The electromagnetic spectrum between microwaves and mid-infrared light is currently defined as the terahertz band (1 THz = 10¹² Hz, corresponding to 0.3 mm wavelength), a spectral region where electronic and optical technologies overlap. Over the years, the word “terahertz” became a moniker encompassing the entirety of this large spectral region, which ranges from 0.3 to 30 THz, for which many interesting applications appeared to be possible. For the scope of this paper, we focus on the opportunity offered by THz Time Domain Spectroscopy (TDS) for plasma diagnostics applications¹ in nuclear fusion.² The fortunate landscape of the optical properties of plasma in the lower part of the THz spectral region, traditionally called Far Infrared Radiation (FIR), has spawned a class of exceptionally successful diagnostics for magnetic confinement plasma devices, such as tokamaks, since the early days of experimental plasma physics. FIR diagnostics use a combination of optical and electronic techniques, often using the same instrument.^3–5 More recently, the opportunities offered by THz-TDS technology have raised a high level of interest in the diagnostic potential of this technique.^6–8 In particular, the wide spectral range of modern terahertz sources,^9,10 detection speed, possibility of using optical fibers to carry the laser beam to the THz emitter and receiver^11,12 for tens of meters, and ease of access to tokamak plasmas make it a very good candidate to potentially replace a number of traditional fusion plasma diagnostics, such as interferometry, reflectometry, and polarimetry,¹ with one single multi-functional instrument.^13,14

Complications and difficulties are in no short supply. THz measurements in plasma physics and nuclear fusion environments will be significantly different from those in laboratory applications, which have been successful in recent years. To start, the “sample” is a plasma device of macroscopic size, not a fragment of material easily handled and positioned. The weakness of the THz signals is another major difficulty, given the low power of THz sources¹⁰ (μW), long optical paths, and heavy dispersion in nuclear fusion experiment plasmas. The critical alignment between the detector and emitter and the polarization issues will require a thorough optical design of long range THz optics to achieve the required measurement precision signal-to-noise ratio (SNR) while maintaining the temporal resolution, which is the time required for a single measurement, compatible with the time scale of the plasma physics experiments (a few milliseconds). The remainder of this paper is organized as follows: in Sec. II, the basic concepts of THz-TDS are presented. Section III is concerned with the principles of FIR and submillimeter wave plasma diagnostics (interferometry, polarimetry, and reflectometry); then in Sec. IV, the design principles of the foreseeable THz-TDS plasma diagnostics are described with particular attention to tokamaks.^15,16 The conceptual design of the THz-TDS multi-functional diagnostic system is described in Sec. V, and in Sec. VI we present a series of targeted laboratory tests that cover most of the experimental issues and constraints of the method. Finally, Sec. VII is devoted to the discussion and an outline of future work. The results are based on a series of experiments performed over time in collaboration between ENEA Frascati, Clarendon Laboratory at Oxford University (UK), and Centro de Investigation Optica (CIO, Leon, Mexico), specifically devoted to the realization of the diagnostic. The International System of Units (SI) will be used throughout this paper.¹⁷ 

The THz-TDS technique is now applied in many scientific and engineering areas, such as semiconductors, materials and components characterization, biophysics, and medicine.¹⁸ Since it is broadly described in the literature, the details will not be repeated in this paper. Let us just recall the basic principles: the interaction of femtosecond infrared laser pulses (λ ∼ 790 nm, duration <100 fs, repetition rate ∼100 MHz) with a photoconductive plate (for example, GaAs) is routinely used to generate extremely broadband (100 GHz up to 10 THz) single-cycle THz pulses with a typical duration between 2 and 3 ps through a process first observed by Auston, called “photoconductive generation of terahertz radiation.”⁹ Such short transients contain as few as one cycle of the electric field oscillation, thus producing the aforementioned broad spectral range: either the THz or the mid-infrared ranges can be covered in their entirety, depending on the choice of the emitter.¹⁹ 

