Optical Frequency Comb Fourier Transform Spectroscopy of $^{14}$N$_2$$^{16}$O at 7.8 {\mu}m

We use a Fourier transform spectrometer based on a compact mid-infrared difference frequency generation comb source to perform broadband high-resolution measurements of nitrous oxide, $^{14}$N$_2$$^{16}$O, and retrieve line center frequencies of the $\nu$$_1$ fundamental band and the $\nu$$_1$ + $\nu$$_2$ - $\nu$$_2$ hot band. The spectrum spans 90 cm$^{-1}$ around 1285 cm$^{-1}$ with a sample point spacing of 3 ${\times}$ 10$^{-4}$ cm$^{-1}$ (9 MHz). We report line positions of 72 lines in the $\nu$$_1$ fundamental band between P(37) and R(38), and 112 lines in the $\nu$$_1$ + $\nu$$_2$ - $\nu$$_2$ hot band (split into two components with e/f rotationless parity) between P(34) and R(33), with uncertainties in the range of 90-600 kHz. We derive upper state constants of both bands from a fit of the effective ro-vibrational Hamiltonian to the line center positions. For the fundamental band, we observe excellent agreement in the retrieved line positions and upper state constants with those reported in a recent study by AlSaif et al. using a comb-referenced quantum cascade laser [J Quant Spectrosc Radiat Transf, 2018;211:172-178]. We determine the origin of the hot band with precision one order of magnitude better than previous work based on FTIR measurements by Toth [http://mark4sun.jpl.nasa.gov/n2o.html], which is the source of the HITRAN2016 data for these bands.


Introduction
Nitrous oxide (N 2 O) is of significant importance for atmospheric physics and chemistry. Despite being only a minor constituent of the Earth's atmosphere, it is a potent greenhouse gas [1], and it contributes strongly to stratospheric ozone depletion [2]. Moreover, global N 2 O emissions are increasing due to human activity [3] and will likely be enhanced by global warming [4]. This calls for the ability to monitor atmospheric N 2 O by spectroscopic means, which requires laboratory studies to provide precise and accurate line parameters. One wavelength range of interest for environmental monitoring is the atmospheric window around 8 µm, where N 2 O absorbs due to its strong ν 1 fundamental band as well as several weaker hot and overtone bands. The 8 µm spectral range has been extensively studied using conventional Fourier transform infrared spectroscopy (FTIR). Guelachvili [5] determined line positions of the ν 1 band, the 2ν 2 band, and several hot bands with uncertainties of one to a few MHz, as well as relative line intensities and linewidths. Levy et al. [6] reported absolute line intensities of the ν 1 and the 2ν 2 bands, while Lacome et al. [7] studied self-and N 2 -broadening of lines belonging to these two bands. Toth [8][9][10][11] measured various N 2 O absorption bands over a wide range of the infrared spectrum and compiled an extensive line list [12], including line positions, intensities, and broadening parameters, which to this day forms the basis for a large part of the entries on N 2 O in the HITRAN database [13]. More recently, FTIR was employed by Wang et al. [14] to measure and assign the spectra of the less abundant N 2 O isotopologues around 8 μm. In an early laser-based spectroscopic study of N 2 O, Varanasi and Chudamani [15] used a tunable diode laser to measure the line intensities of the ν 1 band. A few years later, Maki and Wells [16] determined frequencies of the N 2 O absorption lines in several bands spread over almost the entire infrared spectrum (among them the ν 1 band, 2ν 2 band and hot bands at ~8 μm) with uncertainties in the single MHz range using heterodyne frequency measurements, for use in infrared frequency calibration tables. As quantum cascade lasers (QCLs) became available, Grossel et al. [17] used a continuous-wave (CW) QCL to measure line intensities and air broadening parameters of the ν 1 band, while Tonokura et al. [18] measured CO 2 broadening parameters of the ν 1 band. Moreover, Tasinato et al. [19] employed a pulsed QCL to study collisional processes, and hence pressure broadening, of the ν 1 band in He, Ar, N 2, and CO 2 .
