# SI-referenced formic acid (HCOOH) spectroscopy at the sub ppt level

Maxime Leuliet, Abdessamad Mbardi, Bérengère Argence, Jean-Philippe Karr, Laurent Hilico

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## To cite this version:

Maxime Leuliet, Abdessamad Mbardi, Bérengère Argence, Jean-Philippe Karr, Laurent Hilico. SI-referenced formic acid (HCOOH) spectroscopy at the sub ppt level. 2025. hal-04238089v3

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Preprint submitted on 16 Oct 2025 (v3), last revised 10 Dec 2025 (v4)

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ARTICLE TEMPLATE

SI-referenced HCOOH spectroscopy at the sub ppt level

M. Leuliet<sup>a</sup> and A. Mbardi<sup>a</sup> and B. Argence<sup>a</sup> and J.-Ph. Karr<sup>a,b</sup> and L. Hilico<sup>a,b</sup>

<sup>a</sup>Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-PSL Research University, Collège de France, 4 place Jussieu, F-75005 Paris, France; <sup>b</sup>Université d'Evry Paris Saclay, Université Paris-Saclay, Boulevard François Mitterrand, F-91000 Evry, France

ARTICLE HISTORY

Compiled October 16, 2025

Abstract

We report on a new determination of the ν<sub>6</sub>, *J'* = 21, *K*'<sub>a</sub> = 2, *K*'<sub>c</sub> = 20 ← *J''* = 21, *K*''<sub>a</sub> = 3, *K*''<sub>c</sub> = 19 HCOOH rovibrational line centroid at 9.17 µm with 7.1 Hz statistical uncertainty (1.3 10<sup>-13</sup>), a 100-fold improvement with respect to previous measurements. It involves a Fabry-Perot saturated-absorption spectrometer referenced to the Système International through a frequency comb. We elaborate a detailed line shape model that allows retrieving the transit time width and accurate determination of the pressure and power shifts and broadenings. We also give the air pressure shift as well as the nitrogen and air broadening coefficients.

KEYWORDS

Infrared spectroscopy; ro-vibrational spectroscopy; formic acid; HCOOH; pressure broadening; air broadening; frequency comb; AgGaSe<sub>2</sub>

# 1. Introduction

Optical fiber telecommunication networks enable *Système International* (SI)-referenced
ultra stable optical signals around 1.5 µm to be disseminated from national metrology
institutes to remote laboratories with stabilities better than 10<sup>-15</sup> for integration times
larger than 1 s and long term accuracies better than 10<sup>-19</sup> [1]. These features can be
transferred to all the teeth of optical frequency combs and hence to any laser which
is frequency controlled against the frequency comb, allowing for SI-referenced high
resolution spectroscopy [2, 3].

The ultrastable optical signal disseminated by the REFIMEVE french research in-
frastructure [4] has recently been delivered at Kastler Brossel laboratory on Jussieu
campus in Paris and used for this work. In this paper, we present the optical frequency
synthesis spectrometer built for molecular spectroscopy at 9.17 µm and the measure-
ment campaign performed on the ν<sub>6</sub>, *J'* = 21, *K*'<sub>a</sub> = 2, *K*'<sub>c</sub> = 20 ← *J''* = 21, *K*''<sub>a</sub> =
3, *K*''<sub>c</sub> = 19 formic acid - HCOOH - rovibrational line.

The experimental setup is described in section 2. It presents the SI-referenced CO2 laser system, the saturated absorption cell, the frequency modulation method, the metrology campaign and the resulting spectroscopy signals. In section 2.4 and , we discuss the lineshape model used to adjust the recorded spectra. The results are presented and discussed in section ?? . We obtain a sub ppt (10-12) frequency resolution
on the line center.

## 2. Experimental setup

### 2.1. SI-referenced laser system

The spectroscopy laser is a CO₂ laser frequency controlled against a frequency comb which is optically stabilised to the ultrastable signal referenced to the système international (SI) [2]. The setup is described in Fig. 1. The 1542 nm ultrastable signal disseminated by the REFIMEVE network is phase noise compensated up to the extraction station located next to our laboratory with an expected relative frequency stability better than $10^{-15}$ for integration times larger than 1 s. We use a 102 m non-compensated single mode fiber to bring the ultrastable signal to our lab in the same building, which slightly reduces the frequency stability that remains better than $10^{-14}$. The frequency of the ultrastable signal is $\nu_R$=194 400 008 500 000±20 Hz under standard REFIMEVE operating conditions. We also receive a SI-referenced 100 MHz signal from Paris Observatory. REF This signal is used to phase lock a $f_Q$=10 MHz quartz oscillator used to synchronise all the instruments. The relative stability of this quartz oscillator is better than $5 \times 10^{-13}$ for integration times larger than 1 s.

The laser source is a single mode CO₂ laser operated on the 9R(42) line with up to 800 mW of optical power. The optical cavity is made of a 150 grooves/mm grating in the Littrow configuration for line selection and of a rear mirror mounted on a piezo ceramic split in two zones, a short one for fast frequency control and a long one to compensate slow drifts. The CO₂ laser beam is then split into three parts. The first one allows its frequency control against a 1.5-1.9 µm frequency comb as described in figure 1. The second one is frequency shifted by +128 MHz using a double pass acousto-optic modulator (AOM) instead of a single pass AOM in [5] to probe the HCOOH molecular transition discussed in this paper. The third one is used to control the frequency of a quantum cascade laser to perform $H_2^+$ vibrational spectroscopy, not described in this work.

The optical frequency comb consists of an erbium-doped fibre mode-locked laser emitting around 1.56 µm (Toptica DFC Cor 200) with a repetition rate $f_{rep}$= 200 MHz. Thanks to the use of a difference frequency generation method, the comb is carrier enveloppe offset frequency-free. The comb teeth frequencies are thus given by $p_{f_{rep}}$ where p is the tooth order. The comb can be operated in RF-lock mode and the repetition frequency is referenced to the quartz frequency $f_Q$ or in optical lock mode. In the latter case, the comb is phase-locked to the REFIMEVE ultrastable signal using a beat note between the REFIMEVE signal and the tooth of order $p_0$ of the comb. The beat note frequency is $f_{beat}^R = \nu_R - p_0 f_{rep}$. With $p_0$=974 000 and a $f_{beat}^R$=8.5 MHz, $f_{rep}$ is exactly set to 200 MHz. The frequency stability of the comb, locked on the ultrastable REFIMEVE signal, is thus expected to be at the $10^{-14}$ level or even lower for integration times larger than 1 s.

In RF-lock mode, the feedback loop has a very small bandwidth of less than 1 Hz in order to long term lock the repetition rate without degrading the free running stability of the comb [6]. The RF-locked comb to REFIMEVE beat note line shape exhibit a Voigt profile with a FWHM of 43,5 kHz (24.3 kHz lorentzian and 28.3 kHz gaussian contributions). In optical-lock mode, the comb teeth are tightly locked to the ultrastable signal with a 580 kHz bandwidth, thus transfering the ultrastable signal spectral charateristics to the comb. All the measurements are performed in optical-lock
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Experimental setup for the stabilisation of the CO₂ laser onto the SI-referenced frequency comb. The 9.166 µm CO₂ beam is overlapped with the 1.895 µm part of the frequency comb for sum frequency mixing in a AgGaSe₂ crystal. The shifted comb at 1.565 µm is combined with the original comb at 1.565 µm to produce a beat note on the photodetector. This beat note is used to lock the CO₂ laser via a phase lock loop. Lᵢ: Lenses, Pᵢ: Polarisers, DM: dichroic mirror, BS: beam splitter, FC: fiber collimator, OC: optical coupler, PD: photodetector, PFC: phase/frequency comparator. K: phase/frequency error signal used in Fig. 4.</figcaption></figure>

mode. The optical lock mode improves the spectrometer resolution by a factor 4.8 with respect to the RF lock mode as exemplified in Fig. 2.

An output of the comb is injected in a comb extension (TOPTICA DFC EXT) where it is amplified and sent to a highly non linear fiber to generate an additional comb output at 1.9 µm. The CO₂ laser beam is superimposed and mode matched to the 1.9 µm part of the frequency comb in a 8 mm long AgGaSe₂ crystal. With 87 mW of CO₂ laser radiation and 18.4 mW in 56 nm around 1895 nm we generate about 1.1 µW of sum frequency comb radiation at 1.56 nm, corresponding to a conversion efficiency of 0.7 mW/W². This radiation is spectrally filtered with a 54 nm wide bandpass filter and injected into an optical fiber. An optical delay line ensures the temporal matching between the pulses of the sum frequency comb and the original one. The combs are then mixed in a 75/25% fiber combiner and sent to a fast photodetector for beat note

<figure><img src="image_4.png" /><figcaption>**Figure 2.** Histograms of central frequencies $\nu_c$ determined as explained in Sec. 3 for RF-lock (pink) and optical lock (blue) comb regimes. Pressure: 1 µbar, Intensity: 1.0 V (0.078 mW/mm²), modulation frequency: 700 Hz, modulation depth: 42 kHz.</figcaption></figure>
detection. The original and shifted comb frequencies have the form $\nu_p = pf_{rep}$ and $\nu_q = qf_{rep} + \nu_{CO_2}$. The beat signal contains peaks at $f_{beat}^{CO_2} = \nu_{CO_2} - (p-q)f_{rep}$. The lowest positive beat frequency is close to 64 MHz and is obtained for $p-q = 163\,541$. This comb to comb beat is shown in Fig 3. It has a signal to noise ratio up to 58 dB in a 30 kHz resolution bandwidth (or 53 dB in 100 kHz RBW). It is bandpass filtered, divided by 8 and compared to a reference frequency $f_{ref}$ from a synthesiser using a phase/frequency comparator to obtain a phase error signal. The CO₂ laser servo loop includes a proportional-integral path to the short piezo part, with a loop oscillation frequency of 9 kHz. A second pure integrator slowly acts on the long piezo zone to maintain the fast piezo correction within its working range.

