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Abstract
The mid-infrared (MIR) wavelength coincides with various molecular resonances. In particular, a 13–20 µm wavelength window has fingerprints of unique groups such as organometallic, halogenated, and aromatic bonds. In this work, for the first time, to the best of our knowledge, an on-chip supercontinuum generation (SCG) source based on cadmium telluride (CdTe)/ cadmium sulfide (CdS)/ silicon heterostructure is proposed to extend the on-chip SCG beyond 13 µm (spanning 3.5 to 20 µm). CdTe has an ultra-broad transparent spectral range up to 25 µm, and almost the largest third-order nonlinear coefficient (n2∼ 5×10−17 m2/W at 1.55 µm, 1.3×10−17 m2/W at 9 µm, several times larger than that of silicon) among the MIR materials, making CdTe an excellent candidate for long-wavelength MIR on-chip SCG. The waveguide structure is designed with CdS as the intermediate cladding layer to achieve a low waveguide loss and high mode confinement. A large-core CdTe waveguide is tailored to generate a low and flat dispersion (< 30 ps/nm/km) in a spectral range spanning from 5 to 20 µm, while balancing the large effective nonlinearity and the convenience of coupling. The simulation results solved by the nonlinear Schrödinger equation manifest that the engineered large cross-section waveguide with only 2.5-mm propagation distance broadens the MIR spectrum covering 3.5 to 20 µm pumped by a 9 µm femtosecond laser. Moreover, it is found that good coherence is achieved from the designed MIR waveguide, before severe soliton fission breaks the temporal profile. 5-fold self-compression of the pump pulse down to 1.6 optical cycles is observed while propagating inside the CdTe waveguide. The detailed simulation of the CdTe/CdS/Si waveguide design with the various waveguide parameters, polarizations, pump wavelengths, and pump power are provided. With the SC spectrum covering almost the entire fingerprint regime and the excellent coherence generated from the designed CdTe waveguide, it provides abundant new opportunities for MIR microphotonics.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The Mid-infrared (MIR) spectral region (2.5–25 µm) contains strong and sharp molecular vibrational and rotational absorption lines, namely fingerprint absorptions for most of molecules, which makes MIR wavelengths important for molecular spectroscopic applications such as gas sensing, environmental monitoring, and disease diagnosis. The main molecular fingerprint bands include hydrogen single bonds at 3–4 µm wavelength regime, carbon double bonds located at 5 to 7.5 µm, as well as carbon-oxygen bonds, and carbon-nitrogen bonds at 8 to 11 µm wavelength regime. Extending to longer wavelength ranging from 13 to 20 µm, special molecular groups such as organometallic, halogenated, and aromatic bonds are located [1]. Quantum cascade lasers as the key MIR laser sources have been routinely used in the wavelength window of 13 to 20 µm. However, the emission bandwidth of quantum cascade lasers is relatively narrow, which limits the promotion of quantum cascade lasers for MIR spectroscopic applications with multiple molecular fingerprints. The parametric conversion technique provides another type of coherent MIR radiator. However, besides the complex apparatus, the transmission range of MIR nonlinear crystals hinders the extension of the parametric emission into long-wavelength MIR regime. For example, the major MIR nonlinear crystal such as ZnGeP2, LiGaSe2, and GaSe has the transmission cutoff at 11 µm, 12 µm and 17 µm, respectively [2]. On the other hand, supercontinuum generation (SCG) could offer broadband MIR radiation with special merits of compact size, no requisition of seed and low cost. For example, a single-mode telluride and multi-mode chalcogenide fiber pumped at the wavelength of 7 and 6.3 µm, respectively produces a broadband SC spanning from 1.4 to 16 µm [3,4]. In a further step, integrating the CMOS compatible materials to make the on-chip nonlinear waveguide could enhance the MIR emission efficiency, extend the wavelength coverage, strengthen the design freedom, reduce the radiation threshold, shrink the size, improve the reliability and cut the cost.
