1. Introduction

Harmonic generation in crystals, along with the mixing of light of different frequencies through parametric processes, results in a continuum of accessible laser wavelengths ranging from vacuum-ultraviolet (VUV) [1] over the optical and near-infrared (NIR) [2], down to mid-infrared (MIR) [3] and even the terahertz (THz) regime [4]. On the other hand, high-harmonic generation (HHG) in gases provides the possibility to produce coherent light ranging from extreme-ultraviolet (XUV) up to the soft X-ray regime [5,6]. In-between these spectral regions, a less-explored wavelength regime can be identified, roughly located between the cut-off wavelength of the KBeBO₃F₂ (KBBF) crystal (about 150 nm) and the wavelength regime defined e.g. by the ionization threshold of e.g. Xenon (about 100 nm). The harmonics produced in this regime are often referred to as below-threshold harmonics (BTH). The BTH regime is increasingly gaining attention, brought along by the recent progress in nuclear spectroscopy research [7,8]. In particular, a very special nuclear state of the Thorium-229 isotope, the ^229mTh nuclear isomeric state, is characterized by an exceptionally low transition energy [9,10,11]. While most of the known nuclear transition energies in atomic nuclei are located in energy ranges far above the photon energies of conventional laser sources, recent measurements of the ^229mTh isomeric state revealed that the energy of this transition lies within the 1σ-interval of 7.88 eV < E < 8.16 eV, corresponding to a wavelength interval of 157.34 nm > λ > 151.94 nm in the VUV-range [7]. In a more recent experiment a transition energy of 8.338 eV ± 0.024 eV was measured, corresponding to a wavelength of 148.71 nm ± 0.42 nm [8]. Another remarkable property of the nuclear isomeric state ^229mTh is its long lifetime of the order of 10³-10⁴ seconds [12,13] for charged Thorium-229 in the so-called radiative decay channel. This corresponds to a very narrow transition linewidth in the mHz range or lower [14]. A kHz level line is expected for the so-called internal conversion decay channel occurring in neutral Thorium-229. In order to excite this transition directly, laser light can prospectively be used, requiring a suitable wavelength range, as well as an extremely narrow line-width [14]. A promising laser source is an optical frequency-comb (OFC) [15,16], which consist of many equally spaced narrow-line-width comb-teeth with a spacing defined by the pulse repetition rate f_rep of the laser [16]. Using Ytterbium-based OFCs at 1030 nm wavelength, the most straightforward approach for obtaining an OFC with the required wavelength around 150 nm is 7^th harmonic generation in gases. Such an OFC can prospectively be used to directly probe the nuclear transition of Thorium-229 and immensely narrow down the uncertainties regarding the exact energy of this transition [7]. Thus, the development of a suitable 150 nm OFC source via harmonic generation in gases requires thorough knowledge of the BTH generation process.

BTH generation has been studied in multiple works [17]. Of particular interest has been the question whether the BTH generation process can be described within a perturbative framework, or whether strong-field quantum effects such as quantum electron-path interferences and effects from the ion-core potential play a role in the generation process [18]. Also, the generation of BTH of an OFC within a passive enhancement cavity has been studied [19,20] as well as the coherence properties of such intra-cavity generated extreme-UV comb [21]. However, an experimental study on the achievable harmonic generation conversion efficiencies in the BTH regime in gases has not been reported.

In this work we address this issue and measure the total output pulse energy and efficiency of the 7^th harmonic of a 1030 nm laser source in rare gases, resulting in a wavelength of about 147 nm. In addition, we perform similar measurements for the 5^th harmonic of the 1030 nm source (206 nm) and explore a cascaded harmonic generation scheme, where we generate the 6^th harmonic (172 nm) of the 1030 nm source via the 3^rd harmonic of the 2^nd harmonic (515 nm). The measurements are performed in a phase-mismatched generation scheme arising due to non-favorable dispersion characteristics in the VUV, which limits the harmonic yield. Based on the results of this work, we further provide estimates considering intra-cavity 7^th harmonic generation, revealing a sufficiently high frequency conversion efficiency to match the requirements estimated e.g. for driving nuclear Rabi transitions in a single Thorium-229 ion.