<figure><img src="image_5.png" /><figcaption>**Figure 3.** Beat note between the original and shifted combs around 64 MHz (purple). Blue: analyzer noise floor, green: electronic noise, orange: electronic + shifted comb noise, black line: expected shot noise, red: electronic + comb noise, purple: beatnote signal. Resolution bandwidth RBW = 30 kHz. The side peaks at 580 kHz from the carrier are remnants of the comb to REFIMEVE optical lock.</figcaption></figure>

## 2.2. Saturated absorption setup

The HCOOH saturated absorption setup is shown in Fig. 4. It is performed in a 1.047 m long symmetric Fabry-Perot cavity installed in a vacuum vessel. The mirrors have a 50 m radius of curvature resulting in a beam waist $w_0 = 3.85$ mm. The CO₂ laser is injected into the Fabry-Perot cavity with a linear horizontal polarisation and mode-matched using a telescope. The beam transmitted by the cavity is detected using a liquid nitrogen cooled HgCdZnTe detector and a transimpedance amplifier. The transmitted signal expressed in volts gives a precise relative scale for the intracavity power or intensity but only an approximate absolute scale of 1.48(15) mW/V for the total power $P_o$ and 0.078(8) mW/mm²/V for the intensity on the beam axis $I$ given by $I = 2P_o/(\pi w_0^2)$. The data analysis is made using the precise scale and the results
<figure><img src="image_6.png" /><figcaption>**Figure 4.** HCOOH spectrometer. BS: beam splitter, G: grating, BPF: band pass filter, PZT: piezo transducer, K: error signal from Fig 1. Full lines: laser beams, dashed lines: electronic signals, double dotted line: data transfer.</figcaption></figure>

are converted in physical units with larger uncertainties.

To get stable HCOOH pressure during the measurements, we use a leak valve to inject HCOOH from a small reservoir containing liquid HCOOH and a turbo pump connected to the vacuum vessel through a valve. Both valves are adjusted to get a stable pressure to better than 1% in the vacuum vessel. We have determined the HCOOH single pass absorption by measuring the cavity maximum transmission for different HCOOH pressures and found $7.43 \times 10^{-3} \mu b^{-1}$ corresponding to an absorption of $7.1 \times 10^{-3} m^{-1} \mu b^{-1}$. The empty cavity finesse is 153. The cavity finesse varies from 143 to 119 when we vary the HCOOH pressure from 0.5 to 2 $\mu$bar. Note that HCOOH can be considered to be a pure gas at room temperature with a 98.4% main isotope abundance, a 99.9% abundance in the *trans* rotameric form and with a dimer fraction below 0.1% at 2 $\mu$bar [7], see Sect. C.

To perform spectroscopy, the Fabry-Perot cavity is locked on resonance with the CO₂ laser frequency using first harmonic phase sensitive detection at 33 kHz of the signal transmitted by the cavity. The HCOOH saturated absorption signal is detected using a third harmonic phase sensitive detection (Stanford research SR810). To that end, the reference frequency $f_{ref}$ given by the synthesiser is frequency modulated with a modulation frequency $f_m$ and a modulation depth $\delta f_m$. Since the modulation is transferred to the CO₂ laser frequency by the feedback loop, the actual modulation depth is precisely determined well within 20 Hz analysing the FM sideband heights appearing in the comb to comb beatnote. (or is precisely calibrated by zeroing the first, third and fifth side bands appearing in the comb to comb beatnote.)

## 2.3. Measurement campaign

During a 10-days metrology campaign, we have systematically recorded spectra for four different HCOOH pressures (0.5, 1, 1.5 and 2 $\mu$bar), four different laser intensities (0.5, 1, 1.5 and 2 V), three different modulation frequencies (110, 220 and 330 Hz)
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Experimental spectra averaged on the 5 up and down scans and on the three modulation frequencies. The amplitudes (vertical axis) are normalised by the laser intensity and are then all represented with the same arbitrary unit.</figcaption></figure>

and three different modulation depths (20, 40 and 60 kHz) with five scans up and five scans down totaling 1440 spectra. Each scan is made of 178 steps with 160 small 2.5 kHz frequency steps near the centre of the line and 9 larger steps to probe the wings on both sides and detect a line offset if any. Figure 5 shows 48 spectra averaged on the 5 up and down scans and on the 3 modulation frequencies.

In addition, 120 low intensity (0.1 V) spectra are recorded at 220 Hz modulation frequency for all modulation depth and pressure configurations. They are not included in the line shape analysis but used as a bench mark for line parameter extrapolation at low pressure and low power (square purple points in Fig 8).

The third harmonic phase sensitive detection time constant is set to 30 ms with a -24 dB/octave roll-off and the dwell time between two measurements is set to 340 ms, larger than the PSD settling time of 300 ms. The noise equivalent bandwidth is 2.6 Hz (from SR810 user's manual). For each line, we evaluate the signal to noise ratio (in the 2.6 Hz equivalent bandwidth) as the ratio of the maximum of the signal to the rms of the signal in the wings. It varies from 20 for high pressure, high laser power and low modulation depth up to 363 at low pressure, low intensity and large modulation depth. Right before and right after each scan, the HCOOH pressure and transmitted power are measured. These values are used whenever necessary in the spectrum adjustment procedure. All the recorded data are available on a repository [8].

In order to assess the symmetry of the line, for each spectra we have computed the correlation coefficient of the experimental data $S_{exp}(\nu - \nu_c)$ and the symmetrised data $S_{exp}(\nu_c - \nu)$ which is 1 for even functions and -1 for odd functions of $\nu - \nu_c$. We obtain correlation coefficients between -0.997 and -1 indicating the high degree of symmetry of the line.
## 2.4. Lineshape models

The formic acid line studied here is $ν_6$, ($J'=21$, $Ka'=2$, $Kc'=20$) $\leftarrow$ ($J''=21$, $Ka''=3$, $Kc''=19$). It is a well isolated line within the $\approx$ 40 MHz Doppler width of HCOOH at room temperature, so no line pulling is expected. Spectroscopy is performed in the wavelength modulation regime with modulation and demodulation frequencies of at most 310 and 930 Hz, much smaller than the transit time half width at half maximum (HWHM) limit given by [9]

$$ \Delta\nu_{tr}^{th} = \frac{\sqrt{\ln(2)/2}}{2\pi w_0} \sqrt{\frac{2k_B T}{m}} = 7.92(3) \text{ kHz,} \quad (1) $$

for a line probed by a non-modulated laser at $T = 296(2)$ K. In that case, the experimental lineshape is given by the Fourier coefficients of the intensity transmitted by the Fabry-Perot cavity at the third harmonic of the modulation frequency.

Our goal is to work out a global adjustment of all the spectra recorded during the measurement campaign with a physically justified model involving a reduced set of adjustable parameters and representing well the observed linewidths.

Several authors have analyzed the saturated absorption line shape for a two-level system. An absorption coefficient with a Lorentzian profile is predicted from a third order calculation, i.e. in the case of weak saturation [9]. This profile is distorted by frequency modulation and by Fabry-Perot cavity induced intensity modulation. The resulting profile $S_L$ is presented in section 3. It is used for the determination of the signal offset and a preliminary evaluation of the line centre and effective linewidth of the 1440 spectra recorded during the measurement campaign. Nevertheless, this model fails to interpret the dependence of the observed line widths on pressure and intensity.

To push further the analysis, we use another line shape including saturation effects at all orders, obtained in the rate equation approximation by a number of authors. It is denoted $S_S$ and presented in section 4. For $S_S$ model, we also include line distortion due to unresolved hyperfine structure and Zeeman effect to obtain the $S_S^Z$ line shape. It allows interpreting the measurement campaign for all pressures, all laser intensities and all modulation configuration with a reduced number of adjustable parameters and the latter is used to extract the final results of the measurement campaign.

# 3. Perturbed lorentzian line shape

## 3.1. The model

The Lorentzian profile is perturbed by two effects. The first one is line shape distortion due to frequency modulation described by Arndt [10] (see section A.1 for details) leading to the line shape

$$ S_n(x, m) = \frac{i^n \epsilon_n}{2 m^n} \frac{(\sqrt{(1-ix)^2 + m^2} - (1-ix))^n}{\sqrt{(1-ix)^2 + m^2}} + c.c., \quad (2) $$

where $x = (ν - ν_c)/Δν$ is the normalised laser detuning with respect to the line central frequency $ν_c$, $m = δν_m/Δν$ is the normalised modulation depth, $Δν$ is the effective linewidth and $n$ is the phase sensitive detection order with $ε_0 = 1$ and $ε_{n \ge 1} = 2$. The second effect is combined intensity-frequency modulation (IM-FM) due to the
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Examples of saturated absorption spectra detected in third harmonic. Experimental parameters (modulation frequency, modulation depth, pressure, power) are (a): (220 Hz, 20 kHz, 0.5 µbar, 0.5 V, scan down), B: (220 Hz, 60 kHz, 0.5 µbar, 0.5 V, scan down), C: (220 Hz, 60 kHz, 1.0 µbar, 0.5 V, scan up) and D: (220 Hz, 60 kHz, 2.0 µbar, 2.0 V, scan down). The noise level, estimated from the standard deviation in the wings of the spectra, is given as a percentage of the maximum signal. The profiles are fitted by the $S_{fit}$ model (red solid line). The residuals are represented as a percentage of the maximum signal. Horizontal axis: $ν - ν_{off}$ with $ν_{off}$ = 32 708 391 980 825 Hz.</figcaption></figure>

Fabry-Perot cavity. Since the Fabry-Perot cavity is locked on resonance with the laser frequency, the transmitted intensity varies quadratically with frequency around the transmission maximum, and thus the laser frequency modulation at frequency $f_m$ is turned into an intensity modulation at frequency $2f_m$. As a result, contributions of $S_1$ and $S_5$ type are added to the $S_3$ line shape in the case of third harmonic detection. The line shape including the modulation distortion and IM-FM effects is derived in section A.2, Eq. (A23) and reads:

$$S_L(ν, A_0, A, ν_c, Δν) = A_0 - A \left( \frac{qm^2}{4} S_1(x, m) + \left(1 + \frac{qm^2}{2}\right) S_3(x, m) + \frac{qm^2}{4} S_5(x, m) \right) \quad (3)$$

It depends on four adjustable parameters, the vertical offset $A_0$ of the signal, a global amplitude $A$, the central frequency $ν_c$ that appears in the normalised frequency $x$ and the line HWHM $Δν$ that appears in the normalised frequency and in the normalised modulation parameter $m$. The parameter $q = -4(Δν/Δν_{FP})^2$ defined in Eq. (A8) depends on the Fabry-Perot HWHM $Δν_{FP}$ and hence on the formic acid pressure (see section A.3.1 and Eq. (A28)). Note that in the case of quantum cascade laser (QCL) spectroscopy, the observed lineshapes are asymmetric due to phase/amplitude coupling and $S_2$ and $S_4$ terms appear in the fitting function [11]. In our case, no line asymmetry is allowed with the $S_L$ function. Indeed, thanks to the 128 MHz frequency shift introduced by the AOM (see Fig. 4), the CO₂ laser is swept around the intensity maximum of the CO₂ laser with zero phase/amplitude coupling. We have checked using first harmonic lockin detection that we have almost no detectable intensity modulation at the modulation frequency: the residual intensity modulation is less than $10^{-5}$ of the
intensity at the output of the laser or at the output of the Fabry-Perot cavity.

