Abstract
Light beams with helical phasefronts are known to carry orbital angular momentum (OAM) and provide an additional degree of freedom to beams of coherent light. While OAM beams can be readily derived from Gaussian laser beams with phase plates or gratings, this is far more challenging in the extreme ultraviolet (XUV), especially for the case of high XUV intensity. Here, we theoretically and numerically demonstrate that intense surface harmonics carrying OAM are naturally produced by the intrinsic dynamics of a relativistically intense circularlypolarized Gaussian beam (i.e. nonvortex) interacting with a target at normal incidence. Relativistic surface oscillations convert the laser pulses to intense XUV harmonic radiation via the wellknown relativistic oscillating mirror mechanism. We show that the azimuthal and radial dependence of the harmonic generation process converts the spin angular momentum of the laser beam to orbital angular momentum resulting in an intense attosecond pulse (or pulse train) with OAM.
Introduction
Angular momentum is an intrinsic property of light, with spin angular momentum (SAM) of \(\pm\! \hslash\) per photon carried by circularly polarized (CP) light discussed by Poynting as early as 1909^{1}. At the end of last century it was demonstrated that light beams with helical phasefronts (optical vortices), described by a transverse phase structure of \(\exp (il\phi )\), where \(\phi\) is the azimuthal angle, carry an OAM equivalent to \(l\hslash\) per photon^{2}, and can therefore carry much larger OAM than is possible with SAM alone. Since then, laser beams carrying OAM have been exploited in various applications ranging from optical manipulation^{3}, imaging^{4,5}, and quantum optics^{6,7}, to optical communications^{8,9}.
Currently, OAM beams are generated by introducing azimuthally dependent phase to the initial Gaussian laser with the help of the optical elements, such as spiral phase plates^{10}, spatial light modulators^{11}, gratings^{12}, and \(q\)plates^{13}. XUV/Xray pulses with OAM are of particular interest for certain classes of experiments^{14,15}, preferably in combination with pulse durations in the attosecond regime^{16}. To date highorder harmonics in the XUV with OAM have been observed in laseratom interactions, operating at a moderate intensity (\(\sim\! 1{0}^{14}\,{\rm{W}}\,{{\rm{cm}}}^{2}\)) level^{17,18,19,20,21,22}. With progress made in FEL beams^{23}, producing intense, attosecond, ultrafast XUV pulses with OAM is a challenge that has yet to be met.
The goal of achieving intense attosecond pulses with OAM in the XUV naturally suggests employing the current generation of multiTW or even PW lasers to drive the interaction. In the relativistic regime, the proposed methods to generate intense XUV pulses with OAM mainly utilize vortex laser pulses as the drivers^{24,25,26}, which suffer from a limitation for the driving intensity. Another approach to generating OAM is spintoorbital angular momentum conversion which has been demonstrated in quantum physics^{27} and optics^{13}.
In this article, we show that the SAM of circularly polarized, highpower laser can be transferred to the harmonics carrying OAM via the relativistic oscillating mirror (ROM) mechanism, resulting in a single attosecond pulse with OAM when a fewcycle laser pulse is employed.
