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Abstract
We investigate the phase-matching of the high harmonics (HHG) driven by the circular Airy-Gaussian beams (CAiGB), which abruptly auto-focus and subsequently propagate without diffraction. The results show that the harmonics corresponding to both short and long quantum paths can be well phase-matched after the focusing point of the CAiGB. Therefore, the effective interaction length of HHG for CAiGB is much longer than that for the conventional Gaussian beams with the same size of the waist. Our numerical simulations reveal that the harmonics continuously gain up to 1 cm of the propagation distance. This work provides a route to enhance the conversion efficiency of HHG by the coherent control of abrupt auto-focusing beams.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In the past few years, high harmonics generation(HHG) has captured rising interest in many fields because it can provide a simple way to fabricate table-top soft x rays(XUV) sources [1–9]. Unfortunately, this conventional method is limited in practical applications for the low conversion efficiency. Many efforts have been made to improve the efficiency of HHG, such as phase matching and quasi-phase-matching, which provide an efficient method to optimize HHG efficiency [10–12]. It is worth mentioning that there are also some encouraging findings regarding the generation of high harmonics from solids and plasmas [10,11,13,14]. Phase-matching is an essential concept of optimizing high harmonics [15,16]. Particularly, phase-matching can effectively optimize the high harmonics, resulting in the connection of harmonics generated by single-atom responding at different positions. Thus, the overall HHG can be optimized by this method [15]. Similarly, the corresponding quasi-phase-matching technology has also provided a simple way for enhancing the efficiency of generating high harmonics [17–23]. Based on the concept of quasi-phase-matching, the arrayed form of the media can be changed to retain the regions where the harmonic superposition is enhanced and to remove the regions which generate the reverse phase harmonics, or use other media to reduce the reverse phase harmonics. It can also be achieved a similar effect by periodically changing the driven beam [24,25]. Although quasi-phase-matching provides a new idea for optimizing the generation of harmonics, the efficiency of harmonic generation by quasi-phase-matching is still lower than that of true phase-matching. Therefore, in this paper, we optimize the efficiency of HHG by using a specially constructed beam to effectively control the phase-matching of HHG.
The traditional method of generating high harmonics is to use Gaussian beams to interact with different media [15,26,27]. As the intensity of the driven beam changes, media with different ionization energies are selected to ensure sufficient HHG efficiency [28]. The Gaussian beams have remarkable advantages in generating high harmonics at the macroscopic level, attributing to a good short-trajectory phase-matching after a few millimeters away from the focus point, where the gas cell place can get a harmonic with high quality. However, with the propagation of more than one Rayleigh distance, the intensity of the Gaussian beam sharply decreases to the point where only fewer harmonics can be generated [8,29]. Loosely-focusing intense Gaussian beam can significantly increase the interaction length of HHG to enhance the harmonics signal, but it is still challenging for the laser source. Based on these problems, adjusting the focus area and phase-matching conditions by changing the spatial distribution of the beam seems to be an effective method. The method of using the Bessel-Gaussian beam as the driven beam is a feasible solution to optimize the generation of high-order harmonics [30,31], Furthermore, a two-color Bessel-Gaussian beam also shows impressive results [32]. These results all illustrate the positive influence of using spatially controlled beams on the generation of high harmonics.
In this work, we introduce abruptly auto-focusing beams, named circular Airy-Gaussian beams(CAiGBs) to significantly control the phase-matching of HHG [33,34]. Note that the circular Airy Gaussian beam we use is a linear polarization beam. Thus, the "circular" represent the spatial structure of the beam is circular. Such kind of beams, composed of a series of rings, have the dramatic property of abrupt self-focusing in the neutral media without Kerr nonlinearity: all the rings automatically concentrate in the propagation axis to sharply increase the intensity by one to two orders, and subsequently propagate with weak diffraction. In principle, adopting the abruptly self-focusing beams would significantly reduce the required intensity of the driving source compared to the traditional Gaussian beam. Moreover, the weak diffraction after self-focusing would naturally be conducive to the phase matching of the HHG. Practically, high-intensity CAiGB for HHG could be experimentally produced as follows. First, an 800-nm infrared Gaussian beam is passed a spatial light modulator(SLM) to produce a CAiGB with an additional spatial chirp. Then the beam is focused on an intense CAiGB with the intensity of $8\times 10^{13} \textrm {W/cm}^{2}$. Note that the additional spatial chirp is imprinted to compensate for the phase distribution during transmission of focusing. The intrinsic auto-focusing of the CAiGB merges its main and subsidiary rings to a highly intense spot with an intensity up to $3\times 10^{14} \textrm {W/cm}^{2}$ which is sufficient to drive the HHG with the neon gas. This scheme is consistent with that by Polynkin et al. [35].
