X-ray free-electron lasers (XFELs)1 have been recognized as transformative tools of modern science since the first lasing of the Linac Coherent Light Source (LCLS) in 2009.2 By utilizing the FEL instability when a high-energy and high-quality electron beam wiggles in a magnetic undulator, XFELs can deliver coherent X rays with 1–10 keV photon energy, ∼100 GW peak power, and ps–fs pulse duration, thus enabling applications in many disciplines,3 such as structural biology,4–6 molecular physics,7 and materials science.8 Recently, there has been a desire for high-photon-energy XFELs from the development of fields such as dynamic mesoscale materials science,9,10 high-resolution single-particle imaging,11 nuclear resonance excitation,12 and “diffraction-before-destruction” applications.13 For example, 20 keV hard X-ray photons can mitigate electronic damage in an irradiated sample compared with photons with lower energy, which enables the visualization of the electron density distribution in a nearly damage-free condition.13

The radiation wavelength of an XFEL is determined by the electron beam energy and the undulator as λr=(λu/2γb2)(1+K2/2), while the saturation power is Psat≈ρPb∝K2/3λu2/3γb0, where γ_b is the relativistic factor of the electron beam, λ_u and K are the parameters describing the undulator period and strength respectively, ρ is the growth rate of the FEL instability,1 and P_b is the electron beam power. Shorter radiation wavelengths, corresponding to higher photon energies, can be achieved by using undulators with shorter periods or weaker strengths,14 but both result in a fall in the saturation power. By contrast, beams with higher energy can produce shorter-wavelength radiation while maintaining the saturation power. In current XFELs, electron beams are accelerated to ∼10 GeV in kilometer-long radio-frequency (RF) accelerators. As the acceleration gradient in RF cavities is limited to tens of MV/m, it is costly to improve the beam energy by extending the acceleration structures.

A plasma-based accelerator (PBA)15–17 can sustain an acceleration gradient of ∼GV/cm in meter-long plasmas, and hence is considered to provide a promising route toward the development of compact FEL facilities.18–20 It has been explored at RF accelerator facilities to boost the beam energy in a short distance.21–23 Two-bunch beam-driven plasma wakefield acceleration is mostly adopted, usually referred to as the afterburner, in which one electron beam (witness beam) from RF accelerators is accelerated in the plasma wake excited by another beam (driver beam) from the same RF accelerators.23 The energy-boosted witness beam has the potential to improve the photon energy after passing through a magnetic undulator.24,25 However, since FELs have stringent requirements on beam quality,26 i.e., the normalized emittance ɛ_n ≲ γ_bλ_r/(4π) and the relative energy spread Δγ_b/γ_b < ρ, it remains unclear whether the accelerated witness beams can meet these requirements and eventually radiate coherently to demonstrate the feasibility of improving the FEL photon energy.

In this work, we use theoretical analysis and start-to-end simulations to show that XFEL radiation pulses with nearly fourfold boost in photon energies and maintained peak power can be generated by adding a plasma-based afterburner to current XFEL facilities. As shown in Fig. 1(a), the proposed setup consists of three parts, namely, the PBA stage, the beam transport stage and the radiation stage. In the PBA, the witness beam is accelerated inside the nonlinear plasma wake driven by the driver to double its energy [Fig. 1(b)]. The axial variation of the acceleration gradient increases the slice energy spread, which can be controlled by the current profiles of the two beams and their temporal separation. The degradation of the emittance and the slice energy spread due to nonideal issues, such as the motion of the plasma ions and hosing instability, will be discussed. The witness beam is then transversely matched to the undulator through a magnetic transport line, while the energy-depleted driver is dumped. By properly choosing the beam parameters, the plasmas, and the magnets, a nearly energy-doubled witness beam with more than half of the electrons meeting the FEL requirements can enter the undulator and produce X-ray pulses with ∼22 keV photon energy [Fig. 1(c)] and ∼100 GW peak power. The curved longitudinal phase space distribution of the witness beam provides possibilities to produce two-color X-ray pulse-pairs or large-bandwidth pulses. Such a plasma-based afterburner can be implemented economically in current XFEL facilities to significantly improve the radiation photon energy with a high power, which is beneficial for hard X-ray users from nuclear physics and imaging community.

