# Divergence of High-Order Harmonic Generation by a Convex Plasma Surface
## 3 Simulation and Discussion
To support our experimental results, we set the output of the reflected light field at an angle of 116 degrees to the incident light to obtain the harmonic spectrum. Identical to the experimental setup, we varied the incident angle by rotating the plasma surface around the laser focus, ranging from 40\(\sim\)70 degrees. Figure 3b shows the 5 th-50 th harmonic spectrum reflected by the convex plasma surface at different incident angles. It can be seen from the figure that there are harmonic peaks at the 50\({}^{\circ}\) and 60\({}^{\circ}\) directions (white dashed lines, labeled peak A and peak B, respectively). In contrast, the flat target reflected harmonics with only one peak, as shown by the dashed line in Figure 3d. It can be seen from the simulation that the assumption of convex plasma is in good agreement with the experiment. In the divergent spectrum of the harmonics observed in our experiments, the intensity of the peak at 60 degrees is always greater than that at 50 degrees, which can be interpreted from the distribution of electrons on the plasma surface. Due to the electron heating caused by the pre-pulse, the thermal motion of the electrons manifests as an expansion  
Figure 3: Simulated images of the convex (**a**) and the flat (**c**) plasma surface at time \(t=24T_{0}\). (**a**) The driving laser (red arrow) with relativistic intensity \(a_{0}=8\) is incident on the convex plasma surface at an incident angle of 45 degrees, and the wavefront of the reflected wave mainly has two peaks (grey arrow). (**c**) The simulation setup is the same as (**a**), except for the flat plasma surface, where one can see a peak in the reflected wavefront. The harmonic spectrum was observed at an angle of 116 degrees to the incident light. By rotating the plasma surface, harmonic spectra of reflections from a convex (**b**) and a flat (**d**) plasma surface were observed. perpendicular to the plasma surface, forming a pre-plasma, as shown in Figure 4a. For the plasma target of \(n_{max}=400n_{c}\), the distance from the reflection surface of the fundamental frequency light (\(n_{e}=n_{c}\)) to the initial plasma surface is \(\delta=L\Big{(}r_{\parallel}\big{)}\times\ln(n_{\max}/n_{c})\). We assume that the scale length at \(r_{\parallel}=0\) is \(L\Big{(}r_{\parallel}=0\Big{)}=0.12\lambda_{L}\), so \(\delta\Big{(}r_{\parallel}=0\Big{)}\cong 0.72\lambda_{L}\). Due to oblique incidence, the central reflection point is laterally offset relative to the main pulse by a distance \(\delta\sin\theta\). As a result, the main pulse energy is unevenly distributed on both sides of the reflection axis, as indicated by peaks A (violet) and B (red) in Figure 4a, and the intensity of peak B is always greater than that of peak A. This is confirmed in both our experiments and simulations. Figure 4b shows the intensity of the 22 nd harmonic at different incident angles, and the peaks of its spectra divergence are located at 51 degrees (A) and 60 degrees (B), respectively. The intensity of peak A is about 80 percent of peak B, which is close to 70 percent of the experimental results. Correspondingly, the divergence of the 22 nd harmonic reflected by the flat plasma surface presents a Gaussian structure. Our simulations reproduce well the phenomena observed in the experiments.