Two-color high-harmonic generation in plasmas: efficiency dependence on the generating particle properties

Anna S. Emelina; Mikhail Yu. Emelin; Rashid A. Ganeev; Masayuki Suzuki; Hiroto Kuroda; Vasily V. Strelkov

doi:10.1364/OE.24.013971

1. Introduction

The process of high-order harmonic generation (HHG) of ultrashort laser pulses is widely used as a source of coherent extreme ultraviolet (XUV) emission on multiple frequencies. The main drawback of this approach is very low conversion efficiency. The use of a driving field consisting of the laser field and its second harmonic is one of the reliable methods for enhancing the harmonic yield. The HHG using a two-color pump (TCP) was experimentally studied in gas media using parallel [1–4] or orthogonal [1, 5–9] polarization configurations. Numerous theoretical studies [10–19] addressed different aspects of two-color HHG. In particular, they showed that the role of the second harmonic field is essentially different in the cases when its intensity is much lower than that of the laser field and when the two intensities are comparable. Though more complicated experimentally, the latter case allowed for HHG with very high conversion efficiency (5 × 10⁻⁵ for the 38th harmonic [5]). The first explanation of strong HHG enhancement in two-color orthogonally polarized fields with comparable intensities of the components was given in [5, 6] in terms of the formation of a transient linear polarization of the field, selection of a short quantum path component, which has a denser electron wave packet, and higher ionization rate compared with the single-color pump (SCP) case.

The radiation of 800-nm-class lasers combined with their second harmonic was used at the initial stages of those studies. Various schemes, among which are the generation of the 400 nm wave in a separate channel with further mixing with the 800 nm wave in gases, as well as direct generation of second harmonic (H2) in thin BBO crystals followed with focusing of two beams in the converting medium, were introduced [4, 7]. Experimental [8, 9] and theoretical studies [16] also demonstrated strong dependence of the HHG yield on the relative phase between the fundamental and second harmonic radiation.

The macroscopic aspects of two-color HHG were addressed already in the early theoretical studies [20, 21] and have received much attention very recently [22, 23]. In general, the phase-matching problem in the case of a two-color field is more complicated than in the single-color case because it requires taking into account not only the phase difference between the pump field and the generated one but also the phase difference between the two pump fields which changes during propagation.

The TCP approach has been applied to harmonic generation in laser-produced plasmas (LPP) as well. The phase matching is especially important for the plasma HHG because the free electrons' density is comparable with that of the emitters, potentially leading to non-negligible phase mismatch even for relatively low-density plasmas. The first observation of an increase in the harmonic generation efficiency from narrow (~0.3-mm-long) plasma plumes irradiated by an intense two-color orthogonally polarized fields was reported in [24]. The intensity of the second harmonic was weak (the energy ratio of the laser pulse and H2 was 50:1).

The next stage of two-color HHG is based on the application of mid-infrared (MIR) pulses from optical parametric amplifiers (OPA) combined with their second harmonics. The MIR + H2 scheme, particularly 1300 nm + 650 nm, was successfully applied to both gas [9, 16] and plasma [25] HHG. The higher conversion efficiency into the second harmonic in the mid-infrared region allows for the experimental study of two-color HHG for comparable intensities of the driving field components.

In this paper, we study two-color HHG in an extended (5-mm-long) laser-produced plasma using 1310 nm emission of optical parametric amplifier and its second harmonic of comparable intensities. We observe a significant enhancement of high-order harmonics compared with the SCP; in particular, the highest-order harmonics are very pronounced in the two-color case whereas they are hardly detectable in the single-color case. We compare experimentally the increase of the HHG efficiency in TCP for different plasmas (silver, gold, and zinc) and different harmonic orders. The experimental results are analyzed theoretically. In particular, we find that the commonly used HHG theory, which assumes the 1s ground state of the generating particle, cannot reproduce the experimentally observed HHG spectrum. We therefore suggest the development of the theory using actual quantum numbers for the ground states of the species used for HHG (silver, gold, and zinc ions in our case). Having in mind a significant role of the propagation effects in the extended plasma target, we suggest an approach to calculate the macroscopic HHG signal taking into account the variation of the phase difference between the frequency components of a driving field along the propagation distance. The integrated macroscopic signal calculated using the improved microscopic response reproduces the measured essential dependence of the enhancement on the plasma type, as well as on the harmonic order. Moreover, our theoretical studies allow for clarifying an important aspect of the HHG enhancement in a two-color field which was not discussed earlier. Namely, according to the simple-man HHG model, in the single-color field the majority of the ionized electrons come back to the parent ion with zero energy; we show that in the two-color orthogonally polarized field with comparable intensities of the components, the majority of the electrons can come back with the energy corresponding to the cut-off harmonics. We find the conditions for which this takes place.

