2. Betatron radiation in plasma ion column

The plasma ion column driven by a laser pulse or particle bunch is a place where both intense longitudinal accelerating and transverse restoring forces co-exist (Kostyukov et al. Reference Kostyukov, Kiselev and Pukhov2003). The latter can cause betatron oscillations of off-axis electrons with the fundamental frequency of $\omega _\beta =\omega _p/\sqrt {2\gamma }$ and the wavelength of $\lambda _\beta =2{\rm \pi} c/\omega _\beta$. Here, $\gamma$ is the Lorentz factor of the electron, $c$ is the speed of light, $\omega _p=\sqrt {n_0e^2/m_e\varepsilon _0}$ is the plasma frequency where $e$ and $m_e$ are the electron charge and mass, respectively, $n_0$ is the unperturbed plasma density, and $\varepsilon _0$ is the vacuum permittivity. The strength of the betatron oscillation is usually characterized by the normalized betatron oscillation amplitude $K_\beta$, which is defined as (Kostyukov et al. Reference Kostyukov, Kiselev and Pukhov2003)

(2.1)\begin{equation} K_\beta=\gamma k_\beta r_\beta=1.33\times10^{{-}10}\sqrt{\gamma n_0\,[\mathrm{cm}^{{-}3}]}r_\beta[\mathrm{\mu}\mathrm{m}], \end{equation}

where $k_\beta =k_p/\sqrt {2\gamma }$ is the betatron wave number, $k_p=\omega _p/c$ is the plasma wave number and $r_\beta$ is the betatron oscillation amplitude.

Depending on the strength of $K_\beta$, the BR is mainly categorized into two regimes. The limit of small amplitude near axis betatron oscillations with $K_\beta \ll 1$ is known as the undulator regime. The undulator radiation is narrowly peaked at the fundamental mode ($n=1$) with a wavelength of $\lambda =\lambda _\beta /2\gamma ^2$ (Esarey et al. Reference Esarey, Shadwick, Catravas and Leemans2002). However, if the betatron oscillation amplitude is large enough, i.e. $K_\beta \gg 1$, radiation from different sections of the electron's trajectory will be emitted in different directions, contributing to a wide opening angle $\varPsi =K_\beta /\gamma$ of the radiation cone in the direction perpendicular to the electron oscillation plane. The radiation frequency range also gets wide as high harmonics with finite bandwidth become significant. This regime is known as the wiggler regime.

Finite variations of the plasma wiggler parameter $K_\beta$ for different electrons in the bunch will broaden the bandwidth of each harmonic and lead to an overlap of different frequency spikes. So the betatron spectrum in the plasma ion column appears as a quasi-continuous broadband spectrum, similar to the synchrotron radiation from a bending dipole magnet. This synchrotron-like spectrum has been demonstrated experimentally in laser-plasma experiments (Rousse et al. Reference Rousse, Phuoc, Shah, Pukhov, Lefebvre, Malka, Kiselev, Burgy, Rousseau and Umstadter2004; Kneip et al. Reference Kneip, McGuffey, Martins, Martins, Bellei, Chvykov, Dollar, Fonseca, Huntington and Kalintchenko2010; Fourmaux et al. Reference Fourmaux, Corde, Phuoc, Leguay, Payeur, Lassonde, Gnedyuk, Lebrun, Fourment and Malka2011).

The asymptotic spectrum of the wiggler radiation of a single electron with constant energy is given by (Esarey et al. Reference Esarey, Shadwick, Catravas and Leemans2002)

(2.2)\begin{equation} S_{\gamma,r_\beta}(\omega,\varOmega) \sim\frac{\gamma^2\zeta^2}{1+\gamma^2\theta^2}\left[{\rm K}_{2/3}^2(\zeta)+\frac{\gamma^2\theta^2}{1+\gamma^2\theta^2}{\rm K}_{1/3}^2(\zeta)\right], \end{equation}

where ${\rm K}_{\nu }(\zeta )$ is the modified Bessel function of the second kind, $\zeta =({\omega }/{\omega _c})(1+\gamma ^2\theta )^{3/2}$, $\omega$ is the radiation frequency, $\theta$ is the deflection angle relative to the electron propagation direction and the critical photon frequency $\omega _c$ is defined as (Kostyukov et al. Reference Kostyukov, Kiselev and Pukhov2003)

