Introduction

In plasma-based accelerators (PBAs), an intense laser pulse1 or high-energy charged particle beam2 drives a plasma wake sustaining accelerating fields orders of magnitude higher than those achievable with conventional radiofrequency technology3, offering a path towards highly compact and cost-effective accelerators. Having reached GeV energies in only centimeter scales4,5,6,7,8, femtosecond-long electron bunches with kiloampere current9,10 and micron-level emittance11,12, PBAs are becoming increasingly attractive for applications such as free-electron lasers (FELs)13 or future linear colliders14, which would strongly benefit from reduced size and cost. In particular, there is also a special interest in the production of sub-femtosecond bunches15,16,17 to generate short X-ray pulses for ultrafast science18.

However, despite these major advances, achieving an energy spread below the percent level has remained an issue. This is a problem not only for applications such as FELs, which require a relative energy spread \(\lesssim \)10−3 19, but also for the beam transport after the plasma, which could cause a dramatic emittance growth20 and further reduce the beam quality, limiting any kind of multi-stage acceleration. Although the impact on the emittance can be mitigated by tailored plasma-to-vacuum transitions21 or the use of active plasma lenses instead of conventional quadrupoles22, improving the beam parameters in the plasma is essential for demanding applications. Major efforts have therefore been dedicated towards this objective, including the development of controlled injection techniques23,24,25,26,27 as well as concepts for mitigating the correlated energy spread arising from the steep slope of the accelerating fields. These include beam loading the wake28,29,30, either by the accelerated bunch itself31 or the injection of a secondary one32, the use of modulated33 or tailored34 plasma profiles, the insertion of a magnetic chicane in a multistage PBA35, or taking advantage of the beam-induced wakefields36,37,38,39,40.

Here we describe additional sources of energy spread in PBAs which should also be mitigated, as well as limitations on the minimum achievable bunch length. In particular, we show that the transverse oscillations of beam electrons (known as betatron motion), induced by the strong focusing fields in the plasma wake, can be a significant source of uncorrelated (or slice) energy spread as well as increased bunch length due to path length differences between particles with different oscillation amplitude. Although these effects were already known41,42, no model exists to date for their impact on the beam parameters. We present here a novel analytical theory, validated against numerical simulations with the particle-in-cell (PIC) codes OSIRIS43 and HiPACE44, which accurately describes these phenomena in the assumption of relativistic electrons in a non-evolving wake. This model allows us to understand previously observed limitations, such as a finite energy spread even when the initial bunch length approaches zero45, and define guidelines for minimizing the impact of these phenomena for high-quality electron beams.

Results

Development of a single-particle model

As a first step, a single-particle model capable of describing the coupling of the transverse oscillations into the longitudinal motion is needed. Then, from this model, statistical averages over a particle beam can be analytically obtained in order to determine the impact of this coupling on parameters such as the energy spread and bunch length.

For a relativistic particle moving close to the speed of light c, the longitudinal component of its velocity vz = (v2 − vx2 − vy2)1/2, where v = |v| is the magnitude of the particle velocity v, is limited by c and can be decreased due to motion in the transverse planes, i.e, if vx, vy ≠ 0. Thus, in the speed-of-light frame ξ = z − ct, where z is the longitudinal coordinate in the laboratory frame and t is the time, a particle performing betatron oscillations will experience a slippage towards the back which, at a time t, will be given by \(\Delta \xi (t)=\xi (t)-{\xi }_{0}={\int }_{0}^{t}\,{v}_{\xi }(t)dt\), where ξ0 is its initial position and vξ = vz − c is its longitudinal velocity in the speed of light frame.

Determining the particle slippage thus requires obtaining an expression for vξ(t), which, in turn, depends on the evolution of the transverse components vx and vy. In order to find these expressions, the equations of motion of a relativistic electron in a plasma wakefield have to be solved. These are given by \(\dot{{\boldsymbol{p}}}=-e{\boldsymbol{W}}\), where p = mγv is the particle momentum, m is the electron mass, \(\gamma =1/\sqrt{1-{({\boldsymbol{v}}/c)}^{2}}\) is the relativistic Lorentz factor of the particle, e is the elementary charge and W = (Ex − cBy, Ey + cBx, Ez) is the wakefield. In this expression, Ei and Bi (for i = x, y, z) are the different components of the electric and magnetic fields. Analytical expressions of the wakefields can be found for the linear regime46,47, and several models have been developed for non-linear wakes in the blowout regime48,49. In what follows, the wakefield gradients around the particle, Kx = ∂xWx, Ky = ∂yWy and Ez′ = ∂zWz, will be assumed constant and ∂iWj = 0 for i ≠ j. This is especially well-suited for the blowout regime, where the driver expels all plasma electrons and leaves behind an ion cavity with uniform focusing fields Kx = Ky = mωp2/2e and an approximately constant Ez′ in a wide range of the accelerating phase. The plasma frequency is defined as \({\omega }_{p}=\sqrt{{n}_{p}{e}^{2}/m{\varepsilon }_{0}}\), where ε0 is the vacuum permittivity and np is the unperturbed plasma density. For linear wakes, although these gradients are not uniform, this model can be applied for regions sufficiently close to the propagation axis, where Kx and Ky can be regarded as constant, and if \(\varDelta \xi \ll c/{\omega }_{p}\), so that the longitudinal change in Kx, Ky and Ez′ can be neglected.

