2 Simulations of the current filamentation instability

The onset of CFI occurs when the ratio between the transverse beam size ( $\unicode[STIX]{x1D70E}_{y}$ ) and the plasma skin depth ( $k_{p}^{-1}=c/\unicode[STIX]{x1D714}_{p}$ ), is $k_{p}\unicode[STIX]{x1D70E}_{y}\geqslant 1$ (Roswell & Martin Reference Roswell and Martin1973; Chen et al. Reference Chen, Su, Katsouleas, Wilks and Dawson1987; Su et al. Reference Su, Katsouleas, Dawson, Chen, Jones and Keinigs1987; Sentoku et al. Reference Sentoku, Mima, Kaw and Nishikawa2003; Blumenfeld et al. Reference Blumenfeld, Clayton, Decker, Hogan, Huang, Ischebeck, Iverson, Joshi, Katsouleas and Kirby2007; Allen et al. Reference Allen, Yakimenko, Babzien, Fedurin, Kusche and Muggli2012), where $\unicode[STIX]{x1D714}_{p}=\sqrt{4\unicode[STIX]{x03C0}n_{e}e^{2}/m_{e}}$ is the plasma frequency, $n_{e}$ the background plasma density, $m_{e}$ the mass of the electron, $e$ the charge of the electron and $c$ the speed of light. When the transverse beam size is larger than the plasma skin depth the plasmas return currents can flow through the beam leading to the growth of CFI. If this condition does not hold, i.e. when $\unicode[STIX]{x1D70E}_{y}\leqslant c/\unicode[STIX]{x1D714}_{p}$ , the CFI does not grow (Roswell & Martin Reference Roswell and Martin1973; Chen et al. Reference Chen, Su, Katsouleas, Wilks and Dawson1987; Su et al. Reference Su, Katsouleas, Dawson, Chen, Jones and Keinigs1987; Sentoku et al. Reference Sentoku, Mima, Kaw and Nishikawa2003; Blumenfeld et al. Reference Blumenfeld, Clayton, Decker, Hogan, Huang, Ischebeck, Iverson, Joshi, Katsouleas and Kirby2007; Allen et al. Reference Allen, Yakimenko, Babzien, Fedurin, Kusche and Muggli2012).

In order to illustrate the generation of magnetic fields through the CFI, we start by describing the results from two-dimensional (2-D) OSIRIS PIC simulations (Fonseca et al. Reference Fonseca, Silva, Tsung, Decyk, Lu, Ren, Mori, Deng, Lee and Katsouleas2002b , Reference Fonseca, Martins, Silva, Tonge, Tsung and Mori2008, Reference Fonseca, Vieira, Fiuza, Davidson, Tsung, Mori and Silva2013). The simulations use a moving window travelling at $c$ . The simulation box has absorbing boundary conditions for the fields and for the particles in the transverse direction. The globally neutral fireball beam is initialized at the entrance of a stationary plasma with $n_{e}=10^{17}~\text{cm}^{-3}$ . The plasma is composed of electrons and stationary ions forming an immobile neutralizing fluid background. The initial density profile for the electron and positron fireball beam is given by $n_{b}=n_{b0}\exp (-x^{2}/\unicode[STIX]{x1D70E}_{x}^{2}-y^{2}/\unicode[STIX]{x1D70E}_{y}^{2})$ where $n_{b0}=n_{e}=10^{17}~\text{cm}^{-3}$ , $\unicode[STIX]{x1D70E}_{x}=0.99~c/\unicode[STIX]{x1D714}_{p}=10.2~\unicode[STIX]{x03BC}\text{m}$ and $\unicode[STIX]{x1D70E}_{y}=2~c/\unicode[STIX]{x1D714}_{p}=20.4~\unicode[STIX]{x03BC}\text{m}$ are the bunch peak density, length and transverse waist, respectively. The beam propagates along the $x$ -axis with Lorentz factor $\unicode[STIX]{x1D6FE}_{\text{b}}=5.6\times 10^{4}$ , with transverse velocity spread $v_{th}/c=1.7\times 10^{-5}$ and with no momentum spread in the longitudinal direction. The simulation box dimensions are $L_{x}=8.02~c/\unicode[STIX]{x1D714}_{p}$ and $L_{y}=20.0~c/\unicode[STIX]{x1D714}_{p}$ with a moving window travelling at $c$ along $x$ . The box is divided into $128\times 512$ cells with $2\times 2$ particles per cell.

