Angular momentum oscillation in spiral-shaped foil plasmas

Two types of spiral-shaped foils are investigated for generating significant angular momentum (AM) in plasmas by reflecting a relativistic Gaussian pulse into a vortex laser beam with the same topological charge. This is the first time to find that AM oscillation exists in specific spiral-shaped foils during laser-plasma interaction, while AM oscillation is not observed in other types of foils. Both three-dimensional particle-in-cell simulations and theoretical results have confirmed this finding. AM oscillation is demonstrated to be induced by the asymmetric field on the foil surface, and this asymmetric field can be modulated in order to strengthen or weaken the oscillation amplitude by redesigning the foil surface. AM oscillation is expected to bring insight into radiation, particle heating and other mechanisms with AM effects.


Introduction
Wave fronts can be twisted into vortices in many wave phenomena [1], such as in whirlpools and tornadoes. The phase of an optical vortex (OV) intertwines into multiple helices and remains a singularity at the spatial beam center. An example of this winding light is a Laguerre-Gaussian (LG) laser beam [2], which has a helical phase profile described by a modulation term f ℓ ( ) exp i , where f is the azimuthal coordinate and ℓ is the integer topological charge, i.e. the LG mode order. In 1992, Allen et al [3] first demonstrated that an LG beam carries orbital angular momentum (OAM), which is proportional to the topological charge ℓ. Since that, LG beams have been widely investigated for its unique phase front and OAM, and new approaches to atomic or subatomic scale manipulation [4], ghost imaging [5], quantum entanglement [6] and terabit data transmission [7,8] have been reported. Just as variation in linear momentum produces a force, the variation in angular momentum (AM) produces a torque. Previous researchers trapped absorbing dielectric particles in the dark center of a twisted light beam and set them into rotation. They attributed this to the torque produced by twisted light, thus allowing OAM to be transferred from light to particles [9][10][11][12].
The generation of light with OAM for relatively low intensity (generally below -10 W cm 16 2 ) has been extensively discussed. Nowadays, relativistic laser pulses (above -10 W cm 18 2 ) can be produced [13]. At such intensity, common materials will be destroyed instantly. However, a plasma has high damage threshold as an optical medium, which makes it possible to twist a relativistic Gaussian beam into an OV. Shi et al [14] proposed a 'light fan' scheme, i.e. a spiral-shaped foil plasma, that directly reshaped the phase front of a relativistic Gaussian beam to form an OV. Vieira et al [15] used a plasma as a nonlinear optical medium to amplify a seed laser with OAM via stimulated Raman scattering. Recently, Leblanc et al [16] put forward a kind of transient plasma hologram to experimentally generate vortex beams in diffracted order. One should note that the OAM density of the generated vortex beam can be as high as --0.56 kg m s 1 1 [14]. With such high OAM density and doughnut intensity distribution, a relativistic laser vortex beam can be used for proton and positron acceleration Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
with donut wakefields [17][18][19], high-order vortex harmonics generation [20][21][22][23], vortex gamma-ray and attosecond twisted beam generation [24][25][26], ultrashort twisted particle beam generation [27][28][29][30], and other novel phenomena [31,32]. Recently, when researchers studied the generation of relativistic vortex beams, they found that the plasma carries an equivalent amount of inverse AM after interaction with the laser according to the AM conservation [14]. This balance can be seen as a special OAM transfer mechanism as well. Although the generation of plasma AM is claimed to originate from the torque of the generated vortex beam [14], the exact dynamics that cause the plasma AM are still unclear. However, a plasma with huge AM has great potential in generating very strong longitudinal magnetic fields [33][34][35][36], and may bring insight into laboratory astrophysics [37]. Therefore, an investigation into the evolution of plasma AM during vortex pulse generation is necessary.
In this paper, we investigate the inverse AM obtained by the particles, when two types of structured foils are used to generate a relativistic vortex pulse with specific topological charge by reflecting a relativistic linearly polarized Gaussian pulse. We observe the oscillation of AM for both electrons and protons in simulations for the first time, and not all structured foils experience the AM oscillation process. This unique feature implies that the existence of AM oscillation is directly influenced by the foil structure. Moreover, the net AM of particle is also ascribed to the foil structure, because the incident linearly polarized Gaussian pulse does not possess any AM, neither spin nor orbital. Unlike particles, the laser gradually gains OAM without oscillation during the interaction with both types of structured foils. Both the laser and particle AM have been checked, and the AM conservation basically holds in our simulations. We provide a theoretical explanation of this dynamic process.

