# Reflection of the vortex beam of the relativistic flying mirror

The reflection process was studied using the three-dimensional (3D) particle-in-the-cell (PIC) code EPOCH^{35}. The famous Laguerre-Gaussian (LG_{01}) the mode is used as a typical vortex beam, and it can be expressed by^{36.37}

$$uleft(L{G}_{01}right)=sqrt{frac{2}{pi }}frac{rsqrt{2}}{{w}^{2} left(xright)}mathrm{exp}left[-frac{{r}^{2}}{{w}^{2}left(xright)}right]mathrm{exp}left(ivarphiright)mathrm{exp}left[frac{i{k}_{0}{r}^{2}x}{2left({x}^{2}+{x}_{r}^{2}right)}right]mathit{exp}left[-2i{mathrm{tan}}^{-1}left(frac{x}{{x}_{r}}right)right]{mathrm{sin}}^{2}left(frac{pi t}{2tau}right)$$

when it spreads in *X* direction. Here, (r=sqrt{{y}^{2}+{z}^{2}}); (varphi =mathrm{arctan}(z/y)); ({k}_{0}=2pi /{lambda}_{0}) is the wave number; and the wavelength of the incident beam is ({lambda }_{0}=0.8 mathrm{mu m}). In addition, (wleft(xright)={w}_{0}sqrt{{(x}^{2}+{x}_{r}^{2})/{x}_{r} ^{2}}),(mathrm{where } {w}_{0}=6.25{lambda }_{0}) is the beam waist radius and ({x}_{r}) is the Rayleigh length. The longitudinal profile was composed of ({mathrm{sin}}^{2}(pi t/2tau )) and (tau = 26.7mathrm{ fs}). The dimensionless amplitude ({a}_{0}=e{E}_{0}/{m}_{e}c{omega }_{0}=1)where (e), ({me}), ({omega }_{0}), ({E}_{0})and (vs) are respectively the charge of the electron, the mass of the electron, the frequency of the laser, the maximum amplitude of the laser field and the speed of light in vacuum. In our PIC simulation, a linearly polarized LG_{01} incident beam obliquely on the flying mirror with ({theta }_{in}=pi /6) (compared to the *X* axis), as shown in Fig. 1. Initially, the LG laser was injected from (x=-5{lambda}_{0})and the flying mirror was standing and occupying the domain in (13.75{lambda}_{0})X (15{lambda}_{0}), (-18.75{lambda}_{0})there (18.75{lambda}_{0}), (-10{lambda}_{0})z (10{lambda}_{0}). The electron density of the mirror was 20 ({NC}) to ensure that the laser pulse has been completely reflected^{17.38}here,({NC}) is the critical density. The flying mirror was moving in the –*X*steering at a speed of(v=0.5mathrm{c}). Here we note that the speed of the flying mirror was set to 0.5c instead of a more relativistic speed to save computational resources, i.e. to achieve fast separation of the laser from the target and avoid the influence of digital heating. The simulation box was a (25{lambda}_{0})× (50{lambda}_{0}) × (25{lambda}_{0}) cuboid corresponding to a window of 400 × 800 × 400 cells in the *X*× *there* × *z*direction, and each cell had 10 macroparticles.

The incident and reflected electric field simulation configurations are shown in Figures 2a,b, respectively. Compared to the field before reflection, the distribution of the reflected field changed significantly, including the profile and the direction of propagation. In Fig. 2b, the arrow indicates the direction of the LM of the reflected beam, defined according to the direction of the LM, which must be perpendicular to that of the electromagnetic field. The wavelength of the reflected beam was lower than that of the incident beam and the angle of reflection was lower than the angle of incidence due to the relativistic effect. As shown in Figure 2c, the laser was tightly focused and the longitudinal component of the electromagnetic field could not be ignored; So the *k* -the profile of the spatial spectrum of the reflected field has undergone a spectral broadening. According to the coordinates of the center of gravity of the laser, the wavelength and the emission angle of the reflected beam were ({lambda }_{1}=0.283 mathrm{mu m}) and ({theta}_{re}=0.174) rad, respectively.

