Open Access
Add to BibSonomy
Abstract
The resonance-enhanced harmonics in laser-induced arsenic and selenium plasmas are studied at the quasi-phase-matching (QPM) conditions. We demonstrate that the enhancement of these harmonics was significantly smaller than the one of the neighboring harmonics. Though the enhancement factors of the harmonics in the vicinity of resonance-enhanced harmonics were in the range of 5× to 18×, the resonance-enhanced harmonics were almost unenhanced at QPM conditions. The most probable reason for such restriction in the enhancement of specific harmonics at the conditions of QPM was a stronger influence of free electrons on the phase-matching conditions of the resonance-enhanced single harmonic compared to the QPM-enhanced group of harmonics.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
High-order harmonics generation (HHG) in laser-induced plasmas (LIP) has been notably developed during the last fifteen years. The interest in this process is caused by the specific features of the plasmas allowing the analysis of the spectral characteristics of the medium through HHG. Various methods of harmonics enhancement in LIPs allowed demonstration of the role of the ionic transitions possessing large oscillator strengths (gf) [1], presence of quantum dots and nanoparticles in plasmas [2], quasi-phase-matching (QPM) [3], and two-color pump (TCP) of LIP [4] in the growth of harmonic yield in the extreme ultraviolet (XUV) range. Numerous studies of these processes in various laboratories revealed interesting physical effects, which allowed for increasing the HHG conversion efficiency in LIP [5–33]. The theoretical consideration of those processes in plasma was also frequently reported [34–52]. The above-mentioned methods allowed for achieving the conversion efficiency for a single resonance-enhanced harmonic (REH) in the 62 nm region up to 10−4, particularly in the case of the 13th harmonic (H13) of 800 nm pump [1]. Meanwhile, the conversion efficiency in the shorter-wavelength region of XUV (λ < 30 nm) remains low, which prevents the application of these sources of radiation.
Though gases are not a favorable subject allowing demonstration of the resonance-enhanced harmonics compared with the plasma media, a few examples of such studies, mostly theoretical, are mentioned below. The resonantly-enhanced frequency conversion is well known from low-order frequency conversion processes driven by lasers of moderate intensities. There already have been some (but still quite a few) experimental investigations of resonance enhancements of harmonics in gases via bound atomic states [53]. The authors observed enhancement of particular harmonics for specific laser intensities, e.g. an increase in the yield of the nth harmonic by exciting a dynamically shifted n-photon resonance. Early studies of the theoretical approaches to the resonance enhancement of harmonics were further developed in Refs. [36,38,45].
QPM is an attractive approach to resolving the phase-mismatch problem for the HHG of laser radiation in the XUV range, which has been firstly demonstrated in the case of gas media. QPM was studied both theoretically and experimentally by using the multiple gas jets whose pressure and separation were properly controlled, as well as by using other methods of gaseous medium modulation [54–57]. However, the QPM in multi-jet gases has not achieved further amendments since the first reports of this phenomenon, probably due to the insignificant absolute values of enhancement and the difficulties in the implementation of the QPM technique.
Contrary to that, the realization of the QPM in the laser-produced plasmas during high-order harmonic generation may offer a few prospective applications: easy manipulation of the characteristics of multi-jet plasmas (e.g., jet sizes, distance between jets, electron concentration, etc.); definition of the electron concentration through the measured values of maximally enhanced harmonics and coherence lengths (corresponding to the sizes and distance between the jets at which the group of enhanced harmonics was observed); modification of the QPM using the perforated ablation beams, perforated targets, or interfering heating pulses; and easy tuning of the maximally enhanced groups of harmonics. Presently, none of these prospective applications have been demonstrated using the gas media.
The joint implementation of only two mechanisms of harmonics enhancement is not a new effect, since the examples of such applications were frequently reported in both gas and plasma HHG studies. The examples include a merger of (a) the resonance enhancement of single harmonic and the TCP-induced enhancement of all harmonics in LIPs, (b) QPM-induced enhancement of the group of harmonics and TCP-induced enhancement of all harmonics in LIPs, and (c) nanoparticles- and clusters-induced enhancement of the low-order harmonics and TCP-induced enhancement of harmonics in gases and LIPs. Meanwhile, the difficulty in realization of the QPM effect in the longer-wavelength region of XUV (50–100 nm) where the strong enhancement of REH is frequently reported prevented analyzing all methods of harmonics enhancement in the same spectral range. Correspondingly, it is hard to achieve such a merger of REH and QPM effects in the same spectral range.
Some elements of the periodic table (red squares in Fig. 1) demonstrate the resonance enhancement of single harmonic during propagation of the laser pulses through the LIP. The wavelengths of REH notably differ from each other. Some of them allowed generation of the low-order REH in the case of 800-nm-class lasers thus producing the enhancement of single harmonic in the region above λ = 35 nm. Those include Zn (REH = H11 of 1030 nm pump, λ = 94 nm) [58]), Pb (REH = H11 of 1064 nm pump, λ = 96 nm [59]), Cd (REH = H17 of 1390 nm pump, λ = 82 nm [60]), In (REH = H13 of 800 nm pump, λ = 62 nm) [1]), Ga (REH = H7 of 400 nm pump, λ = 56 nm) [32], Sn (REH = H17, λ = 47 nm) [61]), As (REH = H21 of 800 nm pump, λ = 38 nm [62]), Sb (REH = H21, λ = 38 nm of 800 nm pump [63]). Meanwhile, Mo (REH = H25 of 800 nm pump, λ = 32 nm [64]), V (REH = H27, λ = 29.5 nm [65]), Te (REH = H27, λ = 29.5 nm [66]), Cr (REH = 29, λ = 28 nm [67]), Mn (REH = H33, λ = 24 nm [68]) and Se (REH = H35, λ = 23 nm [69]) allowed generation of the enhanced shorter-wavelength harmonics. The plasma species allowing the generation of REH in the spectral range below 40 nm seem good candidates for the additional enhancement using the QPM. Though the growth of efficiency of those single harmonics compared to the neighboring harmonics is still low (5× to 8×), the additional enhancement using the QPM effect can serve as proof of further amendment of the HHG in LIPs.
Fig. 1. The marked elements of the periodic table, which allow a resonance-induced enhancement of single harmonic during HHG in the plasmas produced on the surfaces of those materials.
To demonstrate a merger of the two processes of harmonics enhancement based on different mechanisms (collective effect in the case of QPM and single-atom effect in the case of RH), the shortest-wavelength REH needs to be used. Among the earlier reported examples allowing a demonstration of the shorter-wavelength REH, selenium can be considered a suitable candidate for the merger of the REH and QPM effects in the same spectral range. Meanwhile, the antimony can also be used for a similar analysis once one applies the longer-wavelength lasers. Below we describe the application of those LIPs for the goal of this research – a merger of REH- and QPM-induced enhancement mechanisms in the shorter-wavelength range of XUV. To clarify a difference in the QPM enhancement in the vicinity of REH, we also analyze two plasmas, silver and indium LIPs, which either do not show the REH effect or demonstrate the enhancement of single harmonic in the longer-wavelength XUV region.