2. Experiment

2.1 Experimental arrangements

Our experimental setup consisted of three parts: (a) Ti:sapphire laser, (b) travelling-wave OPA of white-light continuum, and (c) setup for high-order harmonic generation using propagation of amplified signal pulse of parametric amplifier through the extended LPP.

We used the mode-locked Ti:sapphire laser TSUNAMI (Spectra-Physics Lasers) pumped by diode-pumped, cw laser MILLENIA Vs (Spectra-Physics Lasers) as the source of 803-nm, 55-fs, 82-MHz, 450-mW pulses for injection in the pulsed Ti:sapphire regenerative amplifier with pulse stretcher and additional double-pass linear amplifier TSA-10 (Spectra-Physics Lasers). The output characteristics of this laser were as follows: wavelength 806 nm, pulse duration 350 ps, 10 Hz pulse repetition rate, and pulse energy 9 mJ. This radiation was further amplified in home-made three-pass Ti:sapphire linear amplifier up to 22 mJ. Part of this radiation with pulse energy of 6 mJ was separated from the main beam and used as a heating pulse for homogeneous extended plasma formation using the 200-mm focal length cylindrical focusing lens installed in front of the extended solid target placed in the vacuum chamber, see Fig. 1. The size of the target region where the ablation occurred was 5 × 0.08 mm². The intensity of the heating pulse on the target surface was varied up to E_hp = 4 × 10⁹ W/cm².

Fig. 1 Experimental setup for harmonic generation in metallic plasma.

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The remaining part of the amplified radiation was delayed with regard to the heating pulse in such a way that, after compression and pumping of the OPA, the signal pulse from the parametric amplifier propagated through the LPP 40 ns after the beginning of target ablation. After propagation through the compressor stage the output characteristics of the whole Ti:sapphire system were as follows: pulse energy 8.4 mJ, pulse duration 64 fs, and central wavelength 806 nm. This radiation pumped the OPA HE-TOPAS Prime (Light Conversion). Signal and idler pulses from OPA allowed for tuning in the 1170 − 1620 nm and 1580 − 2650 nm ranges, respectively. In our experiments we used the signal pulses. Most of the experiments were carried out using the ~1-mJ, 70-fs tunable signal pulses. The bandwidth of the pulses was 50 nm.

The signal radiation was used as the driving pulse for HHG in LPP. The intensity of the 1310-nm pulses focused by a 400-mm focal length lens inside the extended plasma was ~2 × 10¹⁴ W/cm². The driving beam was focused, with the focal spot size of 90 μm, into the prepared extended plasma at a distance of ~100 µm above the target surface. The harmonic emission was analyzed by a XUV spectrometer containing a cylindrical mirror and a 1200 grooves/mm flat field grating (FFG) with variable line spacing. The spectrum was recorded on a micro-channel plate detector with the phosphor screen, which was imaged onto a CCD camera. The movement of the micro-channel plate along the focusing plane of the flat field grating allowed for the observation of harmonics in different regions within XUV.

The HHG efficiency in the used LPP (Ag, Au, and Zn) was optimized with respect to the delay between the heating and driving pulses. The harmonic generation efficiency abruptly increased once the delay exceeded 5 ns. In the case of the used particles (with atomic weights of 65, 108, and 197 for Zn, Ag, and Au, respectively), the maximal harmonic yields were observed at 25, 35, and 45 ns delays due to different velocities of the ablated particles. At longer delays for each specific sample, for a fixed distance between the target and the laser beam (~150 μm), the harmonic yield gradually decreased until the entire disappearance at ~150 − 200 ns.