This paper is organized as follows: in section 2 we introduce and compare different schemes for VUV-light generation, which may be suitable for nuclear spectroscopy of Thorium-229. In section 3 the experimental setup and methods are shown. In section 4 we present experimental results on the conversion efficiencies and output pulse energies for 3^rd, 5^th, and 7^th harmonic generation in Argon and Krypton. We discuss the results in the context of a potential experimental setup for Thorium-229 nuclear spectroscopy. In section 5 we conclude our findings.

2. VUV frequency conversion schemes

The production of temporally coherent light at 150 nm in the VUV spectral range is not limited to direct 7^th harmonic generation of 1030 nm laser sources in gas-jets. In view of conducting laser-driven nuclear spectroscopy of Thorium-229 ions, the laser source is required to have a wavelength at around 150 nm, sufficient power of around 10 nW in a bandwidth of 500 Hz or less at the isomeric transition frequency [7]. In addition, high frequency stability over an extended time frame (several minutes or even hours) will be beneficial.

In this section we briefly discuss different methods available. The main focus lies on the practicability of these methods for producing a suitable source in order to probe the nuclear isomer transition of Thorium-229.

2.1 VUV generation in nonlinear crystals

Harmonic generation in nonlinear crystals is a highly developed field and widely used as a frequency conversion technique. Yet the majority of commonly used nonlinear crystals become absorptive in the VUV range below a certain wavelength [22,23]. At the same time, the efficiency of harmonic generation in crystals can greatly profit from a high nonlinearity as well as the non-isotropy of crystals, providing the possibility of achieving phase-matching via dispersion tuning e.g. by angle adjustment of the crystal.

A promising crystal for frequency conversion below 150 nm is KBBF [24–26]. The wavelength-dependent absorption in KBBF shows a cutoff wavelength of 147 nm, down to which it still remains transparent [25]. It has been demonstrated that wavelengths down to 149.8 nm can be produced via sum-frequency generation in a KBBF crystal [1]. There, the fundamental beam with a tunable wavelength range of 749-790 nm and its 4^th harmonic at around 187-198 nm was used to generate the 5^th harmonic via sum-frequency generation in a phase-matched manner. Relative to the 4^th harmonic input of 170 mW, the efficiency of the 5^th harmonic amounted to 2.3 × 10⁻⁵ for 149.8 nm and reached up to 4 × 10⁻⁴ for 158 nm. In this experiment, pulses of 1.6 ns with a comparatively narrow spectral bandwidth of 1.4 GHz were used as the fundamental beam. The 4^th harmonic pulse duration was 0.4 ns, resulting in a pulse duration of 0.4 ns of the 5^th harmonic [1]. It is yet to be shown that more efficient generation of VUV-femtosecond pulses is possible within a KBBF crystal.

2.2 VUV generation in hollow-core fibers

Gas-filled hollow-core fibers (HCF) offer a range of possibilities for nonlinear frequency conversion. They can be used for spectral broadening and pulse compression [27,28], white-light generation [29], as well as cascaded frequency mixing [30]. Most notably, gas-filled HCFs provide the possibility to generate VUV radiation efficiently. Travers and co-workers showed that tunable ultrashort VUV-pulses can be generated in HCFs via the resonant-dispersive-wave (RDW) mechanism [31]. Using Helium-filled fibers with large core sizes of 75-500 µm, ultrashort high-energy pulses of up to a few mJ can be coupled into a HCF. The subsequent formation of solitons and their dispersion dynamics lead to the resonant generation of broadband tunable emission, when linear- and nonlinear dispersion effects balance out [31]. Through the process of RDW-emission, efficiencies of up to 1.8% have been reached for 147 nm generation with a pump wavelength of 800 nm and for 157 nm the efficiency went up to as high as 3.2% [31].

