the fine structure at a reasonable computational cost. The advancing time step is 1.1 fs, and the average number of particles per cell is 30 and 100 for the driver and the witness, respectively.

## 17.1.2 Optimum Bunch Acceleration in the Plasma Section

The driver and the witness have been designed to perform for optimum acceleration in terms of both quality and energy transfer (transformer ratio, $R_T$) [358]. To maximise the energy transfer, the bunches, particularly the witness, have been designed with a triangular [298,359] shape, which is an optimised profile to limit the energy spread growth.

Then the bunches are imported into Architect for the plasma evolution. The aim is to transport and accelerate the witness bunch through the plasma section, preserving the original bunch quality. The witness has also been designed with a triangular current shape to limit the energy spread growth. In addition to the triangular shape, the peak current influences the optimisation of the longitudinal density profile, which minimises the increase in energy spread during the propagation. Bunch shaping with high accuracy is not, however, a trivial task. In our case, a pseudo-triangular shape is produced where the front part of the profile consists of high-current slices for FEL lasing. From the FEL requirements standpoint, it is these high-current slices that need to be transported with as small a phase-space dilution as possible.

Additionally, to maintain the bunch quality and to ensure that both bunches evolve through the plasma with as little deterioration as possible, they have to be
injected at matching conditions. As introduced in Section 7.1, this plasma transverse matching condition is given by

$$ \sigma_{x,\text{matching}} = \sqrt[4]{\frac{2}{\gamma}} \sqrt{\frac{\varepsilon_{n,x}}{\kappa_p}} \qquad (17.1) $$

where $\varepsilon_{n,x}$ is the normalised RMS emittance (in the transverse plane) in mm mrad while $\kappa_p = 2\pi/\lambda_p$ is the plasma wavenumber. Equation (17.1) indicates transverse micron-size bunches. For our case, assuming $n_p = 10^{16} \text{ cm}^{-3}$ as a nominal density background, we calculate $\sigma_{x,D-\text{matching}} = 4 \text{ µm}$ and $\sigma_{x,W-\text{matching}} = 1.55 \text{ µm}$. We have assumed a driver with a normalised emittance of 3 mm mrad and a witness with a normalised emittance of 1 mm mrad.

Two geometrical conditions must also be met. (1) The bunch has to be as symmetric as possible, i.e. $\sigma_x \sim \sigma_y$. (2) The driver length must meet the longitudinal matching condition: $\kappa_p \sigma_{z,D} = \sqrt{2}$ [360] (see also Sect. 7.1). The longitudinal matching condition is used to maximise the transformer ratio and to limit the quantity of driver charge recalled within the self-wake (head-tail acceleration process): $\sigma_{z,D} = 75 \text{ µm}$.

The foreseen design is planned to operate in the so-called weakly non-linear regime, where the electric field induced by the driver bunch has neither a full sinusoidal behaviour nor a full saw-tooth shape. The parameter we used to measure the degree of non-linearity is the reduced charge parameter [361,362], as introduced in Section 7.1, $Q_{rc} = \frac{N_b}{n_p} \kappa_p^3$, with $N_b$ as the electron bunch number (bunch charge divided by the elementary charge). For the foreseen case, $Q_{rc} \sim 0.8-0.9$ for a driver that is carrying a charge of 200 pC. Therefore, we need to leverage on both linear regime scaling laws as well as the non-linear regime scaling laws to identify the best parameter space for the accelerating stage. The injection phase $\varphi_0$, calculated for the maximum accelerating field, in a linear regime can be found as $\varphi_0 \approx \frac{Q_{\text{witness}}}{Q_{\text{driver}}}$. The calculation assumes that a driver bunch produces a linear wake $E_{\text{acc}} = G \cos(k_p \xi)$, while the witness produces a decelerating self-wake $E_{\text{dec}} = -g \sin[k_p(\xi_0 + \sigma_z - \xi)]$, with $G$ and $g$ as the accelerating and decelerating fields, respectively, where $\xi$ is the longitudinal co-moving coordinate. By requiring that the electric field felt by a particle located at the witness front $\xi = \xi_0 + \sigma_z$ experiences the same accelerating field of a particle in the centre of the bunch $\xi = \xi_0$, we retrieve the given solution. Then the injection phase is close to half of the plasma wavelength for the maximum decelerating driver field. Since the witness charge is fixed by the upstream FEL applications, simulations identify that beam loading is compensated for a driver-witness distance of 184 µm, corresponding to a plasma density of $0.55 \times \lambda_p(n_p = 10^{16} \text{cm}^{-3})$.

Our setup, specifically the driver, is characterised not only by a reduced charge factor that is just below 1 but also by a peak density that is about 10 times larger than the plasma background density. At the plasma entrance, the driver core is characterised by $n_b/n_p = 10-15$, where $n_b$ is the peak bunch density. This condition, as discussed in [53], suggests that it is also possible to achieve a transformer ratio larger than 2 with a single bunch. The transformer ratio, for our optimised distance, is estimated to be around 3, while the accelerating gradient is estimated to reach around 1.1 GV/m.

### 17.1.3 Summary of the Simulation Results

In the simulations presented here, density ramps on the order of 0.5 cm are considered, which is experimentally feasible. The length of the ramps is below the betatron
<figure><img src="image_131.png" /><figcaption>**Fig. 17.1.** Bunch and background density after 5 mm propagation within the plasma. The bunch density is plotted with a *plasma* colour map, while the background is plotted with a grey colour map. The longitudinal accelerating electric field on axis is over-imposed with a solid blue line. For the purpose of scaling, $E_z$ is plotted in hundreds of MV/m.</figcaption></figure>

wavelength, assuring that bunch oscillations are not present to increase acceleration robustness.

The witness is delivered at the plasma entrance with the required triangular shape. In the transverse direction, the bunch is relatively symmetric in size and in emittance (see Tab. 17.1), both of which are tuned to the plasma requirements and to preserve the beam brightness. The driver exhibits a weaker degree of geometrical symmetry, and it is delivered at the plasma entrance with a larger parameter than the matching condition because the RF linac has been optimised to deliver the witness bunch specifications at the plasma entrance (more details in Chaps. 12 and 14). The strict requirements for the witness naturally result in the driver to be away from the optimal conditions. However, since the driver will be removed after the acceleration, the lower quality is acceptable. Additionally, at the entrance of the plasma, the front part of the driver, which corresponds to 1/3 of the total beam charge, is highly convergent and will result in expansion within the plasma channel, producing a unique funnel shape which can be seen in Figure 17.1. Nonetheless, the driver is capable of driving a weakly non-linear wake with a maximum effective field that peaks around 2.5–3.0 GV/m (see Fig. 17.1).

