Swept laser interferometry (SLI) is a common technique in many important applications such as distance metrology [1,2], distributed fiber sensing using optical frequency domain reflectometry (OFDR) [3], autonomous vehicle guidance systems using frequency-modulated continuous-wave light detection and ranging (FMCW-LiDAR) [4,5], gas spectroscopy to detect gas pressure using the broadening in linewidth [6,7], integrated photonics for characterizing microresonators [8], and eye disease identification using swept-source optical coherence tomography (SS-OCT) [9]. However, wavelength tuning nonlinearity is considered one of the most challenging obstacles toward achieving the required measurement precision using SLI technology [10,11].

Ideally, an optical frequency ruler with close, exact, and stable frequency spacing can provide a reference for the correction of SLI wavelength scanning nonlinearities. The closer the spacing between the frequency lines, the higher the sampling rate of the reference frequency ruler and hence the better its nonlinearity correction resolution. Several schemes have been proposed to be used as a reference. These schemes include reference Fabry–Perot (FP) interferometer etalons, Mach–Zehnder interferometers (MZIs), optical frequency combs (OFCs), and fiber ring resonators (FRRs) [12–16]. The FP cavities provide accurate and sharp transmission lines; however, the cavity length needs to be extremely large to achieve higher correction sampling points, which is not suitable for integrated photonics systems. The MZI provides a simple solution for correcting the nonlinearities in frequency sweeping systems and can be easily integrated into silicon photonics platforms [4]. However, detecting the peaks of the sine waves generated from the MZI during the frequency sweeping is not precise, which imposes the use of complex and time-consuming phase detection algorithms to improve the correction precision [17]. OFCs provide an accurate way of determining the frequency scanning range and correcting the sweeping nonlinearity [16]; however, they require several RF filters and complex processing to provide closer reference lines than those provided by the repetition rate of the OFCs. In addition, the predetermination of the laser chirp rates proposed in Ref. [16] allows more dependence of the achieved precision on the laser sweeping nonlinearities and the time gap between the calibration and the actual measurement. Despite the advancement in the integrated OFCs, they have several drawbacks since they operate at a high repetition rate (i.e., $>10\mathrm{GHz}$), which increases the number of filters and makes the processing more complex to achieve closer reference lines. In addition, high laser powers are required for their operation due to their high threshold and low conversion efficiency. Moreover, active stabilization is needed to align the laser frequency to the resonance. All of these drawbacks motivate the search for suitable passive solutions. Recently, FRRs have been proposed to generate equidistant narrow sharp transmission lines to act as a robust reference marker for correcting sweeping nonlinearities [14,18]; however, this fiber-based frequency reference is not CMOS compatible and could not represent a solution for the fully integrated on-chip LiDARs.

With the recent advances in integrated photonic components including the development of heterogeneously integrated narrow-linewidth lasers and mode-hop free tunable lasers [19,20], the need for an on-chip CMOS compatible solution for correcting the laser sweeping nonlinearities becomes more crucial. Some on-chip solutions have been proposed for correcting the laser sweeping nonlinearities in LiDAR applications, namely, the on-chip MZI and microresonators [4,21]. Although the on-chip MZI provides fringes with continuous phase information, detecting the phase of the fringes requires computation resources and a time-consuming phase detection algorithm. Even though detecting the sine-wave peaks of the MZI fringes limits the measurement precision [14], on the other hand, the high-Q microresonators provide sharp resonances with easy detection of either the peak or the edges. However, the spacing between the resonances is in the GHz range ($>10\mathrm{GHz}$) which leads to the loss of information in the spacing between the resonances, which limits their precision and applicability in narrow sweeping ranges application. Consequently, an on-chip solution that provides sharp resonance lines with spacing in the MHz range is required.

Another major advancement in integrated photonics platforms is the development of ultralow-loss silicon nitride waveguides and resonators that can reach loss below 0.1 dB/m [22,23], which will lead in turn to the development of high-quality passive photonics structures on chip. In this paper, a spiral resonator (SPR) with a spiral length of 7 m which is fabricated from ultralow-loss silicon nitride is introduced to act as a narrow-spacing frequency ruler for correcting the frequency sweeping nonlinearities in tunable lasers. The SPR is designed in a small footprint and reproducible design, which would greatly expand its outreach and applications of the SLI technique in the integrated photonics platforms. Full characterization of the SPR is made for the parameters that contribute to the accuracy of the frequency ruler such as the frequency spacing between the frequency lines [free spectral range (FSR)], resonator quality factor, resonance linewidth, and the dependence of the line spacing on the temperature variations and wavelength change. Finally, a practical example of using the SPR for the FMCW-LiDAR application will be given to demonstrate the potential of the SPR to enhance the measurement accuracy of these systems.