This means a slow scan time in relation to the modulation period; thus, with the scanner ideally fixed in a given position, the modulation signal runs across many cycles while many more THz pulses occur during one phase when the modulation signal is high, allowing a lock-in average. The fast scanning (∼10 ms) required by plasma diagnostics for fusion experiments is maintained in the above inequalities. It should be noted that the temporal resolution is longer than that of a single scan because, to reach the desired SNR, it is often necessary to average many scans. In any case, the path difference scanning time must remain well above the emitter voltage modulation to enable the lock-in amplifier average of the signal during a single scan. For a preliminary estimate of SNR in THz plasma measurements, we start from the basic assumption that averaging M synchronous measurements of the same physical quantity, the random noise is reduced by a factor 1⁠, while the amplitude of the synchronous signal is, in principle, unchanged.²⁴ Using the standard values in Table I and assuming a duty cycle of the emitter modulation of 50%, the half period will contain 2500 pulses, and consequently the noise will be reduced by a factor of ⁠, an estimate acceptable for laboratory-grade equipment.

Waves are reflected at cutoffs where the refractive index N vanishes and absorbed at resonances (N→ ∞). In the latter case, the local plasma re-emits according to the blackbody laws.²⁷ For completeness, a second hybrid resonance, the Lower Hybrid frequency (LH), should be mentioned. Nevertheless, because it ranges in the MHz region, well away from FIR, it will not be considered in the remainder of this discussion. The above equations open the door to many FIR and millimeter-wave plasma diagnostics in magnetic confinement devices, such as tokamaks, a plasma confinement device based on the combination of a toroidal magnetic field generated by an external coil and a poloidal magnetic field created by the current induced in the plasma ring by a core transformer. Figure 1 summarizes the tokamak configuration and arrangement of the most important FIR diagnostics. Ample and detailed descriptions of tokamak science are available in the literature.^15,16

Radial frequency profiles obtained from Eqs. (1)–(6) for typical tokamak parameters are shown in Fig. 2. They determine the FIR wave propagation inside the plasma. On this basis, many diagnostics have been developed and successfully used: Interferometry exploits the phase shift induced in the probing wave by plasma dispersion. Polarimetry uses plasma birefringence properties to detect the polarization rotation of a FIR beam (Faraday and Cotton–Mouton effects), which can then be used to calculate the magnetic field strength. Reflectometry exploits beam reflection on plasma cutoffs to measure plasma density profiles and fluctuations. Electron Cyclotron Emission (ECE) diagnostic probes the electron temperature by means of blackbody plasma emission at the fundamental frequency in Eq. (2) and its integer harmonics, most important the second. The application of the THz-TDS technique to these diagnostics is discussed in Sec. IV.

THz-TDS offers the unique possibility of directly measuring plasma properties through complete knowledge of the amplitude and phase information carried by the transmitted or reflected THz pulse. By detecting the arrival time of an electric field pulse after its interaction with the plasma, it is possible to determine the plasma refractive index and absorption coefficient (or equivalently the electrical conductivity) directly from the experiment. In addition, the two polarizations of the probing wave can be resolved simultaneously by using photo-conductive polarization-resolving detectors. The diagnostic capability of THz based TDS methods is also more far-reaching than that offered by frequency-domain techniques, such as Michelson interferometry, which is only sensitive to the field amplitude of radiation.³⁰ 

B_‖ is the poloidal magnetic field component parallel to the beam path. By measuring α_Faraday, the convolution between B_‖ and the electron density can be obtained by inversion and then using the density measured along the same line of sight to extract B_‖. For this purpose, interferometry and polarimetry are often combined in tokamaks in a heterodyne-style detection diagnostic^1,32 capable of providing all quantities required to extract the magnetic field. In fusion plasmas, the high densities and temperatures make the plasma magnetic fields comparable to the vacuum field,¹⁶ which makes it essential to properly measure these quantities during the discharge. Although compact, the combined scheme requires additional components, such as beam splitters and additional detectors [Fig. 3(a)]. The use of a THz-TDS spectrometer with polarization-resolving detectors and an emitter and receiver located at the opposite ends of a vertical LOS [Fig. 3(b)] will provide a simplified implementation of the two measurements, as shown by the comparison of the two schemes in Fig. 3. Some of the complications of the classic interferometric arrangement, such as splitting the probe beam into two branches (reference and probe) and then detecting them separately, will be eliminated.

