2.1. The generation of rotating attosecond electron sheets

When an intense circularly polarized Laguerre–Gaussian laser pulse illuminates the droplet, abundant electrons are dragged out of the droplet, forming an isolated dense electron layer, as shown in Figure 1(b). The electron layer has a maximal density of $15{n}_{\mathrm{c}}$ and a duration of 330 as at $t=10{T}_0$ . As the electron layer co-moves with the laser pulse, it is continuously compressed along the longitudinal direction with the duration decreasing to 200 as at $t=30{T}_0$ . Meanwhile, more and more electrons counterclockwise rotate around the optics axis over time. We can also observe the signature of rotation of the relativistic electron sheet from the projection of three-dimensional (3D) electron trajectories in the $yoz$ plane (see Figure S1(a) in the Supplementary Material). These electrons are simultaneously confined by the transverse ponderomotive force, ${F}_{\mathrm{tr}}=\sqrt{2e}{a}_0{m}_{\mathrm{e}0}{\omega}_0c\left(2{r}^2/{\sigma}^2-1\right){\sigma}_0{\left(k{\sigma}^2\right)}^{-1}\exp \left(-{r}^2/{\sigma}^2\right)\cdot {\sin}^2\left[\pi \left(x- ct\right)/2\tau \right]$ , which points to the optical axis and thus pushes the electrons to the center. Here, ${m}_{\mathrm{e}0}$ denotes the electron rest mass and $c$ is the speed of light in vacuum. This finally leads to a small electron divergence angle of approximately ${2}^{\circ }$ at $t=30{T}_0$ , as seen in Figure 1(c). During this process, not only is an isolated attosecond electron sheet generated, but also its density is greatly increased^[ Reference Bulanov, Esirkepov and Tajima^27, Reference Luttikhof, Khachatryan, van Goor and Boller^54, Reference Ferri, Horný and Fülöp^55] and the beam divergence is decreased^[ Reference Kulagin, Cherepenin, Hur and Suk^56]. It is important for the droplet target to align with the laser pulse. To generate a dense isolated attosecond electron sheet, the droplet target should be located in the focal spot, namely in the region of $r<{\sigma}_0/\sqrt{2}$ . Since the droplet target is spherical, the transverse misalignments of the sphere center may affect the results. Therefore, the transverse misalignments should be precisely controlled in experiments with sensing and a control scheme of positions^[ Reference Mizoguchi, Tomuro, Nishimura, Hosoda, Nakarai, Abe, Tanaka, Watanabe, Shiraishi, Yanagiga, Soumagne, Iwamoto, Nagai, Ueno, Suganuma, Niimi, Yabu, Yamada and Saitou^57].

It is interesting to see that this isolated electron bunch shows an initial quasi-monoenergetic distribution, as shown in Figure 1(d), whose cut-off energy is up to 340 MeV at $t=30{T}_0$ . Figure 1(e) illuminates the electron distribution in the phase space $\left({\beta}_x,{\beta}_y\right)$ , which shows that most electrons are located within the zone ${\beta}_x\approx 1$ . The red curve indicates that the longitudinal velocity ${\beta}_x$ of more than 50 $\%$ of electrons is larger than 0.998. Figure 1(f) presents the electron distribution in the phase space $\left({p}_x,\gamma \right)$ . Here, one sees that $\gamma =\sqrt{1+{p}^2}\approx {p}_x$ . Meanwhile, the dephasing rate^[ Reference Robinson, Arefiev and Neely^58]  $R=\gamma -{p}_x=1+\left({q}_{\mathrm{e}}/{m}_{\mathrm{e}0}c\right)\int {E}_x\mathrm{d}\tau <0.125$ for more than 50 $\%$ of the electrons. These electrons co-move forward at a velocity close to $c$ and are phase-locked swiftly by the laser in the longitudinal direction. Since the laser magnetic force counteracts the laser radial electric force, the electrons are subject to null force in the transverse direction, so that the electron layer keeps the intact structure during the propagation. This plays a critical role for bright $\gamma$ -ray emission in the subsequent processes.

Figure 2 (a) Normalized longitudinal electric field ${a}_x$ and (b) Q factor with respect to $\Delta \psi =\psi -{\psi}_0$ . The red, black, blue and green curves correspond to the cases of CEP $=0$ , $\pi /2$ , $\pi$ and $3\pi /2$ , respectively. Evolution of ${\beta}_x$ with respect to $\psi -{\psi}_0$ when the CEP equals (c) $0$ , (d) $\pi /2$ , (e) $\pi$ and (f) $3\pi /2$ in the case of ${a}_{x0}=0.2$ (dashed) and $0.998$ (solid), respectively.