Under these conditions, the equations of motion for a beam electron in a non-evolving wake with a phase velocity vw given by the propagation velocity of the driver, can be written as

$$\dot{\gamma }(t)= {\mathcal E} (t),$$
(1)
$${\ddot{p}}_{x}(t)+{\omega }_{x}{(t)}^{2}\,{p}_{x}(t)=\mathrm{0,}$$
(2)

where \( {\mathcal E} (t)=-\,e{E}_{z}(t)/mc\), \({\omega }_{x}(t)=\sqrt{{{\mathscr{K}}}_{x}/\gamma (t)}\) is the betatron frequency and \({{\mathscr{K}}}_{x}=e{K}_{x}/m\). Since \( {\mathcal E} ^{\prime} =-\,e{E^{\prime} }_{z}/mc\) is assumed constant, then \( {\mathcal E} (t)={ {\mathcal E} }_{0}+ {\mathcal E} ^{\prime} [\Delta \xi (t)+(c-{v}_{w})t]\), with \({ {\mathcal E} }_{0}= {\mathcal E} \mathrm{(0)}\). Using this expression in Eq. (1) yields γ(t) = γ(0)(t) + γ(1)(t), where \({\gamma }^{\mathrm{(0)}}(t)={\gamma }_{0}+{ {\mathcal E} }_{0}t\), with γ0 = γ(0), is the energy evolution assuming constant acceleration and \({\gamma }^{(1)}(t)={\int }_{0}^{t}\, {\mathcal E} ^{\prime} [\Delta \xi (t^{\prime} )+(c-{v}_{w})t^{\prime} ]dt^{\prime} \) accounts for the contribution of slippage and dephasing with respect to the wake50. If \({\gamma }^{\mathrm{(1)}}/{\gamma }^{\mathrm{(0)}}\ll 1\), then \(\gamma \simeq {\gamma }^{\mathrm{(0)}}\) can be assumed in order to solve Eq. (2), for which analytical solutions can be found if ωx(t) is a slowly varying function51, i.e, if \({\dot{\omega }}_{x}/{\omega }_{x}^{2}={{\mathscr{E}}}_{0}/2\sqrt{\gamma {{\mathscr{K}}}_{x}}\ll 1\). This yields the following expressions:

$$x(t)\simeq {A}_{x\mathrm{,0}}\Gamma {(t)}^{-\mathrm{1/4}}\,\cos ({\varphi }_{x}(t)+{\varphi }_{x\mathrm{,0}}),$$
(3)
$${v}_{x}(t)\simeq -\,{A}_{x\mathrm{,0}}{\omega }_{x\mathrm{,0}}\Gamma {(t)}^{-\mathrm{3/4}}\,\sin ({\varphi }_{x}(t)+{\varphi }_{x\mathrm{,0}}),$$
(4)

where \(\Gamma (t)={\gamma }^{\mathrm{(0)}}(t)/{\gamma }_{0}=1+{ {\mathcal E} }_{0}t/{\gamma }_{0}\) is the relative energy evolution, \({A}_{x\mathrm{,0}}=\sqrt{{x}_{0}^{2}+{({v}_{x\mathrm{,0}}/{\omega }_{x\mathrm{,0}})}^{2}}\) is the initial oscillation amplitude in the x plane and ϕx,0 = −arctan(vx,0/x0ωx,0) is the initial phase. The values x0, vx,0 and ωx,0 correspond to the initial transverse position, velocity and betatron frequency. The phase advance \({\varphi }_{x}(t)={\int }_{0}^{t}{\omega }_{x}(t^{\prime} )\,{\rm{d}}t^{\prime} \) is given by

$${\varphi }_{x}(t)\simeq 2\frac{\sqrt{{{\mathscr{K}}}_{x}{\gamma }_{0}}}{{{\mathscr{E}}}_{0}}(\Gamma {(t)}^{1/2}-1).$$
(5)

Analogous expressions can be found for the y plane and are not included here for simplicity.

The solutions to the transverse electron motion can now be used to obtain the longitudinal slippage, \(\varDelta \xi (t)={\int }_{0}^{t}\,({{\boldsymbol{v}}}_{z}(t^{\prime} )-c)\,dt^{\prime} \), by considering \({v}_{z}={({{\boldsymbol{v}}}^{2}-{v}_{x}^{2}-{v}_{y}^{2})}^{\mathrm{1/2}}\simeq |{\boldsymbol{v}}|\,-({v}_{x}^{2}+{v}_{y}^{2}\mathrm{)/2|}{\boldsymbol{v}}|\) and \(|{\boldsymbol{v}}|-c\simeq -\,c\mathrm{/2}{\gamma }^{2}\) for a relativistic electron. Since the time scale of the oscillation amplitude damping, Γ−3/4(t), in Eq. (4) is much longer than the betatron period, the time-average of this expression can be used for the electron transverse velocity. This results in the expression

$$\Delta \xi (t)\simeq \frac{c}{2{{\mathscr{E}}}_{0}{\gamma }_{0}}(\Gamma {(t)}^{-1}-1)+\frac{{A}_{0}^{2}{\mathscr{K}}}{2c{{\mathscr{E}}}_{0}}(\Gamma {(t)}^{-1/2}-1),$$
(6)

where A02 = Ax,02 + Ay,02 and the focusing gradient is assumed to be the same in both planes, i.e \({{\mathscr{K}}}_{x}={{\mathscr{K}}}_{y}={\mathscr{K}}\). The two terms in Eq. (6) account, respectively, for the slippage due to |v| < c and the slippage caused by betatron motion. Most of Δξ occurs initially (at lower energies) and reaches a limit Δξmax as Γ → ∞ given by

$$\Delta {\xi }_{max}=-\,\,\frac{1}{2{{\mathscr{E}}}_{0}}(\frac{c}{{\gamma }_{0}}+\frac{{A}_{0}^{2}{\mathscr{K}}}{c}).$$
(7)

Using Eq. (6) it is now possible to calculate γ(1) and finally obtain γ(t) = γ(0)(t) + γ(1)(t), which takes into account the influence of the particle slippage on its energy. This yields

$$\begin{array}{c}\gamma (t)\simeq {\gamma }_{0}+{{\mathscr{E}}}_{0}t+{{\mathscr{E}}}^{{\rm{^{\prime} }}}\Delta {\xi }_{max}t+\frac{{{\mathscr{E}}}^{{\rm{^{\prime} }}}}{2}(c-{v}_{w}){t}^{2}+\frac{c{{\mathscr{E}}}^{{\rm{^{\prime} }}}}{2{{\mathscr{E}}}_{0}^{2}}\,{\rm{l}}{\rm{n}}\,\Gamma (t)+\frac{{{\mathscr{E}}}^{{\rm{^{\prime} }}}{A}_{0}^{2}{\mathscr{K}}{\gamma }_{0}}{c{{\mathscr{E}}}_{0}^{2}}(\Gamma {(t)}^{1/2}-1),\end{array}$$
(8)

where the third term is a correction to the linear energy gain, which is modified due to slippage, and the fourth one accounts for the influence of dephasing with the wake when vw < c in laser-driven cases. The following terms are higher order corrections which account, respectively, for slippage due to |v| < c and slippage due to betatron motion.

