# Divergence of High-Order Harmonic Generation by a Convex Plasma Surface
## 2 Experimental Results  
The experiments of plasma harmonics are performed on the 1 PW/0.1 Hz laser system of the Shanghai Superintense Ultrafast Laser Facility (SULF) [31].The experimental system mainly consists of a laser, vacuum chamber and home-made flat field grating spectrometer. The SULF facility can deliver a 29.6 fs (full width at half maximum, FWHM) laser pulse with a diameter of 200 mm and central wavelength at 800 nm, and the single pulse on-target energy used in experiments can be up to 12.6J. This experiment was performed before mounting the plasma mirror. The contrast ratio at 80 picoseconds before the main pulse (\(-80\) ps) is about \(2.5\times 10^{-11}\), but the contrast ratio at \(-10\) picoseconds is only about \(1\times 10^{-5}\).  
The schematic diagram in the vacuum chamber is shown in Figure 1a. Through a \(f/3\) off-axis parabolic mirror (OAP, \(f\) = 600 mm, off-axis angle = 26\({}^{\circ}\)), the laser pulse was focused on the polished fused silica plates, whose electron density is \(n_{max}\simeq 400n_{c}\) when entirely ionized, where \(n_{c}\) is the critical density for the incident wave. The focal spot size was 6.0 \(\upmu\)m (FWHM), corresponding to a peak intensity of \(I_{0}=1.6\times 10^{20}\) Wcm\({}^{-2}\) (normalized vector potential \(a_{0}=\sqrt{I_{0}\lambda_{L}^{2}/(1.38\times 10^{18}\) Wcm\({}^{-2}\upmu\)m\({}^{2})}\simeq 8\)) on target. The radiation generated from such geometry was measured with a home-made flat-field grating spectrometer. In the reflection direction, the high-order harmonics passed through a 2 mm wide slit and a 250 nm-thick aluminum (Al) foil filter. The flat field grazing-incidence grating (Hitachi, 1200 lines/mm) was 237 mm away from the slit, and the soft X-ray CCD (Andor, DO940P-BEN) was located at the focal plane of the grating. The horizontal direction in the CCD array represents the wavelength coordinate, which was calibrated by the L3 absorption edge (17.1 nm) of aluminum. Due to the limitation of the experimental layout, the angle between the observation direction of the spectrometer and the incident laser was fixed at 116 degrees. Figure 1b represents the harmonic signal from vertically integral of CCD image.  
In order to observe the divergence of HHG, we adjusted the incident angle by fixing the incident laser and rotating the plasma surface. In general, the so-called divergence of harmonics should be measured by fixing the incident angle of pulses and then adjusting the receiving angle of the detector. This is difficult to achieve experimentally and why we rotated the plasma surface to measure the divergence. It can be seen from Figure 1a that the reflected beam rotated by 2 degrees when the target plane rotated by 1 degree. On the horizontal plane, we rotated the target plane so that the incident angle of the main pulse changed from 42 degrees to 70 degrees. The spectra captured by the CCD arrays indicated that the high-order harmonic signal can be observed in a range of about \(\sim\)56\({}^{\circ}\). Further, we found that the observed HHG signal depends on the angle of incidence, as shown in Figure 2. Unlike the Gaussian type of incident light, the distribution of HHG intensity with an incident angle exhibits a saddle-shaped structure, which we will discuss in the following sections.  
Figure 1: Measurement of spatial distribution of high-order harmonic generation. (**a**) Experimental setup. The angle between the observation direction of the spectrometer and the incident laser was 116\({}^{\circ}\). We rotated the plasma surface to observe the divergence of high-order harmonic generation. The horizontal direction in the CCD array represents the wavelength coordinate. (**b**) Vertically integral of CCD image. The L3 absorption edge of aluminum calibrated the spectra.  
Figure 2: The intensity of high-order harmonics at different incident angles. Different from an incident beam, there are two peaks in the angular distribution, and the peak at 61\({}^{\circ}\) is slightly higher than the one at 48\({}^{\circ}\).