Since only electrons phase-locked simultaneously in the transverse and longitudinal directions are able to be accelerated continuously in the laser field, we may achieve an isolated short electron bunch by manipulating electron phase-locking. Since the laser magnetic force counteracts the laser radial electric force in the transverse direction, electrons can be tightly confined around the laser axis. The transverse velocity of electrons is much less than the longitudinal velocity, as shown in Figure 1(e). Therefore, we ignore the transverse electron motion in the following analysis, which agrees with our particle-in-cell (PIC) simulation results. The electrons’ motion in one-dimensional approximation can be described by $\mathrm{d}{p}_x/\mathrm{d}t=-{q}_{\mathrm{e}}{E}_x$ and $d\left(\gamma {m}_{\mathrm{e}0}{c}^2\right)/\mathrm{d}t=-{q}_{\mathrm{e}}{v}_x{E}_x$ . We have ${a}_x={q}_{\mathrm{e}}{E}_x/{m}_{\mathrm{e}0}{\omega}_0c=-{a}_{x0}{\sin}^2\left[\pi \left(x- ct\right)/2\tau \right]\cos \psi$ , where ${a}_{x0}=2\sqrt{2e}{a}_0/{k}_0{\sigma}_0$ and $\psi ={\omega}_0t-{k}_0x+{\psi}_0$ . For a short laser pulse, ${a}_x$ depends sensitively on the CEP, as shown in Figure 2(a). Here, the longitudinal velocity of electrons can be derived from the equations above:

(1) $$\begin{align}{\beta}_x=\frac{1-{\left(1-{a}_{x0}Q\right)}^2}{1+{\left(1-{a}_{x0}Q\right)}^2},\end{align}$$

where the factor $Q=\left(1/2\right)\sin \psi -\left[\tau /\left(4\tau -2\right)\right]\sin [\left(2\tau -1\right)\cdot \psi /2\tau +{\psi}_0/2\tau ]-\left[\tau /\left(4\tau +2\right)\right]\sin [\left(2\tau +1\right)\psi /2\tau -{\psi}_0/ 2\tau ]+\left[1/\left(8{\tau}^2-2\right)\right]\sin {\psi}_0$ . The $Q$ factor always ranges from −1 to 1 for different ${\psi}_0$ , as shown in Figure 2(b). We can derive the local maximum nodes ${\psi}_{\mathrm{max}}=2 k\pi +\pi /2$ from the equation $\mathrm{d}{\beta}_x/\mathrm{d}\psi =0$ , where $k=0,\pm 1,\pm 2,\cdots$ . Here, $k=0,\, 1,\, 2$ when the pulse duration $2\tau =3{T}_0$ . Obviously, only when $k=2$ and the CEP ${\psi}_0=3\pi /2$ , namely $\psi ={\psi}_{\mathrm{max}}=9\pi /2$ and $\Delta \psi =\psi -{\psi}_0=3\pi$ , $Q=1$ and ${\beta}_x$ reaches the peak ${\beta}_{x, \mathrm{peak}}=\left[1-{\left(1-{a}_{x0}\right)}^2\right]/\left[1+{\left(1-{a}_{x0}\right)}^2\right]$ . When we set a longer pulse duration, the number of local maximum nodes will increase. The CEP ${\psi}_0$ for the peak of ${\beta}_x$ also alters. To phase-lock electrons, ${\beta}_{x,\mathit{\max}}=1$ should be satisfied and we obtain ${a}_{x0}=1$ . Actually, considering the influence of Guoy phase and focal spot variation, ${\beta}_{x, \mathrm{peak}}$ is always less than 1. Therefore, ${a}_{x0}\ge 1$ is required to guarantee the electron phase-locking. As ${a}_{x0}=2\sqrt{2e}{a}_0/{k}_0{\sigma}_0$ , exact solution for the equations reads

