## Abstract

We theoretically analyze the phase-matching of high-order harmonic generation (HHG) in multi-jet plasmas and find the harmonic orders for which the quasi-phase-matching (QPM) is achieved depending on the parameters of the plasma and the generating beam. HHG by single- and two-color generating fields is analyzed. The QMP is studied experimentally for silver, indium and manganese plasmas using near IR and mid-IR laser fields. The theory is validated by comparison with our experimental observations, as well as published experimental data. In particular, the plasma densities and the harmonic phase coefficients reconstructed from the observed harmonic spectra using our theory agree with the corresponding parameters found using other methods. Our theory allows defining the plasma jet and the generating field properties, which can maximize the HHG efficiency due to QPM.

© 2017 Optical Society of America

## Corrections

25 August 2017: Typographical corrections were made to paragraphs 1, 2, and 5–7 of Section 2.1.

## 1. Introduction

Many methods have been implemented in the high-order harmonic generation (HHG) in gases and plasmas to improve the conversion efficiency. In the case of harmonics generated in laser plasma they include the use of nanoparticles and clusters as harmonic sources, the use of various combinations of two- or three-color pumps, the application of extended plasma, the realization of the conditions of resonant amplification of single harmonic, and the formation of quasi-phase matching (QPM) between interacting waves. The latter of these techniques in combination with two-color pumping of plasma have demonstrated their attractiveness by increasing the conversion efficiency in various ranges of the extreme ultraviolet (XUV) region [1]. Most of the previous studies of QPM were performed using conventional Ti:sapphire lasers and their second harmonics as the radiation sources. Meantime, the use of longer wavelengths of the laser sources can lead to a further increase of harmonic yield by applying the multi-jet plasmas compared with the extended homogeneous plasmas. Particularly, the tunable optical parametric amplifiers (OPA) generating in the mid-infrared (MIR) region, and their second harmonics will allow to generate a wide range of wave sum and difference harmonics.

The phase mismatch between the interacting waves suppresses the HHG in the gas and plasma media. The dephasing between the propagating harmonic wave and laser-induced polarization is mainly caused by the dispersion of the medium and becomes essential in the presence of a significant number of free electrons. Due to the difference in the velocity of the waves, at some distance from the beginning of the medium, the phase shift becomes close to *π*. Beyond this distance, called the coherence length, the constructive accumulation of harmonic photons is reversed and the harmonic energy starts to decrease. QPM of interacting waves can significantly reduce the negative impact of phase mismatch as well as to increase the conversion efficiency in different parts of the XUV spectrum.

QPM has been demonstrated in gases [2–7] and multi-jet plasma formations [8]. Though not new for the case of gas harmonics, this approach recently allowed a significant increase of harmonic yield in the case of modulated plasmas. Moreover, the so-called ‘self-probing’ schemes to extract structural and dynamical information about the generating medium from the intensity measurements of HHG, which have already been reported in the case of gas targets, are extended to plasma targets. The implementation of this concept for the case of plasma harmonics widely broadens the field of study due to the overwhelming prevalence of the ablated solid species used in the latter case over the few gas types routinely exploited in conventional HHG.

The key enabling technology for this research is the capability of laser ablation to put atoms and molecules from solids into gas phase at appropriate densities [8,9]. This method of plasma formation also allows the analysis of the relative role of micro- and macro-processes in the enhancement of harmonic yield [10]. Such a concept can be useful for the formation of a group of harmonics when the phase-locking of the harmonics leads to the production of an attosecond pulse train. The comparative analysis of different sources of pump radiation (e.g. fixed-wavelength ones, such as Ti:sapphire lasers and their second harmonics, or those with tunable wavelengths, such as OPA and their second harmonics) allows defining the best schemes for enhancement of HHG in different spectral regions of XUV.