The maximum field is achieved at the bubble closure where the witness is placed. The positioning of the witness would significantly change the bubble structure [186, 298,363]. In the case of interest here, where a witness with a given peak current for FEL applications is desired, positioning the witness closer to the driver and meeting the beam-loading condition allows for better control of the energy spread. The wider bubble width also reduces the chance of quality deterioration. On the downside, such
<table><caption>**Table 17.1.** PWFA bunch parameters at plasma entrance and at plasma exit. The best slice value is also reported.</caption><thead><tr><th>Beam</th><th>Units</th><th>Driver-IN</th><th>Driver-OUT</th><th>Witness-IN</th><th>Witness-OUT</th></tr></thead><tbody><tr><td><strong>Charge</strong></td><td>pC</td><td>200</td><td>200</td><td>30</td><td>30</td></tr><tr><td>σ<sub>x</sub></td><td>µm</td><td>8</td><td>6.4</td><td>1.47</td><td>1.42</td></tr><tr><td>σ<sub>y</sub></td><td>µm</td><td>3.1</td><td>10</td><td>3.17</td><td>1.4</td></tr><tr><td>σ<sub>z</sub></td><td>µm</td><td>52</td><td>50</td><td>3.85</td><td>3.8</td></tr><tr><td>ε<sub>n,x</sub></td><td>mm mrad</td><td>2.56</td><td>4.1</td><td>0.6</td><td>0.96</td></tr><tr><td>ε<sub>n,y</sub></td><td>mm mrad</td><td>4.8</td><td>11.4</td><td>0.55</td><td>1.2</td></tr><tr><td>σ<sub>E</sub></td><td>%</td><td>0.2</td><td>20</td><td>0.07</td><td>1.1</td></tr><tr><td><strong>E</strong></td><td>MeV</td><td>567</td><td>420</td><td>575</td><td>1030</td></tr><tr><td colspan="6"><strong>Best Slice</strong></td></tr><tr><td><strong>Current</strong></td><td>kA</td><td></td><td></td><td>2</td><td>2.0</td></tr><tr><td>ε<sub>n,x</sub></td><td>mm mrad</td><td>N/A</td><td>N/A</td><td>0.59</td><td>0.57</td></tr><tr><td>ε<sub>n,y</sub></td><td>mm mrad</td><td></td><td></td><td>0.58</td><td>0.62</td></tr><tr><td>σ<sub>E</sub></td><td>%</td><td></td><td></td><td>0.011</td><td>0.034</td></tr></tbody></table>

placement will limit the transformer ratio. The central part of the driver that mostly contributes to generating the wake loses about 150 MeV after a 40 cm distance; the witness gains about 460 MeV at the same time. This value indicates that the accelerating gradient is about 1.1 GV/m and that a transformer ratio of about 3 is achieved.

The electron beam properties from the simulations reaching 1 GeV beam energy are shown in Figure 17.2. Note that the rolling slice analysis technique was used here to estimate the witness bunch slice properties. Rather than defining a fixed binning across the longitudinal distribution of the bunch, which may misidentify high- or low-quality portions of the beam if they are split between bins, separate values are defined in this method for the bin size (L) and the step size with which to move the bin across the beam length (dz). Slice values are thus calculated in steps of size dz along the longitudinal beam distribution, while the region of macroparticles taken into account for each slice is defined by the length L≥dz (e.g. corresponding to the cooperation length of an FEL). This provides a larger number of points at which slice parameters are calculated without having to reduce the number of macroparticles considered in each slice, therefore providing a better sampling of the bunch properties.

## 17.2 Numerical Design for the 5 GeV Case

### 17.2.1 Plasma Parameters

Simulations for the 5 GeV case have been performed assuming ideal bunch shapes. The simulations described in this section have been run with a longitudinal and transverse resolution of 1.0 µm with a mesh that allows one to reasonably resolve the witness bunch structure at a moderate computational cost. The advancing time step is 0.59 fs. The drivers and the witness bunches use 200k and 30k particles, respectively.

The simulation includes focussing via a plasma ramp before the plateau; thus, the discretisation corresponds to ≈ 1 particle per cell at the initialisation reaching ≈ 1000 particles per cell at the plateau. The plasma channel is ideal, composed by a ~2.4 m plateau with a plasma density of $n_p = 2.5 \cdot 10^{16} \text{ cm}^{-3}$, preceded by a 1 cm long injection ramp with a density that varies from zero to $n_p$ with a $\cos^2$ shape and a constant transverse dimension of 160 µm.

In this section, less attention has been given to the transverse evolution of the bunch since for ideal bunch shapes, it is always possible to find a matching condition
<figure><img src="image_132.png" /><figcaption>**Fig. 17.2.** Witness phase-space plots at the exit of the plasma-accelerating section (a–c). Longitudinal phase space at the plasma exit, plot of both the driver and the witness (d). Rolling slice analysis for the witness bunch at the plasma entrance (solid line) and at the plasma exit (dashed line). The top panel (e) shows the emittance in both transverse planes as well as the current. The bottom panel (f) plots the energy spread and the current. The corresponding current axis is the right y-axis.</figcaption></figure>

for the preservation of the total transverse emittance [364]. In future full start-to-end simulation studies, where the accomplishment of such conditions is not trivial, the required attention will be dedicated to emittance preservation.

### 17.2.2 Driver Train Structure and Transformer Ratio

We assume a train of driver bunches generated by an X-band RF linac up to the energy of 1.2 GeV. Increasing energy transfer in a PWFA stage requires a driving structure with a higher transformer ratio $R_T$ than the 1 GeV case. This condition can be obtained by utilising a train of drive electron bunches [192]. An energy increase from 1.2 GeV to 5 GeV requires an effective transformer ratio of $R_T \approx 3.2$, whereby “effective” means the transformer ratio is evaluated using the average field acting on
<figure><img src="image_133.png" /><figcaption>Fig. 17.3. Electron bunch and background density after 15 mm propagation within the plasma. The bunch density is plotted with a plasma colour map, while the background is plotted with a grey colour map. The longitudinal accelerating electric field on axis is over-imposed with a solid orange line.</figcaption></figure>

