Abstract
Optical vortices are very attractive because they transport a well-defined orbital angular momentum (OAM) associated with the singularity of the beam. These singular beams, commonly generated in the optical regime, are used in a wide range of applications: communication, micromanipulation, microscopy, among others. The production of OAM beams in the extreme ultraviolet (XUV) and X-ray regimes is of great interest as it allows to extend the applications of optical vortices down to the nanometric scale. Several proposals have been explored in order to generate XUV vortices in synchrotrons and FEL facilities. Here, we study the generation of XUV vortices through high-order harmonic generation (HHG). HHG is a unique source of coherent radiation extending from the XUV to the soft X-ray regime, emitted in the form of attosecond pulses. When driving HHG by OAM beams, highly charged XUV vortices with unprecedented spatiotemporal properties are emitted in the form of helical attosecond beams. In this chapter, we revise our theoretical work in the generation of XUV vortices by HHG. In particular, we illustrate in detail the role of macroscopic phase matching of high-order harmonics when driven by OAM beams, which allows to control the production of attosecond beams carrying OAM.
Keywords
- high harmonic generation
- attosecond pulses
- orbital angular momentum
- nonlinear optics
- ultrafast phenomena
- attosecond helical beams
- extreme ultraviolet vortices
- phase matching
1. Introduction
Light beams are known to carry spin angular momentum (SAM), related to their polarization, or orbital angular momentum (OAM), related to their spatial phase profile [1–3]. Helical phase beams, also called optical vortices, exhibit a transversal spiral phase structure around the beam axis, thus having a well-defined OAM that is characterized by the topological charge,
Schemes for the generation of helical phase beams in the XUV/X-ray regime have been reported in accelerator-based sources, such as synchrotrons and X-ray free-electron lasers (XFELs) [18–22], and in laser-based sources, such as plasma induced in solid surfaces [23, 24] or underdense plasmas [25] and high-order harmonic generation (HHG) in gas-phase media [16, 17, 26–30]. In general, accelerator-based sources produce more intense pulses but they are in the femtosecond timescale (1 fs = 10−15 s) and their spatiotemporal coherence is not always ideal. On the other hand, laser-based sources produce shorter pulses, in the attosecond timescale (1 as = 10−18 s), with higher spatiotemporal coherence. In this chapter, we explore the unique properties of XUV vortices generated via HHG.
HHG is an extreme nonlinear process, driven in an atomic gas-phase medium interacting with an intense IR laser beam. This highly nonlinear interaction results in the emission of coherent harmonic radiation in the XUV and soft X-ray spectral regimes [31, 32]. The HHG radiation presents unique properties, the higher orders being emitted in the form of pulse trains at the attosecond timescale [33–37] or even at the zeptosecond scale if driven by mid-IR lasers [38]. Harmonics with well-defined OAM (OAM-HHG) can be generated using helical phase beams as driving fields. In such scenario, the phase twist is imprinted in the driving IR field—for which a single setup (diffractive mask, for instance) is required [39–41]—and is subsequently transferred to the short-wavelength radiation by nonlinear conversion. The first experiment of OAM-HHG [26] reported the generation of harmonic vortices with topological charge nearly equal to that of the fundamental field, i.e., the spiral phase structures of the harmonics and the fundamental driver beam are similar. Those experimental observations were attributed by the authors to nonlinear propagation effects and not to the OAM-HHG generation process itself. For instance, this finding was unexpected in terms of the present understanding of HHG, in which the harmonic phase scales roughly with the harmonic order [42]. Indeed, the spiral phase structure of the harmonics is expected to be that of the fundamental but multiplied by the harmonic order. In other words, if the driving OAM beam presents a topological charge
Besides the promising applications of generating vortices in the XUV regime via HHG, there are still some fundamental questions that are very interesting to explore. High-harmonic radiation is a macroscopic output resulting from the emission of many atoms, but this observable has its origin in a quantum mechanical (single-atom) mechanism and has indeed a strong correspondence, i.e., what occurs at the single-atom level in a particular spatiotemporal position will affect the macroscopic emission. OAM-HHG is indeed a clear example of this striking effect. Each atom in the target perceives a different transversal phase of the incoming IR field, and even though the atoms do not know the phases of their corresponding neighbors (they are not entangled), this information will be imprinted in the superposition of the photons emitted by the whole sample, giving rise to a transverse phase in the emitted radiation. This fascinating property raises questions such as how this process occurs and which are the significant mechanisms behind the imprinted (OAM) phase in the XUV harmonics. It is in fact a much unexplored field and more research is needed to understand the formation of XUV vortices via HHG.