The precision and accuracy of line position determination can be significantly improved by using optical frequency combs that directly link the optical frequencies to radio frequency standards. To this date, there exists only one set of studies of line positions of N 2 O in the 8 µm range involving a frequency comb. Lamperti et al. [20] referenced a CW QCL tunable around ~8 µm to a Tm:fiber comb at 1.9 µm through a sum frequency generation scheme, and reported the positions of three lines in the ν 1 band with ~63 kHz uncertainty. AlSaif et al. [21] used this comb-referenced QCL to retrieve positions of the P(40) to R(31) lines of the ν 1 band in the Doppler limit with uncertainties below 200 kHz. More recently, they measured the center frequency of the R(16) line with precision below 50 kHz using sub-Doppler spectroscopy [22]. We show that similar performance (in the Doppler limit) can be obtained by employing an 8 µm comb to directly measure the spectrum, removing the need for complex frequency referencing. Moreover, direct frequency comb spectroscopy allows measuring the entire vibrational bands with tens of thousands of comb lines simultaneously (rather than sequentially line by line, as is done with CW lasers), which reduces the influence of drifts. The most promising comb sources for precision spectroscopy in the 8 µm range are based on difference frequency generation (DFG) from a single femtosecond pump source, deriving the signal comb from the pump laser using a nonlinear fiber [23][24][25], or through intrapulse DFG (IDFG) [26,27]. These DFG combs cover wide bandwidths (up to superoctaves for IDFG sources), and are inherently free from carrier-envelope-offset frequency, f ceo , since f ceo is identical for the pump and signal combs and cancels in the DFG process. The lack of f ceo significantly simplifies the absolute stabilization of the DFG-based comb sources. Timmers et al. [26] employed IDFG sources for dual-comb spectroscopy (DCS) with sample point spacing equal to the comb repetition rate, f rep , of 100 MHz. Much denser sampling point spacing was demonstrated by Changala et al. [28], who used a DFG source [24] and a Fourier transform spectrometer (FTS) with comb-mode-width limited resolution to measure the ro-vibrational spectrum of C 60 . The 8 µm range can also be reached through nonlinear conversion in optical parametric oscillators (OPOs), which, however, are not f ceo -free. 8 µm OPOs have been combined with an FTS [29], the DCS approach [30], and an immersion grating spectrometer [31], all with resolution limited by the instrumental broadening rather than the comb mode linewidth. The latter work investigated the pressure broadening of one hot band N 2 O line, but does not report the line positions. It should be noted that frequency combs based on QCLs [32] operate in the 8 µm range. However, their inherently large comb mode spacing is not ideal for precision spectroscopy of molecule problem; the form We u comb em the ν 1 + ν and 1315 we obtain centrifuga 112 absor band orig N 2 O.

Experi
The main a multi-p Behind the OP-GaP crystal, an optical long-pass filter blocks the near-IR radiation emitted by the dualwavelength source while transmitting the MIR idler produced in the DFG process. The latter is collimated and coupled into a multi-pass absorption cell with an absorption path length of 10 m (Thorlabs, HC10L/M-M02). The cell is connected to a vacuum pump, a pressure transducer (Leybold, CERAVAC CTR 101 N), and the sample gas supply (2.98 % N 2 O diluted in N 2 , supplier: Air Liquide AB), as shown in the right corner of Fig. 1.