## 3.2. Results and discussion

We have fitted each of the 1440 scans (labelled by $^{(j)}$) with the $S_L$ profile and obtained individual fit parameters $A_0^{(j)}$, $A^{(j)}$, $\nu_c^{(j)}$ and $\Delta\nu^{(j)}$ and error bars. The offsets $A_0^{(j)}$ are always negligible. We do not observe any systematic difference on the fit parameters when comparing frequency the five scans up and five scans down for the same physical conditions. This absence of drag effect is due to the choice of a sufficiently long dwell time for data recording. Direct adjustment of individual spectra with $S_L$ leads to significant ($\approx 0.8$) Pearson correlation coefficients between the individual line amplitudes $A_0^{(j)}$ and the effective widths $\Delta\nu^{(j)}$. The fits of four chosen spectra with the $S_L$ profile are shown in red in Fig. 6 together with their residuals.

The individual central frequencies $\nu_c^{(j)}$ and the line HWHM $\Delta\nu$ are displayed in Fig. 7 and 8. The different colors correspond to the different modulation frequencies. The horizontal axis are split in three main columns (one for each modulation depth), which are in turn split in four columns (one for each HCOOH pressure), and finally split into four columns for each laser power. At this latter stage, in Fig. 7, the horizontal axis is used to display the individual values of $\nu_c$ whereas in Fig. 8 the horizontal scale locally represents the laser power and the points correspond to the HWHM averaged over the tens scans. The interest of this representation is to show the pressure and modulation depth trends.

<figure><img src="image_9.png" /><figcaption>**Figure 7.** Top frame: central frequencies $\nu_c$ - $\nu_{off}$ and error bars obtained from saturated absorption spectra fitted by the $S_L$ model given in Eq. (3). The central frequencies are plotted as a function of modulation depth, formic acid pressure and intracavity laser power. The different modulation frequencies are represented with different colors: 110 Hz (red points), 220 Hz (blue points), 310 Hz (green points). The thick black curve represents the multilinear fit of $\nu_c$ by Eq. (4) and the horizontal red solid line shows the estimated undisturbed central frequency $\nu_0$ - $\nu_{off}$ with $\nu_{off} = 32\ 708\ 391\ 980\ 825$ Hz. Middle frame: residuals of the multilinear fit, expressed in error bar unit. The red horizontal lines correspond to $\pm 1$ error bar. Bottom frame: standard deviation of the central frequency over the 10 scans performed for each quadruplet of parameter values. The horizontal red dashed line represents the $10^{-12}$ level, relative to the transition frequency.</figcaption></figure>
### 3.2.1. Central frequency analysis

The central frequencies obtained using the $S_L$ line profile for each of the 1440 scans performed with the different modulation frequencies, modulation depths, formic acid pressures $P$, intracavity intensities $I$ and realisations are adjusted by a multilinear fit of the form

$$ \nu_c^{(j)} = \nu_0 + a f_{\text{mod}}^{(j)} + b \delta \nu_m^{(j)} + c P^{(j)} + d I^{(j)}. \quad (4) $$

All the points have an equal weight. The fit is plotted in black in Fig. 7. For most points, the residuals are less than the error bars. This indicates an excellent agreement between the measured central frequencies and the multilinear adjustment, but also indicates that the error bar may be overestimated by the fit with $S_L$ profile.

The $a$, $b$, $c$ and $d$ parameters are given in the second column of Tab. 1. The modulation frequency shift coefficient $a$ is very small and gives a shift of at most 15 Hz for $f_m = 310$ Hz. The modulation depth shift coefficient $b$ is also very small and gives a shift of at most 16 Hz at 60 kHz modulation depth, indicating that the modulation induced distorsion of the line seems to be well deconvoluted. We obtain a preliminary pressure shift coefficient of -13.9(2.6) Hz/μbar and a light shift coefficient of 221(40) Hz/(mW/mm²) which point out the excellent metrological features of the line.

The undisturbed central frequency $\nu_0$ has a statistical uncertainty of 7.1 Hz, i.e. $2.2 \times 10^{-13}$ in relative value. This represents a 100 fold improvement with respect to our previous measurement [12, 13].

<table><caption>**Table 1.** Multilinear fit parameters $a$, $b$, $c$ and $d$ obtained for the multilinear fit function defined in Eq. (4). $d$ is either expressed in Hz/V or in Hz/(mW/mm²) with a slightly larger relative uncertainty.</caption><thead><tr><th></th><th>S<sub>L</sub></th><th>S<sub>S</sub><sup>Z</sup></th><th>unit</th></tr></thead><tbody><tr><td>ν<sub>0</sub></td><td>32 708 391 980 792.9(7.1)</td><td>32 708 391 980 809.5(7.1)</td><td>Hz</td></tr><tr><td>a</td><td>44(18)</td><td>45(18)</td><td>Hz/kHz</td></tr><tr><td>b</td><td>0.265(89)</td><td>0.403(90)</td><td>Hz/kHz</td></tr><tr><td>c</td><td>-13.9(2.6)</td><td>-5.0(2.6)</td><td>Hz/μbar</td></tr><tr><td>d</td><td>17.2(2.6)</td><td>-6.7(2.6)</td><td>Hz/V</td></tr><tr><td>d</td><td>221(40)</td><td>-86(35)</td><td>Hz/(mW/mm²)</td></tr></tbody></table>

### 3.2.2. Linewidth analysis

Figure 8 shows the HWHM $\Delta\nu^{(j)}$ averaged over the 10 scans for each modulation frequency, modulation depth, pressure and intensity. They vary from 30 to 80 kHz and the corresponding $m$ parameters from 0.25 to 2. It shows that the HWHM depends both on the pressure and the intensity, that it does not depend on the modulation frequency but slightly depends on the modulation depth indicating the line distorsion by frequency modulation is not perfectly deconvoluted.

The HWHM of the lorentzian lineshape is expected to vary with the pressure and the intensity as [9]

$$ \Delta\nu^{(j)}(\Delta\nu_{tr}, \gamma_{\text{self}}, I_S^0, \beta) = (\Delta\nu_{tr} + \gamma_{\text{self}}P^{(j)})\sqrt{1 + \frac{I^{(j)}}{I_S(P^{(j)})}}, \quad (5) $$

with $\Delta\nu_{tr}$ the transit time HWHM, $\gamma_{\text{self}}$ the pressure broadening coefficient and $I_S$ the saturation intensity. The saturation intensity $I_S(P) = I_S^0(1 + \beta P)$ linearly depend on the pressure to account for phase changing collisions [14].
<figure><img src="image_10.png" /><figcaption>**Figure 8.** Top: HWHM of the spectra fitted with the $S_L$ profile, averaged over the 10 scans for 144 modulation pressure and intensity configurations. The small purple squares corresponds to low intensity measurement campaign and are not included in the fit. The black solid line corresponds to a global fit of the 144 points with Eq. (5). Bottom: The black line corresponds to three global fits, one for each modulation depth. In both graphs, the horizontal red line shows the expected transit time HWHM and the blue zone indicates the maximum expected hyperfine splitting, which is not resolved.</figcaption></figure>

A global fit of the HWHM with Eq. (5) gives $\Delta\nu_{tr} = 19.6(2.4)$ kHz, which is 2.5 times larger than the expected transit-time width, $\gamma_{self} = 16.4(1.7)$ kHz/µbar, $I_S^0 = 0.83(36)$ V = 64.7(28.9) µW/mm² four times larger than the expected saturation intensity (16 µW/mm²) that can be deduced from the line Einstein coefficient [15] and $\beta = 0.52(27)$ µbar⁻¹. The fit is represented in Fig. 8 (top). It clearly shows that $S_L$ and the linewidth expression in Eq. (5) does not well represent the experimental observations.

We have also fitted the HWHM using Eq. (5) for each modulation depth subsets.
The three fits are represented in Fig. 8 (bottom). While the fits seems to well reproduce the data, they give transit time widths (18.29(49), 19.80(74), 20.63(72) kHz) far from the expected value and saturation intensities (1.58(13), 0.94(12), 0.42(6) V or 123(16), 73(12), 33(6) µW/mm² ) and β parameters (0.224(87), 0.442(86), 0.651(54) µbar⁻¹) that both depend on the modulation depth.

In all cases, the fits do not reproduce the observed low intensity HWHM (purple squares). In conclusion, the perturbed lorentzian line shape, which is valid in the low saturation regime cannot be used to interpret our measurement campaign and extract line broadening coefficients.

Conversion factor = 0.078(8) mW/mm²/V. 16 µW/mm² corresponds to 0.20 V.

## 4. All order saturation lineshape

In this section, we present a new line shape taking into account saturation to all orders in intensity, as well as Fabry-Perot transmission, pressure, modulation, and third-order detection effects.

