Isolated attosecond electron and hard x-ray pulse generation by ultra-intense spatiotemporal vortex laser

The spatiotemporal optical vortex (STOV) laser pulse, characterized by containing a space-time dependent spiral phase structure and intrinsic transverse orbital angular momentum (OAM), has been of much recent research interest for basic physics as well as potential applications in optical communication and manipulation, as well as the attosecond sciences. Here we consider electron acceleration and generation of attosecond hard x-rays by irradiating a thin foil with intense STOV laser pulse. It is found that the affected foil electrons can be trapped and acquire transverse OAM, or synchrotron-like motion, from the STOV, as well as accelerated forward by the transverse and enhanced axial laser electric fields to form a tiny energetic bunch at the front of the laser pulse and emit short ( ∼400 attosecond) high-flux ( >109 photons per pulse) hard (100–1000 keV) x rays.

Since its discovery [1], the vortex laser with axial (or longitudinal, i.e. along the laser propagation direction) orbital angular momentum (OAM) has attracted much research interest in many areas, including high-resolution imaging, optical communication, particle manipulation, quantum information, as well as basic science.Intense vortex lasers with sufficiently high OAM density can exert torque on charged particles and transfer their OAM and energy to the latter.They can thus be useful for generating hollow and tunable-OAM electron beams and high-harmonics radiation, accelerating and focusing ions/positron beams in the bubble regime of laser wakefield acceleration (LWFA), etc [2][3][4][5][6][7][8][9][10][11][12][13][14].
The conventional optical vortex has OAM in the axial direction and locally time-independent optical phase winding in the transverse plane [1].In contrast, the more recently discovered laser pulse with spatiotemporal optical vortex (STOV) [15] has OAM transverse to the axial direction and space-time dependent tightly circulating optical phase [15][16][17][18][19][20][21][22].The STOV itself, which is usually a smoke-ring-like toroidal structure embedded in the waist of a short laser pulse, with space-time dependent light-wave phase and thus energy flux rotation around the toroid's minor circumference, was first observed in short self-focusing laser pulses propagating in air [16].Since then its properties and interaction with charged particles have been investigated [18][19][20][21][22][23].Of interest is that it appears no stringent conditions are required for their existence, and they can occur if the core-to-periphery radial intensity gradient of the laser is sufficiently large, as at the waist of a strongly focusing intense laser pulse.Thus, STOVs could have occurred in many earlier experimental investigations of laser self focusing in matter, but unnoticed since novel diagnostics techniques are needed for their detection [16].The method for producing STOV laser pulse is similar to that for the conventional optical vortex, which is achieved by using spiral phase plates and/or holographic diffraction grating, and then performing inverse Fourier transform to transform the dispersed frequencies of the vortex light into a STOV by using a cylindrical grating-lens pair.However, intense STOV pulse with ≳10 22 W cm −2 intensity is difficult to realize since such high-intensity light can destroy the transmission grating and cylindrical lens.On the other hand, with the rapid advances in laser technology, focused laser intensities ≳10 23 W cm −2 can be expected from the ambitious PW laser programs worldwide, such as those of ELI-NP, J-KAREN, SULF, ZEUS, SEL, ELI-4, etc [24][25][26].We can also expect that by using plasma as reflection media or stimulated Raman scattering [27][28][29], previously invoked for producing axial vortex light [27][28][29][30][31][32], intense STOV can be achieved in the future.
Compact x-ray sources have found applications in many fields.Several schemes for generating brilliant x-rays based on laser-driven energetic electrons have been proposed, including bremsstrahlung emission, betatron radiation in plasma wake or channel, Compton or Thomson scattering, etc [33][34][35][36][37].However, the usually poor quality of laser accelerated electrons restricts the quality of the x-ray radiation.To improve the latter, schemes such as photon energy amplification using undulators in LWFA-based free free-electron lasers [34].With density tailored plasmas, the efficiency of betatron radiation can also be significantly improved [35].High photon number and brightness can also be obtained with direct laser acceleration in carbon nanotube plasma (CNT) through Compton scattering, since the CNT plasma can easily expand to near-critical-density under the effect of the laser prepulse [37].However, it is still difficult to obtain isolated x-ray pulses of short space-time extent, as needed in many applications.