Theoretical model
Since our structured foils are designed to generate relativistic vortex beams with specific topological charge ℓ, the total phase of the reflected beam requires a change of ṕ ℓ 2 within one full annular loop, where the center is a singularity. Generally, two approaches can be put forward. One requires that the phase changes gradually from 0 to ṕ ℓ 2 over the full loop, while the other includes ℓ periodic phase changes from 0 to p 2 . The phase change can be correspondingly modulated base on the thickness of different transverse locations on the foil. These two approaches correspond to two types of spiral-shaped foils with two kinds of thickness design, as shown in figure 1. One type has descending thickness over a clockwise loop (see the bottom row in figure 1), which we name 'Single period spiral-shaped foil (SPF)' in this paper. The thickness of the other type is periodically repeated ℓ times, where each thickness period shows a descending trend (see the left column in figure 1); this type is named 'Multiple period spiral-shaped foil (MPF)'. To ensure the reflected laser pulse has a vortex structure, the foil surface is stepped into ℓ 8 parts, where the thickness interval is l 16. Thus, all these steps combine into an equivalent reflective SPP with total thickness difference of ĺ ℓ 2, generating an ℓ-order laser vortex. As shown in figure 1, the color of each step reveals phase changes of the reflected light in a far-field plane, which also indicates the thickness of each step. Each foil has an additional substrate with The color bars also mark the thickness of each step indirectly, which implies that the thickness of SPF is proportional to ℓ, while the thickness of MPF is fixed. Each foil has a substrate of l 0.5 thickness to ensure the laser pulse is totally reflected. thickness of l 0.5 to ensure complete reflection. Figure 1 shows examples of SPF and MPF, and the thickness variation with azimuth f for different spiral-shaped foils are shown in figure 2.
The incident relativistic Gaussian beam we use here can be written as is the radius of phase front, y = ( ) x x arctan R is the Gouy phase, and x R is the Rayleigh range. For ease of calculation, one thickness period of the foil is simply regarded as a perfectly smooth spiral mirror whose surface inclines with azimuthal position, i.e. the position of the spiral surface along the x axis is lf p = ℓ x 4 .When the incident Gaussian pulse arrives at this surface, the laser field is rearranged by asynchronous reflection. According to the Fresnel equations, we replace each x in equation (1) with x 2 4 , and then the reflected beam can be written as and y¢ = ¢ ( ) x x arctan R are associated with transverse azimuth f. We see that equation (2) is an OV with a topological charge of -ℓ because of the helical f ℓ ( ) exp i phase term. Compared with a standard LG mode, it lacks the radical ( ) ℓ | | r w 2 term which is described as a doughnut-shaped structure. In our cases, due to the continuity of electromagnetic field on the foil surface and the law of independent propagation of light, a phase singularity appears at the center. The phase singularity creates a point discontinuity of electromagnetic field at the center. During propagation, the diffracted field will smooth away the point discontinuity, and create a doughnut-shaped structure [38]. The p ( ) exp i term implies half-wave loss from the denser to thinner medium.
We implement our scheme in a 3D PIC simulation based on EPOCH code [39].
which is close to the value reported in [14].
i i i about one order of magnitude higher than that for electrons. During the interaction process, electrons are easily moved by the laser electromagnetic field, and then drag protons along with them through a charge separation field between them. This process is similar to the radiation process acceleration, but in the azimuthal direction [40]. Thus, the angular velocity of protons is expected to be lower than that of electrons. Both the proton's huge mass  4 1 respectively. One should note that the net AM for protons in an SPF is obviously larger than that in an MPF for the same ℓ, as shown in figures 3(f) and (h) for the = ℓ 2 cases, and similar comparisons for the = ℓ 3 and = ℓ 4 cases. This is because each discontinuity on a structured foil contributes a negative part to total AM. Figure 1 shows that there are more discontinuities in an MPF than those in an SPF for the same ℓ, so the net AM of an MPF is lowered more seriously. Thus, the net AM difference between MPF and SPF is attributed to the discontinuity. Moreover, when the misalignments exist between the incident laser and the foil, the OAM of reflected laser will decrease to some extent. Because the laser spot does not focus on the foil center, the reflected OV becomes unperfect, and finally causes the decrease of the OAM.