Results
Harmonics generation from a selfdented target
The principle of SAM to OAM conversion using the ROM^{28,29,30,31} mechanism is shown in Fig. 1. An intense lefthanded (as defined from the point of view of the observer) circularly polarized Gaussian laser pulse impinges a plane target from the left side. The normalized peak amplitude \({a}_{0}\) of the laser field \({{\bf{E}}}_{{\rm{L}}}\) is \({a}_{0}=e {{\bf{E}}}_{{\rm{L}}} /{m}_{{\rm{e}}}{\omega }_{{\rm{L}}}c=2.8\), corresponding to a laser intensity of \(1.7\,\times 1{0}^{19}\,{\rm{W}}\,{{\rm{cm}}}^{2}\). Here \({m}_{{\rm{e}}}\) is the electron mass, \(c\) is the vacuum light speed, and \(e\) is the electron charge. The laser wavelength is \({\lambda }_{{\rm{L}}}=0.8\,\upmu {\rm{m}}\), the laser period is \(\tau =2.67\,{\rm{fs}}\), the laser angular frequency is \({\omega }_{{\rm{L}}}\) = 2.36 \(\times\) 10\({}^{15}\) rad s\({}^{1}\) and the laser spot radius is \({w}_{0}=4{\lambda }_{{\rm{L}}}\). The details of simulation can be found in the Methods section. The ROM mechanism relies on the modification of the reflected waveform by the relativistic oscillation of the plasma surface. The dominant oscillation modes are at the laser frequency \({\omega }_{{\rm{L}}}\) due to the oblique component of the laser field or at 2\({\omega }_{{\rm{L}}}\) due to the \({\bf{v}}\,\times {\bf{B}}\) force. This generally results in the production of both odd and even harmonic orders^{29}. In the case of normal incidence only the \({\bf{v}}\,\times {\bf{B}}\) force remains, in which case only odd harmonics are produced for linearly polarized lasers. Note that for CP pulses the \({\bf{v}}\,\times {\bf{B}}\) force along the laser axis is slowly varying (i.e. it follows the temporal envelope of the laser^{32}). Thus, for normally incident CP Gaussian laser pulse, the \({\bf{v}}\,\times {\bf{B}}\) force no longer results in surface oscillations and, consequently, the reflected waveform is not modified any more and no harmonics are produced.
However, the radiation pressure along the target normal (mediated by the slowly varying \({\bf{v}}\, \times \, {\bf{B}}\) force) results in rapid target deformation (socalled denting) for sufficiently high intensities^{33,34,35,36}. This breaks the symmetry of the interaction and results in the laser becoming increasingly obliquely incident away from laser axis and therefore the generation of harmonics in the outer parts of the focal spot, while the suppression of harmonic generation on the axis remains. These harmonics are driven by the perpendicular component of the transverse electric field (perpendicular to the local target surface) and result in surface oscillation with a frequency of \({\omega }_{{\rm{L}}}\) and therefore both odd and even harmonics are expected. Our simulations of the interaction of a normally incident CP laser clearly show both odd and even harmonics in the reflected spectrum in Fig. 1b. In accordance with our expectations, the temporal structure after filtering of the lower order (\(n\le 4\)) harmonics shows that a train of attosecond pulses will be obtained (Fig. 1c). The intensity of these attosecond pulses is about \(2.2\, \times \, 1{0}^{15}\,{\rm{W}}\,{{\rm{cm}}}^{2}\), while the intensity of the driving laser pulse is \(1{0}^{19}\,{\rm{W}}\,{{\rm{cm}}}^{2}\).
Harmonics carrying orbital angular momentum
We recall that a characteristic of OAM beams is the vortex phase structure with zero intensity on axis. Since the harmonics in our geometry are only produced offaxis, where the laser is obliquely incident on the deformed target, the resulting intensity profile with zero emission on axis is suggestive of an OAM beam (although this is not a sufficient condition to conclude the presence of OAM). In the present scheme, the laser field component driving the oscillations can be expressed as \({E}_{\perp }={E}_{0}\cos \alpha \sin ({\omega }_{{\rm{L}}}t{k}_{{\rm{L}}}z+\phi )\) (see Methods section), where \({E}_{0}\) is the laser field amplitude, \({k}_{{\rm{L}}}\) is the laser vector, \(\alpha\) is the opening angle of the target, and \(\phi\) is the azimuthal angle of the interaction point. Clearly the oscillating phase of the plasma mirror is related with its azimuthal angle. In Fig. 1d, we plot the electron current in the \(z\) direction, \({J}_{{\rm{z}}}\), for a fixed \(z\) position but at different times. The current \({J}_{{\rm{z}}}\) is directly related to the oscillation velocity \({v}_{{\rm{z}}}\) of the plasma mirror. We see that \({J}_{{\rm{z}}}\) (and therefore \({v}_{{\rm{z}}}\)) is \(\phi\) dependent and rotates with time. Since the harmonic emission time is directly dependent on \({v}_{{\rm{z}}}\), the harmonic phase fronts depend on the azimuthal angle \(\phi\) and surface of equal phase must meet the condition \({\omega }_{{\rm{L}}}t{k}_{{\rm{L}}}z+\phi ={\mathrm{const}}\), corresponding to a twisted phasefront characteristic of OAM beams. In other words, OAM is introduced to the harmonics, as shown in Fig. 1a. The response of the dented target to the laser field is very similar to that of anisotropic and inhomogeneous \(q\)plates^{13}, which also create OAM light beams from an incident beam carrying SAM. A detailed derivation of the OAM generation is presented in the Methods section.
