SI-traceable molecular transition frequency measurements at the 10⁻¹² relative uncertainty level

1. INTRODUCTION

There has been phenomenal progress in the measurements of transition frequencies of extremely cold atoms and trapped ions, where, for example, relative precisions approaching ${{10}^{- 19}}$ have been recently demonstrated in optical lattice clocks [1–4]. Such extraordinarily low uncertainty is being harnessed to transform timekeeping technologies, bound variations in fundamental constants, and to develop advanced methods for control of quantum states. Although the rich internal structure of molecules holds fundamental physical interest, line position measurements in molecular systems are subject to significant technical complications associated with cooling and detection limits, which compromise precision and accuracy. Nevertheless, molecules can provide stringent tests and constraints on fundamental physics questions such as the validity of quantum electrodynamics (QED) [5], determining the proton–electron mass ratio [6,7], and searching for time-reversal violations (the electron’s electric dipole moment) [8–10]. Measured molecular transition frequencies at the kilohertz (kHz) level of uncertainty can underpin secondary frequency standards useful for length metrology and telecommunications wavelength standards [11]. Significant progress has been made recently on molecular cooling [12–14], but high-precision measurements are generally limited to particular molecules or by the limited number density that may readily be cooled with available methods. For classical applications involving particle beams or thermalized gas samples near room temperature, relative precisions at the ${{10}^{- 12}}$ to ${{10}^{- 14}}$ level have been achieved in a number of studies in the mid-infrared (IR) wavelength region from 3–11 µm, via saturated absorption [15–19], two-photon absorption [20], and two-photon Ramsey interferometry [6]. The most accurately determined optical transition frequencies in the near-IR spectral region are typically achieved using saturation laser spectroscopy, with typical uncertainties at the 1 kHz to 100 kHz level [21–29]. Although far less accurate than frequency measurements associated with atomic clock technology, with improved measurements of transition frequencies, energy levels of contributing quantum states can be better determined [30–32], enabling robust tests of ab initio calculations of potential energy surfaces and the calculation of spectroscopic line lists with reduced uncertainty and extended spectral and temperature coverage [33–35].

In the remainder of this article, we begin by surveying previous high-accuracy measurements of molecular line positions that use optical frequency combs (OFCs). We follow by demonstrating OFC-locked (OFCL) cavity ring-down spectroscopy (CRDS) as an appealing alternative to direct absorption and cavity-enhanced saturation spectroscopy techniques for determining line centers. CRDS has been widely adopted for trace gas sensing and molecular line shape studies (see [36] and references therein) given its exceptional sensitivity and immunity to laser intensity fluctuations. Our approach is linear in absorber number density and immune to complications observed in saturation spectra such as uncertainty in the baseline and power-dependent effects. In contrast to the present measurements, previously reported cavity-enhanced approaches, which used OFC-locking [22,23,26], employed nonlinear saturation methods, had substantially larger cavity length modulation amplitudes, did not use tightly phase-locked lasers [37], and could only be tuned sequentially [38].

We also report vacuum line positions for five transitions within the $^{12}{{\rm{C}}^{16}}{{\rm{O}}_2}$ $({{30012}}) \leftarrow ({{00001}})$ combination band which have important applications in remote sensing of greenhouse gases [39] and are in a convenient range for telecommunication standards. We demonstrate a minimum relative combined standard uncertainty in measured transition frequency equal to ${1.1} \times {{10}^{- 12}}$ (nominally 200 Hz) with typical performance at the ${{3}} \times {{10}^{- 12}}$ level. These uncertainties are the lowest reported for near-IR molecular transition frequencies determined by optical absorption [37,38,40–44], including those obtained by Doppler-free saturation methods [22–27,29,45–47]. We note that the uncertainty presented is below those reported for the acetylene ($\lambda = {1.54}\;\unicode{x00B5}{\rm m}$) molecular frequency standard accepted by the Bureau International des Poids et Mesures (BIPM) for practical realization of the meter and secondary representations of the second [11].

2. SI TRACEABILITY OF THE COMB-LINKED FREQUENCY AXIS

In general, OFCs have become a critical tool in the fields of metrology and precision spectroscopy [48], allowing direct traceability of optical frequency standards [49,50] or primary microwave frequency standards [51] to a probe laser. Combs may be employed as a direct probe [52,53], in open-path measurements [54–56], or coupled to a cavity [16,57–60], but are often utilized as a frequency reference for a secondary probe laser. This latter application commonly involves an open-loop configuration, where the heterodyne beat frequency between the probe laser and a given comb tooth is measured [24,40,61–64]. The precision and accuracy of these open-loop techniques may be limited by the linewidth and stability of the probe laser or by the frequency stability of the optical cavity under investigation. Recent work has pursued closed-loop schemes to transfer the frequency accuracy of relatively low-power comb teeth on to single-frequency, higher-power probe lasers.