In this paper, we investigate the phase-matching on the generation of high harmonics driven by the CAiGBs, which shows excellent results under some parameters. Furthermore, we calculate the HHG spectra with the CAiGBs numerically, and find that the harmonics signals can continuously increase to a distance of a dozen of millimeters. The numerical results show a good agreement with the prediction of the phase-matching diagram discussed. We also calculate the harmonics spot at the position 1 m after the focus point through the Hankel transformation [36], and observe that the far-field divergence angle of the harmonics is as small as 1.7 mrad, which means the harmonics have good beam quality at the far-field.
2. Theory
2.1 Dipole phase and phase-matching
The three-step model proposed can describe the generation of high harmonics [7,8]. The strong laser field distorts the Coulomb potential of the nucleus, and then the electrons may tunnel out of the attraction by the nucleus. After obtaining enough energy in the electromagnetic field, with the laser field turns around, the electrons recombine with the ion, and then the energy is radiated in the form of photons. In the actual HHG process, the three-step process occurs periodically, which period is about half the light cycle of the driven beam. Therefore, the radiation process of HHG is also repeated every half a light cycle. In each repetition period, two different quantum trajectories are corresponding to each harmonic. One of the trajectories has an earlier ionization time and a later recovery time and has a more extended movement time in the laser field, called a long trajectory. In the laser field, the trajectory with a relatively short movement time is called the short trajectory. The phase accumulated by the electrons during its propagation in the continuum through these two trajectories and its recombination under the emission of the qth harmonic is called the dipole phase. Here we give the formula for calculating the dipole phase of the qth harmonic [37,38]:
where $i = s, l$ stands for the short and long trajectory. $I$ represents the intensity of the laser field. In this paper, the peak intensity of the laser field is $3\times 10^{14}\textrm {W/cm}^{2}$, and the target medium is neon. In this case, we select two appropriate $\alpha$ values to describe the dipole phase of the long and short trajectories. For convenience of calculation, we approximately believe that the $\alpha$ value of the low-order harmonic is the same as that of the high-order harmonic. For harmonics in the plateau, we use the slopes from the quantum orbit calculations $\alpha _{s}$ = $1\times 10^{-14} \textrm {W}^{-1} \textrm {cm}^{2}$ and $\alpha _{l} = 25 \times 10^{-14} \textrm {W}^{-1} \textrm {cm}^{2}$ [38].While calculating the phase-matching, the influence of the geometric phase needs also be considered. In comparison, the geometric phase is owing to the phase of the driven beam itself. Both the geometric phase and the dipole phase determine the phase of the HHG. Therefore, when studying the generation of high harmonics, the inherent phase of the driven beam and the dipole phase caused by the beam intensity need to be further calculated.
Since the dipole phase of the emitted radiation depends on the local intensity and the atoms in the target are exposed to different local intensities, the atomic and geometric phases of the harmonics generated at each point are different. Consider the overall high harmonics in the gas cell, the spatial intensity and phase distribution of the driven beam felt by each atom have to be taken into account. We introduce the concept of phase-matching to describe the HHG efficiency. To calculate the phase-matching map, we generally need to obtain the wave-vector mismatch. If the wave-vector mismatch is zero, the harmonics can be perfectly phase-matched, and the harmonic generation can be most efficient. Through the result of Balcou et al. [15], the wave-vector mismatch is given by
Generally, the coherence lengths in the order of 1mm can be regard as a good experimental condition.
2.2 Description of CAiGBs
Different from the single-atom response, the spatial structure of the driven beam has a significant effect on the harmonics generation when the laser field interacts with the gas cell, reflected in the intensity and phase. The Gaussian beam has the highest intensity at the waist position, and it tends to diverge after the waist. Meanwhile, the CAiGB is a self-focusing beam with the character of weak diffraction, which indicates the degree of diverges after focusing is lower than the Gaussian beam. Therefore, compared with the situation using the Gaussian beam as a driven beam to generate high harmonics, the CAiGBs have an outstanding advantage in intensity firstly.