Equations (3), (4), (6) and (8) allow for a complete description of the single-particle evolution within the wakefields. In order to test their validity, they have been compared with numerical solutions of Eqs. (1) and (2). Three test cases corresponding to single electrons with different initial transverse offsets are shown in Fig. 1. The electrons in these three cases (Ca, Cb and Cc) have, respectively, x0 = y0 = 1, 3, 5 μm and vx,0 = vy,0 = 0. They are injected with γ0 = 100 and propagate 5 cm within a plasma stage with np = 1017 cm−3 assuming a blowout with \( {\mathcal E} ^{\prime} =-\,{\omega }_{p}^{2}/2c\), \({ {\mathcal E} }_{0}={\omega }_{p}\) and \({\mathscr{K}}={\omega }_{p}^{2}/2\). The wake velocity is directly determined from the group velocity of a laser driver as vw/c = ωl/(ωp2 + ωl2)1/2 if etching effects52 are neglected, where ωl = 2πc/λl, assuming a wavelength λl = 800 nm. The agreement between numerical and analytical solutions is excellent, showing that this single-particle model accurately describes the coupling between longitudinal and transverse dynamics. From these test cases it can also be seen that betatron oscillations with micron-level amplitude can lead to femtosecond-level slippage and energy differences on the percent range.

Figure 1
figure 1

Comparison of the analytical and numerical results for three test particles with different oscillation amplitude in a 5 cm plasma stage: (a) betatron oscillations in x, (b) longitudinal velocity, (c) longitudinal slippage, and (d) the final energy after acceleration. The analytical predictions are all shown in grey, while the numerical solutions have a different color for each particle.

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When considering a particle bunch, these differences in slippage and energy gain between single particles will have an impact on parameters such as the bunch length and energy spread. One immediate consequence will be an increase of the bunch length, which will in turn also lead to increased correlated energy spread if \( {\mathcal E} ^{\prime} \ne 0\). Another implication, more critical for FEL applications, will be an intra-bunch slice mixing which will lead to a growth of the slice energy spread.

Impact on slice energy spread

In order to obtain an expression for the induced slice energy spread, Eqs. (6) and (8) can be used to calculate the energy difference Δγ between two beam particles, p1 and p2, with the same γ0 but different oscillation amplitude, A0,p1 = 0 and A0,p2 ≠ 0, which at some time t are in the same infinitesimally long beam slice at ξs due to the experienced slippage. If \( {\mathcal E} ^{\prime} \) and \({\mathscr{K}}\) are constant along the bunch, then \({ {\mathcal E} }_{\mathrm{0,}{p}_{i}}={ {\mathcal E} }_{\mathrm{0,}s}- {\mathcal E} ^{\prime} \Delta {\xi }_{{p}_{i}}\) for i = 1,2, where \({ {\mathcal E} }_{\mathrm{0,}s}\) is measured at ξs. This allows the particle energy evolution, γpi(t), to be written in terms of the fields at the current slice. Additionally, if \( {\mathcal E} ^{\prime} \varDelta {\xi }_{{p}_{i}}(t)/{ {\mathcal E} }_{\mathrm{0,}s}\ll 1\), then \({ {\mathcal E} }_{\mathrm{0,}{p}_{i}}\simeq { {\mathcal E} }_{\mathrm{0,}s}\) can be assumed for the higher order terms in γpi(t) while maintaining the full expression in the leading linear term. This also means that \(\varDelta {\xi }_{{p}_{2}}(t)-\varDelta {\xi }_{{p}_{1}}(t)\simeq {A}_{0,{p}_{2}}^{2}{\mathscr{K}}({\Gamma }_{s}^{-1/2}(t)-1)/2c{{\mathscr{E}}}_{0,s}\), which, added to the previous considerations, leads to

$$\Delta \gamma (t)={\gamma }_{{p}_{2}}(t)-{\gamma }_{{p}_{1}}(t)\simeq \frac{{\gamma }_{0}{{\mathscr{E}}}^{{\rm{^{\prime} }}}{\mathscr{K}}{A}_{0,{p}_{2}}^{2}}{2c{{\mathscr{E}}}_{0,s}^{2}}\frac{{({\Gamma }_{s}{(t)}^{1/2}-1)}^{2}}{{\Gamma }_{s}{(t)}^{1/2}},$$
(9)

where \({\Gamma }_{s}(t)=1+{{\mathscr{E}}}_{0,s}t/{\gamma }_{0}\). The correlation Δγ ∝ − \({{\rm{A}}}_{0}^{2}\) (if \( {\mathcal E} ^{\prime} < 0\)) within a beam slice arises because, due to slippage, particles with a higher A0 originally come from positions ahead of their current slice. Since \( {\mathcal E} ^{\prime} < 0\), the accelerating fields they have experienced as they slip towards the back are smaller than at their current position, thus leading to a lower net energy gain. Equation (9) therefore shows that although a positive correlation Δγ ∝ A02 exists for particles starting at the same slice, as given by Eq. (8), this correlation is actually negative when looking at particles which have ended in the same slice due to slippage. The induced relative slice energy spread due to this effect can be found from the standard deviation of \(\Delta \gamma /{\bar{\gamma }}_{s}\) if the mean slice energy is assumed to evolve as \({\bar{\gamma }}_{s}(t)\simeq {\bar{\gamma }}_{\mathrm{0,}s}+{ {\mathcal E} }_{\mathrm{0,}s}t\), from what it is found that

