Application of mid-infrared pulses for quasi-phase-matching of high-order harmonics in silver plasma

Rashid A. Ganeev; Anton Husakou; Masayuki Suzuki; Hiroto Kuroda

doi:10.1364/OE.24.003414

1. Introduction

The improvement of high-order harmonic generation (HHG) efficiency through the amendment of phase matching conditions [1] in gaseous and plasma media during propagation of ultrashort laser plasma is aimed in formation of reliable sources of coherent extreme ultraviolet (XUV) radiation for various applications in biomedicine, physical and chemical experiments, spectroscopy, etc. The quasi-phase-matching (QPM) of the interacting waves is a reliable method for the enhancement of harmonic yield [2–13]. The QPM is aimed in improving the relative phase between pump and harmonic waves. Once the envelopes of pulses of these two waves become overlapped over whole range of medium the emission of short-wavelength photons from each emitter accumulates and quadratically increases with medium length. One of methods to fulfill QPM is a modulation of active medium density. Particularly, modification of extended medium onto a group of separated gas or plasma jets allows formation of conditions when, for some group of harmonics, the relative phases of pump and harmonic waves become maintained along the whole set of media bunches. As it has been shown in multi-jet gases [4,5,8,9,13], once the waves depart from the medium their relative phase flips. After that the process of frequency conversion can be efficiently maintained in another bunch of medium. Plasma jet schemes also allow such enhancement [14,15]. In addition, HHG using the QPM in density-modulated nanoparticle composite was recently reported [16].

There are plenty of the methods introduced during studies of gas and plasma harmonics to improve HHG conversion efficiency. In the case of the harmonics generated in the laser-produced plasma plumes, those include the application of nanoparticles and clusters as the harmonic emitters, the commensurate and incommensurate two-color pump of plasmas, the application of extended plasmas, the resonance enhancement of single harmonic, and the quasi-phase-matching of harmonics. Among them the QPM and two-color pump approaches have shown to be the attractive way in improving harmonic yield in different spectral ranges. The combination of these two methods allowed the amendment of harmonics during HHG experiments in laser-produced plasmas (LPP) [17]. Those and most of other refereed studies were carried out using the conventional Ti:sapphire lasers and their second harmonics.

In the meantime, the application of longer wavelength sources may lead to further growth of harmonics in the multi-jet plasmas compared with the imperforated media. The attractiveness of using the longer-wavelength sources for the amendment of plasma HHG has been recently revealed during experiments with nanoparticles [18,19]. One can assume that tunable optical parametric amplifiers (OPA) and their second harmonics are the advanced choice for pumping the modulated LPP to maintain efficient QPM conditions.

In this paper, we show the quasi-phase-matching of a group of harmonics generated in the multi-jet plasma produced by laser ablation of bulk silver target using tunable pulses in the mid-infrared (MIR) region of 1250 − 1400 nm and their second harmonic emission. We report the observation of 12-fold growth of shorter-wavelength harmonics compared with longer wavelength ones in the eight-jet plasma and more than 60 × growth of the 39th harmonic generated in the multi-jet LPP compared with the imperforated extended plasma. We also show the numerical treatment of this effect, which includes the microscopic description of harmonic generation, as well as the macroscopic consideration of HHG including the propagation of pump and harmonic pulses through the medium.

2. Experimental conditions for HHG in silver plasma plumes using tunable 1250 − 1400 nm, 70 fs pulses

Our experimental setup consisted on 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 imperforated and perforated 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 passed linear amplifier TSA-10 (Spectra-Physics Lasers). The output characteristics from this laser were as follows: wavelength 806 nm, pulse duration 350 ps, 10 Hz pulse repetition rate, pulse energy 5 mJ. This radiation was further amplified in home-made three passed Ti:sapphire linear amplifier up to 22 mJ. Part of this radiation with pulse energy of 6 mJ was separated from a whole beam and used as a heating pulse [Fig. 1(a)] 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. The intensity of the heating pulse on the target surface was varied up to E_hp = 4 × 10⁹ W cm⁻². The plasma sizes were 5 × 0.08 mm².