Otherwise, ({lambda}_{1}) and ({theta}_{re}) can be derived from Doppler shift formulas. Using the Lorentz transformation, in the frame of the flying mirror, the angle of incidence ({mathrm{tan}theta }_{mathrm{in}}^{^{prime}}=mathrm{sin}{theta }_{mathrm{in}}/(gamma mathrm{cos}{theta }_{mathrm{in}}+gamma beta )) and the wavelength of the incident beam ({lambda }_{0}^{^{prime}}={lambda }_{0}(1/left(gamma +gamma beta mathrm{cos}{theta }_ {mathrm{in}}right))) with the relativistic factor (gamma=1/sqrt{1-{beta }^{2}}), (beta =v/c). In the same frame, the angle of reflection is equal to the angle of incidence. By inverse Lorentz transformation, in the laboratory reference frame, the reflection angle (mathrm{tan}{theta }_{mathrm{re}}={mathrm{sin}theta }_{mathrm{in}}^{^{prime)}/(gamma { mathrm{cos}theta }_{mathrm{in}}^{^{prime}}+gamma beta ))and the wavelength of the reflected beam ({lambda }_{1}={lambda }_{0}^{^{prime}}(1/left(gamma +gamma beta {mathrm{cos}theta }_ {mathrm{in}}^{^{prime}}right))). For ({theta }_{in}=pi /6) and (beta=0.5, {theta}_{text{re}}=0.176) Rad and ({lambda}_{1}=0.281 {upmu}{text{m}}) were obtained, which are in good agreement with the simulation results.

Considering the angle of reflection, the changes in the beam profile can be well explained. As shown in Fig. 3, the red quadrilateral (ABCD) was chosen in Fig. 2a. The incident beam was reflected along (overrightarrow{{mathrm{AA}}_{1}}) (the direction of the LM), and the angle between the LM and the electric field profile was defined as (varphi)where (mathrm{sin}varphi /mathrm{sin}(varphi +{theta }_{in}+{theta }_{re})=beta mathrm{cos}{theta }_ {in}/mathrm{cos}{theta}_{re}). When the AB aircraft began to interact with the flying mirror, the target was at *X*_{1} and (varphi =0). When the flying mirror was moved to *X*_{2}, the beam is completely reflected. At this time, the AB and CD planes were transformed into A_{1}B_{1} etc_{1}D_{1}, respectively. Therefore, the profile of the reflected field became the blue quadrilateral A_{1}B_{1}VS_{1}D_{1}. According to this analysis, the angle between the LM and the reflected field profile should be (varphi =0.403) rad, which is in good agreement with the simulation result of (varphi =0.424) rad, as shown in Figure 2b.

The above results confirm that the emission angle, wavelength and distribution of the reflected beam observed in the simulation are in line with theoretical expectations. In the next section, we focus on AM and examine the time history of the average OAM per photon of the electromagnetic field.

The average AM carried by a photon was calculated based on field data from PIC simulations using (j={varepsilon }_{0}{int }_{V}rtimes (Etimes B)mathrm{dV}/mathrm{N})where ({varepsilon}_{0}) is the dielectric permittivity of vacuum; *r* is the position vector; *E* and *B* are respectively the electric and magnetic fields; V refers to the entire simulation region for integration; and (mathrm{N}) is the number of photons in the entire laser beam. To accurately calculate the AM, we selected the electromagnetic fields for three periods in the middle part of the laser beam as shown in Fig. 2a. To intuitively display the simulation results, the angles investigated in this study were all angles along the + *X*axis. FIG. 4 shows the temporal evolution of the OAM carried by a photon; ({j}_{x}) and ({j}_{y}) refer to the two components of OAM in the*X*and*there*directions, respectively. To (t=12 T), the selected part of the vortex beam was incident on the left boundary of the simulation box. Before the vortex beam interacts with the flying mirror, ({j}_{x}=0.857) and({j}_{y}=0.491) were constant (t=12 T) at (t=19.5 T). Therefore, the OAM direction of the incident beam was 0.520 rad, which was collinear with the LM ( ({theta }_{in}=pi /6) rad) of the incident beam. To (t=24 T)the selected laser was completely reflected (the same three periods as the incident beam); ({j}_{x}=0.983) and ({j}_{y}=0.147) has remained constant (t=24 T) at (t=30 T). Therefore, the direction of the OAM of the reflected beam was 0.148 rad, thus indicating a non-collinearity with the LM of the reflected beam (according to the aforementioned result of(2.968) radi). The angle between the LM and the OAM is({Theta }_{0.5c}=0.320 mathrm{rad}.) Essentially, in addition to the longitudinal MA in the propagation direction, the transverse MA in the perpendicular propagation direction was generated by the interaction between the vortex laser and the flying mirror. This non-collinear phenomenon results from the profile change of the laser electromagnetic field caused by the double Doppler shift.

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