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Abstract
We present an efficient tunable optical parametric generator (OPG) with its linewidth close to the Fourier transform limit by injection seeding a tunable diode laser. Benefitting from high-peak-power sub-nanosecond (426 ps) laser pumping and a high-gain MgO:PPLN (PPMgLN) crystal, the OPG produced signal peak power up to 0.343 MW at 1638 nm and the total conversion efficiency reached 47.9% at 1-kHz pulse repetition rate. Considering the linewidth limit of short signal pulses (∼ 350 ps), a tunable seeder with the linewidth at hundred-MHz level was applicable. The achieved OPG signal tuning range was 1510–1638 nm with linewidth at GHz level, which is two orders of magnitude narrower than the unseeded OPG. Injection seeding a non-resonant OPG device does not introduce extra cavity feedback electronics that are essential for an optical parametric oscillator (OPO), greatly improving robustness and reducing cost. It is believed such a compact, tunable and costless PPMgLN OPG with high peak power, high repetition rate and relatively narrow linewidth has great significance in lidar, spectroscopy, etc.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical parametric generators (OPGs) play important roles in tunable near- to mid-infrared laser sources, which have good prospects in the applications of material processing, spectroscopy, lidar, etc. Compared with an optical parametric oscillator (OPO), an OPG requires no cavity and possesses the merits of simplicity, flexibility, broad tunability and robustness [1]. High conversion efficiency can be guaranteed in an OPG device with high-gain nonlinear materials like quasi-phase-matched (QPM) periodically poled KTiOPO4 (PPKTP), LiNbO3 (PPLN) and LiTaO3 (PPLT) [2–5]. However, a free-running OPG usually has a broad linewidth since there is no cavity or frequency-selective element and all the signal-idler pairs that close to the phase-matching (PM) condition would be possess high gain, which is an annoying problem to enhance the signal-to-noise ratio (SNR) in detection for certain cases. Compared with the OPO, spectral filtering with gratings or Fabry-Perot (F-P) etalons for OPG is impractical because of the non-resonant structure, and the only effective approach to achieve a narrow-linewidth OPG is injection seeding. Another advantage of injection seeding the OPG over OPO is it’s needless to lock the cavity length as OPG is single-pass and non-resonant.
There are two types of seeding an OPG from the view of seeder. One adopts a pulsed seeder with relatively low pulse energy, which is also called an optical parametric amplifier (OPA). The seeder can be generated from a tunable narrow-linewidth laser oscillator or OPO with complex spectral controlling systems [6,7]. Otherwise, the seeding pulses should be spectrally filtered from an OPG by gratings, etalons etc., inducing unavoidable high optical loss and substantial power reduction [8,9]. The injected seeding pulses should be synchronized with the pump pulses for effective nonlinear interaction, thus the seeder and amplifier sharing an identical strong pump source is a simple and common method. Although injection seeding a narrow-linewidth pulsed laser is complicated, it is a necessary approach to achieve high-energy output pulses at designed wavelengths [10].
Another approach is to directly inject a continuous-wave (CW) seeder into the OPG, such as narrow-linewidth diode or solid-state lasers [11–13], which possess the merits of simple and compact configuration, convenient tuning and robustness. The required seeding power was from microwatt to milliwatt, but once seeded the OPG linewidth can be narrowed by orders of magnitude and even reaches the Fourier transform limit depending on pulse duration. However, CW seeding is currently applied for OPGs mostly with high-repetition-rate picosecond or nanosecond pump lasers and nonlinear crystals like PPLN, orientation-patterned GaAs (OP-GaAs), etc., where the output single pulse energy and peak power were limited to the levels of 10 µJ and a few kW, respectively [11–13].
In this paper, a widely tunable injection-seeded MgO:PPLN (PPMgLN) OPG pumped by a sub-nanosecond (426 ps) microchip Nd:YAG laser is presented, which boosted the peak power to the level of sub-MW at 1-kHz repetition rate. Based on a 50-mm PPMgLN crystal with the grating period of 30 µm, the OPG was continuously tunable in the ranges of 1509–1685 nm (signal) and 3610–2888 nm (idler) through changing the working temperature. By seeding a CW semiconductor laser, the OPG signal linewidth was reduced from hundreds of GHz to less than 5 GHz over the seeder tuning range of 1510–1638 nm, which was approaching the Fourier transform limit of pulse duration of around 350 ps. The narrow-linewidth output power reached 134.1 mW (119.2 mW signal at 1638 nm and 14.9 mW idler at 3036 nm), corresponding to the overall conversion efficiency of 47.9%. The highest peak power at 1638 nm was 0.343 MW, at least two orders of magnitude higher than injection-seeded high-repetition-rate OPG devices reported before.