Ultralow-loss waveguides are required for achieving high-quality factor resonators, especially for meter-scale resonators. Scattering from the rough waveguide walls is the main reason for loss; therefore, waveguides with a large aspect ratio ($\text{width}/\text{thickness}\gg 1$) are desirable for achieving lower propagation loss [24]. By increasing the width of the waveguide, the loss due to the sidewall scattering is reduced and the light propagation inside the ring resonator will be in the whispering gallery mode; and by decreasing the thickness of the waveguide, the scattering loss is minimized due to the lower modal confinement [24]. Figure 1(a) shows the cross-section of the SPR chip, which exhibits the ${\mathrm{Si}}_3{\mathrm{N}}_4$ waveguide width of 10 μm and thickness of 100 nm. It is worth mentioning that, although dispersion can be tailored by increasing the thickness of the ${\mathrm{Si}}_3{\mathrm{N}}_4$ waveguide, thin ${\mathrm{Si}}_3{\mathrm{N}}_4$ waveguides provide lower losses due to their lower scattering loss. In addition, designing a single-mode waveguide is much easier than designing the thick waveguide and CMOS foundry fabrication for thick waveguides ($>400\mathrm{nm}$ of LPCVD silicon nitride) is not readily available. Moreover, the thin waveguide has a lower thermo-optic coefficient than the thicker waveguide since the propagation of mode is mostly in the silicon dioxide clad which has around 2 times less thermo-optic coefficient than that of the silicon nitride. A wide waveguide provides a bending radius greater than 700 μm to avoid the radiation loss caused by a small bending radius of curvature [24]. The upper and lower ${\mathrm{SiO}}_2$ claddings have thicknesses of 2 and 14.5 μm. The cavity of the SPR, having its cross-section area in Fig. 1(a), is designed by connecting two multimode Archimedean spirals with an S-shape bend at the center [25], and connecting the other ends of the two spirals at the outer part. The spiral waveguide is adiabatically tapered from 10 μm to 2.8 μm to match the waveguide width of the S-bend at the center to allow the propagation of only the fundamental mode while damping the higher-order modes. Figure 1(b) shows a simple schematic of the SPR, which shows the structure of the SPR, input port, and drop port. Two similar directional couplers supporting single TE-mode waveguides are used to construct the input and output ports of the ring resonator. As shown in Fig. 1(b), the light is coupled to the spiral resonator at the input port and is coupled out either from the through port or the drop port. The gap width of the directional couplers is tapered to minimize the coupling losses [26]. Figure 1(c) shows a photograph of the SPR compared to one US cent to show the small footprint of the resonator. The fundamental TE-mode profile of the $10\mathrm{\mu m}\times 100\mathrm{nm}$${\mathrm{Si}}_3{\mathrm{N}}_4$ waveguide is plotted in Fig. 1(d) to show the low confinement of the mode due to this thickness. The intrinsic and the loaded quality factors measured for the SPR are measured to be around 75 to 85 million at 1550 nm, as shown in Fig. 1(e).

Several techniques have been developed for the measurement of the FSR of high-quality factor microresonators and Fabry–Perot interferometers. Among those techniques are the frequency difference technique [27], the null method technique [28], and the phase-modulated sideband technique [29]. The last method is commonly used for the measurement of the FSR and relies on referencing the measurement of the oscilloscope time trace to the frequency spacing between the carrier and a modulated sideband. Although this method is commonly used to measure the FSR of microresonators, its accuracy is still insufficient since it still depends on the oscilloscope resolution. Recently, Luo et al. scanned a laser over a fiber ring resonance and phase-modulated its sidebands to near the next resonance [18]. The superposition of the carrier and the sidebands is detected in the time domain and the amplitude of their fast Fourier transform (FFT) is employed as an offline control signal in an iterative process to identify the FSR. Here, a real-time and precise measurement technique for the resonator’s FSR and linewidth is introduced by locking the laser frequency to one of the resonance peaks while sweeping its modulation sidebands on the adjacent resonances, which is schematically demonstrated in Fig. 2(a). The heterodyne interference between the laser and its constantly swept sidebands gives an amplitude-dependent spectrum on the electrical spectrum analyzer centered at the FSR frequency, as depicted in Fig. 2(b). By fitting a Lorentzian curve to the spectrum, the linewidth and the FSR are obtained. The measurement setup consists mainly of a narrow-linewidth laser, a frequency locking system to lock the laser frequency to the central resonance, and a frequency modulation system as demonstrated in Fig. 2(d) and explained in more detail in Section 7 of this paper.