We consider a CAiGB as the initial condition, which can be expressed in the following form [39]:
Using the Fourier transform of the initial distribution, the spatial structure information of harmonic propagation is given. As shown in Fig. 1(b), some parameters of the beam propagating in space are depicted. The line graph in Fig. 1(a) describes the intensity of CAiGB in the initial section along the r direction, which shows that the intensity of the laser field extending from r = 0 to the outside distribution obeys the Airy function. The actual intensity distribution of the initial is shown in Fig. 1(c), which reveals a circular field surround with some low-intensity sidelobes. We use the Fast Fourier transform to propagate the initial field of the CAiGB shown in Fig. 1(a). The results are given in Fig. 1(d), which depict the beam gradually focuses to a point with higher intensity and accompanied by a weak side ring. As shown in Fig. 1(b), the CAiGB beam is slowly shrinking start with a ring shape field and finally focusing to a point. After the focus point, the CAiGB does not continue to return the form of a circle but continues to propagate forward and last for a long distance with a slowly changing intensity, which provides a good region for producing high harmonics.
Fig. 1. (a) The light intensity distribution of a CAiGB. The radius of the main ring is 0.2 mm and the maximum intensity is $8.3\times 10^{13} \textrm {W/cm}^{2}$ at the initial plane. (b) The evolutions of a CAiGB propagating in the free space. (c) and (d) show the intensity distributions at the initial plane and the focus plane.
By changing the radius of the CAiGB initial field, the self-focusing characteristics of the CAiGB also change to a certain extent. Regarding the impact of this on phase-matching, it is discussed in the next section. The CAiGB not only has the characteristics of self-focusing, but its non-diffraction feature also makes it has good potential to be used as a driven beam to generate high harmonics. As shown in Fig. 2, compared with the Gaussian beam, the laser field intensity of the CAiGB under the condition varies with a distance of about one-tenth of the Gaussian beam, which means that when using a CAiGB as a driven beam to generate high harmonics, a long and high enough intensity region can be provided. Herein, it should note that the peak intensity and waist in the focal point of CAiGB are consistent with the Gaussian beam. For the case of the Gaussian beam, due to its fast divergence after focusing, the intensity of the driving field decreases rapidly, which leads to the low HHG efficiency, although in this case the harmonics corresponding to the short trajectory have a good phase-matching condition. Therefore, the CAiGBs seem to have a better harmonic conversion efficiency than Gaussian beams in terms of intensity. However, the effect of phase-matching in generating high harmonics also needs to take into account. In the next section, the HHG of the CAiGBs is further discussed.
Fig. 2. Longitudinal cross-sections of the intensity profiles of a Gaussian beam (left) and a CAiGB(right) in the focus region. The maximum intensity position is defined as the coordinate zero points. Both beams propagate in the z-direction with a 30 $\mu m$ waist radius.
2.3 HHG and propagation model
In this section we introduce the harmonics calculation model to obtain the high harmonics spectrum. The strong-field approximation model (SFA) is applied to calculate the harmonic radiation [16]. The nonlinear dipole momentum can be described as
In this equation, E(t) is the laser field, A(t) is its associated vector potential, $\varepsilon$ is a positive regularization constant. $p_{\textrm {st}}$ and $S_{\textrm {st}}$ are the stationary momentum and quasiclassical action, which values are
The $g(t)$ in Eq. (5) represents the ground-state amplitude:
$w(t')$ is the ionization rate, which is calculated by the Ammosov-Delone-Krainov (ADK) tunneling model [40]:To simulate the collective response of macroscopic gas, we numerically solve the Maxwell wave equation for the laser pulse $E_{t}$ and the harmonics $E_{h}$ fields in cylindrical separately [41],
This propagation model has takes into account both temporal plasma-induced phase modulation and the spatial plasma lensing effects on the driving laser field but except the linear gas dispersion and the absorption of high harmonics during the HHG process, which is due to the low gas density in our numerical calculation [41]. Equations (13) and (14) can be numerically solved with the Crank-Nicholson method as described in Ref. [41].
3. Results and discussion
In the previous section, the self-focusing and non-diffraction properties of the CAiGB have been introduced. From a preliminary point of view, the CAiGB has certain advantages to generate high harmonics than the Gaussian beams. This section gives the phase-matching information of the CAiGB under different initial ring radius, and the harmonic spectral under the condition of CAiGB with the 0.2 mm initial radius is also shown. As described earlier, the change of the initial ring radius can affect the focusing and phase character of CAIGB. Therefore, we choose the CAiGB under different initial ring radius to calculate its spatial intensity and phase distribution and further calculate the wave vector mismatching degree of each point in the space by Eq. (2). Based on the results of the wave vector mismatching degree, the coherence length can be obtained by Eq. (3). Here we show the 21st and 35th harmonics coherence length results in Fig. 3 and Fig. 4.