In this chapter, we revise our theoretical work in the generation of XUV vortices by HHG. We provide a self-explanatory introduction of our theoretical models and we emphasize the most prominent results. In particular, we illustrate in detail the role of macroscopic phase matching of high-order harmonics when driven by OAM beams, leading to helical attosecond beams with unprecedented spatiotemporal properties.
The chapter is organized as follows. First, in Section 2 we introduce HHG, paying special attention to the semiclassical understanding of the process in terms of quantum paths. Second, in Section 3, we present the two theoretical methods that we have developed to simulate the HHG process driven by OAM beams. On one hand, we have developed advanced HHG simulations including quantum single-atom harmonic generation and macroscopic propagation. Our code, which is especially suited for beams without cylindrical symmetry, has been widely contrasted against experiments. On the other hand, we have further developed a simple semiclassical model that allows us to unveil the contribution from different quantum paths in the HHG process. This model is of special interest for the optical community using OAM-HHG, as it allows for qualitative description of future experiments. In Section 4, we present our results of OAM-HHG. We show the main properties (OAM content and divergence) of the generated XUV vortices. In addition, we analyze the phase-matching conditions of HHG driven by OAM, showing that different quantum path contributions to the HHG spectrum can be naturally selected by modifying the target position with respect to the driving beam focus. Finally, we analyze the properties of the temporal emission of the XUV vortices. In particular, we show that different XUV harmonic vortices can be synthesized to produce attosecond helical beams, whose spatiotemporal properties (such as divergence and/or temporal chirp) can be controlled through phase matching.
2. High-order harmonic generation
One of the major challenges since the invention of the laser [43] has been to extend coherent radiation to the extreme regions of the optical electromagnetic spectrum, and, in particular, to higher frequencies. For that purpose, harmonic generation offers a direct route. Although perturbative harmonics were obtained shortly afterwards the invention of the laser [44], nonperturbative harmonics, i.e., those whose efficiency does not decrease exponentially with their order, were not reported until the late 1980s [45, 46]. At the very beginning, this behavior was attributed to multiphoton excitation of atomic subshells, but later, a nice classical interpretation was proposed, relating the harmonic generation with tunneling ionization in the so-called three-step model [47, 48]. Since then, the close interplay between theory and experiments has boosted the development of the field, providing a unique mechanism for the generation of extreme ultraviolet/soft X-ray radiation in the form of attosecond pulses [49–55].
HHG is achieved by focusing an intense femtosecond IR field into a gas target, as depicted in Figure 1. The target, either atomic or molecular, is commonly found in experiments as a gas jet, a gas cell, or a gas-filled capillary. The high nonlinear interaction between the IR field and each atom results in the emission of harmonics of the fundamental IR field, whose frequency extends into the XUV or even the soft X-ray regime. Notably, radiation emitted by the atoms is coherent. Hence, the harmonic signal reaching the detector is strongly affected by the phase matching of the high-order harmonics emitted by each atom of the target. Thus, HHG radiation results from the interplay between the microscopic single-atom emission and the macroscopic superposition of the contributions of all the atoms of the target.
