Self-consistent effects in the ponderomotive acceleration of electron beams

Abstract

In the present work, we extend the results of a previous investigation on the dynamics of electrons under the action of an inverse free-electron-laser scheme (Almansa et al., Phys. Plasmas, vol. 26, 2019, 033105). While the former work examined electrons as single test particles subject to the combined action of a modulated wiggler plus a laser field, we now look at electrons as composing a particle beam, where collective space-charge effects are relevant and included in the analysis. Our previous work showed that effective acceleration is achieved when the initial velocities of the particles are close enough to the phase velocity of the beat-wave mode formed by the laser and the wiggler fields. Electrons are then initially accelerated by a ponderomotive uphill effect generated by the beat mode and, once reaching the phase velocity of the beat, undergo a final strong resonant acceleration step resembling a catapult effect. The present work shows that, under proper conditions, space-charge effects play a similar role as the initial (or injected) velocity of the beam. Even if acceleration is absent when space charge is neglected, it may be present and effective when charge effects are taken into account. We also discuss how far the space charge can grow without affecting the sustainability of the acceleration process.

1. Introduction

Recent results show that efficient electron acceleration can be achieved as one combines an inverse free-electron-laser (IFEL) scheme (Courant, Pellegrini & Zakowicz 1985; van Steenbergen et al. 1996; Ho et al. 2013; Duris et al. 2014; Singh et al. 2022) with a preceding ponderomotive (Shukla et al. 1986; Mendonça 2001; Macchi 2013) step, both stages formed with the very same laser and static wiggler fields (Almansa et al. 2019). The ponderomotive stage requires a wiggler whose amplitude is modulated along the device's axial direction, defined by the co-aligned orientation of both wiggler and laser wave vectors.

As electrons are injected, the non-uniform wiggler enables pre-acceleration by means of an uphill effective ponderomotive potential (Mulser & Bauer 2010; Burton et al. 2017; Ruiz & Dodin 2017; Almansa et al. 2018). Electrons are then resonantly pushed forward by the peaks of the beat wave formed by laser and wiggler, and ultimately accelerated in a catapult fashion to higher velocities than the phase velocity of the beat wave and much higher final energies than the initial. We note that the non-uniform shape of the wiggler both promotes acceleration and allows particles to escape at the far end of the accelerating region. These features contrast with the case of uniform fields where particles remain trapped – undergoing a mixing process – once they fall into the field's troughs (Evstatiev, Morrison & Horton 2005).

As an initial approach, especially if the complexities of the ponderomotive uphill acceleration are considered, one disregards collective effects involving the accelerating electrons and the concomitant back reaction of these particles on the original electromagnetic fields. This is justified if one works with a rarefied electron population, which under these circumstances behaves as a test particle conglomerate.

On the other hand, if one thinks of relatively high-density beams used, for instance, in material processing devices (Davidson & Qin 2001), collective self-consistent effects should be considered.

The purpose of the present analysis is precisely to add collective effects to the description of electron acceleration in the IFEL device described above.

The transverse self-consistent beam dynamics has been largely analysed and understood when beams are transported at constant axial speeds in the absence of accelerating fields. In the present case, we need to take into account the presence of driving non-uniform electromagnetic fields along the axial direction, as commented above. The beam will still be transported along the axis but with varying velocity and density resulting from its non-homogeneous longitudinal profile. Therefore, the theory for uniform transport along the axial direction shall be adapted accordingly to describe how acceleration, collective effects and non-uniform transport relate to each other.

In the following sections, we will introduce the laser and wiggler fields, focusing on their respective polarizations and profiles along the axial direction of the IFEL device. With the rapidly varying fields duly introduced and now adding the slowly varying space-charge collective fields, we shall develop the proper ponderomotive Hamiltonian describing the averaged electron dynamics. In the final step, we solve the self-consistent collective system formed by particles and fields and compare the results with the dynamics of test particles moving under the same circumstances.

The overall analysis is carried out under the assumption that the smooth ponderomotive averaged dynamics is stationary, depending only on the longitudinal and transverse spatial coordinates of a cylindrical geometry.

2. General formalism

2.1. Laser and wiggler fields

As mentioned in the Introduction and discussed in Almansa et al. (2019), the acceleration scheme investigated here depends on the combination of laser and wiggler fields in an IFEL device arrangement where electrons draw energy from a pre-existing finite-amplitude laser mode. We consider the focal region of the electromagnetic fields to be sufficiently larger than the beam radius, so that we can discard transverse features in both the wiggler and the laser. Dissipative forces, like radiation reaction, are in general comparatively small and also neglected here (Landau & Lifschitz 1965; Vranic et al. 2014; Russman, Marini & Rizzato 2022).