We set the energy of the electron beams from the RF accelerators to 8 GeV without loss of generality. Two tri-Gaussian electron beams with a temporal separation propagate into a uniform plasma. The parameters of the electron beams and the plasma are listed in Table I, which are close to those of typical beams in contemporary XFEL facilities. Based on the length of the electron beam driver σ_z,d = 16 μm, we choose a plasma density of n₀ = 7 × 10¹⁶ cm⁻³ to achieve k_pσ_z,d ∼ 1 for effective wake excitation, where k_p = ω_p/c, ωp=n0e2/mε0 is the plasma frequency, m and e are the electron mass and charge, c is the speed of light in vacuum, and ɛ₀ is the vacuum permittivity. The beams are assumed to be transversely matched with the ion column force of the nonlinear plasma wake, i.e., their Twiss parameters27 satisfy β=2γbkp−1≈3.6mm and α = 0. It is challenging to use magnetic focusing elements to achieve this millimeter-class β. The magnetic elements can focus the beam with β = 0.1–1 m at the entrance of the plasma, and we can use an appropriately tailored plasma density upramp to further focus the beam to the needed β.28,29 For example, simulation shows that a density upramp of n(z)=n0/[1+(Lm−z)/lm]2 with a characteristic length l_m ≈ 2.66 mm and a total length L_m ≈ 0.74 m can focus the witness beam from β = 1 m to β = 3.4 mm with its normalized emittance preserved. The length of the plasma is chosen to be L = 1.26 m to maximize the energy extraction of the driver and the energy gain of the witness beam. A heat-pipe oven with lithium vapor can be used as the required plasma source,22 pre-ionized by an ultrafast laser pulse focused by axicon optics.30,31

We use the three-dimensional (3D) quasi-static particle-in-cell (PIC) code QuickPIC32,33 to model the PBA stage. The tightly focused driver beam with a peak current of 2 kA can excite a fully blowout plasma wake, and the subsequent witness beam is located in the acceleration phase, as shown in Fig. 1(b). In a fully blowout wake, an ion column void of plasma electrons is formed with a linear focusing force (Fr=−12mωp2r for the witness beam) and a transversely uniform acceleration gradient.34 The normalized emittance and the absolute energy spread of an infinitely thin axial slice of the witness beam are thus conserved during the PBA. In FELs, electrons within a slice of finite length emit radiation cooperatively. This length is called the cooperation length and is given by l_c ≡ L_satλ_r/λ_u,1 where Lsat=20λu/(4π3ρ) is the saturation length for a self-amplified spontaneous emission (SASE) FEL. The quality of the electrons within one cooperation length is thus essential for FEL. When the witness beam is absent, the acceleration gradient has an approximately linear dependence on ξ,34 causing a growth of the energy spread of a slice with a width of l_c, where ξ ≡ z − ct is the longitudinal coordinate moving with the beams. To quantitatively describe the slice beam properties, we assume that a nearly energy-doubled witness beam (15.2 GeV) with a spot size of 19 μm conducts a SASE FEL in an undulator with λ_u = 2 cm and K=22. The estimated growth rate and the cooperation length are ρ ≈ 0.32‰ and l_c ≈ 158 nm, respectively.

Since the plasma wake evolves little through the entire plasma, we perform one-step QuickPIC simulations to study the profile of the acceleration gradient and its effect on the slice energy spread. The simulation setup can be found in the Appendix. The witness beam with different peak current I, duration σ_z, and delay Δ, defined as the separation between the witness beam center ξ_w and the driver center ξ_d, can modify the profile of E_z through the beam loading effect.35 A 2D parameter scan is performed with delays varying from 58 to 82 μm in steps of 0.48 μm and peak currents increasing from 1.4 to 2.6 kA in steps of 24 A while keeping the charge constant at 103 pC. Since the slice width (∼0.1 μm) is much shorter than the wake wavelength (∼100 μm) and the bunch duration, we derive the relative slice energy spread after the afterburner by assuming a linear variation of E_z and a constant current in each slice asδW≡σWW=σW02+lc23dEzdξL2W0+EzL,(1)where W₀ and σW0 are the initial average energy and the initial energy spread of the witness bunch.

