Intense high-order harmonic vector beams from relativistic plasma mirrors

Vector beams, with a spatial nonuniform polarization distribution, are important for many applications due to their unique field characteristics and novel effects when interacting with matter. Here, through three-dimensional particle-in-cell simulations, we demonstrate that intense vector beams in the extreme-ultraviolet to x-ray spectral region can be generated by means of high harmonic generation (HHG) in the relativistic regime. The vector features of the fundamental laser beam can be transferred to the higher frequency emission coherently during the extreme nonlinear HHG dynamics from relativistic plasma mirrors. The vector harmonic beams can be synthesized into attosecond vector beams. It is also possible to generate vector harmonic beams carrying orbital angular momentum. Such bright vortices and vector light sources present new opportunities in various applications such as imaging with high spatial and temporal resolution, ultrafast magnetic spectroscopy, and particle manipulation.

Figure 1

Electric-field distribution of the incident vector laser beams obtained from 3D PIC simulations. (a) The radially polarized laser ${\mathbf{E}}_r$. (b) The azimuthally polarized laser ${\mathbf{E}}_{\varphi }$. (c) The generalized vector laser beam ${\mathbf{E}}_g={\mathbf{E}}_r+{\mathbf{E}}_{\varphi }$. The color scale shows the intensity pattern in the transverse plane, while the yellow arrows represent the instantaneous local electric-field vectors.

Figure 2

Electric-field distribution of the generated harmonics driven by the (a)–(c) radially polarized, (d)–(f) azimuthally polarized, and (g)–(i) generalized vector laser beams. The lasers are normally incident onto the plasma surfaces. The observation planes are at $x=6.2{\lambda }_L$. The first to third columns correspond to the third (H3), fifth (H5), and seventh (H7) harmonics, respectively. The color scale show the harmonic intensity patterns, with the green arrows denoting the instantaneous local electric-field vectors.

Figure 3

Higher-order transverse modes of the harmonics. Electric-field distribution in the $x\text{−}y$ plane for the (a) third and (b) fifth harmonics driven by radially polarized laser pulses. The black dashed lines at $x=6.2{\lambda }_L$ denote the observation positions in Fig. 2.

Figure 4

(a) The peak intensity of high harmonics as a function of harmonic order $n$. The dashed orange line shows the $n^{-8/3}$ spectral scaling law given by the $\gamma$-spike model of HHG based on relativistically oscillating mirrors. (b) 1D (lineout along the $z=0$ axis) transverse intensity profiles of the fundamental field and the third, fifth, seventh, and ninth harmonics. (c) Isosurface distribution of the high harmonics after spectral filtering by selecting the third-23rd orders, showing an attosecond pulse train structure. These results correspond to the case of radially polarized driving lasers.

Figure 5

(a) Sketch of simulation setup for oblique incidence. The incident vector laser beams propagate along the $x$ axis. The plasma slab is rotated clockwise by ${45}^{\circ }$ around the $z$ axis. Then the reflected pulses propagate along the $y$ axis. (b), (c) The intensity pattern (color scale) and spatial polarization distribution (green arrows) of the third harmonic driven by the (b) radially and (c) azimuthally polarized laser beams observed in the transverse $x\text{−}z$ plane at $y=16.2{\lambda }_L$. (d,) (e) The electric-field distribution of the reflected pulses driven by the (d) radially and (e) azimuthally polarized laser beams observed in the $x\text{−}y$ plane at $z=0$. (f) The local Poynting vectors of the reflected beams (left) and third harmonics (right) driven by radially polarized lasers, showing vector beams carrying a nonzero orbital angular momentum having been generated.

Figure 6

2D cylindrical PIC simulation results. (a) Spectra of the reflected pulses for the radially polarized (RP; 2D cylindrical simulation) and linearly polarized (LP; 1D simulation) driving laser cases. (b) Intensity profiles of the reflected pulses after spectral filtering by selecting the harmonics with wavelength in the range 8–80 nm (harmonic orders 10 to 100). The profiles of the RP case are taken along $r=w_0/\sqrt{2}$.