Near- to mid-IR spectral purity transfer with a tunable frequency comb: Methanol frequency metrology over a 1.4 GHz span

High precision molecular spectroscopy offers exciting perspectives ranging from fundamental physics^1–10 and metrology^11–13 to astrochemistry,¹⁴ remote sensing, and earth sciences.^15–17 Experiments in these domains are often based on frequency measurements of molecular vibrations in the mid-infrared (mid-IR) molecular fingerprint region, therefore generating the need for mid-IR laser sources that are spectrally pure, accurate, and widely tunable.

Here, we report on the upgrade and operation of a quantum cascade laser (QCL) based spectrometer that provides a unique combination of sensitivity, frequency resolution, and tunability of any mid-IR spectrometer to date. An erbium-doped fiber mode-locked optical frequency comb is used to transfer the spectral purity of a metrology-grade ultrastable frequency reference traceable to the International System of Units (SI) from the near-IR to the mid-IR. In addition, we have developed the instrumentation allowing comb modes to be continuously tuned over 9 GHz. This results in a QCL of spectral purity at the state-of-the-art that can be continuously tuned at the precision of the frequency reference over an unprecedented frequency span of 1.4 GHz. These developments constitute enabling technologies for driving the next generation of ultra-high resolution molecular spectroscopy apparatus in the molecular fingerprint region. High spectral purity is key to reaching the resolutions required for resolving the subtlest structures, clusters of blended lines, which constitute unique probes of fundamental processes in molecules. Together with SI-traceability and high sensitivity, it allows systematic effects to be unraveled in order to minimize line center uncertainties. Adding finally wide continuous tuning capabilities to the toolbox of present and future mid-IR photonics is important to facilitate the discovery/resolution of new unreported lines, in particular weak transitions, as demonstrated here, providing in-depth insights into the internal molecular dynamics.

Distributed feedback (DFB) quantum cascade lasers (QCLs) are available over wide ranges of the mid-IR (from 3 to 25 μm), can be tuned over several hundreds of gigahertz, and have continuous-wave (cw) power levels in the milliwatt to watt range near room-temperature. They, however, show substantial frequency noise.^18–29 For the most precise measurements, high-spectral purity and traceability to a frequency standard are both required. This can be achieved by phase-locking to: (i) the mid-IR secondary frequency standard²⁵ (a CO₂ laser stabilized to a saturated absorption line of OsO₄³⁰), resulting in δν = 10 Hz linewidth, 1 Hz stability at 1 s, and accuracy of a few tens of hertz; and (ii) a near-IR metrology-grade frequency reference traceable to primary frequency standards, resulting in ultimate sub-Hz stabilities and accuracies, and linewidth narrowing down to the δν = 0.1 Hz level^26,31 (see also related similar studies, which, however, do not reach such high levels of spectral purity^32,33). Efforts in combining a sub-100 Hz metrology-grade spectral performance and a wide continuous tunability are scarce. We have previously demonstrated, respectively, Δν = 10 GHz and Δν = 0.4 GHz continuous tunability for: (i) the 10 Hz narrow 10.6 μm QCL phase-locked to the mid-IR secondary frequency standard (⁠$∼10−12$ absolute frequency uncertainty)²⁵ and (ii) the 0.1 Hz narrow 10.3 μm QCL ultimately calibrated to primary frequency standards (⁠$∼10−14$ absolute frequency uncertainty).³¹ Another interesting apparatus is the CO₂-laser/microwave-sideband spectrometer demonstrated in Ref. 34, allowing 11.8 GHz wide windows around any CO₂ laser emission line to be covered in a single sweep. This spectrometer shows similarities with our sub-100 Hz metrology-grade equipment^25,35 but uses free-running CO₂ lasers and allows broadband scans at Doppler-limited resolutions only. Note that QCLs can also be phase-locked to a frequency comb controlled with a radio frequency (RF) reference. This enables higher tuning ranges but at the expense of a barely reduced frequency noise (resulting in ∼1 kHz linewidth at best) and a frequency uncertainty limited to around 10⁻¹³ at best by the RF references.^{23,24,36–44} Mid-IR frequency combs, based^45–49 or not^28,49–54 on QCL technologies, are emerging as flexible sources of broadband coherent radiation. They can address a very wide spectral range, via dual-comb⁵⁵ or Fourier transform^56,57 spectroscopy methods,⁵⁸ but a sub-100 Hz metrology-grade performance (linewidth, stability, and accuracy) has yet to be demonstrated for mid-IR combs.

