Effect of the Length and Pressure of a Gas Jet on Optical Harmonics Generation by 4.5-μm Femtosecond Laser Radiation of a Fe:ZnSe Laser System

The effect of the length and pressure of an argon gas jet on low-order harmonics (5th, 7th, 9th, and 11th) generation by 4.5-μm femtosecond laser radiation of a Fe:ZnSe laser system has been studied experimentally. It has been shown that an increase in the length of the generation medium up to the waist length allows one to increase the generation efficiency by a factor of 12. It has also been demonstrated that a change in the length of the gas medium changes the pressure dependence of the energy of the generated radiation because of change in the phase matching conditions, whose correct simulation requires the inclusion of nonlinear propagation effects for pump pulse and generated harmonics.


INTRODUCTION
High- [1] and low-order [2] harmonics generation by femtosecond laser radiation is one of the relevant topics of current research, lying at the junction of laser physics, nonlinear optics, and atomic physics. The main interest is due to the possibility of using coherent radiation of generated harmonics to obtain attosecond electromagnetic pulses [3] and to study the band structure of condensed media [4]. Since the generation process has a high order of nonlinearity, which increases with the harmonic number, and, thereby, the conversion efficiency is relatively low, the generation of harmonics radiation with a desired energy requires the optimization of experimental parameters. Gases [5,6], solids [7], and plasma [8] are currently used as targets for harmonics generation. Harmonics generation in gases [9] is one of the most flexible methods to control experimental parameters since it allows one to control both the microscopic response of a medium by the selection of a gas [10] and the macroscopic response by the selection of the pressure and volume of a gas medium [11].
There are several approaches to optimize the yield of harmonics in the gas medium. The first approach involves multicomponent gas media, where the first gas ensures the nonlinearity of the medium and the second gas is responsible for phase matching [2]. This can most conveniently be implemented near molecular, atomic, and other resonances so that the pump wave frequency is far from resonance (below the first excited level), whereas the optical harmonic frequency is slightly higher than the resonance transition frequency.
The second method of optimization is based on a change in the pressure of the gas medium in order to change the number of atoms involved in the interaction with laser radiation. In addition, because of the dependence of the dispersion properties of the medium on the concentration of atoms, the change in the pressure allows one to control the conditions of phase matching between the pump wave and the corresponding optical harmonic.
The third method of optimization consists of a change in the volume of the gas-medium region involved in generation by changing the length of the medium. The change in the length of the medium, as well as the pressure, makes it possible to control the number of atoms of the medium involved in the interaction and conditions of phase matching. Thus the length of the medium is similar to the pressure in the effect on harmonics generation.
The optimization of the macroscopic generation parameters is particularly important for mid-infrared laser radiation because the microscopic response decreases as [12]. Furthermore, a comparatively low repetition frequency of pulses of high-power midinfrared laser systems [13] also limits the total generated radiation flux. Consequently, the optimization of generation conditions by varying the pressure and λ 5 1/

OPTICS AND LASER PHYSICS
length of the medium is decisive for efficient harmonics generation with mid-infrared infrared radiation.
The discussion of the optimization of the energy yield of harmonics by varying the pressure of the gas jet continues in modern studies of harmonics generation [14,15]. Most studies of the effect of the gas jet pressure on the generation efficiency concern widely used technologically developed near-infrared laser systems, which do not require high pressures of 1-10 bar [16]. In particular, the authors of [17] reached the optimal energy yield of harmonics generated by 800-nm radiation at a pressure of 0.2 and 0.5 bar in argon and molecular hydrogen, respectively. In [18], the optimal pressure of molecular nitrogen and air was 0.33 bar at the same radiation wavelength. Fewer works were devoted to the optimization of harmonics generation by radiation with longer wavelengths of 1.24 μm [19], 3.9 μm [20], 4.5 μm [21]. The problem of optimization of the length of the generating medium is also currently under discussion, particularly together with the problem of the absorption of high-order harmonics [22]. Most works in this field also concern near and mid-infrared radiation sources.
In this work, the problem of optimization of the energy yield of harmonics by varying the pressure and length of the gas medium is studied in the context of harmonics generation by 4.5-μm femtosecond midinfrared radiation of a Fe:ZnSe source [23]. In particular, it is shown that the energy of generated radiation can be optimized by choosing the length of the generating medium equal to the length of the laser beam waist. An increase in the length of the medium directly increases the number of atoms of the medium involved in generation and changes the dependence of the energy of harmonics on the pressure of the gas jet, which is also examined in this work.