The density characteristics of the LPP at the delays of 25 − 45 ns were estimated using the HYADES code [26]. For the heating pulse intensity of 2 × 10⁹ W/cm², the ionization level and the ion density of the silver plasma were estimated to be 0.4 and 2 × 10¹⁷ cm⁻³, respectively.

The harmonic yield from Ag, Au, and Zn plasmas using MIR pulses (1 mJ, 1300 nm) was significantly weaker compared with the 7-mJ, 806-nm pump. Such a decrease of the high-harmonic signal with the driving wavelength was shown earlier in gases [27, 28]. The harmonic cutoff energy in the case of MIR pulses was also lower compared with that in the 806-nm case, contrary to the expectations of the cutoff extension, due to very low conversion efficiency, which did not allow for the observation of harmonics with wavelengths below 50 nm. However, the situation became completely different in the TCP case.

2.2 HHG using single-color and two-color pumps

We installed the 0.5-mm thick barium boron oxide (BBO) crystal (Ɵ = 21°) inside the vacuum chamber in the path of a focused signal beam, see Fig. 1, to generate its second harmonic. The efficiency of conversion into the ~650-nm radiation was ~27%. Two pulses overlapped both temporally and spatially in the extended plasma and allowed for a significant enhancement of odd harmonics, as well as generation of even harmonics with intensity similar to that of the odd ones.

The focusing conditions of TCP also influenced the HHG efficiency in LPP. We found the optimal focusing condition of HHG by varying the focal length of the focusing spherical lens. Maximum harmonic yield was observed when the confocal parameter of MIR beams was approximately equal to the length of the plasma plume (~5 mm).

The absolute value of the energy of harmonics generated from the silver plasma was measured using the method described in [29]. Briefly, the method is based on the measurements of calibrated signals in the ultraviolet and XUV ranges. The measured energy of the harmonics generated in the 40-nm spectral region is 0.04 μJ in the case of Ag plasma. Taking into account the 1-mJ energy of 1310 nm pulses, this gives 4 × 10⁻⁵ conversion efficiency for these harmonics.

A significant enhancement of the harmonic yield in the case of TCP was observed in all the metal plasmas used. Silver plasma allowed for generation of harmonics up to H23 in the case of 1310 nm pump, see Fig. 2(a), thick red curve. The TCP (1310 nm + 655 nm) allowed for odd and even harmonic generation of orders up to the 40s (thin blue curve). The same tendency was observed in the gold plasma, see Fig. 2(b): the harmonics generated using TCP were stronger than those in the SCP case, the harmonic cutoffs were extended up to the orders of the 40s and 50s. The enhancement factor depended on the spectral range and varied from a few units to several tens. Zn plasma demonstrated similar features, see Fig. 2(c).

Fig. 2 HHG in (a) Ag, (b) Au, and (c) Zn plasmas using 1310 nm and 1310 nm + 655 nm pumps.

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The enhancement factor in the case of TCP compared with SCP significantly varied in different ranges of harmonic spectra and also depended on the plasma species. In the case of Ag plasma, the enhancement factor for H21−H23 was in the range of 3 to 5. For higher orders, we did not see pronounced harmonic peaks in the case of SCP in the experimental conditions used in our study, whereas introduction of BBO in the path of the driving beam led to a drastic change of emission pattern in this spectral range, corresponding to at least approximately 50-times growth of harmonic emission of orders of the thirties to forties. We observed similar behavior of the enhancement factor in the case of gold plasma, though the value of this parameter was smaller compared to the case of silver plasma both in the longer- and shorter-wavelength regions. Finally, zinc plasma allowed observation of 5- to 10-fold growth of the conversion efficiency into the lower-order harmonics in the case of TCP, whereas for higher-order harmonics this factor decreased.