2. Experimental setup
The experimental schematic of the injection-seeded narrow-linewidth PPMgLN OPG is shown in Fig. 1. The pump laser was a homemade passively Q-switched microchip Nd:YAG laser composed of a fiber-coupled laser diode (LD) at 808 nm, a 1:1 imaging system, and an Nd:YAG/Cr:YAG composite crystal. The core diameter of the fiber was 400 µm and the numerical aperture (NA) was 0.22. The maximum output power of the LD (BWT Ltd.) was 50 W operating at the pulse repetition rate up to 1 kHz. The composite crystal was diffusion bounded with a 4-mm-thick 1.1 at%-doped Nd:YAG and a 2-mm-thick Cr:YAG (initial transmission T0 = 30%). The entrance face was coated for high transmission (HT) at 808 nm and high reflection (HR) at 1064 nm, while the exit face had partial transmission (PT) of 50% at 1064 nm. Benefitting from a 6-mm short cavity, the full-wave-half-maximum (FWHM) pulse duration was 426 ps when the LD pulse width was 180 µs at the peak power of 30 W. The single pulse energy at vertical polarization (orthogonal to the horizontal plane, shown as double arrows in Fig. 1) behind a Brewster plate (BP) was 0.28 mJ at 1 kHz, or an average power of 280 mW, and the corresponding peak power was 0.66 MW. It can be concluded that the separation between two adjacent longitudinal modes was around 50 pm (14 GHz), thus the laser linewidth of 80–90 pm (21–24 GHz) measured with a Yokogawa AQ6370D optical spectrum analyzer (OSA) should be working close to single longitudinal mode.
Fig. 1. Schematic diagram of the injection-seeded narrow-linewidth sub-nanosecond PPMgLN OPG. One of the insets is the pump beam profile at the center of the PPMgLN crystal along the pump direction, and the other is the seeder power versus wavelength.
The PPMgLN crystal was 1-mm-thick and 50-mm-long to ensure sufficient gain for single-pass nonlinear interaction. Both crystal faces were antireflection (AR) coated for the pump (1064 nm), signal (1350–1800nm) and idler (2400–5000 nm) waves. The crystal was placed in a heating oven which could precisely control the working condition from room temperature to 350 °C with an accuracy of ± 0.1 °C for wavelength tuning. A convex lens (L1, f = 100 mm) could focus the beam diameter (1/e2 of axis intensity) to around 600 µm at the center of the nonlinear crystal along the pump direction. The 1064-nm pump beam had a Gaussian profile, and the beam diameter was less than 700 µm over the entire length of the PPMgLN crystal, measured with a knife edge. The linewidth of the PPMgLN OPG was controlled by seeding a CW Agilent 81642A tunable laser with the tuning range of 1510–1638 nm (strictly speaking, 1510–1640 nm, but the power around 1640 nm was very low as shown by the inset of Fig. 1) and linewidth of around 50 MHz. The seeder was coupled by a 1.0-meter-long single-mode polarization-maintaining fiber connected to a collimator. The vertical polarization component that was effective in the eee-type PM was maximized by rotating the collimator. The collimated seeding beam was around 1 mm in diameter and coupled into the crystal by a dichromic mirror M1, which was coated for HT at 1064 nm and HR at 1500–1700nm at 45° incident angle. Considering the pump beam diameter was around 600 µm, the effective seeding power (within the pump beam size) should be over 50% of the total seeding power. The output signal and idler waves were separated by two filters (M2: HR for pump and signal, HT for idler; M3: HR for pump and HT for signal) for characterization.