### 4.1. Elaborating the lineshape

#### 4.1.1. Cavity transmission signal

The signal $s(t)$ transmitted by the Fabry-Perot is a periodic function of time with period $1/f_m$ and the spectroscopy signal $S_S$ is given by the third harmonic Fourier coefficient in phase with the modulation. Taking into account the transmission coefficient of the Fabry-Perot cavity (Eq. (A25)) and the cavity induced intensity modulation (Eq. (A29)), the signal transmitted by the cavity writes

$$s(t) = I_0 \left( 1 - \frac{\delta\nu_m^2}{\Delta\nu_{FP}^2} \cos^2(2\pi f_m t) \right) \frac{(1-R)^2\tau}{(1-R\tau)^2} \quad (6)$$

where $I_0$ is the injected intensity, $\tau$ is the single pass intensity transmission coefficient of the gas and where the Fabry-Perot HWHM $\Delta\nu_{FP}$ can be expressed from Eq (A24) and (A27) as

$$\Delta\nu_{FP} = \frac{c}{4L} \frac{1 - R\tau}{\pi\sqrt{R\tau}}. \quad (7)$$

The single pass intensity transmission coefficient $\tau$ is linked to the gas absorption coefficient $\kappa$ and the cavity length $L$ by

$$\tau = e^{-\kappa L}. \quad (8)$$

#### 4.1.2. Introducing the Shimoda line shape

For a two-level system, the expression of the absorption coefficient in the strong field regime has been derived by several authors [14]? ] and is given by

$$\kappa^S(\nu, \nu_c, \Delta\nu, G, \kappa_0) = \kappa_0 \frac{1 + \frac{c}{d}}{\sqrt{(c+d)^2 - 4x^2}} \quad (9)$$
where $\kappa_0$ is the low intensity absorption coefficient at resonance, $c = \sqrt{x^2+1}$, $d = \sqrt{x^2+1+2G}$ with $x = (\nu - \nu_c)/\Delta\nu$ and where $G = I/I_{sat}$ is the saturation parameter and $I_{sat}$ the saturation intensity. The linewidth $\Delta\nu$ is determined by the transit time limit and the pressure broadening and is written

$$ \Delta\nu = \Delta\nu_{tr} + \gamma_{self}P. \qquad (10) $$

The saturation effect is taken into account through the parameter $G$.

### 4.1.3. Hyperfine structure

The hyperfine structure (HFS) of HCOOH has been studied by Chardon [16] and Cazzoli [17] in the case of pure rotational spectroscopy in the ground vibrational level and led to the determination of coupling constants for rotational levels close to those probed in this work. In Section B, we define the relevant quantum numbers $F_1$ and $F$ and evaluate the HFS of the line studied here. For linear polarisation excitation, we expect four main lines shifted by $\Delta\nu_{hfs}(F_1, F)=4.6, 0.2, -0.0$ and $-4.4$ kHz with respect to the spin less line position, with close respective amplitudes $W_{hfs}(F_1, F)=0.951, 0.998, 0.998$ and $1$. No cross-over lines are expected since the different lines do not share common HFS sublevels.

The main HFS lines span a 9 kHz interval, comparable to the transit time HWHM limit of the spectrometer. HFS is not resolved but may significantly perturb the single line HWHM analysis presented in Sec. 3.2.2 and has to be taken into account.

### 4.1.4. Absorption coefficient

Although the spectroscopy is carried out in zero magnetic field, the Zeeman structure must be taken into account, as each Zeeman sublevel has a different coupling to radiation and therefore a different saturation intensity. The absorption coefficient of the gas is the sum of the contributions of each Zeeman sublevel and is written as :

$$ \kappa^Z = \sum_{F_1, F, m_F} \kappa^S(\nu, \nu_c + \delta\nu_{hfs}(F_1, F), \Delta\nu_{tr} + \gamma_{self}P, G(F_1, F, m_F), \kappa_0(F_1, F, m_F)). \quad (11) $$

Using the Wigner-Eckhart theorem, the Zeeman line amplitudes $\kappa_0(F_1, F, M_F)$ are written

$$ \kappa_0(F_1, F, m_F) = \kappa_0 W_{hfs}(F_1, F) \langle 10Fm_F | Fm_F \rangle^2, \quad (12) $$

with $\langle 10Fm_F | Fm_F \rangle = m_F/\sqrt{F(F+1)}$ and $W_{hfs}(F_1, F)$ given by Eq. (B4). $F_1$ and $F$ values, frequency shifts and relative weights are given in Tab. B4. Since $|m_F|$ varies from 0 to 22, $\kappa_0((F_1, F, m_F))$ spans a 1 to 500 interval for $m_F \neq 0$.

The reduced absorption coefficient $\kappa_0$ is proportional to the pressure, so we rewrite

$$ \kappa_0 L = \xi P. \qquad (13) $$

$\xi$ has been experimentally determined (see Sect. A.3.1) and we obtain $\xi = 3.338 \times 10^{-3} \mu b^{-1}$.

The linewidth $\Delta\nu$ depends on the pressure and is written $\Delta\nu = \Delta\nu_{tr} + \gamma_{self}P$ where the pressure broadening coefficient $\gamma_{self}$ is supposed to be independent of $m_F$.
The saturation parameter $G(F_1, F, m_F) = I/I_{sat}(F_1, F, m_F)$ depends on the Zeeman transition through the saturation intensity, which we write

$$I_{sat}(F_1, F, m_F) = \frac{I_S^0 + \beta P}{W_{hfs}(F_1, F) \langle 10 F m_F | F m_F \rangle^2} \quad (14)$$

$I_S^0$ is a reduced saturation intensity, which depends on the pressure with a sensitivity coefficient $\beta$ supposed to be independent of the Zeeman sublevel.

### 4.1.5. Lineshape

Injecting equations (7) to (14) into Eq. (6), we obtain the time varying signal $s(t)$ observed at the output of the cavity. Note that the time dependence of $s(t)$ comes from the time dependence of the modulated laser frequency $\nu(t) = \nu_0 + \delta\nu \cos(2\pi f_m t)$ and from the Fabry-Perot induced intensity modulation $I_0(1 - \frac{\delta\nu_m^2}{\Delta\nu_{FP}^2} \cos^2(2\pi f_m t))$ that appear explicitely in Eq. (6) and also in the saturation parameter $G$. As a consequence, the absorption coefficient and hence the Fabry-Perot HWHM are also time-modulated.

The signal transmitted by the Fabry-Perot is a periodic function of time of period $1/f_m$. Since the modulation frequency is small as compared to all the other characteristic frequencies of the system, the spectroscopy signal is proportional to the in-phase third harmonic Fourier coefficient of $s(t)$ with a possible offset $S_0^z$:

$$S_{sat}^z = a \int_{-\pi}^{\pi} s(t) \cos(2\pi 3 f_m t) dt + S_0^z \quad (15)$$

The integral is evaluated numerically using the Python 3.11 quad\_vec function of the scpy.optimise package. The $S_0^z$ line shape depends on the laser frequency $\nu$ and on fourteen parameters listed in Table 2. The two first parameters are the pressure and laser intensity, which are measured for each scan. The third one, the reduced saturation intensity is deduced from the line Einstein coefficient $A_{21} = 1.388 \text{ s}^{-1}$ given in the HITRAN data base using $I_{sat}^0 = 2hc\pi^2 A_{21}/(3\lambda^3) = 2.3 \times 10^{-12} \text{ mW/mm}^2$. $I_{sat}^0$ is extremely small hence $I_{sat}^0 + \beta P$ in Eq. (14) is completely dominated by the pressure dependent term, and $I_{sat}^0$ can be considered as null in the adjustment procedure. The five next parameters have been determined experimentally hence the lineshape depends on six adjustable parameters. The centre frequency, offset and signal amplitude are specific to each recorded spectrum, while the other three are not, and are determined globally.

## 4.2. Results and discussion

### 4.2.1. Adjustment procedure

Directly adjusting the 1440 spectra of the measurement campaign with Eq. (15) requires 4323 adjustable parameters (1440 line centers, offsets and signal amplitudes and the 3 global parameters). To reduce the complexity of the calculations, the adjustments are performed for each of the three modulation depths on 16 spectra obtained averaging the measurement campaign over the 10 scans and 3 modulation frequencies leading to averaged line centers and offsets. These spectra are shown in Fig 5 with one color per modulation depth. Moreover, to drastically reduce the number of degree
<table><caption>**Table 2.** Parameters on which the $S_3^z$ lineshape depends. The seven first ones are determined experimentally.</caption><tr><td>P</td><td>Pressure</td></tr><tr><td>I</td><td>Intensity</td></tr><tr><td>I<sup>0</sup><sub>S</sub></td><td>Reduced saturation intensity</td></tr><tr><td>Δν<sub>m</sub></td><td>Modulation depth</td></tr><tr><td>Δν<sup>0</sup><sub>FP</sub></td><td>Empty cavity HWHM</td></tr><tr><td>η</td><td>FP HWHM pressure dependence</td></tr><tr><td>ξ</td><td>Absorption coefficient pressure dependence</td></tr><tr><td>R</td><td>Mirror reflectivity</td></tr><tr><td>ν<sub>c</sub></td><td>Central frequency</td></tr><tr><td>S<sup>z</sup><sub>0</sub></td><td>Offset</td></tr><tr><td>a</td><td>Signal amplitude</td></tr><tr><td>Δν<sub>tr</sub></td><td>Transit time HWHM</td></tr><tr><td>γ<sub>self</sub></td><td>Pressure broadening coefficient</td></tr><tr><td>β</td><td>Saturation intensity pressure dependence</td></tr></table>

<table><caption>**Table 3.** Optimum parameters for the adjustment by $S_{sat}^Z$ line shape.</caption><thead><tr><th></th><th>20</th><th>40</th><th>60</th><th>kHz</th></tr></thead><tbody><tr><td>δν<sub>m</sub></td><td>7 176 (34)</td><td>6 814 (42)</td><td>8 521 (71)</td><td>Hz</td></tr><tr><td>Δν<sub>tr</sub></td><td>7.732(93)</td><td>17.182(107)</td><td>16.260(128)</td><td>kHz/μbar</td></tr><tr><td>γ<sub>self</sub></td><td>0.08086(84)</td><td>0.06024(68)</td><td>0.05540(95)</td><td>V/μbar</td></tr><tr><td>β</td><td>1.31(13)</td><td>0.97(10)</td><td>0.90(9)</td><td>mW/mm<sup>2</sup>/μbar</td></tr></tbody></table>

of freedom, we have performed the adjustment in two steps. First, for each modulation depth, we use the 16 averaged central frequencies $\bar{\nu}_c^{(j)}$ and 16 averaged offsets $\bar{A}_0^{(j)}$ and adjust the three global parameters using $S_{sat}^Z$ and experimental profiles both normalised to their peak to peak amplitude. That way, the individual signal amplitudes are not required for this step. In the second step, for each modulation depth, we use the three global parameters to fit the individual central frequencies offsets and amplitudes. Those two steps are repeated 20 times to assess the convergence of the fitting procedure. The resulting transit time HWHM, pressure broadening coefficient, and saturated intensity pressure dependence are given in Table 3 for each modulation depth. Finally, these 9 global parameters (3 for each modulation depth) are used to fit the 1440 scans and get final values of the individual line centers and offsets. We now analyse the results and show that the new lineshape $S_{sat}^Z$ very well describe the observations of the measurement campaign.