Fusion-relevant plasmas have electron densities in the range 10¹⁸–10²¹ m⁻³, corresponding to plasma frequency between 10 and 280 GHz. The lower part of the THz-TDS spectrum (1–2 THz) satisfies the linearity condition in Eq. (3) and is suitable for diagnostic applications. The successful implementation of this technique will imply the need to rename the diagnostic since the THz-TDS systems are not, strictly speaking, “interferometric.” In addition, because at the present time polarimetry is being superseded by the motional Stark effect for local magnetic field measurements, the THz-TDS diagnostic application could potentially determine its resurrection.

As an example of the order of magnitude of the quantities involved in the diagnostic design, the relevant parameters can be estimated in the simple case of a compact-size tokamak (such as FTU). With a minor radius of a = 0.3 m, a major radius of 0.935 m, a plasma electron density of n_e = 0.5 × 10²⁰ m⁻³, and a poloidal magnetic field of B_θ = 0.1 T, the refraction index at peak density of 1 THz probing beam is N = 0.997 [Eq. (3)]. The phase shift across the full plasma cross section of L = 0.6 m is [Eq. (7)]  4.03. From Eq. (8) α_faraday ≃7 mrad and from Eq. (9) the maximum refraction angle θ_max ≃ 4.03 mrad.

Reflectometry is a technique based on the detection of a microwave beam reflected at the critical density (or cutoff) plasma layer, where the refractive index N vanishes. Inside the layer, the refractive index becomes imaginary and the wave is damped. For the conservation of energy, the wave is reflected.^1,35 The phase variation of the selected frequency components can be used to diagnose the plasma density profiles or density fluctuations. The details of this technique can be found in the E. Mazzucato review³⁶ and in the seminal Heald and Wharton book.²⁵ A typical reflectometry setup consists of an emitter launching a beam of an appropriate frequency along a horizontal chord on the tokamak equatorial plane and a detector to collect the radiation reflected on the corresponding cutoff layer, as shown in Fig. 1. By using a broad band spectrum probe beam and resolving the phase shift of selected components, the radial density profile can be obtained via Abel inversion. This considers the dispersive effect during propagation across non-zero refraction index plasma layers. The THz-TDS reflectometer is an evolution of the ultrashort pulse radar technique³⁷ but with the broadband spectrum required for the diagnostic obtained through a terahertz switch, with a pulse width of 5 ps rather than a microwave transmitter (a pulse width of 5 ns). It must be noted here that the current THz-TDS technology aims at the higher part of the spectrum (>1 THz). In order to probe the tokamak cutoffs (100–200 GHz, see Fig. 2), the THz emitter and receivers and the optics will have to be tailored to operate at lower frequencies avoiding diffractive losses. This will be easily done by using emitters with larger electrode gaps and apertures.

The prototype diagnostic scheme based on a standard THz-TDS system is discussed here with the help of Fig. 4. The femtosecond laser beam is divided into two sub-beams by a suitable beam splitter: a lower power beam (usually 30%) for the receiver (RX) and the other for the scanning stage, followed by the emitter (TX). The two beams are then coupled from free space into two optical fibers of equal length to reach the RX and TX. The emitter and receiver are GaAs wafers with photo-lithography deposited electrodes, which generate a dipole pattern converted into a Gaussian beam by a Si lens. For tokamaks, additional collimating lenses can be introduced in the optical setup. A suitable material is the Polymethylpentene (Poly-4-methyl-1-pentene),³⁸ often referred to under the trade name TPX^®. The use of long optical fibers poses the issue of compensating for the Group Velocity Dispersion (GVD) with a suitable compression stage because the initially narrow THz pulse will be broadened by the dispersive properties of the fibers.³⁹ This is discussed later in this study. A single GVD compressor for the two beams is desirable, as it minimizes losses and reduces optical design complexity, but at the same time requires the use of two fibers of identical length, which means introducing a long free-air compensation path in one of the two branches. This is in contrast to the requirement of entering the optical fiber as soon as possible after compression, owing to the unavoidable degradation of the laser beam that occurs during compression. This is a potential drawback of the fiber coupled system in plasma diagnostics applications because the THz path in plasma (several meters) is part of the TX branch; therefore, to ensure that the TX and RX paths are equal, a long compensation path is required on the receiver branch. The use of two separate compressors with fibers of different lengths would solve this issue. It is advisable to begin with a single GVD compressor and evaluate the results for the final diagnostic design. Classic path-difference scanning based on a translating stage moving a roof-top mirror may not be suitable for the time resolution of tokamak diagnostics. Alternative solutions are helicoidal rotating mirrors that have been successfully tested in ECE diagnostics with millisecond time resolution²⁹ or dual-laser systems, in which the most popular optical ed electronic configurations are asynchronous optical sampling (ASOPS⁴⁰) and electronically controlled optical sampling (ECOPS⁴¹), based on two mode-locked lasers with slightly different pulse repetition rates, to achieve a time scan between the RX and TX without moving optomechanical components.⁴⁰ 