3. Experimental results and discussions
The tuning characteristics of the free-running (unseeded) PPMgLN OPG is shown in Fig. 2. The tuning ranges were 1509–1685 nm for signal waves and 3610–2888 nm for idler waves, respectively, when the crystal temperature was controlled from 30 °C to 300 °C. Such a signal tuning range could cover the entire tuning range of the seeder, and the range is flexibly extensible by using a multi-period crystal [14]. The predicted tuning curve given in Fig. 2 was calculated from the QPM condition by substituting the temperature-dependent Sellmeier equations of lithium niobite [15]. Generally, the measured results agreed well with the prediction but deviation gradually appeared with the increase of temperature. On one hand, the applicability of the Sellmeier equations might become poor at high temperatures beyond the validated range (≤ 250 °C); On the other hand, there could be inconsistency between the actual and set temperature at high temperatures because of natural heat dissipation.
Fig. 2. Temperature tuning characteristics of the PPMgLN OPG pumped at 1064 nm with the crystal grating period of 30 µm. The experimental signal wavelengths were measured with a Yokogawa AQ6370D OSA and the idler wavelengths were calculated by the energy conservation law.
The output power characteristics of the seeded PPMgLN OPG are shown in Fig. 3. Figure 3(a) depicts the variation of signal power with signal wavelength within the seeder tuning range, when the average pump power was at maximum of 280 mW and the injected seeding power was 1 mW. The results without seeding were also given for comparison. It was strange that the signal power had obvious variation at different signal wavelengths, but the variation was strictly repeatable either with or without seeding. Based on the possible variable factors during wavelength tuning, such a variation should originate from temperature induced mechanical deformation which could change the optical loss when passing through the crystal. Figure 3(b) is the output signal power versus injected seeding power at the same wavelengths as those in Fig. 3(a). From Fig. 3(a) and 3(b) it can be found that the relation between the seeding power and OPG output power is inconspicuous. However, injection seeding would significantly decrease the threshold, which is defined as the pump power required to produce a detectable OPG output power at the level of 1 mW. For example, the measured OPG threshold at the signal wavelength of 1638 nm was less than 40 mW with a seeding power of 1 mW, while it was over 100 mW without seeding. Figure 3(c) shows that a distinct narrow-linewidth signal spectrum could be observed by the OSA (Yokogawa AQ6370D) if pumped at 40 mW and seeded at 1 mW, rather than pure noise without seeding. The seeded in-out characteristics and the related conversion efficiencies are given in Fig. 3(d), at the condition of signal/idler wavelengths of 1638/3036 nm and the PPMgLN crystal working at 270 °C. The maximum total output power was 134.1 mW including the signal power of 119.2 mW and idler power of 14.9 mW, corresponding to an overall conversion efficiency of 47.9%. The powers of the signal and idler waves deviated from the wavelength ratio because the idler divergence was large and only part of the idler power was coupled out. Since the output power and conversion efficiency were still rapidly growing with the increase of pump power, there was still much space to enhance the output power by a stronger pump source. A PPMgLN crystal with larger aperture would also help by improving the idler coupling efficiency.
Fig. 3. Output power characteristics of the PPMgLN OPG. (a) Signal power at different wavelengths with and without seeding. (b) Signal power versus injected seeding power at different wavelengths. (c) Seeded (1-mW seeding power) and unseeded OPG spectra when the pump power was 40 mW. (d) Output power and conversion efficiency versus pump power at signal/idler wavelengths of 1638/3036 nm, when the seeding power was 1 mW.