Figure 1. The interaction of a neutral $e^{-}$ , $e^{+}$ fireball beam having a Gaussian profile with $\unicode[STIX]{x1D70E}_{x}=2\unicode[STIX]{x1D70E}_{y}=20.4~\unicode[STIX]{x03BC}\text{m}$ , peak density $n_{b}=2.7\times 10^{15}~\text{m}^{-3}$ , $\unicode[STIX]{x1D6FE}_{b}=5.6\times 10^{4}$ with a static plasma with $n_{e}=n_{b}$ . (a) Evolution of Transverse magnetic $\unicode[STIX]{x1D716}_{bz}$ (red), Longitudinal $\unicode[STIX]{x1D716}_{ex}$ (green) and transverse electric $\unicode[STIX]{x1D716}_{ey}$ (blue) field energies as function of distance normalized to the initial kinetic energy of the beam $\unicode[STIX]{x1D716}_{p}=(\unicode[STIX]{x1D6FE}_{\text{b}}-1)V$ , where $V_{b}=\unicode[STIX]{x03C0}\unicode[STIX]{x1D70E}_{x}\unicode[STIX]{x1D70E}_{y}$ is the volume of the beam. The dotted line represents the simulation growth rate of the CFI. At time $t=1900.09~[1/\unicode[STIX]{x1D714}_{p}]$ , we show (b) the density filaments corresponds to the electron $e^{-}$ (blue) and positron $e^{+}$ (red) spatially separated from each other (c) the associated transverse magnetic ( $B_{z}$ ) filaments at linear regime after 0.02 m (d) due to space charge radial electric field ( $E_{x}$ ) created.

Figure 1 depicts the growth of the transverse magnetic field energy (a), the beam filaments due to the CFI (b), and the typical electromagnetic field structure (c) and (d). Figure 1(a) shows that the growth of the magnetic field energy as a function of the propagation distance is exponential as expected from the CFI. In figure 1(a), the electromagnetic field is normalized with respect to the initial kinetic energy of the particles $\unicode[STIX]{x1D716}_{p}=(\unicode[STIX]{x1D6FE}_{\text{b}}-1)V_{b}$ , where $V_{b}=(\unicode[STIX]{x03C0}\unicode[STIX]{x1D70E}_{x}\unicode[STIX]{x1D70E}_{y})$ is the volume of the beam. Simulations reveal that the field energy grows at the expense of the total kinetic energy of the fireball beam. The measured simulation linear growth rate ( $\unicode[STIX]{x1D6E4}_{\text{CFI}}/\unicode[STIX]{x1D714}_{p}\simeq 6.0\times 10^{-3}$ ) is within the range predicted by the dispersion relation for the purely transverse CFI (Silva et al. Reference Silva, Fonseca, Tonge, Mori and Dawson2002). As a consequence of the instability, the beam breaks up into narrow (with a width of the order of $0.5\,c/\unicode[STIX]{x1D714}_{p}$ , which correspond to $5~\unicode[STIX]{x03BC}\text{m}$ for our baseline parameters) and high current density filaments. Figure 1(b) shows that these electron–positron filaments are spatially separated from each other. Each filament carries strong currents, which lead to the generation of strong out-of-the-plane (i.e. azimuthal) magnetic fields with amplitudes beyond 20 T. The azimuthal magnetic fields are also filamented, as shown in figure 1(c). Because of their finite transverse momentum, simulations show that current filaments can merge.

As merging occurs, the width of the filaments increases, until the beam filaments width becomes comparable to the initial bunch transverse size. At this point, the CFI stops growing, and no more beam energy flows into the generation of azimuthal magnetic fields. Simultaneously, radial electric fields above $10~\text{GV}~\text{m}^{-1}$ are also generated (figure 1 d) which can be attributed to space charge effects and due to inductive effects. The beam transverse size is larger than the plasma skin depth, which corresponds to the spatial scales where the instability growth rate is maximum (Su et al. Reference Su, Katsouleas, Dawson, Chen, Jones and Keinigs1987). Our simulations show a clear transition from the linear to nonlinear stage of CFI due to the merging of fireball bunch filaments and we expect that the mechanisms leading to the saturation of the magnetic fields in our simulations will not differ from well-known saturation mechanisms, described in Yang, Arons & Langdo (Reference Yang, Arons and Langdo1994), Silva et al. (Reference Silva, Fonseca, Tonge, Dawson, Mori and Medvedev2003), Medvedev et al. (Reference Medvedev, Massimiliano, Fonseca, Silva and Mori2005), Achterberg, Wiersma & Norman (Reference Achterberg, Wiersma and Norman2007).

