# Divergence of High-Order Harmonic Generation by a Convex Plasma Surface
## 3 Simulation and Discussion  
From a geometrical optics point of view, the divergence of light reflected from a flat plasma surface is usually similar to that of the incident laser, as observed in many experiments [20,32]. If a Gaussian incident laser beam is focused onto a plasma surface, the resulting harmonics are also usually Gaussian. In our experiments, the difference in the divergence of the harmonics and incident laser implies the information of the plasma surface. We, therefore, assume that a non-Gaussian harmonic divergence is generated from the uneven plasma surface due to the pre-pulse. Disregarding the energy (\(\epsilon_{\rm e}\)) of the electron oscillating in the magnetic field, the energy of the electrons on the plasma surface oscillating in the electric field of the incident laser is [33]  
\[e_{\rm e}(r)=\left[\gamma(r)-1\right]m_{e}c^{2},\gamma(r)=\left[1+a_{0}^{2}(r )\right]^{1/2} \tag{1}\]  
where \(r\) is a coordinate along the plasma surface, \(m_{e}\) is the electron mass and \(c\) is the velocity of light. For \(a_{0}\ll 1\), \(\epsilon_{\rm e}\propto a_{0}^{2}\); and for \(a_{0}\gg 1\), \(\epsilon_{\rm e}\propto a_{0}\). Therefore, after the Gaussian beam reaches the plasma surface, the spatial distribution of the thermal electron expansion velocity on the plasma surface can be approximated as a Gaussian distribution. In brief, after the action of the pre-pulse, the plasma surface in the center of the focal spot has a larger expansion speed, which induces a larger plasma scale length \(L\). In this case, the pre-pulse acting on the plasma surface is assumed to induce a convex plasma surface profile. When the main pulse is incident on this convex plasma surface, the harmonics emissions will exhibit divergence characteristics different from the flat plasma surface. To verify our hypothesis and support our experimental results, we compared the high-order harmonic spectrum generated by the relativistic laser incident on different shaped plasma surfaces by means of 2D PIC simulation with EPOCH code [34]. The parameters of laser and electron density are similar to our experimental conditions. In the \(x\)-\(y\) plane, the p-polarized light with a relativistic light intensity of \(a_{0}=8\) is incident from the \(y+\) boundary, the wavelength is \(\lambda_{L}=800\) nm, the pulse width is \(3T_{0}\) (FWHM, \(T_{0}\) is laser period), and the beam waist size of the Gaussian beam is \(4\lambda_{L}\) (FWHM). The maximum density of the plasma electrons is calculated from the parameters of silica (mass density \(\rho=2.65\) g/cm\({}^{3}\), relative molecular mass \(M_{r}=60.0830\) g/mol) (assuming that the electrons are fully ionized), and the peak density is about \(400n_{c}\). Due to the presence of the pre-pulse, the plasma surface of the target has an electron density distribution of an exponential gradient profile \(n_{e}(r_{\parallel},r_{\perp})=n_{max}\exp\left[-r_{\perp}/L\left(r_{ \parallel}\right)\right]\), where \((r_{\parallel},r_{\perp})\) are the coordinates parallel and perpendicular to the plasma surface, respectively, and \(L\left(r_{\parallel}\right)\) is the plasma scale  
The simulation results of convex and flat plasma surfaces are shown in Figure 3a,c, respectively, where the incident angle is 45 degrees, \(t=24T_{0}\). The plasma is represented by a grey color bar, the pre-plasma is located on the upper surface, and the thickness of the plasma is \(3\lambda_{L}\), which is sufficient to reflect high-order harmonics. The light field is represented by a red and blue color bar, the red arrow represents the direction of the driving laser, the focal point of the driving laser is at (0, 0), and the gray arrow represents the propagation direction of the reflected wave. From Figure 3a, we can see that the wavefront of the reflected wave generated by the convex plasma surface is clearly divided into two peaks, while for the flat target of Figure 3b, there is only one peak. The difference in the reflected wavefront is caused by the difference in the distribution of the pre-plasma.