1 Introduction

When an atom is illuminated by a short and intense laser pulse, it will respond in a nonlinear manner and generate a phenomenon of high-order harmonic generation (HHG) [1]. This phenomenon has been studied for many years and has become a significant topic in strong-field physics because it can extend the spectrum of the driving laser field into extreme ultraviolet (XUV) radiation with attosecond to femtosecond duration [2]. This source of ultrashort and coherent XUV has been used in numerous applications such as atomic and molecular physics [3], chemistry [4], and photobiology [5]. In recent years, high harmonics have also been used for the external seeding of free electron lasers and other types of soft X-ray lasers amplifiers [6]. To date, it has been demonstrated that a gas-filled hollow-core waveguide (HCW) can generate high harmonics efficiently [7]. In HCW geometry, the laser is guided by glancing-incidence reflection from the walls of the waveguide and propagates with a plane-wave-like mode so that the effective interaction length is extended. Moreover, the photon flux of the harmonics is obviously increased because phase matching between the fundamental laser and the high harmonics can be achieved by adjusting gas pressure in the HCW [8]. However, the phase-matched high harmonics can be generated in HCW geometry only with a laser at a relatively low intensity, which limits the conversion efficiency of the laser light to higher-order harmonics. In previous work, a periodically modulated HCW was used to achieve quasi-phase matching (QPM) to increase the intensity of the HHG at higher photon energies [9]. However, the periodical structure in the HCW is produced with glassblowing technique [10], which presents challenge in the experimental configuration of many studies.

In this study, we used a new geometry—a slab waveguide filled with He gas—to generate high harmonics. We compared the experimental result with those of a similar experiment performed using a long gas cell. There was an obvious extension of HHG for wavelengths between 5 and 20 nm in this new waveguide geometry. We investigated the mechanism that caused the extension of high harmonics emitted from the waveguide by comparing the spectrum obtained in the long gas cell with that obtained in the He-filled slab waveguide under different experimental conditions. This study showed that the newly designed gas-filled slab waveguide which consist of two pieces of fused silica can generate high harmonics efficiently. More importantly, fabrication of the periodical structure on the fused silica surface is easier than that in the HCW. The slab waveguide with a periodical structure has the potential for HHG with quasi-phase matching in the future.

Our experimental setup is shown in Fig. 1. Laser pulses from a kilohertz Ti:sapphire laser amplifier system (3.8 mJ, 75 fs, 800 nm) were focused with a f = 400-mm lens into a gas cell or a slab waveguide. The lens was mounted on a motion stage so that position of focus can be adjusted. The energy of the laser pulse that reached the gas target was 3.5 mJ, and the beam radius at the focal spot was ~40 μm. Therefore, the effective peak intensity at the focal spot was approximately 9.35 × 1014 W/cm2. The gas cell was sealed by two 0.1-mm copper sheets at each ends. In the experiment, the laser beam bored through the copper sheets and interacted with the He in the cell. The slab waveguide was formed by two pieces of fused silica which were placed in parallel with a distance of d = 200 μm. The waveguide was mounted at a vacuum motorized table containing two adjustments. Adjustments include translation and rotation. Thus, the laser could be coupled in the waveguide when the waveguide was inside a vacuum chamber. Both gas cell and slab waveguide were 20 mm long and were backed with helium. As shown in Fig. 1, gas flows into the waveguide through the slit on one side which is perpendicular to the propagation direction of the beam, while the slit on the opposite side is sealed. High harmonics and fundamental wave emitted from the gas cell and the slab waveguide went through two 200-nm zirconium (Zr) filters that block the femtosecond laser pulse. The high harmonics was detected by a XUV spectrometer (Model 251MX, McPherson) which consists of a varied line spacing concave grating (1200 grooves/mm) and a XUV charge-coupled device (CCD) (PI-MTE, Princeton).

Fig. 1
figure 1

Setup for HHG in a gas cell or a gas-filled slab waveguide shown in the insets. M1 and M2 mirrors, L lens, T target, F filter, G grating

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When the laser pulse is focused into the gas cell, it propagates in the free space and interacts with He atoms simultaneously. After careful adjustment of the lens position, the laser was focused in the middle of the medium. The spectrum of HHG from the gas cell where the laser energy was 3.5 mJ and the gas pressure was 5 kPa is shown in Fig. 2 (black curve). Because of the limitation of the spectrometer, only radiations between 5 and 20 nm was detected. Figure 2 shows that the harmonic spectrum from He atoms in free propagation reaches a clear cutoff around 9.2 nm (H87). Similar results have been obtained in many groups [11, 12]. According to semiclassical theory for the motion of an ionized electron during the first optical cycle after it is ionized by laser field [13], the maximum harmonic order is determined by the cutoff law \(h\nu_{\hbox{max} } = I_{\text{p}} + 3.17U_{\text{p}}\), where I p is the ionization potential of the atom and \(U_{\text{p}} \propto I\lambda^{2}\) is the ponderomotive energy for a laser beam of intensity I and wavelength λ. However, in fact, the highest photon energy of harmonics should be lower than that calculated by the cutoff law because of many factors such as phase mismatching between fundamental wave and harmonics in the ionizing gas [14].