In the static wiggler case, the uniformity condition on transverse dimensions has been discussed in more detail in Almansa et al. (2019) and allows us to take the respective vector potential field ${\boldsymbol {A}}_w$ in the form of a right-handed circularly polarized mode

(2.1)\begin{equation} {\boldsymbol{A}}_w = A_{w0} \, {\rm e}^{-({x^2}/{\sigma^2})} \frac{{{\boldsymbol{\hat{y}}} +{\rm i}{\boldsymbol{\hat{z}}}}}{2} {\rm e}^{{\rm i} \theta_w} + {\rm c.c.}, \end{equation}

where $A_{w0}$ represents the peak amplitude of a slowly modulated wiggler displaying a rapidly varying phase $\theta _w=k_w x$; one enforces the slow modulation condition demanding $k_w \sigma \gg 1$, with $k_w$ as the wiggler wave vector and $2 \sigma$ as its extension around $x=0$.

As for the dynamical laser field ${\boldsymbol {A}}_l(x)$, we will not only require an initially large cross-section, but also that its cross-section remains mostly unchanged along the axial direction. This requirement is equivalent to excluding transverse laser beam expansion, therefore holding constant the laser intensity at the central alignment of the system (Elmore, Elmore & Heald 1985). Introducing ${\sigma _{\perp }}_{{\rm laser}}$ as the laser cross-sectional radial dimension, $L$ as the system longitudinal length and $k_l$ as the laser wavevector then, in the absence of dissipative forces involving the laser particle interaction, one can use the slowly modulated wave equation for the laser to show that, if $L \ll k_l \sigma _{{\perp }_{{\rm laser}}}^2$, the laser beam does not appreciably expand transversely. One should also require $L > \sigma _{{{\perp }}_{{\rm laser}}}$ to enforce a beam geometry for the laser, from which follows the expected necessary condition $k_l \sigma _{{\perp }_{{\rm laser}}} \gg 1$ indicating that the cross-sectional dimension should be much larger than the laser wavelength.

Working with a left-handed polarized signal, the above set of conditions allows us to write

(2.2)\begin{equation} {\boldsymbol{A}}_l(x,t)= A_{l0} \frac{{{\boldsymbol{\hat{y}}} -{\rm i} {\boldsymbol{\hat{z}}}} }{2} {\rm e}^{{\rm i}\theta_l} + {\rm c.c.}, \end{equation}

with the laser amplitude constant and the fast phase written in the form $\theta _l \equiv k_l x - \omega _l t+\phi _{{\rm slow}}(x)$. We shall consider a sufficiently underdense electron beam such that $\omega _l/k_l \approx c$, with $\omega _l$ as the laser fast frequency and c as the speed of light. The added slowly varying phase correction $\phi _{{\rm slow}}$ due to the beam's presence can be neglected.

2.2. Collective fields

Let us finally discuss the presence and actions of space-charge collective fields in our model. This is a new feature which is non-existent in the previous single-particle view, where collective effects are absent and where the only low-frequency field results from the ponderomotive action of the laser and wiggler fields preceding the catapulting acceleration discussed earlier.

In our case, we consider a continuous dense particle beam with cylindrical symmetry, where collective low-frequency space-charge fields are generated due to particle–particle interaction.

The primary low-frequency collective field to be considered is the space-charge potential. This low-frequency collective mode has an axial and, importantly, a transverse dependency resulting from the interplay between the finite cross-sectional area of the beam and the boundary conditions surrounding the system. The space-charge potential is denoted by

(2.3)\begin{equation} \varphi=\varphi(r,x), \end{equation}

where we recall the assumed cylindrical geometry characterized by the absence of the azimuthal angle. We shall also consider a thin beam configuration for which the aspect ratio $|\boldsymbol {\nabla }_{\perp }| \gg |\partial / \partial x|$ applies, so the potential can be obtained from a purely transverse form of the Poisson equation

(2.4)\begin{equation} \nabla_\perp^2 \varphi={-} q n/\epsilon_0, \end{equation}

with $x$ seen as a parametric variable, $\epsilon _0$ as the vacuum permittivity, $q$ as the particle's charge and $n=n(x,r)$ as the beam density which will be detailed shortly.

We also observe that the thin beam geometrical aspect remains valid while the displacement of any beam particle along the longitudinal axis $x$ corresponds to a much smaller transverse displacement. This condition, in general, requires high axial relativistic speeds which alone could be sufficient to validate the use of the Poisson equation in its transverse form (Vay 2008; Londrillo, Gatti & Ferrario 2014).

For the given beam density, (2.4) will be solved under the boundary condition $\varphi (x,r=r_w)=0$, where $r_w$ denotes the radius of a conducting wall surrounding the system, which is sufficiently larger than the beam dimensions.

