1. Introduction

Laguerre-Gaussian (LG) beams, which are the most typical representatives of vortex beams, have been widely used in laboratories because of their circular symmetry and availability. These can be obtained using a Gaussian beam through a Q plate [1], spiral phase plate (SPP) [2], and geometric modeling, such as diffraction gratings [3], and spatial light modulators [4]. Vortex beams such as optical tweezers and wrenches, have been widely studied and applied owing to their unique and novel characteristics [5–7]. In recent years, research on vortex beams has advanced to the extreme ultraviolet region through high-order harmonic generation (HHG). For example, Kong et al. [8] used wave mixing to produce an HHG with controllable orbital angular momentum (OAM) values, which could produce a vortex beam with the smallest possible central intensity. This was a new approach to carrier-injected laser machining and lithography. Dorney et al. [9] used a coaxial two-color circularly polarized vortex beam laser to generate high-order harmonic radiation with controllable spin angular momentum and OAM. Rego et al. [10] used two coaxial vortex beams with different OAM to generate a high-order harmonic radiation light source with time-varying angular momentum. The time of existence of a specific angular momentum value in space is on the femtosecond scale, which can be used as a new tool for laser-matter manipulation on attosecond time and nanometer spatial scales. Several studies [11–16] also have used light-field modulation to control the generated high-order harmonic vortex beams, resulting in many novel physical effects.

To achieve strong HHG with OAM, it is necessary to achieve optimized phase-matching (PM) conditions. Many theoretical and experimental studies have been conducted on the PM of HHG for Gaussian beams [17–23]. In recent years, with the trend of vortex beams, the research on the PM of HHG has been pushed to the field of vortex beams. For example, Jin et al. [24] simulated the intensity and phase distribution characteristics of the generated vortex beam in the near and far fields through the quantitive rescattering model and Huygens’ integral combined with the three-dimensional Maxwell’s wave propagation equation. They studied the properties of the high-order harmonic vortex beam generated before and after the focus. A subsequent study [25] investigated PM related to HHG by mixing two LG beams with different topological charges and obtained extreme ultraviolet light with superimposed OAM modes. Hernández-García et al. [26] explored the effect of different quantum path contributions on the PM of HHG vortex beams by combining the quantum path concept in strong-field approximation theory with a simple classical diffraction model. Roman et al. [27] studied the properties of vortex harmonics generated at different focal positions and theoretically investigated the effects of long and short trajectories. The above-mentioned studies were all theoretical studies on the PM of HHG with OAM, but limited experimental research has been conducted to understand this issue.

In addition, it is difficult to maintain perfect vortex spots that are sensitive to the driving laser parameters and free electron/ion disturbances, which can lead to the degeneration of OAM signals. The degree of degeneration can be quantitatively described using the parameters of the dimensional properties of vortex beams, which are [28]: the diameter of the annular beam, ratio of the inner and outer diameter (hereinafter referred to as ‘ratio’) of the annular beam, ring width, beam divergence, and ring area. The diameter and the ratio of diameters are the most easily measured parameters. Therefore, these parameters are commonly used to characterize the properties of vortex beams. For example, Zhang et al. [29] obtained the theoretical values of the diameter and the ratio of an annular beam through theoretical simulations. Reddy et al. [30] obtained the relationship between the outer and inner diameter of a beam and the propagation distance and variation in the divergence angle through theoretical simulations. Therefore, in this study, we chose these two parameters as the dimensional properties to describe HHG with OAM.

In this study, high-order harmonic radiation with the same divergence angle for each order, was generated through an Ar-filled gas cell. By optimizing the focus position, laser energy, and gas pressure, PM of HHG with OAM was obtained. Finally, the dimensional properties of the vortex beam under the PM were experimentally analyzed. Our results contribute to the generation of high-order harmonics with OAM light sources, paving the way for various applications of vortex beams.