$$\frac{{\sigma }_{{\gamma }_{s}}^{\Delta \xi }}{{\bar{\gamma }}_{s}}(t)\simeq \frac{{{\mathscr{E}}}^{{\rm{^{\prime} }}}{\mathscr{K}}{\sigma }_{{A}^{2}}}{2c{{\mathscr{E}}}_{0}^{2}}\frac{{({\bar{\Gamma }}_{s}{(t)}^{1/2}-1)}^{2}}{{\bar{\Gamma }}_{s}{(t)}^{3/2}}\simeq \sqrt{\frac{2{\mathscr{K}}}{{\bar{\gamma }}_{0,s}}}\frac{{{\mathscr{E}}}^{{\rm{^{\prime} }}}{{\mathscr{F}}}_{m}{\varepsilon }_{n}}{{{\mathscr{E}}}_{0}^{2}}\frac{{({\bar{\Gamma }}_{s}{(t)}^{1/2}-1)}^{2}}{{\bar{\Gamma }}_{s}{(t)}^{3/2}},$$
(10)

where \({\bar{\Gamma }}_{s}={\bar{\gamma }}_{s}/{\bar{\gamma }}_{0,s}\), σA2 is the standard deviation of A02 for the particles within the slice and \({ {\mathcal E} }_{0}\) is taken at the slice position. For an on-axis Gaussian bunch with zero average transverse momentum (\(\bar{x}=\bar{y}=0\) and \({\bar{p}}_{x}={\bar{p}}_{y}=0\)), it can be obtained that \({\sigma }_{{A}^{2}}\simeq \sqrt{8{c}^{2}{{\mathscr{A}}}^{2}/{\mathscr{K}}{\bar{\gamma }}_{0,s}}\), where \({{\mathscr{A}}}^{2}=\frac{1}{4}{\sum }_{i=x,y}\,{\varepsilon }_{n,i}^{2}({\beta }_{i}^{4}+{\beta }_{m}^{4})/{\beta }_{i}^{2}{\beta }_{m}^{2}\). The variables εn,i, \({\beta }_{i}={\sigma }_{i}^{2}{\bar{\gamma }}_{0}/{\varepsilon }_{n,i}\) and σi are, respectively, the beam’s normalized emittance53, beta function and rms size in both transverse planes, while βm = c/ω0, with ω0 = ωx,0 = ωy,0, is the matched beta function54 defined by the focusing fields within the plasma. For a cylindrically symmetric bunch, i.e, if εn,i = εn and βi = β, a “mismatch factor” \({ {\mathcal F} }_{m}={(({M}^{4}+\mathrm{1)/2}{M}^{2})}^{\mathrm{1/2}}\) can be defined, where M = β/βm, such that \({\mathscr{A}}={{\mathscr{F}}}_{m}{\varepsilon }_{n}\), which leads to the last expression in Eq. (10). For a matched beam (M = 1) the mismatch factor is \({ {\mathcal F} }_{m}=1\), which also corresponds to its minimum value. The impact of betatron slippage on the slice energy spread will therefore be stronger on mismatched beams. Analyzing Eq. (10) it can be seen that the induced slice energy spread has a fast initial growth until it reaches a maximum when the bunch energy has increased by a factor 9, i.e, when \({\bar{\Gamma }}_{s}=9\). For higher energies, the slice energy spread slowly decreases proportional to \({{\bar{\Gamma }}_{s}}^{-1/2}\). This behaviour arises from \({\sigma }_{{\gamma }_{s}}^{\Delta \xi }\), whose growth rate is initially faster than \({\bar{\gamma }}_{s}\) but gradually slows down due to the reduced slippage at higher energies.

This source of energy spread will act in addition to that due to betatron radiation (BR)55,56, which has also been shown to induce a correlation Δγ ∝ −A02 and generate a slice energy spread given by57 \({\sigma }_{{\gamma }_{s}}^{{\rm{BR}}}/{\bar{\gamma }}_{s}\simeq {C}_{{\rm{BR}}}({\bar{\Gamma }}_{s}^{\mathrm{5/2}}-\mathrm{1)/}{\bar{\Gamma }}_{s}\), where \({C}_{{\rm{B}}{\rm{R}}}=(2{r}_{e}/15{c}^{3}){{\mathscr{K}}}^{2}{\sigma }_{{A}^{2}}{\bar{\gamma }}_{0,s}^{2}/{{\mathscr{E}}}_{0}\) and re = e2/4πε0mc2 is the classical electron radius. As both effects are induced by the same underlying cause (the betatron motion of particles), it is of interest to compare and determine their relative relevance. This can be done by obtaining the intersection points of Eq. (10) with the BR expression, for which perturbation theory is needed. This method yields that the slippage-induced energy spread will initially dominate over BR if the ratio \( {\mathcal R} ={C}_{{\rm{BR}}}/{C}_{\Delta \xi }\ll 0.1\), where \({C}_{\Delta \xi }={{\mathscr{E}}}^{{\rm{^{\prime} }}}{\mathscr{K}}{\sigma }_{{A}^{2}}/2c{{\mathscr{E}}}_{0}^{2}\) is the coefficient in Eq. (10). When this is the case, a transition into a BR-dominated regime occurs at an energy \({\bar{\Gamma }}_{s}={\bar{\Gamma }}_{t}\) which, to first order, can be obtained as \({\bar{\Gamma }}_{t}\simeq { {\mathcal R} }^{-\mathrm{1/2}}\). An illustration of both regimes can be seen in Fig. 2 for a particular set of parameters. As a general rule, although this will vary from case to case, the energy spread due to slippage will typically dominate over BR up to energies on the order of ~10 GeV. This is the energy range in which FELs operate, implying that betatron slippage is the source of slice energy spread that should be minimized for these applications.

Figure 2
figure 2

Comparison of the slice energy spread due to slippage (from Eq. (10)) and betatron radiation (from model by P. Michel et al.57) for the particular case of a 100 MeV bunch with 1 mm mrad normalized emittance injected with matched beta function into a plasma with np = 7 × 1017 cm−3 assuming a wakefield in the blowout regime (\( {\mathcal E} ^{\prime} =-\,{\omega }_{p}^{2}\mathrm{/2}c\), \({ {\mathcal E} }_{0}={\omega }_{p}\), \({\mathscr{K}}={\omega }_{p}^{2}/2\)). Betatron slippage is found to be the dominant effect up to an energy of 44.8 GeV, which is accurately estimated by the \( \sim 47\,{\rm{GeV}}\) value given by the first order approximation \({\bar{\Gamma }}_{t}\simeq { {\mathcal R} }^{-\mathrm{1/2}}\).