3 Role of the peak beam density and beam duration in the growth of current filamentation instability

In the previous section, and for illustration purposes only, we have considered that the total beam density was twice the background plasma density. In this section, we will investigate the propagation of beams with lower peak densities. In order to keep the number of particles constant, we then increase the beam length, such that $\unicode[STIX]{x1D70E}_{x}\geqslant \unicode[STIX]{x1D706}_{p}$ . In these conditions, the OBI competes with the CFI (Bret Reference Bret2009). The OBI can grow when the wave vector is at an angle with respect to the flow velocity direction, and it leads to the generation of both electric and magnetic field components. The maximum growth rate for the CFI and the OBI, assuming the beams are infinitely long, are given by $\unicode[STIX]{x1D6E4}_{\text{CFI}}\sim \sqrt{\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FE}_{\text{b}}}\,\unicode[STIX]{x1D6FD}_{b0}$ and $\unicode[STIX]{x1D6E4}_{\text{OBI}}\sim \sqrt{3}/2^{4/3}(\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FE}_{\text{b}})^{1/3}$ respectively (Bret & Gremillet Reference Bret and Gremillet2006), where $\unicode[STIX]{x1D6FC}$ is the beam ( $n_{b}$ ) to plasma density ( $n_{e}$ ) ratio and $\unicode[STIX]{x1D6FD}_{b0}=v_{b}/c$ is the normalized velocity of the beam. Thus, the ratio between the CFI growth rate and the OBI growth rate is given by:

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x1D6E4}_{\text{OBI}}}{\unicode[STIX]{x1D6E4}_{\text{CFI}}}}={\displaystyle \frac{\sqrt{3}}{2^{4/3}}}{\displaystyle \frac{1}{\unicode[STIX]{x1D6FD}_{b}}}\left({\displaystyle \frac{\unicode[STIX]{x1D6FE}_{\text{b}}}{\unicode[STIX]{x1D6FC}}}\right)^{1/6}. & \displaystyle\end{eqnarray}$$

Equation (3.1) provides criteria for determining which of the two instabilities will dominate. The OBI is almost universally faster than the CFI. However, due to the weak dependence on $\unicode[STIX]{x1D6FE}_{b}/\unicode[STIX]{x1D6FC}$ , given $\unicode[STIX]{x1D6FE}_{\text{b}}\sim 10^{4}$ the OBI will begin to dominate when $\unicode[STIX]{x1D6FC}$ is much smaller 1.

In order to verify this hypothesis, we have carried out additional two-dimensional OSIRIS PIC simulations using the initial set-up described in § 2, varying $\unicode[STIX]{x1D70E}_{x}$ between $2\unicode[STIX]{x1D706}_{p}$ and $10\unicode[STIX]{x1D706}_{p}$ , for which $\unicode[STIX]{x1D6FC}$ varies between $0.0026$ and $1.0$ . In all these cases, our results have consistently shown evidence of the OBI growth.

Figure 2. Interaction of a neutral $e^{-}$ , $e^{+}$ fireball beam with longitudinal size $\unicode[STIX]{x1D70E}_{x}=2\unicode[STIX]{x1D706}_{p}$ and a static plasma by keeping constant beam particle number. (a) Evolution of transverse magnetic $\unicode[STIX]{x1D716}_{bz}$ (red), longitudinal $\unicode[STIX]{x1D716}_{ex}$ (green) and transverse electric $\unicode[STIX]{x1D716}_{ey}$ (blue) field energy as function of distance normalized to the initial kinetic energy of the beam $\unicode[STIX]{x1D716}_{p}$ . The dotted line represents the growth rate of OBI. Dashed red line is the evolution of the out of the plane magnetic field for the condition of the simulation shown in figure 1(a). At time $t=8880.07~[1/\unicode[STIX]{x1D714}_{p}]$ , we show (b) the density filaments corresponding to the electron $e^{-}$ (blue) and positron $e^{+}$ (red) spatially separated from each other (c) the associated transverse magnetic ( $B_{z}$ ) filaments in the linear regime between $x_{1}=0.0551-0.0556~\text{m}$ (d) the space charge separation leads to longitudinal electric field ( $E_{x}$ ).