Fig. 2
figure 2

High harmonics emission from the gas cell and the gas-filled slab waveguide. For both cases, the laser intensity at the focus was about \(9.35 \times 10^{14}\,{\text{W}}/{\text{cm}}^{2}\) and the gas pressure was 5 Kpa. In the case of the slab waveguide, the second-order diffraction can be observed because of higher harmonics generation

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For comparison, Fig. 2 also presents the spectrum generated from slab waveguide filled with He (red curve) on the same scale as that from the gas cell. The signal level obtained for the low harmonic orders (between H47 and H77) in the waveguide is as strong as that in the cell. However, the higher-order harmonics emitted from the waveguide are apparent, even for harmonic orders up to the cutoff (H111) rather than dramatically decreasing starting at H77, as with the gas cell. The result shown in Fig. 2 is similar to that reported in Ref. [15], in which the HHG from an Ar-filled HCW showed an obvious extension compared with the gas jet configuration. The laser intensity, pressure, and length of the waveguide and the cell were same, making it possible to say that under these experimental conditions, the HHG is achieved with higher efficiency in a gas-filled slab waveguide between 5 and 20 nm.

In Ref. [15], the authors attributed the higher harmonic orders generated in the HCW to argon ions. In our experiment, laser intensity at the focal spot is about 9.35 × 1014 W/cm2 at which the laser cannot efficiently drive helium ions to generate high harmonics [16]. Therefore, the extension of the harmonics generated to a higher order in the slab waveguide should not the result of emission by ions. In order to gain some insight into the origin of the different phenomena of the two geometries, we consider the following factors. First, the self-focusing in the waveguide due to the optical Kerr effect can increase laser intensity. On the contrary, the ionization-induced laser defocusing in the gas cell can decrease laser intensity. Because the higher-order harmonics are very sensitive to the laser intensity, the variation of the laser intensity will influence the cutoff region of the high harmonics. Second, phase matching may be improved, which can lead to the extension of the HHG from the waveguide.

For the self-focusing, the critical power P cr can be used to estimate its effect [16]. It is determined by \(P_{\rm cr} = \lambda_{0}^{2} /2\pi n_{2}\) where the λ 0 is the wavelength of the femtosecond laser and n 2 is the Kerr coefficient (\(n_{2} \approx {p} \times 10^{ - 19}\,{\text{cm}}^{2}/{\text{W}}\), where p is the gas pressure in bar). For λ 0 = 0.8 μm and \({p} = 5\,{\text{KPa}}\), \(P_{\rm cr} = \lambda_{0}^{2} /2\pi n_{2} \approx 2 \times 10^{11}\,W\). However, the input pulse has a power \(P_{\text{in}} \approx 4.6 \times 10^{10}\,W\), leading to the ratio \({P}_{\text{in}} /{P}_{\text{cr}} = 0.23\). Therefore, the self-focusing should not occur in the waveguide under our present experimental conditions.

Next, we estimated the effect of ionization-induced laser defocusing in the gas cell. The ADK theory was used to calculate ionization level [17]. The accurate value of the ionization level should be calculated by full quantum theory [18]. Here, we just use the ADK mode to qualitatively analyze the effect of the laser defocusing. In our experiment, the laser intensity of about \(9.35 \times 10^{14}\,{\text{W}}/{\text{cm}}^{2}\) at the focal spot led to an ionization level \(\eta \approx 24\,\%\). Thus, for \({p} = 5\,{\text{KPa}}\), the density of ionized gas atoms \(\rho_{\text{e}}\) was about \(3.3 \times 10^{17}/{\text{cm}}^{3}\). The defocusing length [19], \(L_{D} = \lambda_{0} \rho_{c} /2\rho_{e}\) where \(\rho_{\text{c}}\) is the critical density, was ~2 mm. This is less than the medium length and the confocal parameter of the laser beam. Therefore, the converging light in the range of the Rayleigh length diverges when the distance it travels in the gas cell is more than 2 mm in the gas cell [19, 20].

Phase match of the HHG can be achieved in waveguide using the dispersion of neutral gas atoms to compensate the dispersion of the plasma and the waveguide [8]. However, this method is limited by the critical ionization level above which the dispersion of the neutral atoms cannot balance the dispersion of the plasma [9]. Because the ionization level in our experiment is far larger than the critical ionization level of He (0.5 %) [10], the phase match of the HHG cannot be achieved in the waveguide.

In summary, we considered different factors that may result in the experimental phenomena seen in the two geometries. According to the calculated results, the ionization-induced laser defocusing in the gas cell should be responsible for the result shown in Fig. 2.