Surprisingly, we find that the time evolution of plasma AM is totally different between SPF and MPF cases.  However, the AMs for both protons and electrons do not oscillate in MPF cases for = ℓ 2, 3, 4, as shown in figures 3(f), (j) and (n). This phenomenon implies that the structure of MPF eliminates the AM oscillation effect. We see that both cases for = ℓ 4 do not have AM oscillation, which is worth raising. One should also note that the AM oscillation frequency is the same as the laser field frequency, and thus the AM oscillation may be aroused by the first harmonic of the laser field directly. Unlike the general particle oscillation, the particle AM oscillation acts in a new azimuthal direction. Thus, this effect may have an impact in some physical process aroused by the particle oscillation, such as ultraviolet radiation [41], high harmonic generation [20], particle heating [42,43], and so on, and then transfer AM to the electromagnetic field.

Analysis and discussion
As mentioned above, the AM oscillation is strongly related to the transient laser field which also induces quiver motion for particles in the transverse direction. Thus, it can be expected that the AM oscillation originates from such quiver motion [44], which is confirmed by the following analytical results. However, the charge separation field and the subsequently net AM is not included in our theoretical model, because it is very complicated to describe such field on the step-like foil surface. As an example, we consider electron motion under the interaction of the incident Gaussian pulse and reflected vortex pulse with singularity, as described by equations (1) and (2) respectively. These two pulses simultaneously interact with the periodic helical surface of the foil. On the surface, both pulses are assumed to have the same transverse electric field distribution due to the continuous boundary condition between the foil and vacuum. The motion of an electron can be described by ¶ ¶ where V represents the spatial domain of the foils for integration, which is quite different between SPF and MPF. As a result, we obtain different situations for the SPF and MPF cases, as shown in equation (6). More details can be found in supplementary material, which is available online at stacks.iop.org/NJP/21/043022/mmedia. From this result, we find that the AM of electrons does not oscillate during the interaction in all MPF cases with > ℓ 1, but non-oscillating AM only occurs in some SPF cases with even > ℓ 2. In addition, in the other SPF cases with AM oscillation, the oscillation frequency is equal to the laser frequency w, which implies the oscillating electric field induces AM oscillation during the interaction. In order to confirm the accuracy of calculation of equation (6), we plot the AM curves for electrons in figure 4 with pink lines in the SPF cases for = ℓ 1, 2, 3, 5, where oscillations are predicted to happen. Meanwhile, the PIC simulation results are shown in figure 4 with red stars for the same ℓ, which shows that the theoretical lines from equation (6) and simulation points are in good agreement. However, the PIC results after 80 fs deviate below the theoretical lines, implying the net AM amounts first increase and then saturate. Here we should note our theoretical calculations do not include the nonlinear effects (for instance, the pondermotive force), which are probably the source of the net AM.
As for protons, the Lorentz force in the laser electromagnetic field is just opposite to that of electrons. Thus, we can describe the motion of a proton as  p i are the number density, velocity, and linear momentum of a proton. Based on this, the calculated AM of protons is the same as electrons except an opposite sign.
We can explain the different oscillation situations for SPF and MPF cases in an intuitive way, and we take the cases of = ℓ 2 as examples in the following. Firstly, as shown in figures 5(a) and (d), an MPF and an SPF have different foil structures, so they cause different distributions of electric field on the foil surface  E surf when a Gaussian pulse impinges on either one, as shown in figures 5(b) and (e). Here, the electric field distribution on each step of the foil is cut out as a fan-shaped view, and then all the views are spliced together to obtain  E surf in figures 5(b) and (e). It can be seen that the electric filed on the MPF surface is always symmetric about the foil center, but such symmetry seldom exists in the SPF case. Secondly, one should note that only the electrons within the skin depth are influenced by the electromagnetic field, so the spatial distribution of electron momentum  p e is directly linked with  E . surf Therefore, we see different spatial distributions of current density   where  P e is the electron momentum density integrated along the x direction. Here, the first term can be neglected because the electron momentum is mostly occupied with the y component due to the incident y-polarized laser pulse. In the MPF case, we find the central symmetry in J y or P ey at the S yz plane, that is, , .