As another approach to VUV generation in gas-filled HCFs, a highly cascaded harmonic generation scheme has been put forward. Relying on a four-wave mixing (FWM) process, this method can produce phase-matched cascaded harmonics starting from the 3^rd harmonic of a 1035 nm source up to the 9^th and each harmonic in between [30]. For the 7^th harmonic, efficiencies up to 1.2 × 10⁻⁵ were achieved [30].

Instead of using simple, gas-filled hollow capillaries, one can exploit the benefits of structured photonic-crystal HCF. In particular, gas-filled kagomé-type HCFs exhibit wide dispersion tunability and thus enable flexible tuning of phase-matching e.g. in an FWM scheme [32]. Deep-UV pulses at 266 nm have been generated using such a scheme employing low pump pulse energies in the range of 1 µJ with conversion efficiencies of up to 38% [33], with prospects to reach even shorter wavelengths down to the VUV by choosing the correct pump and seed wavelengths, as well as pressure [33]. This method could enable high-repetition rate, high-power IR sources to be converted into the VUV in a highly efficient manner, provided that the transfer of coherence from the fundamental OFC to the VUV is experimentally verified. Thus, the RDW-based VUV generation in hollow-core fibers constitutes an alternative promising technology regarding a possible application to Thorium-229 nuclear spectroscopy.

2.3 VUV-generation in gases

A more straightforward approach to generating coherent VUV light is the direct generation of below-threshold harmonics in gases. Below-threshold harmonics, such as high-order harmonics, can be generated by directly focusing intense ultrashort infrared pulses into a gas target [18].

The generation of harmonics in gas-jets can additionally be incorporated into a passive enhancement cavity [7,19,20,34–36]. When the carrier-envelope-phase and the repetition rate of a laser source are stabilized, pulses can be coherently superimposed inside a repetition-rate-matched and -locked enhancement cavity [37]. This is particularly suitable for high repetition rate sources, where the pulse energy is rather low, e.g. in the µJ level. With such a source, intra-cavity pulse energies can be enhanced by a factor of more than 100 [20], supporting sufficiently high intensities for generating below-threshold and high-order harmonics in a gas-jet at high repetition rates demanded for frequency combs reaching beyond 10 or even 100 MHz. Out-coupling of the generated harmonics can be achieved efficiently with up to 70% demonstrated out-coupling-efficiency [7].

An important requirement for generating an OFC in the VUV is the transfer of coherence from the source to the harmonic. It has been shown that the harmonic generation process in gas-jets supports coherence times beyond 1 s [21] thus providing promising characteristics for Th-229 spectroscopy [12]. However, the direct generation of the 7^th harmonic of 1030 nm, as well as below-threshold harmonics in general, is expected to lead to comparatively low efficiencies and output pulse energies due to an inevitably phase-mismatched generation process. For high-harmonic generation in gases, contributions to phase matching between fundamental and harmonic light field typically have opposite signs and thus phase-matching is achieved e.g. by tuning the gas pressure [38,39]. In contrast, this approach is not feasible for BTH generation of 1030 nm in most gases as the dispersion contributions stemming from the neutral gas and the generated plasma have the same sign, causing a phase-mismatch which limits the efficiency of this process. In the following sections, we show that despite these limitations, sufficiently efficient below-threshold harmonics can be generated in gas-jets using Argon and Krypton.

3. Experimental setup and methods

In this section we introduce the experimental setup as well as the methods we use in order to obtain the pulse energies and conversion efficiencies of the measured harmonics.