Figure 2(a) shows the on-axis accelerating gradient and the relative slice energy spread of the witness bunch predicted using Eq. (1) for five representative cases: A, B, C, D, and O. Based on the E_z profile experienced by the witness beam, the simulations are classified into three categories:35

1. (i)In cases O, B and D, the wake is well loaded and the accelerating gradient near the center of the beam is approximately flattened (dE_z/dξ ≈ 0), which leads to a small slice energy spread.
2. (ii)In case A, the wake is underloaded and dE_z/dξ > 0 along the beam, resulting in a large slice energy spread.
3. (iii)In case C, the wake is overloaded, with dE_z/dξ < 0 in the middle of the beam and dE_z/dξ > 0 away from the beam center. The relative energy spread reaches its minimum at two turning positions where dE_z/dξ = 0 and has a relatively large value at the beam center.

We define the fraction of the electrons with δ_W ≤ ρ as the useable fraction η, which corresponds to how many electrons can reach FEL saturation in the undulator. Figure 2(b) shows η for different settings of the witness beam. If we either reduce the delay Δ while simultaneously increasing the peak current I, or vice versa, i.e., from O to either B or D, the wake remains well loaded and η barely changes. This is consistent with the beam loading dynamics,35 since a larger (respectively smaller) space charge field is needed to push the sheath electrons of the ion channel outwards to achieve a flat acceleration gradient distribution if the witness beam is closer to (respectively farther from) the wake center. Nevertheless, if we simultaneously reduce or increase Δ and I, the wake becomes underloaded as at A or overloaded as at C. We choose case O for the following start-to-end simulation owing to its relatively high useable fraction (η ≈ 65%) and its robustness to jitters (i.e., far from regions with low η).

We then carry out the full PBA simulation with QuickPIC. Throughout the PBA stage, the plasma wake structure barely evolves, and the witness beam is continuously accelerated until the driver beam is depleted at a plasma length of L = 1.26 m. Figure 3(a) shows the longitudinal phase space of both beams at the exit of the PBA stage. The central energy of the witness beam W_c reaches 15.2 GeV, which is 1.9 times the initial energy. The acceleration gradient is 5.7 GV/m. The inset shows that the slice energy spread of the witness beam is ≲MeV, while the projected energy spread reaches ∼GeV. The current (blue line) and the normalized emittance (black and red lines) of the witness beam at the end of the plasma are preserved as shown in Fig. 3(b). The relative slice energy spread (green line) and η ≈ 65% are close to the predictions from one-step simulation.

As the acceleration continues, the witness beam remains transversely matched with the ion column force, and its spot size is σr=βmεn/γb, where βm=2γbc/ωp.36 At the exit of the plasma, the Twiss parameters and the spot size are β ≈ 4.7 mm, α ≈ 0, and σ_r ≈ 0.25 μm, respectively. This tightly focused beam would diffract quickly after the plasma, which results in a catastrophic emittance growth28 and FEL gain degradation.

Here, a well-designed magnetic transport line is used to adjust the transverse phase space distribution of the witness beam, matching it with the focusing elements along the undulator and maintaining the beam emittance. Meanwhile, a collimator and a magnetic chicane are combined to dump the energy-depleted driver. We use the particle tracking code ELEGANT37 to design the magnetic transport line. It consists of two quadruple triplets and one chicane as shown in Figs. 1(a) and 4(a). The parameters of these elements are given in the Appendix. The evolution of the β function of the witness electrons with energies near W_c is shown in Fig. 4(a), where the black solid line and dashed line represent β_x and β_y in the horizontal and vertical planes, receptively. The maximum β function of these particles along the transport line is 725 m, and the corresponding spot size is 0.1 mm. Figure 4(b) shows the slice properties of the witness beam at the end of the transport line. The Twiss parameters of the electrons near the center of the witness beam (from ξ = −83 to −80 μm) are β_x ≈ 29.4 m, β_y ≈ 26.2 m, α_x ≈ 0.99, and α_y ≈ −1.06. The emittance and the slice energy spread of these electrons are preserved well, with η ≈ 61%.

As shown in Fig. 4(a), a collimator with a radius of 0.5 mm is placed at the beamline position z_l = 2 m, where the driver has a large spot size and divergence and blocks 91% of the driver electrons. Then, a chicane that is capable of separating the energy-depleted driver and the accelerated witness beam in the horizontal plane (x-direction) is placed between the two triplets to remove the rest of the driver. The horizontal separation of the witness beam and the rest of the driver after the second dipole is larger than 2 mm [Fig. 4(a)]. It is calculated that 99.3% of the driver is dumped if the dumping range is set to x > 2 mm, leaving the witness beam unaffected.