In this paper, we present the theoretical description of the QPM in the set of plasma jets. The behavior of the phase mismatch as a function of the propagation distance in the modulated plasma plume is analyzed. Properties of the QPM depending on the number of plasma jets, plasma density, and jet size are studied theoretically and compared both with the previously reported data and the present experimental measurements. We show the results of QPM experiments, which were carried out using different ablated solids. We also compare application of tunable mid-infrared pulses and fixed wavelength 800-nm-class laser for the QPM of harmonics. We demonstrate various peculiarities of QPM, which corroborate the developed theory.

## 2. Theory of the QPM of harmonics in the group of plasma jets

#### 2.1. Basic principles

The phase of the microscopic response at *q*-th harmonic frequency at distance *z* is:

*I*is the driving field intensity,${\varphi}_{q}^{(np)}$ is the phase of the microscopic response as a function of the driving intensity. This phase is traditionally denoted as the “atomic phase” [11–13]. In the present study, however, we are dealing with an

*ionic*medium, so this term would be misleading. So we suggest denoting it with a more general term “non-perturbative phase”, as it originates from the non-perturbative nature of the HHG response [14]. Within certain approximations this phase linearly depends on the laser intensity [15-16] (see also the recent review [17])and the proportionality coefficient $\alpha $ depends on the electronic trajectory.

The phase of the driving field is a sum of the Gouy phase and the term due to the presence of the medium:

where $\Delta k=k-\omega /c,\omega $ is the driving field frequency, $k$ is its wave-vector in the medium. Thus the total phase given by Eq. (1) is:The behavior of these three terms is illustrated in Fig. 1. For the plasma multi-jet target the first term decreases as a function of the propagation distance, because the dispersion due to free electrons overrides that caused by the ions and neutrals. The second term always decreases. The third term decreases before the focus and increases after it [18].

Let *L* is the spatial period of a set of plasma jets. Then the QPM condition for the generation of the *q*-th harmonic is:

*m*is an integer. Within the distance

*L*functions ${\varphi}_{G}(z)$ and

*I(z)*can be considered as slowly-varying ones, so Eq. (4) can be written as:

In the gaseous jets under proper choice of the parameters the three terms can compensate for each other, so the case of *m = 0* takes place. However, this is hardly possible for the plasma jets, because the first two terms decrease ($\Delta k<0$for plasma), and the third one is typically too low to compensate for them. In Fig. 1 we show the case of $m=-1$.

Let us denote

Note that if *l* is the effective thickness of the jet (see Fig. 1)

From Eq. (5) and (6) we find:

To find the order of the most efficiently generated harmonic we consider generation near the laser focus, so

where*b*is the laser beam confocal parameter. In this case Eq. (8) gives

Experimentally we observe only one ${q}_{opt}$ in the HH spectrum, so we have the lowest possible $\left|m\right|$, thus we have $m=-1$. In this case Eq. (10) is written as

When a multi-slit mask with the spatial period *D* is tilted by an angle $\theta $ (see Fig. 1) the period *L* is given by $L=D\mathrm{cos}\theta $. Note that $\Delta {k}_{eff}$ does not change when the mask is tilted (see Eq. (7)) because the plasma density and the l/L ratio does not change. So Eq. (11) predicts the ${(\mathrm{cos}\theta )}^{-1}$ dependence of the order of the most effectively generated harmonic on the tilting angle. In Fig. 2 we present ${q}_{opt}$ measured both in the experiments where $L$ was changed by tilting the mask [9,19], as well as in those done using three different masks [20]. The error in defining ${q}_{opt}$ is naturally plus/minus two harmonics. The experimental data are fitted by ${(\mathrm{cos}\theta )}^{-1}$ and ${L}^{-1}$ dependencies predicted by Eq. (11) (note log-log scale in Fig. 2(b)). One can see that these fits reasonably agree with the experimental data.