3.1 Experimental setup

Our setup mainly consists of the following components: a laser system, an attenuation unit, a vacuum system, a nozzle and gas supply, and a detector, as shown in Fig. 1. The laser system is a commercially available Yb:YLF based laser (PHAROS) with a center-wavelength of 1030 nm. It delivers pulses up to 200 µJ with a repetition rate of up to 100 kHz thus resulting in an average power of 20 W and pulse durations of around 220 fs, which we measured via autocorrelation and frequency-resolved optical gating (FROG) techniques. The laser is equipped with an internal pulse-picking option, which we use to decrease the pulse repetition rate down to 10-20 kHz. Following the laser output, the beam is guided into the attenuation unit, consisting of a λ/2 waveplate and a thin-film polarizer (TFP). The transmitted p-polarized beam is coupled into the vacuum chamber through an anti-reflection (AR) coated window. Inside the vacuum chamber the beam is focused onto the target medium – the gas-jet – by a simple spherical lens of 100 mm or 150 mm focal length. Subsequently after generating harmonics inside the gas-jet, the beams are separated by a CaF₂ prism. After separation the fundamental beam is guided out of the chamber onto a beam dump. The generated VUV-beams transmitted through the CaF₂ prism are detected by a fluorescent plate and the target beam e.g. the 7^th harmonic is guided onto and detected by a UV-photo diode.

 

Fig. 1. Experimental setup used for generation and detection of below-threshold gas harmonics. As a source a commercial 1030 nm laser system with 200 µJ pulse energy and 220 fs pulse duration is used. The light is attenuated by a waveplate and a thin-film polarizer (TFP). The gas target is generated by a glass nozzle, positioned with an xyz stage. After harmonic generation the light is separated by a CaF₂ prism, where the harmonic is aligned onto a UV-photodiode (UV-PD). The signal is read out by a current amplifier (Amp) and an oscilloscope (Osc).

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For pressure-controlled gas supply we use a pressure reducer with a continuously tunable pressure range of 0-18 bar. Besides pulse energy control, the pressure tunability provides control of the other key parameter for the experiment, the particle density of the harmonic generation medium. We use custom-built nozzles with orifice diameters of 50 µm, 100 µm, and 200 µm and a very short converging section at the nozzle tip to minimize gas flow limitations. The choice of nozzle sizes was based on previous works, such as in Ref. [20], where a 100 µm nozzle was used for intra-cavity VUV-OFC generation. Further, the aim was to investigate different nozzle sizes and their effects on the harmonic generation yield. As an example, we show measurements with a 50 µm and a 100 µm nozzle in Krypton for the 7^th harmonic. In case of lower-order harmonics we use larger nozzles with 200 µm, since phase-mismatching effects are less severe and thus the medium length can be chosen longer to achieve higher yield and efficiency. Another key element is the secondary gas supply, which is used to tune the ambient pressure in the chamber. Ambient pressure tuning provides a way to measure the re-absorption of the generated harmonic signal in the gas.

Detection of the VUV harmonics is achieved by a silicon-based calibrated UV photo diode (Opto Diode AXUV576C UV) (UV-PD) with an area of 24 mm × 24 mm. The responsivity of the diode according to the specifications amounts to 0.12 A/W for 147 nm light. The signal of the diode is measured by applying a bias voltage of 4 V and using a SRS570 low-noise current amplifier. In order to reduce stray light and noise on the diode, an appropriate shielding is applied. The calibration of the photodiode and the calculation of the pulse energies from the recorded oscilloscope traces was verified for 1030 nm [40], using the same type of photodiode and the PHAROS laser system we use in our experiment.

3.2 Experimental methods

For VUV characterization, we use a similar detection scheme as used in Ref. [40]. The background-corrected signal trace $S\left( t \right)$ is converted into a pulse energy via [40]