In figure 2, we show an illustrative simulation result considering $\unicode[STIX]{x1D70E}_{x}=2\unicode[STIX]{x1D706}_{p}$ , with $n_{b}=1.274\times 10^{15}~\text{cm}^{-3}$ , for which $\unicode[STIX]{x1D6FC}=0.01274$ . In order to describe the propagation of a longer beam, we have increased the simulation box length. We then increased the longitudinal simulation box length to $L_{x}=63\,c/\unicode[STIX]{x1D714}_{p}$ ( $L_{y}=20\,c/\unicode[STIX]{x1D714}_{p}$ remains identical to that of § 2). The simulation box is now divided into $1024\times 512$ cells with $2\times 2$ particles per cell for each species.

Figure 2(a) illustrates the evolution of the longitudinal and transverse electric and transverse magnetic energy (normalized to $\unicode[STIX]{x1D716}_{p}=(\unicode[STIX]{x1D6FE}_{\text{b}}-1)V_{b}$ , where $V_{b}=(\unicode[STIX]{x03C0}\unicode[STIX]{x1D70E}_{x}\unicode[STIX]{x1D70E}_{y})$ is the volume of the beam). The emergence of oblique modes can be seen in figure 2(b), which shows tilted beam filaments. In a multi-dimensional configuration, the oblique wave vector couples the transverse (filamentation) and longitudinal (two-stream) instabilities resulting in the electromagnetic beam–plasma instability. Unlike figure 1, the simulation results in figure 2 show that the transverse electric field ( $E_{y}$ ) component provides the dominant contribution to the total field energy. The plasma is only weakly magnetized $\unicode[STIX]{x1D714}_{c}/\unicode[STIX]{x1D714}_{p}=0.01$ , much lower than in figure 1(c), where $\unicode[STIX]{x1D714}_{c}/\unicode[STIX]{x1D714}_{p}\simeq 0.6$ , and the CFI does not play a critical role in the beam propagation. The longitudinal and transverse electric fields grow exponentially, as predicted by the linear analysis of the OBI, matching well the simulation results. The growth rate measured in the simulations is $\unicode[STIX]{x1D6E4}_{\text{max}}/\unicode[STIX]{x1D714}_{p}\simeq \unicode[STIX]{x1D6E4}_{\text{OBI}}\simeq 2.1\times 10^{-3}$ , in good agreement with Fainberg, Shapiro & Shevchenko (Reference Fainberg, Shapiro and Shevchenko1970). The OBI generates plasma waves with strong radial electric fields in excess of $1~\text{GV}~\text{m}^{-1}$ (figure 2 d). After 20 cm, the OBI saturates. Despite being limits of the same instability, the electromagnetic beam plasma instability, we will refer to the CFI and the OBI as the manifestation of qualitatively different behaviour of the same instability.

4 Effects of finite beam waist and emittance

Theoretical and numerical studies performed to identify the effect of beam emittance on the growth of plasma instabilities and their saturation (Fonseca et al. Reference Fonseca, Silva, Tonge, Hemker, Dawson and Mori2002a ; Silva et al. Reference Silva, Fonseca, Tonge, Mori and Dawson2002; Shukla et al. Reference Shukla, Stockem, Fiuza and Silva2012) typically assume that the beam is infinitely wide. In this section, we will investigate the role of the beam emittance considering finite beam size effects, in order to make closer contact with laboratory conditions. To study the influence of the beam emittance on the propagation, we first consider the equation for the evolution of the beam waist $\unicode[STIX]{x1D70E}_{y}$ in vacuum (Schroeder & Benedetti Reference Schroeder and Benedetti2011)