### 4.2.2. Transit time broadening

We observe that the fitted transit time broadenings are between 6.8 and 8.5 kHz for the three modulation depths with an averaged value of 7.5(0.9) kHz in good agreement with the expected value of 7.9 kHz.

### 4.2.3. Central frequencies analysis

The offsets $A_0^{(j)}$ are still compatible with 0. The converged central frequencies are shown in Fig. 9. The result of the multilinear fit defined in Eq. (4) are given in Tab. 1.

The central frequency $\nu_0$ is shifted by $\approx 16$ Hz with respect to the preliminary results of section 3.2. This shift appears during the first round of the fitting procedure. It comes from the fact that the $S_{sat}^Z$ lineshape takes into account the different weight of the hyperfine structure components and is slightly asymmetric.

The residuals are no longer systematically smaller than the error bars indicating the error bars are no longer overestimated.
The modulation shift parameters *a* and *b* remain very small indicating the modulation effect are well deconvoluted. The light shift coefficient *d* is comparable to the one obtained in section 3.2. Only the pressure shift coefficient *c* changes from -13.9 to -5.0 Hz/µbar, remaining very small.

### 4.2.4. Linewidth analysis

The adjustment with $S_3^z$ lineshape does not provide individual linewidth $\Delta\nu^{(j)}$ to be compared with the experimental widths obtained fitting the spectra with the $S_L$ profile. From the optimum parameters of the global fit, we have numerically generated lineshapes using Eq. (15) for the different modulation and pressure configuration of the measurement campaign and for intensities varying from 0 to 2.5 V. We have then adjusted those numerical lineshapes with $S_L$ lineshape to determine their HWHM and compare with the HWHM obtained from the experimental spectra. The results are shown in red in Figure 10. We observe that the red curve not only very well match the individual HWHM $\Delta\nu^{(j)}$ but also predicts the low intensity data represented by purple squares, which are not included in the fitting procedure.

<figure><img src="image_11.png" /><figcaption>**Figure 9.** Top frame: central frequencies $\nu_c$ - $\nu_{off}$ and error bars obtained from saturated absorption spectra fitted by the $S_{sat}^Z$ model following the procedure explained in the text. Same representation as in Fig. 7. The thick black curve represents the multilinear fit of $\nu_c$ by Eq. (4) and the horizontal red solid line shows the estimated undisturbed central frequency $\nu_0$ - $\nu_{off}$ with $\nu_{off} = 32\,708\,391\,980\,825$ Hz. Middle frame: residuals of the multilinear fit, expressed in error bar unit. The red horizontal lines correspond to $\pm 1$ error bar. Bottom frame: standard deviation of the central frequency over the 10 scans performed for each quadruplet of parameter values. The horizontal red dashed line represents the $10^{-12}$ level, relative to the transition frequency.</figcaption></figure>

### 4.2.5. Local conclusion

The proposed lineshape $S_{sat}^Z$ well deconvolute the modulation effects and accurately represents the pressure and intensity dependences of the measurement campaign with only three adjustable parameters, thus allowing for reliable determination of the line parameters, in particular of the self broadening coefficient $\gamma_{self} = 18.5(1.0)$ Hz/µbar obtained extrapolating to zero modulation depth the $\gamma_{self}$ determined for each modula-
<figure><img src="image_12.png" /><figcaption>**Figure 10.** Same points as in Fig. 8. The red curve is obtained adjusting the $S_{sat}^Z$ theoretical profiles given by Eq. (15) with the $S_L$ profile of Eq. (3), (see text).</figcaption></figure>

tion depth reported in Table 3. We give a conservative uncertainty of 1 kHz/µbar. Our determination of $\gamma_{self}$ is approximately twice as large as the value reported in HITRAN data base [18, 19], $\gamma_{self}^{HITRAN} = 0.32 \text{ cm}^{-1}\text{/atm} = 9.5 \text{ kHz/µbar}$.

### 4.2.6. Error budget

Table 4 presents the error budget for the line central frequency in the nominal conditions $\nu_{nom}$ defined by $P_{nom} = 1 \mu b$, $I_{nom} = 0.078 \text{ mW/mm}^2$ (corresponding to 1 V), $f_m = 220 \text{ Hz}$ and $\delta f_m = 20 \text{ kHz}$ including the pressure shift, light shift and modulation shifts with respective inaccuracies of 20 nbar, 0.008 mW/mm$^2$, 0.1 Hz and 20 Hz. We also include an inaccuracy source linked to the laser polarisation. Indeed, we have varied the laser polarisation probing the molecules rotating a quarter wave plate inserted before the Fabry-Perot cavity. Central frequency shifts up to 2.5 kHz are observed when the polarisation is varied from linear to circular. We have checked that the laser polarisation is linear to better than 99% analysing the laser polarisation using a 300:1 extinction ratio wire-grid polariser, leading to a at most 25 Hz contribution to the error budget. We obtain $\nu_{nom} = 32 708 391 980 816.7(26.0) \text{ Hz}$ with a relative inaccuracy reduced by a factor of 15 with respect to our previous measurement [12, 13]. This new value can be recommended as a secondary frequency standard at 9.166 µm.

Table 4 also present the error budget for the central frequency extrapolated back to the conditions we used in our previous measurement where the CO$_2$ laser frequency was locked to the HCOOH line with a 5.667(10) kHz modulation frequency and a 100(10) kHz modulation depth and measured with respect to a frequency comb. The agreement is well within the error bars.
<table><caption>**Table 4.** Central frequency $\nu_{\text{nom}}$ under nominal conditions ($P_{\text{nom}} = 1$ µb, $I_{\text{nom}} = 1$V, $f_m = 220$ Hz and $\delta f_m = 20$ kHz) and comparison with previous data. * To compare with our previous measurement reported in [12] and [13] performed with quite different modulation conditions we extrapolate the shifts to 5.667(10) kHz modulation frequency and 100(10) kHz modulation depth.</caption><thead><tr><th>Ref.</th><th></th><th>(Hz)</th></tr></thead><tbody><tr><td></td><td></td><td>32 708 391 980 809.5(7.1)</td></tr><tr><td></td><td>pressure shift</td><td>-14.2(2.6)</td></tr><tr><td></td><td>light shift</td><td>17.1(2.6)</td></tr><tr><td></td><td>$f_m$ shift</td><td>9.9(4.0)</td></tr><tr><td></td><td>$\delta f_m$ shift</td><td>10.6(1.8)</td></tr><tr><td></td><td>polarisation shift</td><td>0.0(25.0)</td></tr><tr><td>This work</td><td>$\nu_{\text{nom}}$</td><td>32 708 391 980 827.6(26.0)</td></tr><tr><td></td><td></td><td>32 708 391 980 809.5(7.1)</td></tr><tr><td></td><td>pressure shift</td><td>-14.2(2.6)</td></tr><tr><td></td><td>light shift</td><td>17.1(2.6)</td></tr><tr><td>*</td><td>$f_m$ shift</td><td>255.0(102.0)</td></tr><tr><td>*</td><td>$\delta f_m$ shift</td><td>26.5(9.3)</td></tr><tr><td></td><td>polarisation shift</td><td>0.0(25.0)</td></tr><tr><td>This work</td><td></td><td>32 708 391 981 094 (106)</td></tr><tr><td>[12, 13]</td><td>$\nu_{2008}$</td><td>32 708 391 981 4(8)*</td></tr><tr><td>HITRAN [19]</td><td></td><td>32 708 394(3)</td></tr></tbody></table>

## 5. Nitrogen and air shift and broadening

In order to determine the nitrogen and air broadening coefficient, we have performed a specific measurement campaign with a modulation frequency of 220 Hz, a modulation depth of 20 kHz, and a HCOOH pressure of 0.5 µbar. We have set the nitrogen or air pressure to 0.5, 1, 2 and 4 µbar and the laser intensity to 0.5, 1, 1.5 and 2 V and recorded 5 scans up and 5 scans down.

We have then performed a global adjustment of the 160 recorded spectra using Eq. 6 to Eq. 15 where we introduce the nitrogen or air broadening coefficients $\gamma_i$ in Eq. (10) and saturation intensity sensitivity coefficients $\beta_i$ in Eq. (14) with

$$ \Delta\nu = \Delta\nu_{tr} + \gamma_{\text{self}}P + \gamma_i P_i \quad (16) $$

and

$$ I_{\text{sat}}(F, m_F) = \frac{I_S^0 + \beta P + \beta_i P_i}{W_{\text{hfs}}(F_1, F) \langle 10 F m_F | F m_F \rangle^2}, \quad (17) $$

where $i=N_2$ or air, and where we use the $\Delta\nu_{tr}$, $\gamma_{\text{self}}$ and $\beta$ parameters for the 20 kHz modulation depth given in Table 3. Table 5 gives the $\gamma_i$ and $\beta_i$ coefficients. The air and nitrogen broadening coefficients are very similar and about 8.1 kHz/µbar, larger than the value of 3 kHz/µbar (0.101 cm⁻¹/atm) reported in HITRAN data base [18–20].

The fitting procedure also gives a first evaluation of the air shift coefficient $\delta_{\text{air}} = -39.0(14.9)$ Hz/µbar = 0.0013(5) cm⁻¹/atm, a very small value to be compared to the null value reported in HITRAN.

