1. Introduction

Tunable X-ray sources from synchrotron radiation facilities have been widely used in material science [1], microtomography [2] and macro-dynamics exploration [3]. However, the number of such huge synchrotron radiation facilities is limited, and they cannot satisfy the increasing demands. In recent years, due to the tremendous acceleration gradient, a table-top betatron radiation source from a laser wakefield accelerator has shown great potential to be the next generation of compact radiation sources [4–8]. In laser wakefield acceleration (LWFA), an ultra-short, ultra-intense laser pulse excites an intense plasma wake that carries the longitudinal acceleration field within the intensity of GV/cm [9–11]. The electrons injected in the wake bucket can be accelerated to high energies with millimeter or centimeter distance. To improve the quality and energy of the accelerated electron beam, many efforts have been performed, such as using ionization injection method [12–18] and multistage coupling acceleration [19–22]. Besides the longitudinal acceleration, the wake can also provide transverse force on the electrons. Off-axis moving electrons can make transverse betatron oscillation, which leads to short wavelength radiations typically in the ultraviolet and X-ray regime [4,8,23].

As is known to all, the radiation intensity, photon energy and radiation brightness are important factors for usual applications. Alternative schemes have been proposed to enhance these factors by optimizing the plasma density, electron energy and amplitude of electron oscillations. For example, by tailoring the plasma profile, the betatron transverse amplitude can be effectively controlled to increase the radiation intensity [24–26], or by using two successive pulses, the betatron oscillations of the accelerated electrons are driven dominantly by the modulation pulse [27,28]. In addition to these parameters, the distributions of spatial profile and polarization are also important for many applications, such as probing of complex structure of molecules [29] and element-specific magnetic properties [30]. Since both the radiation intensity and polarization are closely linked to the trajectories of the accelerated electrons [31], the betatron X-ray radiation can be used to diagnose the dynamics of the accelerated electrons [32,33] and to be a method to analyze the accelerated electrons. On the contrary, to control the polarization of the betatron radiation source, different methods have been proposed to control the electron dynamics, such as using tilted laser pulse front to achieve asymmetric electron injection [34], using composite Laguerre Gaussian pulses to drive a rotating wakefields [35], or using plasma channel to guide the laser and electron propagation [36–40]. However, the experimental operation and tunability of these schemes are still quite a challenge. Recently, by using ionization-induced electron injection and rotating the laser polarization direction, Döpp et al. has successfully obtained a highly stable and tunable X-ray radiation source in experiment [41]. Feng et al. has recently achieved intense circularly polarized X-ray by using a long duration circularly polarized laser ionization injection [42].

In this paper, by using off-axis ionization injection we propose a more flexible scheme to control the electron dynamics and betatron radiation. In our scheme, a drive laser with long wavelength excites a strong plasma wave and an off-axis injection laser is used to control and adjust the trajectories of accelerated electrons. Simulations show that the radiation spatial distribution, polarization and brightness are controllable. In a typical simulation, when the injection laser offset distance matches the electrons’ residual ionization momenta, the electrons can make helix oscillations in the wake, an annular intensity distribution and circularly polarized betatron radiation with peak brilliance of $1.06 \times {10^{19}}\textrm{photon}/\textrm{s}/\textrm{m}{\textrm{m}^2}/\textrm{mra}{\textrm{d}^2}/0.1\%$ bandwidth can be achieved.