After the multi-pass cell, the MIR beam is re-collimated using two lenses and guided to a fast-scanning FTS. The FTS is the same as in Ref. [35], except that the beamsplitter and the detectors have been replaced to extend the operating range to 8 µm. The two out-of-phase interferometer outputs are detected by a pair of thermoelectrically-cooled HgCdTe detectors (VIGO Systems, PVI-4TE-10.6-1x1) in a balanced configuration. The comb beam radius inside the FTS varies between 2 mm at the waist (positioned ~0.5 m after the beamsplitter) and 4 mm at the detectors. Due to the relatively large divergence of the 8 µm beam, the off-axis components give rise to interference fringes that are phase shifted with respect to the interference fringes at the beam center, thus reducing the interferogram contrast. To mitigate this effect, we placed an optical aperture in front of each detector, which yielded an interferogram contrast of ~50% (compared to ~30% without the aperture). The aperture radius (~2 mm) was found empirically as a trade-off between the interferogram contrast and the power level on the detectors (~14 µW). The optical path difference in the FTS is calibrated using a 1556 nm narrow-linewidth CW laser (RIO, PLANEX), whose beam propagates on a path parallel to the MIR comb beam. The CW laser interferogram is registered using an InGaAs detector. The comb and CW laser interferograms are recorded synchronously with an analog-digital converter (National Instruments, PCI-5922) and a LabVIEW program running on a personal computer.
To acquire a spectrum, we first evacuated the cell, purged it with the sample gas, filled it to the desired total pressure, and closed the valve at the input to the cell to isolate it from the rest of the gas system. We stabilized the comb f rep and acquired a set of 50 interferograms. To reduce the sampling point spacing in the spectrum, we stepped f rep in increments of 29 Hz via tuning of the RF generator, and acquired 50 interferograms at each step. In total, we made 14 steps of f rep to scan each comb mode over one mode spacing (f rep ) in the optical domain. We made two measurements, at total pressures of 0.02 mbar and 0.32 mbar, with signal to noise ratio optimized for the ν 1 fundamental band and the ν 1 + ν 2 -ν 2 hot band, respectively. At 0.02 mbar, we made six f rep scans in alternating directions, resulting in a total of 300 interferograms for each f rep setting. One interferogram was acquired in 3.6 s, which yielded a total acquisition time of 4.2 h. At 0.32 mbar, we made two f rep scans with 50 interferograms per f rep step, as well as a final scan with 25 interferograms per step, yielding 125 interferograms per f rep setting and an acquisition time of 1.8 h. To obtain a reference spectrum for normalization, we recorded interferograms with the absorption cell evacuated and the comb locked to the first f rep step. For the measurement at 0.02 mbar, we recorded 150 reference interferograms before and after measuring the N 2 O absorption spectrum, while for the measurement at 0.32 mbar, we recorded 125 reference interferograms before the N 2 O measurement.
We processed the acquired data using a MATLAB code. We first resampled the comb interferograms at the zero-crossings and extrema of the CW laser interferogram and averaged the absolute values of the Fourier transforms of each set of interferograms belonging to one f rep step. We used the method described in Refs [36,37] in order to match the frequency domain sampling points to the comb mode positions at each f rep step and thus minimize the influence of the instrumental line shape. We normalized the averaged spectrum at each f rep step to the averaged reference spectrum, which had been smoothened and interpolated linearly to the sampling points of the pertinent f rep step. Smoothening and linear interpolation was possible since the background features were broad enough to be fully resolved at the comb mode spacing. Using the Lambert-Beer law, we converted each transmission spectrum to an absorption spectrum. To remove the remaining baseline, we fit a model consisting of a sum of a simulated absorption spectrum based on the parameters from the HITRAN 2016 database [13] and a baseline, represented by the sum of a 5 th order polynomial and slowly varying sine terms (periods >6 GHz) to the absorption spectra, and then subtracted the baseline. Finally, we interleaved the averaged and baselinecorrected absorption spectra recorded at different f rep steps to obtain the final spectrum with a sample point spacing of 9 MHz in the optical domain.