Figure 2 presents the phase structure of the first three harmonic orders. As can be seen in Fig. 2a the isosurfaces have a helical structure. The number of intertwined helices depends on the order of the harmonics, e.g., the number of helices \(h\) is \(h=n1\) where \(n\) is the harmonic order. The intensities distribution of each harmonic shown in the second column of Fig. 2 is doughnutlike as expected for a laser beam with OAM. The transverse distributions of the harmonic phase (third column of Fig. 2), indicates that the phases are azimuthally angle dependent. If we plot the phase along the azimuthal angle from \(\pi\) to \(\pi\) for a fixed radius, we obtain sawtooth shaped functions shown in the rightmost column of Fig. 2. The number of peaks of each line corresponds to the \(l\) number in the phase \(\varphi \sim \exp (il\phi )\), or the charge number \(l\) in the OAM of \(l\hslash\). Therefore, we can conclude that the \({n}{{\rm{th}}}\) harmonic carries \((n1)\hslash\) OAM. A detailed analysis can be found in the Methods section. The \((n1)\hslash\) dependence of the OAM on harmonic order \(n\) can be understood simply in terms of the conservation of angular momentum. It is well known that a CP photon carries an SAM of \(\hslash\) and that harmonic generation can be understood as the conversion of \(n\) photons with a frequency of \(\omega\) into one photon with a frequency of \(n\omega\). Since the reflected harmonic field propagating to the \(z\) direction has the same polarization but opposite handedness compared to the driving field (see the calculations in Supplementary Note 1), i.e., the SAM of the emitted photon with a frequency \(n\omega\) is still \(\hslash\) as the incident photon, the conservation of angular momentum implies an OAM of the emitted photon of \((n1)\hslash\).
A single attosecond pulse with OAM
For a target that dynamically deforms in response to the laser pressure, very few harmonics are produced in the leading edge of the laser pulse while the target is still approximately flat. However, this implies that for very short pulses, such as those required to generate a single attosecond pulse, there is insufficient time to achieve appreciable denting. We therefore consider the possibility of employing a predented target. This opens the possibility of controlling efficiency and denting independent from the intensity. Importantly, it also opens up the possibility of generating a single attosecond pulse with OAM, by shooting a fewcycle and intense laser pulse on a predented target^{37,38,39}. Figure 3 presents the generation of a pulse with a duration of 450 attoseconds and with an OAM of \(5\hslash\). The duration of the driving laser pulse is \(5.3\,{\rm{fs}}\). Since the target is predented, this short pulse can generate harmonics efficiently. We filtered from the \({6}{{\rm{th}}}\) to the \(1{6}{{\rm{th}}}\) harmonics and synthesized a single attosecond pulse. In Fig. 3a we present the structure of the single attosecond pulse, by coloring the space points with a threshold \({E}_{{\rm{x}}}\ge 20 \% {E}_{{\rm{xmax}}}\), in which \({E}_{{\rm{xmax}}}\) is the maximum of the filtered field \({E}_{{\rm{x}}}\). One can see a spiral structure during \(t=6.5\tau\) and \(t=7.5\tau\), when harmonics are most efficiently generated. The transverse profile of the filtered field in Fig. 3b clearly shows its OAM with \(5\hslash\). Here only a single value for the OAM is obtained because the intensity of the \({6}{{\rm{th}}}\) harmonic is dominated in the filtered harmonics (see the harmonic spectrum in Supplementary Note 2). The inhomogeneity in Fig. 3b comes from the CEP effect of the fewcycle laser pulse. The temporal evolution of the filtered field at a fixed point \((x,y)\) in Fig. 3c shows the duration of the single pulse is ~450 as. The intensity of this attosecond pulse is about \(1{0}^{17}\,{\rm{W}}\,{{\rm{cm}}}^{2}\), which is two orders of magnitude higher than that of the selfdenting case. While the intensities of the driving laser pulses are the same the amount \(f\) of maximum denting of the targets is different (\(f=0.5{\lambda }_{{\rm{L}}}\) for the predented target and \(f\approx 0.2{\lambda }_{{\rm{L}}}\) for the selfdenting case) resulting in stronger surface oscillations. What should be mentioned here is that, the pulse duration is strongly dependent on the filtering of the harmonic spectrum. Our 3D simulations are numerically limited to low orders and hence overestimate achievable pulse durations. In theory, a shorter pulse with OAM could be expected by using higher harmonic orders. For example, we present the results from 2D simulations with a higher resolution in Supplementary Note 3. It shows that up to \(3{0}{{\rm{th}}}\) harmonic can be clearly observed and an attosecond pulse with a duration of 240 as can be obtained by filtering out the lower orders (\(n\le 15\)) harmonics.
