Optical frequency metrology in the bending modes region

Optical metrology and high-resolution spectroscopy, despite impressive progress across diverse regions of the electromagnetic spectrum from ultraviolet to terahertz frequencies, are still severely limited in the region of vibrational bending modes from 13 to 20 µm. This long-wavelength part of the mid-infrared range remains largely unexplored due to the lack of tunable single-mode lasers. Here, we demonstrate bending modes frequency metrology in this region by employing a continuous-wave nonlinear laser source with tunability from 12.1 to 14.8 µm, optical power up to 110 µW, MHz-level linewidth and comb calibration. We assess several CO2-based frequency benchmarks with uncertainties down to 30 kHz and we provide an extensive study of the v11 band of benzene, a significant testbed for the resolution of the spectrometer. These achievements pave the way for long-wavelength infrared metrology, rotationally-resolved studies and astronomic observations of large molecules such as aromatic hydrocarbons. The lack of widely tunable single-mode lasers in the long-wavelength part of the mid-infrared region has hindered optical metrology of bending vibrational modes. Here, a continuous-wave nonlinear laser source with tunability from 12.1 to 14.8 μm, coupled with a frequency comb, is used to measure bending modes frequencies of carbon dioxide and benzene with accuracy down to 30 kHz.

O ptical frequency metrology requires the capability to measure optical frequencies with high precision and accuracy. A traditional approach to this kind of measurements is the combination of a narrow-linewidth continuouswave (cw) laser probing a molecular transition at high resolution and a second stable laser acting as a frequency reference for the probe laser. Since their invention almost 20 years ago 1 , optical frequency combs have become the primary choice for the reference laser, as their discrete regularly spaced spectrum ensures at once repeatability and calibration of the frequency axis of the probe laser to a primary time reference.
The lack of continuously tunable single-mode lasers in the region from 13 to 20 μm 2 (770-500 cm −1 ) has been a stumbling block for optical metrology and high-resolution spectroscopy in the region of vibrational bending modes 3 , which offers intense and unique spectroscopic fingerprints of a large variety of molecules, including volatile organic compounds of environmental 4 , biomedical 5 and astrophysical interest 6 . Cryogenically cooled lead-salt diode lasers have almost disappeared in recent years 7 . They have been replaced by room-temperature distributed feedback quantum cascade lasers (DFB-QCLs) that offer portability, stable operation and mode-hop-free tuning 8 . However, they are commercially available only below 13 μm 9 , whereas the emerging technology of InAs-based long-wavelength QCLs is still under development and not available for applications 10 . An alternative approach is difference frequency generation (DFG) from cw Ti:sapphire 11,12 or diode 13 lasers, but this is accompanied by extremely low optical powers (10-100 nW range) that impair the acquisition of absorption spectra at a high signal-tonoise ratio (SNR). The limitations of laser technology have established Fourier-transform spectroscopy driven by incoherent light sources as a gold standard for the long-wavelength infrared (LWIR) region, yet not for optical metrology 14,15 and precision spectroscopy studies 16 because of the low resolution (0.0007 cm −1 or 21 MHz in the best cases 17 ) and the lack of absolute calibration for the frequency axis 18 . A powerful solution to both issues is direct comb spectroscopy 19,20 , which exploits millions of laser modes that compose the comb spectrum to directly probe molecular absorption over extremely large bands and with absolute frequency calibration 21 . Thanks to a time-domain counterpart formed by a coherent train of femtosecond laser pulse 20 , frequency combs are ideal sources for efficient nonlinear frequency conversion deep into the mid-infrared (MIR) 22 , also fostered by innovative solutions for laser gain media 23 , nonlinear materials 24 and frequency conversion schemes [25][26][27] . However, it has only been recently that snapshots of entire bands have been acquired by a dual-comb approach 28 at high temporal and spectral resolution in the LWIR, from 6.7 to 16.7 µm (1500-600 cm −1 ) 29,30 . This achievement does not remove, however, the need for cw probe lasers to observe single lines with high accuracy 3,14,15 and extreme speed 31 in a region where the Doppler width and the typical line densities are consistently below the spacing between adjacent comb modes. Furthermore, to the best of our knowledge, no frequency data of metrological quality have been provided so far in the LWIR region.