the witness, including the beam-loading effect. Given the high current of the witness (3 kA), the beam-loading effect generates a critical reduction of the accelerating field and consequently reduces the effective transformer ratio. To avoid this problem, we designed an ideal train of 3 drivers with ramped increasing charge densities. Theoretically, this configuration allows reaching a transformer ratio of $R_T = 2N$, where N is the number of drivers. The bunch length for any bunch is $\sigma_z=33 \ \mu m$, to have the condition for the maximum accelerating gradient $k_p\sigma_z = 1$. The separation between the bunches is constant, with $\Delta\xi = 0.5\lambda_p$, corresponding to $\Delta\xi \approx 106 \ \mu m$. The transverse injection condition is calibrated in order to have bunch focussing inside the ramps, reaching the matching condition (Eq. (17.1)) at the beginning of the plateau (Twiss functions $\alpha_{x,y} \approx 1$, $\beta_{x,y} \approx 22 \ mm$ for all bunches). The transverse emittance of the two bunches is chosen arbitrarily as $\varepsilon_{n,xy} = 1 \ mm \ mrad$. The charges are $Q_1=40 \ pC$, $Q_2=140 \ pC$, and $Q_3=270 \ pC$. They are properly calibrated to obtain a maximum decelerating field within the bunches that is constant for all drivers. In this configuration, without the trailing witness bunch, the transformer ratio is $R_T \approx 7.5$.

### 17.2.3 Bunch Acceleration

The witness is designed with the energy of $E = 1.2 \ GeV$ and an initial uncorrelated energy spread of $\sigma_E = 0.7 \%$. As in the $1 \ GeV$ case, a triangular current shape is chosen to reduce the energy spread growth with an RMS bunch length of $\sigma_z = 3.6 \ \mu m$. The transverse emittance is arbitrarily chosen as $\varepsilon_{n,xy} = 0.7 \ mm \ mrad$, and the Twiss functions are $\alpha_{x,y} \approx 1$ and $\beta_{x,y} \approx 22 \ mm$, to reach the matching condition from equation (17.1) at the beginning of the plateau. The peak field is located at the bubble closure with a value of $E_z \approx 2.7 \ GV/m$ (see Fig. 17.3). The bunch separation between the centroids of the last driver and the witness is set as $0.46 \ \lambda_p$, corresponding to $\approx 97 \ \mu m$ in order to minimise the energy spread growth. With this fixed, the mean accelerating gradient acting on the witness is $E_z \approx 1.6 \ GV/m$. The maximum decelerating field acting on the drivers is $\approx 450 \ MV/m$, leading to an
<figure><img src="image_134.png" /><figcaption>Fig. 17.4. Evolution of the spot size and the transverse emittance of the witness bunch (a), energy and energy spread of the witness bunch (b), phase space of the witness at injection (c), and phase space of the witness at the final step of the plasma target (d). Note that the normalised transverse emittance is defined as $ε_x$ here.</figcaption></figure>

effective transformer ratio of $R_T = 3.5$. The witness is accelerated up to 5 GeV in a 2.4 m long accelerating structure. The evolution of the integrated parameters and the phase space can be seen in Figure 17.4. The slice parameters can be observed in Figure 17.5. The transverse phase-space quality is preserved completely. The energy spread increases to $σ_E ≈ 0.4\%$. The highest energy spread growth is in the tail of the bunch, where the current is at a minimum.

## 17.3 Outlook and Further Research and Development

A further improvement of the beam parameters as presented in the previous subsections will further increase possibilities for a practical use of beam-driven plasma wakefield accelerators. From the studies presented here, it is evident that accelerating the electron beam to energies beyond 5 GeV in a single stage of the EuPRAXIA design may degrade its quality. Therefore, a strong focus for future studies on staging, its challenges, and mitigation techniques is essential.

To this aim, currently, operational R&D beam-driven facilities such as FLASH-Forward and SPARC-LAB have dedicated experiments for the investigation and evaluation of cutting-edge mitigation techniques and diagnostics. These studies are still ongoing. A selection of the most recent developments has been summarised in Section 32.5. The section also discusses some more practical considerations regarding the plasma sources required for a beam-driven wakefield accelerator.
<figure><img src="image_135.png" /><figcaption>Fig. 17.5. Rolling slice analysis in the 5 GeV case for the witness bunch at the plasma entrance (solid line) and at the plasma exit (dashed line). The top panel reports the normalised emittance in both transverse planes as well as the current. The bottom panel plots the energy spread along with the current. The corresponding current axis is the right y-axis.</figcaption></figure>

PWFAs in comparison to LWFAs are especially promising for high-average power free-electron laser applications and linear colliders considering their capabilities in energy transfer efficiency, average power, and repetition rate. It is crucial that future studies at PWFA test installations explore this potential in a complementary fashion beyond the goals set for EuPRAXIA. For example, at FLASHForward, it is possible to drive the plasma wakefield by high-current-density electron beams extracted from the FLASH L-band superconducting RF accelerator at up to a 3 MHz repetition rate with an average power well exceeding 10 kW. Such activities will synergistically complement future EuPRAXIA activities and help to maximise a successful implementation of PWFA technology into operational accelerators for users.

# 18 Magnets, Undulators, and Plasma Lenses

## 18.1 Introduction

Proposed in 1979, the laser-plasma accelerator (LPA) [4] enables the creation of ultra-high accelerating fields capable of generating electron beams with up to several GeV energy, short beam size (fs), excellent emittance at the plasma-vacuum interface (<1 mm mrad), and a high peak current (1–10 kA) within centimetre scale. The concept of the LPA [4] is based on the focussing of an ultra-intense laser pulse used to drive large-amplitude plasma waves that then act as accelerating structures for
particles. Since then, various LPA configurations have been conceived and realised, like the laser wakefield accelerator (LWFA) [4], the plasma beat wave accelerator (PBWA) [4], the self-modulated laser wakefield accelerator [365], wakefields driven by multiple pulses [180], and the highly non-linear regime of electron cavitation [366]. The so-called LWFA bubble (or blowout) regime [367] is very efficient since it preserves the normalised emittance of an accelerated electron bunch as the focussing force can be very large, is linear and provides charges of the order of hundreds of pC with mrad-level divergences, fs bunch durations, and good electron energy tunability. Two stages (injection and acceleration) can be differentiated in LPAs. The injection, where the electrons are introduced into the acceleration structure and the acceleration, is crucial to better control the resulting electron bunch characteristics. Specific acceleration techniques can thus be developed in multiple injection schemes. Ponderomotive injection [368] uses a laser to generate the plasma wakefield via LWFA and a second one to drive electrons into the wakefield, which allows for good injection control. Trapping in the self-modulated LWFA regime [369] uses a single laser to cause self-trapping and the generation of sub-ps electron bunches by trapping background plasma electrons in the wakefield. Colliding pulse injection [370] uses a three-laser system for plasma wave generation; backward and forward injection is used to generate a slow beat wave which heats and traps background electrons that are injected in a fast wakefield. Density transition injection [275] uses a downward transition in the plasma density to induce local self-trapping in the plasma wave, allowing for the generation of well-collimated, short electron bunches with narrow energy spread. Ionisation injection [280] uses a simple scheme where a target medium of low Z with a small proportion of high-Z gas is used, so that in front of an intense laser the outer shell of the high-Z gas is ionised contributing mainly to the plasma wave, and the inner-shell electrons ionised at the intensity peak are easily trapped by the plasma wave. Table 1 shows some experimental results for LPA electrons from the hundreds of MeV to the GeV level.

