Frequency metrology in the near-infrared spectrum of H217O and H218O molecules: testing a new inversion method for retrieval of energy levels

An extensive Doppler-free spectroscopic investigation of the near-infrared absorption spectrum of the H218O molecule was performed, for the first time, with absolute frequency calibration by using a GPS-disciplined fiber-based optical frequency comb. The investigated line pairs belong to the ν1+ν3 band and have been chosen in the wavelength range from 1.38 to 1.41μm with a lambda scheme, so as to share the excited energy level and allow an accurate determination of the rotational energy separations of the fundamental vibrational state. The measurement of the sub-Doppler line-center frequencies, also extended to the H217O spectrum, has been performed with an overall uncertainty of ∼30 kHz, i.e. about three orders of magnitude lower than the HITRAN data set. The retrieved energy separations agree, by less than 80 kHz, with recent findings provided by the so-called MARVEL procedure for spectral data inversion, thus yielding a very stringent test of its accuracy.


Introduction
Molecular spectroscopy aided by the technology of optical frequency combs is nowadays a research subject that is very much alive for several reasons. Absolute frequency measurements of vibration-rotation transitions provide precise information on the potential energy surface and relevant perturbation effects, as recently demonstrated by an extensive study of molecular iodine [1]. Precision measurements on molecular beams or cold molecular samples allow one to check the stability of fundamental constants, such as the proton-to-electron mass ratio [2] or the fine structure constant [3]. Furthermore, narrow molecular resonances serve as high-quality frequency standards in the infrared (IR) region and knowledge of their absolute frequencies is extremely useful for the application of them in telecommunications [4] as well as for calibrating broadband molecular spectra [5]. In this respect, the spectrum of water vapor can play a very important role, being very rich in relatively strong transitions in the near-and mid-IR regions [6].
The water molecule is an important constituent of the Earth's atmosphere, being responsible for about 70% of the known atmospheric absorption of sunlight and the majority of the greenhouse effect. It has been detected in a variety of astronomical objects from the ground as well as from satellite and airborne platforms. Water vapor is also a major product of most combustion processes. Therefore, it is no surprise that the spectrum of water has been the subject of numerous experimental and theoretical studies for decades, for the purpose of achieving a full comprehension of its spectrum. This is an indispensable prerequisite in many research fields, encompassing atmospheric chemistry and physics, climatology, astrophysics, combustion science and free-space telecommunication. With its three nuclei and ten electrons, water is a relatively simple molecule for which highly accurate first-principles quantum mechanical calculations are possible [7]. In particular, very elaborate calculations based upon variational techniques have been recently reported for nuclear motion and electronic energies, and a variety of subtle physical effects arising from a fully relativistic treatment, the breakdown of the Born-Oppenheimer approximation and a quantum electrodynamics approach have also been considered [8]. From the experimental point of view, most spectroscopic investigations have been performed either by Fourier-transform spectroscopy [9,10] or, more recently, by diodelaser-based cavity ring-down spectroscopy [11,12], the absolute calibration of the spectra being obtained by comparison with a few selected molecular references [9]. The frequency uncertainty of the line positions remains, however, relatively high, namely at the 3-30 MHz level, mainly limited by the width of the lines. In this paper, we report the first frequency-combassisted characterization of the near-IR spectrum of the H 18 2 O and H 17 2 O water isotopologues. Sub-Doppler spectroscopy has been extensively performed in the wavelength window between 1.38 and 1.41 µm, probing a large number of line pairs, each of them sharing the excited level in a lambda scheme. To achieve saturation of the selected lines, an extended-cavity diode laser (ECDL) has been frequency locked to a high-finesse optical resonator providing strong enhancement of the power level. Absolute determination of the line-center frequencies was performed by comparing the ECDL frequency, as stabilized against the preselected sub-Doppler line, to a GPS-disciplined Rb microwave standard through a self-referenced frequency comb. The uncertainty of the measured frequencies is found to be of the order of 30 kHz. Such a high accuracy allows one to realize a very stringent test of the methods for inverting experimental transition data and retrieving energy