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Abstract
In current work, spectroscopic properties of Nd3+ ions in titanate-germanate glasses have been studied for near-IR luminescence and laser applications. Near-IR luminescence at 1.06 µm due to 4F3/2 → 4I11/2 laser transition of Nd3+ ions has been examined in the function of TiO2 concentration. Based on theoretical calculations and experimental investigations, several spectroscopic and laser parameters for Nd3+ ions in titanate-germanate glasses were determined and compared to the previous results published for similar glass systems. Our systematic studies indicate that Nd3+ doped glass with molar ratio GeO2:TiO2 = 2:1 presents excellent near-IR luminescence properties and could be successfully applied to laser technology.
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1. Introduction
The first laser action in glass was developed over sixty years ago [1] and initiated successfully the era of laser glass technology. The laser oscillations observed in the near-infrared (NIR) spectral range at about 1.06 µm are associated with the main 4F3/2 → 4I11/2 transition of Nd3+. Further experiments well demonstrated that spectroscopic parameters for the 4F3/2 → 4I11/2 NIR laser transition of Nd3+ ions such as stimulated emission cross section, peak fluorescence wavelength, effective fluorescence bandwidth, calculated and measured lifetimes, transition probabilities, fluorescence branching ratios, and quantum efficiency of excited state varying strongly on glass-host composition [2]. Since then, many excellent papers have been published on optical and near-IR laser properties of Nd3+ ions in numerous inorganic glass systems [3–6]. In particular, near-IR luminescence properties of Nd3+ ions in borate [7–13], phosphate [14–20], silicate [21–23], germanate [24–28], tellurite [28–35], as well as mixed borotellurite [36–38], borogermanate [39] and borobismuthate [40–42] glasses have been examined in details. Special attention has been devoted to non-oxide glasses containing Nd3+ ions [43–46].
Among amorphous matrices, barium gallo-germanate glass belonging to low-phonon oxide glass family presents some advantages, i.e. relatively large glass-forming region, good thermal stability parameters, quite strong chemical and mechanical stability useful for optical fiber drawing and enhanced luminescence properties. It was proposed as a window for high energy laser systems [47]. Moreover, barium gallo-germanate glasses doped with rare earth ions are excellent candidate for optical waveguides [48,49] and solid-state lasers [50,51] operated in the near-IR ranges. Current work is focused on spectroscopic and laser properties of Nd3+ ions in barium gallo-germanate glasses, where GeO2 was partially substituted by TiO2. Their thermal and structural properties were evidenced by differential scanning calorimetry (DSC), X-ray diffraction (XRD), electron paramagnetic resonance (EPR), Raman and infrared spectroscopy in our previously published work [52]. Here, several spectroscopic and laser parameters for Nd3+ ions were determined in function of TiO2 content. The results are presented and discussed for glass samples, where GeO2:TiO2 molar ratio was changed from 5:1 to 1:5. Titanium dioxide plays the role of glass-modifier or glass-former, depending on its concentration. Until now, emission properties of rare earth ions in inorganic glasses with a high content of TiO2 have not been yet studied, to the best of our knowledge. Recently, near-IR emission properties of Nd3+ have been examined in GeO2-PbO-TiO2 [53], TeO2-TiO2-Nb2O5 [54], TeO2-TiO2-WO3 [55], TeO2-TiO2-ZnO [56] and multicomponent PbO-B2O3-TiO2-AlF3 [57] glasses, where amount of titanium dioxide playing the role as glass-network modifier did not exceed 10 molar %. Here, the purpose of our work concerns on the enhanced near-IR luminescence of Nd3+ ions in multicomponent germanate glasses in the presence of TiO2.
2. Experimental details
Glasses with the following chemical compositions xTiO2-(60-x)GeO2-30BaO-9Ga2O3-1Nd2O3 (where x = 0, 10, 20, 30, 40, 45 and 50) were prepared using melt quenching technique. The concentrations of components are given in molar %. The appropriate amounts of metal oxides of high purity (99.99%, Aldrich Chemical Co.) were mixed homogenously together and then melted at 1200°C for 0.45h. Each Nd3+- doped sample was polished for optical measurements.
