1. Introduction

In the past two decades, significant efforts have been focused on improving the generation of extreme ultraviolet (XUV) harmonics by utilizing infrared laser pulses with a high repetition rate [1,2]. These efforts have involved various techniques, such as traditional two-color pump (2CP) schemes for high-order harmonic generation (HHG) in atomic and molecular gases, as well as the mixing of noble gases [3].

The use of dual wavelengths (λ and λ/2) as driving pulses in HHG is one of the promising approaches to improve the efficiency of generated harmonics in the XUV range. This technique allows the generation of stronger high-order harmonics from the plasmas, gases, and solid materials as compared to the single-color pump (λ) at both orthogonal and parallel-polarized arrangements of dual-color driving pulses. Previous studies have shown that the two-color pulses with orthogonal polarization generate stronger harmonics as compared to the parallel configuration. Additionally, 2CP generates harmonics, which are even integers of a fundamental beam (λ) and as strong as the emitted low-order odd harmonics.

The application of this approach reveals new characteristics of resonance-enhanced harmonics related to the strong ionic transitions of the plasma medium, as well as the relation between the phases of the two pumps. This technique allowed probing the even harmonics in the vicinity of resonant transitions [4,5]. Additionally, the use of 2CP has proven effective in enhancing the ionization process of atoms or molecules, which plays a crucial role in the three-step model of HHG [6]. The application of 2CP has been extensively studied both theoretically and experimentally to enhance HHG in gases [7–9]. Consequently, the enhanced XUV harmonics produced in gases through 2CP can serve as a valuable tool for investigating electron dynamics in atoms, molecules, and solids via HHG spectroscopy [10–12]. For instance, enhanced XUV generation in gases using 2CP has been successfully demonstrated by Kim et al. for both orthogonally and parallel-polarized 2CP [13]. Additionally, experimental studies and theoretical calculations have shown a correlation between the polarization state of the driving laser pulses and the HHG yield [14]. These calculations have revealed a variation in the ionization rate for parallel-polarized (approximately 90%) and orthogonally-polarized (approximately 30%) two electric fields of 2CP. These findings are consistent with the Ammosov-Delone-Krainov (ADK) theory, which calculates the ionization rates of gas atoms in strong laser fields [15]. However, the high ionization rate in parallel-polarized fields poses challenges in maintaining phase-matching conditions. Under such conditions, the driving field experiences significant self-phase modulation in the ionized medium, leading to a decrease in HHG conversion efficiency. Typically, the ionization rate (η) exhibits a dependence on the wavelength of the driving laser pulses, which can be approximated as ($\eta \sim 1/\lambda _L^{ - 2}$) where λ_L is the wavelength of laser radiation [16]. One can imagine that if a nonlinear medium consists of one gas, the ionization rate increases by adding the extra laser fields with the shortest wavelength. It leads to the phase-mismatching between the harmonics and 3CP electric fields in the interaction region in the case of highly ionized gases. In that case, the mixing of two media comprising the gas with high-ionization potential and the gas with low-ionization potential might compensate for the ionization and improve the phase-matching conditions for the harmonics in the case of 3CP. Using the single gas medium and 3CP can lead to a larger phase mismatch at increased plas.a densities due to the stronger effect on the harmonic yield at high photon energies [17]. To compensate for phase-mismatch in the highly-ionized medium, the optimization of waveforms of 3CP laser fields requires a complex device, which was suggested by Jin et. al. [18] and experimentally developed by Burger et.al. [19]. This device permits superimposing all three-colors and controlling their relative phase as well as their intensities and polarizations during the process of HHG in gases. In particular, this device with optimized parameters of three-color laser fields and program was used for the generation of enhanced XUV harmonics in pure He and Ne gases [20]. It was shown that the 3CP device offers sufficient flexibility for a variety of future studies on the coherent control of strong-field processes and for time-resolving ultrafast molecular dynamics. The advantage of this method was further demonstrated by comparing the 3CP waveform in a gas cell and a previously optimized two-color waveform in a gas-filled waveguide [21]. The enhanced harmonic continuum spectrum output in the 3CP regime was a result of precise control of delay-time and CEP in between three wavelengths of the driving laser pulse [22–25]