3. Theory

3.1 Microscopic response

Theoretical approach used in this paper is based on the modification [30, 31] of the original analytical HHG theory [32]. In crossed field geometry, when neglecting the electron magnetic drift in a laser field, the dipole moment of the particle can be written as follows:

\begin{array}{l} r (t) = i \int_{0}^{\infty} d τ {(\frac{π}{ε + i τ / 2})}^{\frac{3}{2}} 〈 d (p_{s t} - \frac{A (t - τ)}{c}), E (t - τ) 〉 d^{*} (p_{s t} - \frac{A (t)}{c}) \\ \times \exp [- i S (p_{s t}, t, τ) - \int_{0}^{t} \frac{W (t^{'})}{2} d t^{'} - \int_{0}^{t - τ} \frac{W (t^{'})}{2} d t^{'}] + c . c ., \end{array}

where c.c. denotes complex conjugation; E(t) and A(t) are the total electric field and the total vector potential of a two-color laser pulse, respectively; c is the speed of light; ε is a small regularization parameter; τ is the time of the electron’s free motion in the continuum;

p_{st} (t, τ) = \frac{1}{τ} \int_{t - τ}^{t} \frac{A (t')}{c} d t'

is the stationary point of the quasiclassical action S(p, t, τ); d(p) is the dipole matrix element corresponding to the transitions from the bound state to the continuum plane wave; W(t) is the time-dependent ionization rate. Expressions for S(p, t, τ) and W(t) used for calculations can be found in [31].

In all the calculations in this paper the two components of a laser field are orthogonal and have the following shape:

{\begin{cases} E_{x} (t) = E_{1} (t) = E_{0} \exp [- \ln (4) \frac{t^{2}}{T^{2}}] \exp (- i ω t) + c . c . \\ E_{y} (t) = E_{2} (t) = \sqrt{α} E_{0} \exp [- \ln (4) \frac{t^{2}}{T^{2}}] \exp (- i 2 ω t + i φ) + c . c . \end{cases}

where E₀ is the amplitude of the main component of the laser field (in all the calculations in this paper the fundamental intensity is 10¹⁴ W/cm² unless stated otherwise explicitly); α is the ratio of the intensities of the laser field components; ω = 0.035 atomic units corresponding to the fundamental wavelength of 1300 nm; T is the full width at half-maximum intensity, which is taken equal to 16 cycles; φ is the relative phase. Calculations were performed for various α and φ and for several ionic targets (Ag⁺, Au⁺, and Zn⁺). Note that not only the ionization potential I_p, but also the dipole matrix element d(p) is specific for every target. We have derived a general expression for the dipole matrix element corresponding to the transition from an arbitrary nonmagnetic bound state of the hydrogen-like atom to the continuum plane wave:

\begin{array}{l} d (p) = \frac{i^{l - 1} 2^{2 l + 3 / 2} (l + 1)! γ^{l + 3 / 2}}{π \sqrt{n}} \sqrt{(n - l - 1)! (n + l)! (2 l + 1)} \\ \times \nabla_{p} [p^{l} P_{l} (\frac{p_{z}}{p}) \sum_{m = 0}^{n - l - 1} \frac{{(2 γ)}^{m}}{m! (2 l + 1 + m)! (n - l - 1 - m)!} \frac{d^{m}}{d γ^{m}} \frac{γ}{{(γ^{2} + p^{2})}^{l + 2}}], \end{array}

where n is the principal quantum number; l is the azimuthal quantum number; z indicates the quantization axis; P_l is the Legendre polynomial; γ = q/n, q is the core effective charge (in all the calculations in this paper, q = 2).

Figure 3(a) shows spectra (nm-scale is used here to simplify the comparison with Fig. 2) of silver ion response obtained with the same I_p = 21.48 eV and various expression for d(p). It is clearly seen that the particular form of the dipole matrix element strongly influences the shape of the harmonic spectrum. Note also that the HHG spectrum obtained with the widely used expression for d(p) with n = 1 and l = 0 differs essentially from the experimental one, see Fig. 2(a), thin blue line. In contrast to that, the shape of the spectrum obtained with the actual quantum numbers of the outer electron in Ag⁺ (n = 4 and l = 2) is quite close to the shape of the experimental one. Thus, the proper choice of the expression for d(p) is crucial for the correct reproduction of the HHG spectrum in theoretical calculations. The provided analysis is based on the microscopic response, however, in the macroscopic section of the paper it will be shown that the shape of the spectrum after propagation through the medium in a two-color case is quite close to the shape of the microscopic one obtained with φ = π/2. Therefore, the conclusions on the effect of the matrix element on the shape of the generated harmonic spectrum are valid for the macroscopic response of the medium also. For a better agreement with the experimental results, in all further calculations in this paper the actual quantum numbers of the outer electron and the actual ionization potentials of the target ions are used (which are n = 4, l = 2, I_p = 21.48 eV for Ag⁺; n = 5, l = 2, I_p = 20.5 eV for Au⁺; n = 4, l = 0, I_p = 17.96 eV for Zn⁺) unless stated otherwise explicitly.