Various implementations of probe laser and optical cavity locking have been investigated to produce an experimental scheme allowing for synchronous frequency tunability of lasers and cavity resonances. The use of acousto-optic (AO) feed-forward locking has been explored, with laser tuning accomplished by stepping the OFC repetition frequency [38,65]. Electro-optic modulators (EOMs) have been employed in a similar scheme [26,66], with frequency tuning done through the EOM sideband in place of changing the OFC repetition rate. Pound–Drever–Hall (PDH) [67] schemes have been employed to lock an optical cavity to a probe laser, which may be also locked to an OFC [22,23] or referenced to a secondary clock laser [44]. The probe laser may be locked to a secondary stabilized optical cavity and either referenced [47] or phase locked [25] to the OFC.

The present study requires frequency axes for absorption spectra with metrological quality and traceability to accurately determine vacuum transition frequencies of carbon dioxide. As has been previously noted [38], the use of an OFC to provide frequency axes in absorption spectroscopy also enables long-term measurement reproducibility and deep averaging of spectra, leading to increased sensitivity and potential improvements when uncertainties are limited by statistical effects. The chain of our key experimental components, their relationships, which provide traceability of the optical frequency axis to the International System of Units (SI) dimension of time, and relative uncertainties arising from random effects (Type A evaluations) are shown in Fig. 1. For the present purposes, components of uncertainty in the frequency chain arising from systematic effects (Type B evaluations) are negligible, with the dominant systematic uncertainties in the measured transition frequencies being driven by other physical quantities discussed below. The specified uncertainties in the frequency chain are those arising from random effects (Type A evaluations) and correspond to averaging over the 60-s-long time scale of a single spectrum acquisition. Below, we find that the quadrature sum of these uncertainty components is nominally 90 Hz and is dominated by the specified uncertainty in our Cs clock.

 

Fig. 1. Overview of the frequency chain realized in the present experiment. The indicated relative uncertainties are those arising from random effects (Type A) and correspond to averaging over the 60-s-long time scale of a single spectrum acquisition. All reported Allan deviations and averaging times throughout the text are based on this averaging time scale. Here, ${f_{\rm{RF}}}$ is the 10 MHz reference frequency provided by the Cs clock, ${f_{{\rm OFC},k}} = \pm$${f\!}$_ceo $+ k$ ${f_{\rm{rep}}}$, where ${f\!}$_ceo and ${f_{\rm{rep}}}$ are the OFC carrier-envelope offset frequency and repetition rate, ${f_{\rm{ECDL}}}$ is the probe laser, which is frequency-offset locked to OFC comb tooth $k$, and ${f_{{\rm RDC},q}}$ is the frequency of mode-order $q$ of the ring-down cavity, which is locked to the probe laser frequency ${f_{\rm{ECDL}}}$. Adding all terms in the quadrature, the contribution of these effects to the relative combined standard uncertainty in the average cavity mode frequency is ${4.5} \times {{10}^{- 13}}$. For simplicity, we have not shown the contributions of the phase-lock local oscillator frequency ${f_{\rm{LO}}}$ and the shift frequency ${f_{\rm{EO}}}$ of the EOM discussed in the text.

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Fig. 2. Schematic drawing of the apparatus, showing the comb-locking servo, optical cavity length servo, and electro-optical tuning scheme. PSD, phase-sensitive detector; PD, photodiode; LO, local oscillator; FC, fiber combiner; EOM, electro-optic phase modulator; DAQ, data acquisition.

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The principal optical and electronic components of the experiment are shown in Fig. 2. Here, all driving and measured frequencies are referenced to a 10 MHz radio-frequency (RF) signal provided by the commercial Cs clock. This reference frequency is transferred to an OFC via phase locking of the carrier envelope offset frequency, ${f\!}$_ceo, and repetition rate, ${f_{\rm{rep}}}$. Next, a continuous-wave external cavity diode laser (ECDL) with a single-mode output is offset-frequency locked to the OFC and used to probe the sample using the CRDS technique. To enable mode-by-mode tunability, the probe laser frequency is shifted in increments of the cavity’s free spectral range (FSR) using phase modulation. Finally, a closed-loop servo, which displaces one ring-down cavity mirror, stabilizes the average cavity mode frequency to be coincident with that of the probe laser, thus providing a spectrum frequency axis that is traceable to the Cs clock.