For the purposes of the present investigation, we shall take beams of homogeneous density and sharply defined slowly varying boundary at $r_b=r_b(x)$, a pair of assumptions that will be justified shortly.

Under these conditions, and recalling the restriction to stationary dynamical regimes, the following expression can be obtained for the beam density at any point along the axial direction:

(2.5)\begin{equation} n(x) = \frac{Q_0}{{{\rm \pi} r_b(x)^2 v_x(x)}}, \end{equation}

where $Q_0 \equiv {\rm \pi}r_b(0)^2 n(0) v_x(0)$ measures the unnormalized injected charge flow at $x=0$, and where we introduce the slowly varying axial beam velocity $v_x$ which will be taken as approximately constant over any transverse area of the thin beam. Note that the electron flow, as defined by $Q_0$, implies a smooth, steady-state particle inflow at the beam injection point.

Along the longitudinal axis, however, the density must undergo any necessary changes in order to preserve the constant current flow. With this adjustment, for a given set $Q_0,r_b(x),v_x(x)$, the resulting transversely homogeneous beam density, obtained from expression (2.5), can be inserted in the Poisson equation (2.4), whose solution provides the potential $\varphi$ acting on any point along and across the beam in the form

(2.6)\begin{equation} \varphi(x,r \leq r_b(x))= \frac{v_{\phi}^2 Q_0 }{{4{\rm \pi} v_x(x)}} \left(1+2 \ln \left(\frac{r_w }{r_b(x)} \right)-\frac{r^2 }{r_b(x)^2} \right), \end{equation}

where $v_\phi$ is the carrier phase velocity. From now on, we progressively switch the formalism to dimensionless variables, starting with $(k_w+k_l)(x,y,z) \rightarrow (x,y,z)$, $(k_w+k_l) c t \rightarrow t$, $q\varphi /mc^2 \rightarrow \varphi$ and $n/n_c \rightarrow n$ (where $m$ is the particle mass). Speeds are therefore normalized by the speed of light; in particular, the normalized carrier phase velocity reads $k_l/(k_w + k_l)$. Moreover, $n_c$ is introduced as the critical density for wave propagation in the beam: $n_c= \epsilon _0 m \omega _l^2/q^2$. We use $n_c$ as a convenient parameter to fully render $Q_0$ seen in expression (2.5) into a non-dimensional form.

Note that, since the beam may have a considerable velocity along the axial coordinate, a closely related field to $\varphi$ should be introduced: the axial component of the vector potential, $A_x$. In contrast to the vector potentials connected to the laser and wiggler fields, $A_x$ is a slowly varying field with its source arising from the equally slowly varying axial current density $j_x=q n v_x$.

From the same stationary thin beam assumptions, now, along with the wave equation for the vector potential and with expression (2.6), one obtains

(2.7)\begin{equation} \nabla_\perp^2 A_x ={-}q\mu_0 n v_x = \nabla_\perp^2 \varphi v_x/c^2, \end{equation}

where $\mu _0$ is the vacuum permeability. Equation (2.7) translates into the non-dimensional form as

(2.8)\begin{equation} A_x = \varphi v_x , \end{equation}

with $qA_x/{mc} \rightarrow A_x$, similarly as discussed in Davidson & Qin (2001), and we apply the same normalization condition to the laser and wiggler fields to obtain the overall rule $q{\boldsymbol {A}}/{mc} \rightarrow \boldsymbol {A}$ for the vector potential.

Examining expressions (2.6) and (2.8) we thus see that, under the present assumptions, the slowly varying beam potentials – and the beam itself – are entirely characterized and governed by the variable beam speed and variable beam radius. Our next step is to work out the appropriate self-consistent dynamical equations that will allow us to calculate these quantities.

2.3. Particle and beam dynamics

Now that we have an initial sketch for the basic structure of the EM fields, we proceed to the next step, where the particle and beam dynamics are investigated. This section establishes the theoretical framework essential for the numerical simulations presented in the following section.

Particles are subjected to the action of the rapidly varying laser (${\boldsymbol {A}}_l$) and wiggler (${\boldsymbol {A}}_w$), alongside the space-charge related fields $\varphi$ and $A_x$ self-consistently acting upon the whole beam.

We start by looking into the dynamics of a particle with canonical momentum $\boldsymbol {p}$ with the help of the respective dimensionless Hamiltonian ($H/{mc^2} \rightarrow H$)

(2.9)\begin{equation} H=\sqrt{1+{(\boldsymbol{p}-\boldsymbol{A})}^{2}} + \varphi, \end{equation}

where ${\boldsymbol {A}} = {\boldsymbol {A}}_l+{\boldsymbol {A}}_w+A_x \hat {\boldsymbol {x}}$.