(2.3)\begin{equation} \omega_c=\frac{2}{3}\gamma^3 c r_\beta k_\beta^2. \end{equation}

The critical photon frequency is meaningful since half of the radiation power is emitted above/below the frequency of $\omega _c$. The radiation intensity falls exponentially and becomes negligible for frequencies beyond $\omega _c$. The normalized photon energy spectrum is obtained by integrating (2.2) over all spatial angles ($\theta, \phi$), which has the shape of the universal function for the synchrotron-like emission

(2.4)\begin{equation} S_{\gamma,r_\beta}(\omega)=(\omega/\omega_c)\int_{\omega/\omega_c}^\infty {\rm K}_{5/3}(\omega/\omega_c)\,\mathrm{d}(\omega/\omega_c). \end{equation}

In plasma wakefield accelerators, the spatial scale of the electron bunch is typically of the order of ${\sim }\mathrm {\mu }\mathrm {m}$, which is much larger than the betatron radiation wavelength ($\sim$nm). So the radiation spectrum of a bunch can be simplified as the incoherent summation of the single-electron spectrum (Esarey et al. Reference Esarey, Shadwick, Catravas and Leemans2002; Curcio et al. Reference Curcio, Anania, Bisesto, Chiadroni, Cianchi, Ferrario, Filippi, Giulietti, Marocchino and Petrarca2017b).

Numerical simulations are carried out by the three-dimensional (3-D) quasi-static particle-in-cell (PIC) code QV3D (Pukhov Reference Pukhov2016), which is developed on the VLPL platform (Pukhov Reference Pukhov1999). QV3D calculates the emission of each macroparticle analytically at each time step. The synchrotron radiation is calculated self-consistently from the transverse momentum change of the macroparticle in one time step. This change in momentum determines how many photons are emitted. The momentum of the macroparticle is then updated according to the energy loss to consider the influence of the radiation reaction. The output of the code is the integrated critical photon spectrum per spatial angle, and so the betatron spectrum should be reconstructed by a convolution with the universal function as shown in (2.4).

3. Radiation from the witness electron bunch

For the AWAKE Run 2, the proton self-modulation and the electron acceleration stages are separated into two different plasma cells. The electron-seeded self-modulated proton beam from the first plasma cell is injected into the second plasma cell to drive the acceleration wakefield. The witness electron beam with an energy of 150 MeV is injected in the wakefield right after the proton microbunches. It then accelerates along 10 m plasma aiming at high-quality high-energy electron beam.

A full PIC modelling of the proton dynamics combining with the electron acceleration will be very expensive in time and computer resources. The relativistic proton bunch is quite rigid (with heavy mass and relatively large gamma factor), so the proton bunch after the self-modulation does not change significantly (Adli et al. Reference Adli, Ahuja, Apsimon, Apsimon, Bachmann, Barrientos, Barros, Batkiewicz, Batsch and Bauche2019). Therefore, for simplicity, the acceleration scheme is described by a toy model that was first introduced by Olsen, Adli & Muggli (Reference Olsen, Adli and Muggli2018). The model consists of a single highly rigid proton bunch as the driver and a trailing witness electron bunch in the background plasma as shown in figure 1 (Liang et al. Reference Liang, Xia, Pukhov and Farmer2022). The driver particle mass is magnified by $10^{10}$ times on the base of the real proton mass in the simulation. Since such a dummy proton bunch is highly rigid, the proton-driven wakefield also remains static in its own frame during the propagation except for the dephasing with respect to the witness bunch. This model allows to focus on the acceleration physics of the witness electron beam.

