# Application of Quasi-Phase Matching Concept for Enhancement of High-Order Harmonics of Ultrashort Laser Pulses in Plasmas

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## Abstract

**:**

## 1. Introduction

^{14}and 1.4 × 10

^{14}W cm

^{−2}, for 800 and 1320 nm pumps, respectively. We report the role of the number of coherent zones, sizes of plasma jets, plasma formation conditions, and the characteristics of fundamental radiation on the HHG efficiency at QPM conditions in different ranges of XUV. We demonstrate the TCP-induced QPM at different regimes of laser-plasma interaction. We also present the theoretical approach describing the observed peculiarities of HHG spectra modulation based on the model described in Ref. [18], and compare our calculations with the experimental data.

## 2. Experiment

#### 2.1. Scheme for the Formation of QPM Conditions in Plasma

^{9}W cm

^{−2}and 1.4 J cm

^{−2}, respectively. We used 5-mm-long targets for ablation and formation of extended homogeneous and structured plasmas. The ablation area was 5 × 0.08 mm

^{2}.

^{14}W cm

^{−2}. The driving pulses were focused onto the extended plasma at a distance of ~100–150 µm above the target surface. The harmonic yield was maximized by adjusting the position of target with regard to the optical axis of propagation of two driving pulses and by varying their focusing positions relative to the extended plasma. The harmonic emission was analyzed using an XUV spectrometer containing a cylindrical mirror and a 1200 grooves/mm flat field grating with variable line spacing. The spectrum was recorded on a micro-channel plate detector backed by phosphor screen, which was imaged onto a CCD camera.

^{16}cm

^{−3}) was estimated using the hydrodynamic code HYADES [25]. Similar estimates were reported in [26]. We also estimated the electron density using the relation for QPM (N

_{e}≈ 1 × 10

^{18}/L

_{coh}× H

_{qpm}), where (H

_{qpm}) is the maximally enhanced harmonic of the NIR radiation, and L

_{coh}is the coherence length, which is actually the length of single plasma jet [27]. This approach of determining the electron density provided almost similar value (~1 × 10

^{17}cm

^{−3}), considering the uncertainties of some parameters.

^{0}. The delay between driving and heating pulses was chosen from the point of view of maximal harmonic yield, which means that the driving femtosecond pulse propagated through the most “optimal” part of the plasma plume. The meaning of “optimal plasma” refers to the complex conditions when different impeding processes play a minimal role in the formation of the maximally favorable conditions for HHG.

^{0}and 21

^{0}, respectively). The duration of the driving pulses was sufficiently long to provide the phase matching bandwidth in this crystal for both wavelengths. The position of the focused femtosecond beam was approximately 0.15 mm away from the targets surface. The Rayleigh range of femtosecond beam was 4 mm, i.e., close to the whole length of plasma, thus assuming that the driving beam propagated through the 5 mm long plasma at the condition of almost plane wave-plasma interaction, which was exactly simulated during this study. Thus, the role of Gouy phase was neglected in this study. The peak intensities of 810 nm and 1320 nm pumps in the plasma area were 5 × 10

^{14}and 1.4 × 10

^{14}W cm

^{−2}, respectively.

_{cryst}= d[(n

^{o}

_{ω})

_{group}/c − (n

^{e}

_{2ω})

_{group}/c], where d is the crystal length, c/(n

^{o}

_{ω})

_{group}and c/(n

^{e}

_{2ω})

_{group}are the group velocities of the ω and 2ω waves in the BBO crystal, c is the light velocity, and n

^{o}

_{ω}and n

^{e}

_{2ω}are the refractive indices of the ω (1320 nm) and 2ω (660 nm) waves. The duration of second harmonic pulse at the output of the 0.3-mm-long BBO crystal has a value of ~ 90 fs assuming the application of 1320 nm, 70-fs driving pulses. This value was estimated using t

_{2ω}≈ [(Δ

_{cryst})

^{2}+ 0.5(t

_{ω})

^{2}]

^{1/2}[28]. Hence, the 660 nm beam has longer pulse duration, corresponding to the induced delay and a certain percentage (~50%) of the fundamental pulse duration. The latter results from the fact that the energy of the fundamental radiation is in general not high enough in the pulse wings to effectively generate the second-order harmonic. In the case of 810 nm driving pulses, the influence of group velocity dispersion is larger on the temporal overlap of this radiation and its second harmonic in the plasma area.