It was shown in [1] that when the set of plasma jets is shifted in the transverse direction, ${q}_{opt}$ increases. This result agrees with Eq. (11) because at the periphery of the plasma jets $\left|\Delta {k}_{eff}\right|$ is lower due to lower plasma density; $\Delta {k}_{eff}$ is negative, so the decrease of $\left|\Delta {k}_{eff}\right|$ leads to a decrease in the absolute value of the denominator in Eq. (11), and therefore to an increase of ${q}_{opt}$. From this Eq. the maximal ${q}_{opt}$ (for given *L* and *b*) can be found:

However, the quasi-phase matched generation of the ${q}_{opt}^{(\mathrm{max})}$ -th harmonic is achieved under $\Delta {k}_{eff}=0;$ this condition can hardly be satisfied in the plasma HHG where the dispersion due to free electrons is non-negligible. The experimentally observed ${q}_{opt}$ was typically 2 to 4 times lower than the ${q}_{opt}^{(\mathrm{max})}$.

From the experimentally measured ${q}_{opt}$ we can find $\Delta {k}_{eff}$ using Eq. (11) and thus find the density of free electrons. For the IR frequency the main contribution to $\Delta n$ is due to plasma dispersion, so

where plasma frequency is ${\omega}_{p}^{2}=4\pi {e}^{2}{N}_{e}/m,{N}_{e}$ is the mean electron density. Thus from Eq. (11) and (7) we haveFor the 800 nm wavelength this gives

In [9] the HHG spectra generated in silver plasma were measured for different heating pulse energies, and thus for different plasma densities. Substituting ${q}_{opt}$ found from those spectra in Eq. (15), we reconstruct the electron density as a function of the heating pulse energy, see Fig. 3. In [8] the plasma density (density of ions and neutrals) for different heating pulse energies was calculated numerically via the ITAP IMD code. This density is also shown in Fig. 3. Assuming that the driving laser pulse completely ionizes the neutrals in the plasma plume and to some extent further ionizes ions we can conclude that the electron density in the plasma plume during HHG should be close to that of plasma or slightly exceed it. In Fig. 3, we can see a good quantitative agreement between the electron density reconstructed from the experimental HHG spectra using our theory and the numerical results of the ITAP IMD code.

#### 2.2. Maximization of the generation efficiency under QPM

The harmonic generation efficiency inside every jet is maximized if the jet thickness is equal to the coherence length of this harmonic. This requirement provides additional condition for the generation parameters; note that it is not linked to the QPM condition described by Eq. (4). Below we will find when both conditions are satisfied, i.e. when the coherence length for harmonic ${q}_{opt}$ is equal to the jet thickness:

In this case the harmonic phase shift accumulated inside the jet (due to geometrical and medium dispersion) is -*π*, so the phase shift accumulated outside the jet is also -*π* (see Fig. 1):

This Eq. can be rewritten as

As we have mentioned (see Eq. (12) and comments below it) ${q}_{opt}<{q}_{opt}^{(\mathrm{max})}$ for $\Delta {k}_{eff}\ne 0$ . Thus from Eq. (12) and (18) we conclude that requirement (16) can be satisfied only if $l/L<1/2$. Note that for given $l/L<1/2$ one can satisfy condition (16) for a given number ${q}_{opt}<{q}_{opt}^{(\mathrm{max})}$ by tuning L via tilting the mask (this tilt keeps $l/L$ constant) and tuning the plasma density via varying the heating pulse fluence.

#### 2.3. Width of the QPM-enhanced harmonic group

For the harmonic order $q\ne {q}_{opt}^{}$ we can conclude that (a) the QPM condition (4) is not exactly satisfied, and (b) the ratio of the jet thickness and the coherence length differs from that for the ${q}_{opt}$ -th harmonic. The second item was studied in [9]. However, the first one is much more important if the number of jets *N* is not low. To show this let us consider the phase shifts $\Delta {\varphi}_{q}$ and $\Delta {\varphi}_{{q}_{opt}}$ accumulated by harmonics $q$ and ${q}_{opt}$ at the distance *L*. The QPM for the generation of the *q*-th harmonic is lost when the total phase shift accumulated due to propagation over the multi-jet plasma is $N\Delta {\varphi}_{q}=N2\pi m\pm \pi $, thus

So the QPM is lost for $\Delta {\varphi}_{q}$ which differs just slightly from $\Delta {\varphi}_{{q}_{opt}}=2\pi m$. This difference can hardly influence the generation efficiency within every jet.