In the next step, the Metricon 2010 prism coupler was applied to determine the refractive index of the glass-host at a wavelength of 632.8 nm. Glass samples were then characterized using absorption (Cary 5000 UV-VIS-NIR spectrophotometer, Agilent Technology, USA) and luminescence spectroscopy. For luminescence spectra and decay curve measurements laser equipment was used, which consists of PTI QuantaMaster QM40 spectrofluorometer, tunable pulsed optical parametric oscillator (OPO), Nd:YAG laser (Opotek Opolette 355 LD), double 200 mm monochromators, Hamamatsu H10330B-75 detector and PTI ASOC-10 USB-2500 oscilloscope. Resolution for spectral measurements was ±0.1 nm. Decay curves were recorded and stored by a PTI ASOC-10 [USB-2500] oscilloscope with an accuracy of ±0.5 µs.
3. Results and discussion
Previously published works for titanate glasses [58–60] revealed that they are partly crystallized and present relatively low thermal stability. These glass systems possess crystalline phases usually assigned to different titanates. In particular, it is quite difficult to synthesize thermally stable and fully amorphous systems with high TiO2 concentration. In our case, multicomponent glasses with various molar ratios GeO2:TiO2 from 5:1 to 1:5 were successfully prepared. It is interesting to notice that glass-forming region for the studied compositions is relatively broad. All studied samples are fully amorphous, which was verified by X-ray diffraction. Typical X-ray diffraction patterns for selected glass samples with the absence and presence of titanium dioxide (GeO2:TiO2 = 1:1) are shown in Fig. S1. Further DSC experiments shown in Fig. S2 indicate that the glass transition temperature increases in the presence of TiO2 suggesting less open glass-structure for titanate-germanate glass samples. On the other hand, thermal stability parameter (ΔT = Tx - Tg) is reduced for glass sample, where GeO2 is partially replaced by TiO2 (for more details please refer to Table S1). However, ΔT factor is still above 100°C exhibiting good thermal stability against devitrification and quite large working range during operations for fiber drawing.
Another key parameter is the infrared absorption coefficient αOH for band near 3400 cm-1 assigned to the stretching vibrations of hydroxyl groups (Fig. S3). In general, glass systems with extremely low content of OH- groups are necessary to fabricate active near-IR emitting optical fibers. According to the previous excellent results published for Tm3+ doped barium gallo-germanate glass single-mode fibers [50], the value of αOH is reduced drastically from 3.05 cm-1 for sample melted in air to 0.19 cm-1 for precursor glass prepared under the optimized Reaction Atmosphere Procedure (RAP). In our case, the OH- absorption coefficients are low, because samples were prepared under rigorous technological conditions in special glove-box, in a protective atmosphere of dried argon of high purity. The infrared absorption coefficient for titanate-germanate glass sample is close to 0.23 cm-1 (Table S2) and its value is similar to the results obtained earlier for germanate based glasses [61–63]. Further investigations indicate that that titanium ions at trivalent oxidation state are present in the studied glass samples (Fig. S4). It was also confirmed for Ti3+/Ti4+ doped calcium aluminosilicate glasses [64].
3.1 Absorption spectra and refractive index of the glass-host
Figure 1 shows absorption spectra of Nd3+ ions in glasses with various molar ratios GeO2:TiO2. The following glass parameters, i.e. UV cut-off wavelength, bonding parameter δ and refractive index of the glass-host n in function of TiO2 concentration are also indicated.
Fig. 1. Absorption spectra of Nd3+-doped titanate-germanate glasses (a), UV cut-off (b), bonding parameter (c) and refractive index of glass-host (d) in function of TiO2 content.
The absorption spectra consist of the inhomogeneously broadened lines characteristic for 4f3–4f3 electronic transitions of Nd3+ ions. They are located in the 300–950 nm spectral range. Absorption bands of Nd3+ ions correspond to transitions originating from the 4I9/2 ground state to the higher-lying 4F3/2, 4F5/2, 2H9/2, 4F7/2, 4S3/2, 4F9/2, 2H11/2, 4G5/2, 2G7/2, 4G7/2, 2K15/2, 4G11/2, 2G9/2 and 2P1/2 excited states, respectively. Among observed absorption lines, the 4I9/2 → 4G5/2,2G7/2 transition of Nd3+ located near 600 nm, so-called hypersensitive transition, is the most intense. It follows well the selection rules: |S| = 0, |ΔL| ≤ 2 and |ΔJ| ≤ 2. Its position and intensity is very sensitive to small changes of environment around Nd3+ ions. The absorption measurements for Nd3+ ions clearly indicates that the intensity of this band is the highest for glass with GeO2:TiO2 molar ratio equal to 2:1. Furthermore, the UV cut-off wavelength defined as the intersection between the zero base line and the extrapolation of absorption edge was determined. The absorption edge for titanate-germanate glass is shifted to longer wavelengths with increasing TiO2 concentration.