A. Cs Clock

The Cs clock is a commercial primary frequency standard with a high-performance beam tube specified to provide a relative frequency stability of ${{10}^{- 14}}$ for averaging times of 5 days. The system includes a 5 MHz phase-locked quartz oscillator to improve the short-term phase noise and Allan deviation (${{4}} \times {{10}^{- 13}}$ at 60 s averaging time) of the signal from the Cs frequency standard. The system provides output signals at 10 MHz, which are synchronous with the Cs transition frequency. As discussed below, these clock signals acted as triggers for additional RF signals involved in setting the frequency of the probe laser as well as a frequency counter for measuring the stability of the phase-locked laser.

B. Optical Frequency Comb

We use a commercial octave-spanning (1–2 µm wavelength), self-referenced OFC comprising a femtosecond-pulse-duration fiber laser, highly nonlinear fiber for spectral broadening, and ${\rm{E}}{{\rm{r}}^ +}$-doped fibers for amplification. The OFC operates at ${f_{\rm{rep}}} = {{250}}\;{\rm{MHz}}$ and with ${f\!}$_ceo $= \pm {{20}}\;{\rm{MHz}}$. To quantify the phase stability of the locked OFC, we measured the Allan deviation of ${f\!}$_ceo and ${f_{\rm{rep}}}$, obtaining 0.5 Hz and ${{5}} \times {{10}^{- 5}}\;{\rm{Hz}}$, respectively. Evaluating at the nominal optical frequency of 190 THz gives a Type A evaluated relative standard uncertainty of ${{2}} \times {{10}^{- 13}}$, which is a factor of two better than the manufacturer’s specification for the stability of the Cs clock. Given that the frequency counters used to measure these Allan deviations are referenced to the Cs clock, these results measure the degree to which the OFC copies the Cs clock and are not an absolute measure of the OFC stability.

C. Phase Lock of External Cavity Diode Laser to Optical Frequency Comb

The purpose of the OFC-ECDL phase lock is to map the center frequency, ${f_{\rm{ECDL}}}$, of the ECDL spectrum to a known frequency of a comb tooth. In order to establish this phase lock, a portion of the ECDL light is combined with the OFC on an optical detector (200 MHz bandwidth) to provide a heterodyne beat signal at frequency, ${f_{{\rm{beat}}}}$, which is mixed with a local oscillator at the desired offset beat frequency and measured with a phase-sensitive detector (phase noise floor of ${-}{{165}}\;{\rm{dBc/Hz}}$ at 10 kHz offset). The local oscillator has a frequency ${f_{\rm{LO}}} = {{60}}\;{\rm{MHz}}$ and phase noise of ${-}{{120}}\;{\rm{dBc/Hz}}$. The output of the phase-sensitive detector provides an error signal for a servo controller loop filter operating at a proportional-integral (PI) corner frequency of 100 kHz. The loop filter output feeds back directly to actuate the laser diode current, resulting in a phase lock of the probe laser to the OFC. Suitable RF filters are employed to ensure that only one beat signal is measured between the 250 MHz separation of the OFC teeth. Another portion of the probe laser is directed to a wavelength meter (10 MHz precision) to determine the OFC mode order required for frequency determinations.

We performed a series of measurements to describe the comb phase lock, the complete details of which may be found in the Supplement 1. Briefly, we measured the optical frequency spectrum of the probe laser as well as the frequency spectrum and frequency noise spectral density (NSD) of the heterodyne beat signal between the probe laser and the OFC. The frequency spectrum of the heterodyne beat signal between the phase-locked ECDL and the OFC was measured with an RF spectrum analyzer. This measurement showed that the bandwidth of the phase-locking servo was nominally 150 kHz. We also characterized the phase-locked linewidth with the delayed self-heterodyne interferometric (DSHI) technique [68], as well as by measuring the Fourier spectrum of the heterodyne beat signal between this laser and both the OFC and a narrow single-frequency fiber laser. Both characterization methods demonstrate appreciable narrowing of the phase-locked probe laser, with half-width at half-maximum (HWHM) of no greater than 17 kHz.

To assess the stability of the phase-locked ECDL carrier frequency, we recorded the ECDL-OFC heterodyne beat frequency with a counter triggered by the Cs clock. We found that the phase lock reduced fluctuations in ${f_{{\rm{beat}}}}$ by five orders of magnitude. We measured an Allan deviation of 60 mHz on the time scale of one spectrum acquisition, which corresponds to a relative standard uncertainty of ${{4}} \times {{10}^{- 16}}$ in the difference between the center frequencies of the ECDL and OFC tooth. This uncertainty component is more than two orders of magnitude smaller than the second smallest contribution listed in Fig. 1, making its impact on the combined frequency axis uncertainty negligible.