#### 2.2. QPM in LPPs

^{0}, 20

^{0}and 0

^{0}and creating the 11-, 9-, and 8-jet indium plasmas, with the sizes of single jet of 0.23, 0.28, and 0.31 mm, respectively. Meanwhile, the bottom spectrum shows the HHG from the homogeneous extended indium plasma produced at similar conditions. The collection times of these four spectra and the energy fluencies of target ablation were similar to each other. When the MSM was used, the whole energy of heating pulses interacting with target was two times smaller when compared with the case of extended homogeneous plasma, while the energy fluencies of heating radiation on the target surface were equal to each other. The enhancement factors of these harmonics (three upper spectra) with respect to those generated in the case of imperforated plasma (bottom spectrum) were measured to be 38×, 36×, and 27× respectively. A two-fold decrease of the whole length of multi-jet plasma compared with extended imperforated plasma (2.5 and 5 mm respectively) did not play an important role in the harmonic yield, which underlines the importance of the collective processes over single-atom ones.

_{qpm}) for which conditions of QPM became most favorable.

^{−2}) and the intensity of driving NIR pulses (1.4 × 10

^{14}W cm

^{−2}).

_{coh}≈ 1 × 10

^{18}/(N

_{e}× H

_{qpm}) [1,16]. Here H

_{qpm}is the maximally-enhanced harmonic at QPM conditions and N

_{e}is the electron density in cm

^{−3}. H34 has coherence length of ~0.3 mm at the used fluence (1.2 J cm

^{−2}) of the heating pulses on the surface of In target, and 1.4 × 10

^{14}W cm

^{−2}intensity of the driving NIR pulses. We used the MSM containing slits with the sizes of 0.3 mm and excited the target in such a manner that electron density satisfied the above relation for the H34 at the length of single jet of 0.3 mm. Correspondingly, at another excitation of target the electron concentration of plasma may satisfy to another harmonic order at the fixed sizes of single jet in the MJP.

_{H34}~n

^{2}up to n = 4. Further addition of jets led to deviation from quadratic dependence [Figure 4a] due to some inequality of plasma jets, which led to violation of the optimal phase relations between driving and H34 waves. At these conditions of the imperfect coherent accumulation of nonlinear optical response we were able to observe the 40-fold growth of H34 in the case of 8-jet plasma compared with the single-jet one.

_{qpm}× N

_{e}unchanged at the fixed spatial characteristics of the plasma jets. Figure 4c, showing the log-log dependence of H

_{qpm}on the energy of heating pulses for two different targets (Ag and Al), confirms the relation H

_{qpm}× N

_{e}~ constant.

^{−1}. The intensity of NIR femtosecond radiation used in the plasma area did not exceed 3 × 10

^{14}W cm

^{−2}. At these conditions the phase mismatch between the harmonics emitted at different ends of the 5-mm-long plasma medium can be estimated as 8 cm

^{−1}[34]. Finally, the phase mismatch due to plasma electrons [35] was considerably higher (~50 cm

^{−1}for the 34th harmonic) compared to other components. Thus, at our conditions, the only component influencing the phase mismatch is related to the presence of electrons in the laser-produced plasma. Our studies confirmed that the above-mentioned formula (L

_{coh}≈ 1 × 10

^{18}/H

_{qpm}×N

_{e}) for the coherence length can be used at the conditions of low-density plasma (≤10

^{18}cm

^{−3}), relatively long confocal parameters (b > l

_{plasma}), and relatively low intensities of the driving pulse (<5 × 10

^{14}W cm

^{−2}).

## 3. Theory

^{17}cm

^{−3}, and the density of free electron is equal to ~2 × 10

^{16}cm

^{−3}. Other parameters, such as the intensities of two-color pulses, their durations, and the length of nonlinear medium were chosen to be the same as in experiment.

_{qpm}towards the shorter wavelength region. The position of H

_{qpm}as a function of d is shown in Figure 5b. The results of numerical calculations are shown by open circles. It can be seen that the positions of the center of the enhanced groups of harmonics are inversely proportional to d. Empty and filled squares correspond to experimental data for indium [Figure 2b] and silver (Figure 3) plasmas, respectively. Here we also show some other reported data in the case of silver plasma [40,41,42]. The slopes of the dotted lines (l = −1) correspond to the H

_{qpm}(Ag) ∞ 1/d dependence. The data in the case of indium plasma shows similar tendency [(H

_{qpm}(In) ∞ 1/d], while the graph is slightly shifted with respect to the data and that is due to the different role of two groups of free electrons in phase mismatch at specific ablation conditions for these two metals. A similar shift has been earlier observed in the case of silver plasma ablated at higher energies of heating pulses (see pentagons in Figure 5b; where the data shown was taken from Ref. [40]).