(2) $$\begin{align}{a}_0\ge \frac{k_0{\sigma}_0}{2\sqrt{2e}}.\end{align}$$

Figure 2(c)–2(f) present the evolution of the longitudinal electron velocity ${\beta}_x$ with respect to $\Delta \psi =\psi -{\psi}_0$ for ${a}_{x0}=0.2$ (dashed) and $0.998$ (solid), respectively. The maximum nodes $\Delta \psi =2 k\pi +\pi /2-{\psi}_0$ remain unchanged for different ${a}_{x0}$ . However, the corresponding ${\beta}_x$ quickly rises when ${a}_{x0}$ increases from $0.2$ to $0.998$ . Considering the Guoy phase and multi-dimensional effects, when Equation (2) is satisfied, ${\beta}_x$ approaches the speed of light at these nodes so that the electrons remain phased-locked. Therefore, electron bunches may form at $\Delta \psi =2 k\pi +\pi /2-{\psi}_0$ when the CEP is equal to ${\psi}_0$ . However, when ${\psi}_0=3\pi /2$ , ${\beta}_x$ at the local maximum node $\pi$ and $5\pi$ may be much less than that of the node $3\pi$ , as shown in Figure 2(f). In this case, electrons may be phase-locked only at the node $3\pi$ . In this way, we may generate an isolated electron bunch by setting the CEP to $3\pi /2$ . When the CEP is set to $0$ , $\pi /2$ or $\pi$ , ${\beta}_x$ at two local maximum nodes is much larger than that of another node. When the condition in Equation (2) is satisfied, electrons may pile up at the two nodes. Meanwhile, the number of generated $\gamma$ -ray bunches in the nonlinear Thomson scattering process is determined by the number of electron bunches. The degree of isolation of attosecond electron beams or $\gamma$ -rays is also of importance in the experiments, since non-isolated pulses are not applicable in ultrafast dynamics researches. Thus, the CEP should be set to $3\pi /2$ to generate an isolated $\gamma$ -ray bunch, which is consistent with the one-dimensionless model. Furthermore, for a longer pulse duration, the formation of an isolated electron sheet and an isolated $\gamma$ -ray bunch becomes difficult, since electrons will simultaneously bunch at multiple nodes.

To verify the one-dimensionless model, we carry out a series of 3D-PIC simulations by changing the laser CEP ${\psi}_0$ , and keeping other parameters unchanged. The simulation results are shown in Figure 3. When the CEP ${\psi}_0=0$ , two electron bunches with different density form at $x=7.75{\lambda}_0$ and $x=8.75{\lambda}_0$ . When ${\psi}_0$ rises to $\pi /2$ , the electron bunches are pushed rightwards to $8{\lambda}_0$ and $9{\lambda}_0$ , respectively. Meanwhile, the density and energy distributions of the electron bunches also change. When ${\psi}_0$ further increases to $\pi$ , the electron bunches are pushed to the right-hand side further. The electron bunch at $x=8.25{\lambda}_0$ is subjected to the peak electric field and its energy achieves maximum. When the CEP ${\psi}_0=3\pi /2$ , there is only one isolated electron bunch at $x=8.5{\lambda}_0$ , which is consistent with the one-dimensionless model. In summary, when the CEP ${\psi}_0$ increases to $\pi /2$ , the electron bunch will be shifted rightwards by about $0.25{\lambda}_0$ , and the density and energy are also regulated. In particular, when ${\psi}_0=3\pi /2$ , an isolated dense attosecond electron bunch with higher energy may be generated as electrons are close to the central zone of the laser envelope. We can also increase the number of electron bunches by using a laser pulse with longer pulse duration. It is demonstrated by 3D-PIC simulations that an isolated electron bunch can be produced when the pulse duration $2\tau =3{T}_0$ and the CEP ${\psi}_0=3\pi /2$ , which agrees well with our predictions above.

Figure 3 (a)–(d) Electron density distribution at $t=10{T}_0$ and (e)–(h) corresponding electron energy distribution at $t=30{T}_0$ in the $xoy$ plane when ${\psi}_0=0$ , $\pi /2$ , $\pi$ and $3\pi /2$ , respectively.