Fig. 3. Maps of the coherence length for the 21st harmonics along and perpendicular to the propagation direction driven by CAiGBs. The coherence length from Eq. (3) for the short trajectories [(a)-(c)] is compared with those of the long trajectories [(d)-(f)]. The intensity distributions of the CAiGBs with 800 nm are plotted in (g)-(i), and the peak intensity at the focus z = 0 is $3\times 10^{14} \textrm {W/cm}^{2}$. Left panel [(a),(d),(g)] and middle panel [(b),(e),(h)] and right panel [(c),(f),(i)] for the initial radius of the CAiGBs are 0.2,0.35 and 0.5 mm respectively.
Figure 3 shows the coherence length of the 21st harmonic for the long (left column) and short (middle column) trajectories for various CAiGB beams. We choose a CAiGB with the wavelength of 800 nm and the peak intensity of $3\times 10^{14} \textrm {W/cm}^{2}$ but with the different initial radius. The results show that it can supply a larger focus area under the larger initial ring radius CAiGB condition. However, the phase matching region with high coherence length is also located off-axis, which indicate that the harmonics driven by the large initial radius CAiGB may be a ring-like structure.
Figure 4 displays the same coherence maps as Fig. 3 but for the 35th harmonic. It can be observed that the focal point is positioned a bit forward than the larger radius when the initial radius is small. If the large radius condition is considered, the region with high intensity becomes more extensive. As shown in the phase-matching diagram, the long trajectory phase-matching position is generally near the focus point than the short trajectory phase-matching position. The harmonics produced are mainly originated from the long trajectories when the gas cell is put just after the focus point. Correspondingly, the focus region of a large radius is significantly longer to generate high harmonics when the trajectory is not considered. Therefore, a more efficient harmonic conversion efficiency can be supplied by choosing a CAiGB with a large initial radius. In short, a broad phase-matching region could be obtained by CAiGB compared with the Gaussian beam under some conditions.
The CAiGBs appear to have an excellent harmonic generation efficiency based on the phase-matching diagram. However, it is far from sufficient to conclude from the phase-matching diagram that CAiGBs are superior in generating high harmonics, because the phase-matching is the ideal result in the absence of the ionization of the media. Note that the ionization of the media can significantly impact the generation of harmonics due to the defocusing effect induced by the plasma. Thus, these factors need to be further considered. Therefore, according to the phase-matching diagram, we further calculate the high harmonics signal using the coupling model demonstrated in Fig. 5(a). The Lewenstein model and the ADK ionization rate are calculated in the single-atom response level [16,40], and the nonadiabatic 3D light propagation is further solved to simulate the collective response of macroscopic gas [41]. The driving 800 nm pulse is linearly polarized and assumed to have a sin$^2$ envelope with 5 optical cycles. The radius of the initial ring of the CAiGB beam $r_{0}$ before autofocusing is 0.2 mm. The initial peak density of atoms is $1.4\times 10^{18}/\textrm {cm}^3$ and a 19 mm long gas cell is placed on the laser focus. The peak intensity and radius of the spot for the CAiGB beam at the incident face of the gas medium are respectively $3\times 10^{14} \textrm {W/cm}^{2}$ and 30 $\mu \textrm {m}$, and the peak intensity of the initial plane of the CAiGB is $5.4\times 10^{13} \textrm {W/cm}^{2}$, and the target gas medium is neon. The carrier-envelope phase of the CAiGB pulse is 0. The particle number of neon here corresponds to 45 torr pressure at room temperature. Figure 5(a) describes the variation of the harmonic spectrum as a function of the propagation medium length of the gas cell. The harmonic yields increase continuously within the propagation length as long as 18 mm.
Fig. 5. (a) Map of the spatial integration harmonic spectra changes with propagation direction in the case of 45 torr pressure. (b) The spatial integrating harmonic spectra for the CAiGB and the Bessel-Gaussian beam and the Gaussian beam (in log scale for the y-axis). (c) Variation of the harmonic energy with the length of the gas jet. (d) The spatial integrating harmonic spectra under the pressure of 20, 45, 70, 95 torr gas jet under (z = 19 mm) condition (in log scale for the y-axis).