The laser is locked to the central resonance by slightly dithering the frequency of the laser to generate a dispersion-like signal, which is shown in Fig. 2(c). Proportional-integral-derivative (PID) uses the generated dispersion-like signal to lock the laser frequency to the resonance peak. The laser is additionally amplitude modulated using an electro-optic modulator (EOM) such that the modulation sidebands match the peaks of the next resonances, and the heterodyne beat between the laser central frequency and its sidebands shows maximum amplitude for maximum transmission of the sidebands. On the other hand, when the modulation frequency is detuned from the FSR center frequency, the amplitude of the interference signal decreases due to the decrease in the sideband transmission power. By sweeping the EOM modulation frequency around the sideband frequency using the tracking generator of a spectrum analyzer, while acquiring the beat on the same spectrum analyzer, a transmission peak will be displayed which holds the FSR frequency and resonance linewidth; see Fig. 2(b).

The FSR of the SPR is measured at a temperature of 23°C and a wavelength of 1556.2 nm. However, it is expected that the temperature will be different by the time of application, which will cause a slight change in the FSR by an amount determined by the thermo-optic coefficient of the SPR. In addition, the FSR will change during the sweeping of the laser wavelength by an amount determined by the chromatic dispersion of the SPR. Therefore, it is necessary to predetermine the chromatic dispersion and the thermo-optic coefficients of the SPR before the application, which are not necessary to be that of silicon nitride material, to have accurate measurement results. The chromatic dispersion and the thermo-optic coefficient of the SPR are determined as follows: Chromatic dispersion: Several methods have been proposed for measuring microresonator dispersion, which are summarized in a review article by Fujii et al. [30]. Here, a slightly modified method based on the calibrated reference Mach–Zehnder interferometer method is presented in the paper. An FRR is designed, fabricated, and calibrated to be used as a reference for dispersion measurement. The FRR is designed to have resonances that are similar in FSR and linewidth to that of the SPR, but with much lower dispersion due to the much lower dispersion of the silica-based fibers when compared to that of the ${\mathrm{Si}}_3{\mathrm{N}}_4$ waveguides. As demonstrated in Fig. 3(a), the light from the tunable laser is divided between the SPR and the reference FRR. The wavelength of the tunable laser is swept over steps of 10 nm, while recording the acquired peaks on an oscilloscope. After correction of sweeping nonlinearities using the calibrated FSR of the FRR, the SPR integrated dispersion parameter is calculated for the 10 nm wavelength range from the change in the FSR from that of the center wavelength and plotted against the resonance number, as shown in Fig. 3(b). The second-order dispersion is calculated from the parabolic fitting of the integrated dispersion. Figure 3(c) depicts the group velocity dispersion (GVD) and the chromatic dispersion coefficient (D) over the wavelengths from 1500 nm to 1620 nm. GVD and D are calculated from the change of the FSR with optical frequency as described in detail in Section 7.Thermo-optic coefficient: As the SPR temperature increases, its refractive index will increase which will lead to an increase in the optical length of the SPR and a decrease in its absolute resonance frequency. Provided that the laser frequency is stable during the course of the measurement and the FSR is well known from the measurement in Section 3, the decrease in the resonance frequency can be calculated easily by counting the number of resonance peaks on an oscilloscope as the SPR temperature changes (providing that FSR change is negligible with respect to the absolute resonance frequency change). Therefore, the thermo-optic coefficient ($\frac{\mathrm{d}n}{\mathrm{d}T}=-\frac{n}{\nu }\frac{\mathrm{d}\nu }{\mathrm{d}T}$) is calculated to be $1.08\times {10}^{-5}{\mathrm{K}}^{-1}$ from the frequency change ($\mathrm{d}\nu$) that corresponds to a temperature change ($\mathrm{d}T$), as demonstrated in Fig. 3(d). This thermo-optic coefficient is very near to that of the silicon oxide due to the low confinement of the mode [31]. As an example, a change of 1°C will lead to a change in the FSR by 185 Hz. The thermo-optic coefficient will have a minor effect on the FSR during the measurement but can shift the absolute frequency of the resonance by several FSRs; however, due to the slow nature of temperature change which is much slower than the sweeping speed (1 FSR in 40 μs), this absolute frequency shift will not influence the measurement.