Recent researches have extensively studied the SCG in several integrated systems, such as chalcogenide glass waveguides [5,6], lithium niobate (LN) [7], and periodically-poled lithium niobate (PPLN) [8,9], aluminum nitride (AlN) waveguides [10]. In particular, thanks to the mature semiconductor growth technology of high-quality thin films of silicon (Si) and silicon nitride (Si3N4), as well as the advancement of CMOS integration technology, silicon photonics has been well studied in on-chip devices to realize MIR SCG. For instance, the Si3N4 waveguide pumped in the telecom band achieved a broad spectrum from 2.5 to 4 µm [11]. Octave-spanning SC from 2 to 6 µm has been demonstrated in a Si waveguide [12,13]. However, the transmission of Si drops sharply as the wavelength increasing beyond 8 µm, which prohibits the extension of SC to a longer MIR wavelength. Recently, germanium (Ge), one of the group IV semiconductor is considered as another candidate for the on-chip MIR SCG thanks to the large optical nonlinearity and good transparency up to 15 µm [14–16]. On-chip ultra-wide SCG spanning from 3 to 13 µm is achieved with the Ge-rich graded SiGe waveguide, which pushes the optical mode to the Ge-rich area and reduces the overlap with the Si substrate with strong absorption at long wavelengths [16]. However, till today, there is still no MIR on-chip SCG reported beyond 13 µm, despite important molecular fingerprints are located in the wavelength range of 13 to 20 µm.
Cadmium telluride (CdTe), as an important group II-VI semiconductor, has excellent characteristics as summarized in Table 1. CdTe has an ultra-broad transparency window spanning from 1 to 25 µm covering the entire MIR fingerprint regime, which is a lot broader compared to most of the MIR semiconductors and dielectrics such as Si, Si3N4, Ge, and As2S3. CdTe also has an outstanding nonlinearity with the third-order nonlinear refractive index (${n_2}$) of ∼5×10−17 m2/W at 1.55 µm [17] and ∼1.3×10−17 m2/W at 9 µm [18] which is several times larger than that of Si, and 2-order greater than that of Si3N4 [19]. Moreover, high-quality CdTe epi-layers could be grown by the mature semiconductor deposition technologies, such as sputtering deposition techniques [20]. These excellent properties make CdTe a promising candidate for MIR nonlinear devices, especially at long MIR wavelengths.
Table 1. Summary of optical properties of common MIR materials and SC bandwidths in corresponding waveguides*
In this work, we propose a new on-chip SCG source based on CdTe/Cadmium sulfide (CdS)/Si hetero-waveguide to extend the integrated SCG beyond 13 µm to cover 3.5 to 20 µm, for the first time, to the best of our knowledge. CdS with 15 µm thickness is designed as a barrier layer between the CdTe waveguide and the Si substrate, which keeps the optical mode well confined in the CdTe core. The propagation loss of the designed waveguide is calculated < 3.7 dB/cm at a wavelength of 20 µm. The waveguide with large cross section (8×20 µm2) provides good freedom to tailor the group velocity dispersion (GVD). A flat dispersion profile with anomalous dispersion < 30 ps/nm/km over 5 to 20 µm wavelength range is obtained, by tailoring the dimensions of CdTe and CdS layers. The SC from the CdTe hetero-waveguide is simulated through the generalized nonlinear Schrödinger equation (GNLSE), pumped at 6 and 9 µm with a 200 fs pulse width. 3.5 to 20 µm SC spectrum (at −34 dB) is obtained from a 2.5-mm-long CdTe waveguide, pumped at 9 µm, with superior coherence. Moreover, 5-fold self-compression of the 9 µm pulse down to 47 fs is observed via propagation in the CdTe waveguide with the designed anomalous dispersion. The on-chip coherent radiation covering almost the entire MIR fingerprint regime would be impactful for advanced spectroscopic applications such as MIR dual-comb spectroscopy and multi-dimensional infrared spectroscopy for various molecular groups.
2. Waveguide design and dispersion tailoring
The structure of the proposed CdTe/CdS/Si waveguide is illustrated in Fig. 1(a). Si is chosen as the substrate to be adapted by the Si-based integrated-circuit fabrication technologies. CdTe with the refractive index of n∼2.68 serves as the waveguide core. A layer of CdS with a lower refractive index (n∼2.3) [29] and good transparency in the wavelength range of 1 to 15 µm [30], as shown in Fig. 1(b), is deposited in between the Si substrate and CdTe as the lower cladding layer. It pushes the optical field into the CdTe waveguide core, and reduces the leakage to the Si substrate. It is worth mentioning that the reduction of the transmission of the CdS cladding layer beyond 15 µm has negligible effect on the SCG for the strong field confinement in the CdTe waveguide core, and the short propagation distance, which would be discussed for details in the following.