As discussed earlier, potentials $\varphi$ and $A_x$ appearing in expression (2.9) do not depend on the fast phases $\theta _{l,w}$ and in this sense are thus slowly varying functions of axial and transverse coordinates. Since we are considering a narrow beam which essentially sees plane electromagnetic waves aligned with the axis, we conclude, from the Hamiltonian equation $\dot {\boldsymbol p}_{\perp }=-\boldsymbol {\nabla }_{\perp } H$, that ${\boldsymbol p}_{\perp }$ is a slowly varying coordinate as well.

Our interest is to proceed further in the quest for a fully averaged Hamiltonian capable of describing the ponderomotive dynamics leading to particle trapping and the ultimate resonant – or catapult – acceleration, as commented earlier. We will also see that the averaged variables obtained from the ponderomotive procedure are of central importance to constructing the slowly varying self-consistent fields acting on the particles.

In order to obtain the average form, we proceed along the ensuing general lines.

We shall first recall that although the transverse momentum ${\boldsymbol {p}}_{\perp }$ is a slowly varying variable, neither the full Hamiltonian $H$ nor the axial momentum $p_x$ are – a fact resulting from the presence of the fast varying phases $\theta _l$ and $\theta _w$ in the canonical dynamics. For a generic variable $g$ we thus write

(2.10)\begin{equation} g=\bar{g} + \delta g, \end{equation}

where $\bar {g}$ represent the fast phase average of $g$, and $\delta g$ represents the rapidly varying fluctuations of $g$ due to the presence of the fast phases.

We then shift the scalar potential to the left-hand side of expression (2.9), take the square of both sides of the resulting equality and write all variables in the form (2.10), observing that $\overline {\delta g}=0$.

The final result can be cast as

(2.11)\begin{equation} \bar{H} = \sqrt{1+(\bar{p}_x - A_x)^2+{\boldsymbol{p}}_\perp^2 + \overline{(\delta p_x)^2}- \overline{(\delta H)^2} + A_{l0}^2+A_{w}^2(x)} +\varphi, \end{equation}

where we point out the presence of the averaged quadratic fluctuations related to the axial momentum and to the Hamiltonian itself. Both are crucial for the full description of the dynamics, and we see that, in addition, we will indeed need the vector potential $A_x$ along the $x-$axis and the space-charge potential $\varphi$, which can be calculated from (2.4) and (2.7), respectively.

Calculation of the fluctuations terms $\delta p_x$ and $\delta H$ are based on the canonical integration of Hamiltonian (2.9) over the fast time scales dictated by wave frequencies and wave vectors, with slow variables kept fixed. The procedure is long and deferred to Appendix A.

The various steps leading to the average Hamiltonian emerge in Appendix, and in the conveniently approximated expression (B2)

(2.12)\begin{equation} \bar{H}_{{\rm approx}} \equiv H_0 + H_2, \end{equation}

which is reached if one uses the assumption that the energy associated with the longitudinal motion is much larger than the one associated with the transverse dynamics. In practice, we shall assume $\bar {p}_x \gg \varphi, |{\boldsymbol p}_{\perp }|$ in dimensionless forms, and expand the full $\bar {H}$ up to quadratic terms in $r$ and ${\boldsymbol {p}_{\perp }}$. Here, $H_0$ does not contain $r$ and $\boldsymbol {p}_\perp$, and $H_2$ is quadratic in both variables. We point out that it is precisely the quadratic dependence of ${\bar {H}}_{{\rm approx}}$ on $r$ that allows the use of transversely homogeneous beams as required by expressions (2.5) and (2.6). Spatially quadratic potentials generate spatially linear forces that, in turn, sustain uniformity in the geometry analysed here (Davidson & Qin 2001).

In addition, the energetic contribution of $H_2$ to the full Hamiltonian is closely related to the potential term proportional to $Q_0$, as seen in (B5) of Appendix B. In this context, to maintain consistency with our initial assumptions of a thin beam geometry resulting from velocities along the axis being greater than velocities transverse to the axis, we shall therefore consider limited values for $Q_0$, which is the condition discussed earlier in the context of (2.4). This condition is further discussed following expression (B5).

We next suppress the overbar on average quantities in order to ease off the notation, and solve the dynamics that arises from Hamiltonian (2.12) under a stationary regime, as stated earlier, in two steps. We first focus on the zeroth-order contribution to ${H_{{\rm approx}}}$, $H_0(x,p_x)$, with $\varphi$ to be calculated from expression (2.6) at $r=0.$ From this zeroth order we readily obtain the canonical pair of equations

(2.13)\begin{gather} {\rm d}p_x/{{\rm d}\kern0.07em x}=(1/v_x)\, {\rm d}{p}_x/{\rm d}t ={-} (1/{v}_x)\partial {H}_0 / \partial x , \end{gather}

(2.14)\begin{gather}v_x={{\rm d}\kern0.07em x} /{\rm d}t=\partial {H}_0 / \partial{p}_x, \end{gather}

eventually leading to a radially independent expression for $v_x=v_x(x)$, which we will thus take as our approximation for the beam velocity.