2. Experimental arrangements
The scheme consisted of the vacuum chamber and XUV spectrometer. The heating pulses (HP; pulse duration 250 ps, pulse energy 0.2 mJ, wavelength 800 nm, fluence on the target surface 5 J cm-2, pulse repetition rate 10 Hz) were focused by a cylindrical lens inside the vacuum chamber on the target surface and created the extended (5 mm × 0.4 mm) plasma. The 5 × 5 × 2 mm3 plates of Sb and Se were used as the targets for ablation. Additionally, the silver and indium targets were used as targets to demonstrate a difference in the harmonics distribution with regard to the arsenic, manganese, and selenium LIPs. The insertion of the multi-slit mask (MSM) on the path of the heating pulse led to the formation of the multi-jet plasma (Fig. 2(a)). The driving pulses (DP; pulse duration 65 fs, pulse energy 1.2 mJ, wavelength 800 nm, pulse repetition rate 10 Hz) were focused inside the extended plasma (intensity inside the plasma area 4 × 1014 W cm-2).
Fig. 2. (a) Scheme of HHG using multi-jet LIPs. DP, driving pulse; SL: spherical lens; NC: nonlinear crystal; CL: cylindrical lens; HP: heating pulse; MSM: multi-slit mask; MJP: multi-jet plasma; HE: harmonic emission; XUVS: extreme ultraviolet spectrometer. (b) Experimental setup describing the formation of different multi-jet plasmas in the focal plane of the cylindrical lens (L3) at different positions (1 - 3) of the multi-slit mask. L1 and L2: lenses of a telescope; P: pinhole of a telescope; T: target; HP: heating pulse; DP: driving pulse; HE: harmonic emission; MJP: multi-jet plasma; XUVS: extreme ultraviolet spectrometer, PI: plasma images. Left inset: Image of the multi-slit mask. Right inset: images of extended homogeneous plasma and multi-jet plasma structures obtained at different positions of MSM inside the telescope.
The experiments were carried out using the single-color pump (SCP, 800 nm) or two-color pump (TCP, 800 nm + 400 nm) of plasma. The second wave in that case was second harmonic of 800 nm radiation generated in a thin (0.3 mm) barium borate crystal inserted on the path of the focused driving pulses inside the vacuum chamber. The studies were also carried out using the radiation from the optical parametric amplifier allowing the generation of the tunable near IR pulses (1200–1600 nm). These pulses were used as the driving pulses. The optimal delay between HP and DP (80 ns) was determined by the distance between the axis of DP propagation and the target, as well as the velocity of plasma spreading from the target surface. The generated harmonics were analyzed using the XUV spectrometer [58].
The details of the application of MSM, its image and position, as well as the plasma images, were as follow. The HP propagated through the magnified spatial filter consisted of two lenses (L1 and L2, Fig. 2(b)) and a pinhole to improve the spatial profile and increase the sizes of the laser beam. The growth of laser beam sizes was necessary for the irradiation of the target using the homogeneous line-shaped beam. The HPs were focused using the 200 mm focal length cylindrical lens (L3) inside the vacuum chamber containing an ablating material to create the homogeneous extended plasma plume above the target surface. The intensity of HP on a plain target surface was varied in the range of 5 × 109 W cm-2 - 4 × 1010 W cm-2. The length of all used targets was 5 mm, thus defining the sizes of extended plasma (see the image of homogeneous imperforated plasma in the inset in Fig. 2(b)).
The variation of the shape of spatially modulated HP was carried out by the installation of the MSM on the path of propagation of this radiation. The image of MSM is shown in the left inset in Fig. 2(b). The width of each slit of this mask was 0.3 mm, and the distance between the slits was 0.3 mm. The images of imperforated and perforated plasma formations were captured from the top of the vacuum chamber by a CCD camera. The right inset in Fig. 2(b) shows the line and multi-jet plasmas formed on the target surfaces by the extended focused heating beam and by the installation of the MSM at different positions inside the telescope. One can see the extended homogeneous shape of plasma, as well as the nine-, five-, four-, and three-jet shapes of plasma structures. The width of each jet in different plasma structures was determined using its image, as well as by calculating the geometrical characteristics of the radiation propagated through the MSM placed at different positions inside the telescope. Additionally, the gradual tilting of the MSM placed on position 3 (Fig. 2(b)) allowed the steady growth of the number of jets up to 14.
Once the MSM was introduced on the path of the heating beam using the scheme depicted in Fig. 2(b), the homogeneous expended plasma was transformed, which led to the formation of the multi-jet plasma. The femtosecond DP then propagated through this multi-jet plasma. The confocal parameter of the femtosecond driving beam (10 mm) was larger than the whole length of the plasma (5 mm).
3. Basics of QPM in LIP
Below, the basic description of the quasi-phase-matching is presented. The enhancement of a group of harmonics can be realized at the conditions of the coupling of the interacting driving wave and the harmonic waves for which the phase matching was established. The meaning “quasi” refers to the conditions when a similarity of the group velocities is attributed only to the specific group of harmonics rather than to the whole set of generated emissions.
For HHG, dispersion in the gaseous and plasma media is mostly attributed to the presence of the free electrons appearing in the medium during its ionization. Under this assumption, the coherence length of qth harmonic governs by the relation Lc α (qNe)-1 where Ne is the free electron density [70]. This distance corresponds to the condition when the relative phase of the harmonic with regard to the driving wave changes by π after which the harmonic starts to convert back to the driving wave. So, it is a requirement to not exceed this length of the medium, otherwise, the destructive interference stops the process of HHG. Correspondingly, when the size of medium l becomes equal to Lc, the qth harmonic becomes phase-matched, while other harmonics will be generated at less favorable phase relations with regard to the driving wave. Meanwhile, a few other orders of nearby harmonics will be also at better conditions of generation compared with the significantly lower orders of harmonics. The small dispersion at the low densities of plasmas and gases, which correspond to the concentrations of the order of 1017 particles per cubic centimeter, allows maintaining the conditions close to the phase-matched ones (i.e. quasi-phase-matched conditions) for a group comprising a few higher-order harmonics and a few lower-order harmonics around the maximally enhanced harmonic (qQPM). The relation qQPM α (lNe)-1 [11,18,71,72] indicates the opportunity for variation of the maximally enhanced harmonic by changing the length of medium or electron density. At these conditions, the multiple gas and plasma jets of the size of l can lead to the quadratic growth of HHG conversion efficiency.
This simple relation allows calculating the maximally enhanced harmonic by calculating the electron concentration using different codes and knowing the thickness of a single jet in the multi-jet LIP. The former parameter approximately corresponds to the sizes of a single slit in the MSM transforming the extended plasma to the multi-jet plasma. The widths of slits are commonly varied between 0.2 and 0.8 mm. For the plasma jets of different sizes, the maximally enhanced harmonics will be tuned along the XUV spectrum. Similarly, a decrease in the fluence of HP causes a decrease in electron density, which tunes qQPM towards the shorter-wavelength region of XUV and vice versa.
