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Abstract
We propose to generate vortex high harmonics in the extreme ultraviolet (XUV) with a controllable spatial profile by using a Bessel-Gaussian (BG) beam carrying a nonzero orbital angular momentum (OAM). Such BG beam has quite a different intensity profile at the focus compared to the generally used BG beam without carrying the OAM. We show that the BG beam is capable of generating single-ring structured high harmonics, which is quite different from an Laguerre-Gaussian (LG) beam with a similar intensity distribution at the laser focus. We reveal that favorable phase-matching conditions can be achieved off-axis and away from the laser focus because a single-atom intrinsic phase due to the short electron trajectory can be well compensated by a geometric phase of the BG beam. We thus give a general rule that vortex high harmonics with a single annular profile can be efficiently generated when a gas medium is located at 1.5zred to 2.0zred before or after the laser focus of the BG beam, here zred is a reduced length. We also show the validity of this rule when the BG beam carries a higher OAM. This work is expected to be useful for synthesizing attosecond vortex pulses.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In an optical vortex, the light field displays a helical phase structure, with a phase singularity associated with zero intensity along the axis. Such light field can be characterized by the topological charge $l$, thus it has the azimuthal phase change of $2\pi l$ around its propagation axis, and carries an orbital angular momentum (OAM) of $l\hbar$ per photon, where $\hbar$ is the reduced Planck constant [1]. Since the first demonstration of an OAM beam by Allen et al. [2] in 1992, infrared (IR) and visible light beams carrying the OAM have been intensively studied due to their broad applicative prospects [3–5], such as optical communication [6,7], chiral recognition in molecules [8,9], phase-contrast microscopy [10,11], and so on. Furthermore, generation of vortex light fields in spectral regions of extreme ultraviolet (XUV) and soft X-rays has also attracted a lot of interests [12–16].
To generate vortex fields in the XUV and soft X-rays, one of the effective methods is to transfer the OAM of an IR driving laser to shorter wavelengths through a perturbative nonlinear process, i.e., high-order harmonic generation (HHG) [17–19]. Generation of vortex XUV beams through HHG has been reported by using a gas [13,20] or solid [22] medium. For example, in the first experiment, Zürch and co-workers [23] measured topological charges of vortex high harmonics, which are the same as that of the fundamental driving laser. In 2014, Gariepy et al. [24] experimentally evidenced that the topological charge of each harmonic is $q$ times ($q$ is the harmonic order) that of the driving laser by measuring the transverse phase of vortex harmonics with an interferometry technique. And then, Kong et al. [25] and Gauthier et al. [14] respectively showed that the OAM of XUV beams can be tunable when the wave mixing scheme is applied to the generation of a tabletop high-harmonic source. In 2019, Rego et al. [20] measured a temporal OAM variation along a pulse in the HHG driven by time-delayed pulses with the different OAM, called as self-torque. Troß and Trallero-Herrero presented an experimental technique to mix two LG beams with different OAMs to create two tightly focused laser foci, which can generate high harmonics that interfere in the far field [21]. In 2020, Chong et al. [26] demonstrated a three-dimensional wave packet that is a spatiotemporal optical vortex with a controllable purely transverse OAM. In 2021, Fang et al. [27] studied conversion and modulation of the photon transverse OAM in the HHG driven by spatiotemporal optical vortex (STOV) pulses, and very recently, by designing a streaking-like technique, Chen et al. [28] fully characterized the OAM of the STOV beam. Minoofar et al. [29] explored the generation of OAM-carrying space-time wave packets both experimentally and theoretically, with unique property of a time-dependent beam radius at various propagation distances. Kumar et al. [30] generated a vortex XUV beam with a topological charge of up to 100 through HHG in a noble gas, and characterized it by using a high-resolution Hartmann wavefront sensor. In addition, De Las Heras et al. [31] demonstrated up-conversion of vector-vortex beams from IR to XUV on the grounds of HHG and obtained an increase of the HHG efficiency for drivers with a higher topological charge. As experimentally demonstrated by Géneaux et al. [12], it is crucial to maintain the stable and single-ring structured beam profile for different harmonics to synthesize an attosecond vortex pulse, which can be used to generate the helical electron burst and probe ultrafast electron dynamics.