To enable rapid and broad spectral coverage, we implemented the frequency-agile rapid scanning (FARS) [43] technique involving an EOM driven at frequency ${f_{\rm{EO}}}$ to generate sidebands about the optical carrier frequency at ${f_{\rm{ECDL}}}$. This approach enabled us to tune either the positive or negative first-order sideband between successive cavity modes, while simultaneously using the high-finesse ring-down cavity to reject the locked-ECDL carrier frequency and all other sidebands produced by the EOM. We note that phase coherence between ${f_{\rm{EO}}}$ and ${f_{\rm{LO}}}$ with ${f_{\rm{rep}}}$ and ${f\!}$_ceo was ensured because all of these signals were triggered by the Cs clock reference signal.

D. Frequency Lock of the Ring-Down Cavity

The ring-down cavity used for the molecular absorption measurements acts as a highly selective frequency discriminator by virtue of its narrow high-order resonances. Because relatively small drifts in the cavity length lead to significant shifts in the resonance frequencies, active length stabilization is required. The cavity is a Fabry–Pérot resonator comprising two high-reflectivity ($R = 0.999 976$) spherical mirrors (1 m radius of curvature) with length $L = 1.4\;{\rm{m}}$. This configuration supports transverse electromagnetic (${{\rm{TEM}}_{00}}$) optical resonances, each with an empty-cavity line width (HWHM) given by ${{{\Gamma}}_{\!q}}/(2\pi) = 0.5\;{\rm{kHz}}$. The resonances are separated by the cavity FSR, which is nominally 107 MHz. The phase-locked probe laser is directed in fiber to a fiber amplifier followed by an EOM and then launched into free space and mode-matched to the optical cavity. A piezoelectric lead-zirconate-titanate transducer (PZT) actuator attached to one of the ring-down cavity mirrors adjusts the length to bring the nearest resonant frequency into coincidence with the selected EOM sideband. Spectra are acquired by stepwise tuning of the probe laser frequency through successive modes: $q$, $q + {{1}} \ldots$. Decay events are initiated with a fast, 80 dB extinction-ratio absorptive microwave switch applied to the EOM driving signal. Laser transmission and ring-down decay events are measured by an InGaAs detector with a bandwidth of 8 MHz.

When the laser center frequency and local cavity resonance are near coincidence, the frequency selectivity of the cavity leads to variable-amplitude bursts of laser power transmitted by the ring-down cavity that occur at random time intervals. These bursts are highly sensitive to the instantaneous frequency detuning between the excitation laser and cavity resonance. They are a consequence of the relatively narrow linewidth of the ring-down cavity: a property which makes the transmission signal sensitive to fluctuations in the resonator path length that occur on time scales greater than or equal to the buildup time [69].

The following model for the dynamic response and frequency selectivity of the ring-down cavity provides useful physical insight into the origin of the transmission bursts. First, we note that an arbitrary resonance specified by longitudinal mode order $q$ has angular frequency ${\omega _q}$ and satisfies the condition ${\omega _q}{t_r} = 2\pi q$. Here, ${t_r}$ is the round-trip time of light propagation in the cavity, which equals the inverse of the cavity’s FSR. With regard to the time response of the ring-down cavity to excitation by a laser, it is critical to note that during each transient cavity buildup event the intracavity electromagnetic field involves the sum of two fields: the incident laser field ${\epsilon _c}{e^{i{\omega _c}t}}$ at angular frequency ${\omega _c} = 2\pi ({f_{\rm{ECDL}}} \pm {f_{\rm{EO}}})$ and the transient response (Green’s function) of the cavity to the incident laser field given by ${-}{\epsilon _c}{e^{- {{{\Gamma}}_q}t}}{e^{i{\omega _q}t}}$ and oscillating at the cavity resonance frequency ${f_q} = {\omega _q}/({2\pi})$. Here, ${\epsilon _c} = \frac{{{\epsilon _i}}}{{{t_r}}}\frac{{(1 - R){e^{- \alpha L/2}}}}{{{{{\Gamma}}_q} + i{{\Delta}}\omega}}$ is the field amplitude coupled into the cavity. This quantity depends on the amplitude of the incident field, ${\epsilon _i}$, the round-trip time, ${t_r}$, the absorption coefficient of the medium, $\alpha$, the angular frequency detuning ${{\Delta}}\omega = {{{\omega}}_c} - {{{\omega}}_q}$, and the cavity linewidth ${{{\Gamma}}_q} = {({2\tau})^{- 1}}$, where $\tau$ is the intensity-based ring-down decay time. Importantly, after complete extinction of the excitation laser, the angular frequency of the passively decaying field equals ${\omega _q}$, thus losing all memory of the incident field at ${{{\omega}}_c}$ [70]. Therefore, although off-resonant light from the excitation laser can couple into the cavity during buildup, the light–matter interaction during the ring-down measurement always occurs at the cavity resonance frequency. Further, during the buildup, oscillations in the transmitted signal caused by heterodyne beating between these two fields occur with a frequency of ${{\Delta}}\omega /({2\pi})$, which contributes to an enhancement of the peak transmitted buildup intensity approaching a factor of four when the normalized detuning $| {{{\Delta}}\omega} |/{{{\Gamma}}_q} \gg 1$. This effect partially counteracts the reduction in throughput caused by roll-off with detuning of the Lorentzian cavity resonance given by ${({{{\Gamma}}_{\!q}} + i{{\Delta}}\omega)^{- 1}}$.