Fig. 1. (a) The three-dimensional structure diagram of the CdTe/CdS/Si hetero-waveguide, and (b) the transmission spectra of 3-mm thick CdS and CdTe [30,31]. The reflection loss from the surfaces is subtracted.
The selection of the CdS layer thickness is crucial in reducing the field leakage into the Si substrate, thus the propagation loss, especially for long wavelengths. The propagation loss and mode confinement of the fundamental transverse-electric (TE) mode in CdTe/CdS/Si waveguides, with different thickness of CdS cladding layers are calculated and shown in Fig. 2. The propagation loss is obtained by considering the attenuation coefficient of the different materials of the hetero-structure and the field confinement in the core. The CdTe waveguide core is 8 µm in height and 20 µm in width, and the thickness of CdS layer varies from 10 to 20 µm. As shown in Fig. 2(a), with the 10-µm-thick CdS layer (the blue curve), the propagation loss is low at short wavelengths, and raises up abruptly when the wavelength increasing after 14 µm due to the growth of the mode area at long wavelengths. The loss reaches 8 dB/cm at 20 µm wavelength. Increasing the CdS cladding layer thickness to 15 µm reduces the loss to 3.7 dB/cm at 20 µm wavelength (the black curve). The loss does not exhibit significant drop by further increasing the CdS thickness to 20 µm (the red curve). Thus in this work, to extend the MIR emission into the long-wavelength MIR, in particular in a wavelength range of 16 to 20 µm, 15-µm-thick CdS layer is chosen to achieve a low propagation loss and small mode area thus high nonlinearity. The mode confinement of the CdTe/CdS/Si waveguide with 15-µm-thick CdS layer is calculated and plotted in Fig. 2(b), manifesting the mode confinement factor of 86%, 76%, and 65% for a wavelength of 12 µm, 16 µm and 20 µm, respectively.
Fig. 2. (a) The calculated attenuation coefficient α of the TE mode as a function of wavelength in the CdTe/CdS/Si hetero-waveguide with different CdS layer thicknesses of 10, 15 and 20 µm. (b) The calculated fundamental mode confinement of the CdTe/CdS/Si hetero-waveguide with 15 µm-thick CdS layer. The CdTe waveguide cross-section is chosen as 8×20 µm2, and the TE polarization is adopted for calculation.
The dimension of the CdTe waveguide is deigned to generate a low and flat dispersion as well as good field confinement inside of the CdTe waveguide core. It is found that the dispersion profile and bandwidth can be effectively tailored by controlling the dimensional parameters of the designed large cross-section CdTe waveguide. As shown in Fig. 3, the dispersion and the mode confinement of the fundamental TE mode are calculated by varying the CdTe waveguide thickness and width, with the CdS cladding layer fixed as 15 µm thick. The total dispersion of the waveguide with the CdTe/CdS/Si heterostructure is calculated by solving the GVD of the guided mode,
where $\lambda $ is wavelength, c denotes the speed of light in vacuum, and ${n_{eff}}$ represents the effective refractive index of the mode in the waveguide core. ${n_{eff}}$ and the mode confinement in the waveguide core are simulated by the finite-element method mode solver. The calculated dispersion consists of the waveguide and the material dispersions. The refractive indices of the materials are obtained by the corresponding Sellmeier equations [29,32,33]. The dynamics of dispersion tailoring involves the following three stages in different wavelength ranges. At short wavelengths ($\lambda $ < 4 µm), the field is highly confined in the center of the high-index CdTe core, thus the CdTe material dispersion dominates, resulting a normal dispersion. As shown in Fig. 3(a) and (b), the calculated GVD curves below 4 µm all exhibit normal dispersion for different dimensions of CdTe waveguides. It is also reflected that at short wavelengths the corresponding mode confinement in the CdTe waveguide core is greater than 95% for all the dimensions of waveguides as depicted in Fig. 3(c) and (d). As the wavelength increasing to a range of 4 to 15 µm, the optical field is brought into contact with the CdS cladding layer. Consequently, the dispersion becomes sensitive to the waveguide height at long wavelengths, and the mode transition generates a low and flat anomalous dispersion profile in the middle part of the bandwidth. At longer wavelengths, exceeding ∼15 µm, substantial amount of field leaks into the CdS cladding, which changes the sign of the waveguide dispersion. It compensates the large material dispersion at long wavelength, and leads to the bending of the dispersion curves.Fig. 3. The calculated dispersion profiles (a, b) and corresponding fundamental mode confinement (c, d) of the TE mode as a function of the wavelength with different waveguide structural parameters. In (a, c) the height of CdTe waveguide is varied from 6 to 10 µm and the width is fixed as 20 µm. In (b, d) the width of CdTe waveguide is varied from 15 to 25 µm and the height is fixed as 8 µm. (e-g) The calculated field distribution of the fundamental mode in the designed CdTe waveguide with a width of 20 µm and a height of 6 µm (e), 8 µm (f) and 10 µm (g). 20 µm wavelength and TE polarization are adopted for calculation. The white arrows in the field distribution diagram denote the electric field strength. The complex effective refractive indices ${n_{eff}}$ with the imaginary part representing the attenuation are indicated above the figures. Stronger field leakage into the CdS cladding layer is revealed with thin CdTe waveguide.