On the other hand, in these studies, Laguerre-Gaussian (LG) beam [2] has been often used for fundamental optical vortices, which is the most popular mode. Its unique properties for generating high harmonics have been well documented in a number of theoretical works. For example, Hernández-García et al. [32] theoretically predicted that XUV harmonic vortices are generated and survive to the propagation effect, which have topological charges being equal to the harmonic order times the fundamental one. Rego et al. [33] studied generation of the vortex HHG driven by non-pure LG modes, carrying different OAM contribution, and analyzed the interference signature of spatial modes. Paufler et al. [34] proposed to generate XUV vortices via HHG with two-color counter-rotating LG beams carrying a well-defined OAM. Jin et al. [15] presented detailed phase-matching analysis of HHG driven by the LG beam with a nonzero OAM. We recently revealed that fractional high harmonics with the non-integer OAM generated by a short-duration LG beam can cause change of the helical attosecond pulse train with an azimuthal angle [35]. However, as demonstrated by both experimental and theoretical studies [15,36,37], by using the LG beam, the intensity profile of generated vortex HHG changes from single-ring structure to multiple-ring one if a gas jet is moving from an after-focus position to a before-focus one, and multiple-ring structure is also dependent on the harmonic order. Thus, it is desirable to consider other laser beams to produce vortex HHG with a stable beam profile.
One of the candidate beams is Bessel-Gaussian (BG) beam [38]. As another popular mode, the BG beam can be generated by using output of a laser system to incident onto a tip block, lens, and axicon [41]. It is close to the diffraction-free beam near the focus, has a longer focus depth, and can naturally form an annular beam after long-field propagation [38]. Most notably, it has the ability to overcome the limitation of the Rayleigh range [42,43], which allows obtaining potentially more favorable phase-matching conditions if it is used to drive the HHG process. Thus, the BG beam without carrying the OAM has been employed to generate HHG [44–48]. For instance, some works showed that the two-color gating with BG beams is promising for the generation of attosecond pulses with high-quality beam profiles [49,50] or with extended phase matching conditions [51]. Both Davino et al. [41] and Dao et al. [46] measured high harmonics with Gaussian-like beam profiles by focusing a BG beam into either a thin or a long gas target. Averchi et al. [52] found that the BG beam is able to increase the XUV flux at some higher harmonics compared to the standard Gaussian beam because of the improved phase-matching condition. Furthermore, optical properties of the BG beam can be easily adjusted though a number of controllable parameters, including the beam waist, the OAM, the order of Bessel function, and the angular half-aperture of the core. However, the BG beam carrying the nonzero OAM has been rarely applied for generating vortex HHG. It is interesting to examine whether the BG beam can be used to overcome the sensitivity of the spatial intensity profile of HHG to the gas position with respect to the LG beam focus, and to manage the annular profile for different vortex harmonics.
Our goal in this paper is twofold. First, we would like to identify difference between vortex high harmonics generated by BG and LG beams carrying the nonzero OAM. According to our knowledge, such comparison between two beams has not been performed by others yet. Second, we will show how to utilize the BG beam to maintain the spatial profile of vortex HHG. This paper is arranged in the following. In Sec. 2, we will present the quantitative rescattering (QRS) model for simulating the single-atom induced dipole, propagation equations and Huygens’ integrals for simulating macroscopic vortex HHG, and formulation of BG and LG beams. In Sec. 3, we will first compare free-propagation characteristics of BG and LG beams, and then will show difference of high harmonics in the near and far fields generated by two beams, which will be explained with a map of the coherence length. We will next demonstrate unique advantages of HHG by the BG beam under given laser parameters determined by a general rule, and will finally check vortex HHG driven by the BG beam with a higher OAM. Conclusions will be given in Sec.4.
2. Theoretical methods
To simulate vortex high harmonics generated in a macroscopic gas medium driven by a laser beam with the OAM, one needs to compute both the single-atom induced dipole and macroscopic response from atoms within the laser-gas interaction region, and further propagation and divergence of high harmonics exiting from the gas medium should also be taken into account. Details are discussed in the following.