Fig. 1 (a) Experimental setup for harmonic generation in LPP. (b) Multi-jet plasma formation on the surface of silver target. HP, heating picosecond pulse from Ti:sapphire laser; CL, cylindrical lens; VC, vacuum chamber; T, silver target; MSM, multi-slit mask; MJP, multi-jet plasma.

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The remaining part of amplified radiation was delayed with regard to the heating pulse in such a way that, after compression and pump of OPA, the signal pulse from parametric amplifier propagated through the LPP 40 ns from the beginning of target ablation. After propagation of compressor stage the output characteristics of a whole Ti:sapphire system were as follows: pulse energy 8 mJ, pulse duration 64 fs, central wavelength 806 nm. This radiation pumped the OPA HE-TOPAS Prime (Light Conversion). Signal and idler pulses from OPA allowed tuning along the 1170 − 1620 nm and 1580 - 2650 nm MIR ranges respectively. In our experiments we used the signal pulses. Most of experiments were carried out using the ~1-mJ, 70-fs tunable signal pulses. Pulse bandwidth of tunable pulses was 50 nm.

The signal radiation was used for HHG in LPP. The intensity of 1310-nm pulses focused by 400-mm focal length lens inside the extended plasma was 2 × 10¹⁴ W cm⁻². This driving pulse was focused into the prepared extended plasma at a distance of ~100 µm above the target surface. The plasma and harmonic emissions were analyzed by a 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 with the phosphor screen, which was imaged onto a CCD camera.

Most of experiments were carried out using the two-color pump of LPP. The reason in using double beam configuration was related with small energy of the driving pulse (~0.7 − 1.2 mJ depending on the wavelength of signal radiation). The λ⁻⁵ rule showing a decrease of harmonic yield for the longer-wavelength sources did not allow the observation of strong harmonics from the ~1300-nm pulses. Because of this we used the second-harmonic generation of signal pulse to apply the two-color pump scheme of plasma HHG. 0.5-mm thick BBO crystal (Ɵ = 21°) was installed inside the vacuum chamber on the path of focused signal pulse [Fig. 1(a)]. The conversion efficiency of 650-nm pulses was ~20%. Two pulses were 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 the similar intensity as the odd ones.

We used silver as the target for ablation. The size of the target where the ablation occurred was 5 mm. To create multi-jet plasmas we used a multi-slit mask (MSM). The size of the slits was 0.3 mm with distance between them was also 0.3 mm [Fig. 1(b)]. The MSM was installed between the focusing cylindrical lens and target such to divide the continuous 5-mm-long plasma in ~8 times 0.3 mm plasma jets with ~0.3 mm separation. We were able to increase the number of plasma jets by tilting the MSM, as shown in Fig. 1(b). Particularly, tilting the mask at 45° allowed the formation of 11 jets.

3. Experiment

Below we demonstrate the QPM of the groups of harmonics using the MIR pulses and their second harmonics (1310nm + 655nm). The principles of QPM in LPP have previously been demonstrated using the 800 nm lasers. Here we show the stronger enhancement of the group of harmonics around 39th harmonic (H39), which could be achieved in the case of MIR sources and two-color pump approach.

Figure 2 shows two harmonic spectra using the two-color pump of the silver plasma produced using the fluency of heating pulse of F = 1.0 J cm⁻². Thick red curve shows the harmonic spectrum obtained in the extended imperforated 5-mm-long plasma. A featureless spectrum of gradually decreased harmonics starting from the 15th order up to the 46th order just shows the conventional plateaulike distribution of gradually decreasing harmonics. Once we separated extended plasma on a group of jets by using the MSM placed in front of ablating target, a significant variation of harmonic distribution was observed (thin blue curve). A group of harmonics centered near the H39 was notably enhanced compared with the lower orders. The enhancement factor ~12 × was achieved for the maximally enhanced harmonics compared with the lower-order ones. Furthermore, more than 30 × growth of the 39th harmonic generated in multi-jet LPP compared with extended plasma was achieved. The order of harmonics which satisfied the condition of QPM was larger compared with the case of using 800 nm pump due to lesser dispersion of plasma in the MIR range. The maximally enhanced harmonic order (q_qpm) corresponded to the relation q_qpm = 1.1 × 10¹⁸/(l_jet × N_e), where l_jet and N_e are the length of single plasma jet in multi-jet plume (measured in mm) and electron density of plasma (measured in cm⁻³) respectively. Note that the size of each jet was ~0.3 mm.