From the microscopic point of view, the single-atom HHG process is well described by the three-step model [47, 48], explained at the bottom of Figure 1: (i) an electronic wave packet is ionized by an intense laser via tunneling; (ii) once in the continuum, it is accelerated by the field and driven back to the parent ion; and (iii) finally, upon recollision, it recombines, releasing high-frequency radiation whose energy corresponds to the kinetic energy acquired from the laser field plus the ionization potential. The dynamics of the electronic wave packet in the continuum can be described accurately using classical trajectories, from which it is simple to associate a particular recombination energy with an initial (ionization) time, as we will see in Section 2.2. Interestingly enough, for every half-cycle, there are two different electron paths leading to the same recombination energy, the so-called short and long trajectories. These short and long trajectories appear naturally as path contributions in the quantum framework within the strong-field approximation (SFA) [56, 57]. The phase of the HHG emission depends on the particular path followed by the electron, thus the interference of the emission from different paths may affect the structure of the harmonic spectrum. In addition, the time ordering of the short and long quantum paths gives rise to a positive or a negative chirp, respectively, in the temporal structure of the associated attosecond pulses [58]. We should also mention the possibility of longer quantum paths, evolving over more than a laser cycle, which give rise to higher-order rescatterings [59, 60] (providing zeptosecond waveforms in the high-order harmonic radiation driven by mid-IR pulses [38]). The HHG interferences from short and long quantum paths have been experimentally observed in standard HHG experiments [61]. In the OAM-HHG context, we started the first studies exploring the contributions of the different quantum paths to the harmonics, see Ref. [28]. In this chapter, we will revise the importance of the quantum paths in OAM-HHG.
Macroscopically, HHG in extensive targets can be intricate. Atoms located at different positions in the target emit harmonic radiation whose phase depends on the amplitude and phase of the local driving field. As a consequence, harmonic phase matching plays an essential role, limiting the spatial regions in which the harmonics contribute efficiently. Typically, harmonic phase matching is described in terms of the longitudinal coherence length that corresponds to the distance between two atoms whose emitted radiation interferes destructively, being a critical parameter for the optimization of HHG in macroscopic targets [62]. In addition, transverse phase matching is defined in terms of the transverse coherence length that describes the phase matching of the radiation coming from atoms placed in a plane perpendicular to the propagation axis [63]. Phase matching is essential to understand most of the macroscopic features of HHG radiation and to develop unique sources such as soft X-ray harmonics driven by mid-IR lasers [31]. Harmonic phase matching has been extensively studied in different macroscopic geometries, such as Gaussian beams focused into gas jets or gas cells [62, 64, 65] or transversal Bessel beams propagating in waveguides [66], among others. In Section 4.2, we discuss in detail some of the phase-matching properties of high-order harmonics driven by beams carrying OAM.
2.1. HHG radiation: XUV to soft X-ray radiation emitted in the form of attosecond pulses
Let us now analyze the main properties of the harmonic spectrum generated by an atom irradiated by an intense laser pulse. The harmonic spectrum presents certain peculiar features that have been observed since the earliest experiments and theoretical works [45, 46, 67, 68]. A typical HHG spectrum is composed of few, odd, low-order harmonics whose intensities decrease exponentially, in accordance with the perturbative scaling, followed by a wide region of odd harmonics (the so-called
There is a fundamental interest in extending the cutoff frequency to higher energies. From the dependences of the ponderomotive energy,
On the other hand, HHG offers the exciting perspective of synthesizing XUV pulses of attosecond duration [33, 34]. An attosecond pulse train is obtained by the selection of the higher frequency part of the HHG spectrum, i.e., the
The first experimental measurement of an attosecond pulse train was performed by selecting five consecutive harmonics generated in argon, obtaining 250 as pulses [35]. In addition, isolated pulses with duration of 650 as were produced by spectrally filtering few cutoff harmonics produced by an ultrashort laser pulse [36]. At present, after postcompression, isolated pulses with temporal durations < 100 attoseconds have been measured experimentally [37, 72]. Moreover, there is a great interest to extend the production of isolated attosecond pulses to the soft X-ray regime [73, 74], and there are proposals to produce subattosecond waveforms by using mid-IR driving pulses [38].