In contrast, theoretical studies show that the utilization of non-traditional three-color laser fields (3CP) allows for generating elliptically oriented XUV harmonics with extremely short pulse durations [26]. In their theoretical calculations of HHG in pure noble gases, Habibović et al. reported on the combination of a fundamental driving pulse with the second or third harmonics [27,28]. These calculations provided insights into the prerequisites for generating enhanced HHG with high ellipticities. In addition, the use of the 3CP scheme with high ellipticity of HHG in a mixture of gases could be used to probe molecular structure. In particular, HHG in a mixture of ionization-rate-limited CO₂ and Kr gases provided information on the molecular structure of CO₂ through XUV harmonics generation by using a single-color pump (SCP) scheme [29]. On the other hand, increasing the ionization rate of molecules in 2CP and 3CP schemes of HHG schemes can solve the problems of limiting the ionization rate of molecules at the available maximum laser pulse intensity and provide additional information about the structure of molecules.

The generation of enhanced XUV harmonics can also be explored using mixtures of noble gases in a single gas jet with optimized pressures [30,31]. Optimizing the focusing geometry of the driving laser pulses can contribute to the enhancement of XUV harmonics [32]. For instance, Takahashi et al. reported a significant increase in XUV harmonics intensity by using a single-color laser field with mixing gases that have different ionization potentials [33]. It was shown that the mixing of rare gases with low and high ionization energies can improve the phase-matching conditions of HHG. Correspondingly, one can expect that the simultaneous application of the gas mixture (with low and high ionization potentials) and 3CP schemes for HHG can enhance the intensities of XUV harmonics by improving the dispersion-based phase-matching conditions. This approach can find applications in nonlinear spectroscopy. Furthermore, the enhanced XUV harmonics in the photon energy range of 20–50 eV can be utilized in the transient XUV absorption experiments. This can be particularly useful for investigating coherent electron dynamics in lithium molecules, where the Li K-edge corresponds to the photon energy of 55 eV [34].

In the present work, we conducted experimental studies on the generation of XUV harmonics using femtosecond laser pulses at wavelengths of 1030 nm, 515 nm, and 343 nm. The experiments were performed at a repetition rate of 50 kHz. We utilized a combination of low-order harmonics and the fundamental laser beam to generate XUV harmonics in a mixture of atomic gas species. The use of a mixture of noble gases with varying ionization potentials is of particular interest in optimizing phase-matching conditions. The experimental results presented in this paper provide insights into the generation of XUV harmonics using the 3CP scheme.

2. Experimental arrangements

Femtosecond fiber laser system (Active fiber laser systems) was used to deliver 40 fs driving pulses at a 50 kHz repetition rate regime with an average power of 25 W. The wavelength of the fundamental (ω) laser beam was 1030 nm. The radius (${w_0}$) of the fundamental beam at the focal plane was equal to 40 µm and the Rayleigh range (${z_R} = \frac{{\pi w_0^2}}{\lambda }$., $\lambda $ is a wavelength of fundamental driving pulse) was 4.8 mm. The maximal intensity of the fundamental SCP beam inside the gas jet at these focusing conditions was I_ω= 2.4 × 10¹⁴ W cm^-2. For generation of the second (2ω) and third (3ω) harmonics (SH and TH) from fundamental driving laser pulse, two nonlinear barium borate (β-BBO) crystals were used (Fig. 1(a)). The conversion efficiencies of 2ω in the 0.4 mm thick BBO (type I) crystal was measured to be 8% and the corresponding intensity of 2ω in the focal plane w I_2ω= 7.6 × 10¹³ W cm^-2. The conversion efficiency of 3ω in the 0.1 mm thick BBO (type I) crystal was measured to be 2.5% and the intensity of the focused 3ω beam was I_3ω= 5.4 × 10¹³ W cm^-2. The relative intensities of waves were calculated to be I_ω:I_2ω:I_3ω=1:0.31:0.225. The time delay between two pulses (ω and 2ω) after propagation of the 0.4 mm thick BBO crystal was calculated to be 48 fs. The total time-delay shift between ω and 2ω laser pulses in the case of 3CP could reach up to 60 fs due to the (o_ωe_2ω-e_3ω) type of interaction in the second BBO crystal. The numerical calculations show that the generated TH pulse should be located between the ω and 2ω pulses so that the energy of different waves can be coupled sufficiently in the gas medium [35].