The above equation allows expecting the tuning of the enhancement of a group of harmonics along the whole XUV spectrum. This equation assumes that, by using the longer jets, one can tune the enhanced group of harmonics towards the longer-wavelength region of the enhanced single harmonic in the case of indium plasma (H21 for 1310 nm pumps and H13 for 800 nm pumps). The tuning of qQPM by different means in the case of silver LIP was reported in [73]. Unfortunately, the maximally enhanced harmonic could not be tuned toward λ = 62.4 nm at which the resonantly enhanced 21st harmonic of 1310 nm pump generating in the indium plasma was achieved. The reasons restricting such a tuning will be discussed later.
4. Results
4.1 HHG in extended and spatially modulated silver and indium plasmas
To demonstrate the efficient implementation of the QPM-induced enhancement of harmonics, the studies of the plasmas, that either do not possess the REH effect (silver LIP) or demonstrate this effect in the longer-wavelength spectral range of XUV (indium LIP), were carried out.
Silver plasma has never demonstrated the REH effect while showing attractiveness as a medium for the generation of the groups of QPM-enhanced harmonics in different ranges of XUV. The wavelength of qQPM can be easily tuned by changing the width of slits of the MSM or tilting the mask [72] or changing the fluence of the heating pulses allowing the variation of free electrons concentration in Ag LIP [3]. The important feature of the silver imperforated plasma is the generation of a prolonged plateau-like distribution of harmonics up the H65 of the 800 nm pump. Correspondingly, the formation of the QPM conditions along the higher-order harmonics region can be easily realized in the case of silver LIP.
Figure 3(a) shows two line-outs of the harmonics distribution obtained in Ag LIP. In the case of homogeneous 5-mm-long plasma, the extended harmonics generation with approximately similar intensities down to H46 of 1310 nm emission (blue thin curve) was obtained during TCP (1310 nm + 655 nm) of Ag LIP. As mentioned, no enhancement of specific harmonic order due to resonance effect was observed in that case. After the introduction of the MSM on the path of the heating pulses, which allowed the formation of the five-jet LIP, a significant change in the harmonics distribution was obtained (yellow thick curve). The plasma in both cases was produced by 250 ps pulses at a fluence of 5 J cm-2. In the latter case, the QPM conditions for the effective coupling of the waves of the femtosecond pump and generating harmonics in the 30–50 nm spectral range were created. The maximally enhanced harmonic (qQPM = H36) was ∼5 times stronger than the same harmonic in the case of homogeneous extended plasma. The difference in the harmonic yields in these two cases can be easily distinguished once one compares the raw images of harmonics (see insets in Fig. 2(a)). The harmonic cutoff in the case of the multi-jet plasma was extended up to the 58th order (not shown in this figure).
Fig. 3. (a) Harmonics generation using TCP in silver LIP. The blue thin curve shows the HHG spectrum in the case of 5-mm-long plasma. The yellow thick curve shows the harmonics generation in the multi-jet Ag plasma. One can see the group of enhanced harmonics centered at H36. (b) Harmonics generation using TCP in indium LIP. The blue thin curve shows the HHG spectrum in the case of 5-mm-long plasma. One can see a strong enhancement of H21 in the vicinity of 62 nm. The yellow thick curve shows the HHG spectrum at the conditions of QPM using 5-jet In LIP. The enhancement of harmonics around H33 of the 1310 nm pump did not cause any effect on the REH. (c) Enhancement factors of QPM-enhanced harmonics generated in the multi-jet In LIP.
In the case of extended Ag plasma, a slow decay in harmonics intensities towards the cutoff region is attributed to the influence of the free electrons present in the plasma produced during the ablation of the target by intense laser pulses. The free electrons are rather stronger influencing the higher orders of harmonics by creating the less favorable conditions for the phase matching between the driving and harmonic waves. At equal conditions, the coherent length of harmonic Lc ∝ (q × Ne)−1 decreases with the growth of harmonic order. At a distance equal to Lc, the phase mismatch increases and causes destructive interference between the driving wave and higher orders of harmonics. Because of this, harmonic generation of laser radiation in extended LIP and gas media has long been considered an inefficient process due to the excess in the length of medium over Lc of harmonics. However, once the extended plasma becomes spatially modulated, particularly by dividing the whole length of plasma by a set of plasma jets, the above problem becomes diminished due to the creation of the phase-matched conditions for some of the harmonic and driving radiation.
The next medium used in the first set of HHG studies was the indium LIP. It has been frequently analyzed since the first report of the REH effect in LIP [1]. Both 800-nm-class lasers and near IR optical parametric amplifiers were used to demonstrate strong enhancement of the harmonic generating in the vicinity of the ionic transition (62.2 nm) possessing large gf. In the present case, the application of similar TCP (1310 nm + 655 nm) as in the case of Ag LIP allowed the generation of the strong REH (H21) and some nearby longer-wavelength enhanced harmonics (Fig. 3(b), blue thin curve). The unusual distribution of the harmonics in the vicinity of the strong ionic transition has been explained using the four-step model of REH generation [43]. Meanwhile, the harmonics exceeding the 22nd order are barely seen in this spectrum. These weak harmonics were extended up to the 42nd order.
The creation of the multi-jet indium plasma by dividing the extended plasma into a set of 5 separated jets using the MSM, similar to the one applied during HHG in the multi-jet Ag LIP, significantly increased the intensity of the harmonics in the shorter-wavelength range, especially those close to the 40 nm region (Fig. 3(b), yellow thick curve).
Figure 3(c) shows the enhancement factors of harmonics at the conditions of QPM in indium plasma. This dependence was obtained by dividing the curve shown in yellow color in Fig. 3(b) by the curve shown in blue color in Fig. 3(b). The maximally enhanced harmonic (qQPM = H33) was 26 times stronger than the same harmonic generated in the homogeneous extended plasma. A large group of neighboring harmonics in the 25–55 nm range was also notably enhanced, though less than qQPM. The harmonic cutoff was observed at H51. Similar to silver plasma HHG, the difference in the harmonic yields in the case of imperforated and perforated indium plasma is underlined in the raw images of harmonic spectra (see insets in Fig. 2(b)).
The peculiarity of this QPM experiment in the indium LIP is the absence of the enhancement of the REH (H21), as well as of other longer-wavelength harmonics (Fig. 3(c)). Obviously, the QPM conditions were not fulfilled for those harmonics. Meanwhile, one of the tasks of HHG studies is a search for further enhancement of some specific harmonics, especially those, which were enhanced by resonance-induced effect. In that case, the joint implementation of two mechanisms of harmonics enhancement (REH and QPM effects) could notably increase the yield of this harmonic, which is important for practical needs.
The failure in the merger of two enhancement processes in the same spectral region did not allow for the increase of the yield of resonance harmonic (λ = 62 nm) in the In LIP. QPM was not able to enhance the harmonics in this longer-wavelength region of XUV. The conditions for QPM were chosen in such a way that the harmonics around the 33rd order (39.7 nm) were phase-matched, but not those around the 21st order (62 nm), where the resonance enhancement occurs. Below, we discuss the anticipated approach for combining two methods. We also analyze the problem arising during attempts to shift qQPM toward the longer-wavelength region where the REH effect in the In LIP and some other plasmas is available.