2. Electron injection and dynamics in LWFA

To investigate off-axis ionization injection in LWFA, we carried out a series of three-dimensional (3D) particle-in-cell (PIC) simulations with the OSIRIS code [43]. The schematic view of our proposal is shown in Fig. 1(a). A linearly polarized drive laser with polarization along the z-direction and wavelength of ${\lambda _1}\textrm{ = }2\mathrm{\mu}\textrm{m}$ propagates along x-direction. In order to effectively control the trajectories of the injected electrons, a plasma channel is used to guide the propagation of the long wavelength drive laser and generate a stable wakefield, and the length of the plasma channel is 7mm. The transverse density of the plasma channel satisfies $n(r )= {n_0} + \Delta n{r^2}/r_0^2$, where r is the transverse coordinate, ${n_0}$ is the on-axis electron density, $\Delta n$ is the channel depth and ${r_0}$ is the channel width. We use a matched channel density and width for the driver laser ${n_0}\textrm{ = }\Delta n = 5 \times {10^{17}}\textrm{c}{\textrm{m}^{ - 3}}$, ${r_0}\textrm{ = }{w_1} = 15\mathrm{\mu}\textrm{m}$, where ${w_1}$ is spot radius of the drive laser. The normalized vector potential of the drive laser is ${a_1}\textrm{ = }1.7$, which corresponds to the peak intensity of ${I_1}\textrm{ = }1 \times {10^{18}}\textrm{W/c}{\textrm{m}^2}$. The longitudinal Full-Width-at-Half-Maximum (FWHM) pulse length is ${L_1}\textrm{ = }15\mathrm{\mu}\textrm{m}$. A linearly polarized injection laser with polarization along the z-direction and wavelength of ${\lambda _2}\textrm{ = }0.4\mathrm{\mu}\textrm{m}$ also propagates along the x-direction. The vector potential of the injection laser is ${a_2}\textrm{ = }1.5$, which corresponds to a peak intensity of ${I_2}\textrm{ = }2 \times {10^{19}}\textrm{W/c}{\textrm{m}^2}$. The spot radius and the longitudinal FWHM length are ${w_1}\textrm{ = }4\mathrm{\mu}\textrm{m}$ and ${L_2}\textrm{ = }4\mathrm{\mu}\textrm{m}$, respectively. There are $46\textrm{fs}$ temporal delay and $6\mathrm{\mu}\textrm{m}$ offset (along y-direction) between the two pulses. A moving window is used to run the simulation and ADK tunneling ionization module is used for ionization calculation [44]. The background plasma is made of ionized mixture gas composed of $99{\%}$ Hydrogen and $1{\%}$ Nitrogen. The simulation box size is $80\mathrm{\mu}\textrm{m} \times 80\mathrm{\mu}\textrm{m} \times 80\mathrm{\mu}\textrm{m}$ and is divided into $1600 \times 160 \times 160$ cells with 2 macro-particles per cell for Hydrogen and 8 macro-particles per cell for Nitrogen.

 

Fig. 1. (a) Spatial distributions of the plasma wake (green color iso-surfaces), the injected electron beam (red color points), the electric fields of the drive and injection lasers (blue-red-orange-green iso-surfaces) at $t = 825\textrm{fs}$. (b) Momentum distribution of the accelerated electrons at the ionization time of $t = 132\textrm{fs}$.

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Typical 3D snapshots of the wake structure, the injected electron beam and the two laser pulses at $t = 825\textrm{fs}$ are shown in Fig. 1(a). Projections of x-y and x-z planes show the relative position of the two lasers and the accelerated electron beam. One can see that a long wavelength drive laser with large ponderomotive force and small peak electric field excites a strong plasma wave. Due to the long wavelength, the electric field of the laser is too weak to ionize k-shell electrons of Nitrogen [45,46]. The plasma channel provides a stable propagation for the drive laser and a stable bubble is excited. The offset injected laser with short wavelength is behind the drive laser and it can ionize electrons at off-axis positions. Those ionized electrons are trapped in the first bubble. Due to the tight focus, the injection laser defocuses rapidly. The effective injection length only lasts for about $100\mathrm{\mu}\textrm{m}$. The initial momenta distribution of the ionized electrons at $t = 132\textrm{fs}$ is shown in Fig. 1(b). The momenta of the ionized electrons are initially parallel to the polarization direction of the injection laser, and they are divided into two beams with opposite rotation directions. The initial transverse momenta of the ionized electrons are ${p_ \bot } = \sqrt {1 + a_ \bot ^2({{\psi_i}} )} $ [14], where ${\psi _i}$ represent the laser phase at the position where the electrons are ionized, ${a_ \bot }({{\psi_i}} )$ is the transverse vector potential of the laser at the phase of ${\psi _i}$. Therefore, if one gives certain offset to the injected electrons, a non-zero orbital angular momentum resulting from the residual ionization momenta is obtained, which results in electrons’ helix oscillations. In the transverse direction, the electron motion shows an oscillation with betatron frequency ${\omega _\beta }$ and an amplitude of ${r_\beta }(t )$, where ${r_\beta }(t )\textrm{ = }\sqrt {z_\beta ^2(t )+ y_\beta ^2(t )} $, ${z_\beta }(t )$ and ${y_\beta }(t )$ are the amplitudes in z and y-direction, respectively. The transverse oscillation period is dominated by the betatron frequency as ${\omega _\beta }\textrm{ = }{\omega _p}/\sqrt {2\gamma } $, where $\gamma $ is Lorentz factor of the accelerated electrons, and ${\omega _p}$ is the plasma frequency. Therefore, the betatron wavelength is ${\lambda _\beta }\textrm{ = }{\lambda _p}\sqrt {2\gamma } $. In the simulation, the maximum wavelength of betatron oscillations is about ${\lambda _\beta }\textrm{ = }1400\mathrm{\mu}\textrm{m}$, which is consistent with the simulation result of the maximum rotation period of 1450μm.