Equation (19) leads to the width of the group of QPM-enhanced harmonics $\Delta \tilde{q}=\left|q-{q}_{opt}\right|$ given by

This width was obtained considering generation exactly at the laser focus (Eq. (9)), thus assuming that the size of the multi-jet target is much less than the confocal length. Experimentally, however, this is usually not the case. Let us consider the QPM not exactly at focus. We can see from Eq. (8) that variation of the derivatives $\partial I/\partial z$ and $\partial {\varphi}_{G}/\partial z$ inside the multi-jet target leads to the QPM-enhancement of different harmonics. The derivative of the intensity over *z* is:

From Eq. (21) we have $\mathrm{max}\left|\partial I/\partial z\right|=\frac{{I}_{0}3\sqrt{3}}{4b}\approx 1.3\frac{{I}_{0}}{b}$ ; this maximum is achieved under $z=\pm b/(2\sqrt{3})\approx \pm 0.29b$. Assuming that the multi-jet target occupies at least the volume $-0.29b<z<0.29b$ we find that harmonics $q={q}_{opt}\pm \Delta q$ are QMP-enhanced where

Substituting Eq. (11) we have

For the experimental conditions we used this width exceeds that given by Eq. (20). So we conclude that the experimentally observed width is mainly defined by the variation of the generation conditions inside the multi-jet target and this width should be described by Eq. (25).

From Eq. (25) we can see that $\Delta q$ depends only on the electron density and the driving beam parameters, but does not depend on the spatial properties of the multi-jet plasma (i.e. relative parameters *L* and *l*). This agrees with the experiments [9] and [20] where *L* was changed by tilting the multi-slit mask and using different masks, respectively. Using Eq. (24) we can find $\alpha $ from the experimentally measured width of the enhanced group of harmonics. For the harmonic orders in the range of 33 - 43 using the experimental results presented in [8,9,20] we find $\alpha =2.1$, $2.2$, and $2.7$ [10^{14} W/cm^{2}]^{−1}, respectively. We can see that the values found from the different experiments are remarkably close to each other. Moreover, they are close to the $\alpha $ values found for HHG in gases [21–23] for the shortest quantum path.

#### 2.4. QPM in the case of the two-color pump

Let us consider the generation in the field consisting of the laser field and its second harmonic (SH). The generation of the *q*-th harmonic can be described as a number of elementary processes involving *q _{1}* photons from the laser field and

*q*photons from the SH field, so that

_{2}*q*=

*q*

_{1}+ 2

*q*

_{2}and the total number of the involved quanta

*q*

_{1}+

*q*

_{2}is odd (as the generating medium has central symmetry). Note that

*q*

_{1}and

*q*

_{2}can be positive or negative; usually the SH intensity is lower than that of the laser field, so

*q*

_{1}is comparable to

*q*and |

*q*

_{2}| <<

*q*.

For the case of the two-color driving field the Eq. (1) is rewritten as

*I*

_{1}(

*z*) and

*I*

_{2}(

*z*) are the laser and the SH intensities, respectively.

Let us consider the phases of the generating fields due to the plasma dispersion ${\varphi}_{1,2}^{(pl)}={\displaystyle \underset{0}{\overset{L}{\int}}\Delta {k}_{1,2}(z)dz}$. Taking into account that $\Delta {n}_{2}(z)=\Delta {n}_{1}/4$ (see Eq. (13)) we have ${\varphi}_{2}^{(pl)}={\varphi}_{1}^{(pl)}/2$. Thus near the focus of the beams we have from Eqs. (26) and (4)

*b*

_{1}and

*b*

_{2}are the confocal parameters of the beams. If ${b}_{2}=2{b}_{1}$ (this is so because the wavelength of second pulse is twice that of the main driving pulse) then the latter Eq. is written as:

Thus introducing $Q={q}_{1}+{q}_{2}/2$ we obtain the result formally similar to Eq. (11):