In this paper, we introduce a spectrometer that enables metrological studies in the bending modes region of two symmetric molecules, namely carbon dioxide and benzene. The spectrometer includes a nonlinear laser source, based on DFG between a quantum cascade and a CO 2 laser, covering the 12-15 μm region almost continuously with optical power up to 110 μW, 2.2 MHz linewidth and frequency calibration against a thulium (Tm) comb. We assess several frequency standards as accurate as 30 kHz between 14 and 15 μm based on CO 2 transitions. We also apply the spectrometer to the rich and congested absorption spectrum of the ν 11 bending band of benzene, resolving 729 lines and retrieving the rotational parameters of the excited state with increased accuracy over the state-of-the-art. These outcomes attest to the value of our approach for extending high-resolution spectroscopy and optical metrology to the yet unexplored longwavelength part of the mid-infrared spectrum.

Results
Technical approach. The spectrometer is schematically shown in Fig. 1a. The laser source is based on the DFG process between two MIR sources, namely a cw DFB-QCL and a CO 2 laser, in an orientation-patterned gallium arsenide (OP-GaAs) crystal 24 . The DFG laser provides highly coherent radiation between 12.1 and 14.8 μm (826 and 675 cm −1 ) with optical power up to 110 μW and 2.2 MHz linewidth. The broad spectral range comes from the coarse wavelength adjustment of the CO 2 laser (from 9.23 to 10.71 μm) combined with the fine temperature-based tuning of the QCL (over~4 cm −1 around 5.69 μm) and the fan-out structure of the poling periods of the OP-GaAs crystal (see "Methods" for details). The optical power is 2-3 orders of magnitude higher than previous DFG sources starting from cw Ti:sapphire lasers with GaSe crystals 11,12 . The obtained power agrees within 15% with our calculations using 63 pm V −1 as the effective nonlinear coefficient. Such an excellent agreement reflects the quality of the crystal and a very favourable MIR-MIR interaction that ensures similar diffraction conditions for the interacting beams and thus optimal spatial overlap over a long crystal. The absolute frequency calibration of the DFG radiation comes from the referencing of both QCL and CO 2 lasers to a 100 MHz comb via sum frequency generation (SFG) 32 (see "Methods" for details). As sketched in Fig. 1b, the CO 2 laser remains offset-locked to the nearest comb mode during the measurements, whereas the frequency of the temperature-tuned DFB-QCL is tracked against the comb by real-time fast Fourier-transform (FFT) analysis and barycentre calculation of its beat note. Synchronous acquisition of the beat note and gas transmission allow spectra to be straightforwardly calibrated.
Carbon dioxide metrology. Figure 2a shows an example of a comb-calibrated transmission spectrum for some intense lines of the v 2 band of CO 2 in the 675-689 cm −1 (14.5-14.8 µm) range. Lines are well isolated and in excellent qualitative agreement with the high-resolution transmission molecular absorption database (HITRAN 33 ) simulations. Figure 2b shows three independent experimental traces of the R(14) line, reported in different colours to highlight their reproducibility over the two measurement axes. The presence of spectral gaps is due to the missed calibration that occurs when the comb-QCL beat note approaches either direct current or half the spacing between neighbouring comb modes (50 MHz, see "Methods" for details). With an additional spectral acquisition at a slightly different comb repetition rate, interleaved spectral points could be added to fill the gaps (more details can be found elsewhere 34 ). They do not, however, represent an issue when fitting the data (black line in Fig. 2b): the residuals in Fig. 2c refer to a Voigt model and give an SNR of approximately 300 with a measurement time of 2 s. This translates into a statistical uncertainty of about 140 kHz (4.7 × 10 −6 cm −1 ) on the inferred line-centre frequency, mostly due to the reduced Doppler broadening (30 MHz or 10 −3 cm −1 level) given by the LWIR range and to the negligible collisional broadening at the low pressure used here.
