High-Harmonic Generation and Spin-Orbit Interaction of Light in a Relativistic Oscillating Window

When a high power laser irradiates a small aperture on a solid foil target, the strong laser field drives surface plasma oscillation at the rim of the aperture, which acts as a"relativistic oscillating window". The diffracted light travels though such an aperture contains high-harmonics of the fundamental laser frequency. When the driving laser is circularly polarised, the high-harmonic generation (HHG) process facilitates a conversion of the spin angular momentum of the fundamental light into the intrinsic orbital angular momentum of the harmonics. By means of theoretical modelling and fully 3D particle-in-cell simulations, it is shown the harmonic beams of order $n$ are optical vortices with topological charge $|l| = n-1$. When the driving laser is significantly intense, the HHG is dramatically enhanced through a surface wave breaking effect, which leads to a universal power-law spectrum $I_n\propto n^{-3.5}$ at ultra-relativistic limit, where $I_n$ is the intensity of the $n$th harmonic. This work opens up a new realm of possibilities for producing intense extreme ultraviolet vortices, and diffraction-based HHG studies at relativistic intensities.

Light carries angular momentum as spin and orbital components. The spin angular momentum (SAM) is associated with right or left circular polarisation (± per photon), and the orbital angular momentum (OAM) is carried by light beams with helical phase fronts exp(ilφ) (l per photon), also known as optical vortices, where l is the topological charge and φ is the azimuthal angle [1]. The spin-orbit interaction of light refers to phenomena in which the spin affects the orbital degrees of freedom [2], such as spin-Hall effects [3,4], spin-dependent effects in nonparaxial fields [5] and evanescent waves [6]. Recently, interest in spin-orbit interaction has surged, as it provides physical insight into the behaviour of polarised light at sub-wavelength scales, which is essential in nano-optics and photonics. In addition, spin-orbit angular momentum conservation is also an important concept to produce optical vortices [11], that have rich variety of applications in optical communication [7,8], biophotonics [9], and optical trapping [10].
The production of optical vortices with such methods mostly rely on high-harmonic generation (HHG) in laser-atom interactions, driven by a moderately intense (∼ 10 14 W/cm 2 ) beam [11][12][13][14][15]. The resulting extreme ultraviolet (XUV) vortices are of particular interest for monitoring and manipulating the SAM and OAM of light-matter interactions on the atomic scale, as well as for applications such as nonlinear optics [16,17] and superresolution microscopy [18]. Owing to the remarkable progresses in high-power lasers [19], such advanced light sources open up new possibilities in the relativistic regime (> 10 18 W/cm 2 ) of laser-matter interactions [20][21][22][23][24][25], and can yield fundamental insights into the spin-orbit/orbitorbit angular momentum interactions at ultra-high intensities [26][27][28][29]. It is reported recently [30] that by irradiating a solid foil with a circularly polarised (CP) * longqing@chalmers.se high-power laser, the SAM of the driver can be converted into OAM of the harmonic beams through the relativistic oscillating mirror (ROM) mechanism [31][32][33], giving rise to intense, ultrafast XUV vortices. However, according to the ROM theory [33], HHG is suppressed for CP driver at normal incidence. Therefore the mechanism relies crucially on the pre-denting of the target surface by radiation pressure, and typically produces relativelyweak harmonic intensities [30].
Here we present, for the first time, a semi-analytical theory of HHG based on light diffraction at relativistic intensities.
A new HHG mechanism is identified, which we call relativistic oscillating window (ROW). It allows for producing harmonic beams efficiently with a CP driving laser, and simultaneously, facilitating spin-to-orbital angular momentum conversion that produces ultra-intense XUV optical vortices. The ROW mechanism relies on a high-contrast [34] relativistic laser travelling through a small aperture on a thin foil, with dynamic surface electron oscillation on the rim, driven by the strong laser field. We demonstrate that the diffracted light through such a window contains both even and odd harmonics of the fundamental driving laser frequency; a universal power-law spectrum I n ∝ n −3.5 is produced for any sufficiently intense CP driver, where n is the harmonic order and I n is the intensity of nth harmonic; all the harmonic components (n ≥ 2) are optical vortices with topological charge l = (n − 1)σ, where σ denotes the right (σ = +1) or left (σ = −1) handed circular polarisation of the driving laser.