However, certain ${Q}_{opt}$ corresponds to *several combinations* of *q*_{1} and *q*_{2}, and thus to emission of *several* frequencies

From this we can conclude that adding of the SH field of moderate intensity increases the width of the group of the enhanced harmonics. The higher is the SH field intensity, the larger is the number of the photons from the SH field | *q*_{2}|, which can be involved in the process; so the wider is the group of harmonics enhanced due to QPM. This conclusion can hardly be directly quantitatively checked by the published experimental results because the same laser beam was used to generate the SH field and then to generate high-order harmonics, so the fundamental field intensity was different in single- and two-color pump experiments. However, qualitatively this conclusion agrees with the experimental results. For instance, the harmonics from 22 to 37 were enhanced in the case of two-color pump HHG experiments reported in [19].

## 3. Experimental studies of QPM

#### 3.1. Experimental arrangements

Two laser sources were used for these studies. In both cases, the uncompressed radiation of the Ti:sapphire laser was used as a heating pulse (central wavelength *λ* = 806 nm, pulse duration 370 ps, pulse energy up to *E*_{hp} = 4 mJ) for plasma formation using a 200 mm focal-length cylindrical lens, which focused the pulse inside the vacuum chamber containing an ablating target to create the extended laser-produced plasma (LPP) above the target surface (Fig. 4(a)). The focusing of the heating pulse on the target surface produced the extended non-perforated plasma. The intensity of the heating pulses on a plain target surface was varied up to 5 × 10^{9} W cm^{−2}.

In the first set of studies, the compressed driving pulse from the same laser with the energy of up to *E*_{dp} = 5 mJ and 64 fs pulse duration was used, 45 ns from the beginning of ablation, for harmonic generation in the plasma plume. The driving pulse was focused using a 400 mm focal-length spherical lens onto the prepared plasma from perpendicular direction, at a distance of ~100 *μ*m above the target surface. The confocal parameter of the focused driving beam was 18 mm. The intensity of the driving pulse in the focal area was varied up to 9 × 10^{14} W cm^{−2}. The harmonic emission was analyzed by an XUV spectrometer. A detailed description of XUV spectrometer was given in [9].

In the second set of studies, we pumped the OPA (HE-TOPAS Prime, Light Conversion) by 10 mJ, 806 nm, 64 fs pulses. Signal and idler pulses from the OPA allowed tuning within the 1170 - 1620 nm and 1580 - 2650 nm MIR ranges respectively. The experiments were carried out using ~1-mJ, 70-fs, 1310-nm signal pulses. The spectral bandwidth of these pulses was 50 nm. The intensity of the 1310-nm pulses focused by 400-mm focal length lens inside the extended plasma was ~2 × 10^{14} W cm^{−2}. Most of the experiments in this configuration were performed using two-color pump of the LPP. The choice of the double beam configuration was due to the small energy of the driving pulse (~0.7 - 1 mJ depending on the wavelength of signal radiation). The *I*_{H} ∝ *λ*^{−5} rule (*I*_{H} is the harmonic intensity and *λ* is the driving field wavelength [24,25]) did not allow the observation of strong harmonics from the ~1310-nm pulses. Because of this the second-harmonic generation of the signal pulse was used to implement the two-color pump scheme of plasma HHG. 0.5-mm thick BBO crystal was installed inside the vacuum chamber in the path of the focused signal pulse. The conversion efficiency of the 650-nm pulses was ~20%. Two pulses overlapped both temporally and spatially in the extended plasma and allowed a significant enhancement of odd harmonics, as well as generation of even harmonics with intensity similar to that of the odd ones.

Silver, indium and manganese were used as the targets for ablation. The reasons for choosing these samples are described in the following subsection. The size of the targets where the ablation occurred was 5 mm. To create multi-jet plasmas a multi-slit mask (MSM) was used. The slits of 0.3 mm width were spaced at 0.3 mm (*d* = 0.3 mm and *D* = 0.6 mm, see Fig. 1). The energy of the heating pulse decreased after propagation through the MSM. However the fluence of this radiation on the target surface remained unchanged, since the size of ablated area was also decreased. It means that the electron and plasma densities in the cases of extended homogeneous and multi-jet plasmas were almost equal.

