A driving laser-pulse of moderate intensity ((I lesssim 10^{19}) W cm(^{-2})), linearly polarized in the y-direction, propagates in the longitudingal (x) direction in an underdense plasma (in practice, (n_e) is in the order of (10^{18}) cm(^{-3})) and creates a moderately nonlinear plasma wave. Its first period, the so-called “bubble”, is an ion cavity free of electrons which are expelled by the strong ponderomotive force of the driving pulse. The electron bunch is located in the rear part of the bubble. It is injected transversely (y-direction), either by self-injection, or as is the case in this paper, by controlled injection on the density downramp. A weaker modulation pulse ((I lesssim 10^{18}) W cm(^{-2})) with wavelength (lambda _m) is injected to follow the driving pulse. Its electric field, polarized in the y-direction, still dominates over the electrostatic transverse field of the bubble. The delay between the pulses is chosen in a way that its high-intensity part co-propagates with the electron bunch.

As the modulation pulse propagates within the bubble, its group velocity is approximately equal to the speed of light in vacuum (v_{g,m} lessapprox c). The average longitudinal velocity of an electron in the bunch is lower, due to the relativistic limitation caused by transverse betatron oscillations. The accelerated electrons oscillate transversely on a sine-like trajectory because they gained a considerable transverse momentum dominantly by the fields of the modulation pulse, but also by the injection process and by the electrostatic transverse fields of the bubble. Every periodic increase of their transverse velocity leads to a decrease of their longitudinal velocity. As a result, the modulation pulse steadily overtakes the electron bunch. Consequently, an electron from the bunch experiences the action of a periodically varying transverse component of the Lorentz force as it propagates backward with respect to the modulation pulse.

The transverse electron motion can be described by the equation of motion ({text {d}}p_y/{text {d}}t approx q_e(1-beta _x)E_{0,y,m}cos (k_mxi )), where (q_e) is electron charge, (E_{0,y,m}) is the electric field amplitude of the modulation pulse, (k_mxi) is the phase of the modulation pulse, with (k_m=2pi /lambda _m) being the modulation pulse wavenumber and (xi = x – x_0 – v_{g,m}t) the coordinate co-moving with the modulation pulse. Here, we assumed (|p_x| gg |p_y|), (p_x gg m_ec), and considered the modulation pulse as a plane wave, which is applicable in regions around the propagation axis, where its magnetic field is proportional to its electric field (B_z approx E_y/c). Thus, the electrons flow backward with respect to the modulation pulse and due to the phase dependence of the transverse force, they are periodically pushed in the (pm y-)direction. This effect itself leads to enhancement of the betatron radiation emission in comparison with a standard case without the modulation pulse.

From the positions where (cos (k_mxi ) =0), the absolute value transverse momentum of the electrons decreases and the longitudinal momentum grows; the latter one is largest at the turning points of their trajectory where (p_y=0). Thus, the turning points related to the modulation pulse phase are the same for all electrons of the bunch. Large longitudinal momenta together with low transverse momenta result in a clustering of the bunch electrons in the nests co-moving with the modulation pulse. Alternatively stated: the original electron bunch is microbunched. As the betatron radiation is mainly emitted at the turning points of the electron trajectories, its temporal profile is composed of intensity peaks separated by (lambda _m/2c), i.e. a train of X-ray pulses is emitted and the delay between the pulses is adjustable by choosing (lambda _m).

The effect of microbunching can be understood as a forced betatron resonance. Contrary to previous cases with the modulation by the tail of the plasma wave drive pulse23,25, where the electron beam experiences a long acceleration period before it catches the laser pulse which resulting in limited controllability of the X-ray source, we reach the betatron resonance immediately from the moment of injection.

Numerical simulation

The process of michrobunching and its fingerprint on the betatron radiation signal is studied by means of 2D particle-in-cell (PIC) simulations and their post-processing. A bubble regime configuration with modest laser parameters is chosen for the demonstration of the process. The parameters used in the simulation are the following: plasma electron density (n_0 = 2.5times 10^{18}) cm(^{-3}), driver laser wavelength (lambda _d = 0.8) (upmu)m, waist size (radius at 1/e2 of maximum intensity) (w_0 = 10) (upmu)m, pulse length (FWHM of intensity) (tau =20) fs, and normalized driver laser intensity (a_{0,d} = eE_{0,d}/m_ec omega _0=1.8) which corresponds to intensity (I=6.9times 10^{18};{mathrm{W}};{mathrm{cm}}^{-2}). Its focal spot is located at (x_{f,m}=110) (upmu)m. The modulation pulse has the same fundamental parameters with the exception of normalized intensity, which is (a_{0,m}=0.2), and wavelength (lambda _m = lambda _d/3) corresponding to intensity (7.7times 10^{17};{mathrm{W}};{mathrm{cm}}^{-2}). It is delayed by 58 fs and its focal spot is located at (x_{f,m}=410) (upmu)m. Both pulses are linearly polarized in the (y-)direction.