Figure 1. Schematic of simplified AWAKE Run 2 acceleration stage (Liang et al. Reference Liang, Xia, Pukhov and Farmer2022). The densities of plasma electrons ($n_{pe}$, green), the proton driver ($n_{pb}$, magenta) and the witness electron bunch ($n_{eb}$, black) are shown in contour plots. Particle beams propagate from left to right. Here, $\xi =x-ct$ is the longitudinal coordinate in the co-moving frame. The blue solid line represents the loaded longitudinal wakefield $E_x$, and the red dashed line is the transverse wakefield $W_y=E_y-cB_z$ at the transverse position of $\sigma _{ic}$.

Necessary particle beams and plasma parameters for simulations are presented in table 1. Parameters of the non-evolving driver bunch are set to simulate the quasi-linear wakefield excited by the self-modulated SPS proton bunch train (Olsen et al. Reference Olsen, Adli and Muggli2018). The simulation window co-moving with the particle bunches at the speed of light has the dimension of $(9\times 6\times 6)k_p^{-1}$ and resolution of $(0.05\times 0.01\times 0.01)k_p^{-1}$ in directions of $(x,y,z)$, where $x$ is the longitudinal direction, and $y$ and $z$ are transverse directions. The simulation time step is chosen as $5\omega _p^{-1}$, which is short enough to resolve the envelope evolution of the witness bunch. The number of macroparticles per cell is four for the plasma with fixed ion background and one for the non-evolving proton driver. The witness beam is simulated with $10^6$ equally weighted macroparticles.

Table 1. The AWAKE baseline parameters for simulation.

The waist of the witness bunch should match to the pure plasma ion background at the entrance of the plasma column to prevent beam envelope or root-mean-square (r.m.s.) beam radius oscillation. The matched radial size for a Gaussian beam in the plasma ion column is defined by (Olsen et al. Reference Olsen, Adli and Muggli2018; Litos et al. Reference Litos, Ariniello, Doss, Hunt-Stone and Cary2019)

(3.1)\begin{equation} \sigma_{ic}=\left(\frac{2\epsilon_{n0}^2}{\gamma_{e0}k_p^2}\right)^{1/4}, \end{equation}

where $\epsilon _{n0}=\beta _{e0}\gamma _{e0}\epsilon _0$ is the normalized r.m.s. beam emittance, $\epsilon _0$ is the initial geometric emittance, and $\gamma _{e0}$ and $\beta _{e0}=\sqrt {1-1/\gamma _{e0}^2}$ are the initial mean Lorentz factor and the normalized velocity of the witness beam, respectively. Equation (3.1) gives a matched beam size of $\sigma _{ic}=10.64$ $\mathrm {\mu }\mathrm {m}$ for the parameters in table 1, which results in a normalized peak bunch density of $n_{e0}/n_0\approx 10$ for a Gaussian profile bunch. Since the witness bunch is dense enough, it can further blow out the plasma electrons forming a plasma bubble with linear radial focusing force. The longitudinal wakefield excited by the witness bunch also loads upon the proton-driven wakefield, resulting in a relatively uniform net accelerating gradient along the bunch. The initial delay between the two bunches is fixed to $k_p\Delta \xi =6$ for all cases presented in this work.

The non-ideal conditions are highly likely to happen in the experiment, leading to unexpected results. For instance, during electron beam injection into the plasma, the ideal conditions of the matched radius and the on-axis injection may break down. In the following, these non-ideal cases and their influence on the electron beam dynamics and the relevant radiation emission are explored.

3.1. Influence of mismatched beam radius

In experiment, it is always difficult to precisely match the electron beam to the pure ion channel due to the various errors in the beam transport, e.g. the jitters in the magnetic field, the misalignment of magnets, etc. Additionally, the plasma bubble with longitudinally uniform focusing strength covers only the rear part of the witness beam, while at its head, the plasma focusing strength varies with the sinusoidal quasilinear plasma wave, thereby the matching conditions are not entirely identical for different slices along the witness beam. With different extent of mismatch radius along the beam, the witness beam undergoes a fast and intense envelope expansion and oscillation after being injected into the plasma, especially at the bunch head. This leads to a rapid emittance growth at the initial period of beam propagation until full phase-mixing (Liang et al., Reference Liang, Xia, Pukhov and Farmer2022; Farmer et al. Reference Farmer, Liang, Ramjiawan, Velotti, Weidl, Gschwendtner and Muggli2022). Nevertheless, it is shown that the mismatch of the whole bunch's r.m.s. radius does not necessarily lead to the further degradation of the beam quality. There is actually a wide mismatch tolerance depending on the beam charge and length (Farmer et al. Reference Farmer, Liang, Ramjiawan, Velotti, Weidl, Gschwendtner and Muggli2022).