For each measured line, Fig. 3a (for lines pertaining to the R branch of the v 2 band) and Fig. 3b (for lines pertaining to the Q branch of the v 1 ← v 2 band) show the difference between its absolute line-centre frequency and the value in HITRAN, together with an error bar dominated by statistical uncertainties (see "Methods" for details). Some lines have been intentionally measured several times to improve their line-centre uncertainty, which is below 30 kHz for three lines (see Supplementary  Table 1), and to provide frequency standards in a region where absolute calibration has not been reported so far, to the best of our knowledge. The weighted deviation from HITRAN, averaged over 24 transitions, amounts to a remarkably low 0.34 MHz (1.1 × 10 −5 cm −1 ), which is consistent with the declared HITRAN uncertainty (0.3-3 MHz or 10 −3 -10 −4 cm −1 ). The line centres reported for each line in the Supplementary Table 1 offer a metrological benchmark to calibrate past and future spectra in the 670-720 cm −1 (14-15 μm) region.
Benzene precision spectroscopy. In Fig. 4, we report the experimental spectrum of the R branch of the v 11 band of benzene over 675-688 cm −1 (14.5-14.8 µm). This intense, parallel band is an interesting testbed for the spectral resolution and frequency repeatability of the spectrometer because of the large density of rotational lines. Apart from transitions with quantum number K = 0, 1 and 2, whose separation is smaller than the Doppler width, all K lines are well resolved. This is shown in the insets in Fig. 4 and is particularly evident at large J. Lines from three hot bands, namely v 11 + v 16 − v 16 , v 11 + v 6 − v 6 , and 2v 11 − v 11 , are also present in the recorded spectral region, but the analysis was restricted to the 729 ro-vibration transitions of the ν 11 band, with J up to 38 and K up to 36, since these are mostly isolated, medium  intensity lines. We adopted a weighted least-squares fitting procedure to determine highly accurate term values of the excited rovibration levels, according to the equation:  35 and Eq. (1) with v = 0, E 0 v ¼ 0 and keeping all H centrifugal distortion parameters fixed to zero. A unitary weight was attributed to the isolated lines, whose wavenumber precision is estimated to be 10 −5 cm −1 (300 kHz). The weights of blended or weak lines were reduced to 0.01, and, in addition, for lines with multiple assignments, the weight was further reduced by a factor   Fig. 4 Absorption spectrum of the R branch of the v 11 band of benzene. Experimentally measured transmission at the output of a 0.67 m long cell at a pressure of 13.3 Pa and room temperature. The spectral features have a typical shape, with the K-structure degrading regularly towards higher wavenumbers and an apparent intensity alternation of subsequent lines. The insets show zoomed-in views of R K (6) and R K (33) features (left and right inset, respectively): for the latter, the K-splitting is much more manifest together with intensity alternation of K components, owing to spin statistics, that varies as 10:11:9:14, for K = 6p, 6p ± 1, 6p ± 2, 6p ± 3, respectively, (p = 0, 1, 2, 3…) 41 .
equal to the number of overlapped transitions. Observed and calculated linecenter values, along with the weighting factors, of the benzene v 11 band are provided as Supplementary Data. The parameters obtained from the best fit are listed in Table 1, and compared with those of the ground state. All parameters reported in Table 1 are statistically well determined. The uncertainty of the band origin (E 0 v ) has been reduced by about one third with respect to its value in Hollenstein et al. 36 . The B, C, D J , D JK and D K constants are consistent and close to those of the ground state 35 . The values of B, D J and D JK are more accurate than those in Hollenstein et al. 36 by one order of magnitude. The values of C, D K and H J constants of v 11 = 1 are determined here for the first time. The standard deviation of the fit, 5.1 × 10 −5 cm −1 (1.5 MHz), is one order of magnitude smaller than in the past literature for the same band 36 . At the same time, it is about 10 times higher than for single CO 2 lines due to a combination of factors, namely, the many partially overlapped lines and the difficulty to access the intensity baseline due to the congested spectrum. The absolute calibration of the frequency axis allows, for the first time, the stacking of repeated measurements and the determination of absolute line positions at these wavelengths.