2.2. Quantum path contributions to HHG
A beautifully simple picture to understand the single-atom HHG process is given by the three-step model [47, 48], based on the so-called simple man’s model. In the tunneling regime, one can assume that the ionization process depends only on the instantaneous value of the electromagnetic field, and right after ionization, the electron is located at the coordinate origin with zero velocity. Another assumption of this model consists in considering the dynamics subsequent to ionization as corresponding to a classical free electron in the electromagnetic field, thus neglecting the influence of the Coulomb potential. As a consequence, HHG can be understood in terms of simple semiclassical arguments, as explained at the bottom of Figure 1(a).
We can study in detail the electronic dynamics in the three-step model by integrating the classical equations of motion given by Newton’s law and using the conditions given by the simple man’s model. In Figure 2(a), we have depicted some electron trajectories for different ionization times, in the presence of a monochromatic laser field of
The short and long path contributions also emerge from the quantum framework within the strong-field approximation [56, 57, 75]. The phase of the harmonic emission not only depends on the phase of the fundamental field but also on the particular path followed by the electron. This additional nonperturbative term, the so-called
3. Theoretical treatment of OAM-HHG
The theoretical treatment of HHG admits different levels of description, ranging from the classical to semiclassical and full quantum. For instance, in the previous section, we have shown how HHG can be studied through classical electron trajectories. In this section, we build a theoretical model to describe HHG driven by OAM beams, which takes into account both microscopic and macroscopic physics. We have developed two alternative, full quantum and semiclassical, methods that allow us to describe and have an insight into the microscopic quantum paths involved in the formation of OAM vortices.
Let’s start putting in context the physical scenario of HHG driven by OAM beams. In Figure 3, we present an schematic view of OAM-HHG. A pulsed vortex beam centered at λ0 = 800 nm, a typical wavelength used for HHG, is focused into a gas jet. In this work, we use an argon gas, and the amplitude of the field
The spatial structure of the IR vortex beam is represented by a monochromatic Laguerre-Gaussian beam propagating in the
and
The indices
3.1. 3D quantum SFA theory
We have developed a quantum method to compute HHG including both single-atom (microscopic) and phase matching (macroscopic) physics. In order to take into account macroscopic phase matching, we compute harmonic propagation using the electromagnetic field propagator [78]. To this end, we discretize the target (gas jet) into a set of
where
In the case of intense fields, the computation of the microscopic HHG dynamics of the elementary radiators is not trivial, as the interaction is nonperturbative. Due to the large number of radiators, the use of exact numerical integration of the time-dependent Schrödinger equation becomes extremely expensive. Therefore, the use of simplified models is almost mandatory. In the case of intense fields, S-matrix approaches combined with the strong-field approximation [80–82] are demonstrated to retain most of the features of the HHG process [75, 83]. We use an extension of the standard SFA, hence we will refer it as SFA+, where the total dipole acceleration of the
One of the advantages of this method, which takes into account both microscopic and macroscopic HHG, is that it is well-fitted to nonsymmetric geometries, therefore, it is especially suited for computing HHG driven by singular beams, such as those carrying OAM. The method has been successfully used for describing regular HHG with near- and mid-IR lasers, in good agreement with several experiments [31, 63, 73, 85–88].
For the simulation results presented in this work, we have considered a laser pulse with a well-defined OAM of
3.2. The thin slab model (TSM)
We have developed a simple model, the
The complex beam profile of the fundamental field in the thin slab, placed at a propagation distance
where
We consider the high-order harmonics to be emitted at the thin slab. Let us see how their intensity and phase profiles are related to that of the driving field. In the perturbative regime, the amplitude of the
On the other hand, the phase of the harmonics scales with
Taking into account the previous arguments for the description of the high-order harmonics in the analytic SFA representation, the contribution of the
where
Once we have the description of the
where (
While this equation is valid for a driving field composed of any combination of Laguerre-Gaussian modes, let us now consider a driving field composed of a single OAM mode. In this case, the intensity of the driving does not vary along the azimuthal coordinate, i.e., |
Note that the detected harmonic signal will result from the superposition of short and long quantum path contributions. We remark that Eq. (8) is valid when the fundamental beam is composed by a single OAM mode. For any combination of different OAM modes, one should use Eq. (7) [30].