Fig. 2 Harmonic spectra from the extended homogeneous plasma (thick red curve) and multi-jet plasma (thin blue curve) produced on the silver target using the two-color pump (1310nm + 655nm).

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The definition of maximally enhanced harmonic goes from the definition of the coherence length of this harmonic. More details could be found in [20,21]. Briefly, the plasma dispersion-induced phase mismatch for the qth harmonic is defined by the relation Δk_disp = qN_ee²λ/4πm_eε₀c², where λ is the wavelength of driving radiation, N_e, m_e and e are the density, the mass and the charge of the electron, c is the light velocity, and ε₀ is the vacuum permittivity [22]. The coherence length of this harmonic is L_coh = π /Δk≈π /Δk_disp≈4π²m_eε₀c²/qN_ee²λ. From this expression the coherence length (in mm) at the conditions of using the 1300 nm driving laser could be presented as L_coh ≈1.1 × 10¹⁸/(N_e × q_qpm). This simple formula is useful since it allows defining the electron density by knowing the coherence length, which is in fact the size of single plasma jet at the conditions of the QPM in the multi-jet structure, and the maximally enhanced harmonic order. The decrease of heating pulse fluence on the surface of ablating target should lead to a decrease of electron density, due to less amount of ablation- and tunnel-induced electrons, in the plasma plume followed by the shift of q_qpm towards the shorter-wavelength region. Similarly, one can anticipate that, for the plasma jets of different sizes, the maximally enhanced harmonics will also be tuned along the XUV spectrum. The relation q_qpm = 1.1 × 10¹⁸/(l_jet × N_e) was taken from above formula assuming the equality of the length of single jet (l_jet) and the coherence length of the corresponding harmonic, which had the highest QPM-induced enhancement.

The ablation using larger fluence of heating pulse (F = 1.2 J cm⁻²) led to some change of relative intensities of similar harmonics in the cases of imperforated and perforated LPP. The spectral distribution of harmonics generated in the extended 5-mm-long Ag plasma is shown in the upper panel of Fig. 3. Installation of MSM orthogonally to the optical axis of heating beam led to dramatic enhancement of the harmonics around the H30 (middle panel), similarly to previous case. However, one can see a difference in maximally enhanced harmonics in this and previous cases (H30 and H39). This difference was caused by different concentration of free electrons in the plasma area in these two cases (i.e. at the heating fluencies of 1.2 and 1.0 J cm⁻²). The tilting of MSM at angle 35° increased the number of jets from 8 to 10. A decrease of single jet sizes led to increase of q_qpm in accordance with the above relation (from H30 to H39). The maximally enhanced group of harmonics was tuned towards the shorter wavelength range (bottom panel). One can compare the intensities of those harmonics with the intensities of similar orders in the case of imperforated plasma plume (upper panel). The ratio of these intensities shows the enhancement factor of harmonic emission achieved in 10-jet plasma, which was approximately 40 × in the region of 34 nm.

Fig. 3 Harmonic spectra from silver plasma using two-color pump (1310 + 655nm). (upper panel) Extended imperforated 5-mm-long plasma plume. (middle panel) Eight-jet plasma produced by installation of multi-slit mask orthogonally to the optical axis of propagation of the heating pulse towards the target. (bottom panel) Tilting of MSM at 35° with regard to the axis of heating beam propagation. In that case, the plasma contained ten jets. The size of single jet was 0.25 mm.

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4. Theory

The numerical treatment of the problem under consideration included three components: microscopic description of the HHG process, propagation of the pump pulse, and the propagation of the generated harmonics. The harmonic generation and the corresponding harmonic dipole moment were modeled using the strong-field formalism developed by Lewenstein et al. [23], using the following expression for the dipole moment:

\begin{matrix} d_{H H} (t) = \frac{i e}{2 ω_{0}^{5 / 2} m_{e}} \int_{- \infty}^{t} {(\frac{π}{ε + i (t - t_{s})})}^{3 / 2} d (p_{s t} - e A (t_{s})) \times \\ \times d^{*} (p_{s t} - e A (t)) E (t_{s}) e^{\frac{- i S (t, t_{s})}{ℏ}} d t_{s} + c . c ., \end{matrix}

where

ω_{0}

is the pump carrier frequency,

d (p)

is the dipole moment,

p_{s t}

is the saddle-point canonical momentum,

A (t)

is the vector-potential of the electric field

E (t)