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An effective way for mitigating the slippage-induced energy spread would be to beam-load the wake such that \( {\mathcal E} ^{\prime} \to 0\), thus suppressing the energy spread growth as given by Eq. 10. However, since this condition is difficult to achieve, especially along the whole bunch, other strategies can be derived from this equation. Regarding the accelerating structures, possible strategies include reducing plasma density, since the ratio \({{\mathscr{K}}}^{1/2}{{\mathscr{E}}}^{{\rm{^{\prime} }}}/{{\mathscr{E}}}_{0}^{2}\propto {n}_{p}^{1/2}\), or using hollow plasma channels58, so that \({\mathscr{K}}\to 0\). In regards to the accelerated beam, minimizing the emittance, increasing the initial energy and fulfilling matching conditions (\({ {\mathcal F} }_{m}=1\)) would be beneficial, with these two last points being especially relevant for externally injected beams.

Minimizing this slice energy spread is especially important for PBAs based on chirp compensation schemes32,33,35. This is because although the average \( {\mathcal E} ^{\prime} \) experienced by the beam after acceleration might be 0, its value at any point of the accelerator is not, thus giving rise to the development of a slice energy spread which will remain in the de-chirped beam. It is possible, however, that the alternation between positive and negative \( {\mathcal E} ^{\prime} \) in these schemes might partially mitigate the correlation Δγ ∝ A02 from Eq. (9) and thus the slice energy spread, although never remove it completely. As an additional remark, it should also be noted that this correlation between oscillation amplitude and energy caused by betatron slippage has been proposed as a way of performing beam conditioning for FELs59 by means of a magnetic chicane60. Thus, it might also be beneficial in some cases.

Impact on bunch length and total energy spread of ultra-short bunches

In addition to its impact on the slice energy spread, another immediate consequence of the betatron-induced slippage will be a growth of the bunch length. This is especially relevant for ultra-short bunches where the slippage experienced by the particles is comparable to or bigger than the initial bunch length σz,0. The contribution of the single-particle slippage to σz can be directly obtained from the standard deviation of Eq. (6) by assuming an initially monoenergetic bunch with σz,0 = 0. The bunch lengthening is then found to be

$${\sigma }_{\Delta \xi }(t)\simeq \frac{{\mathscr{K}}{\sigma }_{{A}^{2}}}{2c{{\mathscr{E}}}_{0}}(1-\bar{\Gamma }{(t)}^{-1/2})\simeq \sqrt{\frac{2{\mathscr{K}}}{{\bar{\gamma }}_{0}}}\frac{{{\mathscr{F}}}_{m}{\varepsilon }_{n}}{{{\mathscr{E}}}_{0}}(1-\bar{\Gamma }{(t)}^{-1/2}),$$
(11)

where \(\bar{\Gamma }=\bar{\gamma }/{\bar{\gamma }}_{0}\simeq 1+{ {\mathcal E} }_{0}t/{\bar{\gamma }}_{0}\). The last expression on the right is again obtained for the case of a cylindrically-symmetric Gaussian bunch. The bunch length evolution, when taking into account the impact of slippage, is thus given by σz(t) = (σz,02 + σΔξ2(t))1/2 for initially Gaussian bunches. When this is the case, this expression therefore establishes a limitation on the minimum bunch length achievable in a PBA. This could have a strong impact on the production of high-energy, sub-femtosecond bunches, because although certain techniques to inject such ultra-short bunches into the plasma wake have been proposed15,16,17,31, their sub-femtosecond duration could be lost when further accelerating them. Possible ways of mitigating this lengthening can be found by looking at Eq. (11), which yields the same conditions on the beam emittance, matching and initial energy found for Eq. (10). In this case, however, no benefits from beam-loading or reduced plasma density are expected because \( {\mathcal E} ^{\prime} \) does not play a role in the lengthening and the ratio \({{\mathscr{K}}}^{1/2}/{{\mathscr{E}}}_{0}\) does not depend on the plasma density. Another remark that can be extracted from Eq. (11) is that, as expected from Eq. (7), the lengthening has a finite upper limit when \(\bar{\Gamma }\to \infty \) given by \({\sigma }_{\Delta \xi }^{max}\simeq {(2{\mathscr{K}}/{\bar{\gamma }}_{0})}^{1/2}{{\mathscr{F}}}_{m}{\varepsilon }_{n}/{{\mathscr{E}}}_{0}\).

As a consequence of the bunch lengthening, in cases where \( {\mathcal E} ^{\prime} \ne 0\), the correlated energy spread of the bunch will also increase. This explains previously observed limitations, where convergence to a finite energy spread was found as σz,0 → 045. This limit on the minimum energy spread as the initial bunch length is reduced can be calculated for an initially monoenergetic beam with σz,0 = 0 from the standard deviation of Eq. (8). This leads to

$${\frac{{\sigma }_{\gamma }^{\Delta \xi }}{\bar{\gamma }}(t){\textstyle |}{\textstyle |}}_{{\sigma }_{z,0}\to 0}\simeq \frac{{{\mathscr{E}}}^{{\rm{^{\prime} }}}{\mathscr{K}}{\sigma }_{{A}^{2}}}{2c{{\mathscr{E}}}_{0}^{2}}\frac{{(\bar{\Gamma }{(t)}^{1/2}-1)}^{2}}{\bar{\Gamma }(t)}\simeq \sqrt{\frac{2{\mathscr{K}}}{{\bar{\gamma }}_{0}}}\frac{{{\mathscr{E}}}^{{\rm{^{\prime} }}}{{\mathscr{F}}}_{m}{\varepsilon }_{n}}{{{\mathscr{E}}}_{0}^{2}}\frac{{(\bar{\Gamma }{(t)}^{1/2}-1)}^{2}}{\bar{\Gamma }(t)},$$
(12)

which, in the same way as Eq. (11), also tends to a finite value as \(\bar{\Gamma }\to \infty \). The last expression on the right corresponds again to the case of a cylindrically-symmetric Gaussian bunch.