4. Results: XUV harmonic vortices
Once we have introduced the theoretical methods, we can proceed with the discussion of the main results of OAM-HHG. First, in Section 4.1, we describe the main properties of the XUV vortices that are generated. We will concentrate on their OAM content and divergence. Second, in subsection 4.2, we will analyze how macroscopic phase-matching conditions affect the process of OAM-HHG. We will observe how short and long quantum path contributions can be naturally isolated by adjusting the relative position between the gas jet and the driving beam focus. Finally, in Section 4.3, we will describe the rich spatiotemporal structure of the helical attosecond beams that emerges when several XUV vortices are synthesized.
4.1. XUV vortices: OAM content and divergence
Let us now investigate the main properties of the XUV vortices generated via HHG. In Figure 5, we present the intensity (top) and phase (bottom) far-field angular profiles of the 17th (a), 19th (b), 21st (c), and 23rd (d) harmonic vortices. These results are obtained using the 3D quantum SFA simulations, where we have placed the argon jet at the focus position of the
It is interesting to note that in the OAM-HHG experiments [16, 26, 29] and theoretical works [27, 28] performed to date, the divergence of all the harmonics was found to be similar, in contrast to standard HHG driven by Gaussian beams, where the divergence decreases with the harmonic order. In fact, this result is a consequence of the OAM build-up law
HHG leads to a perfect vortex generation process in terms of its applicability [93], as all these XUV vortices of topological charge
4.2. Phase-matching effects in OAM-HHG
Once we have presented the main properties of the XUV vortices generated via HHG (divergence and OAM content), we study the effect of macroscopic phase matching on their generation. Harmonic phase-matching conditions are known to depend strongly on the position of the target with respect to the focus of the driving field [62]. In particular, in standard HHG experiments with Gaussian beams, short quantum path contributions dominate the detected HHG emission if the gas target is placed after the focus position. However, if the target is placed before the focus, long quantum paths dominate for low divergence angles while the short ones dominate at larger angles [64, 65]. In this section, we discuss the effects of the relative position between the target and the beam focus in HHG driven by helical-phase beams carrying OAM. First, we make use of our TSM approach to disentangle the short and long quantum path contributions under different phase-matching conditions. Second, we perform a time-frequency analysis (TFA) to identify the quantum path contributions in the 3D quantum SFA simulations. More information about the phase matching in OAM-HHG can be found in Ref. [28].
4.2.1. Disentangling quantum path contributions with the thin slab model
In Figure 7, we show the intensity profile (log scale) of the 19th harmonic as a function of the target position along the propagation distance. To this end, the intensity angular profile along the divergence
In order to get a qualitative explanation of the divergence profiles observed in panel (a) of Figure 7, we use the
For better interpretation of the results that can be extracted from Figure 7, we represent in Figure 8 the far-field divergence intensity profile of the 19th harmonic for seven slab positions
As the results presented in Figures 7 and 8 are performed for a thin slab, longitudinal phase-matching effects are neglected, and we can confirm that the behavior presented for different target positions is a direct consequence of transverse phase matching [28, 63]. Note that although in Figure 7(a) we have used an unrealistic 2D gas jet (1 μm -thick), when using a realistic 3D gas jet (500 μm-thick) the results are very similar [28].
4.2.2. Quantum path contributions in the 3D quantum SFA simulations: time-frequency analysis
In order to confirm the qualitative picture given by the TSM for the separation of short and long quantum paths, we perform a time-frequency analysis of the harmonic emission based on the 3D quantum SFA simulations using a realistic 500 μm thick 3D gas jet. The 3D quantum SFA model gives a full quantum description of the HHG emission but it does not provide an insight of the semiclassical picture in terms of quantum trajectories. However, with the help of the TFA, we can resolve the temporal order in which high-order harmonics are emitted, and thus, it allows us to extract relevant information about the quantum path contributions.