We see, however, that, in order to accomplish the calculation of $v_x$, we need expressions for ${\rm d}v_x/{{\rm d}\kern0.07em x}$ and ${\rm d}r_b/{{\rm d}\kern0.07em x}$ as well, since both arise from the space derivative $\partial \varphi /\partial x$ present in $\partial H_0/\partial x$ of (2.13).

A derivation of ${\rm d}v_x/{{\rm d}\kern0.07em x}$ from $v_x$ in (2.14) is straightforward, despite its lengthiness. To calculate ${\rm d}r_b/{{\rm d}\kern0.07em x}$, which appears in (2.13) as well as in ${\rm d}v_x/{{\rm d}\kern0.07em x}$, we need, however, to make use of the second term of $H_{{\rm approx}}$, $H_2$, quadratic in the canonical variables $r$ and $p_r$. The strategy is to obtain the canonical equations for ${\rm d}r/{{\rm d}\kern0.07em x}$ and ${\rm d}p_r/{{\rm d}\kern0.07em x}$ and evaluate both at $r=r_b$ and $p=p_b$, with ‘$b$’ designating the beam boundary.

All in all, we need to numerically solve a system of four coupled dynamical differential equations: one for $p_x(x)$, one for $v_x(x)$, one for $r_b(x)$ and, finally, one for $p_b(x)$. We refer to this procedure as the ponderomotive approach to the problem. We point out that the ponderomotive approach results in the simultaneous calculation of both the longitudinal speed of the beam and the beam envelope (one cannot calculate one without calculating the other). Under the ponderomotive approximation, all particles have the same longitudinal velocity and are surrounded by the common smooth envelope of radius $r_b$.

As soon as the slowly varying potentials $\varphi$ and $A_x$ are obtained in terms of $v_x$ and $r_b$ as in (2.6), we shall proceed to the next step where we relax the ponderomotive approximations involving the effects of laser and wiggler fields on the beam particles. At this point, we can compare the exact dynamics with the averaged ponderomotive approximation, as well as investigate the strong resonant acceleration as particles fall into the troughs formed by the beating of wiggler and laser. In all instances, the role of space charges can be duly identified and analysed.

It should be understood that, when the exact dynamics of single particles deviates from the corresponding ponderomotive approximation, the very idea of a smooth beam is to be taken with caution. We will examine this issue and argue that the smooth beam model is still useful even when acceleration regime due to particle trapping and kicking starts to set in.

3. Numerical analysis

The clearest role of space-charge effects is arguably in the beam's cross-sectional spread along the longitudinal axis. In the absence of any collective effects, the beam reverts to a collection of non-interacting test particles that can even be longitudinally accelerated while still preserving the initial transverse dynamics. As one ‘turns on’ the charge parameter $Q_0$, collective effects and beam spread should be seen.

To discuss the effects of space charge addressed in previous sections, we fix a set of parameters with both the laser field and wiggler amplitudes given by $A_{l,0} = A_{w,0} = 1.2$, (indicative of a moderately relativistic regime of interaction), $\sigma =100$, $v_{\phi }=\sqrt {0.9}$, $v_{y}(0)=v_{z}(0)=0$, $x(0)=-10\sigma$ and $r_w=150$. We consider the parameters $r(0)$, $v_{x}(0)$, and $Q(0)\equiv Q_0$ as free variables to explore the main effects arising from space charge.

In figure 1, we examine a geometrical cut across the beam longitudinal midplane to examine the evolution of the upper and lowermost points of the beam envelope along the longitudinal axis $x$. Discrete choices for the initial beam radii and charges are made, and we keep the initial velocity constant at $v_{x}(0)=0.86$. The dashed blue line represents $Q_0=0.001$ in both panels, and the solid red line likewise represents $Q_0=0.06$. Note that, for a better insight into the beam's dynamics and geometry, we represent the beam radius growth symmetrically as the beam uniformly expands.

Figure 1. Envelope evolution in the $x$ direction as a function of initial charge flow $Q_0=0.001$ (dashed blue line) and $Q_0=0.06$ (solid red line), initial velocity $v_{x}(0)=0.86$ and with radii $r(0)=30$ (a) and $r(0)=60$ (b).