An illustrative example of the phenomena described in this section can be seen in Fig. 3, where an externally injected witness beam is shown at the entry and after 1 cm of plasma, as obtained from an OSIRIS 2D simulation. The induced lengthening as well as the development of a quadratic correlation between energy and oscillation amplitude can be clearly seen. As predicted by Eqs. (8) and (9), this correlation is positive for the overall bunch (Fig. 3b) but negative on the slice level (Fig. 3c).

Figure 3
figure 3

Results from an illustrative 2D OSIRIS simulation of a 100 MeV, 0.7 pC and initially sub-femtosecond Gaussian electron bunch externally injected into the blowout of a laser driver in a plasma with np = 1017 cm−3. More details can be seen in the Methods section. (a) Spatial distribution and current profile of the electron beam at injection and after 1 cm of propagation in the plasma. The development of a curved shape and increased length, as expected from the particle slippage, are clearly visible. (b) Beam energy distribution with respect to the initial oscillation amplitude of the particles. As expected from Eq. (8), those with higher A0 experience a higher energy gain. The analytical prediction of the energy-amplitude correlation as given by this equation is shown by a dashed black line. This curve was computed from the value of the fields in the simulation at the central position of the bunch, i.e., \({ {\mathcal E} }_{0}\simeq 0.60\,{\omega }_{p}\), \( {\mathcal E} ^{\prime} \simeq -\,0.27\,{\omega }_{p}^{2}/c\) and \({\mathscr{K}}\simeq 0.36\,{\omega }_{p}^{2}\). The energy difference Δγtotal was calculated with respect to the average energy of the particles with \({A}_{0}\simeq 0\,{\rm{\mu }}{\rm{m}}\). (c) Distribution of the particle energy with respect to the initial oscillation amplitude within each beam slice. The coloured lines correspond to the distribution within each 20-nm-long slice, while the thin dark lines represent the average. As expected from Eq. (9), the energy-amplitude correlation at the slice level is negative. The analytical prediction was again computed for the central beam slice using the same field values as before. The energy difference within the slice, Δγslice, was calculated with respect to the average energy of the particles with \({A}_{0}\simeq 0\,{\rm{\mu }}{\rm{m}}\) after removing the longitudinal energy correlation (chirp) along the beam.

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Validation against 3D PIC simulations

In order to validate and illustrate the significance of the expressions for the slice energy spread and bunch lengthening, a series of 3D simulations with the PIC codes OSIRIS and HiPACE have been performed. The studied cases consisted on a beam-driven plasma stage providing a 1 GeV net energy gain to an externally injected witness beam, as it would be interesting for FEL applications or multi-staged acceleration. In the OSIRIS simulations, the impact of the slippage effects has been tested for different witness beam parameters and plasma densities, and in all cases beam loading effects could always be neglected. On the contrary, the HiPACE simulations where performed to study the possible mitigating effects of beam loading for a fixed set of plasma and beam parameters, where only the total beam charge was varied. Although only beam-driven cases have been simulated, similar results may apply for a laser driver as long as dephasing effects in the mean beam energy can be neglected, as they are not included in Eqs. (10) to (12).

In all simulated cases, a Gaussian driver with an energy of 1 GeV and a 0.1% spread has been considered. Its dimensions and charge have been defined in terms of the plasma density so that the generated blowout can be scaled with np. For the baseline case in which np = 7 × 1017 cm−3, as used in previous experiments6, this means a transverse size \({\sigma }_{x}={\sigma }_{y}=0.4c/{\omega }_{p}\simeq 2.5\,{\rm{\mu }}{\rm{m}}\), a length \({\sigma }_{z}=c/{\omega }_{p}\simeq 6.4\,{\rm{\mu }}{\rm{m}}\), a normalized emittance \({\varepsilon }_{n}=0.4\,{\sigma }_{x}\simeq 1\,{\rm{\mu }}{\rm{m}}\,{\rm{rad}}\) and a peak electron density \({n}_{b}\simeq 3.78\,{n}_{p}\) for a total charge \(Q\simeq 265\) pC and a peak current \({I}_{p}\simeq 5\,{\rm{kA}}\). The plasma target has a flat-top longitudinal profile and is transversely uniform.

For the studies made with OSIRIS, a Gaussian witness beam is injected on-axis at \({\xi }_{w}\simeq {\xi }_{d}-4.5\,c/{\omega }_{p}\), where ξd is the driver centre. Different emittance values (εn = 0.1, 1 and 10 μmrad) as well as injection energies between 10 MeV and 1 GeV have been tested. The transverse size is matched to the focusing fields so that \({ {\mathcal F} }_{m}\mathrm{=1}\). With regards to the bunch charge and duration, two different sets of parameters have been considered: σz/c = 3 fs with 1 pC for the studies on the slice energy spread, as it could be obtained in the SINBAD facility at DESY61,62, while a shorter bunch with σz/c = 0.1 fs with 0.1 pC was used for the bunch lengthening studies. This shorter bunch duration was chosen to better highlight the impact of lengthening due to slippage on sub-femtosecond bunches. The peak current in both cases is too low to significantly modify \( {\mathcal E} ^{\prime} \) along the bunch, therefore allowing beam loading effects to be neglected. The slice energy spread in all tested beams is dominated by slippage, as \( {\mathcal R} \ll 0.1\) and \(\bar{\Gamma }\ll { {\mathcal R} }^{-\mathrm{1/2}}\). Also, in order to isolate the plasma stage as the only source of energy spread, a zero longitudinal momentum spread at injection has been considered.

In order to compare the simulation results with the analytical theory, the field values in Eqs. (10) and (11) need to be provided. While these could be estimated from analytical models49,63, they were measured directly at ξw from a single simulation with no witness beam, obtaining \({ {\mathcal E} }_{0}\simeq 0.53\,{\omega }_{p}\), \( {\mathcal E} ^{\prime} \simeq -\,0.37\,{\omega }_{p}^{2}/c\) and \({\mathscr{K}}\simeq 0.5\,{\omega }_{p}^{2}\). This allows for a more accurate validation of the model. A view of the plasma wakefields in one of the OSIRIS simulations is shown in Fig. 4. As assumed by the analytical model, a constant \({\mathscr{K}}\) within the blowout and a transversely homogeneous \( {\mathcal E} \) with uniform \( {\mathcal E} ^{\prime} \) along the witness bunch can be seen.