To eliminate drift in ${{\Delta}}\omega$ and to center the mean cavity mode position at the laser frequency such that ${{\Delta}}\bar \omega = 2\pi\! {{\Delta}}\!\bar f = 0$, we use a cavity locking servo similar to that described by [71] and depicted in Fig. 3. To this end, we apply a triangle wave modulation to the PZT mirror assembly with a period of 100 ms and zero-to-peak displacement equal to 0.054 nm. This displacement causes linear modulation in the ring-down cavity mode frequency over the range ${\pm}{7.5}\;{\rm{kHz}}$, at a sweep rate of 300 kHz s${^{- 1}}$. The induced mirror movement sweeps the cavity resonance in the vicinity of the excitation laser frequency and introduces a periodic component to the random transmission bursts. An adjustable threshold is set to convert the strongest bursts to constant-amplitude signals. As illustrated in Fig. 3, the measured centroid of this triggered pulse signal acts as a frequency discriminant to provide an error signal that is proportional to ${{\Delta}}\!f$.

 

Fig. 3. Time dependence of the open-loop ring-down cavity locking error signal (yellow dots) for linear drift (yellow dashed line) of the cavity length. The triangle wave shows the induced modulation (${\pm}{7.5}\;{\rm{kHz}}$) of the cavity mode frequency caused by translation of the ring-down cavity mirror. The green clusters are triggers induced by bursts of transmitted light from the probe laser and produced by a gate and delay generator. Each trigger cluster has a centroid indicated by the blue dashed lines, which gives the detuning of the cavity mode position during each half-cycle of the length modulation. With the loop closed, the servo feedback signal to the PZT actuator is proportional to the indicated error signal.

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Fig. 4. Scheme for locking the mean cavity mode position to the excitation laser frequency. The blue curve represents the efficiency with which the excitation laser couples into the ring-down cavity as a function of the frequency detuning between a single-frequency laser and cavity mode, ${{\Delta}}\!f$, as described in the text. We assumed a linewidth of ${{\Gamma}}\!/\!(2\pi) = 1\;{\rm{kHz}}$, corresponding to when the absorption losses equal the cavity base losses. The calculated coupling efficiency neglects the effect of cavity motion on the dynamics of buildup in the cavity field: an appropriate assumption given that, in the present experiment, the cavity time constant (80 µs) is about 40 times smaller than the time to sweep the cavity mode frequency by one halfwidth. The set of triggered transmission bursts (see Fig. 3) for the unlocked case spans a frequency detuning range that is set by the modulation amplitude of the cavity mirror displacement. The burst threshold can be adjusted upward to reduce the frequency span of sampled bursts, at the expense of reducing the capture range of the servo. With the servo engaged, the distribution in frequency detuning of triggered bursts becomes narrower because fluctuations in the cavity length about the sinusoidal length modulation are reduced by the active stabilization. The symmetry of the coupling spectrum combined with the equal probability of triggered bursts above a given threshold ensure that the average frequency detuning $\to 0$ when the cavity lock is engaged. We note that this description can be generalized to a laser source of non-zero bandwidth by convolving the indicated transmission curve with the laser spectrum, leading to a broadening of the former quantity. Provided the excitation spectrum is symmetric about the center frequency and interrogates a single-cavity mode, the locking scheme will be qualitatively the same as depicted.

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In Fig. 4, we show the coupling efficiency (blue curve) of the incident light as a function of ${{\Delta}}\!f$. This function was derived by summing the two fields described above and evaluating the dependence on ${{\Delta}}\omega\! /\!{{\Gamma}}$ of the peak light intensity coupling into the cavity. In the limit $| {{{\Delta}}\omega} |/{{{\Gamma}}_q} \to 0$, this coupling efficiency reduces to the Lorentzian profile $1/(1 + ({{{\Delta}}\omega} /{{{\Gamma}}_q})^2)$. Upon closing the servo loop by digitally actuating a DC offset voltage applied to the PZT actuator (10 Hz loop bandwidth), the swept cavity mode is stabilized. The effect of closing the servo loop is also shown in Fig. 4, where we observe in our experiments that the spectral span of the triggered bursts is reduced approximately five fold to ${\pm}{1.5}\;{\rm{kHz}}$. Elimination of drift in the mean cavity length causes this narrowing in the distribution of sampled cavity resonance frequencies. Also as shown in Fig. 4, the scheme involves a second set of triggers with a higher threshold (labeled CRDS trigger threshold) to activate the ring-down events. This threshold sets the spectral distribution of cavity frequencies that contribute to the measured ring-down signals from which the absorption spectrum is derived. As this threshold is increased, the spectral range of sampled frequencies will be reduced, thus improving the precision of the cavity lock.