Figure 2(a) shows the evolution trends of the r.m.s. beam radius during the acceleration process for five cases with different initial beam radii. For $\sigma _r/\sigma _{ic}<1$, the bubble formation is much quicker at the beginning. However, the witness bunch is over dense so the emittance pressure (emittance induced defocusing force) exceeds the plasma focusing force. This leads to a rapid expansion of the beam envelope along the whole bunch in the first few tens of centimetres. The transverse expansion of the bunch size is mostly pronounced at the bunch head, because the transverse focusing force in the quasi-linear wake is much weaker than in the bubble. In such cases, the r.m.s. beam radius of the whole bunch after 10 m is larger than the ‘matched’ case. The reduction of the bunch radius with respect to the maximum value is due to the adiabatic damping effect, with the scaling law of $\sigma _r\propto \gamma ^{-1/4}$. For $\sigma _r/\sigma _{ic}>1$, the aforementioned physical process is reversed for the rear part of the beam in the plasma bubble. However, for its head part, there is an ‘optimal’ (quasi-matched) initial r.m.s. radius in the range of $1.25\leq \sigma _{r0}/\sigma _{ic}\leq 1.5$. Below these values, the radius of the head part first sees an expansion before the adiabatic damping. When the initial beam radius exceeds these ‘optimal’ values, the emittance pressure is lower than the plasma focusing force along the bunch, so the entire bunch remains focused during the whole process of acceleration. The large initial bunch radius results in a smaller r.m.s. beam radius quickly, thus a larger average bunch density is obtained. This causes the overloading of the proton-driven wakefield, which then leads to a lower average acceleration gradient. As a consequence, the final energy gain is lower for cases with larger initial bunch radii, as shown in figure 2(b). However, since the variation of the quasi-uniform acceleration gradient due to the mismatch and adiabatic damping effects is relatively small, the difference of the final energy gain for different mismatched cases is also small. The average energy gain, therefore, increases quasi-linearly during the acceleration.

Figure 2. (a) Normalized r.m.s. beam radius $\sigma _r/\sigma _{ic}$ versus the acceleration distance $s$. (b) Dependency of the average Lorentz factor $\left \langle \gamma \right \rangle$ on the normalized initial beam radius $\sigma _{r0}/\sigma _{ic}$, measured after 10 m propagation in plasma. The inset shows the evolution of $\left \langle \gamma \right \rangle$ for the matched case during the beam acceleration.

Figure 3(a) shows the betatron photon energy spectra of the witness electron bunch with different initial bunch radii. The difference is almost negligible for all cases. This is possibly because these betatron spectra are the results of the emission integrated over multiple betatron oscillations up to the diagnostic point at $s = 10$ m. To quantitatively characterize the betatron photon spectra, we can estimate the critical photon energy via the relation (Esarey et al. Reference Esarey, Shadwick, Catravas and Leemans2002)

(3.2)\begin{equation} \hbar\omega_c=\hbar\left\langle \omega\right\rangle/\left(8/15\sqrt{3}\right), \end{equation}