Results
We first demonstrate our scheme with 3D particle-incell (PIC) simulations. The simulation setup and the main results are summarised in Fig. 1: a right-handed CP laser irradiates normally on a solid foil located at x 0 = 4 µm with a small aperture (radius r 0 = 3.3µm) aligned with the laser beam [see Fig. 1(a)], the thickness of the foil is L f = 0.25µm. The intensity of the laser is I 0 ≈ 6.8 × 10 19 W cm −2 , corresponding to a normalised laser amplitude of a 0 ≡ eE 0 /m e cω 0 = 5, where E 0 is the laser field, e, m e , c and ω 0 denote the elementary charge, electron mass, vacuum light speed, and the laser frequency, respectively. The simulation parameters are detailed in Methods. With such intensity, the laser drives strong surface electron oscillations on the rim of the aperture, which modifies the local plasma density as shown in Fig. 1(b). The region with electron density above the critical density (n c = m e ω 2 0 /4πe 2 ≈ 1.1 × 10 21 cm −3 ) is reflective to the laser, therefore the transparent area [indicated by the white dashed lines in Fig. 1 where light can travel through, acts as a "relativistic oscillating window".
The diffraction of light through such an aperture generates high-harmonics of the fundamental driving laser frequency. Figure 1(c) presents a typical HHG spectrum; one can see that both even and odd harmonic orders are generated, and the spectrum has a power-law shape that can be fitted by I n ∝ n −3.5 . The spectrum is obtained as Fourier transform of the fields recorded at an observational plane 11 µm away from the screen (x = x 0 + 11 µm), and is averaged within an opening angle of θ = 30 • .
Each harmonic with order n is then selected by spectral filtering in the frequency range [n − 0.5, n + 0.5]ω 0 , as shown by Fig. 1(d-i). Notably, the spin-orbit interaction of light takes place in the HHG, all harmonics (n ≥ 2) are optical vortices with topological charge l = (n − 1)σ. The sign of l is controlled by the polarisation of the driving laser (σ), which determines the chirality of the oscillating window, and has a profound impact on the orbital degree of freedom of the harmonics.
To understand these results, one must consider the diffraction of light through an oscillating aperture on a screen, for which the boundary condition is crucial. As discussed by Baeva et al. [33], the tangential component of electric field associated with a plasma surface current is negligibly small compared to the laser field E 0 . Therefore it is reasonable to assume that on the rear side of the screen, the tangential components of electric field vanish everywhere except in the aperture, where they can be approximated by that of the incoming laser fields. For the sake of simplicity, we will restrict our attention to the case of a monochromatic plane wave E(x, y, z, t) = U(x, y, z) exp (−iω 0 t) normally incident on the screen, where U(x, y, z) satisfies Helmholtz equation (∇ 2 + k 2 0 )U = 0, and k 0 = ω 0 /c is the laser wavenumber. For a stationary aperture, the solution is given by the generalised Kirchhoff integral [35]: where the integration is only over the aperture on the screen, e n is the unit vector rightward directed normal to the screen, For the situation under consideration in this work, each ds can be considered to be shaken by the laser field, and R becomes time-dependent, leading to nonlinear effects. We now introduce the ROW model, it assumes that the shape of the aperture does not change (rigid window), such that each element ds in the integral Eq. (1) is shifted by the same amount of displacement dR (t). This assumption is valid for weakly-relativistic drivers (laser intensity a 0 ≤ 0.3), we will extend the model to ultrarelativistic intensity (a 0 1) later. The diffracted field is then given by where R (t ) = |R − dR (t )| is the distance measured at retarded time t = t − R (t )/c. Due to the rigid window assumption, R (t ) can be obtained from the electron dynamics on the boundary of the aperture, which depends on the details of laserplasma interaction. Note that Eq. (2) is valid for both circularly and linearly polarised (LP) drivers, however the respective behaviours of R (t ) are distinct. For a LP driver, the window oscillates mostly in the polarisation direction, which produces HHG beams carrying no OAM.
For the purpose of the present work, we will proceed with CP drivers. At weakly-relativistic intensities, the surface electrons are simply shifted antiparallel to the driving laser field, resulting in a harmonic oscillation dR (t ) = −(e y + iσe z )δr 0 exp(−iω 0 t ), where e y (e z ) is the unit vectors in y (z) direction, and the amplitude δr 0 is limited to δr 0 ≤ k −1 0 in this case. The rim of the window is always attached to these oscillating electrons.