Importantly, because the probe laser spectrum (shown in Supplement 1) and cavity transmission resonance are symmetric in frequency detuning, when the cavity lock is engaged, the resulting average error signal tends to zero, consistent with ${{\Delta}}\!\bar f \to 0$. To estimate the associated statistical uncertainty in the average frequency detuning between the cavity and probe laser under locked conditions, we note that our CRDS spectra consist of 60 spectrum points, with each point (200 ring-down decays) acquired nominally in 1 s. Further, we observed approximately 50 lock triggers during the acquisition of each spectrum point, resulting in 3000 triggers per spectrum. Assuming a uniform distribution (equal probability for all triggered pulses above the threshold) and stationary statistics over the time span of a spectrum acquisition, the effective standard deviation of the ensemble of frequency detuning values is estimated to be ${1.5}\;{\rm{kHz/}}{{{3}}^{1/2}}$. Dividing this quantity by the square root of the number of triggers per spectrum gives the standard error of ${{\Delta}}\!\bar f$$.$ Our analysis yields the standard Type A evaluated uncertainty of ${1.5}\;{\rm{kHz/}}{({{3}} \times {{3000}})^{1/2}} = {{16}}\;{\rm{Hz}}$ for the average frequency detuning, ${{\Delta}}\!\bar f$. This result corresponds to a relative standard uncertainty component of ${{8}} \times {{10}^{- 13}}$ in our frequency chain and is approximately five times smaller than that of the Cs clock.

E. Evaluation of the Spectrum Frequency Axis

For the reasons given above, in single-mode CRDS (as realized here), in which a narrowband light source is mode-matched into a ring-down cavity, the intracavity field present after extinction of the source (i.e., during the decay event) oscillates at the local cavity resonance frequency [70] (dictated by the round-trip time of the intracavity light) and not at the excitation frequency. Here, narrowband implies that the width of the excitation spectrum is much less than the cavity’s longitudinal mode spacing, resulting in single-mode excitation. Although detuning between the excitation laser and cavity resonance frequencies affects the coupling efficiency and light throughput, the CRDS decay rates are insensitive to the spectrum of the light source. Taking into account the length modulation of the ring-down cavity discussed above, the correspondence between the OFC and ECDL frequencies, and rewriting the average detuning between the instantaneous cavity mode frequency ${f_q}$ and probe laser frequency as ${{\Delta}}\!\bar f = {\bar f_q}\; - ({f_{\rm{ECDL}}} \pm {f_{\rm{EO}}})$, the value of a spectrum frequency point is assigned to the average mode frequency of the swept cavity given by

 

Fig. 5. (Left panel) Relative frequencies ${{\Delta}}\!{f_0} = {f_0} - {\bar f_0}$ (in kHz) determined for approximately 2300 measurements of the R16e transition of the $^{12}{{\rm{C}}^{16}}{{\rm{O}}_2}$ $({{30012}}) \leftarrow ({{00001}})$ vibrational band, measured at nominally 1.6 Pa and 297.0 K. ${{1}}\sigma$ uncertainty bounds are also present as dashed red lines. (Right panel) Histogram of counts of relative position measurements in 2.5 kHz bins, with overlaid normal distribution.

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3. CAVITY RING-DOWN SPECTROSCOPY MEASUREMENTS OF LINE POSITIONS

All absorption spectra were acquired using the custom-built CRDS system mentioned above and located at the National Institute of Standards and Technology (NIST) in Gaithersburg, Maryland. The system is a modification of a previously described ring-down cavity, the configuration of which may be found in previous publications [63,72]. A schematic of the present spectrometer can be found in Fig. 2. Further experimental details may be found in Supplement 1.