Fig. 3 Spectra of a silver ion response: (a) with α = 1/3, φ = π/2 and various d(p) (see legend); (b) with φ = 0 and various α (see legend); (c) with φ = π/2 and various α (see legend). Results in (b) and (c) are obtained with the actual quantum numbers.

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We have performed calculations with the step π/24 for the variable φ, for several α and for intensities of the fundamental field in the range of 5 × 10¹³ – 2 × 10¹⁴ W/cm² for all the targets. The results for various α and various intensities of the fundamental field are very similar qualitatively and show that for all the targets the addition of the second field leads to an increase of the harmonic yield. The best agreement with experimental results in terms of the width of the harmonic spectra is observed for the intensity of the fundamental field equal to 10¹⁴ W/cm². Figures 3(b) and 3(c) show the spectra ${| r_{Ω} (φ) |}^{2} = {| \int_{- \infty}^{\infty} \ddot{r} (t) \exp (i Ω t) d t |}^{2}$ of the silver ion response for this intensity and several α and φ. Figure 3(b) shows the worst case (φ = 0) and Fig. 3(c) shows the best one (φ = π/2). The harmonic spectra obtained with other values of φ are in between of these two cases. It is clear that the effect of the second field depends strongly on α and φ, and under the optimal conditions the gain can achieve almost 3 orders of magnitude in terms of harmonic intensity. Results for gold and zinc ions are qualitatively similar, but the gain is lower.

The explanation of this enhancement was given in [5, 6] in terms of the formation of a transient linear polarization of the field, selection of a short quantum path component, which has a denser electron wave packet, and higher ionization rate compared with the SCP case. However, the analysis done in [5, 6] is not comprehensive enough to answer all the questions related to the HHG enhancement in TCP. Here we provide a more detailed analysis of the classical electronic trajectories in the two-color laser field.

Due to the orthogonal polarizations of the field components, the electron's motion caused by each of the field components can be treated separately. The electron's motion in the x-direction (due to the fundamental field) in a two-color field is exactly the same as in a single-color one. The motion in the y-direction (due to the second-harmonic field) is synchronized with that in the x-direction at the instant of the electron's release from the particle: both motions start at the origin at the same time with zero velocities. Electron can recombine with the parent ion if it returns close enough to the origin simultaneously in both directions.

Figure 4(a) shows the y-coordinate of the electron at the instant of its return to the origin in the x-direction and the width of the returning wavepacket versus the time of ionization. It is clearly seen that for short enough trajectories with the time of birth exceeding ~0.07 of a laser cycle (as follows from Eq. (3), zero time coincides with the maximum of the oscillation of the fundamental field, see also Fig. 4(b)) the displacement of the electron in y-direction at the instant of recollision is smaller than the width of the returning wavepacket. It means that only these trajectories contribute to HHG in the case of a two-color field and the longer trajectories' contributions are suppressed. A similar conclusion was made in [5, 6]. However, we would like to note that Fig. 4(a) provides more information. The classical picture predicts that the cut-off harmonic is due to the trajectory launched at 0.05 of a laser cycle. It means that almost all short trajectories (0.07 is close to 0.05) can contribute efficiently to HHG under the optimal conditions in the TCP scheme (in contrast to [5, 6], where it was stated that only very short trajectories with travelling time shorter than 0.3 optical cycle contribute to HHG efficiently). That is why the HHG spectrum in the two-color case can be very wide.

Fig. 4 (a) Y-coordinate of the electron at the instant of its return to the parent ion in x-direction (solid line) and the y-width of the returning electron wavepacket (dashed line) as a function of ionization time. The calculations were made using formula proposed in [33, 34]; α = 1/3, φ = 1.4. (b) Time dependence of the absolute value of the total field and field components (see legend); α = 1/3, φ = 1.4. (c) Harmonic intensity vs. relative phase φ for several harmonics (see legend); α = 1/3.