In the following, we describe our high-resolution molecular spectrometer. It exploits a frequency comb to transfer the spectral purity of a 1.54 μm remote SI-traceable optical frequency reference signal to a 10.3 μm QCL. The near-IR reference is operated at the French time/frequency metrology institute (LNE-SYRTE), calibrated there to primary frequency standards, and transferred to Laboratoire de Physique des Lasers (LPL) via a long-haul optical fiber link.²⁶ Compared to our previous work,³¹ the QCL’s tunability has been extended by a factor of more than 3. We use our SI-traceable sub-Hz QCL to conduct sub-Doppler saturation spectroscopy of methanol in a multi-pass cell over an unprecedented continuous frequency span of Δν = 1.4 GHz. This corresponds to a record continuous relative tuning range of Δν/δν ∼ 10¹⁰, with Δν/δν being equal to the ratio of the spectral coverage to the instrumental resolution given by the laser linewidth.

Methanol is found in a wide variety of astronomical sources⁵⁹ and is the second most abundant organic molecule in the earth’s atmosphere.⁶⁰ It is an excellent probe of the physical conditions, the chemistry and the history of these environments. Reliable laboratory spectroscopic data of methanol are thus much needed for interpreting astrophysical and planetary spectra, for air quality monitoring and atmospheric concentration retrieval. Although the simplest of the organic alcohols, it is a slightly asymmetric-top with a hindered internal rotor (torsion), leading to a triple-well internal tunnel dynamics and, therefore, to a rich rotation–torsion–vibration energy structure. Methanol is thus also an important molecule for fundamental spectroscopy,⁶¹ metrological applications and frequency calibration,^31,62 the realization of optically pumped far-IR gas lasers,⁶³ or probing the limits of the standard model; its internal tunnel dynamics makes it one of the most sensitive molecules for a search of a varying proton-to-electron mass ratio.⁸ Here, we measure the resonance frequencies of 14 rovibrational transitions, including very weak lines, some observed for the first time. We demonstrate state-of-the-art frequency uncertainties two to four orders of magnitude improved over previous measurements, at the few kilohertz level (⁠$∼10−10$ in relative value) for the most intense lines and bring to light inaccuracies and gaps in the high-resolution transmission molecular absorption (HITRAN) database. We resolve subtle weak intensity doublets never observed before induced by a combination of the asymmetry and the tunneling dynamics in methanol, demonstrating both the high detection-sensitivity and high resolution of our widely tunable spectrometer and its potential for unraveling subtle internal molecular processes.

Figure 1 illustrates our widely tunable SI-traceable frequency-comb-stabilized-QCL-based high-resolution mid-IR spectrometer. It combines a widely tunable optical local oscillator (OLO) locked to a remote ultrastable frequency reference of LNE-SYRTE, a 10.3 μm cw DFB QCL (Alpes Lasers, LLH package) stabilized to the OLO using an optical frequency comb (OFC) and sum-frequency generation, and a multi-pass absorption cell for carrying out saturation spectroscopy. The reader is referred to Ref. 31 for a detailed description of most of this setup. Note that most of the instrumentation described in this section, in particular all the locking electronics, is home-made.

The widely tunable OLO assembly is shown in Fig. 1(b). The frequency reference ν_ref located at LNE-SYRTE is produced by a ∼1.54 μm fiber laser locked to an ultrastable cavity, which exhibits a relative frequency stability lower than 10⁻¹⁵ between 0.1 and 10 s integration time.^31,64 This reference is calibrated against a combination of a liquid-helium cooled cryogenic sapphire oscillator (CSO) and a hydrogen maser (H-maser), themselves monitored on the atomic fountains of LNE-SYRTE, and its absolute frequency is thus SI-traceable to the primary standards with an uncertainty at the few 10⁻¹⁴ level. This reference signal is transferred to LPL^65,66 through a 43 km long fiber link of the REFIMEVE infrastructure⁶⁵ without any degradation of its stability and absolute uncertainty, thanks to an active compensation of the propagation-induced phase noise.⁶⁷ A 200 MHz ultrastable RF reference f_ref derived from the CSO/H-maser combination (⁠$∼10−15$ stability at 1 s, a few 10⁻¹⁴ absolute frequency uncertainty⁶⁸) is also transferred to LPL through the same 43 km long fiber link using an amplitude modulated auxiliary 1.5 μm laser diode.