Self-injection of electrons in the plasma wakefield does not occur with these parameters if the plasma density is constant. Instead, a plasma density profile is chosen so that controlled injection occurs. In the simulations, the density profile is set in the following way. A 10 (upmu)m long vacuum is located at the left edge of the simulation box, then a 50 (upmu)m linear density up-ramp follows until the electron density reaches (2n_e). Nevertheless, the nature of the presented injection scheme does not depend on the plasma-edge density ramp. Afterwards, a 35 (upmu)m long density plateau follows; then the density linearly drops to (n_e) over a distance of 25 (upmu)m. On this down-ramp, the controlled injection occurs45. The PIC simulations were performed with the epoch code, see the Methods section for details.

Figure 2

Plasma bubble evolution and electron microbunching. Snapshots of the electron density at the injection time ((0.5 ;{mathrm{ps}}) left panel) and during the acceleration process ((1.4 ;{mathrm{ps}}) and (2.3;{mathrm{ps}}), centre and right panels, respectively). The red line in the left panel represents the end of the initial density down-ramp. Only a central part of the simulation box is shown. The upper insets show a zoom of the bunch structure, the bottom insets show a projection of the trapped particles density on the x-axis.

The snapshots of the electron density during the injection and acceleration process are shown in Fig. 2. The density profile in the panel corresponding to the injection time ((t=0.5) ps) suggests that the electron bunch is microbunched immediately after the injection. In later times (1.4 ps and 2.3 ps of simulation), the snake-like structure of the bunch is pronounced.

Figure 3

Electron bunch structure and energy spectrum. (a) Electron density of the trapped electrons (plotted the simulation cells where average kinetic energy of electrons is higher than 10 MeV) and the transverse electric field at (t=2.3;{mathrm{ps}}). (b) Transverse momentum of the trapped electrons. (c) Electron energy spectra at (t=4.0;{mathrm{ps}}) for the cases with and without modulation pulse present.

The detailed view of the electron bunch structure at 2.3 ps is shown in Fig. 3a), together with the transverse electric field. Apparently, the electric field of the modulation pulse dominates over the electrostatic field of the bubble in the region around the axis where the electron bunch is located. The bunch itself has a sawtooth-shape. The distance between the (x-)coordinates of the turning points is (lambda _{m}/2). The peak values of the electron density are located in these turning points.

Figure 3b) shows the positions and transverse momenta of the accelerated electrons. The positions between the peaks of the density bunch profile and the dominant direction of the transverse component of the electron momentum confirm that the electrons propagate backwards in the frame co-moving with the modulation pulse. These findings can be interpreted as the electron bunch as a whole performs snake-like motion in the direction of (-xi). This means that the modulation pulse effectively induces the microbunching of injected electrons and the distance between single microbunches is (lambda _{m}/2) in the longitudinal direction.

The electrons perform betatron oscillations, however, in contrast to standard betatron motion in the case without the modulation pulse, the oscillations are driven dominantly by the modulation pulse. Thus, crucially, the turning points are the same for all of the trapped electrons. In other words, the electron bunch is effectively separated into several equidistant microbunches that are continuously radiating. As a consequence, the observer will receive a modulated betatron radiation signal, comprising of peaks arriving every (lambda _m/2c), as will be shown later.

The electron energy spectrum in time of 4.0 ps just before the structure begins to dephase is shown in Fig. 3c); blue and red lines show the cases without and with the modulation pulse, respectively. The spectra comprise a clear peak which corresponds to the electrons accelerated in the first period of the plasma wave due to the controlled injection. Although, the relative energy spread is rather high. However, for the purpose of betatron radiation generation the energy spread is not a determining factor. The presence of the modulator leads to further electron energy gain compared to the reference case: the electrons receive the energy stored in the modulator by direct laser acceleration46,47. The estimated accelerated charge (electron energy higher than 25 MeV) is about 4 to 8 pC in both cases. There are about 1.3% less electrons trapped when the modulator is present.