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Figure 4(b) shows the absolute value of the total field in the single-color and the two-color cases for the same conditions as shown in Fig. 4(a). One can see that in the two-color case the absolute value of the field is larger. It means that the ionization rate in a two-color field is higher compared with the single-color case due to a very strong dependence of ionization rate on the value of the laser field in the tunneling ionization regime. This increase of the ionization rate was highlighted in [5, 6].

However, neither the quasi-linear polarization of the field between the instants of the electron’s release and return (discussed in [5, 6]), nor the increase of the ionization rate can completely explain the dramatic increase of the HHG efficiency in a two-color field. Indeed, using linearly polarized single-color field with higher intensity one might benefit from both of these factors. However, this is not the case: the HHG efficiency saturates with laser intensity in such a field. To solve this issue, another important feature of the HHG process in a two-color field should be taken into account. This feature is the redistribution of the electronic density in the continuum electron wave packet: the maximum increase of the absolute value of the field and, consequently, the maximum enhancement of the ionization rate is observed for the time interval corresponding to the birth of the short trajectories responsible for efficient HHG in the two-color case, see Fig. 4(a). Summarizing the results shown in Figs. 4(a) and 4(b) one can explain the strong enhancement of HHG yield in a two-color field as follows. There is a family of electronic trajectories that are not significantly deflected by the second harmonic in the orthogonal direction. Fortunately, these are exactly the trajectories whose weights strongly increase due to the change of the tunnel ionization dynamics in a two-color laser field. As a result, the HHG yield in the TCP case is enhanced significantly. Note also that most of the electrons are released at the maximum of the field. In the single-color case, half of them never come back to the origin and the other half produce only low harmonics in the HHG spectrum (in the SCP case the longest trajectories begin at the field maximum and return to the parent ion with almost zero velocities). As a result, only a small fraction of electrons are responsible for HHG. In the two-color case, the total field maximum is shifted so that all the electrons released near the field maximum return to the origin with high energies: the cut-off harmonic is due to the trajectory luanched at 0.05 of a laser cycle, thus very close to the instant when the field is maximal, see Fig. 4(b). Thus, in the two-color case a greater part of the released electrons contribute to HHG rather than just leave the vicinity of the parent ion. The efficiency of HHG can be therefore much higher for the TCP case compared with the SCP one even at the same ionization degree of the gas medium.

From Fig. 4(b) it is also obvious that the change of the relative phase φ would shift the time interval with maximum enhancement of the field absolute value, hence, another group of trajectories (shorter or longer ones) would predominantly contribute to HHG. Different trajectories correspond to different harmonics in the HHG spectrum, hence, the optimal relative phase should be different for different harmonics. For the odd harmonics generated due to the electron motion in the x-direction this conclusion agrees very well with the results of the numerical calculations presented in Fig. 4(c). The even harmonics are generated due to the electron motion in the y-direction, and the optimal relative phase (slightly smaller than π/2) is the same for all of them.

3.2 Macroscopic response

As shown in the previous subsection, the microscopic response is very sensitive to the relative phase φ. This phase changes during the fields' propagation in the medium. Thus, although a very pronounced intensity enhancement of the microscopic response at certain relative phases was found in the previous section, this might not eventually cause a comparable enhancement in the macroscopic signal. Therefore, a complete study of the HHG phase matching in a two-color field is necessary to compare the calculated results with the experimental ones.

Let us consider the phase matching for 1D propagation of the fundamental field and the second harmonic along z-axis. The fields inside a nonlinear medium are:

E_{1} (z, t) = E_{0} (z, t) \exp [- i ω (t - z / c) + i φ_{1} (z)] + c . c .,

E_{2} (z, t) = \sqrt{α} E_{0} (z, t) \exp [- i 2 ω (t - z / c) + i φ_{2} (z)] + c . c .,

where E₀(z,t) is the slowly-varying field envelope. By introducing

t' = t - δ t,

where δt = φ₁/ω, the driving fields given by Eqs. (5) and (6) can be written as

E_{1} (z, t') = E_{0} (z, t) \exp [- i ω (t' - z / c)] + c . c .,

E_{2} (z, t') = \sqrt{α} E_{0} (z, t) \exp [- i 2 ω (t' - z / c) + i φ_{2}' (z)] + c . c .,

where

φ_{2}' = φ_{2} - 2 φ_{1} .