When the initial radius equals $r_b(0)=30$, as shown in panel (a), we can see that if the charge is small, the opening of the beam radius is also small, as expected. However, when the initial charge is relatively large, we observe a significant variation in the beam radius, which at the end of the interaction, with $x\approx 1000$, has a final radius measuring approximately $r_b(x_f)=90$; $3$ times its initial size.

As one increases the initial beam radius, the charge becomes distributed over a larger cross-sectional area, which results in less significant space-charge effects than in the previous case. Indeed, when $Q_0=0.001$, there is virtually no variation in the particles’ radii, and when $Q_0=0.06$ the beam still expands up to an observable larger radius $r_b(x_f)=90$, but with this latter value now representing only a small incremental fraction of its initial size.

Looking at the longitudinal and transverse scales, we point out that the beam envelope smoothly evolves in a much slower rate than it advances along the longitudinal axis, which corroborates the thin beam approach promulgated earlier. We note that not even the length scale associated with the wiggler field's envelope affects the ponderomotive dynamics as particles run through the region $x\sim 0$ with relativistic speeds.

To better understand the effects of both electromagnetic and space-charge fields, a number of individual particles obeying the full dynamics dictated by the newly considered collective effects plus the full wiggler and laser fields can also be included in the simulations. As commented earlier, we deem the ponderomotive approximation to be correct as long as the test particle orbits agree with their averaged counterpart.

A collection of 60 test particles is randomly but uniformly distributed over the initial respective cross-sectional area on the plane $(y,z)$, and their dynamics is henceforth described by the full Hamiltonian (2.9). To mimic the symmetry seen in figure 1, particles launched in the lower (upper) quadrants of the plane $(y,z)$ are represented below (above) the $r_b=0$ axis. Figure 2 reveals how the various radial orbits of the particles evolve along the axis. The various grey curves represent regularly expanding orbits, solid red curves represent beam boundaries and the purple curve represents the central orbit. The sinuous orbits (seen in panel figure 2(e) and discussed below) are represented in blue. We set $r_b(0)=60$ and $v_x(0)=0.86$, with $Q_0=0.001$ in panels (a,b), $Q_0=0.06$ in panels (c,d) and $Q_0=0.12$ in panels (e, f).

Figure 2. Case of $Q_0=0.001$ in (a,b), $Q_0=0.06$ in (c,d) and $Q_0=0.12$ in (e, f); here, $v_{x0}=0.86$ and $v_x$ on the vertical axes represents both the test particles’ velocities and the ponderomotive velocities, obtained from averaging over fast oscillations (green curve) and from (2.12) (black dashed curve). The envelope of the beam is depicted in red. Regular grey curves in the spatial panels $r(x) \times x$ depict the uniformly expanding, non-catapulting beam particles. In contrast, the sinuous blue curves appearing in panel (e) represent the catapulted, highly accelerated particles observed on the right upper side of panel ( f).

In panel (a), we can observe that the test particles flow homogeneously within the region bounded by an approximately constant envelope. The slight inter-crossings of particle trajectories seen are due to the high-frequency jitter driven by the laser and wiggler fields and does not affect our model for the smooth collective space-charge fields.

Complementarily, in panel (b) we refer to the time history of all test particles’ respective velocities, including the central particle – we call them $v^{{\rm test}}_x$ – which  again reveals  the uphill acceleration. Translating the highly oscillating exact velocity $v_x^{{\rm test}}$ into its time average form, one can see either from our previous work (Almansa et al. 2019), or from the green curve representing the average velocity, that the latter closely  follows the upper envelope of the exact $v^{{\rm test}}_x$, with a slightly more pronounced dip at $x=0$. The dip is, however, much less profound than the one associated with the exact velocities $v^{{\rm test}}_x$, which keeps $v_x$ well above the breakup level $v_x=0$, an essential ingredient for thin beam modelling. Particles are accelerated as they enter the region of the first peak in panel (b), decelerated as they approach the coordinate $x=0$, accelerated again near the second peak and decelerated once more to be finally ejected with the very same initial velocity.

At these lowest levels of space charge, and for the considered injection velocity, the overall conclusion is that the beam behaves as in the previously studied case (Almansa et al. 2019) where the particle motion is mostly aligned with the longitudinal axis and the final catapulting acceleration is totally absent.