_{qpm}and its intensity as a function of n (see blue filled circles and red empty squares in Figure 6b). It is seen that the position of H

_{qpm}shifts towards the higher order harmonic region, and the value of the harmonic intensity quadratically grows with the increase of the number of jets. It is worth mentioning that in experiment we also observed the shift of H

_{qpm}with the increase of n, though this shift was less pronounced in comparison with the calculations.

_{qpm}and peak intensity in the case of the two-color pump of MJP. It assumes application of the jets of two different lengths (d

_{1}and d

_{2}). In all calculations the whole length of nonlinear medium was kept the same (5 mm). We took d

_{1}= 0.15 mm and d

_{2}= 0.31 mm and simulated the HHG in the nonlinear medium consisting of 8 jets of d

_{1}and four jets of d

_{2}.

_{1}and d

_{2}jets (i.e., d

_{1}jet −d

_{1}free space −d

_{2}jet −d

_{2}free space −d

_{1}jet −d

_{1}free space…). This group of jets forms the nonlinear medium with the same length as in the previous case (i.e., eight jets of d

_{1}and four jets of d

_{2}).

_{2}becoming divided into two peaks, at H21 and H44, and the third peak remained at the same place corresponding to d

_{1}(H67). To investigate the observed changes of the spectrum more precisely we changed the value of one sort of jets from d

_{1}to d

_{3}= 0.28 mm and formed the nonlinear medium in the same way as in the previous case. The resulting harmonic spectrum is presented by the brown filled triangles in Figure 7. It is seen that the main peak of the spectrum became centered at H34 and the harmonic intensity increased by a factor of 2 in comparison with previous cases.

## 4. Discussion

_{qpm}× N

_{e}(since L

_{coh}≈ 1 × 10

^{18}(H

_{qpm}× N

_{e})

^{−}

^{1}[5,6]). The electron density depends on the ablation conditions and the ionization of the target species by the heating pulse in addition to the tunneling ionization by femtosecond driving pulse. The conditions of ablation are easily adjustable and can be used for variation of N

_{e}and the corresponding H

_{qpm}.

_{e}such that one can assume this technique could be a method to determine the value of electron density of the plasma. Determining the value of electron density of the plasma is a well-recognized problem in plasma physics especially with the case of dynamically changed N

_{e}. Numerous techniques and theories were proposed and realized to resolve this issue. Particularly, in the case of laser-produced plasmas, the code HYADES was used to estimate the electron density of plasma at different regimes of plasma formation. The plasma QPM method allows determining the electron density from direct measurements of two other parameters from the aforementioned relation, and those parameters are the coherence length and the maximally-enhanced harmonic order.

_{qpm}variations since this task has already been analyzed in previous studies [1,16,42]. Particularly, H43 of 800 nm pump was the maximally-enhanced harmonic in the case of 11-jet silver plasma structure. Similar pattern is also seen in Figure 3 (upper panel) in the case of indium plasma pumped by 1310 nm + 655 nm pulses. However, the enhancement in that case was lesser compared with longer wavelength XUV region. As shown in the upper panel of Figure 3, the maximally available cutoff for harmonics extended towards shorter wavelength region once optimal conditions for QPM were realized. This spectrum does not show the cutoff but rather demonstrates the attractiveness of the method when almost invisible harmonics (see bottom panel) appear next to the enhanced ones.

## 5. Conclusions

_{H34}~n

^{2}up to number of jets (n) equal to 4. The quadratic dependence is correct in a wide range of harmonic spectra. However, further deviation of I

_{H34}(n) dependence from the quadratic one can be explained by the violation of equality of the conditions for each next added jet (i.e., non-ideal conditions of propagation through jets, and/or low-order nonlinear optical properties of plasma, etc.). New possibilities for controlling QPM conditions using combined multi-jet plasmas are readily proposed. The results of our experimental and theoretical studies of the QPM conditions provide reliable interpretation of this phenomenon.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Experimental setup for high-order harmonic generation (HHG) in MJP. CL, cylindrical lens; PHB, picosecond heating beam; MSM, multi-slit mask; MJP, multi-jet plasma; T, target; SL, spherical lens; FDB, femtosecond driving beam; HB, harmonic beam; XUVS, extreme ultraviolet spectrometer. Inset: The extended, six-, four-, and single-jet plasma images are shown from the top to bottom. PS, plasma shapes.