When considering implementing a free-electron laser based on electrons that are produced through LWFA [180], several new issues arise. The electron beam being of rather short duration, one should lengthen the electron beam in a chicane for the light not to escape from the electron bunch due to slippage. The electron beam also presents an energy spread which is typically one hundred times larger than the one generally accessible on linear accelerators used for short X-ray FELs [124]. To handle this issue, some electron beam manipulation has to be performed, either prior to the undulator with a demixing chicane [141,371] or directly inside the undulator using a transverse gradient undulator [64].

A relativistic electron passing through a planar undulator of period $\lambda_u$ and peak field $B_u$ is subjected to a sinusoidal trajectory and emits synchrotron radiation because of its acceleration in the transverse plane. The emitted radiation interferes and tends to add constructively depending on the phase lag between the electron and the front of the emitted wave train. This interference produces higher harmonics, leading to a harmonic spectrum. The radiation spectrum in the forward direction is not continuous but nearly monochromatic, i.e. it is composed of narrow spectral lines at well-defined frequencies caused by the harmonics. The resonant wavelength of these harmonics is given by $\lambda = \lambda_u(1 + K_u^2/2)/2\gamma^2$, with $K_u$ as the deflection parameter and $K_u = 0.934\lambda_u(\text{cm})B_u(T)$. A planar undulator leads to a linear polarisation in the plane of the electrons. An elliptical polarised undulator (EPU) produces a periodic magnetic field in both the vertical and horizontal planes, with variable dephasing and helical polarisation.

The periodic magnetic field of an undulator is created by either permanent magnets or electromagnets (superconducting or warm magnets) placed next to each other and separated by equal distance. The various technologies available to build an
undulator for an LPA-based FEL include cryogenic undulators [372,373], superconducting undulators, as well as exotic systems and technologies under development such as microfabricated undulators, plasma undulators, and transverse gradient undulators. For an FEL application, different undulator figures of merit are considered. First, a correct trajectory should be achieved for the radiation of the different period to interfere properly and ensure a proper natural bandwidth. The phase error, caused by the magnetic field errors along the undulator axis, such as variations in the peak field or the period length from one period to another [374,375] (mechanical changes, thermal effects, random effects due to magnetic defects, angular errors, etc.) has to be considered. It can cause a change in the length of the electron trajectory and yields a phase lag between the electron and the photon, resulting in destructive interference. In the framework of EuPRAXIA studies, a particular interest concerns the push towards higher magnetic fields with shorter undulator periods since one aims to minimise the slippage because of the shorter electron bunches produced by laser-plasma acceleration in order to avoid the light from escaping the electron bunch along the progression in the undulator. While reducing the period, one should keep, however, a deflection parameter value slightly larger than 1 to produce harmonics. So typically, one naturally considers cryogenic or superconducting undulators. In addition, depending on the strategy chosen for handling the initial large energy spread, transverse gradient undulators are of particular interest. Then one tries to define baseline undulator parameters to be used for the FEL pilot application of EuPRAXIA.

## 18.2 Specific Magnetic Elements

### Issues to Be Handled for LPA-Based FELs

An LPA-based FEL [60,61,376] appears attractive and conceivable, even though the properties of the beams cannot be directly used for FEL and, as a result, the beams should be first adapted and transported to the undulator.

Electron beam performance should thus fulfill the electron beam requirements imposed by the FEL. For an FEL [111] to be achieved, a relative slice energy spread $\sigma_{\gamma}$ over one coherent length of less than $\sigma_{\gamma}/\gamma < \rho$ is needed, with $\gamma$ as the relativistic factor and $\rho$ as the Pierce parameter. This condition is difficult to achieve since energy spreads on the percent level have been measured for MeV-GeV LPA electron beams. So to use these beams for FEL, the slice energy spread should be reduced, for example, by cutting a part of the initial distribution [377], using a demixing chicane to separate longitudinally the energies and select them [65] or opting for a different approach with the use of transverse gradient undulators (TGU) [64,378]. There is no specific issue regarding the dipoles of the chicane. The transverse gradient undulator has been reviewed in previous EuPRAXIA reports.

A problem arises regarding the transport of these beams towards the undulator. Even though the beam at the plasma-vacuum interface has a low emittance when leaving the interaction region, the normalised emittance suffers a strong deterioration, which makes the beam transport very difficult if not quickly mitigated [377,379]. So as the Twiss beta function $\beta_T = \sigma_x/\epsilon$ (with $\epsilon$ as the geometric emittance and $\sigma_x$ as the transverse beam size) is of sub-µm magnitude in the drift, the transverse beam correlation increases quadratically with distance due to a divergence of the order of mrad and becomes a predominant term in the emittance. The high divergence also causes a constant rapid increase in the beam size. The approach for calculating normalised emittance as $\epsilon_n = \langle\gamma\rangle\epsilon$ used for a drift in conventional accelerators is no more correct for LPA electron beams, and one has to employ the general definition
<table><caption>**Table 18.1.** Typical quadrupole specifications.</caption><thead><tr><th></th><th>LPA</th></tr></thead><tbody><tr><td>Gradient G (T/m)</td><td>200</td></tr><tr><td>G tuneability (%)</td><td>±20</td></tr><tr><td>Gradient homogeneity over 5 mm</td><td>10<sup>-2</sup></td></tr><tr><td>Bore diameter (mm)</td><td>10.5</td></tr><tr><td>Magnetic axis excursion (μm)</td><td>&lt;10</td></tr><tr><td>Length (cm)</td><td>30</td></tr></tbody></table>