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{1}{c^{2}}}{\displaystyle \frac{\text{d}^{2}\unicode[STIX]{x1D70E}_{y}}{\text{d}t^{2}}}={\displaystyle \frac{1}{4}}{\displaystyle \frac{\unicode[STIX]{x1D716}_{N}^{2}}{\unicode[STIX]{x1D70E}_{y}^{3}\unicode[STIX]{x1D6FE}_{\text{b}}^{2}}}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}_{y}$ is the beam radius, $\unicode[STIX]{x1D716}_{N}\simeq \unicode[STIX]{x0394}p_{\text{y}}\unicode[STIX]{x1D70E}_{y}$ is a figure for the beam emittance (corresponding to the area of the beam transverse phase space) and $\unicode[STIX]{x0394}p_{\text{y}}$ is the transverse momentum spread. According to (4.1), the evolution for $\unicode[STIX]{x1D70E}_{y}$ and for sufficiently early times is given by:

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70E}_{y}\simeq \unicode[STIX]{x1D70E}_{y0}\left(1+{\displaystyle \frac{\unicode[STIX]{x1D716}_{N}^{2}~t^{2}c^{2}}{4\unicode[STIX]{x1D70E}_{y0}^{4}\unicode[STIX]{x1D6FE}_{\text{b}}^{2}}}\right)^{1/2}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}_{y0}$ is the initial beam radius. Hence, according to (4.1), the rate at which $\unicode[STIX]{x1D70E}_{y}$ increases is:

(4.3) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{1}{\unicode[STIX]{x1D70E}_{y}}}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D70E}_{y}}{\text{d}t}}={\displaystyle \frac{1}{4}}{\displaystyle \frac{tc^{2}\unicode[STIX]{x1D716}_{N}^{2}}{\unicode[STIX]{x1D70E}_{y0}^{2}\unicode[STIX]{x1D6FE}_{\text{b}}^{2}}}{\displaystyle \frac{1}{\unicode[STIX]{x1D70E}_{y}^{2}}}. & \displaystyle\end{eqnarray}$$

Equation (4.3) indicates that the beam expands in vacuum due to its transverse momentum spread. As the beam expands, $n_{b}$ decreases as $n_{b}/n_{e}\sim (\unicode[STIX]{x1D70E}_{y0}/\unicode[STIX]{x1D70E}_{y})^{2}$ , in three dimensions, and as $(\unicode[STIX]{x1D70E}_{y0}/\unicode[STIX]{x1D70E}_{y})$ , in two dimensions. Because of the reduction of $n_{b}/n_{e}$ , the growth rates for the CFI and for the OBI will also decrease. We then estimate that these instabilities (i.e. CFI and OBI) are suppressed when the rate at which $n_{b}/n_{e}$ decreases is much higher than the instability growth rate. Matching the rate at which the beam density drops, which in two dimensions is given by $(1/\unicode[STIX]{x1D70E}_{y})~(\text{d}\unicode[STIX]{x1D70E}_{y}/\text{d}t)$ , to the growth rate of the instability ( $\unicode[STIX]{x1D6E4}$ ) gives an upper limit for the maximum beam divergence $\unicode[STIX]{x1D703}=\unicode[STIX]{x0394}p_{\text{y}}/\unicode[STIX]{x1D6FE}_{\text{b}}$ (and emittance $\unicode[STIX]{x1D716}_{N}\approx \unicode[STIX]{x1D70E}_{y}(\langle p_{y}^{2}\rangle )^{1/2}$ ) allowed for the growth of the CFI/OBI:

(4.4) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D703}=2\left({\displaystyle \frac{\unicode[STIX]{x1D6E4}\unicode[STIX]{x1D70E}_{y0}^{2}}{L_{\text{growth}}~c}}\right)^{1/2}, & \displaystyle\end{eqnarray}$$

where we have considered that $t\sim L_{\text{growth}}/c$ in (4.4), with $L_{\text{growth}}$ the growth length of the CFI/OBI instability. Equation (4.4) then gives the threshold beam divergence, beyond which the CFI/OBI will be suppressed. It indicates that beams with higher energy can support higher thermal spreads and still be subject to the growth of the CFI because the beam expands slowly in comparison to lower energy beams. Similarly, beams with higher $\unicode[STIX]{x1D70E}_{y0}$ also support higher emittance than narrower beams because of the slower expansion rate.