The equation qQPM ∝ (l × Ne)-1 allows expecting the tuning of the enhancement of a group of harmonics along the whole XUV spectrum. This equation assumes that, by using longer jets, one can tune the enhanced group of harmonics towards the region of enhancement of a single harmonic in the case of indium plasma (H21 for 1310 nm pump). Such a tuning, but only up to the 45 nm spectral region was reported in a few publications (for example [73]). The maximally enhanced harmonics (qQPM) followed the rule when the product l × Ne was maintained constant at a fixed electron density of the plasma.
Meanwhile, during the increase in the size of the single jet, the QPM-related restricting factor starts to play an important role. At a fixed length of extended and homogeneous plasma (5 mm in our case), the increase in the width of a single jet will lead to a decrease in the number of plasma jets appearing on the target surface. For example, the above experiments were carried out using the five 0.5-mm-thick jets of In LIP. These five jets fully cover the whole length of the extended 5-mm plasma (since one has to take into account a 0.5 mm distance between the jets). To shift qQPM from H33 to H21 one has to increase the width of the slit from 0.5 mm to ∼0.8 mm. It means that a smaller number of jets can be created using the division of the 5-mm-long plasma and assuming the conditions when the width of slits becomes equal to the distance between them. The described conditions hardly can be considered as those that pursue the advantages of the quasi-phase-matching effect since to maintain the coupling of the driving and harmonics waves, the QPM has to start playing an important role when the number of jets allows an increase in the number of flips of the relative phases of interacting waves.
Another technique to tune the range of enhanced harmonics by changing the conditions of target ablation leading to the variation of Ne was reported in [3]. The comparative studies of the harmonic spectra generated in the case of the 800 nm pump from the five-jet structure at different fluencies of heating pulses on the target surface were accomplished. Meanwhile like in the previous case, this method did not lead to the tuning of the QPM conditions above the 35 nm spectral region. Here one also would like to mention another impeding factor for HHG caused by the increase in the concentration of free electrons. Though the increase of heating pulse energy leads to a decrease in qQPM [72], this option has its limits. At larger fluencies of heating pulses, a strong and broadband plasma emission appears at these conditions of laser ablation simultaneously with a larger number of free electrons. The intensity of this emission rapidly exceeds the intensity of the harmonics. Plasma emission overlaps the whole harmonic spectrum and hardly allows further separation and application of this mixture of coherent and incoherent XUV radiation. Correspondingly, no information can be retrieved about the characteristics of the harmonic spectrum at these conditions.
4.2 HHG in extended and spatially modulated arsenic, selenium, and manganese plasmas
The REH process is fixed for a few elements of the periodic table (Fig. 1) and cannot be tuned. Correspondingly, the only way to combine the spectral ranges of enhancement mechanisms is to tune the QPM process towards the longer-wavelength region of XUV where the REH effect can be realized, which is impossible for most plasma species (particularly, indium plasma). The merger of the spectral ranges where the REH and QPM mechanisms coincide can be realized in the case of the enhanced single harmonics generated in the shorter wavelength region (i.e. below 45 nm).
This merger of two processes in the 45 nm region was reported in the case of tin plasma using the near-IR driving pulses [74]. Unfortunately, the enhancement factor (2×) of REH in Sn multi-jet LIP was not as high as one can expect from the joint implementation of the two mechanisms of harmonics enhancement. This inefficient enhancement was attributed to the difference in the meaning of “optimal plasma” formation for two cases (REH-induced enhancement and QPM-induced enhancement). The former process requires lesser excitation of the target compared with the latter one to achieve a suitable realization of the enhancement mechanism. Because of this, below we analyze the Se LIP, which allows for the REH effect in the shorter wavelength region (<30 nm). We also demonstrate the enhancement of REH in the Sb plasma when QPM conditions satisfy the requirement of the coincidence of the qQPM and REH.
Figure 4 shows the harmonics distribution in the case of generation in the extended (l = 5 mm) arsenic LIP in the 32–52 nm spectral range using 1326 nm + 662 nm TCP. The choice of the wavelength of near IR radiation is attributed to the coincidence of one of the harmonics in the XUV range with the ionic transition, which in earlier studies of this plasma allowed the demonstration of the REH effect. This harmonic (H35, λ = 37.88 nm) was close to some transitions of SbII. Previously, a 10-fold enhancement of a single high-order harmonic near the 4d105s22p3P2 – 4d95s25p3(2D)3D3 transition of SbII (λ = 37.82 nm) in the case of a low-ionized antimony plume was reported [64]. The used excitation pulses (795 nm) made it possible to increase H21 (λ = 37.7 nm) yield from a narrow Sb plasma. The gf of this transition was calculated as 1.36, which is 6 times higher than those of neighboring transitions [75].
Fig. 4. (a) Harmonics generation in the extended (l = 5 mm) arsenic LIP in the 32–52 nm spectral range using 1326 nm + 662 nm TCP. The enhanced H35 was stronger than the lower-order harmonics down to the H28. (b) An enhancement factor of the QPM-enhanced harmonics in the case of 6-jet Sb LIP. The enhancement was achieved along the whole region of harmonics generated in the 32–52 nm range. The REH showed the smallest enhancement among a set of harmonics from both sides of REH.
The refereed studies [64] did not allow the optimization of the driving pulse wavelength from the point of view of the maximal yield of this harmonic of driving radiation due to the fixed wavelength of Ti: sapphire laser. In the present case, the wavelength of the optical parametric amplifier was tuned in the 1310–1335 nm range to find the strongest yield of the harmonic most closely approaching the abovementioned ionic transition of SbII (H35). It was found that the wavelengths of this harmonic and transition almost coincided with each other (37.88 nm and 37.82 nm, respectively) taking into account the spectral width of H35. The enhanced H35 was stronger than the lower-order harmonics down to the H28 (Fig. 4(a)). The harmonic cutoff was H41.
The application of the QPM concept for this plasma allowed achieving the coincidence of the REH and the central part of the group of QPM-enhanced harmonics. These conditions were achieved in the case of the 6-jet Sb LIP. The number of jets was established by adjusting the position of the MSM in the telescope (Fig. 2(b)). The fluence of HP was 5 J cm-2. Thus the combination of stronger ablation conditions leading to the growth of Ne and the width of a single plasma jet (0.4 mm) led to the formation of the QPM conditions centered in the spectral region of REH of SbII. A division of the lineout of harmonics distribution at these conditions by the lineout shown in Fig. 4(a) allowed the determination of the enhancement factors of all harmonics in the studied XUV range (32–52 nm). The enhancement was achieved along the whole region of harmonics generated in the 33–49 nm range (Fig. 4(b)). The maximal enhancements were observed in the case of H36 (5×) and H34 (6×). Meanwhile, the REH showed the smallest enhancement (∼ 1.5×) among all harmonics in the mentioned spectral range. The resonance-enhanced harmonic (H35), which was 5 and 12 times stronger than the neighboring harmonics (H36 and H34, respectively) in the case of extended plasma (Fig. 4(a)), became almost equal to those QPM-enhanced harmonics, thus showing a notable restriction in the enhancement at the QPM conditions.