 

Fig. 1. (a) Experimental setup for HHG in the gas mixture. AFS is a fiber-based femtosecond laser. FL is a 400-mm focal length focusing lens. The barium borate crystals placed on the axis of the fundamental driving pulse are used for the generation of the low-order harmonics (2ω and 3ω). GJ is a gas tube with a 3 mm inner diameter. S is a slit of the XUV spectrometer. PM is a gold-coated plane mirror, which directs the generated XUV radiation towards the grating. FFG is a flat-field grating for the dispersion of XUV radiation. MCP is a microchannel plate detector of XUV radiation. Inset: the polarization states for ω, 2ω, and 3ω laser pulses inside the gas jet. (b) The relative intensities of the fundamental driving pulse and its harmonics and the delays between three waves.

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The polarization states of a fundamental laser beam and its harmonics are presented in Fig. 1(a). Here the polarization of the fundamental laser pulses is vertically oriented and the polarization states of its low-order harmonics are horizontally oriented with regard to the axis of propagation of the laser beams. Figure 1(b) shows the temporal map of separated pulses after propagation through the 0.4 and 0.1 mm thick BBO crystals. Red, green, and blue curves correspond to the fundamental (ω), SH (2ω), and TH (3ω) pulse shapes. The total sum of the 3CP electric field consisting of the combination of perpendicularly polarized fundamental and harmonics lights can be considered as an elliptically polarized laser field.

The combination of fundamental (λ_ω=1030 nm), (λ_2ω= 515 nm), and (λ_2ω= 343 nm) waves was used for HHG in pure and mixed gases. During the experiment, the gas pressure in the chamber was ∼3.5 × 10⁻³ mbar. The effective length of the gas jet was equal to ∼ L = 3 mm. We used a gas jet with a 3-mm inner diameter, which allows for maintaining weak focusing conditions according to the relation L < b (b is a confocal parameter, b = 9.6 mm). The gas tube was installed at the focal plane of the focusing lens. Different gases (He, Kr, and Xe), as well as equal quantities (50/50) of mixtures of He + Kr and He + Xe, were utilized during these studies. These gas targets were used to generate high-order harmonics during the interaction with the three-color laser pulses.

An XUV spectrometer consisting of a flat-field grating and a micro-channel plate (MCP) detector coupled to a phosphor screen was used for the analysis of the generated harmonics. Additionally, we installed either a gold-coated plane mirror or a gold-coated cylindrical mirror inside the XUV spectrometer. The plane mirror was used to analyze the divergence of the harmonics, thus retrieving the information about their spatial distribution. On the other hand, the cylindrical mirror allowed us to collect the focused images of the harmonics. The XUV harmonics were detected by an MCP and then the images of harmonics distribution were collected by a charged coupled device camera. By employing this experimental setup, we were able to investigate the generation and properties of high-order harmonics in different gas targets and gas mixtures, providing valuable insight into the underlying physical processes.