This section aims to demonstrate the principle of nonlinearity correction in laser sweeping using the transmission peaks of an SPR. Measuring the FSR of a fiber-based unbalanced MZI is targeted in this demonstration since it is the core of several applications such as FMCW-LiDAR, OFDR, and other SLI experiments. A simple schematic for the experiment is depicted in Fig. 4(a). The output from a tunable laser is split between two ports to have 90% of the light to the SPR and the other 10% to the unbalanced MZI. Edge coupling to the SPR chip is made by cleaving the fiber end and using index-matching gel to facilitate coupling and avoid reflections from the air gap. A normal single-mode fiber with a polarization rotator is used here for the experiment to be compatible with other components in the laboratory. The transmission peaks from the SPR and the interference fringes from the MZI are detected using two photodetectors and acquired by two channels of an oscilloscope. The laser is set to have a sweeping period of 1 nm and a sweeping speed of 5 nm/s, which means that the oscilloscope will acquire around 4850 SPR peaks at 200 ms. In Fig. 4(b), two zoomed oscilloscope traces for the MZI (upper) and the SPR (lower) are demonstrated before applying the correction. Two similar double arrow lines are drawn at the MZI trace to demonstrate the slight change in the periods; however, the change in the period can be more severe in other cases. Even with this slight change the FFT of the varying sinusoidal trace shows great broadening in the FSR peak, as shown in Fig. 4(c). After applying a program to correct the laser sweeping nonlinearity, the broadened FSR peak is reduced to a narrow peak, the exact FSR of the unbalanced MZI, as demonstrated in Fig. 4(d). The program is based on referencing the MZI trace to detected sharp peaks with very well-known frequency spacings to resample the time scale of the oscilloscope into frequency scale. The spacing between the SPR peaks is precisely measured by the swept frequency modulation technique demonstrated in Section 3 to be exactly 25.566 MHz. The corrected FSR of the MZI is 40.31 MHz with a shift of 2.25 MHz from the uncorrected FSR. This correction is crucial to several applications such as in FMCW-LiDAR, as will be demonstrated later in the paper.

Precise ranging is important for several applications including metrology, military field, large-scale manufacturing, and autonomous vehicle driving. Several techniques have been commonly used for ranging applications such as the time-of-flight, correlation of femtosecond pulses, mode-locking, and frequency sweeping interferometry [32–36]. Among these techniques, frequency sweeping interferometry allows precise ranging with a cost-effective setup; however, measuring the frequency sweeping range is considered a challenge. With the recent advancement of integrated photonic circuits and integrated tunable lasers, the need for on-chip frequency reference has become of crucial importance. Contrary to the currently used on-chip MZI with a sine-like signal, the sharp slopes and peaks of the spiral resonances determine the correction locations more precisely. Here, a round-trip distance of up to 40 m is measured using the frequency sweeping interferometry. The setup consists of a tunable laser with two output ports, the first is directed to the SPR for nonlinearity correction, and the other port is connected to a collimator. As depicted in Fig. 5(a), the collimator sends the collimated beam from a tunable laser to a retro-reflector which is placed at the target location to which the distance needs to be measured.