In panel (c) we can now observe the particle flow evolution when the initial charge is significantly larger, with $Q_0=0.06$. Notwithstanding the more substantial charge value leading to a final envelope twice as large as the previous one, the flow is still homogeneous, and our initial assumptions are once again corroborated. In addition, we note from panel (d) that all particles are approximately co-moving with the very same velocity, which again advocates in favour of a unique value for the beam velocity, even for the larger $Q_0$. We also note the presence of the uphill acceleration but the absence of the catapulting dynamics. Thus, at this level of the charge factor $Q_0$, the transverse dynamics already exhibits the action of collective effects, but the forward dynamics remains unaffected. Also in panel (d) we add an inset for this case of a larger $Q_0$. The green curve is derived from averaging the rapid oscillations of a test particle's velocity at $r=0$ while the dashed black line directly results from integrating the equations derived from the Hamiltonian given by (2.12). The agreement of the ponderomotive approximation and the time averaging is still solid. In addition, the corresponding panel (c) reveals that, under the same regime, the radius $r_b$ obtained from the ponderomotive approach agrees very well with the exact radial dynamics of test particles launched at the beam border, which can also be seen as an important feature validating the theory.

Then, as we move to panels (e, f), the even larger space-charge effects for $Q_0=0.12$ not only further expand the beam envelope but introduce the catapulting dynamics as well. We just recall that, in the absence of space-charge effects (Almansa et al. 2019), the catapult acceleration was solely driven by the velocity of the injected particles, which panel (a) shows to be insufficient in this case. We thus see that the charge flow parameter $Q_0$ is also a determining factor for acceleration.

We then argue that those particles near the beam border are the first ones to undergo the acceleration process. The background beam core remains largely unaffected, orbits remain dominantly homogeneous without inter-crossing and the beam velocity is dominantly the same for the whole beam. This is a peculiar regime where one can examine the orbits of individual accelerating particles, still using with confidence our collective model to represent the self-consistent space-charge effects.

In fact, with the collective model we can examine the dynamics as a function of the beam radial coordinate, looking at the $r$-dependent terms of $H_2$ in expression (2.12), let us call it ${H_2}_r$

(3.1)\begin{equation} {H_2}_r = \frac{Q_0 r^2(x) }{r_b^2(x)} \frac{(1-v^2_x(x))}{v_x(x)}. \end{equation}

From the expression above, we can see that ${H_2}_r$ becomes more significant near the beam's largest radii. From a perturbative expansion point of view, let us then examine the role of ${H_2}_r$ for particles precisely at the border $r(x)=r_b(x)$. We consider a beam of particles moving forward with a velocity $v_x>0$, which must be less than unity as it is normalized to the speed of light. If the uphill acceleration is present, even if small, as $v_x$ tries to reach higher speeds at any of the two peaks seen in the velocity plots of figure 2(b) for instance (Almansa et al. 2019), the ‘potential energy’ ${H_2}_r$ starts to act as its own magnitude decreases, delivering a forward extra kick proportional to $Q_0$ that helps to pull particles into resonance with the beat mode formed by laser and wiggler. This is seen in figure 2( f) as the clearly distinct group of highly accelerated particles with speeds close to $c$, whose no longer homogeneous spatial orbits correspond to the highly sinuous curves near the boundary at $r_b(x)$, visible as the blue curves in panel (e).

On the other hand, if the uphill acceleration is not significant, typically for lower injection velocities, the downhill valley seen in figure 2 decreases $v_x$, augments, therefore, $H_2$ and may even reflect particles backward. This is precisely what is seen in figure 3. The beam's central body continues to move forward after $x=0$, but some of the particles are already seen moving backward. It is important to emphasize that in the absence of space-charge effects with $Q_0 \rightarrow 0$, the whole beam would keep moving forward. This last case of figure 3 should be seen with reserve since, although $Q_0$ is small, the longitudinal speeds are originally smaller too. These conditions break the beam's longitudinal smoothness at $x=0$ and put us on the verge of the validity of our thin beam approximations, but nevertheless provide a first approach to the case where reflection is present. We also point out that $r_w$ in figure 3(a) was expanded to $r_w=5000$ in order to accommodate the beam; the expansion of the conducting wall radius has very little effect on the dynamics since the potentials depend only logarithmically on $r_w$.

Figure 3. Case of $Q_0=0.12$ and $v_{x0}=0.68$. (a) Evolution of test particle trajectories with the red curves representing the evolution of the beam envelope, and (b) evolution of test particle velocities. In both panels, test particles are depicted in grey, with the purple curve denoting the central particle at $r(x)\approx 0$. Black arrows in (b) indicate the direction the particles are moving in.

In summary, the $H_2$ term contributes a forward extra kick if the uphill acceleration is present and a backward extra kick if the uphill acceleration is negligible.

We examine the charge flow effect on particles at the border of the beam distribution looking at their velocity evolution, as depicted in figure 4. Panel (a) shows the full temporal evolution and panel (b) the corresponding time average. The time average equals the ponderomotive description if one is far enough from the resonance. The figure shows that, when the beam charge flow is low or near zero, i.e. $Q_0=0.001$, the final velocity of the particle remains unchanged from its initial state, as indicated by the solid black curve. However, for a charge flow of $Q_0=0.06$ (red curve), particles at the border experience a significant increment in velocity as they traverse the region where the wiggler field is intense, as predicted by (3.1). However, the increment is not large enough to bring the particle into resonance with the carrier beat wave and the particle eventually returns to its initial state. For a charge flow of $Q_0=0.12$ (blue curve), the space charge is finally strong enough to accelerate some particle into resonance, from which point they are strongly kicked and further accelerated by the beat wave.