EXPERIMENTAL SETUP
The layout of the experimental setup is shown in Fig. 1. A Fe:ZnSe laser system (wavelength is 4.5 μm and the FWHM intensity pulse duration is 160 fs) [23] was used as a femtosecond radiation source. Laser radiation was focused by a lens with a focal length of f = 150 mm on a target placed in an evacuated generation chamber. The 1/e 2 intensity radius of the beam on the lens was approximately 6.5 mm. The measured diameter of the waist after focusing by this lens was (70 ± 10) μm, and the corresponding confocal parameter of the beam at the waist was b = (7.2 ± 1) mm. The target was a laminar argon jet (Fig. 2).
The laser beam waist was located at the center of the tube. The length of the gas jet was specified by the tube diameter L. To study the effect of the length of the interaction medium on harmonics generation, we used two tubes with inner diameters L = 1 and 7.1 mm. The impact on the medium in the tube with the diameter L = 1 mm was carried out only in a small region near the central part of the waist ( ), whereas generation in the tube with the diameter L = 7.1 mm took place in the entire waist region ( ). The generation chamber was continuously evacuated by a vacuum pump, which allowed us to maintain vacuum conditions beyond the tube volume when supplying argon. The energy of the laser pulse in the generation chamber was 1.6 mJ. The radiation of harmonics generated in the gas target was detected behind the interaction chamber by a spectrometer with a detection range of 200-1000 nm.

CALCULATION MODEL
To interpret the experimental results, we considered the macroscopic response of the medium taking into account the phase matching between interacting waves. In the case of the focusing of a Gaussian beam to the volume of the gas jet, the dependence of the energy of the qth harmonic on the jet length and pressure can be expressed as [5,24,25] (1) where ( Fig. 1) p is the pressure of the gas medium in the target, L is the length of the gas medium along the direction of laser beam propagation, and is the phase-matching integral that has the form (2) Here, b(p) = 2z R (p) is the confocal parameter of the generating beam, where z R (p) is the Rayleigh length, and is the mismatch between the wave vectors of the qth harmonic and polarization in the paraxial approximation. It is noteworthy that the contribution of the geometric phase (Gouy phase [12]) of the Gaussian beam to the mismatch between the wave vectors is included in the phase of the complex factor in parentheses in Eq. (2) rather than appearing in . In the case of low-order harmonics generation, the material dispersion of the medium and the dispersion of the generated plasma contribute to the mismatch between the wave vectors. Taking into account the linear dependence of the refractive indices of the medium at the pump and harmonic wavelengths, as well as the linear dependence of the concentration of medium atoms on the pressure [11], the mismatch between the wave vectors as a function of the pressure p can be expressed in the form (3) where is the mismatch between the wave vectors for the qth harmonic at the pressure p = p 0 = 1 bar.
Since argon has normal dispersion in the region of low-order harmonics (at least up to the 31st harmonic at a wavelength of 147 nm) generated in argon by 4.5-μm radiation [26,27], for these harmonics. In this case, contributions from the material and plasma dispersions of the medium to the mis-Δ 1bar > 0 q k match between the wave vectors have the same sign [11] coinciding with the sign of the contribution from the geometric phase [5]. For this reason, the pressure ensuring the phase matching condition = = 0 cannot be chosen. Nevertheless, it is possible to reach a local extremum of the pressure dependence of the energy of harmonics (Fig. 3). Each subsequent maximum is higher, which indicates that the pressure should be increased in order to increase generation efficiency. In practice, the maximum pressure can be limited by nonlinear propagation effects and dispersion broadening of the generating laser pulse. In particular, the nonlinear length L nl [28] in our experiment becomes equal to the length of the medium L = 7.1 mm at the pressure p = 7 bar. Therefore, nonlinear propagation effects occurring at high pressures can lead to the deviation of the calculated dependence from experimental one in this work. Nonlinear effects can be reduced by decreasing the length of the interaction region L.

