Attosecond twisted beams from high-order harmonic generation driven by optical vortices Download: 549次
1 Introduction
Twisted beams, also called optical vortices, exhibit a helical phase structure that imprints orbital angular momentum (OAM) to the beam, in addition to the spin angular momentum associated with the polarization[1–3]. These singular beams, commonly generated in the optical spectral region, have potential technological applications in optical communication, micromanipulation, phase-contrast microscopy, among others[4–7]. The production of twisted beams in the extreme-ultraviolet (XUV) and x-ray regimes is of great interest as it allows for the extension of the applications of optical vortices down to the nanometric scale. In this short-wavelength regime one can drastically reduce the diffraction limit, as well as exploit the selectively site-specific excitation, with an important impact in microscopy and spectroscopy[8–14]. Several proposals have been explored in order to generate x-ray vortices in synchrotrons and FEL facilities[15–18]. However, the generation of twisted beams carrying OAM in the x-ray regime is limited by the availability of efficient optical and diffractive elements.
In the last years, high-order harmonic generation (HHG) has been used to generate XUV vortices by imprinting the phase singularities in the infrared (IR) driving beam[13, 19, 20]. HHG is a unique non-perturbative process, that combines microscopic and macroscopic physics to produce coherent XUV to soft x-ray radiation in the form of attosecond pulses[21–25]. When an intense IR beam is focused into a gas target, the laser–matter interaction in each atom or molecule results in the emission of higher-order harmonics of the driving field. Microscopically, HHG can be understood with simple semiclassical arguments[26, 27]: an electron is first tunnel-ionized from an atom, then accelerated in the continuum, and, due to the oscillatory behavior of the driving field, its recollision with the parent ion leads to the emission of higher-frequency radiation. Interestingly, there are two possible electronic quantum paths leading to the same kinetic energy at the recollision process – and thus to the same harmonic – known as short and long trajectories[28, 29]. Macroscopically, the coherent addition of the radiation emitted from all the atoms in the target – also known as phase-matching[30–32] – plays a relevant role for the efficient production of XUV/soft x-ray harmonics[30, 33]. One can define longitudinal – or transverse – phase-matching as the interference of the harmonic emission emitted from different atoms in the target placed along the longitudinal or transverse direction[34]. Transverse phase-matching is especially relevant in HHG driven by vortex beams due to their involved transverse field structure[35].
If driven by pure vortex beams, the harmonic vortices are generated with a topological charge
In this work we present a detailed theoretical analysis of the generation of twisted attosecond beams in the XUV regime. To this end, we analyze the phase-matching conditions in HHG driven by OAM beams, by means of two theoretical methods: the semiclassical
2 Physical scenario of OAM-HHG
Before describing the theoretical methods, let us analyze the physical scenario of HHG driven by OAM beams. In Figure
The spatial structure of the IR vortex beam is represented by a monochromatic Laguerre–Gaussian beam propagating in the
Fig. 1. Schematic view of HHG driven by OAM beams. An intense IR vortex beam carrying OAM (with $\ell =1$ in this case), is focused into an argon gas jet. The near-field coordinates are ($\unicode[STIX]{x1D70C},\unicode[STIX]{x1D719}$ ). Each atom emits HHG radiation that, upon propagation, results in the far-field emission of XUV vortices with some divergence and azimuth ($\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D711}$ ). In the bottom we show the near-field amplitude (left) and phase (right) of the $LG_{1,0}$ IR mode, with beam waist of $30~\unicode[STIX]{x03BC}\text{m}$ .
3 Theoretical methods
Let us first present the two theoretical methods we use for the theoretical description of HHG driven by OAM beams. First, we present the semiclassical
3.1 TSM
We have developed a semiclassical
We consider the thin slab as a planar source of high-order harmonics, whose intensity profiles at the slab are related to that of the fundamental field. In the perturbative regime the amplitude of the generated radiation is proportional to the power of the fundamental beam amplitude,
Once we have the near-field description of the
3.2 3D quantum SFA simulation
In order to compute OAM-HHG including all quantum dynamics, we use our 3D quantum SFA method to compute HHG including both single-atom (microscopic) and propagation (macroscopic). We compute harmonic propagation using the electromagnetic field propagator[38], where we assume the harmonic radiation to propagate with the vacuum phase velocity, which is a reasonable assumption for high-order harmonics. Propagation effects in the fundamental field, such as the production of free charges, the refractive index of the neutrals, the group velocity walk-off[42], as well as absorption in the propagation of the harmonics, are also taken into account. In order to compute single-atom HHG, the exact calculation requires the integration of the time dependent Schrödinger equation. However, to ease the computational effort required to calculate propagation in a 3D geometry, the use of simplified models is almost mandatory. We use an extension of the standard SFA, hence we will refer it as SFA+, that shows an excellent agreement with the TDSE[43].
One of the advantages of this method, that takes into account both microscopic and macroscopic HHG, is that is well-fitted to non-symmetric geometries; therefore, it is specially suited for computing HHG driven by beams carrying OAM[20, 35, 41].
4 Results
Once we have presented the physical scenario of OAM-HHG and the theoretical methods that we use, we analyze the simulation results. First, using the TSM, we will describe the OAM-HHG emission in terms of short and long quantum-path contributions, in order to identify the macroscopic conditions that allows us to separate them. Second, using the 3D quantum SFA model, we will analyze how the XUV vortices are emitted in the temporal domain, in order to macroscopically select attosecond twisted beams with different spatiotemporal properties.