[194] given by

$$ \epsilon_n^2 = \langle x \rangle^2 + \langle \beta^2 \gamma^2 x'^2 \rangle + \langle x \beta x' \gamma \rangle^2 \quad (18.1) $$

with $x$ and $x'$ as the transverse position and divergence, respectively, $\beta = v/c$, and $\gamma$ as the relativistic factor. If the correlation between energy and transverse position is negligible and assuming relativistic electrons ($\beta=1$), one can write,

$$ \epsilon_n^2 = \langle \gamma \rangle^2 + (\sigma_\gamma^2 \sigma_x'^2 \sigma_x^2 + \epsilon^2) \quad (18.2) $$

with $\sigma'_x$ as the beam transverse divergence. Assuming that the beam is at a waist at the plasma-vacuum interface ($\sigma_x(s) = \sigma'_x s$), with $s$ as the longitudinal beam position, equation (18.2) becomes

$$ \epsilon_n^2 \approx \langle \gamma \rangle^2 + (\sigma_\gamma^2 \sigma_x'^4 s^2 + \epsilon^2). \quad (18.3) $$

This rapid emittance increase is linked to the fact that the betatron frequency of the beam strongly depends on the energy. During the drift, each energy component rotates with a different velocity in the transverse phase space, increasing the area occupied by the bunch, resulting in a position and energy spread dependence of the normalised emittance. The projected normalised emittance of an LPA bunch of percent slice energy spread and mrad divergence causes a rapid increase in normalised emittance in a short drift distance, causing the beam to become emittance dominated and making it difficult to transport. A way to avoid a fatal emittance worsening of the beam is to capture and control the beam using strong focussing magnetic fields as close as possible to the plasma-vacuum interface with conventional accelerator techniques (e.g. the use of FODO with quadrupoles) [61] or active plasma lenses (APL) [380,381]. In both cases, a high magnetic gradient is necessary to control the beam emittance.

## Different Conventional Quadrupole Technologies

### Typically Required Quadrupoles Strength

It is relevant to consider conventional accelerator technology to handle the beam divergence. One can consider typical quadrupole requirements, as indicated in Table 18.1.

### Electromagnetic Technology

Mature resistive magnets, with copper conductors wrapped around soft iron poles, enabling one to tune the strength by simply changing the current with significant associated power consumption, are mostly used nowadays. Superconducting magnets producing very large magnetic fields require a costly helium infrastructure to keep the magnets in a superconducting state at a few Kelvin in temperature.
<table><caption>**Table 18.2.** Characteristics of fixed-gradient quadrupoles.</caption><thead><tr><td>Labs/Projects</td><td>Gradient (T/m)</td><td>Bore diameter (mm)</td></tr></thead><tbody><tr><td>PLEIADES ICS</td><td>550</td><td>5</td></tr><tr><td>FEL LPA</td><td>500</td><td>6</td></tr><tr><td>CLIC</td><td>575</td><td>8.25</td></tr><tr><td>SLAC/Kyoto</td><td>285</td><td>14</td></tr><tr><td>CESR</td><td>25.4</td><td>67</td></tr><tr><td>CESR</td><td>27.5</td><td>67</td></tr><tr><td>ESRF</td><td>82</td><td>24</td></tr></tbody></table>

*Permanent Magnet Technology*

Permanent magnet quadrupoles (PMQs) can offer high fields in a very compact design, free of the need for a power supply and electricity consumption. Wide tunability, high field homogeneity, and cost optimisation are still challenges to be faced.

## Permanent-Magnet-Based Quadrupoles of Fixed Strength

Permanent-magnet-based quadrupoles (PMQs) mainly consist of $Nd_2Fe_{14}B$ magnets because of their high remanent field and coercivity. Several Halbach-based PMQs with a fixed gradient were designed and built recently at facilities worldwide (see Tab. 18.2). Using the strong permanent magnet material NEOMAX, made from $Nd_2Fe_{14}B$ (Sumitomo Special Metal Co, Ltd), a PMQ was fabricated (at Kyoto University/SLAC) with a modified Halbach configuration [382]. A high-gradient minature PMQ is built for a laser-based tabletop free-electron laser at the Mainz electron accelerator (MAMI) [383]. It has a bore diameter of 6 mm with a magnetic length of 17 mm producing a field gradient of 503 T/m. The quadrupole field is defined by four radially magnetised wedges of $Nd_2Fe_{14}B$ with a remanence field of 1.5 T.

In addition, various PMQs have been built at the Cornell Electro-Positron Storage Ring (CESR). A permanent magnet with a 16-sector Halbach system with 1.22 remanence-field $Nd_2Fe_{14}B$ magnets of 67 mm bore diameter and 92 mm magnetic length, providing an integrated gradient of 2.5 T, was built [384]. At PLEIADES ICS, a 16-sector Halbach system of 5 mm bore diameter and maximum gradient of 560 T/m with adjustable focussing by longitudinal translation of the PMQ triplet has been designed and built [385]. At the Department of Physics in Munich, two 12-sector Halbach ring permanent-magnet systems using four radially magnetised magnets of 17 mm (15 mm) length, 6 mm bore diameter, 500 T/m gradient, and 7.5 T (8.5 T) integrated gradient have been built for tabletop FEL application [383]. Within a CLIC-CERN collaboration, a modified 12-sector Halbach permanent magnet with a Permendur mainframe that also works with poles of 100 mm (300 mm), 8.25 mm bore diameter, and 575 T/m gradient has been designed [386]. At ESRF, a compact PMQ with a hybrid-undulator-type design of 24 mm bore diameter and 82 T/m gradient for use in a storage ring has been constructed [387].

## Permanent-Magnet-Based Quadrupoles of Variable Strength

Various original designs were proposed and developed for a PMQ with a variable gradient (see Tab. 18.3) [388]. A modified Halbach-configuration PMQ with a variable gradient has been built in collaboration between Kyoto University and the Stanford Linear Accelerator Center (SLAC) for the final focus lens in a linear collider [389]. The quadrupole consists of an inner and outer ring, where the outer one is rotated to change the integrated gradient. The quadrupole bore diameter is 20 mm, with a magnetic length of 200 mm, generating a maximum gradient of 115 T/m with a tunability of 98 T/m.
<table><caption>**Table 18.3.** Characteristics of variable gradient quadrupoles.</caption><thead><tr><td>Labs/Uni</td><td>Gradient (T/m)</td><td>G (T)</td><td>Bore diameter (mm)</td></tr></thead><tbody><tr><td>ILC</td><td>115</td><td>98.5</td><td>20</td></tr><tr><td>COXINEL</td><td>208</td><td>93</td><td>11</td></tr><tr><td>NLC</td><td>135</td><td>27</td><td>12.7</td></tr><tr><td>SLAC</td><td>115</td><td>102</td><td>13</td></tr><tr><td>ILC</td><td>120</td><td>98</td><td>20</td></tr><tr><td>CLIC beam driver</td><td>60.4</td><td>45.4</td><td>27.2</td></tr></tbody></table>