2. Experimental setup

The experimental geometry is shown in Fig. 1(a). A commercial Ti: Sapphire laser (Coherent) with an output parameter of 8 mJ/45 fs/800 nm/1 kHz was used. The laser was focused by a lens with a focal length of 500 mm, into a gas cell filled with Ar. A soft-edge aperture was used to control the diameter of the incident beam. The SPP was inserted into the optical path to enable the generated IR light to carry the OAM. Selecting an SPP as a device for OAM generation has the advantages of a high energy conversion rate, no need for additional dispersion compensation, and suitability for femtosecond laser sources with good monochromaticity [4]. In this study, an eighth-order SPP was used, and the laser bandwidth is about 40 nm (full-width at half maximum). An almost completely annular vortex beam was generated, as shown in Fig. 1(c). For a more perfect vortex beam, a higher order SPP was deemed required. The beam diameter was adjusted using a soft-edged aperture to prevent the diffraction effect from affecting the beam quality and focusing spot when the laser propagation distance was too long owing to the iris. The generated harmonic OAM beam was focused by a spherical mirror in the horizontal direction, the fundamental frequency laser was filtered by the Al film, and finally, the remaining harmonic signal entered the spectrometer system. The slit of the spectrometer was opened to 1 mm to allow the HHG with the OAM to pass through. The slit size used for the calibration of the harmonic orders was 100 µm. The spectrometer system was placed on a one-dimensional motorized translation stage to scan the horizontal spatial distribution of the signal.

 

Fig. 1. (a) Experimental geometry. SPP: spiral phase plate, RM: reflecting mirror, SM: spherical mirror. (b) The spectrum of HHG with OAM which have been calibrated. The following figure shows the curve data used for the calibration of the harmonic orders. The black spot in (b) is caused by oil pollution of the detector array surface in the vacuum. (c)(d) The intensity distribution and phase momentum information of the incident laser beam.

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3. Results and discussion

Figure 1(c) shows the distribution of the incident laser spot, where a relatively perfect vortex beam can be observed. Figure 1(d) shows the phase information of the incident beam with OAM is passed through a cylindrical lens in which the specific phase information is expressed [9], i.e. the topological charge l = 1. As shown in Fig. 1(b), the 23rd harmonic at 35 nm corresponds to a relatively perfect vortex beam, the corresponding photon number at gas source is about $9 \times {10^9}$, detailed calculation process see Supplement 1. Spectral depression is caused by the imperfect focusing of a spherical mirror. Unlike traditional Gaussian beam HHG, the divergence angle of each harmonic order is the same. According to the results of Géneaux et al. [13], the angular momentum generated by the harmonics satisfies the relationship ${l_n} ={\pm} nl$, i.e., when the nth HHG carried OAM is n times the angular momentum of the fundamental frequency laser, the divergence angles of each harmonic order are the same, which is a unique property of HHG with OAM [31]. As shown in Fig. 1(b), the divergence angle from the 15th to the 25th harmonics is the same. This indicates that the obtained HHG contains an OAM of $nl$. The conversion of OAM from a fundamental frequency laser to HHG photons is very fragile and requires a high beam quality; i.e., this transfer is very sensitive to any effects on the fundamental beam. It should also be noted that this property can only be obtained when the gas cell is placed behind the focus. A detailed discussion is provided below. The vortex spectra before and at the focus, as well as the spectral data used for harmonic order calibration, can be found in Supplement 1.

As shown in Fig. 2, we studied the dependence of the 15th - 25th harmonic signal intensity on the relative position of the gas cell and focus position(See Supplement 1 for complete data). The comparison results of the 15th, 19th, and 23rd Gaussian beam HHG and LG beam HHG are shown. The following results are obtained. First, the 15th harmonic driven with a Gaussian beam shows a single-peak distribution after the focus. With an increase in the harmonic order, the intensity distribution gradually changes to a double-peak distribution before and after the focus. The HHG with the LG beam also exhibits a double-peak distribution in the 15th to 23rd order range. As the harmonic order increases, the range of PM can be gradually expanded before the focus, and gradually narrows after the focus. This is different from the trend of PM of Gaussian beams. Second, for the Gaussian and LG beam HHG, as the order increases, the harmonic intensity after focus is stronger than that before focus, indicating that there is always an optimal PM condition after the focus position, where z=+4. Overall, both the LG beam HHG and Gaussian beam HHG have a double-peak distribution before and after the focus position, and the LG beam HHG can achieve better PM within a smaller range after the focus position.

