Angular momenta (spin or orbital) are recognized as critical light characteristics. The spin angular momentum (SAM,
High Power Laser Science and Engineering, Volume. 10, Issue 6, 06000e46(2022)
Intense harmonic generation driven by a relativistic spatiotemporal vortex beam
Spatiotemporal optical vortex (STOV) pulses carrying purely transverse intrinsic orbital angular momentum (TOAM) are attracting increasing attention because the TOAM provides a new degree of freedom to characterize light–matter interactions. In this paper, using particle-in-cell simulations, we present spatiotemporal high-harmonic generation in the relativistic region, driven by an intense STOV beam impinging on a plasma target. It is shown that the plasma surface acts as a spatial–temporal-coupled relativistic oscillating mirror with various frequencies. The spatiotemporal features are satisfactorily transferred to the harmonics such that the TOAM scales with the harmonic order. Benefitting from the ultrahigh damage threshold of the plasma over the optical media, the intensity of the harmonics can reach the relativistic region. This study provides a new approach for generating intense spatiotemporal extreme ultraviolet vortices and investigating STOV light–matter interactions at relativistic intensities.
Angular momenta (spin or orbital) are recognized as critical light characteristics. The spin angular momentum (SAM,
Concurrent with progress in high-power lasers[6,7], the production of intense spatial vortex beams in the relativistic region (
Apart from longitudinal OAM, theoretical predictions[28,29] and recent experiments[30–32] indicate that light can possess a new class of OAM, tilted or orthogonal to the propagation direction, that is, transverse intrinsic orbital angular momentum (TOAM). In contrast to the conventional spatial vortex beam, such light-carrying TOAM is essentially polychromatic with phase singularity in the spatiotemporal domain, known as the spatiotemporal optical vortex (STOV)[33–35]. Recently, investigations have shown that TOAM can be broadly incorporated into cylindrical vector, partially temporally coherent, diffraction-free Bessel and traditional spatial vortex beams. In the latter cases, a spatiotemporal wavepacket with orientation-controllable OAM[39–41] can be produced by assembling TOAM and longitudinal OAM, which may be applied in optical spanners with arbitrary three-dimensional orientation.
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Accordingly, the interaction of such STOV pulses with matter has been investigated. Novel types of transverse pulse shifts and time delays can be induced when an STOV beam is reflected and refracted at a planar interface, which differs from the spatial deflection effect related to spatial vortices[43,44]. The generation of high harmonics carrying TOAM has also been reported, with beta-barium borate crystals[45,46] and gaseous targets as conversion media. However, the STOV beams involved are restricted to low intensities, considering the limitations of the media damage thresholds. Very recent studies have demonstrated that STOV beams could be focused on subwavelength spatial sizes and femtosecond pulse durations[48,49], exhibiting the ability to yield high-intensity STOV pulses. In addition, they can be produced through the coherent beam combing technique that superposes intense plane waves with different wavevectors. Other plasma-based methods have also been proposed to produce high-intensity vortex beams with tilted or transverse OAM[50–52]. In this regard, it is reasonable to expect that novel nonlinear features and additional application scenarios might arise when the intensities of such STOV pulses become relativistic.
In this paper, we propose and demonstrate spatiotemporal HHG in the relativistic region driven by an intense STOV beam impinging on a solid plasma target, as shown in Figure 1(a). The red torus with wavevector
Figure 1.(a) Schematic of proposed setup. A linearly -polarized spatiotemporal optical vortex (STOV, red torus) pulse with purely transverse orbital angular momentum (TOAM) is incident onto a solid plasma target. Harmonics can be generated in the reflected beam (blue torus). (b) Snapshots of electric field at . (c) Frequency spectrum of (b) generated by performing Fourier transform in the -direction. (d) Time-averaged energy density of the STOV beam. The overlaid white arrows represent the circulated momentum flux. (e) TOAM density with the subtracted propagation term. The red arrow in (b) shows the beam-propagating direction.
