Femtosecond x rays from laser-plasma accelerators

Relativistic interaction of short-pulse lasers with underdense plasmas has recently led to the emergence of a novel generation of femtosecond x-ray sources. Based on radiation from electrons accelerated in plasma, these sources have the common properties to be compact and to deliver collimated, incoherent, and femtosecond radiation. In this article, within a unified formalism, the betatron radiation of trapped and accelerated electrons in the so-called bubble regime, the synchrotron radiation of laser-accelerated electrons in usual meter-scale undulators, the nonlinear Thomson scattering from relativistic electrons oscillating in an intense laser field, and the Thomson backscattered radiation of a laser beam by laser-accelerated electrons are reviewed. The underlying physics is presented using ideal models, the relevant parameters are defined, and analytical expressions providing the features of the sources are given. Numerical simulations and a summary of recent experimental results on the different mechanisms are also presented. Each section ends with the foreseen development of each scheme. Finally, one of the most promising applications of laser-plasma accelerators is discussed: the realization of a compact free-electron laser in the x-ray range of the spectrum. In the conclusion, the relevant parameters characterizing each sources are summarized. Considering typical laser-plasma interaction parameters obtained with currently available lasers, examples of the source features are given. The sources are then compared to each other in order to define their field of applications.

Figure 1

Illustration of the undulator and wiggler limits, at the top and the bottom, respectively. The lobes represent the direction of the instantaneously emitted radiation. $\psi$ is the maximum angle between the electron velocity and the propagation axis ${\overrightarrow{e}}_z$ and $\Delta \theta$ is the opening angle of the radiation cone. When $\psi \ll \Delta \theta$ (undulator), the electron always radiates in the same direction along the trajectory, whereas when $\psi \gg \Delta \theta$ (wiggler), the electron radiates toward different directions in each portion of the trajectory.

Figure 2

Schematic for the calculation of the spatial period $\lambda$ of the radiation emitted toward the direction of observation $\overrightarrow{n}$, forming an angle $\theta$ with the ${\overrightarrow{e}}_z$ direction. At the two positions marked by colored points, the electron radiates the same field amplitude. The distance between these two amplitudes at a given time corresponds to $\lambda$.

Figure 3

In the wiggler limit, the radiation cone points toward the observer during a time $\Delta t$, which corresponds to a duration $\tau$ for the emitted radiation. This is repeated at each period: the observer receives bursts of radiation separated by a time $\lambda /c$. The inset gives the temporal profile of the radiation power seen by the observer.

Figure 4

Principle of the electron acceleration in the bubble regime. The laser pulse expels all plasma electrons out of the focal spot, leaving in its wake an ion cavity of radius $r_b$. The longitudinal force $F_z$ accelerates the self-injected electron bunch in the first half of the bubble, while the bunch decelerates in the second half of the bubble.

Figure 5

Schematic of the betatron mechanism. When an electron is injected in the ion cavity, it is submitted not only to the accelerating force, but also to a restoring transverse force, resulting in its wiggling around the propagation axis. Because of this motion, the electron radiates x rays whose typical divergence is $\theta =K/\gamma$.

Figure 6

Electron trajectory on the left and its gamma factor as a function of the longitudinal coordinate $z$ on the right. The set of initial conditions is $x_i=2\mu \mathrm{m}$, $y_i=0$, $z_i={\zeta }_i=-r_b+0.3\mu \mathrm{m}=-6.42\mu \mathrm{m}$, $p_{xi}=0$, $p_{yi}=0$, and $p_{zi}=25mc$ and the laser-plasma parameters are $n_e=1\times {10}^{19}{\mathrm{cm}}^{-3}$ and $a_0=4$.

Figure 7

X-ray angular distribution calculated from the trajectory of Fig. 6. The color scale represents the radiated energy per unit solid angle.

Figure 8

Transverse trajectory (on the left) and angular profile of the corresponding emitted radiation (on the right, the color scale representing the radiated energy per unit solid angle) for two different cases. A: The trajectory is three dimensional and helical (elliptical in the transverse plane). B: The trajectory is planar. Both electrons are accelerated up to $\gamma \simeq 380$ and the density is $2\times {10}^{19}{\mathrm{cm}}^{-3}$.