An Agilent MSO9254A digital oscilloscope (2.5 GHz bandwidth, 20 GSa/s, and 124.8 ps rise time) was used to record the temporal profiles of the sub-nanosecond pulses detected by an ultrafast InGaAs detector (Max-Ray Photonics PD 12D, 12 GHz bandwidth, 18 ps rise time, and 900–1650 nm wavelength range). The measured full-wave-half-maximum (FWHM) pulse width follows the basic law
where τd,rise is the detector rise time (18 ps) and τosc,rise is the oscilloscope rise time (124.8 ps). The real pulse width τreal should be slightly narrower than the measured value due to the response time of oscilloscope and detector, but the effect was quite limited (around 20 ps inaccuracy for 350 ps pulses) and the measured value could reflect the real pulse duration. Pumped at 280 mW without injection seeding, the maximum output powers were 117.4 and 14.3 mW at signal and idler wavelengths of 1638 nm and 3036 nm, respectively, with a typical pulse width of 344 ps. Once seeded, the pulses get slightly wider and the output power was increased slightly. When the injected seeding power was 1 mW, the signal pulse duration was 348 ps, and the peak powers reached 0.343 MW and 0.043 MW for signal and idler waves considering they had identical pulse widths. The peak powers were more than an order of magnitude higher than those of the injection-seeded high-repetition-rate ps/ns OPGs ever reported [11–13]. The temporal widening effect with the increase of seeding power was related to the rising and trailing edges, where injection seeding could fully exploit the weak part of the pump pulse due to a much lower pump threshold. It is interesting that sometimes we could observe a trailing pulse when the injected seeding power was quite high (e.g., 4 mW), as shown by the dashed line in Fig. 4(a). This was resulted from an unstable 1064-nm pump laser pulse generated occasionally. Only when the injected seeding power was high enough that the weak trailing pump pulse could excite a detectable signal pulse. Figure 4(b) shows the temporal profiles of the residual pump pulse after the OPG process when the pump power was 280 mW and the seeding power was 1 mW. The absolute output voltages from the detector were given directly as the arrangements were all the same except inserting the PPMgLN crystal. The crystal temperatures of 60 °C and 270 °C corresponded to the signal wavelengths of 1527 nm and 1638 nm, and conversion efficiencies of 26.2% and 42.6%, respectively. Significant pump depletion could be observed with the increase of conversion efficiency and the residual pump became obviously weaker.Fig. 4. Temporal pulse profiles of the pump and signal waves. (a) Signal pulse waveforms with different injected seeding powers. (b) Original and depleted pump pulses at different signal wavelengths (1527/1638 nm) and conversion efficiencies (26.2%/42.6%). The incident pump power was 280 mW for all the measurements.
The OPG output spectrum was monitored by a Yokogawa AQ6370D OSA (600-1700nm, 0.02 nm resolution). Firstly, we investigated the variation of output spectrum with different injected seeding power, as shown in Fig. 5(a). The linewidth (FWHM) was around 1.2 nm for the free-running PPMgLN OPG centering at 1518 nm, and the spectrum generally followed a Gaussian profile but the intensity fluctuation was significant at different frequencies. Once seeded at a frequency that was close enough to the OPG peak spectrum, the seeding frequency started to establish advantage in the competition benefitting from a much higher photon degeneracy than the other frequencies originating from noise. In this way, more pump power was consumed by the dominating wavelength and the gain for the side band was depressed. However, pumping at MW-level peak power and using PPMgLN crystals with high nonlinear gain, there was a minimum requirement for seeding power to dominate in the competition. The spectral narrowing effect could be seen with the increase of seeding power, as shown in Fig. 5(a). Note the seeding powers were not the absolute values that contributed in the nonlinear process as the seeding beam size didn’t match the pump beam exactly. When the seeding power was 1 mW, or an estimated seeding intensity around 0.125 W/cm2 considering the effective pump beam diameter of 600 µm, a narrow-linewidth spectrum could be realized. The output spectrum for the seeded OPG at different seeding wavelengths was also studied while the seeding power was fixed at 1 mW. In this case the free-running OPG was tuned to operate at the signal wavelength of 1577 nm by controlling the crystal temperature, and the linewidth was 4.0 nm. The reason that the signal linewidth gets wider at longer wavelengths can be simulated based on the theory of PM tolerance and is reflected by the increased slope (dλ/dT) of the temperature tuning curves in Fig. 2. As shown in Fig. 5(b), only when the seeding wavelength is close enough to the peak of the free-running signal spectrum (1577 nm) can the seeded OPG operate at the narrow-linewidth mode. If the seeding wavelength deviated too much within the gain bandwidth, although a prominent peak emerged at the corresponding wavelength (e.g., at 1575 nm and 1581 nm), the profile of the OPG output spectrum was disordered and unstable.
Fig. 5. Seeded OPG spectra at different seeding conditions. (a) Seeding wavelength fixed at 1518 nm with tunable seeding power from 0 to 1 mW; (b) Seeding power fixed at 1 mW with tunable wavelength from 1571 nm to 1581 nm.