To calculate the diffracted fields, one must solve for the retarded time t = t − R (t )/c numerically according to the motion of the source: However, to explain the spin-orbit interaction in the HHG process, it is sufficient to derive analytically the lowest order of diffracted fields, valid for a 0 1, seen by a distant, paraxial observer, that satisfies (R r = y 2 + z 2 r 0 , δr 0 ). In this case we have R (t ) ≈ R + δr 0 sin(θ) exp(ik 0 R (t ) − iω 0 t + iσφ), where θ = arctan[r/(x − x 0 )] is the diffraction opening angle, and the azimuthal angle φ is measured counterclockwise with respect to the y-axis in the y-z plane [see Fig. 1(a)]. The phase term in Eq. (2) is then 1. Substituting it into Eq. (2) and make use of the Jacobi-Anger identity [36], yield: where R 0 = (x − x 0 ) 2 + y 2 + z 2 is the distance measured from the centre of the aperture, and J n are the Bessel functions of the first kind. Equation (4) shows that HHG beams have the same circular polarisation state as the driving laser, and their phase fronts are all helical, with a topological charge l = (n − 1)σ for the nth harmonic. It agrees well with the main findings from PIC simulations shown in Fig. 1. From a quantum optics point of view, this relation guarantees the conservation of total angular momentum and energy: when n photons at the fundamental frequency are transformed into one photon of nth-order harmonic, their SAM (nσ ) are converted into (n − 1)σ OAM plus σ SAM.
We wish to examine the ROW model in detail for drivers with higher intensity, by comparing the HHG spectra and the diffracted field distribution obtained from the model and the PIC simulations. For this purpose, Eqs. (2-3) are solved iteratively [32]. Figure 2(a) presents a typical solution of Eq. (3) for the distance between the centre of the aperture and an observer located 11 µm away from the screen with θ = 30 • . It shows that due to the time it take for the light to travel from the source to the observer, a harmonic oscillation of the source results in an anharmonic oscillation seen by the observer. This distortion due to retardation is the dominant mechanism to generate the high-harmonics. Figure 2(b) shows the HHG spectra obtained from PIC simulations for weakly-relativistic drivers. The intensities of HHG beams increase dramatically with laser a 0 . In particular, the spectra for small a 0 decays faster than exponentially with harmonic order n, only the secondorder harmonic is visible for a 0 = 0.1. This agrees with Eq. (4) as J n−1 ( ) ∼ ( /2) n−1 /(n − 1)!. As a 0 grows, the spectra asymptotically converges to a power-law shape I n ∝ n α . This trend can be reproduced by our model as indicated by the open circles in Fig. 2(b). However, the model suggests that the power-law exponent α is limited to around α = −8.7, since the amplitude δr 0 should be smaller than k −1 0 . According to Fig. 2(b), this is only true for a 0 ≤ 0.3, because at higher intensities, the electrons that oscillate on the boundary of the aperture may gain enough energy to escape the restoring force of plasma [37][38][39], as shown by the inset of Fig. 2(c). Therefore, the rim of the window can no longer be considered to be attached to these electrons, which significantly modifies the dynamics of the ROW.
This effect is essentially surface wave breaking (SWB) [40], that occurs when the driving force is too strong. To quantify when it should be taken into account, in Fig. 2(c) we plot the ratio of electron number that is emitted from the rim of the aperture divided by the total electron number in the skin layer (N s ≈ 2πr 0 k −1 0 L f n c ), as a function of the laser amplitude a 0 . One can see a surge of electron emission for a 0 > 0.3, as the power-law exponents exceed the prediction from the model based on a harmonic oscillating window (α = −8.7). Therefore, the SWB effect is essential to interpret the HHG process in the ultra-relativistic regime. Figure 3(a) presents a snapshot of typical plasma density distribution near the aperture when SWB occurs. The shape of the window, bounded by the solid curves (red and green), is no longer circular. The SWB effect allows the electrons to excurse far into the aperture, as they travel inwards. The rim of the window on this side (red solid curve), follows the motion of the electrons for about half of one laser cycle, where the maximum displacement is ∼ c × 0.5T 0 = 0.5λ 0 . Afterwards, most of these electrons are emitted away, and the rim of the window falls back to the original boundary as the the local plasma density drops and transparency is restored. On the other side (green solid curve), when the electrons travel towards the plasma bulk, the displacement remains small (∼ k −1 0 ). The diffracted field through such a deformed oscillating window can be calculated by separating the aperture into two parts, A1 and A2. As shown by Fig. 3(a), they are fractions of two rigid windows (represented by the red and green circles, consisting of both solid and dashed lines), which oscillate with different amplitudes δr A1 > δr A2 ≈ k −1 0 . The contributions from each part can then be obtained by integrating Eq. (2) over the area that satisfies r · dR (t ) ≤ 0 and r · dR (t ) > 0 for A1 and A2, respectively.