3. Results and discussion

Figure 2 shows the spectra of the XUV harmonics generated from the interaction of the 1030 nm fundamental femtosecond driving laser pulses with three different gases (He, Kr, and Xe). The spectra exhibit distinct characteristics due to the cutoff law and the wavelength scale dependence of the harmonic order [6]. In Xe gas, we observed a low cutoff position, indicating that the highest harmonic generated was at lower energy compared to the other gases. In Kr and He gases, on the other hand, we observed higher cutoff positions, indicating that higher energy harmonics were generated. Specifically, in He gas, the cutoff position of the harmonic was observed at H73, corresponding to the 73^rd harmonic of the fundamental wavelength. In Kr gas, the cutoff position was at H55, corresponding to the 55^th harmonic, and in Xe gas, the cutoff position was at H51, corresponding to the 51^st harmonic. These cutoff positions were determined under the condition of a SCP intensity of 2.0 × 10¹⁴ W/cm². The cutoff positions provide valuable information about the energy range and the maximum achievable photon energy of the generated harmonics in each gas. The figures demonstrate relatively strong outputs of XUV harmonics in different ranges of high photon energies. In the case of He gas (Fig. 2(a)), we observed strong harmonic yields in the range of 25 to 80 eV of photon energy. This indicates that the He gas was efficient in generating harmonics within this energy range. On the other hand, in Xe gas (Fig. 2(c)), we observed strong XUV harmonics in the low-order range of photon energies, specifically from 15 eV to 25 eV. This behavior can be attributed to the presence of more free electrons in the Xe gas jet. Additionally, laser defocusing in Xe gas led to a lower harmonic cutoff region. The strong harmonic generation in Kr gas (Fig. 2(b)) partially covered both low and high photon energy ranges. This suggests that Kr gas was effective in generating harmonics across a broader energy range compared to the other gases.

 

Fig. 2. HHG spectra generated in pure He, Kr, and Xe gases by using SCP of fundamental (ω) driving laser pulses. HHG spectra generated in He, Kr, and Xe gases show cutoff positions at the H73 (a), H55 (b), and H51 (c) at the intensity I_ω= 2.4 × 10¹⁴ W cm^-2 of linearly polarized fundamental driving laser pulses. The gas pressure inside the chamber was kept at 100 mbar.

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These observations and behaviors of the different gases motivated us to explore the use of their mixtures for the generation of enhanced XUV harmonics in different ranges of photon energies. By combining the gases, we aimed to take advantage of their individual strengths and generate harmonics in a wider range of photon energies. It is important to note that the specific characteristics and behaviors of each gas and their mixtures provide valuable insights for optimizing and tailoring the generation of XUV harmonics for various applications in different energy ranges.

Figure 3 shows the spectra of XUV harmonics generated in the fields of the combined three-color pulses (ω+2ω+3ω) in a 3 mm long gas jet. The gas targets used for this analysis were pure He, Kr, and Xe gases. The spectra provide valuable information about the origin of the high harmonic spectra generated from the 3CP interacting with atoms of each gas target. By analyzing the spectra, we can gain insights into the efficiency and characteristics of HHG in each gas target. The use of pure He, Kr, and Xe gases allows us to isolate and study the specific contributions of each gas to the generation of XUV harmonics in the combined three-color fields. In the case of XUV harmonics generated in pure He gas; we observed the generation of both odd and even harmonics from the combined 3CP. However, we also observed suppression of XUV harmonics intensities in the ranges where the photon energies were lower than 25 eV. On the other hand, in the cases of Kr and Xe gases, we observed the generation of strong odd and even XUV harmonics from the 3CP. This suggests that these gases are more efficient in generating harmonics across a wider range of photon energies compared to the He gas at the conditions of 3CP.

 

Fig. 3. HHG spectra from pure noble gases by using the 3CP comprising the fundamental (ω), SH (2ω), and TH (3ω) laser pulses. HHG spectra generated in He, Kr, and Xe gases show cutoff positions at the H49 (a), H41 (b), and H39 (c), respectively, at the intensities of I_ω= 2.4 × 10¹⁴ W cm^-2, I_2ω= 1.9 × 10¹³ W cm^-2 and I_3ω= 6 × 10¹² W cm^-2 driving laser pulses. The gas pressure inside the chamber was kept at 100 mbar.