It is noted that as increasing the CdTe thickness from 6 to 10 µm, while fixing the width as 20 µm, the bending of the dispersion curve at long wavelength could not be realized when the CdTe thickness reaches 10 µm, as shown in Fig. 3(a). This is attributed to the reduced weight of the waveguide dispersion with thick CdTe. On the other hand, with a thickness of 6 µm, although the second zero-dispersion point is generated, the field confinement drops sharply, as shown in Fig. 3(c). As the verification, the field distributions in CdTe waveguides at 20 µm wavelength with different CdTe thickness are plotted in Fig. 3(e-g), which reveal strong leakage into the CdS cladding with 6 µm thick CdTe waveguide. Thus an 8 µm thick CdTe is chosen as the balance between the low dispersion and high mode confinement. It is worth mentioning that with an 8 µm thick CdTe, a low and flat anomalous dispersion with GVD < 30 ps/nm/km over the wavelength range of 5 to 20 µm is obtained, which is crucial for the broadband MIR SCG extending to the long MIR wavelength.
To design the width of the CdTe waveguide, the dispersion is calculated again by fixing the waveguide thickness as 8 µm, and varying the width from 15 to 25 µm. As shown in Fig. 3(b) and (d), with a 15 µm wide CdTe waveguide, the dispersion is relatively large and the mode confinement is low, in the wavelength range of 10 to 15 µm. When the waveguide width increases to 20 µm, both the dispersion and confinement loss are reduced, and become insensitive to the CdTe width. As the nonlinearity inside the waveguide declines with increasing the mode area, 20 µm is chosen as the CdTe waveguide width to balance the low dispersion, good mode confinement, and high effective nonlinearity.
The propagation polarization of the CdTe/CdS/Si waveguide is also investigated to achieve a flat dispersion and low confinement loss. The dispersion and the field confinement of the fundamental transverse-electric (TE) and transverse-magnetic (TM) modes are numerically calculated and presented in Fig. 4. As shown in Fig. 4(a), TE and TM polarizations exhibit substantial difference in mode confinement, especially at long wavelengths. The TM polarization has a mode confinement of 82% at 12 µm, and drops to 70% and 55% at 16 µm and 20 µm, respectively, which limits the MIR SCG at long-wavelength side. As a comparison, the mode confinement of TE polarization could be improved to 76% and 65% at 16 µm and 20 µm, respectively. The improvement of mode confinement could be seen from the comparison of the calculated propagation loss in TM and TE polarizations, as shown in Fig. 4(b). It is observed that at 20 µm wavelength, as the effective refractive index of the TE polarization is slightly larger than that of the TM mode, TE polarization has less leakage into the air and the CdS cladding of the CdTe waveguide. It is manifested that the propagation loss is reduced from 8.1 dB/cm to 3.7 dB/cm at 20 µm by switching the polarization from TM to TE. In addition, with the tighter mode confinement, the dispersion in TE polarization is lower and flatter compared to that of TM polarization in the middle band wavelength range from 5 to 15 µm, although no second zero-dispersion point is generated, as shown in Fig. 4(a). Therefore, the CdTe waveguide with TE polarization is selected with the better mode confinement, lower loss and flatter dispersion.