In this way, the ROW model can be extended to ultrarelativistic intensities as shown by Fig. 3(b). One can see the exponent of power-law spectrum is increased dramatically by the SWB effect, which allows half of the window to oscillate with a larger amplitude. The HHG spectra for a 0 > 0.3 can be reproduced from the model by adjusting the value of δr A1 . Setting δr A1 = 0.25λ 0 and 0.4λ 0 recovers the HHG spectra from PIC simulations with a 0 = 0.5 and 1 (adjusted to the fifth harmonic), respectively.
In particular, comparing the HHG spectra for a 0 = 2, a 0 = 10 [black and red colours in Fig. 3(b)], and a 0 = 5 [ Fig. 1(c)], one can infer that the power-law exponent is limited to around α = −3.5. This can be easily understood from our model, as the maximum displacement of electron layer is 0.5λ 0 . Thus, setting δr A1 = 0.5λ 0 gives a universal HHG spectrum for any sufficiently strong CP drivers. The results are presented by the black open circles in Fig. 3(b), which agree very well with the PIC simulations. In addition, both the ROW model and PIC  Fig. 3(c).
Finally, the HHG fields can be obtained by filtering the diffracted fields from the ROW model within a certain frequency range. Using the same parameters as in Fig. 1, and setting δr A1 = 0.5λ 0 , the corresponding second, third, and fourth harmonics are presented in Fig. 3(d-f), respectively. Apparently the results confirm the relation l = (n − 1)σ, and the harmonic field distributions agree very well with the PIC simulations shown in Fig. 1(g-i).
In conclusion, we have demonstrated that highharmonic beams are generated due to the diffraction of a high-power CP laser through a small aperture on a solid plasma foil. In this process, the spin angular momentum of the driving laser is converted into orbital angular momentum of the harmonics, giving rise to intense optical vortices in the XUV regime. Three dimensional PIC simulations show that a power-law HHG spectrum I n ∝ n α is produced, and a universal exponent α ≈ −3.5 is observed for sufficiently strong (a 0 ≥ 2) CP drivers. The topological charge of the nth harmonic light is l = (n − 1)σ. The high-harmonic generation and spin-orbit interaction of light stem from the chiral electron oscillation on the rim of the aperture, that act as a "relativistic oscillating window". Based on this picture, a semi-analytical model is developed, which agrees well with the numerical findings.

Methods
Laser-plasma parameters. In the simulation presented in Fig. 1, a circularly-polarised high-power laser beam enters the simulation box from the left (−x) boundary and propagates to the right. The laser field is E 0 = (e y + iσe z )E 0 sin 2 (πt/τ 0 ) exp(ik 0 x − iω 0 t), where 0 < t < τ 0 = 54 fs, E 0 = 16 TV m −1 is the laser amplitude, frequency ω 0 = k 0 c, wavenumber k 0 = 2π/λ 0 with λ 0 = 1µm the laser wavelength. The laser polarisation state is controlled by σ = +1 for RCP and −1 for LCP. In Figs. 2-3, the intensity of laser is changed while other parameters are kept the same.
Note that we have assumed that the laser focus spot is much greater than the size of the aperture, so that the intensity on the edge of the aperture does not depend on the its radius. This is only for the convenience of comparing the PIC results with our model, not crucial for the proposed mechanism to work.
The foil target [assumed plastic (CH)] is modelled by a pre-ionised plasma with electron density n 0 = 30n c , thickness L f = 0.25 µm. The radius of the aperture are r A = 4.0 µm. In order to account for heating by the laser pre-pulse, the inner boundary (r < r A ) is assumed to have a density gradient n(r) = n 0 exp[(r − r A )/h], with scale length h = 0.2µm. which yields an effective radius r 0 = 3.3µm, for which n(r 0 ) = 1n c .
PIC simulations. The 3D PIC simulations presented in this work were conducted with the code epoch [41], and the algorithm developed by Cowan et al. [42] is used to minimise the numerical dispersion. For most of the simulations presented in this paper, the dimensions of the simulation box are x×y×z = 15µm×16µm×16µm, sampled by 2400 × 320 × 320 cells with four macro particles for electrons, two for C 6+ and two for H + ions. A higher transverse resolution x × y × z = 10µm × 10µm × 10µm, sampled by 1000 × 500 × 500 cells, and 14 macro electrons per cell is used to simulate the fine details of density fluctuation in the ultra-relativistic case, presented in Fig. 3(a). A high-order particle shape function (fifth order particle weighting) is applied to suppress numerical self-heating instabilities (see Sec. 5.1 of ref. [41] for details).

Data availability.
The data that support the findings of this study are available from the corresponding author upon request.