where $\hbar \left \langle \omega \right \rangle$ is the mean photon energy. Figure 3(b) shows the total number of photons $N$ emitted during the acceleration which is almost linearly increasing during the beam propagation. Prior research has suggested that the total photon emission of a single electron scales as $N\propto N_\beta K_\beta$ for the wiggler regime and $N\propto N_\beta K_\beta ^2$ for the undulator regime where $N_\beta$ is the number of oscillations (Esarey et al. Reference Esarey, Shadwick, Catravas and Leemans2002; Corde et al. Reference Corde, Ta Phuoc, Lambert, Fitour, Malka, Rousse, Beck and Lefebvre2013a). To illustrate the relative change of the betatron emission, lineouts showing the dependencies of the critical photon energy on the initial bunch radius are plotted in figure 3(c). The critical photon energies are normalized by the value of the matched case. At the early moments, e.g. $s = 0.4$ m, the critical photon energy is nearly proportional to the initial beam radius. During the acceleration, the relative difference of $\hbar \omega _c$ for different cases gets smaller due to the larger contribution of the high energy photon emission near the plasma exit. One can also notice that the critical photon energy of the cases with smaller radii exceeds that of the matched one after $s = 8$ m. This is likely due to the over expansion of the bunch head which results in the betatron oscillations with larger amplitudes.

Figure 3. (a) Photon energy spectrum measured at $s=10$ m. Here, $\Delta \hbar \omega$ is $10^{-3}$ of the photon energy measuring range, given by the horizontal axis. (b) Evolution of the critical photon energy $\hbar \omega _c$ (solid lines) and the total number of emitted photons $N$ (dashed lines) along the acceleration distance $s$. (c) Normalized critical photon energy $\hbar \omega _c$ with respect to the value of matched case versus the normalized initial beam radius $\sigma _{r0}/\sigma _{ic}$.

To reconstruct the beam profile and emittance, the spatial distribution of its emission is needed (Curcio et al. Reference Curcio, Anania, Bisesto, Chiadroni, Cianchi, Ferrario, Filippi, Giulietti, Marocchino and Mira2017a). So it is interesting to look at the spatial distribution of betatron photons. The radiation pattern on the screen is axisymmetric for an axisymmetric electron bunch (Kostyukov et al. Reference Kostyukov, Kiselev and Pukhov2003); therefore, we look at the angular photon distribution with respect to the axial angle $\theta$. The angle $\theta$ represents the ratio between the radial position $r_s$ of betatron photons on a virtual screen and the distance $L_s$ between the screen and the plasma entrance, i.e. $\theta \approx r_{s}/L_s$, as illustrated in figure 4(a). It shows that the betatron photons are confined within a narrow axial angle, with the r.m.s. value of $\sigma _\theta =({1}/{N}\sum _i\theta _i^2)^{1/2}<2$ mrad for photons measured at $s = 2$ m. The photon angular distribution shapes have good agreement with the transverse distributions of the witness beam in different cases. Furthermore, we can see that the evolution trend of $\sigma _\theta$ in figure 4(b) is similar to that of the r.m.s. beam radius in figure 2(a).

Figure 4. (a) Photon angular distribution versus the axial angle $\theta$ measured at $s=2$ m. (b) Evolution of the r.m.s. value of $\theta$ versus the acceleration distance $s$.

3.2. Influence of misaligned off-axis electron injection

In § 3.1, the witness electron bunch is injected on the central axis of the wakefield for ideal acceleration. The on-axis injected electron bunch does not suffer from the possible transverse instabilities like hosing (Huang et al. Reference Huang, Lu, Zhou, Clayton, Joshi, Mori, Muggli, Deng, Oz and Katsouleas2007). This instability can be induced by the off-axis or oblique electron injection and cause a large increase of the beam emittance and even lead to the beam breakup (BBU) if the instability becomes strong enough. Therefore, understanding the influence of misaligned off-axis electron beam injection on the beam dynamics and the corresponding radiation emission would be useful for AWAKE.

The focusing force and therefore the betatron oscillation depend on the transverse position of electrons, so it is expected that the off-axis injection leads to stronger betatron emission with higher photon energy. The polarization of BR can also be enhanced in the preferred oscillation direction, i.e. the offset direction of the bunch (Kostyukov et al. Reference Kostyukov, Kiselev and Pukhov2003). This is due to the fact that the transverse focusing force in the axisymmetric wakefield is pointing towards the axis of the wake structure.