Experimentally, if laser defocusing occurs in the gas cell, the higher harmonic orders, especially for the cutoff region, will be dependent on the gas pressure. Therefore, we measured the dependence of the intensity of high harmonics on the gas pressure in the 20-mm-long gas cell. The laser energy was set at 3.5 mJ, while the gas pressure was varied from 1 to 5 kPa. Figure 3 shows the cutoff region of the harmonics spectra recorded at gas pressure of 2, 3, 4 and 5 kPa. At the low gas pressure such as 2 kPa, the highest harmonic order reaches H97. The cutoff decreases gradually with the increasing gas pressure until it is H87 at 5 kPa. This observed experimental trend is consistent with the result in Ref. [21], where the author used three-dimensional numerical code to simulate the harmonic macroscopic response and thus attributed this phenomenon to ionization-induced laser defocusing. The ionized gas results in a laser focal spot with a larger radius, which then leads to a weaker peak intensity. The higher the gas pressure we used, the weaker the peak intensity of the laser. Therefore, as is shown in Fig. 3, the highest harmonic order for 5 kPa is lower than that for 2 kPa. According to the dependence of the high harmonics cutoff region on the gas pressure in Fig. 3, we can predict that if the gas pressure is low enough that the effect of laser defocusing can be neglected, then the high harmonics cutoff region in the gas cell will be the same as that in the waveguide. However, very low gas pressure results in a low signal level of high harmonics that is too low to be detected by the CCD.

Fig. 3
figure 3

Harmonic spectra at the cutoff region of the 20-mm-long He gas cell measured at different gas pressures driven by a 75-fs laser pulse with a peak intensity of \(9.35 \times 10^{14}\,{\text{W}}/{\text{cm}}^{2}\)

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According to the defocusing length \(L_{\text{D}} = \lambda_{0} \rho_{\text{c}} /2\rho_{e} \approx 2\,{\text{mm}}\), we next consider that if the length of the gas cell is shorten to less than 2 mm, the ionization-induced laser defocusing will be minimized. The harmonic orders obtained in shorter gas cell should be higher than those in 20-mm-long gas cell and would be the same as in the waveguide. Therefore, we replaced the 20-mm-long gas cell with a 1-mm-long gas cell. Figure 4 shows that the emission of the high harmonics in the 1-mm-long gas cell is obviously expended to a shorter wavelength than that in the 20-mm-long gas cell shown in Fig. 2. This experimental result is consistent with our expectation and indicates that laser defocusing was minimized under these experimental conditions. Moreover, Fig. 4 shows that the high harmonics cutoff region of the 1-mm-long gas cell is similar to that of the 20-mm-long waveguide. This further confirms that the decreasing high harmonics cutoff in the gas cell results from the laser defocusing induced by ionized He gas. As a result, similar to HCW geometry, the gas-filled slab waveguide can effectively restrain ionization-induced laser defocusing when a laser pulse propagates inside it.

Fig. 4
figure 4

High harmonics emission from 1-mm-long gas cell (a) and 20-mm-long gas-filled slab waveguide (b) for 5 kPa gas pressure and 75-fs driving laser pulse with peak intensity of \(9.35 \times 10^{14}\,{\text{W}}/{\text{cm}}^{2}\)

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However, the slab waveguide may be thought to guide the laser pulse in only one dimension, resulting in occurrence of the laser defocusing effect in the other dimension. Figure 5 shows that with increasing gas pressure, the trend of the intensity of the high harmonics in the cutoff region in the slab waveguide is similar to that in the 20-mm-long gas cell (Fig. 3). However, compared with Fig. 3, the harmonic orders for the slab waveguide hardly decrease, even with the use of very high gas pressure such as 10 kPa. This implies that the laser pulse is unaffected by laser defocusing in the dimension in which it is not confined by the waveguide boundary. This result may be due to the self-focusing that occurs at higher gas pressure and can compensate for the laser defocusing induced by the ionized gas. This explanation need to be further confirmed in both theory and experiment in future.

Fig. 5
figure 5

Measured harmonic spectra at cutoff region in 20-mm-long He gas-filled slab waveguide for different gas pressure driven by 75-fs laser pulse with peak intensity of \(9.35 \times 10^{14}\,{\text{W}}/{\text{cm}}^{2}\)

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In conclusion, we have experimentally demonstrated that a gas-filled slab waveguide can be used to generate high harmonics. Comparing the spectra of He atoms in a gas cell with those of the He-filled slab waveguide at different experimental conditions, we found that higher harmonic orders were generated in the slab waveguide; however, the harmonic orders in a 1-mm-long gas cell were similar to those in 20-mm-long waveguide. Simple calculations and analysis showed that ionization-induced laser defocusing restrained in the slab waveguide explains these phenomena. The main limitation of this method is the 1D confinement which needs to be improved in future. More experiments and analysis work need to be done to fully understand physical mechanism and process of the HHG in the gas-filled slab waveguide. We plan to change the structure of the boundary of the slab waveguide to achieve high harmonics with higher efficiency. Although this plan is similar to the work of Paul et al. [9] in which they used a modulated HCW to improve the efficiency of the high harmonics, achieving a periodic structure on a flat surface is easier than in a HCW. Furthermore, it is possible to fabricate other structures on a flat surface such as what Zheng et al. [22] proposed to generate high harmonics effectively.