In the TFA of the harmonic emission [94], we select a spectral window in the harmonic spectrum and take its Fourier transform. By shifting the window to cover the entire harmonic spectrum, it is possible to resolve the time in which the different harmonics are generated. Nevertheless, due to the uncertainty principle, we have to be careful when interpreting the results of the TFA, as the width of the spectral window determines the resolution in time; the narrower the spectral window is, the less resolution we obtain in the temporal domain. The TFA has served to reveal unique features of HHG, like for example, the quantum path interferences due to multiple rescatterings in the HHG process [38, 95]. As it is usually observed in HHG calculations, the time evolution of the harmonic emission follows faithfully the distribution of rescattering energies of classical trajectories. As a consequence, from the TFA analysis, we can identify the short (long) quantum paths, as the TFA structures with positive (negative) slope, which give rise to a positive (negative) chirp in the harmonic emission, as commented in Section 2.2.
In Figure 9, we present the HHG spectra as a function of the divergence
When the gas jet is placed before the focus (a), two ring structures can be identified, centered at divergences 3.5 and 6.2 mrad, whose TFA analysis are shown in panels (a1) and (a2), respectively. Whereas the TFA at 3.5 mrad (a1) presents structures with negative slope, at 6.2 mrad (a2), it presents structures with positive slope. As a consequence, long quantum paths are emitted with lower divergence and short ones with higher divergence, in agreement with the results previously obtained with the TSM in Figures 7 and 8. In addition, note that within each half-cycle, the long quantum path structures in (a1) are delayed in time with respect to the short ones in (a2), as expected from the semiclassical HHG theory [96].
On the other hand, when the gas jet is located after the focus (b), a prominent ring centered at 6.2 mrad and two secondary ones centered at 7.9 and 8.8 mrad are observed. The TFA analyses at those three divergences are shown in plots (b1) to (b3), respectively. We note that, in agreement with the TSM of Figures 7 and 8, (b1) is dominated by short quantum path contributions, whereas (b3) by long ones. Interestingly, the annular structure conformed around 7.9 mrad, (b2) presents contributions from both short and long quantum paths.
As a result, longitudinal phase matching, included through the 500 μm thickness of the gas jet, does not modify our previous conclusions in realistic gas jets, and thus the angular separation of highly-charged HHG vortices from different quantum paths is due to the transverse phase matching. In addition, this analysis through the TFA corroborates the adequacy of the TSM approach developed in Section 3.2. A rich scenario of XUV harmonic vortices is obtained, with different intensity structures and temporal properties due to the phase matching of short and long quantum path contributions. Note that long quantum path contributions are hard to observe experimentally due to their lower weight in the overall harmonic emission. Whereas with standard Gaussian beams, long paths have been successfully observed and characterized, the observation of XUV vortices obtained from long quantum path contributions remains unobserved.
4.3. Helical attosecond pulse trains
Finally, we proceed to study how the XUV vortices obtained from HHG are emitted in the temporal domain. As introduced in Section 2, one of the most exciting perspectives of HHG is the possibility of synthesizing XUV pulses of attosecond duration. For the correct synthesis, the spectrum should approximately satisfy two conditions: its structure should approach to that of a frequency comb, in which the harmonic intensities are similar, and the relative phase between the harmonics should be nearly constant. Fortunately, HHG satisfies these two conditions, and as XUV vortices are emitted with similar divergence in OAM-HHG, the radiation of several high-order harmonics can be synthesized to produce attosecond pulse trains.
In Figure 10, we present our simulation results of the temporal evolution of the high-order harmonics produced when the 500 μm-thick gas jet is centered at 2 mm after the focus position. Note that, at this position, short quantum path contributions dominate, and the HHG signal is maximized, as discussed in Section 4.2. We have filtered the low-order harmonics (below 11th) by simulating their transmission through an aluminum filter. First, in panel (a), we present five snapshots of the transverse intensity distribution of the integrated harmonic signal within half a cycle of the fundamental laser pulse, i.e., in a time interval of 1.33 fs. We observe that two well-defined intensity structures rotate in time, with a period of half a cycle.