In this paper, using particle-in-cell (PIC) simulation with EPOCH, we investigate the interaction of STOV-containing ultrashort ultraintense laser pulse with ultrathin foil.It is found that, when the laser pressure force on the electrons can overcome the space-charge field created by the accelerated electrons, the laser can propagate through the foil, trapping and pushing at its front transverse-OAM carrying attosecond electron bunches of high density, high energy, and small volume that radiate intense short-pulse x-ray photons.
Our three-dimensional (3D) PIC simulations were carried out using EPOCH [38].The photon emission process is calculated with a Monte Carlo algorithm [38].The program includes radiation reaction of the electrons as well as feedback between the electrons and photons.A linearly polarized (in the y direction) STOV laser pulse propagating in the x direction impinges normally on the thin foil from the left.The laser pulse is of central wavelength λ = 1µm and period T 0 = λ/c, where c is the vacuum light speed.The initial profile of the STOV laser pulse consists of a series of spatiotemporal Hermite-Gaussian modes: where the HG mode u α;ξ m is given by where H m is the Hermite polynomial of order m, α = y, z, w α (x) = w α (0)[1 + (x/x 0α ) 2 ] 1/2 , w α (0) is the focal waist of the Gaussian laser carrier pulse of frequency ) Thus, we have where ξ = v g t − x can be considered as a local time coordinate and v g ≲ c is the pulse group velocity.The HG pulse in vacuum is stable.The corresponding spiral phase around the focal spot is l tan −1 (y/ξ), and it rotates l times in a 2π cycle.Thus, it can exert a spiral ponderomotive force on electrons, as well as accelerate them.
In the simulations, we set w α (0) = 4λ, w ξ (0) = 4T 0 , laser pulse duration 8T 0 , topological charge l = 1, and laser amplitude a 0 ranging from 70 to 200.The corresponding peak laser intensity (1.37 × 10 18 a 2 0 /λ 2 W cm −2 with λ in units of µm) ranges from 6.7 × 10 21 W cm −2 to 5.5 × 10 22 W cm −2 .The laser is focused at x = 5λ.The thin foil is of thickness l d = 0.1λ and uniform density n e = n c and 50n c , where n c = 1.1 × 10 21 cm −3 is the critical plasma density, and centered at x = 5λ.In Supplemental Materials we have also presented the results for targets of subcritical and gaseous densities, as well as solid density with preplasma.The simulation box is of size 20λ × 30λ × 30λ, with 1500 × 1000 × 1000 grids and 20 electrons and protons each per cell.
Figure 1 shows the laser-foil interaction process for a 0 = 100 and l d = 0.1λ.In this regime, a 0 > π n e l d /(λn c ) [39,40], the laser pressure force can overcome the electrostatic space-charge force of the nearly stationary ions, and expel the energetic electrons from the foil.In fact, as the laser pulse pass through the thin foil, it traps and accelerates most of the foil electrons in its path.In our simulation, the linearly  polarized STOV laser pulse with transverse OAM of l = 1, shown in the panel (a), is normally incident on the thin foil (not shown in this panel for clarity).In the x, y, 0 plane, the light wave electric field has a characteristic fork-like STOV phase singularity.As the focusing laser pulse penetrates though the thin foil, the foil electrons in its path are ponderomotively accelerated and compressed, forming a thin dense electron layer at the pulse front, as can be seen in the panel (b).The x, y, 0 plane projection at the bottom is for the energy density of the higher-energy accelerated electrons.During their interaction with the STOV in the laser pulse, the electrons can directly acquire transverse OAM from the STOV fields, leading to trajectories with transverse spiral components and thus additional focusing and acceleration.As a result, the highest-energy electrons in the front electron layer can emit hard x-ray photons (the purple bunch in the much enlarged inset) with high flux and ultrashort duration.