To further analyze this problem, the plasma allowing shortest-wavelength REH (Se LIP, λ = 22.9 nm) was probed. The shorter-wavelength QPM conditions can be created using a larger number of narrower plasma jets. Correspondingly, one can overcome the problem of insufficient coupling of the REH and DP waves during multiple interactions of waves at suitable phase relations.
The harmonic spectrum generated in the 17–29 nm spectral range in the case of imperforated 5-mm-long Se LIP is shown in Fig. 5 (blue thick curve). The strong resonance-induced enhancement of the 35th harmonic of 800 nm radiation has earlier been reported in a few studies using the narrow selenium plasma. The analysis of different Se-containing plasmas has shown the peculiarities of REH in the case of the molecular Se-contained plasmas [69]. Particularly, H35 was strongly suppressed in the case of ZnSe and HgSe plasmas, while in the case of 0.4-mm-long Se plasma, its enhancement factor with regard to the neighboring harmonics exceeded 30×. In the present studies (5-mm-long Se plasma), this harmonic was only 5 to 6 times stronger than the neighboring harmonics, which showed some suppression of the REH effect in the extended plasma due to the stronger influence of the phase-mismatch on the yield of REH compared to the neighboring harmonics.
Fig. 5. Harmonic spectra in the range of 14–36 nm in the case of HHG in the 5-mm-long Se LIP (blue thick curve) and at the conditions of QPM using the 8-jet plasma (red thin curve).
The difference in the role of phase-matching conditions in the case of the two methods of harmonics enhancement demonstrated in Sb plasma is also attributed to the growing role of the impeding factors preventing the proper coupling of the harmonic and DP waves along the whole length of laser-plasma interaction. The enhanced 35th harmonic was analyzed by different methods of plasma formation. Though it is still not determined which ionic transition is responsible for the REH effect in this spectral range due to the lack of information about the energy structure of ionic transitions in Se, the demonstration of such enhancement at λ = 23 nm allows a merger of REH and QPM effects. The maximal order of generated harmonics was H49.
The application of QPM during these experiments was performed using the 8-jet Se plasma. Again, like in the previous cases, the number of jets was adjusted by moving the position of MSM in the telescope. The width of the single jet in that case was ∼0.3 mm. The maximum enhancement was further fine-tuned toward the 23 nm wavelength, where the REH was observed, by adjusting the fluence of the 250 ps HP. The harmonic spectrum, in that case, was extended up to the 57th harmonic (Fig. 5, red thin curve). The H27 to H47 were enhanced, with the highest enhancement factors (∼6×) observed for the neighboring to REH harmonics (compare relative intensities of H34 and H36 of the 800 nm pump in the case of REH and QPM effects). As for the REH, similar to the previous case (Sb LIP), the enhancement was notably smaller (Fig. 5) and did not exceed 1.25×. Probably, the similar impeding factors as in the case of Sb plasma prevented the stronger enhancement of this harmonic.
The suppression in the gain of the H35 of 1326 nm + 663 nm pump in Sb multi-jet plasma and of the H35 of 800 nm pump in Se multi-jet plasma is attributed to the stronger influence of free electrons on the phase-matching conditions of the resonantly-enhanced single harmonic compared to the QPM-enhanced group of harmonics. This peculiarity was discussed in [65], where a weak enhancement of the resonance-enhanced harmonic in the manganese plasma was observed while trying to simultaneously apply two processes of enhancement. We also observed a small enhancement factor of the REH at the QPM conditions compared to the case of the homogeneous Mn LIP. The REH in the case of this plasma is H33 of the 800-nm-class lasers [25,68]. The enhancement of this harmonic is attributed to its closeness to the region of the giant 3p-3d resonances of Mn II in the range of 23.8–24.3 nm, which possess large gf [76].
5. Discussion
The resonance enhancement and quasi-phase-matching in the vanadium, antimony, indium, and manganese plasmas were compared in Ref. [65]. In those studies, the competition of two processes of harmonics enhancement was studied and the predominance of the QPM gain over the resonance gain has been shown. The enhancements of harmonics were analyzed in different spectral ranges. In the present paper, the resonance-enhanced harmonics in laser-induced arsenic and selenium plasmas are studied at the quasi-phase-matching conditions shifted toward the wavelengths of those harmonics. Taking into account present results and a previous report showing a very small enhancement of resonance harmonic at the QPM conditions [65], we demonstrated the principal difficulties during the merger of the two methods of harmonics enhancement based on the resonance-induced variation of the yield of single harmonic and the QPM-induced variation of the yield of a group of harmonics in the same spectral range. The resonance enhancement and QPM are governed by the individual properties of ions and the collective response of the medium, respectively. This distinction in the mechanisms of enhancement, where the free electrons differently affect the studied processes, does not allow additively enhancing the single resonance-enhanced harmonic like the neighboring ones during the formation of the QPM conditions. This is the main novelty of the present study, which was not underlined in previous publications. The significance of the reported results is related to the optimization of harmonics generation in different spectral ranges where the enhancement processes in laser-induced plasmas play a pivotal role.
The effect of free electrons on the restriction of HHG in gases has been reported in numerous publications (for example, [77–79]). The theoretical models and numerical simulations reported in those studies have shown the basic mechanism suppressing the phase-matching conditions, especially for the higher-order harmonics in the shorter-wavelength region. This mechanism is based on the growing difference in the group velocities dispersion for the driving and harmonic waves in the case of the abundance of free electrons. This difference leads to the phase mismatch between interacting waves and a decrease in the conversion efficiency of the harmonics.
The small concentration of free electrons, as follows from the relation shown in section 3, allows for the maintenance of the phase-matching conditions for as high as possible harmonic orders. The HHG is intimately related to ionization in strong laser fields, which initiates the single atom response, and at the same time limits the macroscopic yield when a too-high density of free electrons in the medium prohibits phase matching and modifies the laser propagation. This short explanation points out the difficulties in developing a possible way to efficiently combine the QPM and resonance processes. The present study just emphasizes this peculiarity of the joint merger of the two above processes. The conversion efficiency of the maximally enhanced harmonics at the conditions of QPM in the case of Sb and Se plasmas was estimated to be 10−5.
The configuration of a few jets allows for creating the conditions when no destructive interference occurs for each harmonic in the vicinity of the maximally enhanced harmonic, for which the coherence length is equal to the length of a single jet. In that case, one can use the term “constructive interference”. This term refers to the case when the dimension of a single jet corresponds to the condition when the relative phase of the harmonic with regard to the driving wave changes by π after which the harmonic starts to convert back to the driving wave. So, it is a requirement to not exceed this length of a single jet, otherwise, the destructive interference stops the process of HHG in the specific spectral region.
The role of the number of jets in increasing the yield of a group of harmonics satisfying the quasi-phase-matching conditions was analyzed in the case of the Sb multi-jet plasma. As was mentioned, the term “quasi” refers to the conditions when a similarity of the group velocities is attributed only to the specific group of harmonics rather than to the whole set of generated coherent short-wavelength emission. Once these conditions are fulfilled, the increase in the number of jets quadratically increases the harmonic yield. The quadratic dependence of the harmonic yield on the number of jets was confirmed in present studies.