From the optical absorption spectra measured for Nd3+ ions in titanate-germanate glasses, bonding parameters (β and δ) were also calculated using the expression δ = [(1- β)/β] × 100%, where β = ∑N = β*/N and β* = νc/νa [65]. In this expression, β is the shift of energy level position (Nephelauxetic effect), N - number of levels used for calculation of β-values, νc and νa are energies of the corresponding transitions in the investigated complex and free-ion [66]. Positive or negative sign for the values of δ suggests covalent or ionic bonding between Nd3+ ions and surrounding ligands. Our calculations indicate that bonding parameter δ increase from 0.281 (glass without TiO2) to 0.665 (glass with molar ratio GeO2:TiO2 = 1:5). It suggests that titanate-germanate glasses exhibit more covalent character in comparison to lead borate glass doped with Nd3+, for which the bonding parameter δ was found to be -0.210 [67]. The results are well correlated with the refractive index, which increases from 1.736 (glass without TiO2) to 1.998 (glass with molar ratio GeO2:TiO2 = 1:5) with increasing TiO2 concentration. Details are given in Table 1.
Table 1. Glass composition, TiO2 content, Nd3+ ion concentration, UV cut-off, bonding parameter (δ) and refractive index of glass-host (n).
3.2 Judd-Ofelt calculations
The standard Judd-Ofelt theory [68,69] was used to calculate radiative transition probabilities for excited states of Nd3+ ions in titanate-germanate glasses. The calculation procedure was carried out using the software OriginPro. According to the standard procedure, the x-axis of absorption spectrum was converted to wavenumbers given in cm-1. Then, the baseline was fitted individually to the each absorption band and the integrated area was calculated. The intensities of bands shown in Fig. 1 are estimated by measuring the areas under the absorption lines using the Eq. (1):
In this equation ε(ν) = A/cl, ∫ε(ν) represents the area under the absorption line, A indicates the absorbance, c is the concentration of Nd3+ ions in mol × l-1 and l denotes the optical path length.
The measured oscillator strengths of transitions were obtained from the absorption bands of Nd3+ ions. The theoretical oscillator strengths for each transition of Nd3+, within 4f3 electronic configuration, were calculated using the following relation (2):
Measured and calculated oscillator strengths for Nd3+ ions in the studied glasses are given in Table 2 (0, 10, 20 and 30% TiO2) and Table 3 (40, 45 and 50% TiO2), respectively. The quality of the fit can be expressed by the magnitude of the root-mean-square (rms) deviation. It is defined by Σ(Pmeas - Pcalc)2. The rms values are also given in Tables 2 and 3.
Table 2. Measured and calculated oscillator strengths (×10−6) for Nd3+ ions in titanate-germanate glasses. Transitions are from the 4I9/2 ground state to the levels indicated. The rms deviation (×10−6) is also given.
Table 3. Measured and calculated oscillator strengths (×10−6) for Nd3+ ions in titanate-germanate glasses. Transitions are from the 4I9/2 ground state to the levels indicated. The rms deviation (×10−6) is also given.
The three Ωt (t = 2, 4, 6) intensity parameters (J-O) were evaluated from the least-square fit of measured and calculated oscillator strengths for Nd3+ ions. They are collected in Table 4.
Table 4. The Judd-Ofelt intensity parameters Ωt (t = 2, 4, 6) × 10−20 cm2.