2.2. The production of isolated attosecond $\gamma$ -rays

In our configuration, such an isolated energetic electron bunch collides head-on with a scattering laser pulse. The scattering pulse is a counterstreaming linearly polarized Gaussian laser pulse with a dimensionless peak amplitude of ${a}_1=45$ . The duration and focal spot radius are ${\tau}_1=6{T}_0$ and ${\sigma}_1=3.5{\lambda}_0$ , respectively. The laser electric field is along the $y$ direction. As the duration ${\tau}_1$ increases, the number, energy and BAM of $\gamma$ -photons also linearly rise (see Figure S8 in the Supplementary Material). Figure 4(a) shows the density distribution of high-energy $\gamma$ -photons with energy of more than 1 MeV situated at $x=33.74{\lambda}_0$ , $33.78{\lambda}_0$ and $33.88{\lambda}_0$ . The peak density and corresponding total number of $\gamma$ -photons are as high as $45{n}_{\mathrm{c}}$ and $2.1\times {10}^{10}$ , respectively. The produced $\gamma$ -photons via nonlinear scattering off electrons show rotations in the same direction as electrons. Figure 4(b) shows the angular distribution of $\gamma$ -photons ( ${E}_{\gamma }>1$ MeV), where $\theta$ denotes the angle between the $x$ -axis and the photon momentum and $\phi$ is the crossing angle between the $y$ -axis and the transverse momentum of photons. In the polarization plane $xoy$ , the maximum emission angle of $\gamma$ -photons is $\sqrt{\theta_{\mathrm{e}}^2+{a}_1^2/{\gamma}_{\mathrm{e}}^2}\approx {4.2}^{\circ }$ , which is consistent with the simulation result of ${4}^{\circ }$ . In the orthogonal plane $xoz$ , the emission angle approaches $\sqrt{\theta_{\mathrm{e}}^2+1/{\gamma}_{\mathrm{e}}^2}\approx {2}^{\circ }$ , in excellent agreement with the simulation results. In classical nonlinear Thomson scattering, it is demonstrated theoretically and experimentally that the $\gamma$ -photons are produced mainly along the polarization direction, and the angular distribution of $\gamma$ -photons usually peaks at $\phi ={0}^{\circ }$ or ${180}^{\circ }$ ^[ Reference Yan, Fruhling, Golovin, Haden, Luo, Zhang, Zhao, Zhang, Liu, Chen, Chen, Banerjee and Umstadter^31, Reference Ji, Pukhov, Kostyukov, Shen and Akli^59]. However, the angular distribution in Figure 4(b) is not consistent with this prediction. The distribution of $\gamma$ -photon divergence is split into two fragments at $\phi ={0}^{\circ }$ or ${180}^{\circ }$ . The average crossing angle between the $y$ -axis and each fragment is $\alpha$ . This characteristic is also obvious at $t=45{T}_0$ (see Figures S1(b) and S1(c) in the Supplementary Material). However, if the colliding laser is replaced by a circularly polarized pulse, the angular distribution will become circularly symmetric, as shown in Figure S2(a) (Supplementary Material). Since the rotating attosecond electron sheets carry BAM, we thus deduce that the splitting of divergence distribution may result from the BAM transfer during the photon emission. In the scattering processes $\gamma$ -ray photons absorb abundant BAM and gain transverse momentum from the relativistic electron layer, which may result in the splitting of divergence distribution. The unique angular distributions of $\gamma$ -ray photons may provide a new signature in the Thomson backscattering experiments to distinguish whether the $\gamma$ -ray beams carry BAM. The BAM transfer during the laser–plasma interaction will be investigated in the following paragraph.

Figure 4 (a) Density distribution, (b) angular distribution, (c) energy distribution and (d) energy spectrum and brilliance of high-energy $\gamma$ -photons with energy of more than 1 MeV at $t=35{T}_0$ . Here, the magenta arrows in (a) denote the average transverse momentum of $\gamma$ -photons at each cell. The black double-headed arrow in (b) denotes the polarization direction.

After the colliding laser is scattered, the photon energy increases by $4n{\gamma}_{\mathrm{e}}^2/\left(1+{a}_1^2/2\right)$ based on simultaneous relativistic Doppler upshift and multi-photon absorption, where $n\propto {a}_1^3$ is the order of harmonics^[ Reference Esarey, Ride and Sprangle^29, Reference Yan, Fruhling, Golovin, Haden, Luo, Zhang, Zhao, Zhang, Liu, Chen, Chen, Banerjee and Umstadter^31]. Figure 3(c) shows that the energy distribution of $\gamma$ -photons is circularly symmetric. Due to ${E}_{\gamma}={p}_{\gamma }c$ , the momentum distribution of $\gamma$ -photons is also circularly symmetric. Since the maximum ${\gamma}_{\mathrm{e}}$ is approximately 680 at $t=30{T}_0$ , this multi-photon process dominates the nonlinear scattering process so that the $\gamma$ -photon energy approaches eventually 100 MeV with an average energy of 18.4 MeV, as shown in Figure 4(d). We also calculate the photon brilliance with respect to different photon energy by assuming the root mean square (rms) values of the transverse size of $1.7\;\mu \mathrm{m}\times 1.7\;\mu \mathrm{m}$ . It is shown that the peak brilliance of the generated isolated attosecond $\gamma$ -photons is up to $9.3\times10^{24}$ photons s ${}^{-1}$ mrad ${}^{-2}$ mm ${}^{-2}$ per 0.1 $\%$ bandwidth at 4.3 MeV.