The detailed analysis of this dependence has been reported in the first demonstration of QPM in plasma [3]. The anticipated featureless shape of harmonic spectra from a single 0.4-mm-long Sb plasma jet was similar to the one observed in the case of the 5-mm-long plasma. With the addition of each next jet, the spectral envelope was drastically changed, with the 34th harmonic intensity in the case of the six-jet configuration of Sb plasma becoming almost a few ten times stronger compared with the case of a single-jet plasma. One can expect the n2 growth of harmonic yield for the n-jet configuration compared with the single jet once the phase mismatch becomes suppressed, as was shown in the case of HHG in the multi-jet gas targets [54,55]. This relation gives the expected enhancement factor of 36 in the case of a six-jet medium, which was close to the experimentally measured enhancement factor in the case of the arsenic plasma. Notice that the maximum enhancement factor may decrease from the ideal value of n2 at the conditions when the absorption processes are turned on, or in the case of unequal properties of the jets (i.e. free electrons concentration), which can arise from the heterogeneous excitation of the extended target by the Gaussian heating pulses.
6. Conclusions
These studies aimed to determine whether it is possible to additively enhance the specific harmonic using two effects (REH and QPM). We have shown that in the case of the LIP, which does not demonstrate REH properties (Ag plasma), the significant enhancement factor in the shorter-wavelength region of XUV was obtained for all harmonics at the conditions of QPM. Correspondingly, neither Ag nor many other elements of the periodic table are suitable for the comparative studies of the addictiveness of the abovementioned processes. A similar is true for the LIP, which showed the REH effect in the longer-wavelength XUV region (62 nm in the case of the In plasma) due to the difficulties in the creation of the QPM conditions for the lower-order harmonics.
The application of the plasmas allowing demonstration of the REH effect for the shorter wavelengths (38 nm and 23 nm in the case of the Sb and Se LIPs, respectively) led to a decrease of the enhancement factor of REH compared to the neighboring harmonics at the conditions of QPM. The difference in the QPM-induced enhancement factors of REH and neighboring harmonics is especially underlined once the quasi-phase-matching became adjusted toward the resonance harmonics in arsenic and selenium plasmas. At these conditions, the REHs do not follow the QPM-induced enhancement, contrary to other harmonics in those spectral ranges. Meanwhile, the latter harmonics demonstrated the enhancement factors of the order of 5×. The reason for the failure in the enhancement of REHs at QPM conditions for other neighboring harmonic orders is related to the difference in the influence of the free electrons in LIPs on the two methods of harmonic enhancement. The distinction in the mechanisms of enhancement, when the free electrons in plasmas differently affect the studied processes, does not allow additively enhancing the single REH like the neighboring ones during the formation of the QPM conditions. The yield of REHs remained almost intact thus demonstrating the absence of the QPM effect on the harmonics enhanced due to their closeness to the ionic transitions possessing large oscillator strength.
Funding
European Regional Development Fund (1.1.1.5/19/A/003); World Bank Project (REP-04032022-206).
Disclosures
The author declares no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. R. A. Ganeev, M. Suzuki, M. Baba, et al., “Strong resonance enhancement of a single harmonic generated in extreme ultraviolet range,” Opt. Lett. 31(11), 1699–1701 (2006). [CrossRef]
2. R. A. Ganeev, G. S. Boltaev, V. V. Kim, et al., “Effective high-order harmonic generation from metal sulfide quantum dots,” Opt. Express 26(26), 35013–35025 (2018). [CrossRef]
3. R. A. Ganeev, M. Suzuki, and H. Kuroda, “Quasi-phase-matching of high-order harmonics in multiple plasma jets,” Phys. Rev. A 89(3), 033821 (2014). [CrossRef]
4. R. A. Ganeev, C. Hutchison, A. Zaïr, et al., “Enhancement of high harmonics from plasmas using two-color pump and chirp variation of 1 kHz Ti:sapphire laser pulses,” Opt. Express 20(1), 90–100 (2012). [CrossRef]
5. L. B. Elouga Bom, F. Bouzid, F. Vidal, et al., “Correlation of plasma ion densities and phase matching with the intensities of strong single high-order harmonics,” J. Phys. B 41(21), 215401 (2008). [CrossRef]
6. H. Singhal, V. Arora, B. S. Rao, et al., “Dependence of high-order harmonic intensity on the length of preformed plasma plumes,” Phys. Rev. A 79(2), 023807 (2009). [CrossRef]
7. H. Singhal, R. A. Ganeev, P. A. Naik, et al., “Study of high-order harmonic generation from nanoparticles,” J. Phys. B 43(2), 025603 (2010). [CrossRef]
8. Y. Pertot, L. B. Elouga Bom, V. R. Bhardwaj, et al., “Pencil lead plasma for generating multimicrojoule high-order harmonics with a broad spectrum,” Appl. Phys. Lett. 98(10), 101104 (2011). [CrossRef]
9. L. B. Elouga Bom, Y. Pertot, V. R. Bhardwaj, et al., “Multi-µJ coherent extreme ultraviolet source generated from carbon using the plasma harmonic method,” Opt. Express 19(4), 3077–3085 (2011). [CrossRef]
10. L. B. Elouga Bom, S. Haessler, O. Gobert, et al., “Attosecond emission from chromium plasma,” Opt. Express 19(4), 3677–3683 (2011). [CrossRef]
11. A. H. Sheinfux, Z. Henis, M. Levin, et al., “Plasma structures for quasiphase matched high harmonic generation,” Appl. Phys. Lett. 98(14), 141110 (2011). [CrossRef]
12. C. Hutchison, R. A. Ganeev, T. Witting, et al., “Stable generation of high-order harmonics of femtosecond laser radiation from laser produced plasma plumes at 1 kHz pulse repetition rate,” Opt. Lett. 37(11), 2064–2066 (2012). [CrossRef]
13. Y. Pertot, S. Chen, S. D. Khan, et al., “Generation of continuum high-order harmonics from carbon plasma using double optical gating,” J. Phys. B 45(7), 074017 (2012). [CrossRef]
14. S. Haessler, L. B. Elouga Bom, O. Gobert, et al., “Femtosecond envelope of the high-harmonic emission from ablation plasmas,” J. Phys. B 45(7), 074012 (2012). [CrossRef]