To further show the advantage of the CAiGB for HHG, We also present the numerical results for the Gaussian and Bessel-Gaussian beams for comparison. As mentioned above, such intense CAiGB is produced by an initial Gaussian beam using a SLM and a lens. We can remove them and use another lens with much shorter focal length to tightly focus the initial Gaussian beam. The focal length of the lens is carefully chosen to ensure the same peak intensities of the CAiGB and the compared Gaussian beam. Assume 2 mm of the beam waist of the initial Gaussian beam, the focal length of Lens2 is calculated to be 0.3 m. The comparing scenario with the Bessel-Gaussian beam is the similar but with a conical lens, and the focusing half-angle $\gamma$ = $0.5^\circ$. The harmonic spectra with these three beams at z = 19 mm are shown in Fig. 5(b). Note that the target neon cell is placed at the focus point for these three beams. It can be seen that the harmonics yields with the CAiGB is much higher than the other two beams. We further calculate the total harmonic yield as a function of the propagation distance for these three cases, as shown in Fig. 5(c). The harmonic energy for the CAiGB case is continuously enhanced with the increasing of the propagation distance and tends to saturate until 18 mm of the cell length, while those of the Gaussian and Bessel-Gaussian cases reaches saturation at about 8 mm of the cell length. These results indicate that using CAiGBs can significantly improve the harmonic conversion efficiency compared with the traditional scheme using Gaussian or Bessel-Gaussian beams, since the saturation length for CAiGBs is twice longer than those of the two comparing beams. In addition, we further provide the integral spectra of the harmonics with CAiGBs at z = 19 mm under different pressures in Fig. 5(d), it is clear that the spectral structures for different pressures are nearly the same, and the pressure only affects the harmonic yields. The results agree well with the conclusion given by the phase-matching diagram discussed above.
As shown in Fig. 5(c), the benefit of CAiGB becomes significant after a few millimeters, which would be a very stressed condition for gas harmonic experiment. It is worth mentioning that the spatial scale of the beam propagation and the corresponding phase matching is flexible for the above results. One can change the parameter of the CAiGB for a shorter interaction length where the benefit of CAiGB becomes significant. For instance, we can reset the characteristic parameter characterizing the beam width $w_{0}^{\prime } = 0.01 \textrm {mm}$ and the radius of the initial ring of the CAiGB beam $r_{0}^{\prime } = 0.1 \textrm {mm}$, and then we can obtain the self-similar map of coherent length but with half scale. In this case, the saturation length for CAiGB is approximately 9 mm, while those for the comparing Gaussian and Bessel-Gaussian beams are as short as 4 mm.
Based on the conclusion above, the CAiGBs can well improve the efficiency of HHG. However, the far-field case of the harmonics also needs to be considered because the far-field spatial characteristics are essential for its applications. Therefore, we perform a Hankel transformation to calculate the spatial distribution of the 35-45th harmonics 1 m away from the focal point, and the result is shown in Fig. 6 [36]. Figure 6(a) and (b) depicts the far-field spatial distribution of the harmonic driven by CAiGB, which show a Gauss-like structure. Compared with the Gaussian beam result shown in Fig. 6(c) and (d), it indicates that the harmonic of CAiGB have better energy concentration than that of the Gaussian beam. The divergence angle of the far-field center point for the CAiGB case is as low as 1.7 mrad, which implied that the harmonics are well collimated and have a good beam quality at the far-field position. The result indicate the harmonics driven by CAiGB are advantageous for many potential applications such as nonlinear studies and plasma diagnostics.
Fig. 6. (a) The far-field spatial profile of the 35-45th harmonics driven by the CAiGB. (b) The one-dimensional situation of the far-field spot. (c) and (d) are the 35-45th harmonic far-field spatial profiles but under the Gaussian beam condition.
4. Conclusion
In summary, we adopted the circular Airy-Gaussian beams to generate high-order harmonics. The phase-matching of the harmonics has been detailly investigated. Due to the abrupt auto-focusing and the non-diffracting propagation of the beams, the phase-matching of the harmonics can be well controlled, and the generated harmonics signals can continuously increase to a distance of a dozen of millimeters. Moreover, the divergence angle of the harmonics of the far-field spot is as small as 1.7 mrad, which achieves a good beam quality of the harmonics. These findings pave the way to the macroscopic control of the high harmonics using the types of abruptly auto-focusing beams.
Funding
Natural Science Foundation of Guangdong Province (2019A1515011172); National Natural Science Foundation of China (11874019).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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