 

Fig. 2. The dependence of the intensity of HHG on the relative position of the gas cell and the focus position, (a)-(c) represent the results of the Gaussian beam, and (d)-(f) represent the results of the LG beam. (a)(d), (b)(e), and (c)(f) represent the 15th, 19th and 23rd harmonic order, respectively. The 17th and 21st harmonic are not shown in the figure. The negative position indicates that the gas cell is in front of the focus.

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We compare these results with the simulation results in Ref. [24], and the experimental and simulation results are in good agreement; i.e., the intensity and phase of the generated higher-order harmonics are more sensitive to the position of the gas cell relative to the laser focus. At the same time, the LG and Gaussian beams have the same position such that there exists a PM position with the strongest signal behind the focus. This is because the allowable coherence length of the short trajectory behind the focus is longer, which strengthens the generation of harmonics in the medium. This is exactly the same for an HHG with an LG beam. In [24,26,27], a multi-ring structure with varying focus positions was obtained theoretically and experimentally. If the gas jet is in front of the laser focus, the intensity mode often exhibits several rings, and the gas jet consists primarily of a single-ring structure behind the focus. When the gas jet is in the laser focus, the intensity distribution typically exhibits the highest number of rings. Theoretical analysis suggests that the appearance of multiple ring structures is due to the coherent superposition of long and short trajectories. However, multiple ring structures at different focal positions are not observed in our experimental results. We believe that the following reasons may contribute to this deviation: in Refs. [24,26,27], both experimental and theoretical simulations were conducted using gas media with a length of ≤0.5 mm, while in our experiment, a gas cell with a length of 2.0-3.0 mm was used, which has a longer effective interaction length. Similarly, in Ref. [11], a gas cell with a length of around 2.0 mm was used, and no obvious multi-ring structure was observed in the generated harmonics. In Ref. [15,16], a gas cell with a length of 10 mm was used to generate vortex beams, and no significant multiring structures were observed. Therefore, we believe that the single-ring structure observed in our experiment may be related to the gas cell or the influence of different quantum trajectory on the radial mode. For example, according to Ref. [27], by controlling the position of the laserfocus, the contribution of the short and long orbits can be adjusted, thereby changing the distribution of the radial mode of high-order harmonics. This phenomenon can be explained by the nonlinear effects and atomic dipole phase during the process of HHG. However, we are currently unable to conduct similar theoretical analyses on the above issues.

As shown in Fig. 3, considering the 13th - 25th harmonics, we tuned the laser intensity and gas pressure to achieve optimal PM conditions of HHG with OAM. In Fig. 3(a), the relationship between the changes in laser intensity and the harmonic order is shown. At this time, the slit was opened to a width of $100{\;\ \mathrm{\mu} \mathrm{m}}$, and a portion of the vortex beam signal was intercepted. The gas pressure was fixed at 40 torr. As the laser intensity increases, the harmonic intensity increases, and when the laser intensity reaches ${\sim} 1.1 \times {10^{14}}W/c{m^2}$, the harmonic intensity reaches its maximum value. When the laser intensity exceeds the saturation intensity, the generated harmonic intensity weakens, and the circular structure of the spectrum also undergoes rapid degradation. The dependence of the harmonic intensity on the LG beam is consistent with the process of a Gaussian laser beam producing HHG [19,32]. The degradation of the circular structure is due to the enhanced disturbance of a large number of free electrons in the OAM transfer process. Figure 3(b) shows the relationship between the pressure changes and harmonic order. As the harmonic order increases, the pressure required for the optimal PM condition increases. This is also consistent with the changing trend of Gaussian beam HHG [23,24].

 

Fig. 3. The dependence of vortex harmonic intensity with (a)laser intensity and (b)gas pressure.