levels of small molecules. In particular, a comparison is made with the so-called MARVEL procedure, standing for Measured Active Rotational-Vibrational Energy Levels [13], which relies on the combined use of spectroscopic data and first-principles theoretical calculations and has been recently applied for a critical and extensive evaluation of the water IR spectrum by an international task group sponsored by IUPAC [14]. The separation between the investigated energy levels is found to be in excellent agreement with that provided by MARVEL: over a set of nine line pairs, the average deviation is in fact below 80 kHz, very close to our experimental accuracy. Figure 1 shows the experimental setup adopted for the absolute frequency characterization of the near-IR spectrum of H 18 2 O and H 17 2 O. An ECDL, operating in a Littrow configuration with an emission wavelength tunable from 1.38 to 1.42 µm and a maximum output power of 35 mW, is used for Lamb-dip spectroscopy of the investigated lines. A 10% fraction of the laser beam is injected into an 86 mm long build-up resonator with an empty cavity finesse of 4450. Another fraction is combined with the output of an optical frequency-comb synthesizer on a 125 MHz bandwidth photodiode to produce the beating signal used for absolute frequency measurements. The remaining fraction is sent to a 7-digit wavemeter (Burleigh, model WA1500) for coarse frequency calibration. The Pound-Drever-Hall (PDH) technique [15] is used to lock the laser to the high-finesse cavity. For this purpose, two servo loops are used, a slow one (∼1 kHz bandwidth) acting on the extended cavity length and a fast one (∼1 MHz bandwidth) controlling the injection current. With such a setup the laser linewidth is narrowed down to 30 kHz in the short term (10 ms), as demonstrated in a recent work in which further details of this part of the experimental setup are provided [16].

Experimental setup
The optical frequency-comb source is based on an amplified, commercially available, erbium-doped fiber laser (Toptica, model FFS1550) with a pulse repetition rate ( f rep ) of 100 MHz. Its output is coupled to a highly nonlinear fiber that generates a supercontinuum (SC) spanning more than one octave at wavelenghts from 1 to 2.2 µm. Two long-wavelength pass filters (LWP) are used to separate the SC into three spectral intervals. The central part (wavelengths between 1.2 and 1.7 µm) is reflected from LWP2, recombined with the ECDL output, spectrally narrowed by a grating-lens-slit assembly and then sent to the photodiode for frequency counting. The long-wavelength (above 1.7 µm) and the short-wavelength (below 1.2 µm) parts of the SC are separated by LWP1 and exploited for active stabilization of the comb carrier-envelope offset frequency ( f CEO ) in a self-referencing scheme. In particular, the long-wavelength part is frequency doubled in a 1.8 cm long periodically poled lithium niobate crystal and, after being collinearly recombined with the short-wavelength side in a f − 2 f interferometer configuration, it is spectrally filtered and focused on a fast detector to produce a beat note at f CEO . By servo-controlling the pump diode current of the fiber oscillator, the detected beat note is locked to a GPS disciplined Rb oscillator (Symmetricom, model 4410), acting as a master clock for all electrical loops with a relative accuracy better than 10 −12 .
The stabilization of the comb repetition rate ( f rep ) is achieved by servo-controlling the fiber oscillator cavity length through a piezoelectric transducer (fine control) and a step motor (coarse control). The error signal of this servo-loop is obtained by comparing the f rep signal given by the internal detector of the laser with the 100 MHz low-phase-noise signal coming from a quartz oscillator, also referenced to the Cs-clock primary standard through the GPS-disciplined Rb oscillator.

Experimental results
Examples of saturated absorption spectra, for the two isotopologues, are shown in figure 2. They were obtained by scanning the cavity resonance frequency over the absorption line and by recording the cavity transmission with a 1 kHz bandwidth low-noise InGaAs photodiode. An 80% enriched 18 O water vapor sample (with a nominal purity better than 99.9%) was used, the remaining 20% being H 17 2 O. The total gas pressure inside the optical cavity was kept constant at about 10 Pa, as measured by a high-accuracy capacitance pressure gauge. The contrast of the Lamb-dip with respect to the Doppler-broadened envelope varied between 5 and 15%, The latter spectrum shows that the intracavity power was sufficiently high to enable the saturation of the weaker H 17 2 O line (whose intensity is a factor of 8 smaller, as compared to the H 18 2 O line), the frequency separation between the two lines being equal to ∼ 25 MHz. The good signal-to-noise ratio, resulting from dc detection in a 1 kHz bandwidth, demonstrates the excellent performance of the PDH locking loops.