An adjustable PMQ prototype has been designed and built in collaboration between CERN and the STFC Daresbury Laboratory [390]. The design incorporates a novel method of adjustment based on moving the permanent magnets. The field is generated by four permanent magnets at an angle of 40° to the horizontal, and each pair is attached by a wedge-shaped bridge of ferromagnetic material, where the gradient strength is varied by moving the bridge and permanent magnets vertically away from the centre to increase the gap. The bore diameter is 27.2 mm, with a magnetic length of 241 mm providing a maximum gradient of 60.4 T/m with a tunability of 45.4 T/m. A lower-strength version is also designed with the same bore diameter but with a 194 mm magnetic length. The maximum gradient attained is 43.8 T/m, with a tunability of 40.9 T/m.

Other designs of PMQs with variable gradients have been proposed and developed, such as a design (SLAC-Fermilab collaboration) based on four permanent magnet blocks and four soft iron poles, with the possibility to retract linearly the permanent magnet blocks, enabling an integrated gradient variation between 7 T and 68.7 T (gradient between 13 T/m and 115 T/m) for a 6.5 mm bore radius [391]. A “super hybrid quadrupole” concept (28 T/m, 7 T integrated gradient, and 27.5 mm bore radius) combining permanent magnetic material, coils, and soft magnetic material with 30% variability has also been built [392]. A high-reliability prototype quadrupole for the Next Linear Collider (NLC) of 12.7 mm bore with a range of integrated strengths from 0.6 T to 132 T, a maximum gradient of 135 T/m, and an adjustment range of 0 to -20% with a 1 µm magnetic centre stability has been built and fully tested [393].

A new patented [394] concept of compact hybrid permanent magnet quadrupoles of variable focussing strength for accelerators has been developed in the frame of the ERC COXINEL project for handling the large divergence of the electron beams produced by LPA while aiming at FEL application [141]. These so-called QUAPEVAs provide a large gradient (up to 200 T/m) and a wide tunability range (~45 %) [140,395-398]. They are mounted on translation tables for fine alignment and combine a Halbach-type quadrupole surrounded by four motorised rotating cylindrical magnets, enabling the gradient adjustment and plates behind for magnetic field shielding. Seven systems have been built and measured. The measurements confirmed the expected performance and successfully enabled the LPA electron transport for COXINEL [65]. The magnetic centre excursion, while varying the gradient, is kept within the ±10 µm range.

## Prospects Offered by Active Plasma Lenses

First discussed by Baker and Panofsky in 1950 [399], active plasma lenses (APLs) have been extensively used for ion beams, for example by using z-pinch discharges [400]. Now with the necessity of a transport strategy for LPA electron beams, the use of APL is already being seriously considered to substitute quadrupoles [380,381]. The high gradient fields of the order of kT/m, high tunability, and radial focussing symmetry allow for cm-scale focal lengths for MeV and GeV electron beams.
Recent works comparing state-of-the-art permanent magnet quadrupole triplets and active plasma lenses [401] have been carried out at LBNL in California, USA. A PMQ triplet positioned 17 cm away from the LPA source and 1.73 m from the spectrometer entrance with a 25 mm bore, 39, 54.5, and 25.5 mm lengths, 51.9, -50.9, and 51.3 T/m gradients in the x-direction, and -51.3, 51.0, and -51.1 T/m gradients in the y-direction is compared with an APL positioned 24 cm from the LPA source of $L_{APL}=1.5$ cm length and a $R_{APL}=500$ µm radius and with different densities.

Experiments and simulations show a lower energy dependence (chromaticity) in the APL-based line. Given the limited acceptance of around ±2 mrad, the electron beam wings for the APL are cut. The APL aperture also reduces the effective source divergence to approximately 1.8 mrad RMS and makes the charge throughput more sensitive to pointing fluctuations. Other effects on the electron beam caused by the APL walls lead to changes in charge density and a slight increase in emittance for large beam sizes with respect to the APL.

APLs favour smaller beams, but the effect can be reduced with low-density APLs of $n < 10^{17} \text{ cm}^{-3}$, short capillaries, and low beam densities (larger longitudinal length beams). A study of an APL transport line design that keeps the emittance degradation to 0.4 µm is being researched [401], two-stage LPA setups using APL are being used [380], and numerous experiments to characterise APL and their capabilities for electron bunch transport are being carried out in numerous laboratories [305,381].

## Conclusion

The transport of an LPA electron beam requires strong focussing right after the plasma. The technological solutions based on permanent magnet quadrupoles have been presented. The first results of comparing active plasma lenses with quadrupoles are also discussed. For EuPRAXIA, in view of robustness, it appears that the solution based on permanent magnet quadrupoles of variable gradients and variable strengths is the most suitable, since it enables the compensation for alignment errors of the components of the transfer line and of the electron beam pointing changes [65].

## 18.3 Review of State-of-the-Art Short-Period Undulators

### Development of State-of-the-Art Cryogenic Undulators

#### *Permanent-Magnet-Based Undulators*

Permanent magnet undulators (PMUs) [402] are able to function at room temperature and attain a fair magnetic field depending on the magnet material. PMUs are typically on-air devices with comparably large undulator gaps, which limits the on-axis magnetic field. Most pure permanent magnet undulators (PPMUs) use the Halbach geometric design [403]. Introducing poles between the magnets of a permanent magnet undulator makes it a hybrid type and enhances its magnetic peak field [404]. Poles are usually made of Vanadium Permendur material, i.e. cobalt-iron-vanadium alloys which saturate rather easily and have a high permeability and very high flux density at saturation.