#### 3.2. Enhancement of the groups of harmonics

In the case of homogeneous plasma target the silver plasma allows generation of almost ideal plateau-like distribution of harmonic intensities: the decrease of the latter starting from the 13th order (for 806 nm pump wavelength) is very slow up to the cut-off region (~H61). Therefore the multi-jet target setup for silver plasma allows a clear observation of the quasi-phase-matching effect in different regions of XUV by variation of the plasma modulation parameter. This parameter actually represents the size of a single jet in the multi-jet plasma formed by installation of the MSM between the focusing cylindrical lens and the target (inset in Fig. 4(a).

Below we show some experimental results, which can be compared with those obtained with the theoretical approach developed in previous section. Initially, we show the plasma shapes used for harmonic generation in the silver multi-jet plasmas produced at different modes of excitation of the target. Left picture in Fig. 4(b) shows the extended (5-mm-long) homogeneous plasma (upper panel), eight-jet plasma produced with MSM installed between the cylindrical lens and the target (middle panel), and nine-jet plasma produced as the MSM is tilted at *θ* = 25° (bottom panel). The picture in the right shows the images of three-, four- and five-jet plasmas produced as the MSM is placed in different positions inside the telescope installed in front of the cylindrical lens.

Figure 5(a) shows the harmonic spectra generated using the 806 nm pump in the extended (thin curve) and eight-jet (thick curve) silver plasmas produced by installation of the MSM (*θ* = 0°). In the latter case we can see an enhancement of the group of harmonics centered at H33. The enhancement factor for the H33 compared to the extended (5-mm-long) plasma was measured to be 11. The width of the group of harmonics can be used to find the phase coefficient *α* for the generation of this harmonics using Eq. (24). The found value of $\alpha =2.5$ [10^{14} W/cm^{2}]^{−1} agrees with those found in the previous sections while analyzing published experiments on QPM in plasma.

Stronger enhancement was observed in the case of two-color pump using the 1310 nm + 655 nm radiation (Fig. 5(b)). The two spectra shown in this figure represent the harmonic distributions captured under similar experimental conditions (energy of heating pulse, collection time of the harmonic spectra). Thin curve demonstrates the harmonic distribution obtained in the case of 5-mm-long silver plasma (see also [19]). The application of the MSM tilted at 35° drastically changed this distribution (thick curve). The group of harmonics centered around H37 was notably stronger compared with previous case (the enhancement factor of the maximally enhanced H37 was ~18), whereas the intensity of the lower-order harmonics decreased due to worsened conditions of the phase matching.

To prove the role of QPM in the observed peculiarities of harmonic spectra it would be straightforward to investigate the intensity of harmonics as a function of the number of plasma plumes. This experiment would also confirm the involvement of the coherent accumulation of harmonic yield along the whole length of divided nonlinear optical medium, since the signature of QPM is a quadratic growth of harmonic yield with the growth of number (*n*) of coherent zones contributing to the signal. The number of heating areas on the target surface was shielded step-by-step to create different numbers of plasma jets.

In Fig. 5(c), we show three HHG spectra in the case of 1-, 2-, and 5-jet silver plasmas using the 806-nm pump and similar conditions of the heating radiation. The anticipated featureless shape of harmonic spectra from the single 0.3-mm-long plasma jet was similar to those observed in the case of 5-mm-long plasma [compare the thin curve of Fig. 5(a) and the harmonic spectrum generated from the 0.3-mm-long single jet shown in Fig. 5(c)]. With the addition of each next jet, the spectral envelope was notably changed, with the 35th harmonic intensity in the case of five-jet configuration becoming almost 20 times stronger compared with the case of single-jet plasma. One can expect the *n*^{2} growth of harmonic yield for the *n*-jet configuration compared with the single jet once the phase mismatch becomes completely suppressed. This estimation gives the expected growth factor of 25 in the case of five-jet medium compared with 0.3-mm-long single-jet plasma, which was close to the experimentally measured enhancement factor of 20. Notice that the maximum enhancement factor may decrease from the ideal value of *n*^{2} at the conditions when absorption processes are turned on, or in the case of unequal properties of the jets, which can arise from the heterogeneous heating along the extended target [8].

