# Divergence of High-Order Harmonic Generation by a Convex Plasma Surface
## 1 Introduction  
The technique of chirped pulse amplification (CPA) [1] could generate ultra-intense laser pulses with intensities ranging from \(10^{14}\) to \(10^{23}\) Wcm\({}^{-2}\) in the laboratory [2]. For an intensity greater than \(10^{18}\) Wcm\({}^{-2}\), the interaction of laser light with matter is dominated by the relativistic character of the electrons. With the increase in laser intensity, the interaction of the laser with over-dense plasma to generate high-order harmonics [3] efficiently has attracted increasing research interests, such as the attosecond lighthouse [4, 5], laser-plasma electron acceleration [6], and X-ray nonlinear optics [7]. In the future, relativistic laser and plasma surface interaction is a hopeful route to achieving quantum electrodynamic critical fields [8, 9] and zeta-second pulses [10]. High-order harmonic generation (HHG) is an important route to a bright coherent XUV/X-ray source, which has been investigated in many media, such as gases [11, 12], clusters [13] and laser-irradiated plasma surfaces [14, 15, 16]. The harmonic generation of gas needs to avoid strong ionization of the gaseous medium, so the laser intensity is usually limited below \(10^{16}\) Wcm\({}^{-2}\). In contrast, HHG from plasma surfaces has no laser intensity limitation and has significantly higher conversion efficiency for high photon energy sources, e.g., X-rays. In experiments, due to the ablation of the target caused by the pre-pulse, the plasma surface will inevitably expand into the vacuum, forming a density gradient profile called pre-plasma. If this density profile can be expressed as \(n(x)\), where \(x\) is the spatial coordinate, \(L=n(x)/|\
abla n(x)|\) is the plasma scale length. The electron density gradient scale length of the pre-plasma has a decisive influence on the HHG. For interactions between the laser field and the plasma with a sharp density profile \(L<0.02\lambda_{L}\), where \(\lambda_{L}\) is the wavelength of driving pulses, the electrons on the plasma surface are first pulled out of the plasma by the laser field, and then pushed back into the dense plasma by the reversed laser field. These electrons, called vacuum-accelerated Brunel electrons [17,18], carry away energy into dense plasma and generate harmonics named coherent wake emission (CWE) [19,20]. For a relativistic intensity laser interacting with a long density gradient of \(L\) less than \(0.2\lambda_{L}\), the incident laser field is reflected around the critical surface, where the electrons behave as relativistic oscillations. The HHG is attributed to a periodic Doppler effect induced by the relativistic oscillations of the electrons, called a relativistic oscillating mirror (ROM) [10,21,22]. As for a plasma scale length L larger than \(0.2\lambda_{L}\), an excessively long pre-plasma will induce parametric instability and greatly reduce the coherence of the harmonics [23]. The optimal scale length of pre-plasma to generate harmonics exists within a range [24] from 0 to \(0.2\lambda_{L}\), where the mechanism of harmonic generation transitions from CWE to ROM [25]. For example, a laser with a high contrast ratio will form a sharp pre-plasma, and the plasma scale length \(L\) will exceed a threshold only at the center of the laser spot, resulting in an annular CWE source [26]. A curved plasma surface created by laser radiation pressure can focus the harmonics [27,28,29], which means a shaped plasma surface can be used to control the harmonic divergence [30]. In this letter, we report the HHG driven by a 12.6J 800 nm laser pulse at the laser peak intensity of \(I_{0}=1.6\times 10^{20}\) Wcm\({}^{-2}\). We find an asymmetric saddle-shaped structure of the harmonic divergence, which can be explained by the harmonics reflected from a Gaussian convex plasma surface formed by a Gaussian pre-pulse with a similar spatial distribution as the main pulse. This is confirmed by 2D particle-in-cell (PIC) simulations, deducing a density scale length of \(\sim\)0.12\(\lambda_{L}\) at the center of the plasma surface.