The full exploitation of THz-TDS in the vast landscape of plasma diagnostic applications requires a systematic study of each individual measurement to demonstrate its effectiveness in experimental situations as close as possible to the expected ones and to obtain the parameters required for the final diagnostic design.

The photoconductive and electro-optical detection schemes were directly compared using the same spark-gap emitter shown in Fig. 5(a) (identified by the code FR-05) with two different detection systems: the photoconductive GaAs detector in Fig. 5(b) and an electro-optical setup [Fig. 5(c)]. The overall setup is shown in Fig. 5(d). The emitter FR-05 is made of two parallel conductive strips separated by a 400 μm gap, deposited on a 3 mm thick semi-insulating GaAs substrate. It was aligned with the pump laser beam by minimizing the electrical resistance between the strips.²¹ The emitter was polarized with a square-wave modulated voltage (∼100–200 V), generating a field of 5 kV/cm across the 400 μm gap, bringing the semiconductor close to the spark emission threshold, which is triggered by the laser pulse electric field. In our case, the optimum resistance obtained during the alignment of the emitter photoconductor was 335 kΩ at 120 V bias voltage and 26 kΩ at 200 V.

The photo-conductive detector (identified as FD-12) is made of three gold electrodes deposited on a 0.4 mm GaAs substrate, with annealing at 600 °C. The gap between the ground (common) contact and each of the two electrodes in the X–Y direction [Fig. 5(b)] was 12 μm. Only one polarization was used for this test, although this arrangement has the potential to simultaneously measure the two polarizations. The gap was aligned with the laser using the same resistance minimization technique used for the emitter. The laser pulse, divided by a beam splitter, enables the detector gap during the THz pulse, so a current proportional to the THz intensity (electric field) flows in the gap. It was measured using a lock-in amplifier (signal recovery model 7280), set in current mode, with a sensitivity of 2 nA and phase ϕ = 0.47° after reset to maximize the X component of the signal.

Electro-optical sampling exploits the variation in birefringence induced in a non-linear medium (GaP crystal with 110 orientation) by the incoming electric field. By measuring the intensity of the two polarizations resulting from birefringence in the crystal (for example, with a pair of balanced photodiodes), the intensity of the THz pulse electric field can be inferred.⁴² The measurement parameters are summarized in Table II.

Data were analyzed using a Python script to calculate the Fast Fourier transform, perform data selection, differentiate when required, smooth, and normalize the spectra. The results, displayed in Fig. 6, show that the shape of the normalized spectra is consistent when the better spectral resolution of the EO system is considered.

The expected widening of a 30 fs laser pulse (corresponding to 100 nm bandwidth) with a fiber length of 5 m is 120 × 5 × 100 ∼ 60 000 fs, which means that THz-TDS generation is impossible. To compensate for this GVD, we tested the widely adopted scheme shown in Fig. 7. With systematic scanning of the distance d, the optimum compression was found for d = 27 mm (Fig. 8), obtaining a full width at half maximum (FWHM) of 542 fs, corresponding to a compression factor of >1000. This value is consistent with the estimate d ≃ 50 mm obtained from F. Ellrich paper,¹² while taking into account the different characteristics of the optical fibers and the extra positive dispersion occurring in the system, for example in the triplet collimator and in the gratings glazing. The compressed pulse length achieved in this test (∼500 fs) is still marginal for THz generation and detection but constitutes good experimental proof of the feasibility of the technique, given the ample optimization margins of the system. In addition, for THz plasma diagnostics we are interested in the lower part of the THz spectrum (below 2 THz) because it will be possible to start with a wider laser pulse (>100 fs), which will result in a narrower spectral band and reduced dispersion; hence less compensation is required.