**Figure 2.**Raw images of harmonic spectra from (

**a**) extended homogeneous (top to bottom) indium, silver and gold laser-produced plasmas (LPPs) pumped by 810 nm, 50 fs pulses and (

**b**) multi-jet indium LPP produced using tilted MSM with the single plasma jet sizes of (top to bottom) 0.23, 0.28, and 0.31 mm pumped by two-color NIR + visible (1320 nm + 660 nm), 70 fs pulses. The peak intensities of 800 and 1320 nm pumps in the plasma area were 5 × 10

^{14}and 1.4 × 10

^{14}W cm

^{−2}, respectively, while the intensity of the second harmonic of 1320 nm pump was ~1 × 10

^{13}W cm

^{−2}. Maximally enhanced harmonics (H34, H31, and H28) are clearly distinguished for each of these plasma configurations. Bottom spectrum corresponds to the HHG from the homogeneous extended indium plasma produced at similar conditions.

**Figure 3.**Raw images of harmonic spectra from the multi-jet silver LPP produced using tilted MSM with single plasma jet sizes of (top to bottom) 0.25, 0.31, and 0.35 mm pumped by two-color (1310 nm + 655 nm), 70 fs pulses. The intensities of 1310 nm and 655 nm pumps in plasma area were 1.3 × 10

^{14}and 1 × 10

^{13}W cm

^{−2}, respectively. Maximally enhanced harmonics (H43, H36, and H31) are determined for each of these plasma configurations. Bottom spectrum corresponds to homogeneous extended silver plasma. Similar conditions of plasma ablation using the same energy fluence of heating pulses and data collection times are applied for these four cases.

**Figure 4.**(

**a**) Dependence of H34 intensity on the number (n) of jets of the indium MJP. The multi-jet indium LPP was produced using the tilted MSM with the single plasma jet sizes of 0.23 mm and pumped by two-color near infrared (NIR) + visible (1320 nm + 660 nm), 70 fs pulses. The peak intensity of 1320 nm pump was 1.4 × 10

^{14}W cm

^{−2}. The dashed line shows the slope l = 2 of the experimental dependence up to n = 4. (

**b**) Harmonic distribution obtained from indium MJP in the case of propagation of driving pulses (1360 nm + 680 nm) through the plasma [bottom spectrum, similarly to the case shown in the upper panel of Figure 2b] and calcium quarter-wavelength plate and plasma (upper panel, see text). (

**c**) Influence of heating pulse energy on the maximally enhanced harmonic in the case of silver (filled spheres) and aluminum (empty circles) MJPs.

**Figure 5.**(

**a**) Harmonic distribution from silver MJP in the case of propagation of two-color driving pulses (1310 nm + 655 nm) through the plasma of ~5 mm length containing the jets with the sizes (d) equal to 0.15 mm (black empty squares), 0.23 mm (red filled circles), 0.31 mm (blue empty triangles), 0.45 mm (pink filled spheres), and 0.8 mm (green empty rhombs). (

**b**) Influence of the jet sizes on the order of maximally enhanced harmonic H

_{qpm}in the case of silver MJP. Black empty circles correspond to theoretical calculations and green filled squares correspond to experiment with Ag plasma. Triangles, stars and pentagons correspond to earlier reported data for silver plasma [40,41,42]. Here we also show the experimental data for indium plasma (blue empty squares). The slopes of all dependences correspond to l = −1.

**Figure 6.**(

**a**) Harmonic distribution in the case of silver MJP during propagation of two-color pulses (1310 nm + 655 nm) through the plasmas consisting of different number of jets. Single jet sizes were 0.23 mm. (

**b**) H

_{qpm}and its intensity as the functions of n (blue filled circles and black empty squares).

**Figure 7.**Harmonic distribution in the case of propagation of two-color pulses (1310 nm + 655 nm) through silver MJP of 5 mm length consisting of different combinations of two groups of jets (see text).

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**MDPI and ACS Style**

Ganeev, R.A.; Stremoukhov, S.Y.; Andreev, A.V.; Alnaser, A.S.
Application of Quasi-Phase Matching Concept for Enhancement of High-Order Harmonics of Ultrashort Laser Pulses in Plasmas. *Appl. Sci.* **2019**, *9*, 1701.
https://doi.org/10.3390/app9081701

**AMA Style**

Ganeev RA, Stremoukhov SY, Andreev AV, Alnaser AS.
Application of Quasi-Phase Matching Concept for Enhancement of High-Order Harmonics of Ultrashort Laser Pulses in Plasmas. *Applied Sciences*. 2019; 9(8):1701.
https://doi.org/10.3390/app9081701

**Chicago/Turabian Style**

Ganeev, R. A., S. Y. Stremoukhov, A. V. Andreev, and A. S. Alnaser.
2019. "Application of Quasi-Phase Matching Concept for Enhancement of High-Order Harmonics of Ultrashort Laser Pulses in Plasmas" *Applied Sciences* 9, no. 8: 1701.
https://doi.org/10.3390/app9081701