In order to better see the acceleration of electrons trapped in the rather complex space-time dependent phase of the STOV fields, it is instructive to first look at its electromagnetic field structure.Figure 2 is for the distributions of the l = 1 STOV laser fields and energy density of the accelerated electron bunch.One can see that in vacuum the axial laser field E x is only about 10% of the transverse field E y .Moreover, figure 2(a) (also in figure 1(a)) shows that E y has a phase singularity, around which the field distribution is tilted (resembling that in the 'attosecond lighthouse' [41]).The tilting field phase due to the STOV leads to a space-time phase asymmetry in the otherwise periodic laser wave fields, thereby allowing electrons trapped in the local wave potential well(s) to move faster than the local wave field and accumulate at the pulse front.As mentioned, these electrons can acquire transverse OAM from the STOV and spiral forward in the transverse and axial laser fields.It should be emphasized that only electrons that are suitably in phase with the STOV fields can be trapped and acquire transverse OAM for additional forward acceleration and rotation, as well as untrapped and re-trapped (in the next phase) in the process.The energetic electrons in the layer at the front of the laser pulse shown in figure 2(c) originate from a foil region x ∈ 5 ± 0.1λ, r ∼ 3λ (more precisely, y = −2.12λand z ∈ ±2λ; see also figure 3(a)) traversed by the toroidal STOV in the waist of the passing laser pulse.As a result, a series of attosecond electron bunches of high density (∼10n c ), high energy (∼800 MeV), and small radius (∼0.15λ) are generated, as can be seen in figure 2(c) at x ∼ 9.5λ.
Particle tracing is used to follow the motion of typical electrons that gain high energies.Figure 3(a) shows that these electrons, originating from the lower part (same y but different z) of the STOV-affected target area, are accelerated forward in the axial direction, as discussed above and can also be seen in figure 2. The energy gained by an electron from the different laser field components, namely W j = −e ´t 0 v j E j dt, where j = x, y, z, are shown in figure 3(b).One can see that besides by the ponderomotive force, electrons are also directly accelerated by the transverse fields as they acquire transverse OAM from the STOV.In fact, figure 3 shows that at the pulse front their momentum and energy are mainly in the transverse motion.
Figure 3(c) for the ratio of the transverse-to-axial momentum versus energy shows that the electrons can also gain transverse OAM as they gain energy from the STOV. Figure 3(d) shows that the electron energy from the axial motion saturates as the electrons become dephased from the axial field E x as they gain forward speed and move faster that the local light wave phase speed.More details can be found in Supplementary Materials (see figure S1).
The electrons are accelerated and deflected by the intense STOV fields, and hundreds-keV to MeV photons are emitted via synchrotron radiation, as shown in figure 4 for the space-time dependence of the energy densities of the energetic electrons and emitted photons in the x, y plane.Closer examination of the panels (a)-(c) shows that the electron bunch as a whole also rotates in the x, y plane.
Figure 5 is for the properties of the x-ray photons.Panel (a) shows that isolated attosecond hard x-ray pulses of high brilliance (up to 2.01 × 10 25 photons s −1 mm −2 mrad −2 per 0.1% BW at photon energy of 100 keV) are emitted.The mechanism for attosecond pulse generation here is somewhat similar to that of coherent synchrotron emission [42,43], except that here the radiation is from the tightly spiralling electrons in the STOV, rather than from the electrons sharply reflected by the space-charge fields at the foil walls.Panels (b) and (c) show that the corresponding photon number is ∼5.8 × 10 9 and the average duration of the x-ray pulses is ∼400 a.s.Neglecting quantum effects, from classical synchrotron emission theory [44][45][46] one can  show that our simulation results are consistent with these theoretical estimates of the dependence of the photon yield, synchrotron radiation power, and overall efficiency, respectively, on laser amplitude.It may be of interest to note that photons with similar (but much less energetic) properties can also be generated by STOV laser interaction with subcritical-density and gaseous targets (see supplementary materials).However, the latter has to be of sufficient length in order to have enough electrons for the STOV to trap and accelerate.