Here we look at several cases in which the witness bunch has minor offsets from the axis: $\Delta r_0/\sigma _{ic}=0$, 0.35, 0.71 and 1, where $\Delta r_0=\sqrt {\Delta y_0^2+\Delta z_0^2}$ is the combined initial offset of the transverse beam centroid, and $\Delta y_0$ and $\Delta z_0$ are the initial offsets in the $y$- and $z$-directions, respectively. For the second and third cases, the beam offsets are only in the $y$-direction. For the last one, equal offsets of $0.71\sigma _{ic}$ in both $y$- and $z$-directions are considered, which correspond to a combined offset of $\Delta r_0=\sigma _{ic}$ along the azimuthal angle $\phi ={\rm \pi} /4$ in the $y$–$z$ plane. These minor offsets allow to investigate the BR from the witness bunch without significant charge loss due to the transverse instabilities. Additionally, similar cases with a higher charge of 400 pC are also studied. Higher charge can compensate the beam emittance growth induced by the minor offset at the injection point, as the bubble formation is much quicker for the high charge bunch (Farmer et al. Reference Farmer, Liang, Ramjiawan, Velotti, Weidl, Gschwendtner and Muggli2022).

Figure 5 shows that when the radial offset is larger than $0.71\sigma _{ic}$, the beam radiates with higher critical photon energy than the on-axis injection. For smaller offsets, the amplification in photon energy is not significant. Similar trends can also be found in the radiation of the 400 pC witness bunch. This may suggest a safe range for driver-witness misalignment at the injection point. In figure 5(a), one can also see that the 400 pC bunch can produce more BR than the baseline, which is mainly due to more charge contributing to the radiation process. Additionally, figure 5(b) shows that the critical photon energies of cases with 400 pC charge are generally lower than the baseline, which is essentially due to the lower average accelerating wakefields after beam loading.

Figure 5. (a) Betatron photon energy spectrum at $s = 10$ m for different injection offsets. Both the baseline witness beam and higher charge beam (400 pC) are considered. (b) Corresponding critical photon energies versus the beam offsets.

Figure 6 shows the 2-D photon angular distribution over the two spatial angles $\theta$ and $\phi$, and the 1-D projection on the azimuthal angle $\phi$. As expected, the enhancement of the radiation occurs in the direction of the initial offset, e.g. $\phi ={\rm \pi} /4$ and $-3{\rm \pi} /4$ for the offsets of $+0.71\sigma _{ic}$ in both $y$- and $z$-directions as shown in figure 6(a). We also notice that the off-axis injection reshapes the photon density distribution with respect to the axial angle $\theta$. In the offset direction, the photons fall in a wide range of the angle $\theta$, while in other directions, the betatron photons are confined radially within a much narrower $\theta$ angle (Claveria et al. Reference Claveria, Adli, Amorim, An, Clayton, Corde, Gessner, Hogan, Joshi and Kononenko2019). Prior works have shown that for an electron oscillating in a plane along the propagation direction, the typical opening angle of the radiation cone scales as $\varTheta \sim K_\beta /\gamma$ in the trajectory plane, while in the vertical direction, it scales as $\varTheta \sim 1/\gamma$ (Wang et al. Reference Wang, Clayton, Blue, Dodd, Marsh, Mori, Joshi, Lee, Muggli and Katsouleas2002; Rousse et al. Reference Rousse, Phuoc, Shah, Pukhov, Lefebvre, Malka, Kiselev, Burgy, Rousseau and Umstadter2004). In figure 6(b), we can find that the enhancement of the radiation along the initial offset direction leads to the reduction of photon emission at other angles of $\phi$, compared with the on-axis injection. This enhancement is also stronger for cases with higher charge. This feature may allow us to deduce the beam misalignment direction via the betatron diagnostics at the plasma exit.

Figure 6. (a) Two-dimensional photon angular distribution of the baseline witness bunch with offsets of $0.71\sigma _{ic}$ in both $y$- and $z$-directions. The inset shows the spherical coordinates used to represent the photon spatial distribution. (b) The 1-D dependencies of photon numbers on the azimuthal angle $\phi$. Here, $\phi =0$ and $\pm {\rm \pi}$ represents the ${\pm }y$ directions, respectively, and ${\pm }{\rm \pi} /2$ are the ${\pm }z$ directions. The photon densities are normalized by the value of the on-axis injection for both the 120 pC and 400 pC beams. The azimuthal distribution of the on-axis injection is represented by the horizontal green dash–dotted line.