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Regarding the degradation of the annular beam structure, we believe it can be explained from the perspective of phase matching. When the incident laser intensity exceeds the saturation intensity, it indicates that the gas medium exceeds the optimal ionization rate, resulting in an increased number of free electrons. Additionally, according to the factors affecting the phase matching of HHG, when the ionization rate of the electrons is too high, the contribution of the free electron term becomes dominant and cannot be compensated by the other terms. As a result, the phase mismatch increases, leading to degradation of the spatial quality of the generated harmonics and rapid attenuation of the signal intensity. We are currently unable to accurately quantitatively describe this process, and an appropriate theoretical explanation is also a subject for future research. For the generation of high-order harmonics with a vortex beam, it can be regarded as incorporating the transfer of OAM into the HHG process [27,33]. The physical process of HHG can be described by a three-step model [34], where the traditional model consists of ionization, acceleration, and recombination steps. For the generation of HHG with vortex beams, it is only necessary to introduce the transfer of OAM during this process [33]: the photon carrying OAM is transferred to the electron, the free electron accelerates and gains energy in the laser field, and at the same time, the transfer of OAM also occurs. Then, the electron then combines with the parent ions to produce the corresponding HHG, and at the same time, its carried OAM is also transferred to the harmonic photons.

Finally, we experimentally analyzed the dimensional properties of the vortex harmonics under PM conditions, and Fig. 4 shows the dimensional properties of the vortex beam. Here, R represents the diameter, ${R_{out}}$ represents the outer diameter, and ${R_{in}}$ represents the inner diameter. We measured the dimensional properties of the annular beam at three different positions, i.e., before focus z = −4, at focus z = 0, and after focus z=+4. From the measurement results in Table 1(raw data shown in Supplement 1), we can observe that the diameter of the annular beam generated after focus is the same for each harmonic order, and the ratio changes with the order. This is because the PM effect combined with the carried OAM makes the generated harmonics in a single-ring structure having the same diameter for all orders [24]; meanwhile, the OAM of each order is different, and the ratio is only a function of the topological charge [29], thus, the value of the ratio varies with harmonic order. Then, the diameter and ratio of the generated harmonic also change with different orders before and at focus, and there is no obvious pattern. Except for the 15th harmonic at the focus, the ratio of each order before and at focus is larger than that after focus. However, the diameter R before and after focus is basically the same. Combining with Fig. 6 in Ref. [24] (hereinafter referred to as Fig. 6, not shown in the article), we further analyzed the results in Table 1. In Fig. 6(d)(e), for the after focus region of harmonic orders, the long trajectory can only achieve phase matching in a very narrow region, while the coherence length of the short trajectory is significantly stronger than that of the long trajectory (as shown in Fig. 6(a)(b)). This indicates that in the vortex LG beam HHG, the short trajectory dominates, which is the main reason for the same divergence angle of all generated harmonics in Table 1. This also explains why the beam diameter data of all orders after focus in Table 1 are consistent. In Fig. 6(c-e), for the focus and before focus positions of each order, the phase matching of the long trajectory splits into two regions, and the coherence length is longer than that of the long trajectory after focus, indicating that its contribution ratio increases. Combined with the change of the diameter of the annular beam at the focus and before focus positions in Table 1, the coherence of the short and long trajectories is superimposed, resulting in different orders of diameters no longer being consistent. In Fig. 6(e)(f), at the focus and before focus positions, the radial position of the phase matching of the long trajectory gradually increases, which may cause the outer diameter of the annular beam to increase at the focus and before focus positions, while the change in the inner diameter is small, that is, the ratio of outer diameter to inner diameter will increase. This is consistent with the rule that the ratio after focus in Table 1 is almost smaller than that at the focus and after focus positions.

 

Fig. 4. Dimensional properties of the LG beam.

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Table 1. The diameter and the ratio of the outer and inner diameters of generated harmonic at different focus positions.

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4. Conclusion

In conclusion, high-order harmonic radiation with OAM in the extreme ultraviolet region is generated in Ar gas. By measuring the signal intensity before and after the focus, it is proved that there is an optimal PM position after the focus, and the divergence angles of all harmonic orders at this position are the same which means that the OAM carried by the nth harmonic may be n times the fundamental frequency of the laser. The variation in signal intensity at the optimal PM position with pressure and energy is studied. It is found that its pattern is similar to that of the traditional Gaussian HHG. However, the LG beam HHG is more easily destroyed compared to the traditional Gaussian beam. Finally, by measuring both the diameter and the ratio at different focus positions, the influence of PM on the dimensional properties of vortex beams is obtained, and the degree of PM conditions can be inferred from the measurement of the dimensional properties.