Discussion
In conclusion, we have found a route to converting PW class lasers to harmonics containing OAM and this is also compatible with generating an intense single attosecond pulse with OAM in the relativistic regime. The natural conversion of SAM to OAM makes this method more simple and more straightforward than other proposed methods^{24,25,26} in the relativistic regime. Our results may open exciting opportunities for the applications employing intense XUV attosecond pulses carrying OAM. For example, the nearrelativistic intensity of the vortex harmonics enables the applications such as transferring OAM of light to atoms^{40}, generating twisted gamma photons by Compton scattering^{41} and manipulation of relativistic vortex cutter^{42}. The attosecond duration of the isolated OAM pulse makes it an ideal tool for probing the chiral interactions^{43} on the subfemtosecond timescale and the ultrafast dynamics of spin and orbital moments in magnetic materials^{44}.
Methods
Principle of the transfer from SAM to OAM
The harmonics generation here is interpreted by the relativistic oscillating mirror (ROM) model^{28,29,30,31}, in which the target (actually plasma) surface serves as a mirror to reflect the light. When the laser field drives the surface, it will simultaneously oscillate with the same frequency of the laser. The field reflected by the oscillating surface contains highorder harmonics. The reflected laser field observed at \((t,z\, < \, 0)\) was emitted at a retarded time \({t}_{{\rm{ret}}}=tZ({t}_{{\rm{ret}}})/c+z/c\) from the oscillating source, where \(Z({t}_{{\rm{ret}}})\) is the position of the source at time \({t}_{{\rm{ret}}}\). In the case of small curvature of the target, the electric field at the observer can be approximately expressed as^{29}
where \(\eta\) is the reflection factor. As it turns out, it is the surface motion \(Z(t)\) that generates the harmonics. Because the normally incident laser here is circularly polarized, the \({\bf{v}}\, \times {\bf{B}}\) force has no high frequency oscillations and therefore, it will not contribute to the harmonics generation. Only the perpendicular component \({{\bf{E}}}_{{{\perp }}}\) of the transverse electric field \({{\bf{E}}}_{{\bf{0}}}\) will mainly oscillate the electron layer of the target surface (the longitudinal electric field coming from the tightly focusing has a weak effect here, as discussed in Supplementary Note 4). The transverse electric field can be expressed as \({{\bf{E}}}_{{\bf{0}}}={{\bf{E}}}_{{{\perp }}}+{{\bf{E}}}_{{{\parallel }}}\), where \({{\bf{E}}}_{{{\perp }}}\) is perpendicular to the target surface, and \({{\bf{E}}}_{{\boldsymbol{\parallel }}}\) is parallel to the target surface. From Fig. 4, one can get \({{\bf{E}}}_{{{\perp }}}\) at an arbitrary position (\(z\), \(\alpha\), \(\phi\)) on the target surface as
with \(\alpha\) the open angle of the cone target, \(\phi\) the azimuthal angle and \(r\) the radius in the transverse profile. Driven by this field \({{\bf{E}}}_{{{\perp }}}\), the electron surface originally at \(z\) moves as
where \({\varphi }_{0}(r)\) is the relative phase between the driving field and the surface oscillation, and it is a function of field strength^{29}.
