2.1. Laser and wiggler fields

As mentioned in the Introduction and discussed in Almansa et al. (Reference Almansa, Russman, Marini, Peter, de Oliveira, Cairns and Rizzato2019), 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 Reference Landau and Lifschitz1965; Vranic et al. Reference Vranic, Martins, Vieira, Fonseca and Silva2014; Russman, Marini & Rizzato Reference Russman, Marini and Rizzato2022).

In the static wiggler case, the uniformity condition on transverse dimensions has been discussed in more detail in Almansa et al. (Reference Almansa, Russman, Marini, Peter, de Oliveira, Cairns and Rizzato2019) 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 Reference Elmore, Elmore and Heald1985). 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 Reference Vay2008; Londrillo, Gatti & Ferrario Reference Londrillo, Gatti and Ferrario2014).

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 (Reference Davidson and Qin2001), 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 Reference Davidson and Qin2001).

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.