In summary, we have proposed a scheme for producing isolated ultrabright attosecond x-ray pulses from the interaction of an intense STOV laser with a thin foil in a regime where the laser pressure force on the accelerated electrons can overcome the electrostatic space-charge force created by their expulsion, or a 0 > π n e l d /(λn c ).The STOV with transverse OAM presents a new path for manipulating the electron dynamics in LPI.As a result, ultrashort isolated energetic transverse-OAM-containing electron bunches that efficiently radiate x-rays with similar properties are produced.Such electron and radiation pulses can be useful in applications requiring ultrahigh spatial and temporal resolution, such as for time-resolved probing and/or manipulation of nuclear processes with small cross sections.

Figure 1 .
Figure1.Schematic of the laser-foil interaction based on the 3D PIC simulation result.(a) Isosurfaces (red and green) of the directions of the linearly polarized STOV laser electric field and their (x, y, 0)-plane projection at t = 10T0, when most of the laser pulse has passed through the thin foil.Here the foil at x = 5λ is not shown for clarity.(b) Density isosurfaces (in greenich yellow shades) of the more energetic electrons, as well as the (x, y, 0)-plane projection at the bottom, showing the energy density, rainbow-colored from red (more energetic) to blue (less energetic, in arbitrary units) of the higher-energy electrons in the accelerated layer.The inset shows a density isosurface (purple) of the hard x-ray photons emitted by the accelerated electrons.In the interaction, the electrons in the laser's path are trapped, compressed, and accelerated forward by the ponderomotive force of the passing laser pulse into the target-backside vacuum.They also acquire transverse OAM from the embedded STOV and have spiral-like trajectories.They can thus move forward somewhat faster than the local laser fields and pile up at the pulse front and survive for several Rayleigh lengths.In contrast with the traditional LWFA blowout regime, the density and energy density distributions of the accelerated electrons are bi-centered, as well as space-time asymmetric due the STOV action.See figure2and text for more details.

Figure
Figure Field structure of the l = 1 STOV laser pulse and the corresponding electron energy density distribution at t = 10T0.Profiles of the (a) transverse (Ey) and (b) axial (Ex) electric fields (normalized by meωc/e).(c) Energy density distribution (the color bar is in units of nc MeV) of the strongly accelerated electrons originating from the foil region 4.9λ < x < 5.1λ, r ∼ 3λ, (more precisely, y = −2.12λand −2λ < z < 2λ, as can be seen in figure 3(a).)For comparison, also shown in (c) are the transverse and axial laser fields at y ∼ −2.12λ (or r = 3λ and θ = π, dashed line) obtained from the linear theory.Note that the averaged Ey is slightly positive and the averaged Ex is null before but becomes significantly positive after the STOV.

Figure 3 .
Figure 3. (a) Typical trajectories of electrons accelerated by the STOV laser pulse to high energies.The electrons originate from the foil, as described in the caption of figure 2. (b) Evolution of the energy gain Wx (red curves) in the axial direction, and Wy (blue curves) and Wz (green curves) of electron motion in different directions.Note that some of the blue and green curves overlap, and electrons not in proper phase with the STOV do not gain energy in the z-direction motion.(c) The transverse-to-axial momentum ratio py/px versus the total energy gain γ.One can see that the electron energy gain is mainly in their transverse motion.(d) Evolution of the energy gain of the electrons.

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Figure 5. Properties of the emitted hard x rays.(a) Brilliance B (in units of photons s −1 mm −2 mrad −2 per 0.1% BW) versus the photon energy, where the inset is for the photon density in the x, y, 0 plane at t = 8T0.Dependence of the (b) photon number, (c) peak power with duration, and (d) laser-to-photons conversion efficiency on the laser amplitude a0 for foil thickness l d = 0.1λ.