2 Simulations and results
The proposed scheme is confirmed via two-dimensional (2D) PIC simulations using the EPOCH code because the spatiotemporal characteristics can be fully described in 2D spatial geometry. The near-paraxial and quasimonochromatic
The normalized amplitude is
Figures 1(b)–1(d) show snapshots at
This field structure leads to a donut intensity distribution in the propagating
With the circulation of the canonical momentum, it is intuitive to calculate the TOAM of the beam using
To verify harmonic generation, we performed a 2D Fourier transform for the field at
Figure 2.(a) Two-dimensional high-harmonics spectra of reflected beam . The white line is a one-dimensional spectrum at . The inset in (a) shows the magnified third-harmonic spectrum region. (b)–(d) Field distributions of the (b) first, (c) third and (d) fifth harmonics. (e)–(g) TOAM densities and momentum fluxes of (b)–(d), respectively.
Next, we extracted the fields of the first-, third- and fifth-order harmonics using the inverse Fourier transform, as shown in Figures 2(b)–2(d). The reflected fundamental-frequency beam,
To understand the production of spatiotemporal harmonics in detail, we studied the interaction between the STOV beam and the plasma target. According to the well-known ROM model[53–55], when an intense linearly polarized laser pulse imprints on a solid foil, the foil surface oscillates with twice the frequency of the incident pulse because the ponderomotive force it receives is
In Figures 3(a)–3(c), we present three typical oscillating patterns of electron spikes at
Figure 3.(a) Spatial–temporal-coupled relativistic oscillating mirror (ST-ROM). Three typical oscillating patterns are shown: (a) , (b) and (c) . (d) Spectrum of the ST-ROM. The black dashed curve represents the local center angular frequency. The black line in (a)–(c) represents the density contour of at which the beam is reflected. The spatiotemporal singularity reaches the plasma surface at approximately , denoted by the triangle in (a).
Accompanied with the electron oscillation, the STOV beam periodically exchanges TOAM with both the electrons and protons in the plasma target. The TOAM of electrons only oscillates around zero because of their instant responses to the electromagnetic field of the driving beam. However, the protons dragged by the electrons through a charge separation field accumulate a negative net TOAM. According to the conservation law of angular momentum, an immediate consequence is the loss of TOAM in the reflected beam, as shown by the TOAM densities in Figures 2(e)–2(g). In addition, this indicates that one can use a heavy-ion plasma target to mitigate TOAM losses in the spatiotemporal harmonics. Our simulation results with immobile ions confirmed that each photon carries an average TOAM of about
The use of plasma materials resolves the damage threshold issue of normal optical media, enabling the generation of high-power EUV STOV light sources. Figure 4 shows the scaling of the TOAM per photon and the energy conversion efficiencies of the spatiotemporal harmonics driven by the STOV beam with intensities from
Figure 4.(a) TOAM values per photon of harmonics for STOV drivers with (square), 4 (circle), 6 (right-hand triangle) and 8 (left-hand triangle). The blue dashed line is a linear fit of the average TOAM for each order harmonic. (b) Energy conversion efficiencies with fitted lines of the power law of harmonics.
The proposed scheme can also operate at oblique incidence, which is necessary for practical experiments. Figure 5 presents the harmonic results driven by the s-polarized STOV beam with an incident angle of
Figure 5.Spectra of (a) and (b) components driven by a -polarized STOV beam with incident angle of . Field distributions of the (c) second and (d) third harmonics. (e), (f) TOAM densities of (c) and (d), respectively. The reflected beam propagates in the -direction.
In conclusion, we have demonstrated that relativistic spatiotemporal high harmonics are generated when a high-power STOV beam carrying TOAM irradiates an over-dense plasma target. The frequencies of the STOV beams are spatially diverse. Thus, it drives the spatial–temporal-coupled ROM to radiate spatiotemporal harmonics. During this process, the TOAM of the driving beam is transferred to the harmonics. Benefitting from the ultrahigh damage threshold of the plasma, the intensity of the generated harmonics approached the relativistic region. Therefore, our proposed scheme provides a promising method for producing spatiotemporal EUV vortices with extreme intensities. In addition, the direction reversal of the TOAM during the reflection suggests that it might be more efficient to deposit TOAM into the plasma than conventional longitudinal OAM. It would be interesting to examine the effects of TOAM in other relativistic STOV beam and plasma interaction scenarios.
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Lingang Zhang, Liangliang Ji, Baifei Shen. Intense harmonic generation driven by a relativistic spatiotemporal vortex beam[J]. High Power Laser Science and Engineering, 2022, 10(6): 06000e46
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Category: Research Articles
Received: Oct. 10, 2022
Accepted: Nov. 23, 2022
Published Online: Jan. 9, 2023
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