At LPL, a laser diode of frequency ν_OLO ∼ 1.54 μm is used as an OLO. Two sidebands are generated in the OLO signal using an electro-optic modulator [EOM₁ in Fig. 1(b), MPZ-LN-20, Photline]. EOM₁ frequency $fEOM1$ is precisely controlled using a home-made phase-jump-free microwave synthesizer based on a current driven Yttrium Iron Garnet (YIG) oscillator with a continuous tuning range of 9 GHz (7.5–16.5 GHz).³¹  Figure 1(a) shows a simplified scheme of the synthesizer. A 1 GHz signal is generated using a RF tracking oscillator (TO) referenced to a 10 MHz signal (stability at 1 s and relative frequency uncertainty better than 10⁻¹² and 10⁻¹³, respectively⁶⁹) synthesized from the remote LNE-SYRTE 200 MHz reference f_ref. A 0–400 MHz direct digital synthesizer (DDS, AD9858, Analog Devices) uses the 1 GHz signal as a reference. After division by 32, the YIG frequency is mixed with the doubled DDS frequency. The resulting phase error signal is converted into correction signals [using phase-locked loop PLL₁ in Fig. 1(a)] applied to the YIG oscillator’s current, of which the frequency is thus locked to the DDS with a frequency ratio of 64. This allows not only the spectral performance of the local 10 MHz reference to be transferred to the resulting microwave signal, but also the synthesizer frequency to be tuned over 9 GHz without any phase jumps by adjusting the DDS frequency. As illustrated in Fig. 1 and detailed below, the QCL is stabilized to the OLO via a series of three cascaded phase-locked loops. The wide tunability and, crucially, the phase-jump-free nature of the YIG-based synthesizer are key to allowing a broad but also continuous tunability of the stabilized QCL and, therefore, demonstrate both broadband and ultra-high resolution spectroscopy.

A f_rep ∼ 250 MHz repetition rate erbium-doped fiber optical frequency comb (Menlo Systems, FC1500) is used to transfer the spectral performances of the remote frequency reference and the wide tunability of the OLO to the QCL.

As demonstrated in Refs. 26 and 31, the QCL frequency (i) exhibits a stability at the level of the remote reference signal, below 0.03 Hz (10⁻¹⁵ in relative value) for averaging times from 0.1 to 10 s, and a linewidth of δν ∼ 0.1 Hz at 100 ms timescales; and (ii) is SI-traceable and known within a total uncertainty better than 4 × 10⁻¹⁴ after 1 s averaging time.

Scanning the OLO frequency thus allows us to continuously tune f_rep and, in turn, the QCL frequency at the precision of the remote reference. In a previous study,³¹ we have demonstrated a QCL continuous tunability over a span of ∼400 MHz, limited by the course of the PZT actuator. With the additional capability provided by the stepper motor control, scanning the OLO carrier frequency over its entire 9 GHz span (limited by the YIG tunability) enables the QCL, which is both ultrastable and SI-traceable, to be continuously tuned over a range of 1.4 GHz (given by the ∼6.7 mid-IR to near-IR wavelength ratio). However, as explained in  Appendix A, if the time delay between the SFG and original combs is not canceled, tuning f_rep over the extended spans resulting from the use of the stepper motor comes with a deterioration of the pulses’ time overlap and, in turn, of beat-note signal Δ₃’s signal-to-noise ratio, preventing us from carrying out long scans. Time delay cancellation is achieved by adding fiber length in order to balance the optical paths (see the details in  Appendix A). The combination of tunability and spectral purity in our spectrometer yields a Δν/δν ∼ 10¹⁰ continuous span, setting, to our knowledge, a new record both for the mid-IR QCL and for the near-IR comb teeth.

As illustrated in Fig. 1(e), the stabilized QCL has been used to carry out saturated absorption spectroscopic measurements in a multi-pass absorption cell (Aerodyne Research, model AMAC-36, 182 passes). With a 20 cm long distance between two astigmatic mirrors, the cell provides an effective absorption length of 36.4 m and allows us to perform spectroscopic measurements of weak lines at low pressures.

The QCL beam is split into two using an 80/20 beam splitter (BS₁). Around 8 mW is needed to phase-lock the laser to the comb, while about 2 mW remains for the spectroscopy. Using BS₃, the beam is further split into a pump beam and a probe beam, which are coupled into the multi-pass cell (incident powers of ∼1.3 and ∼0.7 mW, respectively) for conducting saturation spectroscopy. The mirrors’ reflectivity leads to a ∼20% transmission after 182 passes, which, in turn, results in pump and probe powers that vary inversely by almost an order of magnitude through the cell. This also goes with an ∼50% total power (sum of pump and probe powers) variation from pass to pass and to an intra-cell total averaged power of 0.81 mW (averaged over the 182 passes). After exiting the cell, the probe beam is detected using a liquid-nitrogen-cooled mercury–cadmium–telluride photodetector [MCT in Fig. 1(e)]. Undesirable interference fringes as typically observed with multi-pass cells⁷⁰ are averaged out by a vibrating mirror M₂ held in a piezo-actuated mount and shaking the multi-pass cell assembly with a fan.

Frequency modulation (FM) of the QCL is used to improve the signal-to-noise ratio. This is done by modulating the frequency of the DDS used as a reference for PLL₄ (to lock the QCL to the comb) at a frequency of 20 kHz (well within PLL₄’s 250 kHz bandwidth). The signal detected by the MCT photodetector is fed to a lock-in amplifier for demodulation.