The J-O parameters Ωt (t = 2, 4, 6) are quite well correlated with thermal results obtained by DSC method (Fig. S2). The Ω4 and Ω6 parameters are found to be a structure-dependent. In contrast to germanate glass without TiO2, the Ω2 and Ω6 parameters are higher than values of Ω4 suggesting that titanate-germanate glasses exhibit more rigidity and confirm less open glass structure from DSC measurements. Moreover, the magnitude of Ω4/Ω6 named as spectroscopic quality parameter, is found to be less than unity for glasses where the 4F3/2 → 4I11/2 transition at 1.06 µm is the intense lasing transition. Glass systems with more than unity of Ω4/Ω6 factor demonstrate the intense lasing 4F3/2 → 4I9/2 transition of Nd3+ at 0.89 µm. It was well presented and discussed in details for numerous inorganic glasses containing Nd3+ ions [70]. In our case, all values of Ω4/Ω6 factor are less than unity, except glass without TiO2, which indicate that titanate-germanate glass is promising to achieve lasing condition for the 4F3/2 → 4I11/2 (Nd3+) near-IR luminescence channel at about 1.06 µm.
The Judd-Ofelt parameter Ω2 indicates the degree of covalency between Nd3+ ions and surrounding ligands. Also, it reflects the asymmetry of the environment around Nd3+ ions. It is interesting to see that the Judd-Ofelt parameter Ω2 for glass with molar ratio GeO2:TiO2 = 2:1 is the highest (Ω2 = 10.67 × 10−20cm2) than values obtained for other studied Nd3+ doped glass samples. It suggests that the bonds between Nd3+ ions and ligands existing in glass sample with GeO2:TiO2 = 2:1 are highly covalent in character and the results are comparable to the values Ω2 equal to 10.43 × 10−20cm2 obtained for Nd3+ ions in Li2O-Ta2O5-ZrO2-SiO2 glasses [71] and 10.26 × 10−20cm2 for Nd3+ doped oxyfluorosilicate glass [72].
In the next step, the values of Ωt (t = 2, 4, 6) were used to calculate the radiative transition probabilities AJ, luminescence branching ratios β and radiative lifetimes τrad. The radiative transition probabilities AJ for excited states of Nd3+ ions from an initial state J to a final ground state J’ were calculated using the following expression (3):
The total radiative emission probability AT involving all the intermediate terms is given by the sum of the AJ terms. Thus, radiative lifetime τrad of an excited state is the inverse of the total radiative emission probability given by the following relation (4):
Luminescence branching ratio β is related to the relative intensities of transitions from the excited state to all lower-lying states of Nd3+ and given by expression (5):
The calculated radiative transition probabilities AJ and the luminescence branching ratios β for Nd3+ ions in titanate-germanate glasses are calculated. The results for the studied glass samples are shown in Table 2 (0, 10, 20 and 30% TiO2) and Table 3 (40, 45 and 50% TiO2), respectively. In all cases, the luminescence branching ratios for the 4F3/2 → 4I11/2 transition at 1.06 µm are the highest and the values of β are between 46% to 52%, depending on TiO2 concentration. The radiative transition probability is the highest for glass with GeO2:TiO2 = 2:1 and its value for the 4F3/2 → 4I11/2 transition at 1.06 µm is close to AJ = 2787 s-1.
3.3 Near-IR luminescence spectra and their decays
Figure 2 presents near-IR luminescence spectra of Nd3+ ions in titanate-germanate glasses. Luminescence bands at 0.89 µm, 1.06 µm and 1.33 µm correspond to 4F3/2 → 4I9/2, 4F3/2 → 4I11/2 and to 4F3/2 → 4I13/2 transitions of Nd3+. They are schematized on energy level diagram (Fig. 3).
Spectroscopic studies indicate that the intensities of luminescence bands of Nd3+ ions are enhanced with increasing TiO2 concentration in glass composition. The most intense near-IR emission band located near 1.06 µm is related to the main 4F3/2 → 4I11/2 laser transition of Nd3+. It was examined in details in function of TiO2 concentration. Figure 4 presents near-IR emission band at about 1.06 µm varying with TiO2 content. Some spectroscopic parameters such as the emission peak wavelength and the emission bandwidth for the 4F3/2 → 4I11/2 transition of Nd3+ are also schematized. It is clearly seen that the 4F3/2 → 4I11/2 transition of Nd3+ ions is shifted to longer wavelengths with increasing TiO2 content. The near-IR emission band is shifted from 1064 nm (glass without TiO2) to 1070 nm (glass sample with molar ratio GeO2:TiO2 = 1:5), respectively. The luminescence bandwidth referred as full width at half maximum (FWHM) is nearly independent on TiO2 concentration and its value is close to 41.5 ± 1 nm.