To describe the dynamics of the two electron beams with opposite angular momenta, the collective and single electron behaviors of the accelerated electrons are shown in Fig. 2. The dynamics of electrons in the case where the injection is on-axis propagating are also given for comparison. Figure 2(a) shows the energy spectra of the accelerated electrons at t=23ps, and the inset in (a) shows the injected electron charge as a function of offset distance. One can see that a quasi-monoenergetic electron beam with peak energy of 234 MeV and energy spread of 13.7% can be obtained for the case with offset of 6μm, which is slightly higher than that of the case without offset (peak energy of 213MeV and energy spread 13.1%). This is because that the larger offset distance makes the injected electrons fall to the positions with deeper wake potential and stronger wakefield, and they are accelerated by a stronger longitudinal accelerating electric field. Furthermore, the injected electron charge increases with the increase of offset distance (1.4pC for the case without offset and 1.6pC for the case with offset of 6μm), but the difference is small. Figures 2(b) and 2(c) show the evolution of the center of the beam defined by the average of the absolute value of their transverse positions and typical electron trajectories for different offsets of the injection laser. One can see that the electrons almost oscillate exclusively along the z-direction for the case without offset. The oscillation direction is the same as the polarization direction of the injection laser. For the case with offset of $6\mathrm{\mu}\textrm{m}$, the electrons also have an oscillating component in the y-direction due to the initial injection offset. The electron beams can make collectively helix oscillations in the first bubble.

 

Fig. 2. (a) Energy spectra of the accelerated electrons at t=23ps. The inset in (a) shows the injected electron charge as a function of offset. (b) The evolution of the center of the beam defined by the average of the absolute value of their transverse positions. (c) Typical electron trajectories for different offsets of the injection laser. The projections of the trajectories are also plotted. (d) Evolutions of typical electrons’ orbital angular momenta.

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The orbital angular momenta (OAM) of typical electrons for different offsets are shown in Fig. 2(d). The angular momentum is conserved with a steady state approximation [32], so that the OAM of the accelerated electrons can be given by $L(t )= y(t ){p_z}(t )- z(t ){p_y}(t )\sim {y_0}{p_{{z_0}}} - {z_0}{p_{{y_0}}}$, where ${y_0}$, ${z_0}$ and ${p_{{y_0}}}$, ${p_{{z_0}}}$ represent the initial transverse coordinates and transverse momenta of the ionization injected electron. The electron shows a two-dimensional transverse oscillation for the case of $L = 0$, which corresponds to the black line in Fig. 2(d). For the case of $L \ne 0$, a three-dimensional helix oscillation appears.

3. Far-field radiation pattern and polarization control

We studied the far field radiation pattern and polarization distribution by using a post-processing code Virtual Detector for Synchrotron Radiation (VDSR) [47]. In the simulation, we randomly selected 200 super-particles from the PIC simulations and traced their trajectories. The temporal interval of each trajectory tracing is dt=0.0165fs. The radiation detection region covers a maximum polar angle of ${2^ \circ }$, which is divided into 20 parts in the radial direction $\mathrm{\theta }$ and 180 parts in the azimuthal direction $\phi $.