The amount of re-absorbed light depends on the wavelength, gas type, ambient pressure of the residual gas and the propagation length. We measure the re-absorption of 7^th, 6^th, and 5^th harmonic corresponding to wavelengths of about 147 nm, 172 nm and 206 nm in the corresponding gases, in which the harmonics are generated. The process of absorption of light in a medium after propagating a distance $z$ can be modeled by the simple exponential law $\textrm{I}\left( {\textrm{p},z} \right) = \textrm{I}\left( {0,0} \right)\,\textrm{exp}\left( { -{\kappa pz}} \right)$ [41], where $I$ is the intensity, $p$ the pressure, z the propagation length and $\kappa$ the absorption coefficient we want to measure. Measurements of the absorption coefficients require control over either the propagation length, or the pressure of the medium. By measurement of the harmonic signals at different pressure levels, the wavelength and gas-type dependent coefficient $\kappa$ can be extracted via linear regression on semi-logarithmic plot of the measured signal strengths, enabling us to estimate the signal strength before propagation through the residual gas medium.

It is important to note that re-absorption, especially in the VUV range, strongly depends on the purity of the gas [42]. In the experiments we use commercially available (Krypton) and in-house produced (Argon) gases with a specified purity of 99.999%. The re-absorption of VUV-light strongly depends on the constituents of the remaining contaminants of the gas [42]. Since we do not know the composition of these contaminants we conclude that the re-absorption measurements are only applicable to the specific batch of gases we use in the experiment. In Table 1 the measured absorption coefficients $\kappa$ are shown for the gases and the respective harmonics.

Table 1. Losses of harmonics and gas re-absorption parameters^a

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Re-absorption naturally occurs during the harmonic generation process within the generation medium – i.e., the gas jet – itself. The gas jet density, however, is expected to vary only little when the ambient chamber pressure is changed. Consequently, our re-absorption measurements do not take re-absorption inside the gas jet into account. Considering that upon installation of suitable differential pumping units, re-absorption behind the gas jet can be avoided, we are mainly interested in the possible harmonic yield directly after harmonic generation medium.

In addition to re-absorption in the residual ambient gas, we must also consider the losses which occur due to the CaF₂ prism used to separate the harmonics. The prism surface is uncoated and thus reflection losses on the surfaces can be simply calculated using Snell’s law. Absorption of the harmonics within the CaF₂ prism are accounted for using reported transmission data for CaF₂ [43].

4. Results

In this section, we present the results of 7^th harmonic, as well as 5^th harmonic and cascaded 6^th harmonic generation measurements.

4.1 7^th harmonic of 1030 nm

The conversion efficiency of a 7^th harmonic generation process in gas jets depends on the pressure of the gas applied on the nozzle — or more precisely the particle density — as well as the pulse energy and the focal geometry of the fundamental beam. In our experiment, we choose two gases as nonlinear media, Argon and Krypton at a fixed focal geometry. The beam is focused down with a lens of 100 mm focal length, resulting in an estimated maximum peak Intensity of around 2 × 10¹⁴ W/cm² (if beam-distorting effects such as plasma defocusing are neglected). Another important parameter is the length of the medium, which is determined by the nozzle orifice diameter. In this experiment we use two different nozzles with 50 µm and 100 µm diameter.

Figure 2 shows the 7^th harmonic generation results for a 100 µm nozzle in Argon and Krypton. The plots depict the measured output pulse energies and efficiencies as a function of laser input pulse energy and backing gas pressure applied on the nozzle. In both cases the measurements were conducted under similar experimental conditions and only the gas was exchanged.

 

Fig. 2. Results for the 7^th harmonic output pulse energy (a, c) and conversion efficiency (b, d) in Argon (a,b) and Krypton (c,d) with a 100 µm nozzle orifice diameter. The measurements are taken as a function of pressure and input pulse energy of the fundamental 1030 nm source. Missing datapoints are indicated in grey.

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In case of Argon (Figs. 2(a) and 2(b)), the maximum 7^th harmonic pulse energy reaches 1 nJ at the highest laser pulse energy and gas pressure. The output VUV pulse energy is not fully saturated and exhibits an increasing trend at maximum input parameters. On the other hand, the efficiency (Fig. 2(b)) reaches a maximum of 0.57 × 10⁻⁵ at an input of about 150 µJ laser pulse energy.