The incoherent synchrotron radiation (ISR)38 and coherent synchrotron radiation (CSR)39 emitted by the electrons during their passage through the dipoles may degrade the beam quality. For the parameters considered here, the normalized emittance growth caused by ISR and CSR is estimated to be ∼1 nm, which is negligible. As shown by the current profile in Fig. 4(b), the head and tail of the witness bunch with a large energy chirp are notably compressed, while the middle part of the beam with a small chirp is weakly compressed.

We choose a planar undulator with a period λ_u = 2 cm and K=22 (corresponding to a peak magnetic field of 1.5 T) as the radiator, and the resonant wavelength is λr=0.56Å for the 15.2 GeV beam. We first use Xie’s fitting formula40 which includes the effects of the beam emittance, energy spread, and radiation diffraction to optimize the β function of the beam. Numerical calculations show that a high saturation power of 11 GW and a short gain length of 3.4 m are achieved when β = 28 m. The corresponding 3D growth rate is ρ_3D ≈ 0.27‰. This optimized β function is used for the transport line design in the last section and the design of the focusing elements along the undulator (see the Appendix).

When studying high-photon-energy FELs, the quantum effect may degrade the FEL performance. When emitting photons, the quantum recoil may kick electrons out of the FEL resonant bandwidth. This effect is characterized by the quantum recoil parameter41q_λ = ℏω_r/(ργ_bmc²), where ℏ is the reduced Planck constant and ω_r = 2πc/λ_r. Its value is q_λ ∼ 10⁻³ ≪ 1 for the parameters studied in this work, which indicates that the quantum recoil effect is negligible. Quantum diffusion, on the other hand, is induced by continuous spontaneous undulator radiation (SUR), which can be divided into the classical part and the quantum part.42 The former describes the average energy loss due to SUR, given by43dγb/dz=−13reγb2(2π/λu)2K2, where r_e = 2.818 × 10⁻¹⁵ m is the classical electron radius. For the presented case, the relative energy loss rate is −(1/γ_b)dγ_b/dz ≈ 2.2 × 10⁻⁵ m⁻¹, causing the beam energy to move away from the resonant bandwidth after tens of meters of undulator. A tapered undulator44 with a longitudinally varying K can compensate for this energy loss. Although the classical part can be tackled by tapering, there are few ways available at present to cope with the quantum part, which comes from the random fluctuations of the SUR photon energy and increases the beam energy spread. Theoretical analysis43 gives d[(Δγb/γb)2]/dz≈3.9×10−10m−1 for the case presented. This corresponds to an additional energy spread of 0.22‰ at an undulator length of z_u = 120 m, which is still less than ρ. The radiation power growth in simulations is similar whether this additional energy spread is turned on or off.

We model the FEL process with the 3D FEL code GENESIS45 (see the Appendix). Figure 5(a) shows the power gain over the undulator in three different cases. In the first case (blue line), the quantum diffusion from the SUR is not considered, and the radiation saturates at z_u = 60 m with P_sat ≈ 10 GW, which is consistent with Xie’s formula. The second case (yellow line) shows that quantum diffusion suppresses the saturation power, since P_sat ≈ 0.2 GW. A properly tapered undulator is used in the third case (orange line) to compensate for the energy loss caused by quantum diffusion, and the power at z_u = 60 m is similarly to that in the first case. The tapered undulator can extract more energy from the electron beam after saturation.46 For example, the radiation peak power is 101 GW at z_u = 120 m.

Figure 5(b) shows the temporal profile of the radiation pulse at z_u = 120 m in one representative SASE run. The pulse duration is ∼20 fs, and the total energy is 0.26 mJ, corresponding to 7.6 × 10¹⁰ photons. The radiation spectrum is shown in Fig. 5(c). Both the temporal and spectral distributions of the FEL pulse contains many random spikes, which is a characteristic of the SASE mode.47 A Gaussian fitting gives a central photon energy of 21.9 keV, which agrees with the resonant wavelength, while the relative bandwidth is 1.6‰, five times larger than the bandwidth in typical SASE, where the bandwidth is approximately equal to the growth rate. This large bandwidth can be explained by the energy chirp of the electron beam. Figure 5(d) shows the Wigner-Ville distribution of the radiation pulse,48 where a positive chirp (photon energy increasing from head to tail) is presented, which is quantitatively consistent with the longitudinal phase space distribution of the electron beam [the inset of Fig. 3(a)], since the photon energy in FEL is proportional to γb2. The bandwidth can be further increased by fine-tuning the witness current profile.