15. M. Kumar, H. Singhal, J. A. Chakera, et al., “Study of the spatial coherence of high order harmonic radiation generated from preformed plasma plumes,” J. Appl. Phys. 114(3), 033112 (2013). [CrossRef]
16. S. Haessler, V. Strelkov, L. B. Elouga Bom, et al., “Phase distortions of attosecond pulses produced by resonance-enhanced high harmonic generation,” New J. Phys. 15(1), 013051 (2013). [CrossRef]
17. H. Singhal, P. A. Naik, M. Kumar, et al., “Enhanced coherent extreme ultraviolet emission through high order harmonic generation from plasma plumes containing nanoparticles,” J. Appl. Phys. 115(3), 033104 (2014). [CrossRef]
18. N. Rosenthal and G. Marcus, “Discriminating between the role of phase matching and that of the single-atom response in resonance plasma-plume high-order harmonic generation,” Phys. Rev. Lett. 115(13), 133901 (2015). [CrossRef]
19. M. A. Fareed, N. Thire, S. Mondal, et al., “Efficient generation of sub-100 eV high-order harmonics from carbon molecules using infrared laser pulses,” Appl. Phys. Lett. 108(12), 124104 (2016). [CrossRef]
20. M. A. Fareed, S. Mondal, Y. Pertot, et al., “Carbon molecules for intense high-order harmonics from laser-ablated graphite plume,” J. Phys. B 49(3), 035604 (2016). [CrossRef]
21. M. Oujja, I. Lopez-Quintas, A. Benítez-Canete, et al., “Harmonic generation by atomic and nanoparticle precursors in a ZnS laser ablation plasma,” Appl. Surf. Sci. 392, 572–580 (2017). [CrossRef]
22. M. A. Fareed, V. V. Strelkov, N. Thire, et al., “High-order harmonic generation from the dressed autoionizing states,” Nat. Commun. 8(1), 16061 (2017). [CrossRef]
23. M. Wostmann, L. Splitthoff, and H. Zacharias, “Control of quasi-phase-matching of high-harmonics in a spatially structured plasma,” Opt. Express 26(11), 14524–14535 (2018). [CrossRef]
24. Z. Abdelrahman, M. A. Khohlova, D. J. Walke, et al., “Chirp-control of resonant high-order harmonic generation in indium ablation plumes driven by intense few-cycle laser pulses,” Opt. Express 26(12), 15745–15756 (2018). [CrossRef]
25. M. A. Fareed, V. V. Strelkov, M. Singh, et al., “Harmonic generation from neutral manganese atoms in the vicinity of the giant autoionization resonance,” Phys. Rev. Lett. 121(2), 023201 (2018). [CrossRef]
26. M. Kumar, H. Singhal, and J. A. Chakera, “High order harmonic radiation source for multicolor extreme ultraviolet radiography of carbon plumes,” J. Appl. Phys. 125(15), 155902 (2019). [CrossRef]
27. M. Singh, M. A. Fareed, A. Laramée, et al., “Intense vortex high-order harmonics generated from laser-ablated plume,” Appl. Phys. Lett. 115(23), 231105 (2019). [CrossRef]
28. J. Liang, Y. H. Lai, W. Fu, et al., “Observation of resonance-enhanced high-order harmonics from direct excitation of metal nanoparticles with femtosecond pulses,” Phys. Rev. A 102(5), 053117 (2020). [CrossRef]
29. S. R. Konda, V. R. Soma, M. Banavoth, et al., “High harmonic generation from laser-induced plasmas of Ni-doped CsPbBr3 nanocrystals: implications for extreme ultraviolet light sources,” ACS Appl. Nano Mater. 4(8), 8292–8301 (2021). [CrossRef]
30. S. R. Konda, Y. H. Lai, and W. Li, “Investigation of high harmonic generation from laser ablated plumes of silver,” J. Appl. Phys. 130(1), 013101 (2021). [CrossRef]
31. W. Fu, J. Wang, J. Yu, et al., “Extension of high-order harmonic generation cutoff from laser-ablated tin plasma plumes,” Opt. Express 31(10), 15553–15563 (2023). [CrossRef]
32. M. Singh, M. A. Fareed, V. Birulia, et al., “Ultrafast resonant state formation by the coupling of Rydberg and dark autoionizing states,” Phys. Rev. Lett. 130(7), 073201 (2023). [CrossRef]
33. J. Mathijssen, Z. Mazzotta, A. M. Heinzerling, et al., “Material-specific high-order harmonic generation in laser-produced plasmas for varying plasma dynamics,” Appl. Phys. B 129(6), 91 (2023). [CrossRef]
34. D. B. Milošević, “High-energy stimulated emission from plasma ablation pumped by resonant high-order harmonic generation,” J. Phys. B 40(17), 3367–3376 (2007). [CrossRef]
35. R. A. Ganeev and D. B. Milošević, “Comparative analysis of the high-order harmonic generation in the laser ablation plasmas prepared on the surfaces of complex and atomic targets,” J. Opt. Soc. Am. B 25(7), 1127–1134 (2008). [CrossRef]
36. I. A. Ivanov and A. S. Kheifets, “Resonant enhancement of generation of harmonics,” Phys. Rev. A 78(5), 053406 (2008). [CrossRef]
37. I. A. Kulagin and T. Usmanov, “Efficient selection of single high-order harmonic caused by atomic autoionizing state influence,” Opt. Lett. 34(17), 2616–2618 (2009). [CrossRef]
38. V. Strelkov, “Role of autoionizing state in resonant high-order harmonic generation and attosecond pulse production,” Phys. Rev. Lett. 104(12), 123901 (2010). [CrossRef]
39. D. B. Milošević, “Resonant high-order harmonic generation from plasma ablation: Laser intensity dependence of the harmonic intensity and phase,” Phys. Rev. A 81(2), 023802 (2010). [CrossRef]
40. M. F. Frolov, N. L. Manakov, and A. F. Starace, “Potential barrier effects in high-order harmonic generation by transition-metal ions,” Phys. Rev. A 82(2), 023424 (2010). [CrossRef]
41. P. V. Redkin and R. A. Ganeev, “Simulation of resonant high-order harmonic generation in three-dimensional fullerenelike system by means of multiconfigurational time-dependent Hartree-Fock approach,” Phys. Rev. A 81(6), 063825 (2010). [CrossRef]
42. M. Tudorovskaya and M. Lein, “High-order harmonic generation in the presence of resonance,” Phys. Rev. A 84(1), 013430 (2011). [CrossRef]
43. M. V. Frolov, N. L. Manakov, A. M. Popov, et al., “Analytic theory of high-order-harmonic generation by an intense few-cycle laser pulse,” Phys. Rev. A 85(3), 033416 (2012). [CrossRef]
44. J. M. Ngoko Djiokap and A. F. Starace, “Resonant enhancement of the harmonic-generation spectrum of beryllium,” Phys. Rev. A 88(5), 053412 (2013). [CrossRef]
45. V. V. Strelkov, M. A. Khokhlova, and N. Y. Shubin, “High-order harmonic generation and Fano resonances,” Phys. Rev. A 89(5), 053833 (2014). [CrossRef]
46. I. J. Kim, C. M. Kim, H. T. Kim, et al., “Highly efficient high-harmonic generation in an orthogonally polarized two-color laser field,” Phys. Rev. Lett. 94(24), 243901 (2005). [CrossRef]