The plasma density of different arc discharge spectroscopic lamps was measured using THz-TDS.⁴⁶Using a simple model⁷ of the THz pulse transmission, the plasma electron density n was estimated to be consistent with the spectral characteristics of the lamps.

As detailed in Sec. IV, the aim of reflectometry diagnostics is to locate the position of cutoff reflecting layers. On this basis, an estimate of the attainable reflectometry spatial resolution was made with a series of measurements of stacks of plain paper sheets (from 1 to 4), made with the Picometrix T-Ray 5000 system described in Sec. VI C. A THz beam splitter was inserted between the emission and detection heads, perpendicular to each other. In this scheme, the THz pulse passes through the beam splitter and reaches the reflecting sample. After reflection on the sample, a fraction of the signal will reach the detection head.

The results shown in Fig. 10 are as follows: with one layer there was no measurable difference from the reference pulse, that is, no detectable optical path variation being the refraction index of paper in the THz range between 1.2 and 1.5. The distance between the paper sheets can be estimated from the spacing between the corresponding peaks in the THz pulse, using the propagation time in free air: 0.3 mm in 1 ps. The separation between two sheets estimated with this technique is 0.7 mm for the first two and 1.3 mm for the second. The convolution of the Fabry–Perot étalons maxima was evident in the reflected radiation spectrum (Fig. 10). The correct understanding of this effect is a key point in plasma diagnostics design, where there are many optical components that can potentially create étalons.

Another demonstration of the potential of this type of measurement was obtained using the same reflectometry system to map, through a 3D scan, the internal structure of a more complex sample (an electronic board), as shown in Fig. 11.

Polarizing optical components based on an inductive grid of metal wires^47,48 are widely used in FIR plasma diagnostics applications because of their high efficiency in the lower part of the terahertz spectrum. According to Marcuvitz et al.⁴⁹ (p. 285), an inductive wire-grid polarizer (WPG) is an infinitely extended grating formed by small metallic obstacles (wires) with axes parallel to the electric field. Alternatively, when the electric field is perpendicular to the wires, the WGP is called capacitive. The WGP parameters are the wire gauge (diameter) (g) and the period of the wire grid (a) (i.e., the spacing between two corresponding parts of the wires). They define the wire spacing as d = a − g. The most important parameter of a WGP is the extinction ratio, which is defined as the ratio of the transmitted power for THz wave polarization perpendicular to the wires in the grid to that parallel to them. For wavelengths larger than the WGP step (a ≪ λ) and a WGP arrangement such that d ≪ a, the grid acts almost ideally, reflecting waves with an electric field E parallel to the wires and transmitting them when E is perpendicular to the wires. In other words, the extinction ratio is very large. For our tests, we used a standard 1-inch, custom made WGP (QMC Instruments QWG/RT) for plasma diagnostics applications. It was made of free standing tungsten wires of g = 10 μm diameter, equally spaced by a = 25 μm center-to-center. With these parameters, inequality a ≪ λ translates into f ≪ 12 THz. This is in good agreement with the full transmission spectral response measured with a nanowire detector setup^50,51 for three relevant orientations of the polarizer, as shown in Fig. 12, which displays a close-to-ideal response up to ∼1.5 THz. Subsequently, nonlinear transmission and noise become dominant (see caption for details).

The same system was used to determine the extinction ratio of the WGP. Figure 13(a) shows the optical arrangement of this measurement, where a THz pulse passes through the WGP (with vertically oriented wires) and is then collected by the horizontal and vertical channels in the nanowire detector. In the experiment, the polarizer was placed close to the emitter, and the wires orientation was fixed. The incident THz polarization was allowed to rotate from 0° to 90° by rotating the emitter. Figure 13(b) shows the dependence on the emitter rotation angle of the peak-to-peak current in the THz pulse detected by our nanowire detector with and without the wire-grid polarizer inserted in the system. No significant spectral response change was observed with or without the polarizer on the horizontal detection channel (Fig. 13, red line and dots), except for the amplitude transmission ratio; therefore, the broad band measurements made with the THz pulse peak are representative of the response of the component. The accuracy of this basic test setup is  (9 mrad), and with reasonable technical improvements, the value of 7 mrad required by plasma diagnostics (see Sec. IV) can be easily achieved.