Saturated absorption spectroscopy of methanol (purchased from VWR Chemicals, $≥99.8$% purity, see  Appendix B for more details on the sample purity) is carried out by tuning the QCL frequency in a series of discrete steps. To cancel out the frequency shifts induced by the limited detection bandwidth,^71,72 we have carried out frequency scans in both directions with increasing and decreasing frequencies and only consider the averaged spectrum of pairs of up- and down-scans (see below). Figure 2(a) shows a saturated absorption spectrum of methanol averaged over five such pairs spanning the full 1.4 GHz tuning range of our system (from ∼971.312 to ∼971.357 cm⁻¹) after demodulation on the first harmonic. It has been recorded at a pressure of 1.5 Pa with a frequency step of ∼15 kHz and a step duration and lock-in amplifier time constant both of 10 ms. The SI-traceability detailed in Sec. II B allows us to retrieve the absolute frequency scale using Eq. (4). Although ν_ref is measured and made available every second at LNE-SYRTE, its value varies by a few hertz at most over the duration of our scans. Therefore, for each pair of scans, we fix ν_ref in Eq. (4) to its value measured halfway through the scans.

The spectrum exhibits 14 rovibrational transitions of methanol, which belong to the P branch of the ν₈ C–O stretch vibrational mode.⁷³ The red solid line is a fit to the data used as a guide to the eye. Each line is fitted with the first derivative of the sum of a Lorentzian and a Gaussian profile to model the saturated absorption and Doppler (oscillations in the baseline) contribution, respectively. The transitions reported in the HITRAN database⁷⁴ falling in the spectral window covered are shown as blue lines. As an example, Fig. 2(b) shows a zoom-in on the saturation feature of the P(A, co, t = 0, K = 2⁻, J = 33) methanol line around 971.319 cm⁻¹ (in this work, we adopt the notations of Ref. 73 for the spectroscopic assignment of methanol transitions; see also  Appendix C). It exhibits a signal-to-noise ratio of ∼370 and an ∼760 kHz full width at half maximum of the free-of-FM-induced-distortion Lorentzian underlying the line shape. The latter is a combination of transit time (⁠$∼120$ kHz) pressure (⁠$∼330$ kHz) and power broadening.

To achieve a reasonable signal-to-noise ratio, the FM amplitude has been set at 250 kHz, resulting in a line shape slightly distorted compared to the typical Lorentzian profile. Furthermore, the residual amplitude modulation associated with FM and the power variation over a scan can both contribute to an asymmetry of the line shape. We fit our data to a model described in  Appendix D that takes into account these complex distortions. It is based on a direct absorption profile introduced by Arndt⁷⁵ and Schilt et al.⁷⁶ that we have extended to the analysis of saturation spectra. Exploiting this model has allowed us to use larger FM amplitudes, directly resulting in an increase in signal-to-noise ratio.

To determine the line-center frequencies of the 14 resolved transitions, we first select a spectral range of ∼6 MHz around each. We perform the following “pair by pair” analysis already established in Ref. 31. We average each pair of up- and down-scans resulting in five averaged spectra for each of the 14 transitions. To all data points of an averaged spectrum, we assign the same experimental error bar, calculated as the standard deviation of the residuals obtained after fitting a second-order polynomial to a small portion of the averaged spectrum far from resonance. To each of the five averaged spectra of a given transition is fitted the sum of the line profile described in  Appendix D and of a second-order polynomial to account for the baseline (see  Appendix E). The absolute frequency and associated statistical uncertainty of each line are estimated by calculating the weighted mean and weighted standard error of the five fitted center frequencies, with the weights determined from the fits’ error bars. Unlike in our previous work,³¹ we have conducted measurements at a single pressure (1.5 Pa) and power (0.81 mW intra-cell averaged power corresponding to a saturation parameter slightly larger than 1 for the most intense lines probed), preventing us from deducing zero-power and collision-free transition frequencies. However, we estimate the resulting overall pressure- and power-shift to be of the order of 10 kHz. The reader is referred to Refs. 31 and 69 and to the summary given in  Appendix F for a detailed description of the line positions’ uncertainty budget, which results in a systematic uncertainty of 5.4 kHz on the frequency of all the resonances studied here. Table I lists the line center frequencies of the 14 transitions recorded at a pressure of 1.5 Pa and an intra-cell averaged power of 0.81 mW. 1-σ total uncertainties are quoted into parentheses. The absolute frequencies of three unambiguously assigned high intensity transitions have been determined with a sub-10 kHz total uncertainty (2.1 × 10⁻¹⁰ at best in relative value), an improvement of ∼2000 over previous measurements based on the Fourier-transform IR (FTIR) spectroscopic technique.^73,74 All other absolute frequencies are determined with a total uncertainty ranging from ∼10 to ∼170 kHz, to be compared to the typical 15 MHz FTIR uncertainty.⁷³ 