## 6. Conclusion

We have presented a thorough study of the $\nu_6$, $J' = 21$, $K'_a = 2$, $K'_c = 20 \leftarrow J'' = 21$, $K''_a = 3$, $K''_c = 19$ HCCOH line using saturated absorption in a Fabry-Perot cavity, third harmonic phase sensitive detection and a CO₂ laser referenced to the *Système*
<table><caption>**Table 5.** Air pressure shift coefficient $\delta_{\text{air}}$, nitrogen and air broadening coefficients $\gamma_i$ and pressure dependence of the saturation intensity $\beta_i$, and comparison with HITRAN data.</caption><thead><tr><th></th><th></th><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td>This work</td><td>δ<sub>air</sub></td><td>-39.0(14.9)</td><td>Hz/μbar</td><td>0.0013(5)</td><td>cm<sup>-1</sup>/atm</td></tr><tr><td>HITRAN</td><td>δ<sub>air</sub></td><td>0</td><td>Hz/μbar</td><td></td><td></td></tr><tr><td>This work</td><td>γ<sub>N2</sub></td><td>8.056(83)</td><td>kHz/μbar</td><td></td><td></td></tr><tr><td>This work</td><td>γ<sub>air</sub></td><td>8.121(81)</td><td>kHz/μbar</td><td></td><td></td></tr><tr><td>[20]</td><td>γ<sub>air</sub></td><td></td><td></td><td></td><td></td></tr><tr><td>HITRAN</td><td>γ<sub>air</sub></td><td>9.6(1.0)</td><td>kHz/μbar</td><td></td><td></td></tr><tr><td>This work</td><td>β<sub>N2</sub></td><td>0.0400(19)</td><td>Hz/V</td><td>0.51(6)</td><td>Hz/(mW/mm<sup>2</sup>)</td></tr><tr><td>This work</td><td>β<sub>air</sub></td><td>0.0444(21)</td><td>Hz/V</td><td>0.57(7)</td><td>Hz/(mW/mm<sup>2</sup>)</td></tr></tbody></table>

International using a frequency comb optically locked to the ultrastable signal disseminated by the REFIMEVE network.

We have carried out a metrology campaign with a systematic study of the pressure, intensity and modulation effects on the line shape and line center. We have shown that a line shape based on a modulation-distorted Lorentzian profile is not able to represent the metrology campaign. We have presented a new lineshape that takes into account the absorption coefficient for saturated absorption to all orders in intensity, the hyperfine and Zeeman structure of the transition, the Fabry-Perot transmission and Fabry-Perot induced intensity modulation. This lineshape allows modeling the 144 configurations of the metrology campaign with only three adjustable parameters and residuals smaller than XX% of the signal. We extract the line centre at the 30 Hz (10<sup>-12</sup>) inaccuracy level and re-define the $\nu_6$, $J' = 21$, $K'_a = 2$, $K'_c = 20 \leftarrow J'' = 21$, $K''_a = 3$, $K''_c = 19$ HCOOC line as a secondary frequency standard with an inaccuracy of 25 Hz. We also give updated values of the pressure shift and light shift as well as the pressure and intensity broadening coefficients. The new determination of pressure self broadening coefficient is twice as large as the present HITRAN value. We also give the first determination of the nitrogen and air pressure shift and pressure broadening coefficient of the line.

## Acknowledgement(s)

We acknowledge the strong support of « RENATER, réseau national pour l'enseignement et la recherche – www.renater.fr » and REFIMEVE network and team for providing the ultrastable optical signal. REFIMEVE is a French research infrastructure funded in the frame of PIA3 (ANR-21-ESRE-0029).

## Disclosure statement

An unnumbered section, e.g. \*section\*{Disclosure statement}, may be used to declare any potential conflict of interest and included in the non-anonymous version before any Notes or References, after any Acknowledgements and before any Funding information.

## Funding

This work was supported by LABEX Cluster of Excellence FIRST-TF (ANR-10-LABX-48-01), within the Program « Investissements d'Avenir » operated by the French National Research Agency (ANR); The authors have received financial support from
Agence Nationale de la Recherche (grants ANR-19-CE30-0029 HYMPE and ANR-24-CE30-00 ULTIµOS), from région Ile de France DIM QUANTIP and DIM SIRTEQ (Coqteil). Support from the CPER project COMB’IdF (Convention #22008311 of Région Ile-de-France and Convention #2022-IDF-P2 of Ministère de l’Enseignement Supérieur, de la Recherche et de l’Innovation) is gratefully acknowledged.

**Should we thank COMOMET? ça ne mange pas de pain.** The project (23FUN04 COMOMET) has received funding from the European Partnership on Metrology, co-financed from the European Union’s Horizon Europe Research and Innovation Programme and by the Participating States.

## Notes on contributor(s)

An unnumbered section, e.g. \section*{Notes on contributors}, may be included in the non-anonymous version if required. A photograph may be added if requested.

## Nomenclature/Notation

An unnumbered section, e.g. \section*{Nomenclature} (or \section*{Notation}), may be included if required, before any Notes or References.

## Notes

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## References

[1] F. Guillou-Camargo, V. Ménoret, E. Cantin, O. Lopez, N. Quintin, E. Camisard, V. Salmon, J.-M. Le Merdy, G. Santarelli, A. Amy-Klein, P.-E. Pottie, B. Desruelle, and C. Chardonnet. First industrial-grade coherent fiber link for optical frequency standard dissemination. *Applied Optics*, 57(25):7203, aug 2018.

[2] Bérengère Argence, Bruno Chanteau, Olivier Lopez, Daniele Nicolodi, Michel Abgrall, Christian Chardonnet, Christophe Daussy, Benoît Darquié, Yann Le Coq, and Anne Amy-Klein. Quantum cascade laser frequency stabilization at the sub-hz level. *Nature Photonics*, 9(7):456–460, jun 2015.

[3] R. Santagata, D. B. A. Tran, B. Argence, O. Lopez, S. K. Tokunaga, F. Wiotte, H. Mouhamad, A. Goncharov, M. Abgrall, Y. Le Coq, H. Alvarez-Martinez, R. Le Targat, W. K. Lee, D. Xu, P.-E. Pottie, B. Darquié, and A. Amy-Klein. High-precision methanol spectroscopy with a widely tunable si-traceable frequency-comb-based mid-infrared qcl. *Optica*, 6(4):411, April 2019.

[4] https://www.refimeve.fr/index.php/en/.

[5] Franck Bielsa, Albane Douillet, Tristan Valenzuela, Jean-Philippe Karr, and Laurent Hilico. Narrow-line phase-locked quantum cascade laser in the 9.2 µm range. *Opt. Lett.*, 32(12):1641–1643, Jun 2007.

[6] Andrew A. Mills, Davide Gatti, Jie Jiang, Christian Mohr, Will Mefford, Livio Gianfrani, Martin Fermann, Ingmar Hartl, and Marco Marangoni. Coherent phase lock of a
9 µm quantum cascade laser to a 2 µm thulium optical frequency comb. *Optics Letters*, 37(19):4083, September 2012.

[7] J. Vander Auwera, K. Didriche, A. Perrin, and F. Keller. Absolute line intensities for formic acid and dissociation constant of the dimer. *The Journal of Chemical Physics*, 126(12):124311, mar 2007.

[8] Maxime Leuliet and Laurent Hilico. *Formic acid (HCOOH) MIR spectroscopy data*, 2024.

[9] C. J. Bordé, J. L. Hall, C. V. Kunasz, and D. G. Hummer. Saturated absorption line shape: Calculation of the transit-time broadening by a perturbation approach. *Physical Review A*, 14(1):236-263, jul 1976.

[10] Rolf Arndt. Analytical line shapes for lorentzian signals broadened by modulation. *Journal of Applied Physics*, 36(8):2522-2524, aug 1965.

[11] Stéphane Schilt, Luc Thévenaz, and Philippe Robert. Wavelength modulation spectroscopy: Combined frequency and intensity laser modulation. *Applied Optics*, 42(33):6728, 2003.

[12] Franck Bielsa, Khelifa Djerroud, Andrei Goncharov, Albane Douillet, Tristan Valenzuela, Christophe Daussy, Laurent Hilico, and Anne Amy-Klein. Hcooh high-resolution spectroscopy in the 9.18 µm region. *Journal of Molecular Spectroscopy*, 247(1):41-46, 2008.

[13] Laurent Hilico, Franck Bielsa, and Anne Amy Klein. Erratum to hcooh high-resolution spectroscopy in the 9.18 µm range. *Journal of Molecular Spectroscopy*, 393:111773, mar 2023.

[14] Koichi Shimoda. *High-Resolution Laser Spectroscopy*. Springer Berlin Heidelberg, 1976.

[15] Franck Bielsa. *Spectroscopie vibrationnelle à deux photons de l'ion H<sub>2</sub><sup>+</sup>: développement d'une source laser à 9,166µm*. Thèses, Université Pierre et Marie Curie - Paris VI, October 2007.

[16] Jean-Claude Chardon, Claude Genty, Daniel Guichon, and Jean-Gérard Theobald. rf spectrum and hyperfine structure of formic acid. *The Journal of Chemical Physics*, 64(4):1437-1441, feb 1976.

[17] G. Cazzoli, C. Puzzarini, S. Stopkowicz, and J. Gauss. Hyperfine structure in the rotational spectra of trans-formic acid: Lamb-dip measurements and quantum-chemical calculations. *Astronomy and Astrophysics*, 520:A64, sep 2010.

[18] A. Perrin and J. Vander Auwera. An improved database for the region of the formic acid spectrum. *Journal of Quantitative Spectroscopy and Radiative Transfer*, 108(3):363-370, dec 2007.