Figure 4
figure 4

(a) 3D visualization made with VisualPIC66 of one of the OSIRIS simulations. The case shown is that of a witness beam with εn = 10 µmrad and np = 1017 cm−3, taken at a time \(t\simeq \mathrm{816/}{\omega }_{p}\). (b) and (c) show the longitudinal field \( {\mathcal E} \) and focusing gradient \({\mathscr{K}}\) of the same simulated case along the central slice (y = 0) and on axis (black line).

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The final simulation results, displayed in Fig. 5, show a strong dependence of the generated slice energy spread on the plasma density and initial beam parameters, reaching values in excess of 10−2 (or 1%) for certain configurations, way above FEL requirements. Similarly, the increase in bunch length can be on the femtosecond level for typical emittance and energy values, showcasing the difficulty of achieving GeV-class, sub-femtosecond bunches. No variation of the bunch length with the plasma density is observed, as expected from Eq. (11). The analytical model shows excellent agreement with the simulations over several orders of magnitude, with expected discrepancies in some of the higher emittance cases. The differences in Fig. 5(b) arise from the large slippage experienced by particles with higher oscillation amplitude, which causes the transverse distribution of many beam slices to be a truncated Gaussian and thus leads to a smaller energy spread than predicted. The discrepancies in Fig. 5(c,d) arise from the longitudinal energy correlation, neglected by Eq. (11), which induces a slight longitudinal bunch compression due to velocity differences between head and tail. In addition to the final slice energy spread and bunch length values, the analytical model also accurately predicts the evolution of these parameters during propagation within the PBA, as shown in Fig. 6. The discrepancies in the energy spread evolution in Fig. 6(a) arise from the initial dephasing between the bunch and plasma wakefield due to the low initial energy (10 MeV), which leads to an experienced accelerating field \( {\mathcal E} > { {\mathcal E} }_{0}\) and, thus, a lower energy spread than predicted by Eq. (10). Furthermore, the initial oscillations on the slice energy spread at the beginning of the simulation correspond to the betatron oscillations of the witness beam, which lead to an oscillation in vz, as seen previously in Fig. 1(b). When the vz of the beam particles is minimum (maximum), more (less) slippage occurs and the energy spread growth is faster (slower). This also leads to small oscillations on the bunch length, as seen in Fig. 6(b). The impact of the vz oscillations on the particle slippage had to be neglected in the theoretical model in order to obtain the analytical expression in Eq. (6). Thus, this effect is not reproduced by the analytical curves shown in Fig. 6.

Figure 5
figure 5

Comparison between simulation results and the analytical curves from Eqs. (10) and (11) for a 1 GeV plasma stage, corresponding to an acceleration length of \( \sim 3240\,{\omega }_{p}/c\) in all cases. The influence of injection energy, (a,c) and plasma density, (b,d) on the generated slice energy spread and bunch lengthening is displayed. The energy spread of the FLASH FEL at DESY (\( \sim 2\times {10}^{-4}\))67 is shown for reference. Only results from simulations with no significant numerical noise and in which the witness beam was sufficiently narrow to fit within the wake have been included.

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Figure 6
figure 6

Evolution of the relative slice energy spread (a) and bunch duration (b) on two of the simulated cases. Excellent agreement between analytical model and simulations is shown, including the predicted energy spread maximum at \(\bar{\Gamma }=9\).

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In order to explore the potential of beam loading for mitigating the generated slice energy spread, an additional set of simulations was performed with HiPACE. The same driver parameters as in the OSIRIS simulations and a plasma density of 7 × 1017 cm−3 were used. The witness beam parameters were fixed to an initial energy of 100 MeV, εn = 1 μmrad and σz/c = 3 fs, while its total charge was varied from 1 to 100 pC. The final energy spread of the central bunch slice after a 1 GeV energy gain is shown in Fig. 7 for the different bunch charges. Only the central slice energy spread is displayed in this case because the effect of beam loading on \( {\mathcal E} \) varies along the bunch. The analytical points were calculated from the average value during acceleration of the fields at the center of the bunch.

Figure 7
figure 7

(a) Effect of beam loading on the on-axis longitudinal electric field in all simulated cases for different total bunch charge. The ideal case which minimizes \( {\mathcal E} ^{\prime} \) is shown in black, while those in which \( {\mathcal E} ^{\prime} < 0\) are shown in blue tones and those in which beam loading inverts the slope (\( {\mathcal E} ^{\prime} > 0\)) are displayed in red. The fields are shown at t = 500/ωp. (b) Final central slice energy spread of the bunch with respect to its total charge as obtained from the HiPACE simulations. The value of the energy spread is calculated after a 1 GeV energy gain in the plasma. The effect of beam loading is clear, providing a minimum energy spread for a charge of 40 pC, which corresponds to a peak current of ~5.3 kA.

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The results show again good agreement between theory and simulations, and that beam loading can significantly reduce the development of slice energy spread (up to a factor of ~20 in this case). For the particular set of parameters under consideration, the energy spread minimum is achieved for a total bunch charge of 40 pC (corresponding to a peak current of ~5.3 kA) and grows again for higher charges because beam loading overcompensates \( {\mathcal E} ^{\prime} \), which becomes positive. Peak currents in this range of parameters have been experimentally demonstrated9,10 and, if optimized, could therefore provide an effective way of mitigating the slice energy spread growth. Discrepancies between the analytical curve and the simulations appear, however, around the energy spread minimum. This is because, in theory, the slice energy spread due to slippage could be completely suppressed for an ideally beam-loaded case in which \( {\mathcal E} ^{\prime} =0\) along the whole bunch. This, however, requires an optimized trapezoidal current profile30, while the bunch in the simulated case is Gaussian and therefore does not experience a uniform \( {\mathcal E} ^{\prime} \). Thus, even if \( {\mathcal E} ^{\prime} =0\) at the center, a certain energy spread will develop in the central slice due to the slippage of particles originally coming from slices ahead, where \( {\mathcal E} ^{\prime} \ne 0\). In addition, other effects apart from particle slippage can lead to increased slice energy spread in beam-loaded cases, such as transverse variations of the accelerating fields in the linear and weakly non-linear regimes64.