, and

S (t, t^{'}) = \int_{t_{s}}^{t} (I_{p} - \frac{1}{2} {[p - e A (t^{'})]}^{2} / m_{e}) d t^{'}

is the classical action with

I_{p}

being the ionization potential. No approximation of zero velocity of the ionized electron was used, which allowed for the accurate description of the spectra also well below the harmonic threshold. Analytic expressions for dipole transition moments of the hydrogen-like atoms

d (p) = i 2^{7.25} {[ℏ ω_{0} m_{e}^{2} I_{p}]}^{1.25} π^{- 1} p / {(p^{2} + α)}^{3}

with

α = 2 \sqrt{m_{e} I_{p}}

were utilized, and ADK formalism was used to describe the ionization rate:

\frac{\partial ρ}{\partial t} = (\frac{3 m_{e} e^{4}}{π ℏ^{3} n_{*}^{4.5}}) {(\frac{4}{ε n_{*}^{4}})}^{2 n_{*} - 1.5} e^{- \frac{2}{3 n_{*}^{3} ε}},

with

ε

being the electric field in the atomic units and

n_{*} = \sqrt{2 I_{p} ℏ^{2} / (m_{e} e^{4})}

.

For the propagation of the pump pulse we have numerically solved a (1 + 1)D unidirectional propagation equation without relying on the slowly-varying envelope approximation, which allowed to include both pulses in the same formalism:

\frac{\partial E (z, ω)}{\partial z} = i β (z, ω) E (z, ω) + \frac{i ω^{2}}{2 c^{2} ε_{0} β (z, ω)} P_{N L} (z, ω),

where

β (z, ω) = ω n (z, ω) / c

is the wavenumber and the

P_{N L} (z, ω)

is the nonlinear polarization, determined by

P_{N L} (z, t) = ε_{0} χ_{3} E (z, t) - e ρ d_{e} - 2 I_{p} \int_{- \infty}^{t} \frac{1}{E (z, t^{'})} \frac{\partial ρ (z, t^{'})}{\partial t} d t^{'},

where

χ_{3}

is the third-order susceptibility and

d_{e}

is the electron displacement with

{\ddot{d}}_{e} = - e E (z, t) / m_{e}

.

The defocusing of the pulse can be ignored at the considered propagation lengths, as they are shorter than the Rayleigh length. The contribution to the phase mismatch of both linear refractive index of silver plasma and the ionized electrons to the refractive index, together with corresponding dispersion terms to all orders, were included in the simulation. In addition, loss of the pump beam due to ionization as well as the Kerr nonlinearity was included in the simulation, though these two effects do not play a major role in the pulse evolution. For the details of the numerical treatment see [16]. Finally, the propagation of the high-order harmonics is governed by the (1 + 1)D propagation equation, which incorporates a high-harmonic polarization $P_{H H} = N e d_{H H}$ using harmonic source dipole moment $d_{H H}$ calculated as described above, as well as harmonic reabsorption by the silver plasma.

In Fig. 4(a), we present the map of the spectrum evolution for parameters reproducing the parameters of experiment shown in Fig. 3, as detailed in the caption. One can see that for most harmonic numbers, periodic modulation of the intensity is predicted, and the output power remains low. However, in the spectral range around the 19th harmonic the intensity steadily increases, as a result of the quasi-phase-matching achieved by the modulation of plasma density with a whole period of 0.6 mm, which contained a filled zone (0.3 mm) and empty zone (0.3 mm). Roughly 11 periods of the modulation fit in the propagation length, which would theoretically correspond to roughly two orders of magnitude enhancement of the harmonic intensity at the quasi-phase-matched number in comparison to non-quasi-phase-matched harmonic numbers. In the simulation, we achieve a factor of roughly 60. Note that both even and odd harmonics are generated due to two-color excitation. In comparison to the experiment, we predict the QPM at the harmonic order of 19 while relatively broad range of QPM around harmonic order of 30 is observed in the experiment (Fig. 3, second panel). The difference can be explained by the contribution of the plasma to the index of refraction, which rather sensitively depends on the pump field and makes accurate reproduction of the numerical conditions somewhat challenging.