Achieving a short-period undulator design with a sufficiently high magnetic field sets requirements for the magnetic material. Shortening the period implies a magnet-size reduction that results in a lower magnetic field. The magnetic remanence increase is at the expense of the coercivity (i.e. resistance against demagnetisation). So in-vacuum undulators [405–407] which avoid the beam pipe undulator gap limitation
<table><caption>**Table 18.4.** Typical characteristics of permanent magnets used for undulators.</caption><thead><tr><td>Magnets</td><td><i>B</i><sub>r</sub> (T)</td><td><i>H</i><sub>c</sub> (kA/m)</td></tr></thead><tbody><tr><td>SmCo<sub>5</sub></td><td>0.8–0.9</td><td>2400</td></tr><tr><td>Sm<sub>2</sub>Co<sub>17</sub></td><td>1.05–1.15</td><td>1700–2000</td></tr><tr><td>Nd<sub>2</sub>Fe<sub>14</sub>B</td><td>1.0–1.4</td><td>1000–2600</td></tr><tr><td>Pr<sub>2</sub>Fe<sub>14</sub>B</td><td>1.1–1.37</td><td>1000–1300</td></tr></tbody></table>

were adopted to reach a small gap with a sufficient magnetic field in placing the magnetic arrays in vacuum. The mechanical design should thus handle the magnetic forces due to larger fields and be vacuum compatible. The carriage can consist of a metallic base and a frame on which two external girders are fixed. A system of rods and flanges enables one to connect the internal girders where the magnet arrays are installed to external girders fixed to the carriage, which is designed to handle the magnetic forces. The carriage is equipped with motors for the movement of the gap (entrance and exit) and eventually for the movement of the offset to adjust vertically the magnetic axis of the undulator to the electron beam axis. Linear and rotated encoders are currently used to read the absolute gap. In addition, a vacuum chamber and pumping system should provide a good vacuum. Usually, the in-vacuum undulators are baked, so the magnet grade should handle being heated; tests with an unbaked in-vacuum undulator showed that beam conditioning can improve rapidly the vacuum [408]. The undulator vacuum chamber is connected to the standard chamber with specific RF tapers [409] for preserving a proper value of the impedance with or without water cooling. A liner (generally in Ni-Cu) is laid on the magnet arrays to prevent the heat load from the image current because of wakefields or up-stream synchrotron radiation [410].

## Magnet Choice

Permanent magnets are characterised by their remanence $B_r$ (strength of the magnetic field), coercivity $H_c$, energy product $BH_{max}$ (density of magnetic energy), and Curie temperature $T_c$ (temperature at which the material loses its magnetism). Permanent magnets [411,412] used for undulators combine rare-earth (RE) ferromagnetic elements with incomplete f-shells and transition metals with d-shells such as iron, nickel, and cobalt. The rare-earth magnets present a crystalline structure with a very high magnetic anisotropy (stable alignment of the atoms), enabling an easy magnetisation along one direction and a high resistance along the other. High magnetic moments at the atomic level combined with the high anisotropy result in a high magnetic field strength. The typical performances of SmCo$_5$ [413], Nd$_2$Fe$_{14}$B [414–416], and Pr$_2$Fe$_{14}$B magnets are presented in Table 18.4.

Magnets' resistance to demagnetisation [417,418] and heat budget are an issue, in particular for in-vacuum undulators, for which intermediate grades of Nd$_2$Fe$_{14}$B ($B_r \le 1.26$ T; $H_c = 1900$ kA/m) could be used. A choice of Nd$_2$Fe$_{14}$B with high coercivity avoids demagnetisation at UHV baking and radiation damage. A small inclusion of Dysprosium also allows for a larger coercivity. Typically, one should consider a coercivity larger than 1000 kA/m to avoid demagnetisation at room temperature and larger than 2000 kA/m to prevent it at 120°C (393 K), i.e. in baking conditions.

The idea of cooling down RE$_2$Fe$_{14}$B rare-earth permanent magnets, which increases the remanent field and coercivity, was proposed [419], leading to the concept of cryogenic undulators. For example, typical temperature coefficients for RE$_2$Fe$_{14}$B are -0.11%/°C for the remnant field and -0.58%/°C for the coercivity, i.e. decreasing the temperature by a factor of two enables one to increase the remanent field by 10% and the coercivity by more than 50%. As the increase in coercivity is larger than
the one of the remanent field, one can even take a magnet grade that is less resistant at room temperature but presents a higher remanent field.

Measurements [407,420] of remanent field and coercivity for different grades of $RE_2Fe_{14}B$ permanent magnets versus temperature were performed. The grades for each material slightly differ in field remanence and coercivity but show similar behaviour as a function of temperature. For Neodymium grades at low temperature (130–140 K), the remanent field starts to decrease because of the so-called spin re-orientation transition (SRT) phenomenon [421–423], which exhibits a negative dependence of remanent fields against temperature below 130–140 K due to a change in the preferred direction of the magnetisation with respect to the easy axis of magnetisation. In contrast to the Praseodymium grades, the remanent field continues to increase at low temperatures down to 30 K. These magnets can be cooled down to 30 K, and attain a high coercivity (~7300 kA/m) and high field remanence (1.7 T) [424,425]. The coercivity of all grades maintains increasing with lower temperature [426].

### Cryogenic Permanent-Magnet-Based Prototype Undulators

Several cryogenic permanent magnet undulator (CPMU) prototypes were built at different locations. At SPring-8 [427], a 40×15 mm period $Nd_2Fe_{14}B$ system has been built and optimised, with 3.3° and 3.2° RMS phase errors at 300 K and 130 K, respectively. The temperature control at 140 K was enabled thanks to sheath heaters. It then appeared quite attractive to use $Pr_2Fe_{14}B$ magnets for being able to operate at lower temperature, thus with a larger magnetic field and coercivity. The first prototype of 8×14.5 mm period NSLS [428] using $Pr_2Fe_{14}B$ magnets (NEOMAX 53CR) and Vanadium-Permendur poles has been measured in the Vertical Test Facility at liquid nitrogen and He temperatures with a slight increase in the RMS phase error at lower temperature (3.1° at room temperature and 3.5° at 77 K). The second system [429] developed at NSLS-II, using a grade of $Pr_2Fe_{14}B$ magnet that can be baked (NEOMAX CR47) led to a higher field than the previously employed grade (at 80 K: 1.12 T for the CR47 and 1.22 T for the CR53). Several CPMU prototypes were built at SOLEIL [430]. The first one [431], a 4×20 mm period hybrid $Nd_2Fe_{14}B$ system, shows a 11.5% increase in the magnetic field between room temperature and a cryogenic temperature of 140 K, the operation temperature. The second one [432] (4×18 mm period) and the third one [433] (4×15 mm period) of the $Pr_2Fe_{14}B$ hybrid type (NEOMAX CR53), take advantage of the absence of SRT phenomena. The magnetic field grows by 13% between room temperature and a cryogenic temperature of 77 K. A 20×9 mm period ($Pr,Nd_2)Fe_{14}B$ (Vacuumschmelze /Vacoflux50) cryogenic undulator [434,435] with $Co_{49}Fe_{49}V_2$ poles with a saturation magnetisation of 2.35 T, built jointly by the Helmholtz-Zentrum Berlin and the Ludwig-Maximilian-University München (LMU), shows an increase in the remanence by 20% and of the peak field at the fixed gap of 2.5 mm by 11%, with a partial saturation of the pole pieces from 300 K to 30 K. The second prototype with modified poles exhibits a larger field [436] and it enabled the observation of synchrotron radiation [437] using the MAMI-B beamline with a 855 MeV beam.