In Krypton we observe an efficiency maximum at lower input energies at around 110 µJ, reaching up to 1.0 × 10⁻⁵ and a maximum output pulse energy of about 1.25 nJ as shown in Fig. 2. Corresponding results for the 50 µm nozzle in Krypton are displayed in Fig. 3, where the efficiency reaches around 1.1 × 10⁻⁵ and the output pulse energy 1.47 nJ.

 

Fig. 3. Results for the 7^th harmonic output pulse energy and conversion efficiency in Krypton for a smaller nozzle of 50 µm diameter. (a) shows the experimental data for the output 7^th harmonic pulse energy and (b) the efficiency derived from the output pulse energy. Missing datapoints are indicated in grey.

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While the maximum UV pulse energy in Krypton is only slightly higher than in Argon, the maximum efficiency is roughly twice as high. Since it appears that the VUV output energy is still not in full saturation in the case of Argon (Fig. 2(a)), the maximum pulse energy output could potentially reach a similar value as the one in Krypton. On the other hand, the maximum conversion efficiency already reaches its peak and would only decrease with higher input IR energies.

Harmonic generation in both Argon and Krypton exhibit a similar qualitative behavior in terms of harmonic efficiency as a function of input pulse energy (intensity) and gas pressure. At lower intensities we observe an increase in efficiency, which saturates at a certain intensity and decreases at higher intensities. Similarly, when tuning the pressure (particle density) from lower to higher values, we first observe an increase in harmonic efficiency until the signal saturates. For different nozzle sizes in the Krypton measurements, we observe a saturation of the efficiency which is shifted to higher pressures for smaller nozzle sizes as visible in Fig. 3.

In the case of HHG in gases, maxima for photon flux and efficiency are reached when the phase-matching condition is fulfilled. Phase-matching can be achieved via pressure tuning until an intensity-dependent phase matching pressure is reached [38,39]. In contrast, for BTH generation in gases, phase-matching is constantly deteriorating with increasing intensity and pressure. However, our results indicate that the nonlinear signal growth can easily dominate the negative effect of an increasing phase-mismatch in the low pressure/intensity regime. A signal maximum can likely be attributed to multiple effects such as signal depletion at high intensity when a large ionization fraction is reached, plasma defocusing as well as to limitations of the macroscopic signal buildup arising due to re-absorption of the generated harmonics at high gas density.

To illustrate the severity of phase-mismatch in the 7^th harmonic generation process we can estimate the coherence length ${L_{\textrm{coh}}}$ of the harmonic generation process at the efficiency maxima in our measurements in Fig. 2. The coherence length is defined by the length where the relative phases of the fields shift by π, i.e. ${L_{\textrm{coh}}} = \pi /{\Delta }k$. The wave-vector mismatch ${\Delta }k$ between fundamental and harmonic is comprised of several contributions, namely the Gouy phase-mismatch, neutral gas dispersion and plasma dispersion, as well as the dipole phase [38]. Disregarding the dipole phase we calculate the coherence lengths using the Sellmeier equations with the data for Argon and Krypton provided in Ref. [44,45] using the Perelomov-Popov-Terent’ev (PPT) theory [46] for determining ionization, taking into account linear laser pulse propagation. For Argon ${L_{\textrm{coh}}}$ is roughly 7.8 µm at the efficiency maximum (Fig. 2(b) at 150 µJ and 18 bar) on the optical axis. The ionization ratio at the peak of the pulse amounts to roughly 6%. For Krypton (Fig. 2(d) at 100 µJ and 11 bar) we calculate a similar value of 9 µm, with a similar ionization of around 6.5%. Thus, the estimated coherence lengths at the peak efficiencies are about an order of magnitude shorter than the medium length. This shows that although the generation process is highly phase-mismatched, increased number of atoms in the gas as well as more intensity can still counteract the effects of phase-mismatch up to certain input pressures and pulse energies.