Fig. 4 (a) The map of the harmonic spectrum as a function of the propagation length for plasma density modulated with period of 0.6 mm. The two-color 70-fs pump pulses are considered, with peak intensities of 2 × 10¹⁴ W cm⁻² and 0.54 × 10¹⁴ W cm⁻², centered at 1310 and 665 nm, correspondingly. The silver plasma with density of 9.4 × 10¹⁶ cm⁻³ and density modulation depth of 0.9 is assumed. (b) The map of the harmonic spectrum as a function of the propagation length for plasma density modulated with period of 0.45 mm. Other parameters are the same as in Fig. 4(a).

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To exemplify the sensitive dependence of the generated spectra on the period of the modulation, we have repeated the simulations with a modulation period of 0.45 mm, as shown in Fig. 4(b). One can see that the maximum of the spectrum has shifted to the harmonic order of 26. This result is in accordance with the relation that the quasi-phase-matched harmonic number is inversely proportional to the period, with expected harmonic order of 25. This offers a possibility to control the harmonic spectrum by changing the modulation by, e.g., tilting the mask, as explained in the experimental part of this paper.

The output spectra are presented in Fig. 5 for both single-color (1310 nm) and two-color pumps. By adding second-harmonic pump, roughly one order of magnitude increase of the harmonic amplitude in the QPM region is achieved. Note that, as one can see in Fig. 5(a), the intensity contrast is much less pronounced outside of the quasi-phase-matched region, which indicates the intricate role of the second harmonic pump in the HHG process. On the other hand, the difference of efficiencies is quite homogeneous in the quasi-phase-matched region as shown in Fig. 5(b) except for fast modulation. Also, we note that even a second-harmonic generated with a smaller efficiency of 6.5%, as presented by the green short-dashed curve in Fig. 5(b), is sufficient to notably increase the efficiency of HHG.

Fig. 5 The output spectra with (blue dashed) and without (red solid) second-harmonic pump pulse. In (a), the overview spectrum is shown, while in (b) the zoom-in around the quasi-phase-matched frequency is demonstrated. In (b), the green short-dashed curve is for the second harmonic conversion efficiency of 6.5%. Other parameters are as in Fig. 4.

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Finally, in Fig. 6 the temporal profile of the QPM harmonics in the spectral range from 17th to 23rd harmonic for the parameters considered in Fig. 4 is presented. One can see a periodic pattern in the temporal profile which results from the subcycle harmonic bursts. In general, absence of maximum at some time moment favored by the quasi-phase-matching is predicted.

Fig. 6 The output temporal profile of the harmonics in the spectral range from 17 to 23 harmonics for the parameters considered in Fig. 4.

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5. Conclusions

In conclusion, we have shown the QPM of a group of harmonics generated in the multi-jet plasma produced by laser ablation of bulk silver target using tunable pulses in the region of 1250 - 1400 nm and their second harmonic emission. We have reported the observation of 12-fold growth of shorter-wavelength harmonics compared with longer wavelength ones in the eight-jet plasma and more than 60 × growth of 39th harmonic in multi-jet LPP compared with extended plasma. We have also shown the numerical treatment of this effect, which included microscopic description of the harmonic generation, propagation of the pump pulse, and propagation of the generated harmonics. This theoretical treatment has shown the qualitative agreement with the observed peculiarities of modulated harmonic spectra from the multi-jet plasma.

Acknowledgments

R. A. Ganeev acknowledges support from the FY2015 JSPS Invitation Fellowship for Research in Japan (# L-15537). A. Husakou acknowledges support from German Research Council, project HU 1593/2-1.

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Application of mid-infrared pulses for quasi-phase-matching of high-order harmonics in silver plasma

Abstract

1. Introduction

2. Experimental conditions for HHG in silver plasma plumes using tunable 1250 − 1400 nm, 70 fs pulses

3. Experiment

4. Theory

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Equations (4)

Optics Express