The radiation intensity distributions for different injection laser offsets are compared in Figs. 3(a) and 3(b). For the case without offset, the radiation intensity profile is consistent with the injected electron trajectory (black line in Fig. 2(d)). The radiation is mainly along the z-direction, and the maximum radiation intensity is located at azimuthal angle of $\phi \textrm{ = }{180^ \circ }$ and polar angle of $\theta \textrm{ = 0}\textrm{.6}{\textrm{3}^ \circ }$. However, the radiation intensity shows annular profile for the case with injection laser offset of $6\mathrm{\mu}\textrm{m}$. The key parameter to describe the betatron radiation is the strength parameter $K = {r_\beta }{k_p}\sqrt {{\gamma / 2}} $, where ${k_p}\textrm{ = }{\omega _p}/c$ is the plasma wave number, c is the speed of light in vacuum. In the case with offset of $6\mathrm{\mu}\textrm{m}$, the average energy of the accelerated electrons is $\gamma \textrm{ = }458$, and the minimum oscillation amplitudes in the transverse directions are ${y_{\beta \min }} \approx 1\mathrm{\mu}\textrm{m}$ and ${z_{\beta \min }} \approx 1.5\mathrm{\mu}\textrm{m}$. Therefore, the strength parameters are ${K_{y\min }} \approx 2$ (in the y-direction) and ${K_{z\min }} \approx 3$ (in the z-direction), respectively. As a result, most of the radiation propagates along the characteristic angles ${\theta _{y\min }} \sim {{{K_{y\min }}} / \gamma } \sim {0.25^ \circ }$ and ${\theta _{z\min }} \sim {{{K_{z\min }}} / \gamma } \sim {0.37^ \circ }$. The smaller radiation angular width around the emission angle of ${\theta _t} = {1 / \gamma } \sim {0.12^ \circ }$ leads to the annular intensity distribution, as shown in Fig. 3(b). One can also see that the region with higher radiation intensity is within the angle range of ${135^ \circ } < \phi < {225^ \circ }$. This is because the two electron beams with opposite angular momenta intersect around that region, and the curvature radius there is minimum. Meanwhile, electrons trajectories of the two beams end in that azimuth angle range, which corresponds to the red line in Fig. 2(c), and the electrons with higher energy can radiate more photons.

 

Fig. 3. Distributions of X-ray intensity and polarization for different injection offsets. (a) and (c) correspond to the case without offset. (b) and (d) correspond to the case with offset of $6\mathrm{\mu}\textrm{m}$. The radiation spectra of the accelerated electrons for the case with offset at azimuthal angle $\phi \textrm{ = }{180^ \circ }$ and polar angle $\theta \textrm{ = }{0.32^ \circ }$ (e), azimuthal angle $\phi \textrm{ = }{90^ \circ }$ and polar angle $\theta \textrm{ = }{0.52^ \circ }$ (f).

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The polarization properties of the betatron radiation for different injection laser offsets are shown in Fig. 3(c) and Fig. 3(d). Firstly, the polarization direction degree is defined as ${P_d} = {{{I_z}} / {{I_t}}}$, where ${I_z}$ is the radiation intensity in the z-direction and ${I_t}$ is the total radiation intensity. For the case without offset, the radiation is almost linearly polarized along the z-direction. However, the polarization direction degree changes periodically for the case with offset of due to the variation of the electron’s transverse velocity and acceleration directions. A circularly polarized X-ray source can be achieved when the transverse trajectories are helix like [29]. The radiation spectra at two different directions for the case with offset of are shown in Fig. 3(e) (azimuthal angle $\phi \textrm{ = }{180^ \circ }$ and polar angle $\theta \textrm{ = }{0.32^ \circ }$) and Fig. 3(f) (azimuthal angle $\phi \textrm{ = }{90^ \circ }$ and polar angle $\theta \textrm{ = }{0.52^ \circ }$). The radiation intensity in the z-direction is stronger for ${P_d} > 0.5$ (blue line in Fig. 3(e)), and it is stronger in the y-direction for ${P_d} < 0.5$ (red line in Fig. 3(f)).