Fig. 4. (a) The calculated group velocity dispersion (GVD) (solid curves) and mode confinement (short dashed curves) for the TE (black) and TM (red) fundamental polarizations of the designed CdTe waveguide. The long dashed horizontal plots the zero-dispersion line. (b) The calculated attenuation coefficient $\alpha $ as a function of wavelength for TE and TM polarizations. (c) The field distributions of the fundamental modes at 20 µm wavelength with TE and TM polarizations. The complex effective refractive indices neff with the imaginary part representing the attenuation are indicated above the figures. (d) The calculated effective mode area Aeff and the effective nonlinear parameter $\mathrm{\gamma}$ as a function of the wavelength in TE polarization. The CdTe cross-section of 8 × 20 µm2 and CdS thickness of 15 µm are chosen for the calculation.
To understand the nonlinearity of the designed CdTe waveguide, the effective nonlinear parameter is plotted in Fig. 4(d) by
where ${\omega _0}$ is the laser angular frequency, and ${n_2}$ represents the Kerr refractive index of CdTe [17,18]. ${A_{eff}}$ denotes the effective mode area by3. Supercontinuum generation and self compression
SCG of MIR femtosecond pulses in the designed CdTe/CdS/Si waveguide with the engineered anomalous dispersion is simulated by solving the GNLSE [34].
The left and right side of Eq. (4) models the linear propagation and nonlinear effects, respectively, with A as the slowly-varying envelope of the wave, $\alpha $ as the linear loss and ${\beta _k}$ as the dispersion coefficients. The Raman term in the equation is ignored as the interaction length in the waveguide is very short. It is worth noting that as the Raman coefficient of CdTe is unknown, we tried to use the Raman coefficient of chalcogenide glass for the simulation as a comparison (not shown here), and no detectable change is observed in either the SC spectrum or the coherence of the output emission. The symmetrized split - step Fourier method is employed to solve the GNLSE in both time and frequency domains [33].
Simulated SC spectra pumped at 6 and 9 µm wavelengths with the peak powers of 10 and 15 kW, and input pulse width of 200 fs are compared and showed in Fig. 5. The CdTe waveguide has a designed dimension of 8 µm in thickness and 20 µm in width, with 15 µm thick CdS cladding layer on the Si substrate. Together with the selected TE polarization, small and flat anomalous dispersion, low waveguide loss and a decent effective nonlinearity at long MIR wavelengths are obtained for the broad SCG extending beyond 13 µm. As shown in Fig. 5(a-c), pumped at 6 µm which is closer to the zero dispersion wavelength, the spectra are broadened to 13 µm (−30 dB) with a 3.6-mm propagation distance in the engineering CdTe waveguide. However, further extending to a longer MIR wavelength is difficult as the energy is consumed by self-phase modulation in the wavelength range of 6 to 11 µm. On the other hand, by selecting a longer pump wavelength at 9 µm, thanks to the low and flat dispersion, the SC wavelength is pushed to 20 µm (−34 dB) to cover the entire MIR fingerprint regime. The optimal propagation distance corresponding to the broadest SCG in the designed CdTe waveguide is reduced to 2.5 mm due to the shortened soliton fission distance pumped at 9 µm.
Fig. 5. The simulated SC spectra in the CdTe waveguide pumped at 6 µm (a-c) and 9 µm (d-f) wavelengths. The pump peak power in (a, d) and (b, e) is 15 kW and 10 kW, respectively. (c, f) present the input spectra. The propagation distances of 6 µm and 9 µm are 3.6 mm and 2.5 mm, respectively.
The spectra and temporal evolution dynamics of the SCG in the CdTe waveguide are simulated and presented in Fig. 6. The spectra are broadened through self-phase modulation and soliton fission processes. The injected pump pulse undergoes an initial stage of nearly symmetric spectral broadening and temporal compression through self-phase modulation in the waveguide with designed anomalous dispersion. Subsequently the pulse is broken up with strong asymmetric spectral broadening due to the soliton fission and the presence of higher-order dispersion. The continuous broadening of the SC spectra towards 20 µm is resulted by the flat anomalous dispersion profile of the designed waveguide at long wavelengths. As compared in Fig. 6, the soliton fission occurs at 2.9 and 2 mm, pumped at 6 and 9 µm, respectively, with 15 kW peak power. The shortening of the soliton fission distance pumped at 9 µm is due to the larger second-order dispersion at 9 µm than that of 6 µm. Therefore, shorter propagation distance is required for a pump wavelength of 9 µm to achieve the broadest SCG.