The data shown in Fig. 2 demonstrate both the high detection-sensitivity and high resolution of our spectrometer. An illustration of this double asset is the resolution of two weak doublets [see the insets of Fig. 2(a)], which exhibit a splitting of 2.80 ± 0.11 and 1.67 ± 0.19 MHz, respectively. The use of a multi-pass cell allows relatively high-J (J > 30) rovibrational lines to be probed at low pressures required for ultra-high resolution measurements, including weak lines belonging to hot-bands (torsional excitation t ⩾ 1). We have thus been able to demonstrate sub-10 kHz uncertainties for the strongest transitions shown here, exhibiting intensities of several 10⁻²² cm⁻¹/(mol cm⁻²) two to three orders of magnitude weaker than the most intense lines of this vibrational mode and $∼100$ kHz uncertainties for lines as weak as a few 10⁻²³ cm⁻¹/mol cm⁻².

As shown in Fig. 2(a), our spectrum is a very rich source of information. First of all, we resolve 14 lines where the HITRAN database lists only nine (of which two pairs reported as degenerate). Several transitions have, to our knowledge, never been reported elsewhere. Based on HITRAN, we are, in fact, able to unequivocally assign only four of the lines (see Table I). Five weak transitions (including one doublet) that could not be assigned are indicated with blue arrows and most probably belong to hot-bands with torsional excitation t ⩾ 2. As discussed in the following and summarized in Table I, the five remaining measured resonances are tentatively assigned to the five other HITRAN transitions. As shown in Fig. 2(c), around 971.328 cm⁻¹, we resolve one weak line labeled (2) and two intense lines labeled (1) and (3). We tentatively assign these to transitions P(E, co, 1, 1, 33), P(E, co, 0, 5, 33), and P(A, co, 0, 6⁺, 35), without being able to decide which one is which. Those three transitions are, indeed, listed in Ref. 77 at the same degenerate frequency of 971.3288 cm⁻¹. In HITRAN, however, and as shown in Fig. 2, only P(E, co, 1, 1, 33) and P(E, co, 0, 5, 33) are reported as degenerate at 971.328 94 cm⁻¹, while P(A, co, 0, 6⁺, 35) is ∼150 MHz blueshifted at 971.333 77 cm⁻¹. It is, however, unlikely for the latter to correspond to one component of the doublet observed at ∼971.334 cm⁻¹ (based on intensities, this doublet looks very much like a single structure, not like two fortuitously quasi-coincident lines). P(A, co, 1, K = 10^±, 35) [red arrow in Fig. 2(a)] corresponds to a so-called closed-lying K-doublet that splits for A-symmetry lines due to the combination of the slight asymmetry and internal tunnel dynamics of the methanol molecule.⁷⁷ It is reported as degenerate in the HITRAN database, but no matching resonance is observed around. Even at the high resolution demonstrated here, we do not expect to resolve such K-doublets for K ⩾ 7 (here K = 10), but enhanced splittings are possible as a result of mixing of the considered transition upper states that belong to the C–O stretching vibrational mode with a closely lying state from another vibrational mode.⁷⁷  P(A, co, 1, K = 10^±, 35) could then well be assigned to either of the two observed resolved doublets. Based on intensities, we tentatively assign it to the strongest doublet around 971.334 cm⁻¹. All certain and tentative assignments are finally summarized in Table I.

Our data are a source of information much richer than what is currently available in the literature. It has yet to be fully exploited to build a more accurate spectroscopic model of methanol. As seen, it brings to light some inaccuracies and gaps in the HITRAN database. In addition, our measured and unequivocally assigned resonance frequencies are redshifted by 5–20 MHz with respect to previous FTIR measurements at the origin of the current HITRAN edition line list⁷⁴ (see Fig. 2 and Table I). These shifts are about three orders of magnitude larger than our uncertainties and more than an order of magnitude larger than the HITRAN frequency accuracies reported to be between 30 and 300 kHz. It is consistent with deviations between FTIR and saturation spectroscopy data previously observed in the C–O stretch of methanol and is attributed to FTIR spectrometers’ calibration imperfections.^31,62,78,79 Our data also offer new information on weak hot-band transitions for which molecular databases remain largely incomplete. Exploiting this in refined models of methanol is essential for atmospheric quantification. Moreover, mixing between near-degenerate levels of two different modes is known to lead to collision-induced population transfer from one vibrational mode to another. Doublets exhibiting enhanced splittings such as those resolved in our work, a signature of this type of mixing, may thus help understand how molecules transfer among different modes and give insights into thermal equilibration in gases.⁷⁷ 

Other saturation spectroscopy measurements of the C–O stretch of methanol exist. Almost 700 frequencies have been measured with an accuracy of ∼100 kHz using a CO₂-laser spectrometer,^{34,62,78–80} however, mostly in the Q and R branches, with a few low J lines in the P branch. To our knowledge, there are only two other methanol frequencies that have been measured with an uncertainty comparable to the present work: our previous measurement³¹ and a weak unassigned line⁸¹ around 947.7 cm⁻¹.