Basic theoretical explanations to the above experimental results can be found from the solutions of coupling wave equations of the OPG process under a collinear plane-wave interaction model [16]. The OPG can be regarded to originate from the interaction between the input pump and noise signal/idler waves. Given a seeder exists, then the initial noise signal is substituted with the injected seeding beam. The first example considered a quite low pumping intensity of 30 MW/cm2, the PPMgLN crystal worked at 197 °C corresponding to the signal wavelength of 1571.12 nm, the initial noise signal was at the level of 10−5 W/cm2, and if a seeder was applied the injected seeding intensity was 0.125 W/cm2 (equivalently, seeding power around 1 mW). As shown in Fig. 6(a), the seeded OPG would generate signal waves with profiles quite different to the unseeded case. The output power of the unseeded OPG reflected by the area integration enclosed by the spectrum curve and the horizontal axis, is much lower than that of the seeded case. That is the reason why injection seeding could significantly decrease the so-called OPG threshold in experiment. In addition, the seeded OPG has a narrow signal linewidth with SNR of over two orders of magnitude, while the unseeded OPG signal spectrum is much broader and represented as the small-signal nonlinear gain profile. It should be noted that the calculated gain bandwidth is narrower than the experimental results in Fig. 5(a), because the simulation ignored noncollinear interactions which actually play remarkable roles in non-resonant nonlinear devices like OPG. Noncollinear PM not only expands the signal bandwidth, but also raises the required seeding power to establish advantage at the seeding wavelength. The second example increased the pump intensity to 100 MW/cm2 when the seeding intensity was 0.5 W/cm2 (seeding power around 4 mW). We compared the output spectrum of three cases: unseeded, seeding at 1571.12 nm (at peak gain) and seeding at 1570.82 nm (far from peak gain), as shown in Fig. 6(b). The efficiency of the unseeded OPG can be comparable to the seeded one, but the output power spreads over a wide wavelength range. If the seeding wavelength is close to the peak of the gain spectrum at 1571.12 nm, the seeded photons would quickly consume the pump power and dominate the nonlinear process compared with the noise photons. Therefore, the conversion at other signal wavelengths is greatly depressed, leading to a narrow spectrum with SNR over five orders of magnitude. However, if the seeding wavelength is tuned away, e.g., seeding at 1570.82 nm, the gain here is too low to enable high conversion efficiency, then adequate pump power is left to amplify the noise phonons around 1571.12 nm and the it fails to produce a pure narrow spectrum, corresponding to certain experimental cases in Fig. 5(b).
Fig. 6. Simulation of the unseeded and seeded OPG signal spectra. (a) Low pump intensity at 30 MW/cm2 and seeding at 0.125 W/cm2; (b) High pump intensity at 100 MW/cm2 and seeding at 0.5 W/cm2.
Figure 7 compares the OPG signal spectra with and without injection seeding in a more intuitive form. Note the spectra in Fig. 7(a) can’t reflect the true SNR because the OPG signal was directly scattered into a fiber-coupled OSA without any collecting optics. Evidently, seeding a CW narrow-linewidth laser greatly compressed the OPG signal linewidth and enhanced the SNR, which is of great importance in applications of lidar, spectroscopy, etc. The narrow-linewidth signal output could cover the whole tuning range of the seeder. Directly given by the OSA as shown in Fig. 7(b), the signal linewidth varied from 0.015 nm to 0.036 nm at different wavelengths, that is, approximately compressed by 43 (1510 nm) to 343 (1638 nm) times compared with the free-running mode. The linewidth fluctuation could be caused by random environmental change or mechanical instability, as we believe mode hopping of the pump laser has little effects on a single-pass OPG owning a gain bandwidth of hundreds of GHz and a much narrower seeding linewidth than the signal. Taking the output wavelength at 1626 nm for example, the linewidth of 0.015 nm (1.72 GHz) was already below the nominal resolution of the OSA and the real value could be even narrower. From the other side of view, the Fourier transform limit of the signal linewidth was around 1.27 GHz calculated by Δτ × Δν ≥ 0.441 [17], where Δτ and Δν were the pulse duration and linewidth of a pulsed laser. Therefore, the measured results were approaching the linewidth limit of sub-nanosecond laser pulses. Such a tunable parametric source with GHz linewidth does have sufficient resolution for non-gas-phase spectroscopy applications.