Below we compare these experimental results with our theory. Let *A* is the ratio of the fundamental wavelengths in these two experiments, *A* = 1310nm/806nm. Similar conditions of producing the plasma and fundamental beam focusing allows assuming that $\Delta {k}_{eff}^{(1310nm)}=A\Delta {k}_{eff}^{(806nm)}$ (see Eq. (13)) and ${b}_{}^{(1310nm)}=A{b}_{}^{(806nm)}$ . Knowing from Fig. 5(a) that ${q}_{opt}^{(806nm)}=33$ we find $\Delta {k}_{eff}^{(806nm)}$ from Eq. (11), substitute $\Delta {k}_{eff}^{(1310nm)}=A\Delta {k}_{eff}^{(806nm)}$ in Eq. (30) and find that for the conditions of Fig. 5(b) ${Q}_{opt}=25$. Taking into account that the second harmonic field is 5 times weaker than the fundamental one we assume that the typical number of the second harmonic photons involved in the process should be also approximately 5 times less than that of the photons from the fundamental field. So in Eq. (31) we have ${q}_{2}={q}_{1}/5$ and finally we obtain from this Eq ${q}_{opt}\approx 32$. This result reasonably agrees with experimentally observed ${q}_{opt}=37$ (see Fig. 5(b)). Overall, in agreement with our theoretical results presented in section 2.2, the group of QPM-enhanced harmonics is remarkably wider in the case of the two-color pump.

Indium plasma has an attractive feature allowing the observation of the joint influence of micro- and macro-processes on the harmonic efficiency. From the very beginning of plasma HHG studies, this plasma demonstrated the largest enhancement of a single (H13 of the 800-nm-class lasers) harmonic. The reported enhancement factors were close to 80 [26,27]. In our studies we used this plasma to demonstrate the QPM effect for the harmonics far from the spectral region (61 nm) where the resonance-induced harmonic occurred.

In the present studies, we also analyzed the variation of this spectrum while introducing the modulation of the extended (5 mm) indium plasma, similarly to the method described in the case of silver plasma. Particularly, a proper choice of the plasma conditions and pump wavelength can further optimize the QPM process using the described configuration. We were able to increase the enhancement factor of QPM harmonics by applying the wavelength of OPA allowing stronger emission (1310 nm), while exciting the indium target in such a manner that allowed us to maintain the QPM conditions for the group of lower-order harmonics (Fig. 6). The two-color pump (1310 nm + 655 nm) of this modulated plasma containing eight jets resulted in the enhancement factors of 40 to 50 for the group of harmonics centered at H35 (*λ* = 37.4 nm). Some other features of the influence of micro- and macro-processes on the high-order harmonic generation in the laser-produced plasmas were recently analyzed in [10].

Finally, we demonstrate the observation of the QPM effect in the manganese plasma. The peculiarity of this material is the closeness of the QPM- and resonance-enhanced harmonics, which is in contrast to the case of indium plasma. The advantage of Mn plasma from the point of view of highest harmonic cut-off (i.e. above 100th orders) was reported in [28,29], though the reasons of this peculiarity were not yet clearly explained. Note that the harmonic yield from this plasma was not as high as from some other plasma species. The use of a few-cycle pulses (3.5 fs) [30] has allowed the observation of a single (33rd) broadband harmonic.

In the present studies, the Mn plasma produced by 370 ps pulses allowed generation of harmonics of 806 nm radiation with orders up to nineties. A weak excitation of target by 370 ps heating pulses (*F*_{hp} ≈0.2 J cm^{−2}) led to plateau-like harmonic generation up to the cut-off (27th harmonic) similar to the one defined from the three-step model of HHG [31] for the singly charged Mn ions. Earlier, a growth of the harmonic intensity for the harmonics above H31 was attributed to the influence of the giant 3p - 3d resonances of manganese ions [28]. H27 - H31 were significantly suppressed compared with the higher-order harmonics starting from H33.