2.1 Quantitative rescattering (QRS) model for simulating the single-atom response
In the single-atom response, each atom interacts with a local electric field at a given spatial position of the OAM beam. The laser-atom interaction can be treated under the dipole approximation [53]. For a linearly polarized laser, the single-atom induced dipole moment $D(t)$ can be accurately calculated by using the QRS model [54,55]. Based on the three-step model, QRS improves the strong-field approximation (SFA), and can give the HHG spectrum nearly as being accurate as solving the three-dimensional (3D) time-dependent Schrödinger equation (TDSE) [56]. In the QRS, the length gauge is employed, and the induced dipole moment $D(\omega )$ (in the frequency domain) can be written as [54,57]
where $N$ is the ionization probability taken at the end of the laser pulse, $d(\omega )$ is the complex photorecombination (PR) transition dipole matrix element, and $W(\omega )$ is the complex returning electron wave packet. QRS replaces the plane wave used in the SFA [58] with an accurate scattering wave in the calculation of the PR transition dipole matrix element, while the electron wave packet remains the same as that in the SFA. Thus, one can practically obtain the induced dipole moment as2.2 Macroscopic propagation of the high-harmonic field
To take into account of macroscopic response, one of standard ways is to solve the three-dimensional Maxwell’s wave equation of the high-harmonic field in the gaseous medium. We assume that the fundamental beam with the OAM is not modified within the ionized gas medium, i.e., it propagates as in free space. This is valid when the laser intensity is low and the gas pressure is low. By employing a moving coordinate frame ($z' = z, t' = t - z/c$), propagation of the vortex high-harmonic field in the frequency domain is governed by (in a Cartesian coordinate) [15]
Here $\hat {F}$ is the Fourier transform operator acting on the temporal coordinate, and $P_{\text {nl}}(x,y,z',t')$ is the nonlinear polarization, which can be calculated as
withHere $n_{\text {0}}$ is the neutral atom density, $n_{\text {e}}(x,y,z,t)$ is the free electron density, and $w(\tau )$ is the tunnel ionization rate. The induced dipole moment $D(x,y,z,t)$ caused by the local laser field is computed by using the QRS model.
An operator-splitting method is used to solve the Eq. (3). Advance of the electric field from $z'$ to $z' + \Delta z'$ is separated into two steps:
The high-harmonic field obtained at the exit plane of the gas medium ($z' = z_{\text {out}}$) is called a near-field one.
2.3 Far-field harmonic emissions
Near-field harmonics are propagated and divergent in vacuum until reaching the detector. These harmonics far away from the gas medium are called far-field ones. Far-field harmonic emissions can be obtained from near-field ones by using the Huygens’ integral under paraxial and Fresnel approximations (in a Cartesian coordinate) as
2.4 Fundamental Bessel-Gaussian (BG) and Laguerre-Gaussian (LG) beams
We first consider a BG beam under paraxial and slowly varying transverse amplitude approximations, its electric field can be expressed in a cylindrical coordinates as [38–40]
We then give the field expression of the LG beam in the following [15]:
Note that above expressions of the two beams are written in the moving frame, thus the propagation factor of $\exp (-ik_0z)$ in the rest frame is omitted.
3. Results and discussion
3.1 Comparison of free-propagation characteristics of BG and LG beams
We first compare spatial profiles of BG and LG beams carrying the same OAM. A typical example of such comparison of two beams at $z' = 0$ and $z' = {+\infty }$ is presented in Fig. 1. For the BG beam, the topological charge $l$ = 1, the beam waist $w_0$ is 60 $\mu$m, and the angle $\theta$ is 20 mrad. For the LG beam, $l=1$, $p=1$, and the beam waist $w_0$ is 25 $\mu$m. The laser central wavelength is 800 nm. The temporal pulse keeps the same for both beams, with the full-width-at-half-maximum (FWHM) duration of 10 optical cycles (26.7 fs) and the carrier-envelope phase (CEP) of 0. In Fig. 1, spatial intensity and phase distributions of two driving beams at the central wavelength are shown. Above parameters are chosen to ensure that main rings of two beams at the focus are overlapped for easy comparison. In Fig. 1(a), laser intensity distributions at the focus (near field) when $x$ = 0 $\mu$m are plotted for two beams. Two beams are fully overlapped at the main peak, but the BG beam indicates typical multiple node structure while the LG beam only has one node determined by the $p$. To further characterize the driving laser beam with oscillations in the intensity profile [60], we define its spot size according to Summers et al.’s [61]. The radius of the spot size is defined when 80${\% }$ of total laser energy fall within it. With chosen parameters, the radius of the spot size is 45 $\mu$m (37 $\mu$m) for the LG (BG) beam at the focus ($z'$ = 0). When two beams propagate in vacuum till the far field, they show quite different spatial features, as shown in Fig. 1(b). The LG beam maintains its node structure while only one peak structure appears in the BG beam. A complete comparison of spatial distributions of the intensity and phase of BG and LG beams are displayed in Figs. 1(c-j). The multiple node feature of the BG beam at the focus can also be clearly seen from its phase distribution in Fig. 1(h). After free propagation, the BG beam becomes an annular one as shown in Fig. 1(i), and there is no phase jump in its phase distribution in Fig. 1(j). For the LG beam, its intensity and phase distributions are maintained from the near field to the far field, as shown in Figs. 1(c-f). Therefore, even carrying the OAM, the BG beam itself shows a single-ring spatial profile in the far field.