Then we can rewrite the reflected electric field in Eq. (1) as
where we dropped the relative phase \({\varphi }_{0}(r)\) and made the approximation \({Z}{^{\prime} }(t)\simeq Z(t)\). Employing the JacobiAnger identity^{45}, we can get the Fourier expansion of Eq. (4) as
where \({{\rm{J}}}_{n}\) is the Bessel functions of the first kind. The last step stands when \(\epsilon \ll 1\). From Eq. (5) we can find that the \({(n+1)}{{\rm{th}}}\) harmonic carries an OAM with \(n\hslash\).
Particleincell simulations
The 3D simulations have been performed on JURECA^{46} at Jülich Supercomputing Centre and Tianhe2 at Guangzhou National Super Computer Center using 3D Particleincell (PIC) code LAPINE^{47,48}. The envelope of the incident laser pulse is shaped with a function of \({\sin }^{2}(\pi t/{T}_{0})\,(0\le t\le {T}_{0})\), with \({T}_{0}=25\tau\) for the selfdenting case and \({T}_{0}=4\tau\) for the predenting case. The target employed in the selfdenting case is plane, with a thickness of 0.8 \(\upmu {\rm{m}}\) and a density of \(2{n}_{{\rm{c}}}\), where \({n}_{{\rm{c}}}=1.1\times 1{0}^{21}{\lambda }_{{\rm{L}}}(\upmu {\rm{m}})\,{{\rm{cm}}}^{3}\) is the critical density. The shape of the target surface in the predenting case is designed as \(z=f\exp [({x}^{2}+{y}^{2})/{w}^{2}]+{z}_{0}\), with \(f=0.5{\lambda }_{{\rm{L}}}\) and \(w=4{\lambda }_{{\rm{L}}}\). The target is placed at \({z}_{0}=6{\lambda }_{{\rm{L}}}\). A detector is placed at \({z}_{{\rm{d}}}={\lambda }_{{\rm{L}}}\) to collect the reflected field. The size of the simulation box is \(24{\lambda }_{{\rm{L}}}({\rm{x}})\times 24{\lambda }_{{\rm{L}}}({\rm{y}})\times 10{\lambda }_{{\rm{L}}}({\rm{z}})\) corresponding to grids \(1200({\rm{x}})\times 1200({\rm{y}})\times 1000({\rm{z}})\), with 8 macroparticles per cell. The resolution \({\rm{d}}{\rm{z}}=0.01{\lambda }_{{\rm{L}}}\) will limit the order number of the harmonics that can be observed <20.
Data availability
The data that support the findings of this study are available from the corresponding authors upon request.
References
 1.
Poynting, J. H. The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light. Proc. R. Soc. Lond. A Contain. Pap. Math. Phys. Character 82, 560–567 (1909).
 2.
Allen, L., Beijersbergen, M. W., Spreeuw, R. J. C. & Woerdman, J. P. Orbital angular momentum of light and the transformation of LaguerreGaussian laser modes. Phys. Rev. A 45, 8185–8189 (1992).
 3.
Padgett, M. & Bowman, R. Tweezers with a twist. Nat. Photon. 5, 343–348 (2011).
 4.
Maurer, C., Jesacher, A., Bernet, S. & RitschMarte, M. What spatial light modulators can do for optical microscopy. Laser Photon. Rev. 5, 81–101 (2010).
 5.
Fürhapter, S. et al. Spiral interferometry. Opt. Lett. 30, 1953–1955 (2005).
 6.
Leach, J. et al. Quantum correlations in optical angleorbital angular momentum variables. Science 329, 662–665 (2010).
 7.
Malik, M. et al. Multiphoton entanglement in high dimensions. Nat. Photon. 10, 248–252 (2016).
 8.
Wang, J. et al. Terabit freespace data transmission employing orbital angular momentum multiplexing. Nat. Photon. 6, 488–496 (2012).
 9.
Willner, A. E. et al. Optical communications using orbital angular momentum beams. Adv. Opt. Photonics 7, 66 (2015).
 10.