We present our progress in the development and operation of a widely tunable SI-traceable frequency-comb-stabilized high-resolution spectrometer potentially covering the 8–12 μm spectral window. A mode-locked frequency comb and a metrological fiber link are used to transfer the spectral purity and SI-traceability of a 1.54 μm frequency reference located at LNE-SYRTE to a 10.3 μm QCL located at LPL. The QCL exhibits a sub-10⁻¹⁵ relative frequency stability from 0.1 to 10 s and a linewidth of ∼0.1 Hz at 100 ms, and its frequency is SI-traceable with a total relative uncertainty better than 4 × 10⁻¹⁴ after 1 s of averaging time. In addition, we have developed the instrumentation allowing comb modes to be continuously tuned over 9 GHz, resulting in a continuous tunability of 1.4 GHz for the QCL, a more than threefold improvement compared to previous measurements at such levels of spectral purity. This tuning range can potentially be increased by transferring the full 9 GHz tunability of comb modes directly to the QCL or by using a commercially available 40 GHz telecom electro-optic modulator together with a home-made microwave synthesizer of wider tunability, potentially giving access to a continuous span of Δν/δν > 10¹¹.

We have carried out saturation spectroscopy of methanol in a multi-pass absorption cell. We report the line-center frequencies of 14 transitions of methanol in the P branch of the ν₈ C–O stretch vibrational mode, including very weak transitions belonging to hot-bands, some observed for the first time. We demonstrate record global uncertainties ranging from few kilohertz to $∼100$ kHz—three to four orders of magnitude improved over previous measurements—depending on the line intensity. We expose manifest discrepancies with the HITRAN database, provide information on hot-bands essential to atmospheric quantification, and resolve subtle weak intensity doublets never observed before induced by a combination of the asymmetry and the tunneling dynamics in methanol. The 10⁻¹⁰ level frequency accuracy achieved here, which surpasses traditional high resolution Fourier transform spectrometers by three orders of magnitude, thus allows molecular databases to be filled with increasingly accurate parameters. This is essential for improving spectroscopic models of molecules of atmospheric or astrophysical interest, such as methanol. In this context, sub-Doppler spectroscopy potentially gives access to line positions, transition dipole moments, pressure broadenings and shifts as well as corresponding beyond-the-Voigt speed dependent parameters. Conducting saturation spectroscopy in a Fabry–Pérot cavity rather than in a multi-pass cell using our sub-Hz frequency-comb-stabilized QCL would result in improved mid-IR measurements in methanol with improved uncertainties to the 10⁻¹¹–10⁻¹² level.²⁶ This is important for (i) refining constraints on the variation in the proton-to-electron mass ratio μ by comparing laboratory data to the spectra of cosmic objects⁸ and (ii) unraveling the hyperfine structure in view of using methanol as a magnetic field tracer in star-forming regions.⁸² In addition, the sub-10⁻¹⁵ level of mid-IR laser spectral purity achieved is important for pushing back the limits in ultra-high resolution molecular spectroscopy. It could be exploited to reach 10⁻¹²–10⁻¹³ mid-IR frequency uncertainties on simple and calculable trapped hydrogen molecular isotopologues (H$2+$⁠, HD⁺, D$2+$⁠, etc.) for providing stringent tests of quantum electrodynamics or precise measurements of μ, or to search for physics beyond the standard model by looking for fifth forces or extra dimensions at the molecular scale.^5,83 Finally, the mid-IR spectral performance reported in the present work is mandatory for reaching uncertainty levels at the 10⁻¹⁴–10⁻¹⁵ level or beyond, i.e., required to test the parity symmetry by measuring the tiny energy differences between chiral enantiomers^3,9,84 or to refine direct and model-free constraints on possible current-epoch variations of μ,^7,9,85 for which methanol is a promising candidate molecule.

This project received funding from the EMPIR program co-financed by the Participating States and from the European Union’s Horizon 2020 research and innovation program, through EMPIR Project No. 15SIB05 “OFTEN.” This work was supported by the Region Île-de-France in the framework of DIM Nano-K and DIM SIRTEQ, the Agence Nationale de la Recherche project PVCM (Grant No. ANR-15-CE30-0005-01), the LabEx Cluster of Excellence FIRST-TF (Grant No. ANR-10-LABX-48-01), the EquipEx Cluster of Excellence REFIMEVE+ (Grant No. ANR-11-EQPX-0039), CNRS, and Université Sorbonne Paris Nord. D. B. A. Tran was supported by the Ministry of Education and Training, Vietnam (Program No. 911).