High-resolution spectra
Figure 2(a) shows the absorption spectrum of N 2 O measured at 0.02 mbar, dominated by the P and R branches of the ν 1 fundamental band centered at 1285 cm -1 . A second progression of lines, one order of magnitude weaker, belongs to the ν 1 + ν 2 -ν 2 hot band that is split into two components designated by their rotationless parity (e/f). To improve the signal-to-noise ratio (SNR) of these lines, we made a measurement at a higher pressure of 0.32 mbar. Figure 2(b) shows a vertical zoom of the hot band in this second measurement, where the lines belonging to the ν 1 band are plotted in gray for clarity. In addition to the strong P and R branches of the hot band, there is a weaker Q-branch around 1290 cm -1 . Due to minor water impurities in the cell, a few water lines are visible [e.g., around 1260 cm -1 in Fig. 2(a)], some of them accompanied by slight baseline distortions caused by water absorption in the ambient air. The maximum SNR of the lines in the ν 1 band in Fig. 2(a) is ~420, while for the hot band in Fig. 2(b) it is ~200. The SNR is limited by the detector noise. In (b), the spectrum is vertically zoomed in to show the ν 1 + ν 2 -ν 2 hot band, while the lines belonging to the fundamental ν 1 band are plotted in gray.

Line by line fitting
To retrieve the line center positions of the individual absorption lines, we fit Voigt line shapes to them. We applied the fit to lines with line strengths exceeding a threshold of 2 × 10 -20 cm -1 /(molecule/cm 2 ) for the ν 1 band and 2 × 10 -21 cm -1 /(molecule/cm 2 ) for the hot band (based on the values from the HITRAN database [13]). These criteria translate to a minimum SNR of ~30. In the Voigt model, we fit the intensities, the Lorentzian widths, and the center frequencies, together with a linear baseline. We fixed the Doppler widths to the theoretical values calculated at 23 ºC, which are around 71 MHz (FWHM). We used a nonlinear fitting routine based on the Levenberg-Marquardt algorithm, and we chose a fitting range of ±426 MHz for each line, equal to roughly ±6 times the FWHM. Lines separated by less than 9 times the Doppler FWHM were fitted simultaneously in one window. Such cases occurred only for the hot band since the line pairs with different parity overlap in parts of the spectrum. Lines separated by less than twice the Doppler FWHM were excluded from fitting. Weak interfering lines from other N 2 O bands and isotopologues within the fit window were included in the fit with their parameters fixed to the HITRAN values if they exceeded 2% of the line strength threshold. Weaker interfering lines were ignored since they were below the noise floor. We kept the line fits with a quality factor (i.e., the ratio of the peak of the line and the standard deviation of the residuum) larger than 30, thereby eliminating some lines that were affected by baseline distortions, e.g., due to interfering water absorption. Figure   1240  3 shows fits to the R(14) line of the ν 1 band [ Fig. 3(a)], and to a pair of lines belonging to the R(4) transition of the hot band with e-and f-parity [ Fig. 3(b)]. The residuals displayed in the lower panels show no significant structure. The linewidth at the pressures used is dominated by the Doppler broadening, and the pressure broadening is expected to be ~90 kHz for the ν 1 band (at 0.02 mbar) and ~1.5 MHz for the hot band (at 0.32 mbar), as calculated using the pressure broadening parameters from the HITRAN database [13]. The fitted Lorentzian FWHMs are larger by on average 4.2 MHz than the expected values, with no clear J-dependence. We assume that this discrepancy is due to an instrumental broadening, and we made a number of tests to investigate its origin. The characterization of the f rep locking stability described in the experimental section excludes it as a possible cause. To rule out that the broadening stemmed from the drift of the CW reference laser wavelength during the few-hour long measurement, we replaced the RIO diode laser with a CW Er:fiber laser (Koheras Adjustik E15) stabilized to an Er:fiber frequency comb, but this yielded no reduction of the broadening. We also found no dependence of the broadening on the size of the apertures in front of the detectors. We therefore suspect that the broadening is caused by the linewidth of the modes of the 8 µm idler comb, but we have no means of measuring it at this time. Since the line broadening appears symmetric, as indicated by the flat residuals in Fig. 3, we assume it does not skew the retrieved line center positions.