Figure 9

Betatron radiation spectrum, integrated over angles and corresponding to the trajectory of Fig. 6, in number of photons per 0.1% bandwidth and per electron.

Figure 10

Electronic density in the $z\mathrm{\text{−}}y$ plane averaged over $5\mu \mathrm{m}$ in the $x$ direction. Densities are given in ${\mathrm{cm}}^{-3}$. $z$ is the distance from the beginning of the first density ramp. Different steps of the simulation are shown. First the bubble forms, then injection starts, a whole bunch of accelerated electrons is created inside the bubble, and finally the bunch comes out of the bubble and is scattered. The dynamics of the bunch is such that the trailing electrons have a much higher oscillation amplitude than leading electrons.

Figure 11

Normalized electron beam spectrum before scattering at $z\simeq 1800\mu \mathrm{m}$.

Figure 12

Trajectory, energy, and radiated power of a single electron before scattering. The tube shows the particle’s trajectory. Its radius is proportional to the particle’s energy and the color scale measures the instantaneous radiated power of the particle. The orbit of this particle is in the polarization plane.

Figure 13

Projection in the $z\mathrm{\text{−}}x$ plane of trajectories and energies of 500 particles from injection until scattering. The laser is polarized in the ${\overrightarrow{e}}_x$ direction and modulates the particle trajectories. Note the different scales in $x$ and $z$ axes.

Figure 14

Trajectories and radiated power of the same 500 particles as Fig. 13. The laser is polarized in the ${\overrightarrow{e}}_x$ direction.

Figure 15

Angular profile of the emitted radiation.

Figure 16

Spectrum of the x-ray radiation emitted by all injected macroparticles.

Figure 17

Betatron x-ray beam (round spot), produced by the 28.5 GeV SLAC electron beam propagating through a plasma of density $\sim 2\times {10}^{13}{\mathrm{cm}}^{-3}$, and recorded by a fluorescent screen imaged onto a visible CCD camera. From 277.

Figure 18

Experimental setup for the generation and observation of betatron radiation. A 50 TW/30 fs class laser is focused onto the front edge of a gas jet. Electron spectra are obtained by deflecting electrons with a magnet and using a LANEX phosphor screen imaged onto a CCD. Betatron radiation is recorded on axis using a charged coupled device sensitive up to 10 keV (CCD X). A filter (Be, Al, etc.) is placed before the CCD to block infrared light.

Figure 19

Left: A raw image of the angular distribution of the betatron radiation beyond 3 keV recorded by the CCD in a single shot. Right, divergence (full width at half maximum) of radiation as a function of density. The dotted line shows the dependence of $\theta$ on $n_e$ for constant $r_{\beta }$ and $\gamma$ according to the equations of Secs. 4a, 4b.

Figure 20

X-ray signal (proportional to radiated energy beyond 1 keV per unit solid angle) as a function of plasma electron density. Each point corresponds to an average value over ten shots. The dotted line corresponds to the results obtained using 3D PIC simulations. From 219.

Figure 21

Raw electron spectra for three different densities and fixed laser parameters. Horizontal axis, electron energy; vertical axis, exit angle; and color scale, number of counts. This latter gives an indication of the beam charge. No electron is observed below the trapping threshold ($n_e=7.5\times {10}^{18}{\mathrm{cm}}^{-3}$), a quasimonoenergetic electron beam is produced just above ($n_e=1.25\times {10}^{19}{\mathrm{cm}}^{-3}$), and a broadband spectrum is recorded for a higher density ($n_e=1.5\times {10}^{19}{\mathrm{cm}}^{-3}$). The threshold density is not the same as in Fig. 20 because the laser parameters are different.

