# Divergence of High-Order Harmonic Generation by a Convex Plasma Surface
## 4 Conclusions  
To conclude, we report high-order harmonic generation on a plasma surface with the relativistic light of intensity \(a_{0}=8\). The results show that the high-order harmonic divergence has a saddle-shaped structure. Further analysis shows that a Gaussian gradient distribution formed on the plasma surface before the arrival of the main pulse can lead to this saddle-shaped structure of divergence on the harmonic spectrum. We compared the divergence of harmonics generated by a relativistic laser incident on plasma surfaces with different shapes using PIC simulations and confirmed that the divergence of harmonics from a Gaussian gradient plasma surface coincides very well with experiments. Our work shows that the shape of the plasma surface has a very important effect on the generation of high-order harmonics, which provides useful insights into the generation and application [27; 28; 29; 35] of higher brightness high-order harmonics and the tools to control the emission of high-order harmonics in the future, e.g., natural focused harmonics.  
Figure 4: **(a)** Schematic diagram of a relativistic laser incident on a convex plasma surface. Due to the plasma expansion, the \(n_{e}=n_{c}\) interface is shifted by a distance \(\delta\) in the direction perpendicular to the target, which causes the reflection center to be shifted by \(\delta\sin\theta\) relative to the main pulse. Therefore, the Gaussian main pulse is divided into two parts, A and B, after reflection, where A has less energy than B. **(b)** The intensity of the 22 nd harmonic generated by the convex (blue) and flat (red) plasma surface at different incident angles.  
**Author Contributions:** Conceptualization, writing--original draft preparation, C.Y.; data curation and visualization, C.Y. and C.Z.; methodology, writing--review and editing, C.Y., C.Z., Y.T. and Z.Z. ([PERSON]); investigation, C.Z., Y.Z., D.Z., J.G. ([PERSON]), Y.B., R.Q., J.Q., J.G. ([PERSON]) and Z.Z. ([PERSON]); supervision, project administration and funding acquisition, C.Z., Y.T. and Z.Z. ([PERSON]). All authors have read and agreed to the published version of the manuscript.  
**Funding:** This work has received funding from the National Natural Science Foundation of China (grants Nos. 91950203, 11922412, 11874372, 11874374), Shanghai Pilot Program for Basic Research--Chinese Academy of Science, Shanghai Branch, Key Research Program of Frontier Sciences, Chinese Academy of Science and Youth Innovation Promotion Association of Chinese Academy of Sciences, as well as funding from Young Talents Support Project of China Association for Science and Technology, China Postdoctoral Science Foundation (grant No. 2021M/03329), Shanghai Sailing Program under Grant 22 YF1455000. PIC simulations were run using EPOCH codes, which was in part funded by the UK EPSRC grants EP/G054950/1, EP/G056803/1, EP/G055165/1 and EP/ M022463/1.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data supporting the findings of this study are available from the corresponding author upon reasonable request.  
**Acknowledgments:** The authors acknowledge Shanghai Supercomputer Center for providing computing resources.  
**Conflicts of Interest:** The authors declare no conflict of interest.