Fig. 1. Comparison of intensity and phase distributions of BG and LG beams. Transverse intensity profiles (a) in the near field ($z'$ = 0) and (b) in the far field $(z' = {+\infty })$: LG (black lines) versus BG (red dotted lines). (c-j) Spatial intensity and phase distributions of LG (middle row) and BG (bottom row) beams in the near and far fields. The intensity has been normalized, and the phase is defined from -$\pi$ to $\pi$.
3.2 Comparison of spatial profiles of high harmonics by BG and LG beams
We next compare high harmonics driven by BG and LG beams in Fig. 2. The driving beams have been shown in Fig. 1. Spatial intensity distributions of two selected harmonics in the plateau and one harmonic at the cutoff are plotted. Both near- and far-field harmonics generated by two beams are considered. In simulations, a uniformly distributed Ar gas jet with the length of 1 mm is used, the center of the gas jet is placed 6 mm after the laser focus for the BG beam, and this value is 2 mm for the LG beam. The peak intensity at the center of the gas jet is fixed at 1.5 $\times$ 10$^{14}$ W/cm$^2$ for both beams. Note that the gas jet is located in regions away from the laser focus of the BG beam, thus the intensity at the focus should be very high to ensure that the intensity in the annular part is high enough to drive the HHG process. We calculate that the pulse energy of the BG beam is 2.5 mJ, which is achievable with commercial Ti-Sapphire lasers nowadays. For the BG beam, all harmonics exit from the gas medium (at the near field) exhibit the feature of the single-ring intensity distribution, see Figs. 2(a1-c1). We also plot spatial phase distributions of vortex high harmonics in Figs. 2(d-f). From rapid change of the phase along the azimuth angle, it indicates that the OAM of HHG is $q$ times of the fundamental driving one, where $q$ is the harmonic order. This is consistent with what has been measured by Gariepy et al. [24] for HHG driven by an LG beam. After free propagation in vacuum (at the far field), annular structure is remained for different harmonic orders, with about the same divergence, see Figs. 2(g1-i1). In contrast, when the driving beam is LG, multiple-ring structure can be identified from intensity distributions both at the near and far fields for three harmonics under investigation, see Figs. 2(a2-c2) and Figs. 2(g2-i2), respectively. Although both BG and LG beams have similar intensity profiles in the laser-gas interaction region, as shown in Figs. 1(c) and (g), they present a significant difference in generating the HHG. This indicates that phase matching of HHG dramatically changes by varying the driving laser beam. For the BG beam, annular structure of harmonics in the far field is caused by free propagation of similar structure in the near field, so we then look for origins of annular structure of near-field harmonics.
Fig. 2. Intensity distributions of high harmonics generated by BG (Left) and LG (right) beams in the near field (at the exit plane of the gas medium) and in the far field (far away from the gas medium), respectively. First row: 15th-order harmonic (H15); second row: H23; and third row: H31. Harmonic intensities in each figure are normalized independently. Phase distributions of near-field high harmonics by the BG beam are plotted in (d)-(f). The topological charge (or the OAM) of HHG can be read from rapid change of the phase, which is defined within [-$\pi$: $\pi$].