The nonlinear polarization P(z,t') induced by the fields (8), (9) can be written as

P (z, t') = \int_{Ω} P_{Ω} (φ_{2}') \exp [- i Ω (t' - z / c)] d Ω + c . c .,

where P_Ω(φ₂') = Nr_Ω(φ₂')/Ω²; N is the density of the generating particles. The calculation of the microscopic response spectrum r_Ω(φ₂') for given relative phase φ₂' between the driving fields was described in the previous section. Substituting Eqs. (7) and (10) in Eq. (11) we find

P (z, t) = \int_{Ω} P_{Ω} [φ_{2} (z) - 2 φ_{1} (z)] \exp [- i Ω (t - z / c) + i \frac{Ω}{ω} φ_{1} (z)] d Ω + c . c .

The intensity of the macroscopic response is (here we assume that the refraction index for the high harmonic field is unity)

I_{Ω} = N^{2} {| \int_{0}^{L} r_{Ω} [φ_{2} (z) - 2 φ_{1} (z)] \exp [i \frac{Ω}{ω} φ_{1} (z)] d z |}^{2} .

Considering the dependence of the phases φ₂ and φ₁ on z we take into account the presence of the initial relative phase δ:

φ_{1} (z) = z ω (n_{1} - 1) / c,

φ_{2} (z, δ) = z 2 ω (n_{2} - 1) / c + δ,

where n₁ and n₂ are the refraction indices for the fundamental field and the second harmonic.

The macroscopic response after the averaging over the initial relative phase (in our experimental conditions this phase randomly changed from shot to shot) is found to be

{\bar{I}}_{Ω} = \frac{1}{π} \int_{0}^{π} d δ ​ N^{2} {| \int_{0}^{L} r_{Ω} [φ_{2} (z, δ) - 2 φ_{1} (z)] \exp [i \frac{Ω}{ω} φ_{1} (z)] d z |}^{2},

where the phases φ₂ and φ₁ are given by Eqs. (13) and (14).

Below we assume that

n_{1} - 1 = 4 (n_{2} - 1) .

This is the case for the plasma dispersion under the conditions when the free electron dispersion is described within the Drude model and the atomic or ionic dispersion is neglected; the latter assumption is valid for the ionization rate exceeding 10-20%, thus, it is definitely applicable to our experimental conditions. Moreover, Eq. (16) also describes the geometrical dispersion for a certain focusing.

The theoretical results for the behavior of the macroscopic signal versus the propagation distance in single- and two-color cases are shown in Fig. 5. One can see that the signal of the qth harmonic in the single-color case goes down to zero after propagation over the doubled coherence length $L_{q}^{c o h}$ . In contrast, the signal in the two-color case does not diminish to zero. This is a consequence of strong dependence of the microscopic response (both its amplitude and phase) on the relative phase in the integral (12). Indeed, due to this dependence only a limited range of the relative phases (and, accordingly, a limited range of z) contributes to the integral, preventing vanishing of the integral which would be the case if the variation of the phase in the exponent was 2π.

Fig. 5 The macroscopic signals of (a) H25 and (b) H35 as functions of the propagation distance in the single-color (dotted line) and two-color (solid lines) cases calculated for silver plasma. The signals for the two-color fields with I₂_ω = I_ω/3 (α = 1/3) and I₂_ω = I_ω (α = 1) are multiplied by 10⁻¹ and 10⁻², respectively.

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The calculations of the macroscopic signals as functions of the propagation length for Zn⁺ and Au⁺ in the two-color case show that their behavior is qualitatively similar for all the conditions considered in our calculations: the signal of a certain harmonic reaches a maximum at approximately 1.5-2 coherence length for this harmonic (calculated for a single-color case) and then oscillates but does not go down to zero for the considered propagation length.

In Fig. 6 we present the spectra calculated for a given propagation length for different plasma species. Figure 6(a) shows that the shape of the spectrum after propagation through the medium in a two-color case is quite close to the shape of the microscopic one obtained with φ = π/2, see Fig. 3(c). The reason for this similarity is the strong dominance of the response from the regions with the optimal phase delay in the total response of the medium. Another consequence of this fact is that the signal of a certain harmonic in a two-color case does not go down to zero for any propagation length, as it was shown before in Fig. 5. From the comparison of Figs. 2(a) and 6(a) it is obvious that a good agreement between experimental and theoretical results can be achieved only with the use of the actual quantum numbers of the outer electron. Thus, the macroscopic analysis confirms the conclusion done in the previous section about the crucial role of the expression for d(p).