The increase of the heating pulse fluence up to 1.0 J cm^{−2} caused a significant change of harmonic distribution in the case of imperforated 5-mm-long manganese plasma. Thin curve in Fig. 7 shows the generated spectrum, which points out the presence of only low-order harmonics (up to H29) in the case of imperforated Mn plasma, with some weak traces of higher orders. Replacement of the single extended nonlinear medium with the group of separated components of this medium (i.e. eight 0.3-mm-long plasma jets) drastically modified the generated harmonic spectrum (thick curve in Fig. 7). A group of strongly enhanced harmonics (H33 - H51) prevailed in this spectral distribution. Notice that the resonance-enhanced harmonic (H33) was also enhanced, to some extent, due to QPM. The total enhancement factor of this harmonic was only ~4 (more accurate measurement of this factor was problematic because of the low signal from the extended plasma target). Nevertheless, we can see that H33 is more intense than the neighbor harmonics (H31 and H35). The enhancement of the single harmonic is typical for resonance-induced enhancement. Thus the enhancement of H33 is due to both micro- and macro-processes.

In our QPM experiments, the length of the active material, i.e. the whole plasma length was twice shorter (2.5 mm) than the length of extended imperforated plasma (5 mm). From one hand, it would make more sense if the comparisons were made for the multijet plasmas and 2.5-mm-long imperforated plasma. However, from other hand, we tried to not modify the conditions of experiments. The only variation was the insertion of the MSM on the path of heating beam. Moreover, the comparative studies of imperforated 5- and 2.5-mm-long plasmas showed that the harmonic yield in the former case was ~1.3 time stronger, which also points out the involvement of phase-mismatch processes for such extended plasmas. It means that the comparison of the enhancement factor of harmonic yield in the case of multijet plasmas with regard to the imperforated 2.5-mm-long plasma will be even larger compared with the measured values of this parameter with regard to the 5-mm-long plasma.

## 4. Conclusions

In conclusion, we presented the theory of HHG quasi-phase-matching in multi-jet plasma plume for the case of one- and two-color generating fields. Theoretical dependence of the most effectively generated harmonic order (${q}_{opt}$) on the spatial period of the jets (*D*) and tilting angle of the mask producing the jets (*θ*) agrees with published experimental results. Free-electron density reconstructed with our theory from the published experimental spectra quantitatively agrees with the published results of numerical calculation of this density using molecular dynamics simulations. The phase coefficient *α* found using our theory from several published experiments and from experiments reported in this paper are remarkably close to each other and to the values found for HHG in gases for the shortest quantum path. The theoretical dependence of ${q}_{opt}$ on the fundamental wavelength reasonably agrees with our experimental results for HHG in silver using 806 nm and 1310 + 655nm. Thus a good agreement between our theory and different experimental results validates our theory. Quasi-phase-matching in the vicinity of the resonant harmonic in manganese shows that harmonic yield can be enhanced through both these processes. Our theory can be used to find optimal parameters of the multi-jet plasma and the generating beams to maximize the HHG macroscopic signal in future studies.

In our experiments, as well as in the published ones *l/L* = 1/2 was used. According to Eq. (18) *l/L*<1/2 should be used to optimize QPM. Experimental verification of this prediction is a natural outlook of these studies. Note that taking into account the mask thickness for the high tilting angle we have *l/L*<1/2 even for a mask with *d/D* = 1/2 because the slits are effectively more narrow than in the case of an ideal thin mask presented in Fig. 1.

## Funding

Russian Science Foundation (Grant No. 16-12-10279).

## Acknowledgments

R.A.G. thanks H. Kuroda and M. Suzuki for support during experimental studies. V.V.S. thanks S. A. Maiorov for fruitful discussions.

## References and links

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