The authors have no conflicts to disclose.

D. B. A. Tran: Formal analysis (equal); Investigation (equal); Validation (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). O. Lopez: Investigation (equal); Methodology (equal); Writing – review & editing (equal). M. Manceau: Investigation (equal); Supervision (equal); Writing – review & editing (equal). A. Goncharov: Investigation (equal); Supervision (equal); Writing – review & editing (equal). M. Abgrall: Investigation (equal); Methodology (equal); Writing – review & editing (equal). H. Alvarez-Martinez: Investigation (equal). R. Le Targat: Investigation (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal). E. Cantin: Investigation (equal); Writing – review & editing (equal). P.-E. Pottie: Investigation (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal). A. Amy-Klein: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal). B. Darquié: Conceptualization (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Supervision (lead); Validation (lead); Visualization (equal); Writing – review & editing (lead).

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

The wide tuning range of the stabilized QCL may be altered if the time delay between the SFG and original combs is non-zero, as a result of a propagation length difference ΔL. Even if, for a given value $frep0$ of the repetition rate, the pulses of the two combs are well time-overlapped by using a short delay line (few tens of nanoseconds), there can exist a delay between combs $ΔT=nΔL/c=m/frep0$ corresponding to an integer number m of the time between two adjacent pulses, i.e., when the kth pulse of the SFG comb, with k being an integer, overlaps with the (k + m)th pulse of the original comb (we assume here propagation in a medium of refractive index n, with c being the speed of light). Tuning f_rep away from $frep0$ results in a gradual loss of the overlap and in a time separation between the SFG and original comb pulses of $δpulse≃mδfrep/frep02$ for small variations δf_rep of the repetition rate. To avoid degrading the signal-to-noise ratio of the two-comb beat-note used to lock the QCL, δ_pulse should remain smaller than the combs’ pulse width (a few hundreds of femtoseconds) during a scan.

As shown in Fig. 3, we have set up an experiment to measure the delay between the pulses of the original comb and the additional output (after the wavelength shifting fiber) used to generate the SFG comb. We lock f_rep to the OLO as detailed in the main text, and we detect the two beat-note signals at frequencies f₁(t) and f₂(t) between the OLO and the closest pth tooth of the original, respectively, additional comb. A Π-type RF frequency counter is used to count f₁(t) and f₂(t) (1 ms gate time), and the frequency difference Δf(t) = f₂(t) − f₁(t) is processed. If ν_OLO is kept fixed, Δf(t) = 0 (f_rep is the same for both combs). During a scan, however, as a consequence of the propagation time difference ΔT, the original and additional combs will be detected with a slightly different value for f_rep. A linear scan $frep(t)=frep0+α×t$ then leads to Δf(t) = Δf = pα × ΔT constant in time and, thus, to a phase difference between the pth teeth of the two combs after a scan of duration t given by Δϕ(t) = Δf × t.

As expected, this corresponds to a gradual loss of the overlap between comb pulses, corresponding to a time separation per unit length difference and OLO frequency change of 25 fs m⁻¹ GHz⁻¹. The blue curve in Fig. 4 shows δ_pulse(t) during a back-and-forth 9 GHz OLO frequency scan, before cancellation of the time delay between the original and additional combs. It is determined using Eq. (A1) after numerically integrating Δf(t). We observe that δ_pulse varies by $∼2.25$ ps during the scan and conclude that ΔL ≃ 10 m.

To compensate the delay between the two combs, we have thus added a 10 m-long fiber to the original comb. We repeat the measurement of δ_pulse after the adjustment. The result shown in red in Fig. 4 confirms a good cancellation of the time delay between the original and additional combs. Residual pulse separations of a few tens of femtoseconds are measured during the scan, which is smaller than the pulses’ width. Such fluctuations most probably arise from differential temperature fluctuations between the two paths.

We have purchased our methanol from VWR Chemicals (methanol $≥99.8$% purity, AnalaR NORMAPUR® ACS, ref. 20847.240). According to the supplier, the most abundant impurities in our methanol sample are ethanol and water at 0.1% and 0.03%, respectively. Any other impurities are at the 10 ppm level maximum. Methanol is contained in a glass tube connected to the vacuum system. Before each experiment, we typically pump on it for about a minute to evacuate any potential gas phase impurities. To our knowledge, no water lines are predicted in the spectral window covered by our spectrum. No other hydrocarbon lines are predicted either, except a few weak to very weak lines of ethene, which do not match with our spectrum. 0.1% of ethene in our sample, which is highly improbable, would not even allow us to detect the strongest of those C₂H₄ lines. After comparing the ethanol and methanol cross sections provided in the HITRAN database, we conclude that 0.1% of ethanol would result in ethanol lines three orders of magnitude weaker than the strongest observed methanol lines, well below our detection sensitivity. We are thus confident that all the lines probed in this work correspond to methanol rovibrational transitions.