ey ey
This condition leads to a zero for the integration, so we obtain = AM 0.
x During the interaction, the central symmetry always exists in both  E surf and  J , so the electron AM is slow varying without any oscillation. In the SPF case, such symmetry in J y or P ey is broken, so the integrated electron AM has a surplus and oscillates with the laser field. As stated above, we find that the broken central symmetry in the electric field on the foil surface results in the AM oscillation for electrons. The analysis for protons is similar as electrons, so we can draw the same result.
Since AM oscillation for both proton and electron depend on the structure of the foil surface, we can redesign the foil surface to strengthen or weaken AM oscillation, and meanwhile properly generate an OV with specific topological charge. Figures 6(b) and (e) show that vortex pulses with topological charges of −1 and −3 are also generated using the redesigned spiral-shaped foils shown in figures 6(a) and (d), and the foil thicknesses along the azimuthal direction are different from SPF. In the = ℓ 1 case, part of the surface is lowered by l 0.5 compared with the corresponding SPF case to ensure the vortex structure of the reflected pulse. Based on this, which part of the surface is lowered determines the electric field distribution on the foil surface and its central symmetry. As a result, the oscillation strength of the particle AM is much smaller than that in the SPF case (see figures 3(d) and 6(c)) because the central symmetry is increased. In the = ℓ 3 case, one part of the surface is lowered while the other part is lifted by l 0.5 compared with the corresponding SPF case, and the oscillation becomes stronger than that in the SPF case (see figures 3(l) and 6(f)) due to the reduced central symmetry. These results further convince us that heavier asymmetry induces stronger oscillation, thus providing an effective method to modulate the AM oscillation strength for potential applications and gaining a deeper physical understanding regarding the origin of AM. In addition, more discontinuities added onto the foil surface decreases the net AM of protons (see figures 3(l) and 6(f)), which coincides with our aforementioned comparison between the net AM in SPF and MPF.

Conclusion
In summary, two types of structured spiral-shaped foils (SPF and MPF) are investigated for generating relativistic vortex pulses with specific topological charge ℓ using an ultra-intense femtosecond Gaussian pulse. The vortex pulse carries massive OAM. According to AM conservation, an equivalent amount of AM is transferred to the foil and results in rotation. However, the AM of the foil is not always smoothly increasing. Under specific conditions, this process is accompanied by oscillation. The AM oscillation pattern is analyzed for SPF and MPF, and the consistency of PIC simulations and theoretical calculation shows that the oscillation originates from particle quiver motion and central symmetry of the laser field on the foil surface.
As stated above, AM oscillation is a dynamic process originating from the collective quiver motion of all electrons or protons in the foil, and such oscillation is found to depend on the foil structure, or precisely, the central asymmetry of the electric field on the foil surface. This finding convinces us that the nature of AM oscillation is a mere asymmetry problem, and it is unrelated to net AM of particle. Whether the particle AM of a specific foil oscillates or not can be known by making asymmetry analysis. Due to the oscillatory characteristic of this phenomenon, we can expect that new AM effects may happen in radiation emitted by electrons, particle heating, and other mechanisms with potential applications. In addition, AM primarily carried by protons implies a similar process as ion radiation pressure acceleration mechanism. For radiation pressure acceleration, both the laser electric field and pondermotive force initially combine to affect electrons, and then accelerate protons through a charge separation field between the electrons and protons. Similar to this longitudinal acceleration, the net AM of the protons predicts an angular acceleration and subsequently rotation of the protons. For instance, when the net AM of proton does not oscillate in the MPF case, the foil may be continuously and steadily driven to rotate with extremely large torque. As the net AM in the foil grows with topological charge ℓ nearly linearly to the current extent, we expect realization of an ultrafast microscopic motor driven with an ultra-intense laser. Moreover, it is also predictable that the interaction between a relativistic laser pulse and a spiral-shaped foil will result in twisted high harmonics, as we use a vortex oscillation mirror model on the foil surface [20].