### State of the Art of Cryogenic Permanent-Magnet-Based Full-Scale Undulators

The construction of full-scale devices to be installed for beamlines has started at ESRF [438–441] with a 2 m long full-scale 18 mm period $Nd_2Fe_{14}B$-magnet (NEO-REM 595t) hybrid CPMU. The peak field is increased by 6% when cooled down from 273 K to 150 K at a gap of 6 mm. The RMS phase error slightly increases from room temperature (4.8°) to 150 K (5.7°), because of a residual longitudinal temperature gradient. It is the first full-scale (2 m length) CPMU to be built and installed for operation with an electron beam and a liquid nitrogen closed loop for
<table><caption>**Table 18.5.** Width of magnets and poles for different periods and coefficients of the fitting curves, a, b and c, shown in Figure 18.2. The magnetic field is fitted with the equation $B_{peak} = a \exp(b(\frac{g}{\lambda_u} + c(\frac{g}{\lambda_u})^2))$, where g is the magnetic gap.</caption><thead><tr><th>Period (mm)</th><th>Magnet Width (mm)</th><th>Pole Width (mm)</th><th>a</th><th>b</th><th>c</th></tr></thead><tbody><tr><td>18</td><td>6.5</td><td>1.25</td><td>3.74306</td><td>-4.0533</td><td>0.69459</td></tr><tr><td>15</td><td>5</td><td>1.25</td><td>0.389558</td><td>-4.0222</td><td>0.52895</td></tr><tr><td>12</td><td>4</td><td>1</td><td>3.98608</td><td>-4.0874</td><td>0.67293</td></tr><tr><td>10</td><td>3.5</td><td>0.75</td><td>3.5308</td><td>-3.6476</td><td>0.40497</td></tr></tbody></table>

cooling. A second CPMU has been built and installed at ESRF. Two additional $Pr_2Fe_{14}B$-based hybrid undulators are under construction.

At the Paul Scherrer Institute [442–444], a full-scale 1.7 m long 14 mm period CPMU using $Nd_2Fe_{14}B$ (Hitachi NMXS45SH) magnets and Vanadium-Permendur poles, cooled with LN2, had been measured with SAFALI (Self-Aligned Field Analyser with Laser Instrumentation). The measured phase error of 1.1° is similar to the one measured at room temperature, thanks to an in situ correction method.

SOLEIL [445,446] had built and measured the first $Pr_2Fe_{14}B$ (CR53)-based full-scale hybrid cryogenic undulator (2 m long, 18 mm period) cooled down to 77 K with LN2. The phase error at a 5.5 mm gap at room temperature of 2.8° RMS increases up to 9° at 77 K, but has been corrected down to 3° by shimming the rods. It is the first $Pr_2Fe_{14}B$ full-scale cryogenic undulator installed in a synchrotron radiation facility, in use at the Nanoscopium long beamline. SOLEIL is in the process of building three new cryo-ready devices with a different $Pr_2Fe_{14}B$ grade with an enhanced coercivity (1912 kA/m): a second 2 m long U18 CPMU that has been successfully installed for use at COXINEL [447,448] at room temperature as well as a 3 m long U15 CPMU [433] enabling a 1.65 T at 77 K for a 3 mm gap.

At DIAMOND [449], a 17.7 mm period full-scale $Nd_2Fe_{14}B$ (Vacodym 776TP)-based hybrid CPMU has been built by Danfysik [450]. As the temperature is decreased from 300 K to 157 K, the field is increased by 7.03% at a 4 mm gap, while for a gap of 10 mm the increase in field is 8.69%. At 157 K, the RMS phase error is measured to be 3.5° at a gap of 4 mm.

Based on earlier prototypes, HZB [451–453] has built one full-scale cryo-cooled CPMU of dimensions 175×17 mm and with a gap of 5 mm [454] and is building another one with a 15 mm period length and 2 mm gap, investigating two cooling concepts based on liquid nitrogen and single-staged cold heads, respectively. $(Pr,Nd)_2Fe_{14}B$ magnets (Vacuumschmelze) treated with a grain boundary diffusion process for an enhanced stability and Co-Fe poles are used. The gap size is measured using an optical micrometer. CPMU15 is developed for a plasma-driven FEL experiment in close cooperation with Hamburg University.

The National Synchrotron Radiation Research Center (NSRRC) [455] is developing a 22 mm period CPMU. A CPMU magnetic measurement bench is developed and tested, with a carriage and optical components being redesigned to improve the reproducibility.

A 2 m long cryogenic undulator with 140 periods of length 13.5 mm is to be built for the High-Energy Photon Source Test Facility (HEPS-TF) in Korea [456]. This undulator consists of $Pr_2Fe_{14}B$ magnets and will be cooled down to liquid nitrogen temperatures (80 K), reaching a magnetic field of 1 T at a gap of 5 mm.
<figure><img src="image_136.png" /><figcaption>**Fig. 18.1.** (a) RADIA model of a cryogenic undulator with a 12 mm period: the remanent field is 1.57 T, the dimensions (x, z, s) are the following: half-poles: 56×1×26 mm³, magnets: 50×4×30 mm³. (b) Magnetic field of this undulator for a 3 mm gap.</figcaption></figure>

*Magnetic Fields Generated by Cryogenic Permanent-Magnet-Based Full-Scale Undulators for EuPRAXIA*

The peak field-to-gap relationship of undulators with different periods have been computed using the RADIA software [457]. The model is presented in Figure 18.1 (top).

The peak field versus the gap for a 12 mm period has been computed using the RADIA software, as shown in Figure 18.1 (bottom).

Figure 18.2 depicts the deflection parameter versus the undulator gap and period. The magnetic field is fitted with the following equation: $B_{peak} = a \exp(b \frac{g}{\lambda_u} + c(\frac{g}{\lambda_u})^2)$, where g is the magnetic gap, and a, b, c are the fitting coefficients (see Tab. 18.5). Typically, one can get 1.65 T on axis with a U15 at a 3 mm gap, or 1.52 T on axis with a U15 at a 3 mm gap.