4. Radiation from the seeding electron bunch

In the first plasma cell of the AWAKE Run 2 experiment, an electron bunch seeds the proton self-modulation when it runs ahead of the long proton bunch (Muggli et al. Reference Muggli, Guzman, Bachmann, Hüther, Moreira, Turner and Vieira2020). The electron seeded self-modulation will offer better control on the wakefield's phase and amplitude than the self-modulation growing from random noise. In this section, simulations are performed to study the seeding electron beam dynamics and its relevant betatron spectrum features. Parameters are generally similar to that used in the preliminary electron seeding experiment (Verra et al. Reference Verra, Zevi Della Porta, Pucek, Nechaeva, Wyler, Bergamaschi, Senes, Guran, Moody and Kedves2022). The electron beam has an initial energy of 18.5 MeV, charge of 250 pC, radial size of $\sigma _{r0}=0.2\,\mathrm {mm}$ and duration of ${\sigma _t=5\,\mathrm {ps}\sim 1.5\,\mathrm {mm}}$. The plasma density is $n_0=2\times 10^{14}\,\mathrm {cm}^{-3}$. Other simulation environment settings are similar to those in the previous section.

Some of the main bunch statistics, such as the r.m.s. radial size $\sigma _r$, total normalized beam energy $W/W_0$ and the relative energy spread $\Delta \gamma /\left \langle {\gamma }\right \rangle$, are shown in figure 7. The fast increase of the radial r.m.s. beam size $\sigma _r$ in the first 0.5 m is due to the defocusing of seeding electrons at the head and the tail of the bunch where the local plasma focusing force is too weak to compensate the emittance-induced defocusing effect. This results in approximately 25 % charge leaving the simulation window without significant deceleration. After that, the remaining electrons get focused by its wakefield which leads to the reduction of the beam size, and hence the increase of the seed wakefield amplitude. These electrons are widely distributed over decelerating and accelerating phases after $s=1$ m, which leads to a huge increase of the beam energy spread, but no significant net deceleration. The energy of electrons at the bunch head is quickly depleted in the plasma after approximately 5 m, so they start to slip backwards into the defocusing phase, which results in a significant bunch size expansion and charge loss. As a result of this second-stage charge loss, the remaining electron bunch is cooled down as shown by the decrease of the energy spread. The overall beam energy also sees a larger decrease.

Figure 7. Evolution of the seeding electron bunch with respect to its propagation distance $s$ in the plasma. Here, $\sigma _r$ is the r.m.s. beam radius, $W=Q\left \langle \gamma \right \rangle$ is the total energy stored in the bunch, where $Q$ and $\left \langle \gamma \right \rangle$ are the charge and the mean Lorentz factor, and $\Delta \gamma /\left \langle \gamma \right \rangle$ is the relative energy spread.

Figure 8 shows the betatron photon energy spectra and the photon angular distributions of the seeding electron beam. The integrated spectra are measured over different locations along the plasma cell. It can be seen that the photon energy spectra and the corresponding angular distributions do not evolve too much during the beam propagation. Calculation shows that the critical photon energy of the seeding beam radiation is decreasing in this process, but the change is less than 10 % for critical energy at $s = 1$ m (12.9 meV) and $s= 10$ m (11.7 meV). Similarly, the r.m.s. angle $\sigma _\theta$ also slightly decreases from 4.2 mrad (at $s=1$ m) to 3.8 mrad (at $s=10$ m). In fact, the increasing contribution of the low energy photon emission at a latter time from a larger portion of focused electrons oscillating in low amplitudes leads to the decrease of the critical photon energy and the r.m.s. angle. However, the maximum photon energy sees a small increase which probably is because of the presence of a fraction of accelerated and defocused electrons.

Figure 8. (a) Integrated BR spectra measured over different locations in the plasma. (b) Corresponding 1-D photon angular distributions with respect to the angle $\theta$.