Fig. 7. Comparison of the seeded and unseeded OPG signal spectral characteristics. (a) Typical OPG signal spectrum profiles at 1510 nm, 1551 nm, 1591 nm and 1638 nm; (b) OPG linewidth directly measured by the OSA.
Here we would like to discuss further on the difference of OPG and singly resonant OPO devices (provided the pump pulse is temporally long enough to pump OPO, e.g., ∼10 ns or longer), concentrating not on efficiency but on spectra, as shown in Fig. 8. The most essential difference between two devices is whether there is a cavity, which decides only certain near-equidistant discrete frequencies (longitudinal modes) is self-consistently reproduced after each round trip. Considering a free-running parametric device with single-frequency pumping, the non-resonant OPG generates signal and idler frequencies continuously distributed in the gain bandwidth, but the OPO spectrum shows as discrete lines (Fig. 8(a)). Once seeded by a single-frequency signal, both the OPG and OPO signal waves strongly dominate at the injected light frequency and both idler waves should be also single-frequency from energy conservation (Fig. 8(b)). If the parametric device is unseeded and pumped by a multi-frequency laser with much narrower linewidth than the parametric gain bandwidth (Fig. 8(c)), the OPG and OPO spectra are almost the same to those of single-frequency pumping. Injection seeding can still enable single-frequency signal generation even when pumped by a multi-frequency laser (Fig. 8(d)), but the accompanying non-resonate idler wave carries all the wideband properties of the pump laser [18]. Discrete cavity mode is an intrinsic character of a resonant OPO device and it is the major difference compared with a single-pass OPG device.
Fig. 8. Comparison on spectral characteristics of OPG and OPO devices. (a) Free running and single-frequency pumping; (b) Injection seeded and single-frequency pumping; (c) Free running and multi-frequency pumping; (d) Injection seeded and multi-frequency pumping.
There are a few advantages for injection-seeded OPG over OPO devices. Firstly, As the OPO has a feedback resonator, the resonant signal cannot be maintained at single frequency indefinitely, because other unseeded modes will build up eventually from noise later in the pump pulse when the pump pulse duration is relatively long for an injection-seeded OPO [19], regardless of single-frequency or multi-frequency pumping. The onset is aggravated under larger pump energy and if the seeding wavelength is not centered on the peak of the spectral gain profile. The OPG suffers much less even if the peak pump power is quite high. Secondly, a complicated electronic feedback system is essential for an injection-seeded OPO to control the cavity length synchronously with the seeding wavelength, and it can be very difficult if there is a wide tuning range. For an OPG device, however, the feedback electronics are needless and the only required extra component is a tunable seeder. Last but not least, injection-seeded OPG compromises on the mechanical stability and temperature controlling accuracy because it is less sensitive to PM condition and optical path compared with OPO, greatly improving the robustness and reducing the cost.
4. Conclusion
Narrow linewidth accompanied with wide tunability and high peak power have long been desirable performances for laser sources in scientific research, lidar, industrial and environmental sensing. However, convenient, efficient, and economic methods for such a device have often proved to be elusive. In this paper, a widely tunable narrow-linewidth high-peak-power OPG based on a PPMgLN crystal was demonstrated by injection seeding a CW tunable diode seeder. The OPG operated at the repetition rate of 1 kHz, had high peak power (0.343 MW at 1638 nm), high conversion efficiency (47.9% in total) and narrow linewidth approaching the Fourier transform limit of sub-nanosecond signal pulses at 348 ps. It should be the most simple and costless method for narrow-linewidth frequency converters, as there is no need of a high-performance seeder with the linewidth at kHz or even narrower. In addition, a multimode pump source is suitable and cavity controlling electronics are not required for injection-seeded OPG compared with injection-seeded OPO. It is important to note that only signal waves at the seeding wavelength inherits the narrow-linewidth feature and the idler wave in the mid-infrared range is beyond control. Even if a single-longitudinal-mode sub-nanosecond pump laser is used, the idler wave should have a wider linewidth than the signal as the linewidth of sub-nanosecond pump pulses cannot be ignored. The signal linewidth achieved in this work was at the level of a few GHz and the tuning range was over 100 nm, which is sufficient for spectroscopy applications for solid or liquid phases, a good option in high-SNR hyperspectral lidar, and can also be used as a seeder for OPA.
Funding
National Natural Science Foundation of China (62175184).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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