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Notice that the experimental generation of high harmonics in gases using the 3CP has not yet been reported. The only study of the implementation of 3CP for HHG was the analysis of this process in laser-induced plasma at the conditions of the sum and difference frequencies generation using the 806 nm and tunable near-infrared (1300–2000nm) laser pulses [36]. At optimal conditions for the laser–plasma interaction, the intensities of the sum and especially the difference components are almost equal to the intensities of neighboring harmonics attributed to the strong 806 nm pump. Additionally, the demonstration of high-order sum and difference frequency generation using mixtures of tunable near-infrared pulses from an optical parametric amplifier and an 800-nm class Ti:Sapphire laser in extended (5-mm-long) laser-induced graphite plasma using the 2CP and 3CP have shown the generation of the combinations of interacting waves [37]. It was demonstrated that difference frequency mixing schemes are more efficient than sum frequency mixing due to better phase matching. This is because, in difference frequency mixing, the geometrical phase shifts because the two colors have opposite signs, and therefore partly compensate for each other, leading to better phase matching. In contrast, in sum frequency processes, the geometrical phase shifts have the same sign and therefore add, leading to poorer phase matching.

The temporal overlapping of the two-color pulses has been extensively analyzed in previous studies [38,39], as it is a crucial parameter in the high harmonic generation (HHG) process using 2CP or 3CP laser fields. The theoretical evaluations of this process have been reported by Habibović et al. [27], providing valuable insights into the expected behavior and efficiency of harmonic generation using 3CP laser fields. In our study, we specifically used temporally and spatially overlapped three-color driving laser pulses at different wavelengths.

The comparison of the intensities of XUV harmonics was provided in Figs. 2, 3 and 4. Figure 2 shows the spectra of the XUV harmonics generated from the interaction of the 1030 nm fundamental femtosecond driving laser pulses with three different gases (He, Kr, and Xe). Figure 3 shows the spectra of XUV harmonics generated in the fields of the combined three-color pulses (ω+2ω+3ω) in a 3 mm long gas jet with the similar 100 mbar pressure of used gases.

 

Fig. 4. (a) Raw images of HHG in mixed gases He + Kr at the total pressure of 100 mbar with 50/50 ratio of each gas, respectively. The upper panel is 3CP and the bottom panel is SCP. (b) Raw images of HHG in mixed gases He + Xe at the total pressure of 100 mbar with a 50/50 ratio of each gas, respectively. The upper panel is 3CP and the bottom panel is SCP. (c) Dependence of the critical ionization level on the ionization potential of the noble gases. For extrapolation of the data for 1030 nm driving laser pulses (black-solid line) we used data from Ref. [16]. The purple and green squares show the possible value of the critical ionization levels of Xe and Kr gases for 1030 nm driving laser pulses, respectively.

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Figure 4 shows the raw images of harmonics in mixed gases, specifically He (50 mbar) + Kr (50 mbar) (upper panel) and He (50 mbar) + Xe (50 mbar) under the influence of the 3CP (upper and middle panels). In the total pressure of 70 mbar of mixed gases, we observed a clear spectrum of XUV harmonics generated from mixed (He (35 mbar) + Kr (35 mbar) and He (35 mbar) +Xe (35 mbar) gases at the 3CP scheme of HHG. In this case, we observed lower intensity and cutoff of the XUV harmonics. By increasing pressure of mixed gases, we observed increasing of intensity of XUV harmonics due to the constructive phase-matching conditions at the high pressure of the mixed gases. In general, the yields of HHG for pure gases, initially increase with increasing pressure reaching the maximum, and then they decrease with the further increase of gas pressure [40,41]. It has been also demonstrated that in the presence of a second pulse, the destruction of the phase-matching conditions was apparent. From our observation, we conclude that the increase of pressure of the mixture of gases can improve phase-matching conditions by compensating free-electron dispersion from heavy Xe and Kr gases and atomic dispersion from light He gas. The observed enhancement factors are related to the difference in the ionization level of different gases mixed in the interaction area with a pressure of 50 mbar of each gas. It follows from Fig. 4, that the HHG in the 3-mm gas jet comprised the light He and heavy Kr and Xe gases demonstrate stronger harmonics compared to our previous work, where the diameter of the gas jet was 1-mm [42].