Fig. 4. Near-IR luminescence bands due to 4F3/2 → 4I11/2 laser transition of Nd3+ ions in titanate-germanate glass (a), emission peak wavelength (b), emission bandwidth FWHM (c) and stimulated emission cross-section (d) in function of TiO2 content.
The luminescence bandwidth and the radiative transition probability AJ calculated for the 4F3/2 → 4I11/2 transition of Nd3+ (Tables 5 and 6) were applied to obtain the peak stimulated emission cross-section σem using the following relation (6):
Table 5. The calculated radiative transition probabilities AJ and luminescence branching ratios β for Nd3+ ions in germanate glass without TiO2 and titanate-germanate glasses (0, 10, 20 and 30% TiO2).
Table 6. The calculated radiative transition probabilities AJ and luminescence branching ratios β for Nd3+ ions in titanate-germanate glasses (40, 45 and 50% TiO2).
The peak stimulated emission cross-section σem belongs to the important spectroscopic parameters, which designate the potential laser performance of the glass-host. The emission cross-section for the 4F3/2 → 4I11/2 transition of Nd3+ is relatively large for glass with molar ratio GeO2:TiO2 = 2:1 (σem = 4.41 × 10−20cm2) compared to the other values, which were calculated for the studied glass samples (Fig. 4). Details are also given in Table 7.
Table 7. The emission peak wavelength λp, the luminescence bandwidth FWHM, the radiative transition probability AJ for the 4F3/2 → 4I11/2 transition of Nd3+, the stimulated emission cross-section σem and gain bandwidth σem × FWHM varying with TiO2 concentration.
In the next step, luminescence decay curves for the upper 4F3/2 laser state of Nd3+ ions were measured and analyzed in function of TiO2 concentration. Decay curves were quite well fitted to nearly single-exponential function given below (7):
Figure 5 shows luminescence decays from the 4F3/2 state of Nd3+ ions in multicomponent titanate-germanate glasses. Luminescence lifetimes and quantum efficiencies varying with TiO2 content are also schematized. Luminescence decay curve analysis suggests that the 4F3/2 lifetimes of Nd3+ ions depend slightly on TiO2 concentration. Their values are shorter compared to glass sample without TiO2.
Fig. 5. Luminescence decays from the upper 4F3/2 laser state of Nd3+ ions in multicomponent titanate-germanate glasses (a), measured luminescence lifetime (b) and quantum efficiency (c) in function of TiO2 content.
The radiative lifetime τrad calculated from the Judd-Ofelt theory using Eq. (4) and the measured lifetime τm were applied to obtain the quantum efficiency for the 4F3/2 → 4I11/2 laser transition of Nd3+ ions using the following expression (8):
Among the studied samples, the quantum efficiency for the 4F3/2 → 4I11/2 transition of Nd3+ ions is the highest for glass with GeO2:TiO2 = 2:1 (Fig. 5) and its value is close to η = 88%. Details are also given in Table 8.
Based on measurements of luminescence spectra and their decays it can be concluded that glass with molar ratio GeO2:TiO2 equal to 2:1 has the highest stimulated emission cross-section (σem = 4.41 × 10−20cm2) and quantum efficiency (η = 88%) for the 4F3/2 → 4I11/2 laser transition of Nd3+ ions. It suggests that titanate-germanate glass (GeO2:TiO2 = 2:1) with 1 mol% Nd3+ can be successfully used for near-IR laser applications. At this moment, it should be also mentioned that nominal activator concentration is in a good agreement with the actual neodymium content, which was determined from the chemical analysis (Table S3). In particular, the stimulated emission cross-section for the 4F3/2 → 4I11/2 transition of Nd3+ in glass with GeO2:TiO2 = 2:1 is comparable with the values σem ranging between 4 and 4.75 (in 10−20cm2) obtained for similar inorganic laser glasses [72–80].
Table 8. The calculated radiative lifetime τrad, the measured lifetime τm, the quantum efficiency η, the stimulated emission cross-section σem and figure of merit σem × τm varying with TiO2 concentration.
Finally, the stimulated emission cross-section (σem), the luminescence bandwidth (FWHM) and the measured lifetime τm for the 4F3/2 → 4I11/2 transition of Nd3+ were applied to calculate the laser parameters: gain bandwidth (σem × FWHM) and figure of merit FOM (σem × τm). These laser parameters varying with TiO2 concentration are presented in Tables 7 and 8, respectively.