Generation of 7^th harmonic in gases is not only limited by phase-mismatching effects, but also by the single-atom-response of the medium. In order to reach higher VUV conversion efficiencies, lower-order processes can be used to generate light at the desired wavelength. Techniques for shifting the wavelength such as optical serrodyne frequency-shifting [47] could potentially be employed to shift the fundamental wavelength such that, for example, the 6^th harmonic yields a wavelength around 150 nm. The 6^th harmonic can then be generated in a cascaded scheme, where the 3^rd harmonic is generated from the 2^nd harmonic of the fundamental. In the following section, we measure the output pulse energy and efficiency of lower order harmonics, namely the 5^th harmonic resulting in 206 nm wavelength, as well as the 3^rd harmonic of the 2^nd harmonic, which results in 172 nm. We show that the conversion efficiencies are significantly larger than in the case of the 7^th harmonic.

4.2 5^th harmonic and cascaded 6^th harmonic of 1030 nm

Following our 7^th harmonic generation experiments we additionally measure the 5^th harmonic and cascaded 6^th harmonic output pulse energies and conversion efficiencies. In order to generate the 6^th harmonic, we use the 2^nd harmonic output of the PHAROS laser system at 515 nm and generate the 3^rd harmonic in Argon. In both cases we use a nozzle of 200 µm orifice diameter.

In Fig. 4 the measured data is shown for the 5^th harmonic. Again, the 5^th harmonic pulse energy (Fig. 4(a)) and efficiency (Fig. 4(b)) are plotted as a function of nozzle backing pressure and input IR pulse energy. The experimental conditions are similar to the 7^th harmonic experiments, only the nozzle orifice diameter is changed to 200 µm, as well as the prism re-adjusted for better beam separation. For the 5^th harmonic of 1030 nm we measure a maximum pulse energy of 8.8 nJ and a maximum conversion efficiency of 0.78 × 10⁻⁴. Similar to the 7^th harmonic, the efficiency maximum appears at an input pulse energy of 100-110 µJ.

 

Fig. 4. Experimental results for the 5^th harmonic (206 nm) output pulse energy (a) and corresponding conversion efficiency (b) in Krypton for a nozzle with 200 µm orifice diameter.

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In addition, we observe a second peak in efficiency at lower pressures at roughly 10 bar and slightly lower input energy (Fig. 4(b)).

In the cascaded 6^th harmonic experiment we chose Argon as the nonlinear medium and a nozzle orifice diameter of 200 µm. The results are displayed in Fig. 5. Here, the second harmonic output of the laser centered at 515 nm is guided onto the target with a maximum pulse energy of 100 µJ at the target. We measure a pulse length of 178 fs at the output of the laser. In this setup we choose a looser focal geometry of 150 mm focal length. Considering the given beam parameters from the laser system we calculate a peak intensity up to 1.3 × 10¹⁴ W/cm² in the focus in vacuum.

 

Fig. 5. Experimental results for the cascaded 6^th harmonic (172 nm) output pulse energy (a) and corresponding conversion efficiency (b) in Argon for a nozzle with 200 µm orifice diameter.

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Here we measure a maximum of 280 nJ output pulse energy and a maximum efficiency of 0.3%. The efficiency is directly calculated in relation to the 2^nd harmonic input pulse energy. Considering 2^nd harmonic conversion efficiencies of 1030 nm sources reaching up to 70% [48], a total cascaded 6^th harmonic efficiency of about 0.2% can be considered realistic.

The general behavior of 5^th and 6^th harmonic signal with increasing intensity and pressure reflects similar trends as observed for the 7^th harmonic. In the 5^th harmonic (Fig. 4) however, we observe a double efficiency maximum with increasing gas pressure. Effects like pressure-dependent phase-matching and/or nonlinear laser pulse reshaping at high pressures can lead to multi-peak structures for HHG as reported in earlier works [49]. A detailed analysis of this effects for BTH goes beyond the scope of this work.