3.3 Analysis of phase-matching conditions of vortex harmonics generated by the BG beam
To understand spatial distributions of the harmonic intensity in the near field by the BG beam, we calculate coherence lengths of HHG in space. The coherence length is defined as [15,62]
Here $\delta \textbf {k}_{q}(r,\phi,z')$ is phase mismatch of wave vectors between the $q$th harmonic and the BG beam, and its norm along the direction of $q \textbf {k}_0 + \textbf {K}$ is with $k_{q} = q \omega _{0}/c$, the wave vector $\textbf {k}_{0}$ of the fundamental BG beam, and the wave vector $\textbf {K}$ of the intrinsic atomic phase in the single-atom response. Calculated spatial distributions of the coherence length for three selected harmonics are shown in two left columns in Fig. 3. Since the gas medium used in this work has a length of 1 mm, a coherence length of at least 1 mm is required for good phase matching. We use "white" color to represent coherence lengths greater than or equal to 1 mm. We show intensity distributions of the driving laser in Fig. 3(f) for reference. In this figure, the maximum intensity is set as 2.0 $\times$ 10$^{14}$ W/cm$^2$ to highlight intensity distributions of the BG beam in the butterfly shaped areas. We define a reduced length $z_{\text {red}} = w_0/\theta$ with the unit of $\mu$m/mrad (or mm) for easy comparison. For all harmonics, spatial distributions of the coherence length are almost symmetric with respect to $z_{\text {red}}$ = 0 for short trajectories. Two plateau harmonics (15th harmonic, i.e., H15, and H23) have good phase-matching regions ("white" areas) off axis and away from the laser focus, which overlap with two major peaks of the BG beam (fenced in green shadow areas for laser intensities from 1.4 to 1.6 $\times$ 10$^{14}$ W/cm$^2$) as shown in Figs. 3(a) and (c). For long trajectories, some phase-matching regions are located off axis and close to the focus plane, but laser intensities are relatively weak in these regions, see Figs. 3(b) and (d). Thus, for plateau harmonics, long-trajectory harmonic emissions are not expected to survive during propagation. Single annular profiles of plateau harmonics in the near field are dominated by short-trajectory emissions when the gas jet is put 6 mm (or 2.0$z_{\text {red}}$) after the laser focus. Compared to long-trajectory phase-matching, the cutoff harmonic (H31) exhibits the similar feature on the map of the coherence length, but with a longer length (or a weaker color), see Fig. 3(e). This also leads to the single annular profile of the cutoff harmonic in the near field when the gas jet is located after the laser focus. Note that similar maps of the coherence length for HHG driven by a BG beam without the OAM have been presented by Averchi et al. [52]. They have identified that favorable phase-matching areas are located close to the axis, which are quite different from off-axis ones in this work. Furthermore, good phase matching occurs in the off-axis (or butterfly shaped) regions, in which laser intensities are much lower than that at the focusing areas (indicated by dark color) as shown in Fig. 3(f). In Fig. 4, we compare the macroscopic harmonic spectra when the gas jet is located at three different positions. One can see that the conversion efficiency of HHG doesn’t vary too much when the gas position is moved in the range of [$z_{\text {red}}$, 2$z_{\text {red}}$], and the best one can be obtained when the gas jet is put at 1.5$z_{\text {red}}$ after the laser focus.Fig. 3. Left: Maps of the harmonic coherence length along radial and propagation directions driven by the BG beam. Intensity distributions of the BG beam away from the focus are plotted in (f) and $I_0$ = 1.0 $\times$ 10$^{14}$ W/cm$^2$. First row: H15; second row: H23; third row: H31. For plateau harmonics (H15 and H23), coherence lengths are shown in terms of short or long electron trajectories. H31 is at the cutoff. Note that the "white" area indicates that the coherence length is longer than 1 mm. Right: (g)-(j) show the geometric phase of the driving laser, the single-atom intrinsic phase, and the total phase of H23 along the propagation axis (in the units of reduced length $z_{\text {red}}$) at two radial positions $r$ = 20 and 100 $\mu$m. Panels (k) and (l) show derivatives of the total phase of H23 with $z$ at $r$ = 20 (solid yellow lines) 100 $\mu$m (solid green lines), respectively.
Fig. 4. Simulated macroscopic spectra by integrating yields of vortex harmonics over the exit plane of the gas medium driven by the BG beam. The peak intensity at the focus is fixed [as shown in Fig. 3(f)] and the gas jet is put at three positions behind the laser focus (indicated by the reduced length $z_{\text {red}}$).