We identify three sources of uncertainty of the fitted line positions. The dominating component is the fit precision, which is in the range of 50-600 kHz. The second contribution stems from the uncertainty of the effective CW reference laser wavelength, which depends on the alignment and divergence of the two beams in the FTS, as well as variations of the refractive index of air that fills the FTS. Using the method described in Refs [35,37] for determination of the reference wavelength and its effect on the retrieved line positions, we estimate its contribution to the uncertainty to be 60 kHz. The third source of uncertainty are the pressure shifts. The pressure shifts of the lines in the hot band at 0.32 mbar are on average 10 kHz (according to HITRAN) and we subtracted them from the fitted line center frequencies. We conservatively assumed the uncertainty of the pressure shift to be as large as the shift itself. The pressure shifts of the lines in the fundamental band at 0.02 mbar are lower than 3 kHz, i.e., more than one order of magnitude below the next largest contribution to the uncertainty budget (originating from the effective reference laser wavelength). We thus neglected pressure shifts in the analysis of the fundamental band.

Line positions of the ν 1 band
The line centers retrieved by fitting to 72 lines in the ν 1 fundamental band are presented in Table 1. The uncertainties (160 kHz on average) are calculated by summing in quadrature the fit precision and the contribution from the reference laser wavelength, described in the previous section. The black dots in Fig. 4(a) show a comparison of the fitted line centers to those from the HITRAN database. A clear systematic discrepancy is visible, though remaining within the HITRAN frequency uncertainty of 3 MHz. Furthermore, this systematic tendency closely follows that reported in a recent work by AlSaif et al. [21], using a comb-referenced externalcavity QCL, shown by the red circles. The root-mean-square discrepancy between the line centers retrieved in that study and in this work is 240 kHz.
Similarly to what was done by AlSaif et al., we fit a model based on the effective ro-vibrational Hamiltonian (see Eq. (1) in [21]) to the retrieved line positions using a Levenberg-Marquardt algorithm implemented in MATLAB. We weighted the lines by the inverse squares of their uncertainties. We fit the vibrational term value G v , the rotational constant B v , and the quartic and sextic centrifugal distortion constants D v , H v of the vibrationally excited state (fixing L v to 0) and fixed the ground-state rotational and centrifugal distortion constants to those determined by Ting et al. [38]. The fitted upper-state constants of the ν 1 band are presented in Table 2, and compared to those obtained by AlSaif et al. [21]. For further comparison, we include the constants reported by Tachikawa et al. [39] from sub-Doppler measurements of the ν 3 -ν 1 and ν 3 -ν 2 bands performed by heterodyning two fluorescence-stabilized lasers. Except for G v , the retrieved upper-state constants from our work and Ref. [21] agree within their combined 1σ uncertainties; the G v constants agree within the combined 3σ uncertainty. The agreement with Tachikawa et al. is noticeably worse for all constants except B v . Note however that their values resulted from fits to two other bands and might be influenced by factors not present in the two other studies, in which the ν 1 band was measured directly. Figure 4(b) shows a comparison of the line centers measured in this work to those calculated from the fitted spectroscopic constants (black dots). The weighted standard deviation (observed -calculated) of the 72 fitted lines is 3.2 × 10 -6 cm -1 (96 kHz). The red circles and blue triangles show a comparison to the line positions calculated using the constants from Refs [21] and [39] respectively. Considerable disagreement with the latter is apparent mainly as a 690 kHz offset. On the other hand, the discrepancy between our model and the one from AlSaif et al. remains within the uncertainties of the measured line positions in the studied wavenumber range. We thus provide an independent support of the accuracy of the line positions and constants reported by AlSaif et al. [21], using a different measurement technique. Fig. 4 (a) The line positions of the ν 1 fundamental band from this work (black dots) and from Ref. [21] (red circles) relative to those from the HITRAN database [13]. (b) The line positions obtained by line-by-line fitting (black dots) relative to the simulation based on the upper state constants from this work. The red circles and blue triangles indicate line positions simulated using constants from Ref. [21] and Ref. [39] relative to the simulation from our work.