As expected the transmitted cross-polarized THz radiation detected by the horizontal channel agreed well with the cosine function with respect to the incident THz polarization, whereas the transmitted co-polarized THz radiation detected by the vertical channel was close to zero. The insertion loss of a wire-grid polarizer is defined as the ratio of the transmitted power for THz polarization perpendicular to the wire grids to the transmitted power in the absence of the polarizer. In this case, it was 54%. Furthermore, we observed a small but measurable response signal from the vertical channel when the incident THz polarization was perpendicular to the wire grids. This is due to the leakage of the cross-polarized component of the THz radiation after the polarizer. Such leakage is commonly frequency dependent and increases quickly when the incident THz radiation is above 1 THz, limiting the extinction ratio of the polarizer. The extinction ratio of the WGP was much lower than that of our nanowire detector, which was found to be 40.

The work presented in this article encompasses several years of intensive laboratory research aimed at developing a diagnostic tool for harsh tokamak plasma environments based on THz-TDS technology. To this end, an extensive study of the operational parameters of a THz-TDS plasma diagnostic system was conducted. Interferometry, reflectometry, and polarimetry have been discussed in detail to obtain a full set of instrumental requirements.

Photo-conductive detectors, potentially allowing simultaneous dual-polarization measurements, were studied in depth and compared with a standard THz-TDS electro-optical setup. Valuable experience has been acquired regarding the assembly, alignment, and use of these devices, and the optical arrangement of a plasma diagnostic head working in the Gaussian beam mode has been devised.

Optical fiber operation is essential in plasma diagnostics; hence, we developed and tested a tailored pulse compression stage based on a negative-dispersion grating pair. The device was successfully tested at Clarendon, and an operational set of parameters was defined.

A complementary line of work was developed at the CIO laboratory using commercial equipment with single polarization photo-conductive antennas. The application of this technique to plasma diagnostic measurements have been studied in many cases (polarimetry, refraction index, and reflectometry) to obtain useful insights from techniques performed at the CIO for materials and heritage science.

The last few decades of nuclear fusion research have been dominated by quasi-optical and microwave techniques. Although powerful, while approaching next-generation fusion devices, these techniques start showing some limitations, especially in terms of plasma access, SNR, calibration, and reliability. In this framework, through a series of laboratory tests, we outlined the building blocks of the novel THz-TDS multi-functional plasma diagnostic. To transfer this technique from the laboratory table to the nuclear fusion environment, several key issues will have to be resolved. Neutron emission can affect the hardware near the machine; this is a problem shared with all the plasma facing diagnostics, and the solution developed for it can be applied to THz-TDS as well. Noise induced by mechanical vibrations with loss of alignment (see Sec. IV) is specific to each machine and can be contained with an appropriate GBM design of the optics. Electrical pickup disturbance is a common issue for detectors, tackled with the appropriate choice of lock-in frequency after in situ tests of the noise spectrum.

A powerful network of laboratories has been created, each providing a specific set of skills and techniques. The resulting synergy will be the propulsive force for the final step in the realization of the diagnostic for the next step fusion devices.

We are pleased to thank the people who contributed to this work, particularly M. Johnston for the useful discussions providing advice, encouragement, and continuous support, and for making his Oxford facilities available; E. Castro-Camus for hosting and guiding one of the authors during his Ph.D.; J. Boland for her constructive comments; G. Rocchi and V. Piergotti for precious and continuous technical support; and E. Giovannozzi for his valuable help with Python coding. In no particular order, we acknowledge S. Jones, D. Damry, H. Joyce, J. Lloyd-Hughes, R. Milot, C. Xia, and P. Pattinson.

The authors have no conflicts to disclose.

G. Galatola Teka: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Project administration (lead); Resources (lead); Software (lead); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). K. Peng: Data curation (equal); Formal analysis (equal); Investigation (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). M. Alonzo: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). F. Bombarda: Conceptualization (supporting); Data curation (supporting); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). C. L. Koch-Dandolo: Data curation (equal); Investigation (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). L. Senni: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). A. Taschin: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Software (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). M. Zerbini: Conceptualization (lead); Data curation (lead); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (lead); Project administration (lead); Resources (equal); Software (equal); Supervision (lead); Validation (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.