We achieve coarse laser frequency tuning by adjusting the ECDL grating, while we take fine steps with an EOM, as depicted in the FARS schematic [43]. Frequency steps of arbitrary size may be taken, limited by the linewidth of the laser. In practice, however, it proves advantageous to make frequency steps of the probe laser at the nominal FSR of the cavity, such that ${f_{\rm{EO}}} = \Delta q \times {f_{\rm{FSR}}} + {f_{\rm{offset}}}$, where $\Delta q$ is the change in ring-down cavity longitudinal mode order and ${f_{\rm{offset}}} = {f_q}({\Delta q = 0}) - {f_{\rm{ECDL}}}$. This step size ensures that there is minimal travel of the PZT-mounted cavity mirror, which minimizes dead time during stepwise tuning between cavity modes. All subsequent results shown employ frequency steps of one ${f_{\rm{FSR}}}$ unless noted otherwise. The small displacement of the cavity mirror required to follow changes in the probe laser frequency of one mode order also mitigates variation in the mirror base losses associated with slight displacement of the cavity optical axis. To produce spectra with frequency steps ${\lt}{f_{\rm{FSR}}}$, after the tuning range of the EOM is exhausted (here, roughly 40 GHz, employing the positive and negative sidebands of a 20 GHz modulator) or the desired spectral coverage is achieved, the process is repeated by shifting ${f_{\rm{offset}}}$ and interleaving the resulting spectra until the desired spectral density is achieved. To acquire spectra spanning more than 40 GHz, the phase lock of the probe laser to the OFC lock is broken and reacquired after tuning the grating of the ECDL. This procedure is rendered robust by always choosing a new ${f_q}$ such that ${f_{\rm{beat}}}$ lies within the bandwidth of the RF filter, which may easily be calculated. However, for the low-pressure spectra reported below, we chose a spectral bandwidth of 6 GHz, corresponding to 60 spectral points and more than 30 times the profile half-width.

A. Low-Uncertainty Measurements of $^{12}{{\rm{C}}^{16}}{{\rm{O}}_2}$ Transition Frequencies

As a demonstration of the performance of the CL-CRDS technique, we performed repeated measurements of ${{\rm{CO}}_2}$ line positions in the Doppler-broadened domain. To illustrate the benefits of the long-term averaging enabled by the highly reproducible frequency axis achieved with this technique, we made 2235 measurements of the R16e $({{30012}}) \leftarrow ({{00001}})$ $^{12}{{\rm{C}}^{16}}{{\rm{O}}_2}$ transition located near ${6359.9}\;{{\rm{cm}}^{- 1}}$. Type A analysis yielded a statistical uncertainty in the line center of $u({\bar f_0}) = 208\;{\rm{Hz}}$, corresponding to a relative uncertainty of ${1.1} \times {{10}^{- 12}}$ based on the standard error of the measurement ensemble. The Type A measurement standard error is based on the ensemble of 2235 spectra and implicitly includes statistical variations caused by the finite signal-to-noise ratio (SNR) of the spectra as well as by the frequency chain and temperature drift.

These spectra were acquired in several sets of experiments over roughly four months, each with unique sample charges and pressures. The elapsed data acquisition time for all spectra was approximately 3 days. The results are given in Fig. 5. No systematic trends with time can be observed, nor is there any discernable offset between the various experiments. We note that the standard deviation of the measured line positions is 9.8 kHz, shown as ${{1}}\sigma$ bounds in Fig. 5, and nominally the same as the average fit uncertainty for individual spectra reported by the spectrum fitting algorithm. Further, the Kolmogorov–Smirnov test [73] demonstrates that the fitted transition frequencies were consistent with having been sampled from a normal distribution ($p\;{\rm{value}} \lt 0.05$), indicating excellent long-term reproducibility of the CL-CRDS determinations of vacuum line positions. The Allan deviation of the fitted transition frequencies has a fitted slope of 0.505(10), consistent with ${n^{- 1/2}}$ averaging, and does not show a turn around, which would indicate a departure from stationary statistics.

A typical measured spectrum and least-squares fit of a nearly Doppler-limited Voigt profile to a measured spectrum are shown in Fig. 6. The fitted profile, which was obtained by floating the Doppler width, line center, and line area, agrees well with the observed line shape, as evidenced by no systematic residuals and no evidence of asymmetry. Speed dependence and narrowing due to velocity-changing collisions are not expected to be significant at the pressures considered here [61]. For this spectral fit, we used the calculated Lorentzian width based on the HITRAN 2016 self-broadening coefficient (30.18 MHz/kPa) [69]. Interfering transitions within ${\pm}$${{5}}\;{\rm{GHz}}$ of the target with an intensity ratio ($S\!/\!{S_{{\rm{interference}}}} \gt {\rm{SNR}}$) were fitted using parameters from HITRAN 2016 to mitigate potential systematic asymmetry. The spectrum exhibits a peak SNR (peak absorption/baseline noise level) of 30,000:1 with noise-limited fit residuals. Experiments and numerical simulations described below indicate that using frequency steps equal to one ${f_{\rm{FSR}}}$ fully captures the Doppler profile to provide a quantitative determination of the linewidth and line center. We also carried out the measurements at a much smaller sampling step size equal to 5 MHz and achieved equivalent measurement precision after accounting for the total number of measured spectra and points per spectrum. However, in this case, undesirable asymmetries in the measured profile resulted because of the longer spectrum acquisition time and changes in the base losses as the cavity length is tuned. For these reasons, we conclude that spectra acquired in steps of one ${f_{\rm{FSR}}}$ are optimized for measurement precision and accuracy.