Sueda, K. et al. LaguerreGaussian beam generated with a multilevel spiral phase plate for high intensity laser pulses. Opt. Express 12, 3548–3553 (2004).
 11.
Heckenberg, N. R., McDuff, R., Smith, C. P. & White, A. Generation of optical phase singularities by computergenerated holograms. Opt. Lett. 17, 221–223 (1992).
 12.
Biener, G., Niv, A., Kleiner, V. & Hasman, E. Formation of helical beams by use of PancharatnamBerry phase optical elements. Opt. Lett. 27, 1875–1877 (2002).
 13.
Marrucci, L., Manzo, C. & Paparo, D. Optical spintoorbital angular momentum conversion in inhomogeneous anisotropic media. Phys. Rev. Lett. 96, 163905 (2006).
 14.
Van Veenendaal, M. & McNulty, I. Prediction of strong dichroism induced by X rays carrying orbital momentum. Phys. Rev. Lett. 98, 157401 (2007).
 15.
Patchkovskii, S. & Spanner, M. Nonlinear optics: high harmonics with a twist. Nat. Phys. 8, 707–708 (2012).
 16.
Hentschel, M. et al. Attosecond metrology. Nature 414, 509–513 (2001).
 17.
Zürch, M., Kern, C., Hansinger, P., Dreischuh, A. & Spielmann, C. Strongfield physics with singular light beams. Nat. Phys. 8, 743–746 (2012).
 18.
HernándezGarcía, C., Picón, A., Román, J. S. & Plaja, L. Attosecond extreme ultraviolet vortices from highorder harmonic generation. Phys. Rev. Lett. 111, 083602 (2013).
 19.
Gariepy, G. et al. Creating highharmonic beams with controlled orbital angular momentum. Phys. Rev. Lett. 113, 153901 (2014).
 20.
Géneaux, R. et al. Synthesis and characterization of attosecond light vortices in the extreme ultraviolet. Nat. Commun. 7, 12583 (2016).
 21.
Gauthier, D. et al. Tunable orbital angular momentum in highharmonic generation. Nat. Commun. 8, 14971 (2017).
 22.
Dorney, Kevin M. et al. Controlling the polarization and vortex charge of attosecond highharmonic beams via simultaneous spinorbit momentum conservation. Nat. Photon. 13, 123–130 (2019).
 23.
Ribič, P. R. et al. Extremeultraviolet vortices from a freeelectron laser. Phys. Rev. X 7, 031036 (2017).
 24.
Zhang, X. M. et al. Generation of intense highorder vortex harmonics. Phys. Rev. Lett. 114, 173901 (2015).
 25.
Vieira, J. et al. Amplification and generation of ultraintense twisted laser pulses via stimulated Raman scattering. Nat. Commun. 7, 10371 (2016).
 26.
Denoeud, A., Chopineau, L., Leblanc, A. & Quéré, F. Interaction of ultraintense laser vortices with plasma mirrors. Phys. Rev. Lett. 118, 033902 (2017).
 27.
Nagali, E. et al. Quantum information transfer from spin to orbital angular momentum of photons. Phys. Rev. Lett. 103, 013601 (2009).
 28.
Bulanov, S. V., Naumova, N. M. & Pegoraro, F. Interaction of an ultrashort, relativistically strong laser pulse with an overdense plasma. Phys. Plasmas 1, 745 (1994).
 29.
Lichters, R., MeyerterVehn, J. & Pukhov, A. Shortpulse laser harmonics from oscillating plasma surfaces driven at relativistic intensity. Phys. Plasmas 3, 3425 (1996).
 30.
Baeva, T., Gordienko, S. & Pukhov, A. Theory of highorder harmonic generation in relativistic laser interaction with overdense plasma. Phys. Rev. E 74, 046404 (2006).
 31.
Thaury, C. et al. Plasma mirrors for ultrahighintensity optics. Nat. Phys. 3, 424–429 (2007).
 32.
Macchi, A., Cattani, F., Liseykina, T. V. & Cornolt, F. Laser acceleration of ion bunches at the front surface of overdense plasmas. Phys. Rev. Lett. 94, 165003 (2005).
 33.