Figure 4. Time evolution of velocity for particles at the border of the beam distribution assuming different initial charge flows. Panel (a) shows the full velocity, and panel (b) depicts the averaging over the rapid oscillations observed in the test particle's velocity. In both panels, the dotted green line represents the carrier phase velocity. The figures demonstrate that, for higher charge flow, the final velocity of the test particle changes from unperturbed at $Q_0=0.001$ (black curve) to perturbed at $Q_0=0.06$ (red curve) and finally to accelerated at $Q_0=0.12$ (blue curve).

In figure 5, we examine a particle beam with an initial radius $r_0=60$ and different initial velocities, $v_{x0}=0.82, 0.84$ and $0.86$, depicted by the green squares, blue triangles and red points, respectively. In the figure, the energy gain, denoted as $\tilde {\gamma }(t_f)- \gamma _0$, is plotted against the initial charge $Q_0$, where the mean energy, $\tilde {\gamma }$, at the final time $t=t_f$ is defined as $\tilde {\gamma }(t_f)=(1/n)\sum _{i=1}^{n}\gamma _i(t_f)$. The figure shows that the energy gain tends to saturate as one increases the initial charge. Saturation occurs at different points for each initial velocity: for $v_{x0}=0.86$, saturation occurs approximately at $Q_0=0.4$, for $v_{x0}=0.84$ at $Q_0=0.6$ and for $v_{x0}=0.84$, saturation is only observed for larger charges above $Q_0=1.2$.

Figure 5. Saturation of energy gain as a function of initial beam velocity. Green squares, blue triangles and red dots represent particles with initial velocities $v_{x0} = 0.82$, $v_{x0} = 0.84$ and $v_{x0} = 0.86$, respectively. Solid curves are fitting curves to guide the eye.

The saturation behaviour is attributed to a shifting position of the resonant particle group as the initial charge escalates. At zero charge, no resonance is present under the present circumstances. However, as $Q_0$ grows, at a certain point particles on the beam border $r_b$ are first kicked with the exact amount of energy into resonance, which results from the fact that the kick's amplitude, proportional to $r^2$, is the most intense at the border. This is the case previously discussed along with figure 2( f).

From this point on, as $Q_0$ keeps growing the resonance locus plunges deeper towards the central region of the beam in order to preserve the extra-kick amplitude given by the product $Q_0 r^2$. In this cyclic behaviour, as a new group of particles goes into resonance, the previous adjacent group falls off resonance. This interchange between limited resonant particle groups explains the complex dynamics manifested in the figure. In particular, it explains why the energy gain remains the same once installed; the carriers of the energy gain change in position but not in number.

The solid curves represent fitting curves to guide the eye. Fluctuations in the values are related to the particle's initial position, which, in turn, is determined randomly.

4. Conclusions

In the present paper, we studied the role of self-consistent space-charge effects on the acceleration of particle beams under the external action of laser and wiggler fields in an IFEL configuration (Almansa et al. 2019).

The work extends previous results on the acceleration of single isolated particles and examines the role of the beam density in the process.

If one initially suppresses charge densities, efficient acceleration can be reached for convenient choices of the particle's initial velocity, the wiggler and laser intensities and the phase velocity of the beat mode formed by laser and wiggler.

As the charge density of the beam increases, one would perhaps expect a detrimental effect on acceleration. What we saw, however, is that charge effects are less obvious.

In the more conventional case, where uphill acceleration effects are absent or sufficiently small, it is true that endowing the system with a non-zero charge density does indeed break the beam flow sooner than in the single-particle case. However, when uphill effects are present and sufficiently strong, space-charge effects actually help the forward catapulting acceleration.

We modelled the laser field as a non-expanding beam, which is an appropriate model when the radiation beam's cross-section area is large enough. It is, however, a matter of rising interest to use applications involving tightly focused radiation beams where the wave front is small and involves non-planar geometries (Singh et al. 2022), a topic that presents itself as a relevant extension of the present analysis.

Our analysis focuses on tenuous and sufficiently short electron beams in which the radial dynamics is moderate enough to dispense with the need for external focusing mechanisms. Higher-density beams for which factor $Q_0$ is larger, or even much longer beams, may require the mentioned external control (Davidson & Qin 2001), which opens yet another front where the appropriate techniques shall be added to the present line of analysis.