Betatron radiation spectrogram

Figure 4 shows the spectrograms, i.e. both temporal and energy profiles of the betatron radiation, with and without the modulation pulse; for details see the Methods section. Four different cases are presented: a) the case when the modulator is not present, (b) with (lambda _m = lambda _d) and (a_{0,m}=0.6), (c) with (lambda _m = lambda _d/3) and (a_{0,m}=0.1), and (d) with (lambda _m = lambda _d/3) and (a_{0,m}=0.2). The results presented in Figs. 2 and 3 correspond to case (d).

Figure 4

Spectrograms of the betatron radiation emitted by the electrons. Temporal and energy profiles are shown for a reference case without a modulator and for three different modulator pulse cases. The signal close to (t = 0) corresponds to the front of the bunch and arrives first at the detector. The inset in the panel (d) shows the electron energy distribution within the bunch. It displays a matrix of the average electron energy in cell; only the cells with average energy over 10 MeV are shown. Both temporal and energy profile of emitted X-rays are correlated with the inner structure of the bunch. Note that the x-axis is reversed.

All the signals are approximately 10 fs long, corresponding to a bunch length of (approx) 3.5 (upmu)m shown in Fig. 3. Nevertheless, while the signal is continuous in the case without the modulator (Fig. 4a), the modulated signals (Fig. 4b–d) exhibit trains of ultrashort pulses. Moreover, the spectrograms show that the betatron radiation critical energy is also modulated in time. In average, the energy of radiation is considerably higher when the modulator is present. The inset in panel (d) confirms the correlation between the energy distribution of electrons within the bunch and the temporal and energy profile of emitted X-rays.

Figure 5 shows the temporal profiles of betatron radiation. Whereas the blue curve belonging to reference case (a) does not vary significantly, the other three curves (b–d) show several clear peaks. The red curve represents the case (b); three dominant peaks are present. The peak-to-peak distances is between the first and the second and the second and the third dominant peaks are 1.35 fs and 1.29 fs, respectively. This is in good agreement with the theoretically expected value (lambda _m/2c=1.overline{3}) fs. The green curve corresponds to case (c). The signal comprises of more than thirteen clear peaks. The peak-to-peak distance is (0.46 ± 0.02) fs (estimated by Fourier transform of signal) and is in good agreement with the expected value of (lambda _m/2c=0.overline{4}) fs. Such a feature can be interpreted as a betatron radiation pulse train coherence with respect to the modulation pulse.

The radiation peaks themselves are even shorter, the FWHM of the brightest one at 2.65 fs is 140 as. There is a considerable continuous background, the pulsed signal to noise ratio is about 5:1. This ratio could be significantly improved by employing a transmission filter which effectively cuts the low energy parts of the spectra.

The inset of Fig. 5a contains the last case (d). The signal is an order of a magnitude more intense than the other cases. It is bunched, with a signal-to-noise ratio of better than 20:1. Again, Fourier transform of this signal shows that the fundamental period is (0.45 ± 0.01) fs, and the FWHM of the brightest peak at 1.92 fs is 100 as.

Figure 5

Energy spectra for the emitted betatron radiation. (a) Temporal profile of the betatron radiation for a reference case without a modulator and for three different modulator pulse cases. The inset, corresponding to the case (lambda _m = lambda _d/3) and (a_{0,m}=0.2) is an order of magnitude more intense than the other cases. (b) On-axis time-integrated energy spectra of emitted X-rays and the critical energy of the emitted signal for the four cases (a) no modulator, (b) (lambda _m = lambda _d) and (a_{0,m}=0.6), (c) (lambda _m = lambda _d/3) and (a_{0,m}=0.1), and (d) (lambda _m = lambda _d/3) and (a_{0,m}=0.2.)

The number of electrons within the bunch differs by less than 3.5% between all four compared cases.The estimated total energy within the pulse train is 0.10 nJ in case (a). It increases greatly when the modulator in present: it is 0.45 nJ, 0.65 nJ, and 2.2 nJ in cases (b–d), respectively. The increase is caused partly by the higher energy of the electrons and partly by the higher amplitude of betatron oscillations.

Finally, the time-integrated energy spectra on axis for all the cases (a–d) are shown in Fig. 5b, including information about the critical energy of the emitted signal in all cases. The critical energy of the case (d) is 5.3(times) higher than in the reference case (a).

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