To confirm our theoretical findings, we performed additional two-dimensional simulations using fireball beams with relativistic factors $\unicode[STIX]{x1D6FE}_{\text{b}}=700,1050,1400$ (the lower $\unicode[STIX]{x1D6FE}_{\text{b}}$ factors used now, in comparison to § 2, to minimize the computational requirements). We use $\unicode[STIX]{x1D70E}_{x}=0.22~c/\unicode[STIX]{x1D714}_{p}=11.7~\unicode[STIX]{x03BC}\text{m}$ and $\unicode[STIX]{x1D70E}_{y}=10~c/\unicode[STIX]{x1D714}_{p}=530~\unicode[STIX]{x03BC}\text{m}$ with peak density $n_{b0}=10~n_{e}=10^{15}~\text{cm}^{-3}$ . For each case, we varied the transverse temperature $\unicode[STIX]{x0394}p_{\text{y}}=\unicode[STIX]{x1D6FE}_{\text{b}}\unicode[STIX]{x1D703}=1,3,5,7,10$ and $20$ in order to determine the threshold beam spread for the occurrence of instability. We note that we have used the classical addition of velocities in the beam thermal spread initialization in order to more clearly identify the dependence of evolution of the instabilities with emittance.

Figure 3. (a) Temporal evolution of the transverse magnetic field energy for different beam emittance (b) Thermal velocities as function of Lorentz factor $\unicode[STIX]{x1D6FE}_{\text{b}}$ , filamentation suppressed for higher thermal velocities. The red line represents a linear fit of the simulation data. At time $t=705.60~[1/\unicode[STIX]{x1D714}_{p}]$ , panels (c,d) show beam filaments for thermal velocities $\unicode[STIX]{x0394}p_{\text{y}}=1,10$ from 2-D PIC simulations.

Figure 3(a) shows that the magnetic field energy decreases with increasing transverse momentum spread. Figure 3(a) also shows a transition in the evolution of the magnetic energy between $\unicode[STIX]{x0394}p_{\text{y}}=10$ , where the magnetic field (B-field) still grows at the end of the simulation, and $\unicode[STIX]{x0394}p_{\text{y}}=20$ , where the B-field decreases with propagation distance. According to (4.4), using $L_{\text{growth}}\sim 0.037~\text{m}$ and $\unicode[STIX]{x1D6E4}_{\text{CFI}}\sim 1.657\times 10^{11}\text{s}^{-1}$ , we obtain the threshold $\unicode[STIX]{x1D703}\sim 0.12$ for the shutdown of the instability. This is in good agreement with figure 3(a). The instability is suppressed for higher beam emittance because the beam to plasma density ratio drops faster than the growth rate of the instability, hence the beam expands in the transverse direction and does not have enough time for the CFI to grow. The threshold for emittance should be taken as only a guide to design experiments, and provides an upper limit required to observe the instabilities. The emittance should thus be much smaller than the values predicted in the model.

Figure 3(b) depicts the dependence of the threshold beam emittance on the fireball beam energy. Figure 3(c,d) shows the positron density for two simulations, where all the parameters are kept constant, except for the beam emittance. In particular, in figure 3(c) a beam emittance of $\unicode[STIX]{x0394}p_{\text{y}}=1$ , much smaller than the threshold value given by (4.4), has been considered. In this case, the CFI develops, leading to the filamentation of the beam (see figure 3 c) and to the exponential growth of the magnetic field energy (see figure 3 a, red curve). However, in the second case (figure 3 d) a higher beam emittance $\unicode[STIX]{x0394}p_{\text{y}}=10$ is considered. This suppresses the growth of the magnetic field energy (see figure 3 a, orange curve). As a result, the beam expands before the development of the CFI. These results show that the growth of CFI can only be achieved if the beam emittance is sufficiently small.

5 Effect of beam energy spread

In typical laboratory settings (Sarri et al. Reference Sarri, Poder, Cole and Schumaker2015), electron–positron fireball beams can contain finite energy spreads. It is, therefore, important to evaluate the potentially deleterious role of the energy spread in the growth of CFI. In this section, we then present simulation results with finite longitudinal momentum spreads. We consider that the central beam relativistic factor is $\unicode[STIX]{x1D6FE}_{\text{b}}=700$ , and compare two simulations with $\unicode[STIX]{x0394}p_{\text{x}}/\unicode[STIX]{x1D6FE}_{\text{b}}=0.13$ and $\unicode[STIX]{x0394}p_{\text{x}}/\unicode[STIX]{x1D6FE}_{\text{b}}=0.29$ ( $\unicode[STIX]{x0394}p_{\text{x}}$ is the longitudinal momentum spread). All other simulation parameters are similar to those described in § 4.