To quantitatively analyze the harmonic efficiency in the mixed gases under the 3CP scheme of HHG, we determined the enhancement factor of the harmonic yields at similar conditions by mixing gases in the 3CP laser field. We observed an enhancement factor of approximately 7 in the intensity of XUV harmonics in the photon energy range of 20-35 eV when using the 3CP scheme compared to the case of a single-color (1030 nm) laser. However, we did not observe a similar enhancement of XUV harmonics in the case of the He + Xe mixture under the 3CP scheme of HHG. This suggests that the interaction between the 3CP laser fields and the mixed gases has a varying effect on the harmonic generation efficiency, depending on the specific combination of gases.

The analysis of the XUV harmonics generated in mixed noble gases using the 3CP provides valuable insights into the enhancement factors and efficiency of harmonic generation in these gas mixtures. To explain the trends observed in our results, we performed extrapolation curves for ionization rates of different gases at the different driving laser pulses. The ionization rate of gases can be determined by the critical ionization level, which is the threshold power required for ionizing gas atoms. This critical power, denoted as P_c, can be approximated by the equation ${P_c} \approx \frac{{\lambda _L^2}}{{8\pi {n_0}{{\underline{n} }_2}}}$, where ${n_0}$ is a refractive index of the medium, and ${\underline{n} _2}$ is a nonlinear refractive index of the driving laser pulse [16]. When the driving laser pulse reaches the critical power, the ionized gases contribute to the Kerr nonlinearity-based self-focusing effect, which leads to a change in the phase-matching of harmonics. The nonlinear variation of the refractive index is caused by the absence of ionized electrons in the initial stage of HHG within the core. These electrons remain free for a longer duration compared to the duration of the laser pulse. The critical power of the driving laser pulse determines the ionization rate of gases, which in turn affects the self-focusing effect and phase-matching change in harmonics generation. The presence of ionized electrons and their prolonged free state contribute to the nonlinear variation of the refractive index during the HHG process.

In Fig. 4(c), the extrapolation curves of the critical ionization level (η) for noble gases in strong laser fields are shown for different wavelengths [16]. Typically, η exhibits a dependence on the wavelength of the driving laser pulses, which can be approximated as ($\eta \sim 1/\lambda _L^{ - 2}$) where λ_L is the wavelength of laser radiation. To obtain the extrapolation curves, the calculated data for the critical ionization levels of He, Ne, and Ar gases were extrapolated for the field’s of 1300 nm and 800 nm laser pulses [16]. Using this data, a new curve was plotted for 1030 nm driving laser pulses, taking into account the differences in wavelengths.

Additionally, the dependence of the ionization level for noble gases on their ionization potential is presented, which can be approximated as $\eta \sim 1/I_p^{ - 4}$, where I_p is the ionization potential of the atoms. This relationship describes how the ionization level varies with the ionization potential for a specific wavelength of the driving laser pulses. Overall, Fig. 4(c) illustrates the extrapolation curves for the critical ionization levels of noble gases in strong laser fields at different wavelengths. It also demonstrates the dependence of the ionization level on the ionization potential for a given wavelength of the driving laser pulses.

To estimate the values of the critical ionization levels for Kr and Xe gases at a certain wavelength of the driving laser pulses, one can use the extrapolated data for He, Ne, and Ar gases. Since the critical ionization.levels generally decrease with increasing wavelength of the driving laser pulses, one can apply this trend to estimate the values for Kr and Xe gases. Furthermore, when considering mixtures of gases like He + Kr and He + Xe, the ionization levels can be influenced by applying 2CP or 3CP. These pumping techniques can increase the ionization levels, particularly for gases with low ionization potentials such as Xe and Kr, where the ionization level depends on the wavelength of laser pulses ($\eta \sim 1/\lambda _L^{ - 2}$). Using the extrapolated data and considering the effects of 2CP or 3CP, we estimated the critical ionization levels for the pure gases (Kr and Xe) as well as for the mixtures of these three gases (He + Kr and He + Xe) at a wavelength of 1030 nm.