Figure 22

Top: Total number of photons obtained experimentally within the spectral bandwidths determined by filters: $25\mu \mathrm{m}$ Be ($1<E<10\mathrm{keV}$), $25\mu \mathrm{m}\mathrm{Be}+40\mu \mathrm{m}\mathrm{Al}$ ($4<E<10\mathrm{keV}$), and $25\mu \mathrm{m}\mathrm{Be}+25\mu \mathrm{m}\mathrm{Cu}$ ($6<E<10\mathrm{keV}$). From 219. Bottom: Averaged spectrum of the radiation obtained by photon counting, with a fit (in green line) by the function $S(\omega /{\omega }_c)$ with $\hslash {\omega }_c=5.6\pm 1\mathrm{keV}$. From 83.

Figure 23

Synchrotron spectrum fitted from transmission values through six different filters. The fit uses the synchrotron function $(\omega /{\omega }_c{)}^2{K}_{2/3}^2(\omega /{\omega }_c)$ instead of $S(\omega /{\omega }_c)$, and the fitted critical energy is $29\pm 13\mathrm{keV}$. From 131.

Figure 24

Bee imaged with the betatron x-ray beam with an edge line out indicated by the rectangular area: (a) one laser shot; (b) 13 laser shots. (c) Line out of the images taken at the indicated rectangular area. From 82.

Figure 25

Schematic of the undulator source. A laser-plasma accelerated electron beam is injected in a conventional undulator. Electrons oscillate transversally and emit x rays due to this motion.

Figure 26

Electron trajectories. The orbit is plotted as a solid line for the undulator case ($K=0.2$) and dashed line for the wiggler case ($K=2$). At the entrance, the electron has $\gamma =1000$ and $\overrightarrow{p}=\sqrt{{\gamma }^2-1}{\overrightarrow{e}}_z$ in both situations. The undulator or wiggler is ten periods long.

Figure 27

On the left is represented the angular distributions of the radiation (radiated energy per unit solid angle) and the corresponding spectra are shown on the right (in number of photons per 0.1% bandwidth, per electron, and per shot). The undulator case, $K=0.2$, is on the top whereas the wiggler case, $K=2$, is on the bottom. In the first situation, three different solid angles of integration have been considered to highlight the stretching of the spectrum. In the wiggler situation, the integration is performed over the total angular distribution.

Figure 28

Schematic of the nonlinear Thomson scattering source. A free electron is submitted to a relativistic laser pulse ($a_0\gg 1$). It is simultaneously accelerated and wiggled, and emits XUV radiation due to this motion. This illustrates the motion of an electron in the case of a linearly polarized laser pulse.

Figure 29

Top: The electron orbit in an intense linearly polarized laser pulse with $a_0=5$ (solid line) and $a_0=10$ (dashed line). Bottom: The $\gamma$ factor as a function of the longitudinal position for each case. The electron is initially at rest and is submitted only to the linearly polarized laser pulse whose temporal profile is Gaussian with 20 fs duration (FWHM).

Figure 30

Spatial distribution of the nonlinear Thomson scattering, corresponding to trajectories of Fig. 29. The radiated energy per unit surface in the plane situated at 1 m from the source and perpendicular to the propagation axis is represented. On the top is the case $a_0=5$ and on the bottom the case $a_0=10$.

Figure 31

Nonlinear Thomson scattering spectrum, integrated over angles and corresponding to the trajectories of Fig. 29. The number of photons per 0.1% bandwidth for one electron is displayed for the two cases: $a_0=5$ in the solid line and $a_0=10$ in the dashed line. The spectrum near the maximum angle of emission is represented in the inset for the case $a_0=10$.

Figure 32

Experimental setup for the nonlinear Thomson scattering experiment. Radiation is produced from the interaction of a laser pulse with $a_0\sim 5$ and a helium gas jet. Radiation is collected using a grazing incidence spherical mirror and spectrally analyzed using a transmission grating. Filters are used to block the infrared light and to select a specific spectral range. To scan the angular distribution of the radiation, the whole system is mounted on a rotation stage centered on the gas jet.