Figure 4. (a,b) Temporal evolution $(t=705.60~[1/\unicode[STIX]{x1D714}_{p}])$ of positron density exhibits the filaments for energy spread $\unicode[STIX]{x0394}p_{\text{x}}=90,200$ ; (c) (colour) the spectrum of two different energy spreads; (d) the solid line represents the simulation growth rate of the CFI for different beam energies.

Figure 4(a,b) shows the temporal evolution of the beam electron density for $\unicode[STIX]{x0394}p_{\text{x}}/\unicode[STIX]{x1D6FE}_{\text{b}}=0.13$ (figure 4 a) and for $\unicode[STIX]{x0394}p_{\text{x}}/\unicode[STIX]{x1D6FE}_{\text{b}}=0.29$ (figure 4 b). The initial energy spectra of these two beams are shown in figure 4(c). Figure 4(d) shows the comparison of the magnetic field energy evolution. The blue curve shows the growth of magnetic field energy generated by the fireball beam with energy spread $\unicode[STIX]{x0394}p_{\text{x}}/\unicode[STIX]{x1D6FE}_{\text{b}}=0.29$ , while the red curve is associated with the lower energy spread $\unicode[STIX]{x0394}p_{\text{x}}/\unicode[STIX]{x1D6FE}_{\text{b}}=0.13$ .

The simulation growth rate ( $\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D714}_{p}\simeq 2.8\times 10^{-2}$ ; shown by the dotted black line in figure 4 d) is consistent with the theoretical growth rate predicted for the purely transverse CFI (Pegoraro et al. Reference Pegoraro, Bulanov, Califano and Lontano1996; Inglebert et al. Reference Inglebert, Ghizzo, Reveille, Sarto, Bertrand and Califano2012). We notice that one to one comparisons with the simulation growth rate are non-trivial. We speculate that the differences are probably due to the finite beam size and non-uniform density profile in the transverse directions. We have performed additional simulations, which confirmed that numerical results converge to theoretical predictions as simulations considered progressively wider beams. In fact, to the best our knowledge, the excitation of these modes for finite length finite width modes has not been investigated. Thus, this work also motivates further theoretical developments that could predict the exact growth rate for non-uniform density profiles. However, this discussion does not prevent the main purpose of this section, namely, the energy spread naturally present in laser-produced fireball beams will not prevent the CFI occurring.

6 Summary and conclusions

In summary, the growth and saturation of an ultra-relativistic beam propagating through plasma have been investigated using particle-in-cell (PIC) simulations. We have shown that short fireball beams, i.e. beams shorter than the plasma wavelength, interacting with uniform plasmas lead to the growth of the CFI. For typical parameters available for experiments, the instability can generate strong transverse magnetic field of the order of MGauss. We found that in typical laboratory settings, the incoming fireball beam filamentation saturates after 10 cm of propagation for the SLAC parameters (Muggli et al. Reference Muggli, Martins, Vieira and Silva2013), while for the laser–plasma generated fireball beam produced recently (Sarri et al. Reference Sarri, Poder, Cole and Schumaker2015) it saturates after 4.4 cm.

We have demonstrated that the beam density needs to be higher than the background plasma density to suppress the growth of the competing OBI instability, which leads to the growth of electrostatic modes (instead of electromagnetic). Beams with lower peak densities will then drive the OBI, which results in tilted filaments and the generation of mostly electrostatic plasma waves. We have also shown that the beam emittance needs to be minimized, reducing transverse beam defocusing effects, which can shut down the CFI or the OBI if the beam defocuses before these instabilities grow. We have also extended our numerical studies to investigate the effect of finite fireball energy spreads on the growth of CFI, and showed that the energy spreads of currently available fireball beams allow for the growth of CFI in the laboratory.

In conclusion, we have identified the factors for the generation of strong magnetic fields via CFI. We expect that the results will influence our understanding of astrophysical scenarios, by revealing the laboratory conditions where these effects can be studied.