depending on the transition dipole moment of the selected line, while its width (full-width at half maximum, FWHM) was measured to be in the range between 2 and 3 MHz, mostly resulting from the pressure broadening effect. The investigated lines are listed in table 1. All vibration-rotation transitions belong to the ν 1 + ν 3 band and have a central wavelength between 1.38 and 1.41 µm. Their saturation intensity, in our operation conditions, ranges between 2.5 × 10 6 and 2.5 × 10 7 W m −2 . The H 18 2 O lines were carefully selected within the reach of our ECDL so as to form nine different line pairs with a lambda scheme, each of them sharing the excited energy level.
The involved levels are labeled using the standard asymmetric top notation J K a,K c , where J is the familiar quantum number of total angular momentum, K a and K c the projections along two of the three principal axes of inertia, the axes being distinguished on the basis of the corresponding moment of inertia, from the smallest (A-axes) to the largest (C-axes).
For a given absorption line, the ECDL emission frequency is locked to the central frequency using the procedure reported in [16]. Briefly, the cavity mode frequency is locked to the center of the Lamb-dip by using the wavelength modulation spectroscopic method. The cavity resonance is dithered by means of a sinusoidal signal at 2 kHz with a modulation depth of about 0.5 MHz. Being constantly locked to the cavity, the laser frequency follows this modulation. A lock-in amplifier performs a synchronous processing of the cavity transmitted signal, thus allowing one to retrieve an error signal given by the first derivative of the Lamb-dip. Such a signal, once properly integrated, is used to actively control the cavity length through the piezoelectric actuator that is mounted on the cavity input mirror. As a result of an Allan variance analysis of residual laser frequency fluctuations at the lock-in amplifier output, a relative frequency stability of the order of 10 −13 for an integration time of 1 s is typically measured. Absolute frequency determinations are carried out by comparing the water-stabilized laser frequency to the self-referenced optical frequency-comb synthesizer. The beat note between the laser radiation and the closest comb tooth exhibits a signal-to-noise ratio of 40 dB in a 100 kHz resolution bandwidth, which is sufficient for direct counting. Figure 3 shows an example of repeated beat frequency ( f beat ) measurements for the H 18 2 O 2 1,2 → 1 1,1 line, covering a time span of about 1 hour, with a 1-s gate-time. The power spectral density of the beat note is shown in the inset, along with a fit to a Gaussian function that yields a FWHM of ∼ 600 kHz. This relatively large value is consistent with the depth of the sinusoidal modulation that is superimposed on the laser frequency by the cavity length dithering. The standard deviation of the data set of figure 3 is equal to 3.4 kHz. The small drift that is clearly evidenced can be explained in terms of drifts in the pressure and electronics offsets of the locking loops that shift the locking point. By removing such a drift, the standard deviation reduces to only 2.8 kHz. The absolute frequency can be retrieved using the equation ν 0 = n f rep ± f beat ± f CEO . The comb order, n, is easily provided by direct measurement of the laser wavelength by means of the wavemeter, whose accuracy is better than half the comb mode spacing. On the other hand, the correct signs can be determined by slightly varying f rep and f CEO and observing the consequent variation of f beat .
In order to quantify the measurement repeatability in the long term, the absolute frequency of a predefined line, specifically the H 18 2 O 2 2,1 → 1 2,0 line, was determined from several measurement runs performed over a time period of 2 weeks. Each data set encompassed 50 frequency samples of the beat note and, after each run, the chamber containing the optical resonator was evacuated and refilled at the same water pressure. In this dead time, the laser and the cavity were unlocked and the offsets properly adjusted, before locking again. The standard deviation of the mean values was found to be about 15 kHz, which includes the statistical uncertainty in the measurement of the center frequency as well as small systematic deviations (type B errors) arising from sample-handling effects and from lock-to-lock sequences. Another source of type B error is given by the pressure shift, which is expected to be of the order of a few kHz Pa −1 for the investigated lines [17]. Nonetheless, even adopting a very pessimistic approach, it should be sufficient to multiply the 15 kHz uncertainty by a factor of 2, in order to take into account the frequency shift due to the H 18 [18], can be neglected, being estimated to be well within the 30 kHz uncertainty. Concerning the influence of the dither amplitude (namely, the modulation depth) on the linecenter frequency, we estimated it to be at the kilohertz level, on the basis of the high symmetry of the Lamb-dip and of its derivative profile. The measured frequencies are reported in the second column of table 1. A comparison with the HITRAN database [19] is also performed . In this respect, our data represent an improvement, by more than three orders of magnitude, in the accuracy.