RESULTS AND DISCUSSION
In this work, we studied the effect of the length and pressure of a laminar argon gas jet exposed to 4.5-μm femtosecond laser radiation on low-order harmonic (5th, 7th, 9th, and 11th) generation. To interpret the experimental results in the frame of the macroscopic response of the medium, we used the model described above. As already mentioned, the jet length was specified by the tube diameter L. Two tubes with the diameters L = 1 mm (data from [21]) and 7.1 mm were used. The spectra of harmonics generated at a pressure of 10 bar are presented in Fig. 4. According to Fig. 4 and Table 1, an increase in the length of the gas medium to the value L = 7.1 mm close to the waist length b = 7.2 mm raises the generation efficiency by more than an order of magnitude (by a factor of ≈12 for the fifth harmonic). The match of the length of the gas medium with the waist length led to an increase in the number of atoms of the medium involved in the interaction, which in particular allowed us to detect the eleventh harmonic (Fig. 4b). A further increase in the length of the medium will also increase the number of atoms involved in generation, which should raise the generation efficiency. In practice, as in the case of increasing pressure considered above, the maximum length of the medium will be limited by either the dispersion or the nonlinear length.
The number of atoms involved in generation increases with the length of the medium because the volume of the interaction region grows. Indeed, the volume of the interaction region is the volume of the beam caustic in the range (Fig. 2). In ∈ − [ /2; /2] z L L the Gaussian beam approximation, this volume can be represented in the form (4) where (5) is the radius of the Gaussian beam at the point z and is the radius of the Gaussian beam at the point . The substitution of Eq. (5) into Eq. (4) and the calculation of the resulting integral give (6) Substituting the experimental parameters r 0 = 70 μm (1/е 2 intensity level), z R = 3.6 mm, and L = 1 and 7.1 mm into Eq. (6), one can show that the ratio of the corresponding interaction volumes is V 7.1 mm /V 1 mm ≈ 10, which is close to the ratio of generation efficiencies in these two cases η 7.1 mm /η 1 mm ≈ 12 for the fifth harmonic. The slight difference between these two ratios can be caused by four reasons. The first reason is the presence of gas atoms outside the generation tube in the form of a jet ejected from the ends of the tube, which effectively increases the length of the interaction region L. Second, this difference can be due to the effect of the generated plasma on the beam propagation in the waist region. The third reason is the deviation of the beam propagation law in the experiment from the propagation law of an ideal Gaussian beam (i.e., in the experiment). Four, this difference can be attributed to nonlinear propagation effects. Thus, the generation efficiency increases with the length of the generation medium because the volume of the interaction region increases.
The harmonics generation efficiencies at a pressure of 10 bar and lengths L = 1 and 7.1 mm are given in the first and second rows of Table 1, where the maximum generation efficiency at a pressure 16 bar and the length L = 7.1 mm is presented in the third row.
It is remarkable that the maximum generation efficiency of 10 -5 for the fifth harmonic in our experiment at a pressure of 16 bar is only a factor of 2 lower than the generation efficiency in Xe at a pressure of 45 bar in [2], which can be explained by both a higher nonlinearity of Xe compared to Ar [29] and a much higher  gas pressure. The generation efficiency two orders of magnitude higher than that in this work can be reached in the mixture of Xe and CO 2 at a pressure of 50 bar, which can be explained by the effect of the СО 2 resonance near a wavelength of 4.3 μm and by a much higher gas pressure.
An increase in the length of the medium not only raises the generation efficiency but also changes the dependence of the energy of harmonics on the pressure of the gas jet in the pressure range under consideration. In particular, the dependence of the energy of the fifth harmonic on the pressure of the gas jet in the 1-and 7.1-mm-long media is sigmoidal and exponential, respectively (Fig. 5). This change in the dependence of the energy of the fifth harmonic is explained by the effect of phase matching conditions expressed mathematically in terms of the integral . Taking into account the sum of the material and plasma 5 ( , ) F p L mismatch between the wave vectors = (see Eq. (3)), as well as the geometric phase specified by the confocal parameter of the beam b = 7.2 mm, this dependence can be calculated for the central part of the laser beam r = 0 (see Eq. (1)). Figure 5 presents the results of this calculation, where the contribution of the material dispersion was calculated directly (Δk 5disp1bar = 12 m −1 ) taking into account the refractive index of argon for the generating wave calculated by the Sellmeier formula from [27] and the refractive index at the wavelength of the fifth harmonic taken from [27]. The plasma dispersion contribution was used as an approximation parameter and was determined from the best fit of dependences in Fig. 5 as Δk 5plasma1bar = 21 m −1 .
Deviation from the experimental results is assumingly due to nonlinear propagation effects changing x t y 2 = 0.997 R the spatiotemporal shape of the generating pulse [21], which are ignored in the model given by Eq. (1). In particular, the nonlinear length L nl becomes equal to the length of the medium L = 7.1 mm already at a pressure of the gas jet of 7 bar; with a further increase in the pressure up to 16 bar, the nonlinear length decreases to 3 mm.
Thus, an increase in the length of the generation medium L both raises the generation efficiency because the number of atoms of the medium involved in generation increases and changes the dependence of the energy of harmonics in the fixed pressure range because the phase matching conditions change. The correct simulation of the pressure dependence of the energy of harmonics requires the inclusion of nonlinear propagation effects for generating radiation.

CONCLUSIONS
To summarize, the effect of the length and pressure of an argon gas jet on low-order harmonics generation by 4.5-μm intense (up to 10 14 W/cm 2 ) femtosecond laser radiation of a Fe:ZnSe laser system has been studied. It has been shown that an increase in the length of the generation medium up to the waist length of laser radiation allows one to increase the generation efficiency by a factor of about 12 because of an increase in the number of atoms of the medium involved in the interaction. In addition, a change in the length of the generation medium changes the phase matching conditions, which, in particular, changes the dependence of the energy yield of harmonics on the pressure of the gas jet. To correctly describe this dependence, it is necessary to take into account the nonlinear propagation effects for the pump pulse and generated harmonics. Thus, the results demonstrate the possibility of optimizing the harmonics generation efficiency in the gas jet by choosing the optimal length of the interaction region and indicate that nonlinear optical effects should be taken into account to quantitatively interpret experimental data.

CONFLICT OF INTEREST
The authors declare that they have no conflicts of interest.
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