[19] I.E. Gordon, L.S. Rothman, R.J. Hargreaves, R. Hashemi, E.V. Karlovets, F.M. Skinner, E.K. Conway, C. Hill, R.V. Kochanov, Y. Tan, P. Wcisło, A.A. Finenko, K. Nelson, P.F. Bernath, M. Birk, V. Boudon, A. Campargue, K.V. Chance, A. Coustenis, B.J. Drouin, J.-M. Flaud, R.R. Gamache, J.T. Hodges, D. Jacquemart, E.J. Mlawer, A.V. Nikitin, V.I. Perevalov, M. Rotger, J. Tennyson, G.C. Toon, H. Tran, V.G. Tyuterev, E.M. Adkins, A. Baker, A. Barbe, E. Canè, A.G. Császár, A. Dudaryonok, O. Egorov, A.J. Fleisher, H. Fleurbaey, A. Foltynowicz, T. Furtenbacher, J.J. Harrison, J.-M. Hartmann, V.-M. Horneman, X. Huang, T. Karman, J. Karns, S. Kassi, I. Kleiner, V. Kofman, F. Kwabia-Tchana, N.N. Lavrentieva, T.J. Lee, D.A. Long, A.A. Lukashevskaya, O.M. Lyulin, V.Yu. Makhnev, W. Matt, S.T. Massie, M. Melosso, S.N. Mikhailenko, D. Monde lain, H.S.P. MĀ₁₁ller, O.V. Naumenko, A. Perrin, O.L. Polyansky, E. Raddaoui, P.L. Raston, Z.D. Reed, M. Rey, C. Richard, R. Tóbiás, I. Sadiek, D.W. Schwenke, E. Starikova, K. Sung, F. Tamassia, S.A. Tashkun, J. Vander Auwera, I.A. Vasilenko, A.A. Vigasin, G.L. Villanueva, B. Vispoel, G. Wagner, A. Yachmenev, and S.N. Yurchenko. The HI-TRAN2020 molecular spectroscopic database. *Journal of Quantitative Spectroscopy and Radiative Transfer*, 277:107949, jan 2022.

[20] J. Notholt, F. Cappellani, H. Roesdahl, and G. Restelli. Absolute infrared band intensities and air broadening coefficient for spectroscopic measurements of formic acid in air. *Spectrochimica Acta Part A: Molecular Spectroscopy*, 47(3-4):477-483, January 1991.
# Appendix A. Perturbed Lorentzian profile

In this appendix, we derive the expression of the perturbed lorentzian profile $S_L$. We first summarise Arndt derivation of modulation distorted phase sensitive detection signals and extend Schilt derivation of combined intensity modulation (IM) and frequency modulation (FM) to the case of $2f_m$ IM where $f_m$ is the frequency of the modulation frequency.

## A.1. Arndt calculation

A Lorentzian line profile centered at $\nu_c$ is given by

$$t_0(x) = \frac{1}{1 + x^2} \qquad (A1)$$

with $x = (\nu - \nu_c)/\Delta\nu$ the reduced laser frequency. When a FM is applied, the instantaneous reduced frequency is

$$x(t) = x_0 + m \cos(2\pi f_m t) \qquad (A2)$$

and the time dependant spectroscopic signal is the periodic function of time

$$s(t) = t_0(x(t)) = t_0(x_0 + m \cos(2\pi f_m t)), \qquad (A3)$$

that can be expanded in Fourier series as

$$s(t) = \sum_{n=0}^{\infty} S_n(x_0, m) \cos(2\pi n f_m t). \qquad (A4)$$

Phase sensitive detection signals are given by the Fourier coefficients $S_n(x_0, m)$ of $s(t)$. Arndt [10] has shown that

$$S_n(x_0, m) = \frac{i^n \epsilon_n}{2 m^n} \frac{(\sqrt{(1-ix_0)^2 + m^2} - (1-ix_0))^n}{\sqrt{(1-ix_0)^2 + m^2}} + c.c., \qquad (A5)$$

with $\epsilon_0 = 1$ and $\epsilon_{n>0} = 2$.

## A.2. Fabry-Perot induced IM-FM

For diode laser or quantum cascade laser sources, FM can be obtained modulating the injection current. This method results in combined and phase shifted IM-FM at the same frequency. The effect of IM-FM on line shape has been studied by Schilt in [11].

In this work, we use a CO₂ laser operated very close to the maximum of the emission line. The FM at $f_m$ is applied through the piezo-driven laser cavity length. The spectroscopy signal comes from the transmission of the Fabry Perot cavity, which is locked on resonance with the laser frequency. We have checked that this scheme results in a transmitted (and hence intracavity) IM at frequency $2f_m$ due to the quadratic behaviour of the Fabry-Perot transmission around resonance. No IM at $f_m$ was found. We extend Schilt calculation to this case.
The instantaneous frequency and laser intensity are written

$$ \nu(t) = \nu_0 + \delta\nu_m \cos(2\pi f_m t) \qquad (A6) $$

$$ I(t) = I_0 (1 + qm^2 \cos^2(2\pi f_m t - \Psi)), \qquad (A7) $$

where q is the IM parameter

$$ q = -(\Delta\nu/\Delta\nu_{FP})^2 \qquad (A8) $$

calculated in section A.3. It depends on the Fabry-Perot HWHM $\Delta\nu_{FP}$. $\Psi$ represents a possible phase shift between IM and FM. Note that $\Psi$ has been introduced in the intensity expression leading to much simpler calculation than in [11].

The spectroscopic signal transmitted by the cavity now reads

$$ s(t) = I_0 (1 + qm^2 \cos^2(2\pi f_m t - \Psi)) t_0(x_0 + m \cos(2\pi f_m t)). \qquad (A9) $$

The Fourier components of $s(t)$ can be expressed with those of $t_0(x_0 + m \cos(2\pi f_m t))$. Writing

$$ s(t) = \sum_{n=0}^{\infty} S_{np} \cos(2\pi n f_m t) + S_{nq} \sin(2\pi n f_m t), \qquad (A10) $$

we obtain, for $n \ge 3$

$$ S_{np} = \left(1 + \frac{qm^2}{2}\right) S_n(x, m) \qquad (A11) \\ + \frac{qm^2}{4} \cos(2\Psi)(S_{n-2}(x_0, m) + S_{n+2}(x_0, m)), $$

$$ S_{nq} = \frac{qm^2}{4} \sin(2\Psi)(S_{n-2}(x_0, m) - S_{n+2}(x_0, m)). \qquad (A12) $$
Due to side effects, the formulas for $n = 0, n = 1$ and $n = 2$ are different and given by

$$S_{0p} = \left(1 + \frac{qm^2}{2}\right) S_0(x_0, m) + \frac{qm^2}{4} S_2(x_0, m) \cos(2\Psi), \quad (A13)$$

$$S_{0q} = 0, \quad (A14)$$

$$S_{1p} = \left(1 + \frac{qm^2}{2}\right) S_1(x_0, m) + \frac{qm^2}{4} \cos(2\Psi)(S_1(x_0, m) + S_3(x_0, m)), \quad (A15)$$

$$S_{1q} = \frac{qm^2}{4} \sin(2\Psi)(S_1(x_0, m) - S_3(x_0, m)), \quad (A16)$$

$$S_{2p} = \left(1 + \frac{qm^2}{2}\right) S_2(x_0, m) + \frac{qm^2}{4} \cos(2\Psi)(2S_0(x_0, m) + S_4(x_0, m)), \quad (A17)$$

$$S_{2q} = \frac{qm^2}{4} \sin(2\Psi)(2S_0(x_0, m) - S_4(x_0, m)). \quad (A18)$$

Phase sensitive detection with a phase $\phi$ leads to the theoretical profile

$$S_{n,th}(x, m) = S_{np}(x, m) \cos(\phi) + S_{nq}(x, m) \sin(\phi). \quad (A20)$$

In the case of third harmonic detection, the theoretical line shape is

$$S_{3,th}(x, m) = A_1 S_1(x, m) + A_3 S_3(x, m) + A_5 S_5(x, m), \quad (A21)$$

where

$$\left\{ \begin{aligned} A_1 &= \frac{qm^2}{4} \cos(\phi - 2\Psi) \\ A_3 &= \left(1 + \frac{qm^2}{2}\right) \cos(\phi) \\ A_5 &= \frac{qm^2}{4} \cos(\phi + 2\Psi). \end{aligned} \right. \quad (A22)$$

The coefficients $A_1, A_3$ and $A_5$ depend on $q, \phi$ and $\Psi$, which are known experimentally. In this work, the highest modulation frequency $f_m=310$ Hz is much smaller than the cavity HWHM (600 kHz) so the IM is in phase with the FM such that $\Psi = 0$. $\phi$ is chosen in order to maximise the detected signal amplitude, and corresponds to $\phi = \pi$ in our case. This leads to the lineshape fit function

$$S_L(x, m) = A_0 - A \left( \frac{qm^2}{4} S_1(x, m) + \left(1 + \frac{qm^2}{2}\right) S_3(x, m) + \frac{qm^2}{4} S_5(x, m) \right) \quad (A23)$$

where $A_0$ is introduced to account for a signal offset and where the amplitude $A$ is positive.
### A.3. Cavity width and intensity modulation

#### A.3.1. Pressure dependent cavity width

We consider a Fabry Perot cavity made of two identical ideal mirrors of reflectivity $R$ and transmission $T = 1 - R$. Absorption by HCOOH in the cavity is described by the single pass intensity transmission factor $\tau$. The cavity finesse and maximum transmission are given by

$$F = \pi \sqrt{R\tau}/(1 - R\tau) \qquad (A24)$$

and

$$T_{max} = (1 - R)^2 \tau / (1 - R\tau)^2. \qquad (A25)$$

They are linked by $F^2 = F_0^2 T_{max}$. We have measured an empty cavity finesse $F_0 = 140$, corresponding to $R = 0.9778$.