Figure 33

On the left is represented the angular dependence of the radiated energy recorded experimentally by rotating the whole detection system. The spectrum of the radiation on axis is given in logarithmic scales on the right of the figure and the inset shows the total number of photons integrated over several spectral bands determined by the choice of filters (the horizontal extension of the lines indicates the spectral band).

Figure 34

Schematic of the all-optically driven Thomson backscattering source. A laser-plasma accelerated electron beam is injected in a counterpropagating laser pulse. Electrons oscillate transversally and emit x rays or $\gamma$ rays due to this motion.

Figure 35

Electron trajectories. The orbit is plotted as the solid line for the undulator case ($a_0=0.2$) and as the dashed line for the wiggler case ($a_0=2$). Before the interaction, the electron has $\gamma =200$ and $\overrightarrow{p}=\sqrt{{\gamma }^2-1}{\overrightarrow{e}}_z$ in both situations. The counterpropagating laser pulse has a Gaussian temporal profile with FWHM duration of 20 fs and a central wavelength of 800 nm.

Figure 36

On the left are represented the angular distributions of the radiation (radiated energy per unit solid angle) and the corresponding spectra are shown on the right (in number of photons per 0.1% bandwidth, per electron, and per shot). The undulator case, $a_0=0.2$, is on the top whereas the wiggler case, $a_0=2$, is on the bottom. In the first situation, three different solid angles of integration have been considered to highlight the broadening of the spectrum. In the wiggler situation, the integration is performed over the total angular distribution.

Figure 37

Principle of the all-optical Thomson source based on the combination of a laser-plasma accelerator and a plasma mirror.

Figure 38

Spectrum of the Thomson backscattering radiation measured using a set of copper and aluminum filters. The inset is the x-ray beam angular profile measured using a phosphor screen.

Figure 39

Schematic of the FEL amplification process. A typical profile for the electric field amplitude $E_0$ along the undulator axis in linear scale (solid line, left $y$ axis) and logarithmic scale (dashed line, right $y$ axis).

Figure 40

Longitudinal profile of radiation power as a function of the position inside the undulator obtained from Genesis simulations. The corresponding electron beam and undulator parameters are indicated in Table . Top: X-FEL 1. Bottom: X-FEL 2. The position relative to the electron beam center is defined as $s=c{t}_e-ct$, where $t_e$ and $t$ are, respectively, the arrival time of the electron beam center and of the radiation, at a given position $z$ in the undulator.

Figure 41

Peak brightness for different types of x-ray sources: high harmonic generation from relativistic laser and overdense plasma interaction (triangle and line) [experiment conducted with the Vulcan Petawatt laser system at Rutherford Appleton Laboratory (RAL) by 63], $K\alpha$ radiation from laser-produced plasmas (star) [experiment conducted at the Laboratoire d’Optique Appliquée (LOA) by 215], nonlinear Thomson scattering (NTS) from relativistic laser and underdense plasma interaction (diamond and line) [experiment conducted at LOA by 249], conventional undulator radiation from laser-plasma accelerators (square) [experiment conducted with the ATLAS laser system at the Max-Planck-Institut für Quantenoptik (MPQ) by 88], betatron radiation from laser-plasma accelerators (pentagon and line) [experiment conducted with the Hercules laser system at the University of Michigan (U-Mich) by 131], Thomson backscattering (TB) radiation from laser-plasma accelerators (circle and line) [experiment conducted at LOA by 247], undulator radiation from conventional accelerators (dotted line) [data from the European Synchrotron Radiation Facility (ESRF); see http://www.esrf.eu/Accelerators/Performance/Brilliance], femto-slicing-based undulator radiation from conventional accelerators (dash-dotted line) [data from the femto-laser station at the Paul Scherrer Institut (PSI); see http://www.psi.ch/sls/microxas/femto and 18] and Thomson backscattering (TB) from conventional accelerators (dashed line) [experiment conducted at the Lawrence Livermore National Laboratory (LLNL) by 3]. The repetition rate and the pulse duration used in the peak brightness estimation are given in the legend, and permit one to convert peak brightness into average brightness.