Fig. 6 The macroscopic spectrum calculated for (a)-(b) silver, (c) gold, and (d) zinc plasma in the single- and two-color cases for a given propagation distance $L = L_{21}^{c o h}$ . Results in (a) are obtained with α = 1/3 and various d(p) (see legend).

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The gain due to the two-color field is presented in Fig. 7. One can see that for given α the highest gain is achieved for silver, in agreement with the experiment. For α = 1/3 the typical gain obtained in the calculations for silver is 10-30, which is in reasonable agreement with the experimental value (about 50). Comparing the results for the same medium (silver) and different intensities of the second harmonic (namely, for α = 1/3 and α = 1) one can see that the gain grows dramatically with increasing α: for α = 1 it is approximately an order of magnitude higher than for α = 1/3. This very pronounced dependence of the gain on α can explain some disagreement between the theoretical results for α = 1/3 and the experimental ones, having in mind the limited accuracy of experimental measurement of α and possible modulation of this quantity in time and space.

Fig. 7 The gain in harmonic intensity in the TCP case compared to the SCP one for different plasmas.

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Moreover, from Fig. 7 one can see that the enhancement in silver plasma is more pronounced, in agreement with the experiment, see Fig. 2(a). For gold and zinc the enhancement for different harmonics is more similar. Experimentally this is the case for gold, see Fig. 2(b), although for zinc even some decrease of the enhancement is observed, see Fig. 2(c).

4. Conclusions

In this paper we studied HHG by laser-produced silver, gold, and zinc plasma plumes using 1310 nm emission from optical parametric amplifier and its second harmonic. This approach allowed to increase significantly the high-order harmonic yield in comparison with the single-color pump. The highest increase of more than an order of magnitude is found for silver plasma; the enhancement is higher for higher-order harmonics. These experimental tendencies, as well as the gain values, are reasonably well reproduced in our calculations. The agreement was achieved due to the development of the HHG theory both in microscopic and macroscopic aspects.

We find that the microscopic HHG spectrum calculated using the widely used assumption that the ground state of the generating particle is the 1s state dramatically differs from the experimental one. In contrast to that, the shape of the spectrum obtained with the actual quantum numbers of the outer electron is quite close to that of the experimental one. Thus, the proper description of the ground state is crucial for the correct reproduction of the HHG spectrum in theoretical calculations.

The pronounced dependence of the microscopic response on the relative phase between the frequency components of a two-color field strongly influences the properties of the macroscopic signal because the relative phase changes during the propagation. Due to this dependence, only a limited range of the relative phases contributes to the phase-matching integral (12), preventing the vanishing of the harmonic macroscopic signal after propagation over the doubled coherence length (calculated for this harmonic in the single-color field).

Our theoretical studies of the microscopic response clarify the important aspect of the HHG enhancement in the two-color orthogonally polarized fields with comparable intensities of the components. In a single-color field, half of the electronic trajectories never come back to the origin and the majority of the other trajectories return with low velocities. As a result, only a small fraction of electrons are responsible for HHG. In contrast, in the two-color case (with properly chosen relative phase) the total field maximum is shifted so that all the electrons released near the field maximum return with high energies. Though deflected in the orthogonal direction by the second field, the returning electron wave packets are large enough to recombine nearly as efficiently as in the single-color case. Thus, enchanced population of the trajectories recombining with high energy in conjunction with comparable recombination efficiency leads to the increase of the HHG yield in a two-color field.

Acknowledgments

This research was partly funded by RFBR (grant N 16-02-00858 and N 16-02-00527). The studies presented in section 3.2. were supported by RSF (grant N 16-12-10279).

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Two-color high-harmonic generation in plasmas: efficiency dependence on the generating particle properties

Abstract

1. Introduction

2. Experiment

2.1 Experimental arrangements

2.2 HHG using single-color and two-color pumps

3. Theory

3.1 Microscopic response

3.2 Macroscopic response

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (7)

Equations (17)

Optics Express