47. V. V. Strelkov and R. A. Ganeev, “Quasi-phase-matching of high-order harmonics in plasma plumes: theory and experiment,” Opt. Express 25(18), 21068–21083 (2017). [CrossRef]
48. I. S. Wahyutama, T. Sato, and K. L. Ishikawa, “Time-dependent multiconfiguration self-consistent-field study on resonantly enhanced high-order harmonic generation from transition-metal elements,” Phys. Rev. A 99(6), 063420 (2019). [CrossRef]
49. J. M. Ngoko Djiokap and A. F. Starace, “Origin of the multi-photon regime harmonic-generation plateau structure,” Phys. Rev. A 102(1), 013103 (2020). [CrossRef]
50. A. I. Magunov and V. V. Strelkov, “Intensity of resonant harmonic generated in the multiphoton ionization regime,” Phys. Wave Phen. 28(4), 369–374 (2020). [CrossRef]
51. M. A. Khokhlova, M. Y. Emelin, M. Y. Ryabikin, et al., “Polarization control of quasimonochromatic XUV light produced via resonant high-order harmonic generation,” Phys. Rev. A 103(4), 043114 (2021). [CrossRef]
52. S. Y. Stremoukhov, “Quasi-phase matching of high harmonics driven by mid–IR: Towards the efficiency drop compensation,” J. Opt. Soc. Am. B 39(4), 1203–1208 (2022). [CrossRef]
53. E. S. Toma, P. Antoine, A. de Bohan, et al., “Resonance-enhanced high-harmonic generation,” J. Phys. B 32(24), 5843–5852 (1999). [CrossRef]
54. J. Seres, V. S. Yakovlev, E. Seres, et al., “Coherent superposition of laser-driven soft-X-ray harmonics from successive sources,” Nat. Phys. 3(12), 878–883 (2007). [CrossRef]
55. A. Pirri, C. Corsi, and M. Bellini, “Enhancing the yield of high-order harmonics with an array of gas jets,” Phys. Rev. A 78(1), 011801 (2008). [CrossRef]
56. V. Tosa, V. S. Yakovlev, and F. Krausz, “Generation of tunable isolated attosecond pulses in multi-jet systems,” New J. Phys. 10(2), 025016 (2008). [CrossRef]
57. K. O’Keeffe, T. Robinson, and S. M. Hooker, “Quasi-phase-matching high harmonic generation using trains of pulses produced using an array of birefringent plates,” Opt. Express 20(6), 6236 (2012). [CrossRef]
58. R. A. Ganeev, M. Suzuki, S. Yoneya, et al., “Resonance enhancement of harmonics in laser-produced Zn II and Zn III containing plasmas using tunable mid-infrared pulses,” J. Phys. B 49(5), 055402 (2016). [CrossRef]
59. G. S. Boltaev, R. A. Ganeev, I. A. Kulagin, et al., “Resonance enhancement of the 11th harmonic of 1064 nm picosecond radiation generating in lead plasma,” J. Opt. Soc. Am. B 31(3), 436–442 (2014). [CrossRef]
60. R. A. Ganeev, V. V. Kim, J. Butikova, et al., “High-order harmonics generation in Cd and Pd laser-induced plasmas,” Opt. Express 31(16), 26626 (2023). [CrossRef]
61. M. Suzuki, M. Baba, R. Ganeev, et al., “Anomalous enhancement of single high-order harmonic using laser ablation tin plume at 47 nm,” Opt. Lett. 31(22), 3306–3308 (2006). [CrossRef]
62. R. A. Ganeev, H. Singhal, P. A. Naik, et al., “Single harmonic enhancement by controlling the chirp of the driving laser pulse during high-order harmonic generation from GaAs plasma,” J. Opt. Soc. Am. B 23(12), 2535–2540 (2006). [CrossRef]
63. M. Suzuki, M. Baba, H. Kuroda, et al., “Intense exact resonance enhancement of single-high-harmonic from an antimony ion by using Ti:sapphire laser at 37 nm,” Opt. Express 15(3), 1161–1166 (2007). [CrossRef]
64. V. V. Kim, G. S. Boltaev, M. Iqbal, et al., “Resonance enhancement of harmonics in the vicinity of 32 nm spectral range during propagation of femtosecond pulses through the molybdenum plasma,” J. Phys. B 53(19), 195401 (2020). [CrossRef]
65. R. A. Ganeev, V. V. Kim, G. S. Boltaev, et al., “Joint manifestation of quasi-phase-matching and resonance enhancement of harmonics in laser-induced plasmas,” Opt. Continuum 1(5), 1098–1116 (2022). [CrossRef]
66. M. Suzuki, M. Baba, R. A. Ganeev, et al., “Observation of single harmonic enhancement due to quasi-resonance conditions with the tellurium ion transition in the range of 29.44 nm,” J. Opt. Soc. Am. B 24(10), 2686–2689 (2007). [CrossRef]
67. R. A. Ganeev, M. Suzuki, M. Baba, et al., “Harmonic generation in XUV from chromium plasma,” Appl. Phys. Lett. 86(13), 131116 (2005). [CrossRef]
68. R. A. Ganeev, T. Witting, C. Hutchison, et al., “Isolated sub-fs XUV pulse generation in Mn plasma ablation,” Opt. Express 20(23), 25239–25248 (2012). [CrossRef]
69. R. A. Ganeev, “High-order harmonics generation in selenium-containing plasmas,” Photonics 10(7), 854 (2023). [CrossRef]
70. M. Hentschel, R. Kienberger, C. Spielmann, et al., “Attosecond metrology,” Nature 414(6863), 509–513 (2001). [CrossRef]
71. L. Zheng, X. Chen, S. Tang, et al., “Multiple quasi-phase-matching for enhanced generation of selected high harmonics in aperiodic modulated fibers,” Opt. Express 15(26), 17985–17990 (2007). [CrossRef]
72. R. A. Ganeev, V. Tosa, K. Kovács, et al., “Influence of ablated and tunneled electrons on the quasi-phase-matched high-order harmonic generation in laser-produced plasma,” Phys. Rev. A 91(4), 043823 (2015). [CrossRef]
73. R. A. Ganeev, G. S. Boltaev, B. Sobirov, et al., “Modification of modulated plasma plumes for the quasi-phase-matching of high-order harmonics in different spectral ranges,” Phys. Plasmas 22(1), 012302 (2015). [CrossRef]
74. R. A. Ganeev, “Influence of micro- and macro-processes on the high-order harmonic generation in laser-produced plasma,” J. Appl. Phys. 119(11), 113104 (2016). [CrossRef]
75. R. D’Arcy, J. T. Costello, C. McGuinnes, et al., “Discrete structure in the 4d photoabsorption spectrum of antimony and its ions,” J. Phys. B 32(20), 4859–4876 (1999). [CrossRef]
76. H. Kjeldsen, F. Folkmann, B. Kristensen, et al., “Absolute cross section for photoionization of Mn+ in the 3p region,” J. Phys. B 37(6), 1321–1330 (2004). [CrossRef]
77. P. Balcou, P. Salières, A. L’Huillier, et al., “Generalized phase-matching conditions for high harmonics: The role of field-gradient forces,” Phys. Rev. A 55(4), 3204–3210 (1997). [CrossRef]
78. C. M. Heyl, C. L. Arnold, A. Couairon, et al., “Introduction to macroscopic power scaling principles for high-order harmonic generation,” J. Phys. B 50(1), 013001 (2017). [CrossRef]
79. R. Weissenbilder, S. Carlström, L. Rego, et al., “How to optimize high-order harmonic generation in gases,” Nat. Rev. Phys. 4(11), 713–722 (2022). [CrossRef]