The aforementioned scaling of the photon energy and the saturation power show that using undulators with shorter period λ_u or smaller K can further improve the photon energy at the cost of reduced power and photon number. For instance, 44 keV photons with a saturation power of 5 GW can be produced by using the same accelerated witness bunch but with an optimized β-function of ∼80m and a 120-m-long undulator with λ_u = 2 cm and K = 1.7. In this case, the pulse duration is reduced to ∼13 fs and the pulse energy decreases to 29 μJ, corresponding to 4 × 10⁹ photons.

Although the plasma ions are usually treated as immobile and have a uniform distribution in plasma wakefield accelerators, the space charge fields of tightly focused driver and witness beams can pull ions inwards and cause a nonuniform density distribution, which would modify the electromagnetic field distribution of the nonlinear wake and degrade the beam quality.49,50 In the simulations with 2 kA driver and witness beams, the density perturbation of the lithium ion at the witness beam center is 0.16n₀, and the induced emittance and energy spread growth are negligible. If the currents of both beams are increased to 10 kA, the ion density perturbation reaches 2.5n₀ at the center of the witness beam. Nevertheless, the growth in emittance caused by the ion motion is only ∼10%,50 and the growth in relative slice energy spread in the useable fraction is ∼10⁻⁵, which have little impact on the FEL performance according to the Xie’s formula. This indicates that for the typical beams in current XFEL facilities, the beam quality degradation induced by the lithium ion motion can be neglected.

Hosing instability is another possible mechanism that may destroy the beam quality in realistic experiments. If there exists a misalignment between the beams or the beams are asymmetric, the coupling between the beams and the plasma wake results in an exponential amplification of the initial small misalignment or the asymmetries and degrades the beam quality.51 To illustrate the effect of the hosing instability on FEL performance, we assume that the witness beam has an initial x-direction offset δ_x relative to the driver and present the emittance growth at the end of the PBA from simulations and the predicated 3D FEL growth rate in Fig. 6. When δ_x varies between 0.1σ_r and σ_r, the emittance growth in both directions and the ρ_3D reduction are approximately linear, where σ_r is the initial witness beam spot size. Specifically, if δ_x = σ_r, then ρ_3D decreases by a factor of three, which indicates that the saturation length increases by the same factor and the saturation power decreases by the same factor squared.

The ability to control the longitudinal phase space distribution of the witness beam through the PBA enables more operation modes of XFEL. For instance, if the PBA works in the overloaded case (case C in Fig. 2), there are two narrow and separated regions that satisfy the FEL energy spread requirement. Furthermore, the acceleration gradients of these two regions are different. This beam can thus produce two ultrashort pulses with two colors when radiating in the undulator. The temporal and spectral separations of these two pulses can be adjusted by the current profile and the delay of the witness beam.

In conclusion, through start-to-end simulations and theoretical analysis, we have demonstrated the feasibility of a plasma-based afterburner to improve the photon energy at current XFEL facilities. The beam loading effect can flatten the axial acceleration gradient profile and suppress the growth in slice energy spread. Simulations show that more than half of the witness beam can satisfy the XFEL requirements after a meter-long plasma if the current profile and the delay are appropriate. The nearly energy-doubled accelerated witness beam can generate ∼22 keV XFEL pulses with ∼10¹¹ photons, which may enable critical applications in advanced imaging and nuclear physics.

Acknowledgment. This work was supported by the National Grand Instrument Project No. SQ2019YFF01014400, the Natural Science Foundation of China (Grant Nos. 12375147, 12435011, 12075030), the Beijing Outstanding Young Scientist Project, Project for Young Scientists in Basic Research of Chinese Academy of Sciences (YSBR-115), the Beijing Normal University Scientific Research Initiation Fund for Introducing Talents No. 310432104, and the Fundamental Research Funds for the Central Universities, Peking University. The simulations were supported by the High-Performance Computing Platform of Peking University and the Tianhe new generation supercomputer at the National Supercomputer Center in Tianjin.