In this work, we adopt the notations of Ref. 73 for the spectroscopic assignment of methanol transitions. The P(σ, co, t, K, J) resonances studied here are all ΔJ = −1, ΔK = 0, and Δt = 0 transitions between the ground vibrational state and the first excited state of the C–O stretch vibrational mode ν₈, with σ, t, K, and J, the torsional symmetry (A or E), the torsional mode (ν₁₂) quantum number, and rotational quantum numbers of the lower state (J is the total orbital angular momentum quantum number and K is the quantum number for the projection of the total orbital angular momentum onto the molecular symmetry axis). K-doublets of A symmetry have ± superscript on K to distinguish the A⁺ or A⁻ component of the doublet. For E symmetry levels, K is a signed quantum number with positive (respectively, negative) values corresponding to levels often denoted as E1 (respectively, E2).

The QCL frequency is modulated at a frequency f_m with a FM amplitude m. The instantaneous QCL frequency can be written as ν_QCL(t) = ν_c − m cos(2πf_mt + Ψ) with ν_c being the optical carrier frequency and Ψ being a phase shift between the frequency modulation and the associated intensity modulation (see below). To achieve a reasonable signal-to-noise ratio, the FM amplitude has been set at m = 250 kHz, comparable to the measured resonances half-width-at-half-maximum of $∼380$ kHz, resulting in a line shape slightly distorted compared to the typical Lorentzian profile. Furthermore, the residual amplitude modulation associated with FM and the power variation over a scan can both contribute to an asymmetry of the line shape. The power frequency dependency can result from (i) the QCL gain curve, (ii) the underlying Doppler envelop (potentially non-trivial in case of neighboring lines not resolved in the Doppler regime; see Fig. 2), or (iii) multi-pass-cell-induced residual interference fringes not fully averaged out (see the text).

We fit each averaged pairs of up- and down-scans of a given transition to the sum of the line profile described in  Appendix D and of a second-order polynomial to account for the baseline (dominated by the underlying Gaussian contribution on the narrow frequency span used for fitting). All line shape parameters are left free in the fit except the FM amplitude fixed to 250 kHz. The doublets are fitted together using a single second-order baseline, and, apart from line intensity factors and center frequencies, line shape parameters are set to be equal for the two components. We apply the same procedure for the pair of lines (1) and (2) [see Fig. 2(c)]. As an example, Fig. 5 shows the average of one pair of up- and down-scans of the P(A, co, 0, 2⁻, 33) rovibrational transition of methanol (gray points), the fit of the line shape model to the data (red solid curve), and the corresponding residuals.

The reader is referred to Refs. 31 and 69 for a detailed description of the line positions’ uncertainty budget. We only briefly list here the systematic effects and associated uncertainties: (i) we estimate a conservative upper bound on the frequency shift resulting from gas lens effects, wavefront-curvature-induced residual Doppler shifts, the second-order Doppler shifts, Zeeman effects, black-body radiation shifts, and the photon recoil doublet to be 5 kHz and take this as the corresponding uncertainty; (ii) we assign a conservative 2 kHz systematic uncertainty to the inaccuracy of our model for the line shape (see below); (iii) our 0.02 Pa pressure measurement accuracy and the 5% specified accuracy of the power meter result in pressure- and power-shift-induced systematic uncertainties of ∼0.1 and ∼0.5 kHz, respectively³¹ (pressure and power fluctuations, respectively, smaller than 0.02 Pa and 10% during a scan and from one to another result in frequency fluctuations of <0.1 and ∼1 kHz, respectively, which are already included in our statistical uncertainty); and (iv) the 1 Hz uncertainty on the mid-IR frequency scale is dominated by the 4 × 10⁻¹⁴ uncertainty on the LNE-SYRTE frequency reference value ν_ref used to retrieve the absolute frequency scale using Eq. (4).

We use the three unambiguously assigned highest intensity lines shown in Fig. 2(a) to determine the systematic uncertainty resulting from the inaccuracy of our model for the line shape [point (ii) above]. We fit each averaged pair of up- and down-scans with the sum of the spectral line shape described in  Appendix D and a polynomial of order k with k ranging from 2 to 5. For all such averaged pairs, we find the standard deviation of the four frequencies resulting from the corresponding fits to be less than 2 kHz, which we take as our conservative systematic uncertainty.