In order to visualize the spatial structure of the HHG emission, we plot in panel (b) the spatiotemporal evolution of a given HHG intensity. We observe that a helical attosecond pulse train is obtained [27], i.e., an attosecond pulse train delayed along the azimuth
The attosecond helical beams obtained from OAM-HHG were theoretically predicted in [27] and have been recently measured experimentally using the RABBITT technique [29]. Note that the carrier-envelope phase of the driving field is imprinted in the attosecond helical beam along the azimuth. If the pulse duration of the driving field is restricted to few cycles, the number of pulses of the attosecond helical beam changes along the azimuthal direction, from an isolated pulse, to two pulses [27].
Finally, note that helical attosecond pulse trains with different divergence and chirp can be obtained by properly selecting the phase-matching properties to maximize the emission from short or long quantum path contributions [28]. Thus, our results show the possibility of generating helical attosecond pulse trains with different spatiotemporal structures that could be selected depending on the application. In these types of helical beams, which can also be generated using relativistic laser-matter interaction [23, 98], not only the phase but also the intensity profile possess a helical structure. As a consequence, these beams are able to exchange OAM in situations where standard Laguerre-Gauss beams cannot, opening an entirely new light-matter interaction regime [97].
5. Conclusions and outlook
Extreme ultraviolet OAM beams with high spatiotemporal coherence are now produced via HHG. When focusing an intense infrared OAM beam into a gas target, the HHG process combines the microscopic quantum mechanics with the macroscopic physics to imprint OAM into higher-order harmonics of the driving field. For instance, HHG maps the OAM properties of the driver into the harmonic vortices. When driven by pure vortices, the
Harmonic phase matching in the generation of XUV vortices allows for the selection of their spatiotemporal properties. Since beams carrying OAM have a very unique transverse structure, transverse phase matching plays a fundamental role in the macroscopic emission. With the help of a thin slab model, we unveil the role of different quantum path contributions in the generation of XUV vortices. For instance, the relative position between the gas jet and the beam focus allows for the spatial selection of XUV vortices produced from short and/or long quantum path contributions in the HHG process.
It has been recently shown that if OAM-HHG is driven by a combination of vortex beams with different topological charge, the generated XUV vortices exhibit a rich OAM content, arising from the nonperturbative behavior of HHG [30]. For instance, experimental methods to produce vortex beams with well-defined OAM may be imperfect, and such beams with several OAM contributions are naturally produced. Even when those imperfections are small, the sensibility of the nonperturbative nature of the HHG makes them visible. Another example where nonpure OAM beams are present is in the case of beams with fractional OAM, which can be generated with spiral phase plates or spatial light modulators [99], or using conical refraction [100, 101].
In all these situations, the OAM dynamics are much more complex than those presented in this review. On the other hand, it has been recently shown that spin angular momentum can be transferred to the XUV harmonics in the HHG process [102], producing circularly polarized harmonics and attosecond pulses [87, 88, 103, 104]. New scenarios, in which orbital and spin angular momenta add a new degree of freedom to light-matter interaction, open an exciting route for the next generation of high-resolution, ultrafast, XUV/X-ray diagnostic tools for fundamental studies and applications.
6. Acknowledgements
C. Hernández-García acknowledges support from the Marie Curie International Outgoing Fellowship within the EU’s Seventh Framework Programme for Research and Technological Development (2007–2013), under REA Grant Agreement No. 328334. C. Hernández-García, J. San Román, and L. Plaja acknowledge support from Junta de Castilla y León (Project SA116U13, SA046U16) and MINECO (FIS2013-44174-P, FIS2016-75652-P). A. Picón acknowledges financial support of the U.S. Department of Energy, Basic Energy Sciences, Office of Science, under Contract No. DE-AC02-06CH11357.
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