We then analyze why phase-matching conditions of HHG are different from short and long trajectories for the BG beam. There are two main contributions to the phase of the high-harmonic field: (i) the harmonic order $q$ times the spatial geometric phase $\phi$ of the BG beam, i.e., $q\phi$; (ii) the intrinsic phase of the single-atom response $\alpha I$, with the laser intensity $I$ and with different coefficients $\alpha$ for short- and long-trajectory harmonics in the plateau or for cutoff harmonics. We take an example of H23. We isolate different contributions to the phase, and plot them in Figs. 3(g-j) at two radial positions $r$ = 20 and 100 $\mu$m. The total phase as a function of the reduced propagation distance is also shown in these figures. Derivatives of the total phase with the propagation distance are plotted in Figs. 3(k) and (l) for short- and long- trajectory harmonics, respectively. For the short trajectory, $q\phi$ is the main contribution to the phase since the intrinsic phase of the induced dipole is very small, as shown in Figs. 3(g) and (i). And the phase change along the propagation distance is very gentle and maintains as a small value, see Fig. 3(k), indicating the favorable phase-matching condition. For the long trajectory, the induced-dipole phase is very large at $r$ = 20 $\mu$m and it becomes relatively small at $r$ = 100 $\mu$m while $q\phi$ is only slightly modified by varying the radial position, see Figs. 3(h) and (j). Its derivative of the total phase has a larger value, and differs much with the radial position, as shown in Fig. 3(l). This means the phase-matching condition of the long-trajectory harmonic is worse, which is dramatically changed in space.
From above results, we conclude that the single annular profile of the harmonic is due to the good phase-matching condition of short trajectories, which is mostly determined by the geometric phase of the BG beam. It can also be seen in the map of the coherence length that the single-ring structured harmonic can be effectively generated when the gas target is properly located, for example, the gas jet is put at $\pm$ 1.5$z_{\text {red}}$. For comparison, in Fig. 5, we show similar maps of the coherence length and phase analysis of vortex HHG driven by the LG beam discussed before. One can clearly see that the geometric phase of the LG beam is different from that of the BG beam, leading to quite different maps of the coherence length. From Fig. 5, the spatial profile of HHG in the right column of Fig. 1 can be understood. Both short- and long-trajectory emissions contribute to plateau harmonics, and generation of the cutoff harmonic is limited in a small region close to the axis. In Fig. 6, we further show intensity distributions of vortex high harmonics in the near field generated by the LG beam when the gas jet is located before the laser focus. Compared to results in Figs. 2(a2-c2), for each harmonic, spatial profile is changed significantly by varying the position of the gas jet respect to the laser focus. This is quite different from the BG beam. By comparing Figs. 2(a1-c1) with Figs. 9(a-c) (below), the spatial profile of vortex HHG doesn’t change much when the gas jet is moved from an after-focus position to a before-focus one. From these discussions, we can conclude that the geometric phase is unique if the mode of the driving laser beam is given.
Fig. 5. Similar to Fig. 3 except that the HHG is driven by the LG$_{1,1}$ beam and the propagation axis is in the units of mm.
Fig. 6. Intensity distributions of vortex high harmonics in the near field generated by the LG$_{1,1}$ beam (as shown in Fig. 1) when the gas jet is put at 2 mm before the laser focus.
3.4 General rule for single-ring structured harmonics generated by the BG beam
In the above, we only gave one example of HHG by the BG beam. Next, we will examine the general rule of the gas position and laser parameters for generating single-ring structured harmonics by using the BG beam. We choose different laser parameter combinations: $w_0$ = 60 $\mu$m, $\theta$ = 20 mrad, leading to $z_{\text {red}}$ = 3 mm; and $w_0$ = 40 $\mu$m, $\theta$ = 20 mrad and $w_0$ = 60 $\mu$m, $\theta$ = 15 mrad, resulting in $z_{\text {red}}$ = 2 and 4 mm, respectively. The peak intensity at the center of the gas jet is fixed at 1.5 $\times$ 10$^{14}$ W/cm$^2$. In Fig. 7, we show simulated intensity distributions of H23 in the near and far fields. The gas jet is put at different positions after the laser focus. When the gas jet is put at $z_{\text {red}}$ after the laser focus, for different parameter combinations, multiple discontinuous "white" (or good phase-matching) areas exist along the radial direction for short-trajectory emissions, see Fig. 3(c), thus multiple-ring structure can be seen in the intensity distribution of the harmonic in the near field, see Figs. 7(a1), (d1), and (g1). The intensity distribution also exhibits multiple rings in the far field after free propagation, as shown in Figs. 7(a2), (d2), and (g2). If the gas jet is located at 1.5$z_{\text {red}}$ or 2.0$z_{\text {red}}$ after the laser focus, the single-ring intensity profile appears in both the near and far fields for different BG parameters, but with a different radius, see other figures in Fig. 7. Thus, the single-ring structured intensity profile of the harmonic can be readily obtained by using the BG beam when the gas jet is put at the reduced position between 1.5$z_{\text {red}}$ and 2.0$z_{\text {red}}$ even though laser parameters are greatly varied.