The reflected beam from the reflector is made to interfere with the light reflected from a reference arm at the photodetector (PD2). As the laser wavelength of the tunable laser is swept, a sinusoidal signal is generated at the second channel of the oscilloscope and the reference peaks of the SPR are generated at the first channel with similar curves to those shown in Fig. 4(b). After using the calibrated SPR to correct for the sweeping nonlinearity of the tunable laser, the round-trip distance is calculated by taking the FFT of the corrected signal. The FFT of the signal before and after applying the correction is very similar to the signal shown in Figs. 4(c) and 4(d), respectively, but shifts in frequency for each measured distance. For each distance, at least 10 measurements are taken so that the standard deviation can be calculated. Figure 5(c) shows the standard deviation of the round-trip distances up to around 40 m without using the SPR trace for nonlinearity correction for 10 nm sweeping range, which reaches around 4 m for the longest measured distance. In contrast, when applying the nonlinearity correction using the SPR trace, the standard deviation reaches less than 270 μm for the longest measured distance of 40 m as shown in Fig. 5(d), which means an enhancement of 4 orders of magnitude in precision. Figure 5(b) demonstrates the degradation of the precision with the decrease of the frequency sweeping range, which reflects that even with a range of 4 GHz, the precision is still under 1.5 mm, unlike other publications that use the MZI or ring resonators as a reference where the precision is in the few cm levels [4,21,37,38]. Table 1 demonstrates a comparison between the precision achieved in this work and the precision achieved by other works that use integrated MZI or ring resonators as a reference for mitigating the laser sweeping nonlinearity. Although different types of lasers with intrinsic linewidths down to 22 Hz are used in these experiments [21,37], the precision of determining the distance depends mainly on the reference used for correcting the laser sweeping nonlinearity.Table 1.

Comparison between the Precision Achieved by FMCW-LiDARs Referenced to the SPR in This Work Referenced to MZI in Other State-of-the-Art Publications^a

In a work published by Lihachev et al. [21], they locked a laser to a ring resonator with an attached actuator for active control of the resonator length. They tuned the length of the ring resonator to linearly sweep the laser over only 1.2 GHz limited by the locking range, which enabled them to obtain 12.5 cm precision in measuring 10 m distance. In Ref. [37], a short delay-line self-heterodyne Mach–Zehnder interferometer is used to generate a chirped frequency while sweeping the laser over up to a frequency range of 4.2 GHz. Hilbert transformation of the chirped frequency is performed to deduce the instantaneous frequency. They take the previous distortions to calculate a function to correct for the next sweeps. An iterative learning algorithm is performed of up to 15 iterations before the real measurement to obtain a linear FMCW. A LiDAR experiment is performed using this technique for ranging up to 31 m with 4.6 cm precision. The same technique has been used by Sayyah et al. [38] to correct for the laser sweeping nonlinearity to enable 16.7 cm measurement accuracy for the 75 m range. In another paper by Martin et al. [4], they used a Mach–Zender interferometer with an external 5 m fiber delay line to have a detectable FFT signal from a laser sweeping range of 525 MHz; however, they reported a precision of 28 cm in measuring a distance of 60 m. Here, we used the spiral resonator to measure a round-trip distance of 40 m with a precision of 100 μm when the laser sweeping range was 10 nm and a precision of 1.5 mm when the sweeping range was 4 GHz, which demonstrates the significant improvement in precision when the 7 m spiral resonator is used. This range depends only on the space available inside the laboratory and can be extended. The extension of the range depends mainly on the linewidth of the laser used since narrower linewidths assure high coherence ranges. For the work published by Xiang et al. [19] for on-chip lasers, Lorentzian linewidths down a few Hz could be achieved.

The 7 m spiral resonator was fabricated on a 200 mm diameter silicon substrate at a commercial CMOS foundry [22]. The 100 nm thick stoichiometric ${\mathrm{Si}}_3{\mathrm{N}}_4$ waveguide layer was deposited via LPCVD on top of a 14.5 μm thick thermal oxide. The top cladding consists of a 2 μm thick layer of ${\mathrm{SiO}}_2$ deposited by LPCVD using a TEOS. The wafers are annealed at 1150°C for over 20 h to drive out residual hydrogen in the LPCVD-deposited layers, drastically reducing the absorption loss.

Measuring the quality factor of the SPR is performed by sweeping the tunable laser (Santec TSL510) over steps of 1 nm each while detecting the transmission from the SPR. To correct for the laser sweeping nonlinearity, a ring fiber resonator is designed to have similar FSR and resonance linewidth to that of the SPR. An oscilloscope (Keysight-DSO3012A) with 250,000 of record length is used to acquire the transmission peaks (around 5000–6000  peaks in 1 nm) of each of the SPR and the ring resonator. A computer program uses the fiber ring peaks with the known spacing on the first oscilloscope channel to convert its time scale into a uniform frequency scale. A fitting is applied to each SPR resonance peak to calculate the parameters needed to calculate the intrinsic and the loaded quality factors.