4.1 Selecting different quantum-path contributions in OAM-HHG
The HHG spectrum results from the coherent addition of different quantum-path contributions. At the single-atom level, short and long quantum-path harmonic contributions are always present, although the later ones are less intense due to their longer excursion time. However, as they exhibit different harmonic phases, macroscopic phase-matching differs for each quantum-path contribution. As a consequence, different phase-matching conditions have been typically applied to macroscopically select the contributions from a single path. For example, when driving HHG by Gaussian beams, it is well known that if the gas jet is placed before the beam focus position, short quantum-path contributions dominate in the on-axis far-field profile. Moving to off-axis detection angles, long quantum-path contributions become prominent. In contrast, if the gas jet is placed after the focus position, short quantum-path contributions dominate the HHG spectrum over the whole far-field profile[31, 32]. As a result, the relative position between the gas jet and the beam focus can be used to select quantum-path contributions, and thus to modify the temporal properties of the attosecond pulses emitted[44]. Note that these phenomena has been successfully explained in terms of longitudinal phase-matching. In other words, the longitudinal phase variation along the gas jet, which depends strongly on the relative position between the Gaussian beam focus and the gas jet, favors the efficient emission of different quantum-path contributions. However, if HHG is driven by OAM beams, this behavior changes completely[35], as we will discuss here in detail.
Here, with the help of our TSM we will disentangle the presence of short and long quantum-path contributions in HHG driven by OAM beams. As it has been done when driving HHG by Gaussian beams, in OAM-HHG we scan the relative position between the gas jet and the beam focus to modify the phase-matching conditions. Note that in the TSM, longitudinal phase-matching effects are neglected as a thin target is assumed. Thus, any macroscopic effect in the harmonic emission is unequivocally related to transverse phase-matching. In Figure
First, we observe that the yield of the short quantum-path contributions is more intense than the long ones, as expected form the shorter excursion time. Second, whereas the harmonic profiles from short quantum-path contributions are almost symmetric when placing the gas jet with respect to the focus position, those from the long contributions are completely asymmetric. For instance, in Figure
As a consequence, we can conclude that the emission of the 19th harmonic vortex shown in Figure
Fig. 2. Spatial intensity profile of the emitted 19th harmonic for a slab placed at seven near-field positions, from $z_{t}=-3~\text{mm}$ (left) to $z_{t}=3~\text{mm}$ (right), calculated with the TSM considering (a) short$+$ long, (b) short and (c) long quantum-path contributions. Whereas short quantum-path contributions exhibit similar intensity and structure independently of the near-field slab location, long ones are more intense if the slab is placed before the focus position. As a consequence, a rich vortex structure profile is obtained depending on the relative position between the gas jet and the beam focus.
Fig. 3. Radial intensity profile of the emitted 19th harmonic as a function of the near-field slab position, calculated with the TSM considering (a) short$+$ long, (b) short and (c) long quantum-path contributions. As depicted in Figure 2 , the dependence of the profile of short quantum-path contributions with the slab position is almost symmetric with respect to the focus, whereas that of the long ones is completely asymmetric. As a consequence, the relative position between the gas jet and the beam focus serves as a knob control to select harmonic vortices with short or long quantum-path contributions.
For illustration purposes we present in Figure
Fig. 4. Attosecond twisted beam structures of the emitted harmonic radiation obtained with the 3D SFA quantum model for a multi-cycle driving laser pulse of $\unicode[STIX]{x1D70F}_{P}=15.4~\text{fs}$ (a, b) and a few-cycle driver of $\unicode[STIX]{x1D70F}_{P}=3.8~\text{fs}$ (c, d). The harmonics are generated in a $500~\unicode[STIX]{x03BC}\text{m}$ argon gas jet placed 2 mm before (a, c) and after (b, d) the focus position. On the left we show the attosecond twisted beam structure, whereas on the right we show transverse intensity snapshots at four different time instants within a half-cycle, $0.12T,0.25T,0.38T$ and $0.5T$ (where $T$ is the laser period). The contribution from long quantum-path contributions is indicated in yellow.
4.2 Attosecond twisted beams
Harmonic vortices generated through OAM-HHG exhibit similar far-field divergence. This result, that has been observed both theoretically[20, 35] and experimentally[13, 19, 36], is a consequence of the simple OAM build-up rule
To this end, we use the 3D quantum SFA model. The driving laser beam, spatially represented as an
In Figure
Finally, we show the attosecond twisted beams driven by a few-cycle laser pulse
5 Conclusions and outlook
We have presented a detailed analysis of the macroscopic synthesis of attosecond twisted beams. With the help of a semiclassical
Our results open an exciting perspective for the application of attosecond twisted beams. First, we note that although we have reported harmonic vortices in the XUV regime, their energy content could be extended towards the soft x-rays if driving beams with longer, mid-IR[33], or shorter, ultraviolet[46] wavelengths were used in HHG, or by using other scenarios, such as plasmas[47–49], solid targets[50, 51], or free electron laser facilities[52]. In the attosecond twisted beams both the phase and the intensity profile exhibit a helical structure. Thus, they are perfect candidates to exchange OAM in situations where not only the phase possesses a helical structure (like standard OAM beams), opening an entire new light–matter interaction regime. New regimes, in which the non-perturbative behavior of HHG imprints a relevant signature in the generated attosecond twisted beams[41], or where harmonics with both OAM and spin angular momentum are generated[53], remain to be explored.
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Article Outline
Carlos Hernández-García, Laura Rego, Julio San Román, Antonio Picón, Luis Plaja. Attosecond twisted beams from high-order harmonic generation driven by optical vortices[J]. High Power Laser Science and Engineering, 2017, 5(1): 010000e3.