Fig. 7. Intensity distributions of H23 generated by the BG beam when the gas jet is placed at $z_{\text {red}}$, 1.5$z_{\text {red}}$, and 2.0$z_{\text {red}}$, respectively. Results are shown both in the near field (left) and in the far field (right). Laser parameters: beam waist $w_0$ = 40 $\mu$m, angle $\theta$ = 20 mrad (first row); $w_0$ = 60 $\mu$m, $\theta$ = 20 mrad (second row); and $w_0$ = 60 $\mu$m, $\theta$ = 15 mrad (third row). Harmonic intensities in each figure are normalized independently.
We also check spatial distributions of the coherence length of H23 with BG beam parameters: $w_0$ = 40 $\mu$m, $\theta$ = 20 mrad and $w_0$ = 60 $\mu$m, $\theta$ = 15 mrad, respectively. Results from short- and long-trajectory harmonic emissions are plotted by using the reduced length along $z'$ in Fig. 8. One can see some similar features to those in Figs. 3(c) and (d) are present. For short-trajectory harmonic emissions, good phase-matching regions ("white" color) are also emerging off axis and away from the laser focus. While for long-trajectory ones, phase-matching regions are limited, and they don’t synchronously overlap with spatial distributions of considerable driving intensities. This clearly explains that single-ring structured harmonics generated using two sets of BG beam parameters owe to favorable phase-matching of short-trajectory emissions achieved off axis for the reduced propagation distance between 1.5$z_{\text {red}}$ and 2.0$z_{\text {red}}$. In Figs. 3 and 8, spatial distributions of the coherence length are nearly symmetric with $z'$ = 0 for short-trajectory harmonic emissions (or cutoff harmonic emissions). One thus may expect that high harmonics with single annular profiles could be generated by locating the gas jet between −2.0$z_{\text {red}}$ and −1.5$z_{\text {red}}$. We put the gas jet at 2.0$z_{\text {red}}$ before the laser focus, and present intensity distributions of some selected harmonics under different BG laser parameters in Fig. 9. For BG beam parameters of $w_0$ = 60 $\mu$m and $\theta$ = 20 mrad, all spatial profiles of H15, H23, and H31 show single-ring structure in the near and far fields, see Figs. 9(a-c) and Figs. 9(f-h), respectively. The far-field ring has about the same radius for different harmonic orders. For another two sets of BG parameters, we select H23 for investigation, and its intensity distributions in the both near and far fields display single annular structure, as shown in Figs. 9(d) and (e) and Figs. 9(i) and (j), respectively. The radius of the ring in the far field is different, which is determined by initial BG parameters. All these behaviors above are the same as those when the gas jet is put after the laser focus.
Fig. 8. Maps of the spatial coherence length of H23 driven by BG beams: $w_0$ = 40 $\mu$m, $\theta$ = 20 mrad [(a) and (b)] and $w_0$ = 60 $\mu$m, $\theta$ = 15 mrad [(c) and (d)]. The results are plotted for short and long trajectories, respectively.
Fig. 9. Spatial intensity distributions of high harmonics by the BG beam in the near field (upper row) and in the far field (lower row) when the gas jet is located at 2.0$z_{\text {red}}$ before the laser focus. Laser parameters: $w_0$ = 60 $\mu$m, $\theta$ = 20 mrad [(a)-(c) and (f)-(h)]; $w_0$ = 40 $\mu$m, $\theta$ = 20 mrad [(d) and (i)]; and $w_0$ = 60 $\mu$m, $\theta$ = 15 mrad [(e) and (j)].