Hörlein, R. et al. Dynamics of nanometerscale foil targets irradiated with relativistically intense laser pulses. Laser Part. Beams 29, 383–388 (2011).
 34.
Zepf, M. et al. Measurements of the hole boring velocity from Doppler shifted harmonic emission from solid targets. Phys. Plasmas 3, 3242 (1996).
 35.
Wilks, S. C. et al. Absorption of ultraintense laser pulses. Phys. Rev. Lett. 69, 1383 (1992).
 36.
Dromey, B. et al. Diffractionlimited performance and focusing of high harmonics from relativistic plasmas. Nat. Phys. 5, 146–152 (2009).
 37.
Geissler, M. et al. 3D simulations of surface harmonic generation with fewcycle laser pulses. New J. Phys. 9, 218 (2007).
 38.
Rykovanov, S. G. et al. Intense single attosecond pulses from surface harmonics using the polarization gating technique. New J. Phys. 10, 025025 (2008).
 39.
Heissler, P. et al. Fewcycle driven relativistically oscillating plasma mirrors: a source of intense isolated attosecond pulses. Phys. Rev. Lett. 108, 235003 (2012).
 40.
Picón, A. et al. Transferring orbital and spin angular momenta of light to atoms. New J. Phys. 12, 083053 (2010).
 41.
Chen, Y. et al. \(\gamma\)Ray beams with large orbital angular momentum via nonlinear compton scattering with radiation reaction. Phys. Rev. Lett. 121, 074801 (2018).
 42.
Wang, W. P. et al. New optical manipulation of relativistic vortex cutter. Phys. Rev. Lett. 122, 024801 (2019).
 43.
Cireasa, R. et al. Probing molecular chirality on a subfemtosecond timescale. Nat. Phys. 11, 654–658 (2015).
 44.
Boeglin, C. et al. Distinguishing the ultrafast dynamics of spin and orbital moments in solids. Nature 465, 458–461 (2010).
 45.
Cuyt, A., Petersen, V., Verdonk, B., Waadeland, H. & Jones, W. B. Handbook of Continued Fractions For Special Functions (Springer, 2008).
 46.
Forschungszentrum Jülich. JURECA: generalpurpose supercomputer at Jülich Supercomputing Centre. J. LargeScale Res. Facil. 2, A62 (2016).
 47.
Xu, H. et al. Parallel programming of 2 1/2dimensional PIC under distributedmemory parallel environments. Chin. J. Comput. Phys. 19, 305 (2002).
 48.
Lei, B., Wang, J., Kharin, V., Zepf, M. & Rykovanov, S. \(\gamma\)Ray generation from plasma wakefield resonant wiggler. Phys. Rev. Lett. 120, 134801 (2018).
 49.
Zacharov, I. et al. “Zhores”  Petaflops supercomputer for datadriven modeling, machine learning and artificial intelligence installed in Skolkovo Institute of Science and Technology. Open Engineering 9, 512–520 (2019).
Acknowledgements
We are grateful to Dr. Vasily Kharin, Prof. Baifei Shen for fruitful discussions. This work was supported by the National Natural Science Foundation of China (NSFC 11674341), the Helmholtz Association (Young Investigator Group VHNG1037), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB1603) and Chinese Academy of Sciences President’s International Fellowship Initiative (Grant No. 2018VMC0012). We gratefully acknowledge the computing time granted by the John von Neumann Institute for Computing (NIC) and provided on the supercomputer JURECA at Jülich Supercomputing Centre (JSC), and by the National Supercomputer Center in Guangzhou. Part of the simulations was performed using the Skoltech CDISE supercomputer “Zhores”^{49}.
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J.W.W. and S.G.R. designed the concept. J.W.W. and S.G.R. carried out the simulations and calculations, and J.W.W. drafted the manuscript. S.G.R. and M.Z. contributed to analysing of the results and writing of the manuscript. All authors discussed the results and commented on the manuscript.
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Wang, J.W., Zepf, M. & Rykovanov, S.G. Intense attosecond pulses carrying orbital angular momentum using laser plasma interactions. Nat Commun 10, 5554 (2019). https://doi.org/10.1038/s41467019133571
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