In a perturbative picture, the generation of 7^th, 5^th and cascaded 6^th harmonic are processes that depend on the nonlinear 7^th, 5^th and 3^rd order responses of the medium, respectively. Hence, we expect a strong increase in signal strength when reducing the nonlinear order. This is reflected by our measurements: the output pulse energies increase by an order of magnitude from the 7^th to 5^th and from the 5^th to the 3^rd harmonic of the 2^nd harmonic. The same conclusion can be drawn for the efficiency. We see an increase of more than an order of magnitude for each step down in the harmonic order. Table 2 summarizes the input parameters used, as well as maximum pulse energies and conversion efficiencies for each measurement.

Table 2. Input parameters and key results^a

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4.3 Discussion of results regarding a 150 nm OFC for nuclear spectroscopy of Thorium-229

In view of the requirements for nuclear spectroscopy of Thorium-229, we can follow Ref. [7] and estimate if the measured 7^th harmonic efficiency can be deemed sufficient to build a laser for driving the given nuclear isomeric transition. We base our estimates on demonstrated approaches for VUV comb production considering intra-cavity harmonic generation. Assuming an OFC source at 1030 nm central wavelength, 200 fs pulse duration, 65 MHz repetition rate, a laser average power of 84 W, as reported e.g. in Ref. [50], and a passive enhancement cavity with a realistic enhancement factor of 100 [18], an intra-cavity peak intensity > 1 × 10¹⁴ W/cm² can be reached. Taking into account our measurements of the conversion efficiency of 7^th harmonic generation reaching a maximum of 1.1 × 10⁻⁵, we can conservatively assume an efficiency of 0.1 × 10⁻⁵, considering a reduced efficiency due to intra-cavity intensity clamping, which typically limits the intensity which can be reached inside an intra-cavity gas jet compared to single-pass experiments. For perturbative harmonic generation, the Fourier-transform-limited bandwidth of the generated 7^th harmonic pulses can be estimated to yield around 6 THz, considering a VUV pulse duration of around 75 fs. Considering a non-zero dipole phase for below-threshold harmonic generation in gases [51], the generated harmonic is expected to be chirped, leading to a broader bandwidth. For simplicity reasons, we therefore take a bandwidths safety factor of two into account, leading to a bandwidth of 12 THz. Dividing the bandwidth by the repetition rate, which defines the spacing of two comb-teeth in the spectrum, yields a number of comb-lines of almost 2 × 10⁵. Further assuming beam transport losses of 75% including out-coupling losses [52], we end up with an available power per comb tooth of about 10 nW, which is equivalent to the power considered in Ref. [7], providing multiple options for direct laser-driven Th-229 spectroscopy. Consequently, we can conclude that the requirements specified in Ref. [7] for driving the nuclear transition of a single Th-229 ion with a 150 nm VUV-frequency comb can be fulfilled. Furthermore, we demonstrated that by exploiting lower-order harmonic generation, such as the 5^th or 3^rd harmonic, the conversion efficiency can be greatly enhanced, opening opportunities for alternative frequency conversion routes for efficient 150 nm production.

5. Conclusion

In this work we study below-threshold harmonic generation in Argon and Krypton using 1030 nm pulses as the fundamental wavelength. We measure the output pulse energies as well as conversion efficiencies of the 7^th (147 nm), 5^th (206 nm), as well as the 3^rd harmonic of the 2^nd harmonic (172 nm). We show that within the parameter range of our measurements, we can reach conversion efficiencies up to 1.1 × 10⁻⁵ for the 7^th harmonic in Krypton. We estimate that this value is sufficient for nuclear spectroscopy experiments of charged Thorium-229. Furthermore, we confirm that lower order harmonic generation in gases, such as the 3^rd or 5^th harmonic, results in orders-of-magnitude higher conversion efficiencies. The phase-mismatched generation conditions are identified as the main limiting factor for efficient generation of BTH in the here discussed wavelength regime.