Fig. 6. The temporal (a, c, e, g) and spectral (b, d, f, h) evolution of the MIR SCG in the CdTe waveguide. 6 µm and 9 µm pump wavelengths are compared in (a-d) and (e-h), and 15 kW and 10 kW peak power of the pump pulse is employed in (a, b, e, f) and (c, d, g, h), respectively. The propagation distance in the CdTe waveguide pumped at 6 µm and 9 µm is optimized as 3.6 mm and 2.5 mm, respectively, to generate the broadest the SC spectra.
The degree of optical coherence of the simulated SC is determined by calculating the first-order mutual coherence degree $g_{12}^{(1 )}$ according to the following formula [13]:
Fig. 7. The wavelength-dependent first-order degree of coherence of the SC from the CdTe waveguide calculated from 50 simulated spectra pumped at 6 µm (a, b) and 9 µm (c, d) wavelengths. The peak power of 10 kW (a, c) and 15 kW (b, d) are used for both the pump wavelengths to reveal the coherence of the SC output with different pump power. 3.6 and 2.5 mm propagation distance is chosen for the pump wavelength of 6 and 9 µm, respectively, corresponding to the broadest SC spectra.
Besides the broadband SCG, the MIR pulse experiences self-compression in the engineered CdTe waveguide. By choosing a 1.9-mm propagation distance pumped at 9 µm, pulse splitting is supressed, and symmetric spectral broadening is almost completely supported by self-phase modulation. As shown in Fig. 8, the 200 fs input pulse is compressed to 47 fs corresponding to ∼ 1.6 optical cycles centered at 9 µm wavelength.
Fig. 8. (a) The input pulse is compressed from 200 fs to 47 fs after propagating with a distance of 1.9 mm in the designed CdTe waveguide, with a 9 µm, 15 kW pump. (b) The corresponding SC spectrum reveals symmetric broadening.
4. Conclusions
In summary, we propose a new integrated waveguide by employing CdTe which has the transparent window of 1 to 25 µm as well as the largest third-order nonlinearity among the common MIR materials as the waveguide core medium to generate SC in the long MIR wavelength range. CdS is employed as a cladding layer to enhance the confinement of the optical field in the CdTe waveguide core. The dimensions of the CdTe waveguide are designed to realize a low and flat anomalous dispersion. Pumped at 9 µm, the SC spectrum spanning over 3.5 to 20 µm at −34 dB corresponding to 2.5 octave is generated, covering almost the entire molecular fingerprint regime. The pulse self-compression down to 1.6 optical cycles is also observed propagating in the CdTe waveguide. This is the first design of an on-chip MIR SCG extending beyond 13 µm wavelength, to the best of our knowledge. It is worth mentioning that this proposal has solid practical fabrication foundations based on the semiconductor fabrication technologies, and both high quality CdTe and CdS could be grown by mature techniques such as magnetron sputtering, electrochemical, and close-spaced sublimation deposition [36]. This work demonstrates that the CdTe/CdS/Si hetero-waveguide is an excellent on-chip platform of coherent MIR SCG for special molecular sensing (like molecules with organometallic, halogenated and aromatic bonds) and more advanced molecular spectroscopic applications such as multi-dimensional IR spectroscopy. Furthermore, CdTe has both the large second and third-order nonlinearity (second-order nonlinear coefficient: ∼109 pm/V at 1.064 µm [22] and third-order nonlinear coefficient: 5×10−17 m2/W at 1.55 µm, 1.3×10−17 m2/W at 9 µm), thus the proposed design could be extended to other CdTe-based integrated devices based on other nonlinear effects such as four-wave mixing, parametric conversion and Raman shift, for more breakthroughs in the field of ultra-broadband MIR microphotonics.
Funding
National Natural Science Foundation of China (62075144); Engineering Featured Team Fund of Sichuan University (2020SCUNG105).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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