For radiation source, besides the distributions of radiation intensity and polarization, the photon energy and radiation brightness are also key factors for applications. At the crossing time of the two separated electron beams, the transverse area of a helix motion is about $4.74\mathrm{\mu}{\textrm{m}^2}$, the electron beam length is about $4\mathrm{\mu}\textrm{m}$ and the accelerated beam charge is 1.6pC. Therefore, the peak radiation brightness of the annular radiation is about $1.06 \times {10^{19}}\textrm{photon}/\textrm{s}/\textrm{m}{\textrm{m}^2}/\textrm{mra}{\textrm{d}^2}/0.1\%$ bandwidth, and the peak position of the radiation spectrum is approximately $378\textrm{eV}$, as shown in Fig. 3(e). To verify that our scheme can produce high energy radiation, we also make an accurate two-dimension simulation with offset of 10μm and the polarization direction of the injection laser along the y-direction instead of the 3D simulation due to the limitation of the computational resource. Meanwhile, we increase the intensity of the injection (a₂=1.9) and drive (a₁=1.9) lasers and the length of plasma channel (8mm). Figure 4(a) shows the average energy of the accelerated electrons as a function of the acceleration distance. We can see that the electrons can be accelerated to higher energies, and the maximum average energy is 360MeV at the acceleration distance of 8mm. Figure 4(b) shows the radiation spectra of the accelerated electrons for 2D case, higher energies resulting in higher photon energies. It is shown that a peak photon energy of ∼2.6keV can be achieved. In addition, the incoherent radiation intensity is proportional to the beam charge. The charge in the current 3D simulation is very small, but it can be easily increased. It can be increased by increasing the ionization injection length, tthe concentration of the high Z gas and the intensity of the injection laser. It means that by varying the laser plasma parameters one can tune the peak photon energy, intensity and polarization of the X-ray radiation.

 

Fig. 4. 2D-PIC simulation results. (a) The average energy of the accelerated electrons as a function of the acceleration distance. (b) The radiation spectra of the accelerated electrons at azimuthal angle $\phi \textrm{ = }{0^ \circ }$ and polar angle $\theta \textrm{ = }{0.42^ \circ }$.

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In this scheme, besides adjusting the offset distance, the radiation intensity and polarization distributions can also be tuned by varying the polarization of the injection laser. Figure 5(a) shows the radiation intensity distribution for a fixed offset of $6\mathrm{\mu}\textrm{m}$, and the polarization direction of the injection laser is left-handed rotated with rotation angle of ${45^ \circ }$. One can see that the radiation intensity shows an elliptical shape, which shows similar pattern as the distributions of the electrons’ transverse trajectories. The maximum radiation intensity is located at azimuthal angle $\phi \textrm{ = }{180^ \circ }$ and polar angle $\theta \textrm{ = }{0.32^ \circ }$, and the polarization distribution changes accordingly (shown in Fig. 5(b)). Figures 5(c) and 5(d) show the transverse momenta and trajectories distributions of typical electrons for different polarization directions of the injection laser. One can see that the radiation intensity distribution corresponds exactly to the transverse momenta and transverse trajectories of the electrons (red lines). Meanwhile, the transverse momenta increase and the oscillation amplitude of the betatron motion decreases. We also show the case with the polarization direction of the injection laser along the y-direction as a comparison. The radiation intensity distribution is similar to the case without offset (Fig. 3(a)), and the radiation is linearly polarized along the y-direction (not shown here).

 

Fig. 5. Radiation intensity (a) and polarization (b) distributions for the case with offset of $6\mathrm{\mu}\textrm{m}$, where the polarization direction of the injection laser is left-handed rotated with rotation angle of ${45^ \circ }$ along the propagation direction. Transverse momenta (c) and transverse trajectories (d) of typical electrons for different polarization directions.

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4. Conclusion

In summary, we have studied the X-ray radiation based on laser wakefield acceleration using off-axis ionization injection. By controlling the injection laser’s offset distance and polarization direction, the dynamics of the accelerated electron beam can be adjusted, leading to a high tunability on the distributions of the radiation intensity and polarization. When an appropriate injection laser parameter is used, the injected electrons make collective spatial helix oscillation, and an annular intensity distribution and circularly polarized radiation with peak brilliance of $1.06 \times {10^{19}}\textrm{photon}/\textrm{s}/\textrm{m}{\textrm{m}^2}/\textrm{mra}{\textrm{d}^2}/0.1\%$ bandwidth can be obtained. Such a compact controlled X-ray source will provide future applications in the study of complex structure probes.