An NKT Photonics (Koheras-Adjustik) single-frequency fiber laser is coupled to the SPR using a normal single-mode fiber with index-matching gel to avoid reflections. The light from the drop port is coupled to a single-mode fiber connected to the 200 MHz bandwidth photodetector. The laser frequency is locked to one of the SPR resonances by modulating the fiber laser piezo input using a lock-in-amplifier (SRS MODEL SR850) to generate the dispersion-like error signal, shown in Fig. 2(c). The error signal is needed to lock the laser to the resonance using a laser servo-controller (Newport-LB1005) which is connected to a function generator (SRS-DS345) to generate the sawtooth signal for sweeping the laser over the resonance. A spectrum analyzer (Anritu MS 2712E) with a sweeping generator is used to drive an electro-optic modulator so that the generated sidebands are swept over the next resonances. The beat between the center frequency and the sideband is detected on the photodetector, as shown in Fig. 2(b). The center of this spectrum represents the FSR of the SPR and the linewidth represents the resonance linewidth. The FSR of the SPR is measured 10 times to be 25.566 MHz with a standard deviation of 14 kHz.

The fiber ring resonator is used as a reference for measuring the SPR chromatic dispersion since the chromatic dispersion of the silica-based fiber is well known from previous measurements using the optoelectronic-oscillation technique by the same authors [39]. In addition, it is expected that the silica-based fiber has much lower chromatic dispersion than the silicon nitride waveguides. The FSR of the fiber ring resonator is measured using the modulation sweeping technique discussed previously at a wavelength of 1556 nm to 27.3 MHz. The FSR is then corrected for the chromatic dispersion at each wavelength from 1520 nm to 1630 nm using the values obtained from the fiber chromatic dispersion measurement. The laser is swept over 10 nm, to ensure a wide enough range to observe the change in chromatic dispersion change, while recording the peaks in two channels of an oscilloscope. The oscilloscope (Yokogawa DLM 2022) should have a large record length (6.25 million) to enable the acquisition of a large number of peaks at the 10 nm sweeping range (around 50,000 peaks) with sufficiently high resolution. A computer program is used to compare the spacing between the resonances of both the SPR and the calibrated SRR to deduce the deviation of the SPR resonance spacing from that of the SRR. The integrated dispersion parameter is calculated for the 10 nm wavelength range from the change in the FSR from that of the center wavelength using the relation $D_{\mathrm{int}}=({\omega }_{\mu }-{\omega }_o)-\mu D_1$, where ${\omega }_o$ is the center frequency, $\mu$ is the number of resonances from the center frequency, $D_1=2\pi \cdot {\mathrm{FSR}}_o$, and ${\mathrm{FSR}}_o$ is the free spectral range of the SPR at the center. The integrated dispersion [$D_{\mathrm{int}}(\mu )$] is plotted against the resonance number ($\mu$), as shown in Fig. 4(b). The second-order dispersion is calculated from the parabolic fitting of $D_{\mathrm{int}}(\mu )$ using the relation of $D_{\mathrm{int}}(\mu )=0.5D_2{\mu }^2$. The GVD (${\beta }_2$) is calculated from the experimental results from the variation of the free spectral range [$\mathrm{FSR}(\omega )$] with the frequency, ${\beta }_2=\frac{1}{L}\frac{\mathrm{d}}{\mathrm{d}\omega }(\frac{1}{\mathrm{FSR}(\omega )})$, and $L$ is the length of the spiral. Then, the SPR GVD (${\beta }_{2\_\mathrm{SPR}}$) is extracted by subtracting the fiber GVD from the calculated values. The chromatic dispersion coefficient ($D$) can be calculated from $D=-\frac{2\pi c{\beta }_{2\_\mathrm{SPR}}({\omega }_n)}{{({\lambda }_n)}^2}$, where $c$ is the speed of light in free space. Hence, $D$ is calculated at the center angular frequency (${\omega }_n$), corresponding to the central wavelength (${\lambda }_n$), of each 10 nm wavelength span over the wavelength range from 1500 nm to 1620 nm.

Acknowledgment. We acknowledge Mario Paniccia and Avi Feshali for assistance in managing the foundry run. O. Terra acknowledges the support of the Fulbright scholar program.