The single pass intensity transmission factor is given by

$$\tau = e^{-\kappa L} \qquad (A26)$$

where $\kappa$ is the gas absorption coefficient and $L$ the cavity length. Since $\kappa$ is proportional to the pressure $P$, we write $\kappa L = \xi P$ hence $\tau = e^{-\xi P}$. We have measured the maximum transmission for $P = 0.5$ to $4 \mu\text{bar}$ by steps of $0.5 \mu\text{bar}$ at low intensity and fitted the results with Eq. (A25) and (A26), leading to $\xi = 6.676/2 \times 10^{-3} \mu\text{bar}^{-1} = 3.338 \times 10^{-3} \mu\text{bar}^{-1}$. For pressures between $0.5$ and $2.0 \mu\text{bar}$, $\tau$ varies from $99.8$ to $98.7\%$, the maximum transmission from $86.6$ to $59.4\%$ and the finesse from $130$ to $108$. The cavity HWHM is given by

$$\Delta\nu_{FP} = \frac{c}{4FL}. \qquad (A27)$$

It varies approximately linearly from $501$ to $603$ kHz and can be approximated by

$$\Delta\nu_{FP} = \Delta\nu_{FP}^0 + \eta P \qquad (A28)$$

with A REVERIFIER $\Delta\nu_{FP}^0 = 477$ kHz and $\eta = 73.0$ kHz/$\mu\text{bar}$. Eq. (A28) is used in the fitting process by $S_L$ to take into account the pressure dependent cavity width for each scan.

#### A.3.2. Cavity induced intensity modulation

When the cavity is locked on resonance with the laser frequency, the frequency modulation of the laser is turned into intracavity intensity modulation. Both the FM amplitude (at most $60$ kHz) and the FM frequency ($f_m = 310$ Hz) are much smaller than the cavity width $\Delta\nu_{FP}$. The IM is thus determined by the quadratic behaviour of the intensity build-up close to resonance. Writing the FM $\nu(t) = \nu_0 + \delta\nu_m \cos(2\pi f_m t)$, the intracavity intensity is given by

$$I(t) = I_{max} \left( 1 - \frac{\delta\nu_m^2}{\Delta\nu_{FP}^2} \cos^2(2\pi f_m t - \Psi) \right) \qquad (A29)$$
where $I_{max}$ is the maximum intensity. Using the frequency modulation parameter $m = \delta\nu_m/\Delta\nu$, we write $I(t) = I_{max}(1 + qm^2\cos^2(2\pi f_m t))$ where the IM parameter $q$ is given by

$$q = - \left( \frac{\Delta\nu}{\Delta\nu_{FP}} \right)^2 . \qquad (A30)$$

## Appendix B. HCOOH Hyperfine Structure

To the best of our knowledge, HCOOH hyperfine structure has only be investigated for a limited number of rotational levels in the vibrational ground state of the molecule, first by Chardon [16] and reconsidered by Cazzoli [17]. The spin of the proton attached to the oxygen (resp. carbon) atom are denoted $I_1$ (resp. $I_2$). The coupling scheme is $\mathbf{J}+\mathbf{I}_1 = \mathbf{F}_1$ and $\mathbf{F}_1+\mathbf{I}_2 = \mathbf{F}$. Hyperfine structure splits each HCOOH rotational level into four sublevels given by

$$E_1 = \frac{M_1 + M_2}{2(J + 1)} + \frac{\delta}{2(J + 1)(2J + 3)} \qquad (B1)$$

$$E_{2,3} = -\frac{M_1 + M_2 + \delta}{4J(J+1)} \pm \frac{1}{4J(J+1)} [(M_1 + M_2 + \delta)^2 + 4J(J+1)(M_1 - M_2)^2]^{1/2} \quad (B2)$$

$$E_4 = -\frac{M_1 + M_2}{2J} + \frac{\delta}{2J(2J-1)}. \qquad (B3)$$

In these expressions,

* $E_1$ corresponds to $F = J + 1$, $E_2$ and $E_3$ to $F = J$ and $E_4$ to $F = J - 1$.

* $M_{1,2} = \sum_{g=a,b,c} M_{gg1,2} \langle J_g^2 \rangle$ where $M_{gg1,2}$ are the spin rotation constants, the label 1 (resp. 2) corresponding to the hydrogen bounded to the oxygen (resp. carbon) atom. We take the average between the experimental and theoretical values given in [17] (see Table B1). $\langle J_g^2 \rangle$ are the averaged values of $J_g^2$ along the principal inertia axis. They are given in [16] for $J=2$ to 14 and small values of $K_a$ and $K_c$. One can remark that $\langle J_a^2 \rangle \approx K_a^2$ and that $\langle J_b^2 \rangle$ is quadratic in $K_c$ following the law $\langle J_b^2 \rangle = 0.78(5)K_c^2 + 3.55(43)K_c + 1.12(75)$. Finally, $\langle J_c^2 \rangle$ is given by $J(J+1) - \langle J_a^2 \rangle - \langle J_b^2 \rangle$. Extrapolated values of $\langle J_g^2 \rangle$ and $M_{1,2}$ to the quantum numbers relevant to this work are given in Table B2.

* $\delta$ is the spin-spin interaction matrix element. It depends on J. $\delta$ values given in [16] for J from 2 to 14 can be extrapolated with a quadratic law of the form $1.84(7)J^2 - 43(6)$. The extrapolation gives $\delta = -769(27)$ kHz for $J=21$.

The resulting hyperfine splittings are given in Table B3. Figure B1 shows the stick spectrum with weights given by the square of the relative dipole matrix elements obtained using standard angular algebra formula (last equation in [16]):

$$W_{\text{hfs}} = (2F'_1 + 1)(2F_1 + 1)(2F' + 1)(2F + 1) \begin{Bmatrix} J & F'_1 & 1/2 \\ F_1 & J & 1 \end{Bmatrix}^2 \begin{Bmatrix} F'_1 & F' & 1/2 \\ F & F_1 & 1 \end{Bmatrix}^2 \qquad (B4)$$

It shows that the main hyperfine components obtained for $\Delta F = 0$ and $\Delta F_1 = 0$ form a narrow quadruplet with peaks located at $\Delta\nu_{hfs}=-4.4, -0.0, 0.2$, and $+4.6$ kHz from the hyperfineless transition frequency, with respective weights given in Tab. B4.
<table><caption>**Table B1.** *M*<sub>ggi</sub> coefficients taken from [17]. Hydrogen 1 is bounded to the oxygen hydrogen 2 is bounded to the carbon atom.</caption><thead><tr><th>H</th><th>M<sub>aai</sub></th><th>M<sub>bbi</sub></th><th>M<sub>cci</sub></th></tr></thead><tbody><tr><td>1</td><td>-6.904</td><td>0.7755</td><td>-1.305</td></tr><tr><td>2</td><td>-6.9275</td><td>1.0385</td><td>-0.8105</td></tr></tbody></table>

<table><caption>**Table B2.** Extrapolated averaged values of the square of angular momentum components and extrapolated *M*<sub>1</sub> and *M*<sub>2</sub> coefficients.</caption><thead><tr><th>J</th><th>K<sub>a</sub></th><th>K<sub>c</sub></th><th>⟨J<sub>a</sub><sup>2</sup>⟩</th><th>⟨J<sub>b</sub><sup>2</sup>⟩</th><th>⟨J<sub>c</sub><sup>2</sup>⟩</th><th>M<sub>1</sub></th><th>M<sub>2</sub></th></tr></thead><tbody><tr><td>21</td><td>2</td><td>20</td><td>4</td><td>387(11)</td><td>71(11)</td><td>180(9)</td><td>317(11)</td></tr><tr><td>21</td><td>3</td><td>19</td><td>9</td><td>353(10)</td><td>100(10)</td><td>81(9)</td><td>223(11)</td></tr></tbody></table>

The position of the peaks mainly depends on *M*<sub>1</sub> and *M*<sub>2</sub> values with 0.2 to 0.3 kHz contributions to the uncertainties but only hardly depends on *δ* with less than 0.02 kHz contributions. The quadratic sum of all contributions gives an inaccuracy of about 0.5 kHz on the line positions.

## Appendix C. HCOOH dimerisation

HCOOH dimerisation equilibrium is characterised by the equilibrium constant *K*<sub>p</sub> defined

$$K_p = \frac{P_{\text{HCOOH}}^2}{P_{(\text{HCOOH})_2}} \qquad (\text{C1})$$

*K*<sub>p</sub> has been measured by different authors and techniques and the values range between 226 and 361 Pa at room temperature [7] and references therein. Introducing the monomer and dimer molar fractions *x* and 1 − *x* and the total pressure *P*<sub>tot</sub>, Eq. C1 leads to

$$x = \frac{-K_p + \sqrt{K_p^2 + 4K_p P_{\text{tot}}}}{2P_{\text{tot}}} \approx 1 - \frac{P_{\text{tot}}}{K_p} \qquad (\text{C2})$$

for small pressures with respect to *K*<sub>p</sub>. With *K*<sub>p</sub> between 226 and 360 Pa and total pressures less than 2 µbar = 0.2 Pa, the dimer fraction at room temperature is less than 0.1%.

<table><caption>**Table B3.** Extrapolated hyperfine structure splittings in kHz.</caption><thead><tr><th>J</th><th>K<sub>a</sub></th><th>K<sub>c</sub></th><th>E<sub>1</sub></th><th>E<sub>2</sub></th><th>E<sub>3</sub></th><th>E<sub>4</sub></th></tr></thead><tbody><tr><td>21</td><td>2</td><td>20</td><td>10.9</td><td>3.3</td><td>-3.0</td><td>-12.3</td></tr><tr><td>21</td><td>3</td><td>19</td><td>6.5</td><td>3.6</td><td>-3.1</td><td>-7.7</td></tr></tbody></table>
<table><caption>**Table B4.** *F*,*F'*, *F*₁ and *F*₁' quantum numbers, HFS shifts Δν<sub>hfs</sub> and relatives intensities *W*<sub>hfs</sub> of the four main hyperfine lines used in Eq. (11).</caption><thead><tr><th></th><th>F = F'</th><th>20</th><th>21</th><th>21</th><th>22</th></tr></thead><tbody><tr><th>F<sub>1</sub> = F'<sub>1</sub></th><td>20.5</td><td>20.5</td><td>21.5</td><td>21.5</td></tr><tr><th>Δν<sub>hfs</sub></th><td>4.6</td><td>-0.0</td><td>0.2</td><td>-4.4</td></tr><tr><th>W<sub>hfs</sub></th><td>0.91</td><td>0.96</td><td>0.96</td><td>1</td></tr></tbody></table>

<figure><img src="image_13.png" /><figcaption>**Figure B1.** Stick spectrum of the fourteen hyperfine transitions in units of the reduced transition matrix element.</figcaption></figure>