Discussion
Overall, the metrological approach presented here considerably expands the portfolio of laser tools to probe molecules in the gas phase with high resolution and accuracy. This is proactive for the study of large molecules through the unique fingerprint features offered by the LWIR spectral region, for example molecules of the BTEX (benzene, toluene, ethylbenzene, xylenes) family that are of high interest for environmental modelling applications. In the case of benzene, which lacks a permanent dipole moment and, therefore, cannot be detected by pure rotational spectroscopy, infrared bands such as the ν 11 explored here represent the clue to assess their presence in remote spatial environments, such as planetary atmospheres 6,37,38 . On another front, by the use of a sub-hertz linewidth comb and phase-locking loops for pump and signal lasers, the spectrometer could reach the level of stability and spectral purity needed for tests of fundamental physics on molecular samples 39,40 in a spectral region not explored so far.

Methods
Difference frequency generation. The pump radiation in the DFG process is provided by a DFB-QCL from Alpes Lasers, emitting at around 5.69 μm. Tuning its temperature from 20°C to −10°C results in continuous spectral coverage from 1757 to 1761 cm −1 (i.e., over 4 cm −1 ) with an optical power from 17 to 48 mW, respectively. The signal radiation is generated by a CO 2 laser (L20GD, Access Laser) operating in a pulse modulation mode ("super-pulse mode") with a duty cycle of 20% and a modulation frequency of 10 kHz. Its emission can be finely piezo-tuned over few tens of megahertz and coarsely tuned by grating over~80 lines, from 9.23 to 10.71 μm, with peak powers varying from 3 to 27 W depending on the line. The wavelength combination of the two lasers provides access to a 12.1-14.8 μm (675-826 cm −1 ) wavelength range for the idler beam, apart from 10 cm −1 gaps at 12.6 and 14.4 μm (794 and 694 cm −1 ) and a 30 cm −1 gap at 13.3 μm (752 cm −1 ) due to the discontinuities between adjacent CO 2 lasing bands. The OP-GaAs crystal used for DFG is 35 mm long and equipped with poling periods from 183 to 203 μm in a fan-out structure, which ensures phase matching over the entire tuning range available for the nonlinear laser source. The width of the crystal is 20 mm, which is sufficiently large to discard any efficiency drop due to non-uniform phase-matching conditions across the beams. In fact, the poling period changes by <0.2 μm, which is far below the phase matching bandwidth of 0.7 μm, over beam diameters of 120 and 200 μm for pump and signal, respectively. As expressed in terms of wavenumbers, the phase-matching bandwidth amounts to 4.2 cm −1 , which implies that there is no need for adjustment of the crystal position while tuning the QCL. The optical power generated by the DFG considerably depends on both the QCL temperature and the CO 2 lasing line, but it remains above 20 µW (peak power) if the CO 2 emission is chosen among the 70% most intense lines, as it is the case for all measurements shown here.