The choice of a strong fundamental driving pulse for the estimation of ionization level was related to the non-perturbative nature and typical spectrum of HHG. For tunnel ionization of the electron from the atom, the intensity of driving pulses should be I_ω > 1 × 10¹⁴ W cm^-2, when the electric field of the driving laser pulse becomes comparable to the electric field of the gas atom. The growth of the ionization rate is common for the parallel-polarized (approximately 90%) and orthogonally-polarized (approximately 30%) two electric fields of 2CP [14]. In the case of 3CP with orthogonal and perpendicular components of driving laser pulses, the ionization rate can be varied significantly.

During HHG using 3CP, the estimated values of the critical ionization levels of He, Kr, and Xe are approximately ${\eta _{He}}\sim 0.3\; \%$, ${\eta _{Kr}}\sim 4.08\; \%$, and ${\eta _{Xe}}\sim 6.9\; \%$. The total ionization rate for He + Kr and He + Xe can be changed thus creating different conditions for interaction of laser pulses with the medium. The addition of Kr and Xe gases with higher critical ionization levels will increase the overall ionization rate of the mixture. This change in the ionization rate can have an impact on the phase-matching conditions of the nonlinear medium.

The constructive accumulation of coherent radiation along the beam propagation direction during HHG relates to the phase-matching conditions [43]. This accumulation corresponds to the condition Δk = k_q−qk₀= 0, where Δk is the phase mismatch factor, k_q is the q^th harmonic wavevector, and k₀ is the wavevector of fundamental driving pulse. The phase mismatch for a focused Gaussian beam can be expressed as follows:

Here the first term is a geometric phase-mismatch and the second term is a dispersion phase-mismatch for atomic and ionized gases. At low gas pressure and low intensity of the driving laser pulses, the dispersion parameter of these gases is almost similar [44]. However, the growing pressure or intensity can increase the probability of ionization of these gases, which can change the phase-matching conditions. Taking into account the ionization level and ionized gas parameter, the second term of phase-mismatch between the driving fundamental laser pulses and the XUV harmonics, Δk, is a sum of contributions from the neutral atom and free-electron dispersions as well as from geometric dispersion, which is in our geometry corresponds to the weak focusing conditions [45,46]:

The dispersion phase-matching conditions depend on the ionization level of the ases present in the mixture. Therefore, the increased ionization rate resulting from the addition of Kr and Xe gases will affect the dispersion phase-matching conditions of the nonlinear medium. It is important to consider these changes in the ionization rate when analyzing the dispersion phase-matching conditions and the generation of XUV harmonics in the mixture of gases. The increased ionization rate can have significant implications for the overall behavior and properties of the generated harmonics. Probably, these conditions led to a significant enhancement of harmonics in the case of the He + Kr mixture and 3CP.

4. Conclusion

In summary, we have conducted a study on the generation of harmonics in mixed gases using three-color laser fields at a repetition rate of 50 kHz. Our observations revealed an enhancement in the intensity of harmonics when employing three-color laser pulses in the mixed-gas target. This enhancement was observed across a wide spectral range of high-energy photons. To further investigate this phenomenon, we utilized a combination of ω+2ω+3ω in the second nonlinear crystal. As a result, we obtained a 7× enhancement factor for the harmonics in the photon energy range of 20-35 eV when using the 3CP scheme compared to the case of a single-color (1030 nm) laser. We also noticed the suppressed harmonics in the gas mixture when subjected to the three-color laser fields. This suppression can be attributed to the destructive interference of harmonics caused by the increased ionization rate of the gas atoms.

Our findings suggest that the He-Kr gas mixture, under the influence of three-color laser fields, can serve as a strong coherent XUV source. This is due to the variation in the ionization rate of the mixed noble gases and the utilization of a combination of three-color laser fields, which improves the phase-matching conditions for the XUV harmonics. These results have significant implications for the development of efficient and coherent XUV sources. Such sources can find applications in various fields, including spectroscopy, imaging, and ultrafast science.