Discussion

The presented analytical model and simulation results show that the impact of the betatron-induced slippage could be very significant on beam parameters such as the slice energy spread (up to ~1%) and bunch length (growth on the femtosecond range). These effects should therefore be minimized, and guidelines for their mitigation can be derived from the analytical expressions together with the simulated cases. These show that an effective method of suppressing the slippage-induced slice energy spread growth is the use of beam loading (so that \( {\mathcal E} ^{\prime} \to 0\)), while another possible solution would be performing the acceleration in hollow plasma channels (\({\mathscr{K}}\to 0\)). Otherwise, in order not to rely on these two options, the simulations show that for achieving a slice energy spread <10−3, as typically required by FELs, a plasma density \(\lesssim {10}^{17}\,{{\rm{cm}}}^{-{\rm{3}}}\), a normalized emittance \(\lesssim 1\,\mu {\rm{m}}{\rm{r}}{\rm{a}}{\rm{d}}\) as well as an initial energy \(\lesssim 100\,{\rm{MeV}}\) become necessary. The same considerations apply regarding the production of sub-femtosecond bunches, although beam loading or lower plasma densities do not provide any benefit in this case. For applications requiring multiple accelerating stages, such as plasma-based particle colliders, these results imply that the slice energy spread due to slippage would mostly be generated in the first accelerating stage, decreasing at higher energies and eventually being dominated by the emission of betatron radiation.

Methods

3D Particle-in-Cell simulations with OSIRIS

The 3D PIC in simulations corresponding to the results shown in Figs. 4, 5 and 6 where carried out with the OSIRIS code43 using a standard finite differences field solver65. Due to the long propagation distance within the plasma, a co-moving simulation box was considered. The simulation grid on the cases shown in Figs. 5 and 6 consisted on 146 × 106 × 106 cells and had dimensions of 8.66 × 6.30 × 6.30 in units of ωp/c. In the graphs shown in Fig. 4 used for visualizing the fields, the simulation grid was slightly longer, with 170 × 106 × 106 cells and dimensions of 10 × 6.30 × 6.30 in units of ωp/c. This was done in order to show the tail of the wakefields, which was otherwise cut off. The time step in all cases was chosen to fulfil the stability criterion65. The number of particles per cell was 16 for the beam driver, 1 for the plasma and between 64 and 4096 for the witness beam, depending on the case, in order to have a meaningful total number of particles. The initialization of the electron beams was done using the n_accelerate method in OSIRIS, which progressively increases the longitudinal particle momentum over the specified number of time steps in order to obtain almost self-consistent electromagnetic fields before the beam enters the plasma.

3D Particle-in-Cell simulations with HiPACE

The 3D PIC in simulations corresponding to the results shown in Fig. 7 where carried out with the HiPACE code44, which is based on the quasi-static approximation. The simulation grid consisted on 320 × 106 × 106 cells and had dimensions of 8.66 × 6.30 × 6.30 in units of ωp/c. A constant time step size of 5 was used. The number of particles per cell was 16 for the beam driver, 3 for the plasma and 512 for the witness beam in order to have a meaningful total number of particles. The initialization of the particle species in HiPACE is performed by computing the self-consistent fields in each time step by means of fast Poisson solvers.

2D Particle-in-Cell simulation with OSIRIS

The data displayed in Fig. 3 showcasing the bunch lengthening and energy spread development comes from a single 2D simulation with OSIRIS of a laser-driven PBA. This simulation considered a plasma density np = 1017 cm−3, a Gaussian laser pulse with a peak power of 1.5 PW and a total energy of 25 J, a peak normalized vector potential a0 = 4, a wavelength λ = 800 nm, a spot size w0 = 54 μm and a FWHM duration τ = 15 fs. The witness bunch has a Gaussian profile based on achievable parameters by the ARES linac at SINBAD61,62, featuring a total charge of 0.7 pC, an initial energy of 100 MeV, a duration σz/c = 0.78 fs, a transverse size with σx = 5.0 μm and σy = 5.3 μm, an emittance εx = 0.70 μmrad and εy = 0.74 μmrad and an energy spread \({\sigma }_{\gamma }/\bar{\gamma }=0.37 \% \). This simulation was also performed with a standard finite differences field solver65 considering a co-moving simulation box with 5000 × 500 cells and dimensions of 5.95 × 29.76 in units of ωp/c. The time step was chosen to fulfil the stability criterion65. The number of particles per cell was 8 for the plasma and 64 for the witness beam. The initialization method of the witness beam is the same as in the 3D case described above.

Numerical solution of the single-particle equations of motion

As shown in Fig. 1, the equations of motion for a single particle (Eqs. (1) and (2)) were solved numerically to compare against the analytical expressions. These equations were solved with a Runge-Kutta method of order 4 with a time step of 0.01 fs. The fields experienced by the particles take into account dephasing effects assuming a laser driver with λ = 800 nm.

Calculation of the slice energy spread

Due to the limited amount of particles in a PIC simulation, calculating \({\sigma }_{\gamma }^{s}/\bar{\gamma }\) for an infinitesimally short slice as in Eq. (10) is not possible. However, this quantity can be estimated for a finite slice in which the longitudinal energy correlation is removed. The uncorrelated energy distribution can then be obtained as γunc = γ − δ(\(\xi -\bar{\xi }\)), where δ is the slope of the correlation. Additionally, since δ might change along the bunch, the bunches are divided into n slices in which \({\sigma }_{\gamma }^{s}/\bar{\gamma }\) is calculated. From there, for the cases shown in Figs. 5 and 6, the slice energy spread of the whole bunch is obtained as the weighted average of the value on each slice. This process is repeated by increasing the number of slices until convergence to a value is reached while still having a significant amount of particles per slice. For the cases to study the effect of beam loading shown in Fig. 7, the value of the slice energy spread only given for the central slice and not as an average.