Comb referencing and spectra acquisition. Pump (5.7 μm) and signal (9.2-10.7 μm) lasers are referenced to the same 1.9 μm Tm:fibre frequency comb through two independent SFG processes, in AgGaSe 2 and ZnGeP 2 (ZGP) crystals, respectively. The SFG processes shift the comb frequency v n = f ceo + nf rep (f ceo and f rep being carrier-envelope-offset and repetition frequency of the comb, respectively) by the pump (v p ) and signal (v s ) frequency, thereby generating two replicas of the comb around 1.4 μm (v n,p = v n + v p ) and 1.6 μm (v n,s = v n + v s ) for pump and signal, respectively. When these replicas are superimposed with a coherent continuum (v m = f ceo + mf rep ) generated from the original comb and extending down to 1.4 μm, a beat note is extracted, f beat ¼ ± ðν n;p=s À ν m Þ ¼ ± ðν p=s À m À n ð Þ f rep Þ that straightforwardly links the unknown pump/signal frequency to an integer number of f rep , independently of f ceo . In our case, f rep is stabilized to a global positioning system (GPS)-tracked low-noise radiofrequency (RF) signal at 100 MHz while f ceo is left free running. From an analysis of their beat notes (Supplementary Figs. 1 and 2), QCL and CO 2 lasers give an almost Gaussian instrumental line shape full width at half maximum (FWHM) of 1.6 and 1.5 MHz, respectively, which overall determine a spectral resolution limited to 2.2 MHz (see Supplementary Note 1). The relatively large contribution from the CO 2 laser comes from the adopted super-pulse mode, and it could be suppressed by choosing a cw emission followed by external modulation, at the expense of DFG power. The CO 2 beat note is locked to an RF local oscillator via piezo-feedback to the grating position, whereas the QCL beat note is measured in real time by a 100 MS s −1 digital acquisition board, followed by an FFT conversion at every 1024 samples and a barycentre calculation via fieldprogrammable gate array (FPGA, PXIe-7961 FPGA board and NI-5781 add-on, National Instruments). The same board digitizes the lock-in output in order to synchronize horizontal and vertical axis of the measurement. An example of raw data for QCL beat note and gas absorption is given in Supplementary Fig. 3. The acquisition of a 4 cm −1 large spectrum is typically accomplished in 12 min (tuning rate of 180 MHz s −1 ) to prevent laser frequency changes by more than its linewidth over the integration time of 10 ms set for the lock-in. A higher speed would be possible with a faster modulation of the CO 2 laser and a correspondingly smaller integration time. Spectra larger than 4 cm −1 , as those in Figs. 2 and 4, are acquired piecewise, due to the need of changing the CO 2 laser frequency and correspondingly adjusting the phase-matching condition for DFG. No tuning of the phase-matching angle is necessary for the SFG processes because of the much larger phase-matching bandwidths given by the shorter interaction length (6 mm) and the type of phase matching (birefringence rather than quasi-phase matching). The assignment of the comb mode order for correct calibration of the frequency axis is straightforward because of the low uncertainty of both CO 2 and C 6 H 6 tabulated lines, well below the comb mode spacing (100 MHz).
Uncertainty budget. A 240 MHz-broad spectrum of a single isolated line, such as that shown in Fig. 2b, is typically acquired in 2 s and encompasses 170 spectral points: these are almost evenly distributed in 5 groups separated at every 50 MHz, namely at every half f rep , by 18 MHz blanks due to the missed tracking of the comb-QCL beat note. In these conditions, the absorption noise (3 × 10 −3 on the most intense lines) and the frequency uncertainty on each spectral point (29 kHz) account for about 70% and 5%, respectively, of the observed 140 kHz root mean square (rms) deviation of the line centre for repeated measurements. The missing contribution comes from distortions of the spectral baseline that change from measurement to measurement and are not properly fit out. Repeated measurements were performed on most lines to reduce the statistical uncertainty on the line frequency. A Gaussian-like distribution was found, justifying an inverse square law between statistical uncertainty and number of independent measurements. This number is specified in the last column of Supplementary Table 1 for each line. The statistical uncertainty dominates over systematic effects, due to the negligible impact from the instability of the GPS-based frequency standard (2 kHz over 2 s), the inaccuracy of the absolute pressure gauge and of the pressure shift coefficients used to extrapolate line centres to zero pressure (<2 kHz, thanks to the low pressure), and the inappropriateness of the Voigt fitting model (<3 kHz, due to the negligible asymmetry of line profiles in the low-pressure collisional regime). Spectroscopic parameters (cm −1 ) for the ground state and for the v11 = 1 excited state of benzene. Standard uncertainties (1σ) in parentheses refer to the least significant digits. Ground state parameters are taken from Doi et al. 35 .

Data availability
The datasets generated during the current study are available from the corresponding author on reasonable request.