Line positions of the ν 1 + ν 2 -ν 2 hot band
The positions of the 112 fitted lines of the P and R branches of the hot band are given in Table 3. The uncertainties include all three contributions described in Section 3.2 summed in quadrature. Their mean is 230 kHz. Figure 5(a) shows the center frequencies compared to those listed in the HITRAN database (black dots) and reported by Maki and Wells [16] (red cicles), where the plotted uncertainty is from our work only. The missing lines between 1277 cm -1 and 1283 cm -1 were excluded from the fitting procedure due to the overlap of the e-and f-parity lines. A second gap between 1288 cm -1 and 1294 cm -1 is due to too low SNR close to the band center.
The discrepancies with respect to HITRAN have a slight J-dependence; the agreement is excellent at the band center, and worse at higher J numbers, but all deviations are within the HITRAN uncertainty of 3 MHz. The discrepancy with respect to Maki and Wells is larger and has a clearer J-dependence, but it is within the 3σ uncertainty (2.3 MHz) reported in their work.
We modeled the hot band with the same effective Hamiltonian as the fundamental band, using a common vibrational term value G v for the e-and f-component, and independent B v , D v and H v constants. We restricted the fit to the G v , B v and D v constants of the upper states, while fixing the upper state H v constants as well as all the lower state constants. (Floating the H v constants resulted in uncertainties comparable to their absolute values.) We took the values for the fixed constants as well as the initial values for the fitted ones from Ref. [12], which is the source of the line positions listed in HITRAN. We applied the fit to all 112 lines weighted by the inverse squares of their uncertainties. Figure 5(b) shows the measured line positions relative to the fit (red solid circles and blue solid squares), where the weighted standard deviation (observed -calculated) is 7.0 × 10 -6 cm -1 (210 kHz). Also indicated are the line positions calculated using the constants from Toth [12] for the e-and f-parity lines (red open circles and blue open squares, respectively). Table 4 shows the band origins, calculated as ν 0 = G v ' -G v ", where G v ' and G v " are the vibrational term values of the upper and lower states, and the B v and D v constants obtained in this work compared to those from Ref. [12]. The uncertainty in the band origin, ν 0 , from Toth was calculated as the in-quadrature sum of the uncertainties of G v ' and G v " reported in Ref. [10] (both equal to 1 × 10 -5 cm -1 ). We note the agreement with Toth in the band origin and the improvement of its precision by one order of magnitude. The significance of the discrepancy in the remaining constants is difficult to evaluate, since no uncertainties are reported on these constants in Ref. [12]. The accuracy of the model also depends on the fixed lower state constants, which are presumably known with lower certainty than their ground state counterparts.

Conclusions
We demonstrated a Fourier transform spectrometer based on a compact fiber-based DFG optical frequency comb source and a multi-pass cell capable of measuring low-pressure spectra with center frequency precision of the order of 100 kHz in the atmospheric window around 8 µm. The DFG comb source is offset-frequency-free, which implies that an RF lock of the repetition rate is sufficient to achieve absolute stability of the comb mode frequencies. This, in combination with the FTS with sub-nominal resolution sampling-interleaving scheme, provides spectra with a calibrated frequency axis. We verified the accuracy of the retrieved line positions of the ν 1 fundamental band of N 2 O by demonstrating excellent agreement with the results of an independent highprecision measurement by AlSaif et al. [21] using an external-cavity QCL referenced to a frequency comb. We reached a comparable precision in the retrieved line positions using a system with reduced experimental complexity. We also retrieved line positions of the ν 1 + ν 2 -ν 2 hot band with up to one order of magnitude improved precision compared to previous studies [5,12,16]. This allowed us to determine the band origin with an uncertainty reduced by one order of magnitude compared to the FTIR-based work of Toth [12]. Longer averaging times will further increase the SNR and hence the precision of the retrieved line positions. Furthermore, the wavelength coverage of the DFG comb can be expanded within the 7-9 µm range by changing the poling period of the crystal and tuning the soliton signal [25]. Our work opens up precision measurements of entire bands of various molecules of interest in atmospheric science and astrophysics, such as methane, ammonia, sulphur dioxide or methanol, in this fingerprint spectral region.