From literature data [72–81] it is well-known that the gain bandwidth and the figure of merit are really important to identify the laser glass-host and near-IR broadband amplification. The gain bandwidth is defined as the range of frequencies into which the optical amplification can occur, whereas the laser threshold is evaluated by the figure of merit FOM [82]. In general, high gain medium with low threshold pump power is required. Both parameters σem × FWHM and σem × τm exhibiting relatively high values for the 4F3/2 → 4I11/2 near-IR transition of Nd3+ are necessary to generate laser action in inorganic glasses.
Previously published works for Nd3+ ions in glass systems containing lead give interesting results [77–79]. For lead tungsten tellurite glass [77] the gain bandwidth is extremely high, but the figure of merit is found to be smaller. The opposite situation is observed for lead phosphate glass [78], where value of σem × FWHM is found to be rather smaller, but product of σem × τm known as the figure of merit is relatively higher. For lead fluorosilicate glass [79], both values are large compared with other reported glasses. It clearly suggests that both laser parameters depend significantly on glass composition.
Our near-IR luminescence investigations clearly demonstrate that the highest values of gain bandwidth (σem × FWHM = 1.85 × 10−25cm3) and figure of merit (σem × τm = 67.91 × 10−25cm2s) were obtained for Nd3+ ions in multicomponent glass with molar ratio GeO2:TiO2 = 2:1. They are compared to some Nd3+- doped laser glasses published previously [72–81]. The results are summarized in Table 9.
Table 9. Comparison of the stimulated emission cross-section (σem), gain bandwidth (σem × FWHM) and figure of merit (σem × τm) in different laser glasses doped with Nd3+.
In our case, the gain bandwidth for the 4F3/2 → 4I11/2 near-IR transition of Nd3+ ions in glass with GeO2:TiO2 = 2:1 is found to be larger than for other lead-free glasses reported in Table 9. However, the figure of merit (FOM) is smaller compared to oxide and oxyfluoride phosphate based glasses [80–81]. The product σem × τm for our sample (GeO2:TiO2 = 2:1) is similar to the values obtained for glasses based on ZnO-TeO2 [75] and B2O3-Na2O-NaF [76], respectively. It can be concluded that both parameters σem × FWHM and σem × τm for the 4F3/2 → 4I11/2 transition of Nd3+ are promising and glass (GeO2:TiO2 = 2:1) could be potentially useful for lasing action compared along with reported laser glasses. Therefore, Nd3+- doped titanate-germanate glass with GeO2:TiO2 = 2:1 is suitable for near-IR laser gain active media operating at 1.06 µm.
4. Conclusions
In this work, Nd3+ doped titanate-germanate glasses have been examined for near-IR laser applications. Multicomponent glasses with different molar ratios GeO2:TiO2 were prepared and then studied experimentally using photoluminescence spectroscopy. Theoretical calculations were also obtained using the Judd-Ofelt framework.
Based on absorption and luminescence spectra, and decay curve measurements, several spectroscopic and laser parameters for Nd3+ ions in glass varying with TiO2 concentration were determined. Systematic investigations indicate that Nd3+ doped glass with GeO2:TiO2 = 2:1 presents strong near-IR luminescence at 1.06 µm corresponding to the 4F3/2 → 4I11/2 transition. The spectroscopic and laser parameters for the main 4F3/2 → 4I11/2 transition of Nd3+ ions are as follows: the stimulated emission cross-section σem = 4.41 × 10−20cm2, the measured lifetime τm = 154 µs, the quantum efficiency η = 88%, the gain bandwidth σem × FWHM = 1.85 × 10−25cm3 and figure of merit known as σem × τm product = 67.91 × 10−25cm2s, respectively. They are attractive compared with some reported glass systems and commercially-available laser glasses documented in literature. Therefore, we suggest that Nd3+- doped titanate-germanate glass with molar ratio GeO2:TiO2 equal to 2:1 is suitable for near-IR luminescence at 1.06 µm and could be useful for solid-state laser applications.
Funding
Narodowe Centrum Nauki (2018/31/B/ST8/00166).
Disclosures
The Authors declare no conflict 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.
Supplemental document
See Supplement 1 for supporting content.
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