## 3 Results and Discussion
### \(\gamma\)-ray Pulses Generation  
Electrons escaping from the target surface contribute to the high-energy photon generation through NCS, before which, the electrons experience further accelerations under a different mechanism. As shown in Figure 3b,d, after the time \(t_{2}\), the energy increases mainly due to the positive work carried out by the \(x\)-direction electric field, namely, transiting from the incident laser action before \(t_{2}\) to the reflected electromagnetic field action. Meanwhile, the momentum mainly increases in the \(y\)-direction (Figure 3c), suggesting that the electrons are mainly under the vacuum laser acceleration. At time \(t_{4}\) and \(t_{5}\), one can observe the dramatic variation in momentum \(p_{x}\), which is also reflected from the shake of the electron's trajectories and works carried out by the electric field. This momentum variation is comprehensible for the scattering between electrons and the counter-propagating attosecond electromagnetic pulse. Followed by the NCS, quantities of high energy photons are generated to constitute an attosecond \(\gamma\)-ray pulse. Figure 4 demonstrates the photons' distribution, as well as characteristics, after all of the scatterings. \(\gamma\)-ray pulses asymmetrically distribute in the whole area, and a typical density profile along the pulse propagation direction shows a pulse duration \(\text{FWHM}\approx T_{0}/10=267\text{as}\) (Figure 4b); in addition, the maximal photon energy can extend to 6 MeV (Figure 4d). The peak brilliance \(>10^{22}\) photons/(s-mm\({}^{2}\)-mrad\({}^{2}\)-0.1%BW). Furthermore, the directionality of photon propagations is also investigated, as shown in Figure 4c, where the photons generally propagate along the azimuthal angle around \(\pm 90^{\circ}\) and \(\pm 45^{\circ}\). The former is rational since the electron slices and the reflected electromagnetic field both propagate vertically before heading into collisions. With regard to the latter, it is probably due to the momentum modulation caused by the incident laser. Under the additional vacuum laser acceleration of the incident laser toward the positive \(x\)-axis direction, \(p_{x}\) of the escaped electrons gets enlarged, yielding the electrons, and EM pulses collide at an acute angle at around \(45^{\circ}\). As a potential light source of the \(\gamma\)-ray, this fantastic directionality suggests that the direction of the \(\gamma\)-ray pulses can be manipulated by adjusting the cone target's open angle. Figure 2: (Color online). (**a**) The distributions of the electric field perpendicular to the target \(E_{\perp}\) (normalized to \(E_{0}\)) and electron density \(n_{e}\) (normalized to \(n_{c}\)) at \(t=15T_{0}\). The mini figure on the right shows the distributions of \(E_{\perp}\) and \(n_{e}\) at \(x=8.75\lambda_{0}\) (the black dotted line), in which, the yellow dotted line represents \(E_{\perp}\) and the blue dotted line represents \(n_{e}\). There are three electron bunches near \(y=10\lambda_{0}\), \(y=11\lambda_{0}\) and \(y=13.6\lambda_{0}\) in the vacuum at the inner side of the target. (**b**) Spectrum of HHG at \(13.25T_{0}\), conforms to the power-law of ROM \(I\propto(\omega/\omega_{0})^{-8/3}\). The inset is the distribution of the reflected field \(E_{x}\). Another characteristic of the \(\gamma\)-ray pulses lies in the asymmetries of the spatial (Figure 4a) and angular (Figure 4c,d) distribution. As a matter of fact, one can find that the pulses with \(p_{y}>0\) and \(p_{y}<0\) are alternately generated, which can be originated to the escape asymmetry of the electron slices (Figure 2a). Equation (1) shows the \(\pi/2\) phase difference between the \(E_{\perp}\) in the upper side (\(y>12\lambda\)) and lower side (\(y<12\lambda\)), suggesting that the electron slices are alternately generated from the two sides with a \(\pi/2\) phase difference. \(\gamma\)-ray generations influenced by this asymmetry may become macroscopically insignificant for the long incident laser pulse.