We therefore can summarize the general rule for generating single-ring structure harmonics by using the BG beam: the gas jet should be located at 1.5$z_{\text {red}}$ to 2.0$z_{\text {red}}$ before or after the laser focus, here $z_{\text {red}}$ is the reduced propagation distance.
3.5 Spatial profiles of vortex high harmonics by the BG beam with $l$ = 2 and corresponding phase-matching analysis
We finally check vortex high harmonics generated by the BG beam with the topological charge $l$ = 2. We choose the beam waist $w_0$ is 60 $\mu$m, and the angle $\theta$ is 20 mrad, resulting in the reduced length $z_{\text {red}}$ = 3 mm. The gas jet is put 6 mm after the laser focus, i.e., at 2.0$z_{\text {red}}$. We show spatial intensity distributions of selected harmonics in the both near and far fields in Figs. 10(a-f). They all exhibit the feature of single-ring structure. And the radius of each ring is almost maintained the same as that generated with the $l$ = 1 in Fig. 2. We then show maps of the harmonic coherence length in Figs. 10(g-k). Comparing to Figs. 3(a-e) with the $l$ = 1, general features of phase-matching conditions are very similar. For plateau harmonics, good phase-matching of short trajectories still occurs off axis and away from the laser focus. And this good phase-matching region remains for the cutoff harmonic. For reference, the laser intensity distribution of the BG beam with the $l$ = 2 is plotted in Fig. 10(l). One can see that the maximum intensity ring is shifted up a little bit when the topological change of the fundamental laser is increased, in comparison with Fig. 3(f). Therefore, we can summarize in the following: (i) even though the topological charge of the driving BG beam is increased, the general rule for generating single-ring structured harmonics is still valid; and (ii) increasing the topological charge of the fundamental only increases the OAM of the high harmonic, not the radius of the harmonic beam. This is also quite different from the LG beam, for which the radius of the resulted harmonic beam is proportional to $\sqrt {|l|}$ [12,15].
Fig. 10. Similar figures to those in two left columns in Figs. 2 and 3 except for the driving BG beam with the topological charge $l$ = 2.
4. Conclusions
In summary, we demonstrated unique properties of the Bessel-Gaussian (BG) beam with the nonzero orbital angular momentum (OAM) for generating vortex HHG with a single annular profile. Note that this BG beam with the zero intensity at the center around the laser focus is quite distinct from the BG beam without the OAM, which has been often used to generate HHG with a Gaussian-like beam profile. We first compared BG and Laguerre-Gaussian (LG) beams with the similar intensity profile at the laser focus and showed that they have quite different free-propagation behaviors. We next compared spatial profiles of vortex high harmonics driven by using two beams with the OAM interacting with the Ar gas, when the laser intensity in the laser-gas interaction region maintains low. The BG beam always generates the single-ring structured high harmonics in the near and far fields while the LG beam with the radial node can generate the near- and far-field high harmonics with multiple spatial rings. We also performed detailed phase-matching analysis by calculating spatial maps of the harmonic coherence length. It reveals that the geometric phase of the BG beam can effectively cancel out the single-atom intrinsic phase of short trajectories, leading to favorable phase-matching areas off axis and away from the laser focus. And it turns out that such areas are overlapped with effective intensity distributions of the driving BG beam, thus generating vortex HHG with single-ring intensity structure in the gas medium. We then summarized a general rule for generating single-ring structured harmonics by using the BG beam. This rule is given in terms of the reduced length $z_{\text {red}}$, defined as the ratio of the beam waist and the angular half-aperture of the cone, and states that the gas medium should be located at 1.5$z_{\text {red}}$ to 2.0$z_{\text {red}}$ before or after the laser focus. We finally checked that the general rule is still valid when the BG beam carries a higher OAM and single-ring structured high harmonics are present with about the same radius. In addition, we showed that spatially distributed high harmonics have a similar divergence radius, indicating all harmonics are mostly overlapping in space. Taking advantage of this unique property, one can synthesize an attosecond vortex pulse by spectral filtering a few harmonics [12,32,37]. The resulted annular beam profile of the attosecond pulse and the spatial overlap of different harmonics are maintained even after focusing through an optical system in most applications. Such vortex harmonics may also be useful in other fields, such as optical manipulation, imaging, and so on.
Funding
National Natural Science Foundation of China (12274230, 91950102, 11834004); Funding of NJUST (TSXK2022D005); Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX23_0443).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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