Test of the MARVEL energies and line positions
The absolute frequencies of table 1 also provide reliable information on the rotational terms of the (0, 0, 0) vibrational level of the H 18 2 O electronic ground state. In fact, the frequency separation between preselected rotational levels can be retrieved from those line pairs forming a lambda scheme. The results are reported in table 2 and compared with the values taken from [14], coming from the application of the MARVEL procedure. They agree remarkably well with our values. In fact, the absolute deviations that are listed in the fourth column of table 2 exhibit a root-mean-square value of about 76 kHz, which is close to our experimental accuracy. It must be mentioned that the levels involved in this study are particularly well determined by the MARVEL procedure, all of them being classified as A+ in [14] (the grade A+ meaning that the level is involved in more than five lines and that there are at least two independent experiments probing this energy level), with a quoted uncertainty smaller than 10 MHz. However, even though limited to these types of levels, our investigation seems to provide a strong validation of the methodology adopted in the MARVEL procedure for inverting experimental transition data to give energy levels. In fact, if a similar test is carried out for the HITRAN database (see the last two columns of table 2), it turns out that the root-mean-square deviation amounts to about 1237 kHz, which is nearly a factor of 16 larger. Surprisingly, the same level of agreement is not found for the line-center frequencies, as clearly shown in the last column of table 1, in which the absolute deviations of our values with respect to those coming from [14] are listed. This is particularly evident for the two lines involving the 3 2,2 level of the ν 1 + ν 3 vibrational state. In fact, a comparison between our values and those of [14] yields a root-mean-square deviation of about 37 MHz, which reduces to 15 MHz when removing the measurements on the line pair sharing the 3 2,2 level. In any case, better agreement is found with the HITRAN data set, the root-mean-square deviation being equal to only 9 MHz. Therefore, the results allow us to conclude that the MARVEL procedure, while being extremely reliable and accurate for the retrieval of energy terms belonging to the (0, 0, 0) state, does not ensure the same performance for the energies of the (1, 0, 1) vibrational state, probably because of the limited accuracy of the employed spectroscopic data. The extension of our investigation to many other absorption lines, involving different excited vibrational levels, could thus be of utmost importance for further refining the prediction capability of MARVEL and contributing to a more accurate knowledge of the H 17 2 O and H 18 2 O energy diagram. In this respect, it is useful to recall that the knowledge of water energy levels is of primary importance in order to calculate by explicit summation the internal partition function, Q, as well as relevant thermodynamic quantities including the specific heat capacity, the Gibbs enthalpy function, the Helmholtz function and the entropy for such a relevant molecular target [20].

Conclusions
Precision Lamb-dip spectroscopy was performed for H 17 2 O and H 18 2 O molecules using a highfinesse build-up cavity in conjunction with a self-referenced fiber-based optical frequencycomb synthesizer. This relatively complex and advanced apparatus enabled the first extensive and highly accurate investigation of the near-IR spectrum of the water isotopologues. A relative accuracy of about 10 −10 was achieved in the absolute determination of the line-center frequencies, specifically those of 20 vibration-rotation transitions of the ν 1 + ν 3 band. Concerning the energy separations between rotational levels of the H 18 2 O fundamental vibrational state, the retrieved values agree extraordinarily well with data provided by the recently developed MARVEL algorithms for determining energies and line positions of the main water isotopologues, based on the joint use of available experimental data and firstprinciples theoretical calculations. The reported results pave the way to further theoretical and experimental studies on the IR spectrum of water and its isotopologues, such as those aimed at searching for ortho-para switching transitions in the water IR spectrum [21]. A further outcome of our work is the establishment of new optical frequency standards in the spectral window between 1.38 and 1.41 µm that can be useful for calibration purposes in high-resolution molecular spectroscopy.