 

Fig. 6. Typical CL-CRDS single-spectrum at a pressure of $p = {1.6}\;{\rm{Pa}}$ and nominally $T = {296.6}\;{\rm{K}}$ of the R16e $^{12}{{\rm{C}}^{16}}{{\rm{O}}_2}$ $({{30012}}) \leftarrow ({{00001}})$ transition, with a spectrum sampling density of 107 MHz. Lower panel, spectral residuals of Voigt profile fit.

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We have evaluated uncertainty components of our measurement caused by systematic effects (Type B analysis), and a complete uncertainty budget may be found in Table 1. These uncertainties are dominated by uncertainty in the pressure shifting coefficients. We have determined the ${{\rm{CO}}_2}$ partial pressure of each spectrum using the integrated line area along with recently determined high-accuracy molecular line intensities [74] with relative combined standard uncertainties better than 0.1%. The ${{\rm{CO}}_2}$ partial pressure is determined using the relation $p = A k_B\!T\!/\!(Sc)$), where $A = \int\! \alpha (f){\rm d}\!f$ is the integrated line area, $S$ is the line intensity, $T$ is the measured temperature, $c$ is the speed of light, and ${k_B}$ is the Boltzmann constant. The line intensities are corrected to the standard reference temperature of 296 K using the appropriate partition functions and lower-state energies given in HITRAN 2016 [75]. We simultaneously recorded the cell pressure with a capacitance manometer (1.33 kPa full scale), which provides a confirmation of the spectroscopic determination and evaluation of any possible leaks into the cell. This approach gives an offset-free measure of pressure along with independent measures of the individual pressure shifting components. At these low pressures, the uncertainty associated with spectroscopic pressure determination is well below those achievable by conventional pressure gauges and simultaneously immune to offset bias in both the ${{\rm{CO}}_2}$ partial pressure and air leaks, which may be determined by relative changes in gauge pressure from the start of each measurement. The pressure uncertainty determined here is, in fact, below the stated uncertainty of the NIST primary transfer standard typically employed for gauge calibrations.

Table 1. Uncertainty Budget for the Unperturbed Line Positions ${f_0}$ of the $({{30012}}) \leftarrow ({{00001}}$) $^{12}{{\rm{C}}^{16}}{{\rm{O}}_2}$ Vibrational Band

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The vacuum line positions are determined using the appropriate pressure shifting coefficients reported by Devi et al. [76], the details of which may be found in Table 1. A more comprehensive discussion is presented in Supplement 1. At the pressures used here (near 1 Pa), the total combined pressure shifts are on the order of a few kHz. Both the self- and air-shift terms are determined and corrected for individual spectra. We measured a leak rate of atmospheric air into the cell of ${0.179}\;{\rm{Pa}}\;{\rm{da}}{{\rm{y}}^{- 1}}$, which for a typical 16 h overnight average contributes an uncertainty of 3 Hz. To mitigate growth of this uncertainty, we evacuated the cell and refilled with fresh ${{\rm{CO}}_2}$ samples after no more than 5 days. Adsorption of ${{\rm{CO}}_2}$ to the cell walls, observed as a linear decrease in the integrated spectral line area over time, was spectroscopically measured to occur at a rate of ${{5}} \times {{10}^{- 5}}\;{\rm{Pa}}\;{\rm{mi}}{{\rm{n}}^{- 1}}$. The uncertainty contributed to the pressure shift by this effect is noted in Table 1 as “pressure change (single spectrum).” We spectroscopically probed for the presence of water vapor in the cell, from either desorption from the cell or leaks, and found no detectable water at our detection limit of 0.01 Pa (for the ${{\rm{H}}_2}{\rm{O}}$ transition located at ${6334.250}\;{{\rm{cm}}^{- 1}}$, using the self-shifting term from [77]), setting an upper bound of 24 Hz (with 100% uncertainty) on the possible contribution of water shifting.

We have investigated the contributions of DC and AC Stark shifts, first-order Doppler shifts from moving cavity mirrors, and the second-order Doppler shifts. The magnitudes and uncertainty contributions may be found in Table 1. A more complete description on how these terms were determined may be found in Supplement 1.

We have also determined the line positions of several other transitions in the $({{30012}}) \leftarrow ({{00001}})$ $^{12}{{\rm{C}}^{16}}{{\rm{O}}_2}$ band, ranging from P24e to R16e, the results of which can be found in Table S1 in Supplement 1 along with their corresponding combined uncertainties. All reported uncertainties are below 600 Hz, with the lowest values reaching nearly 200 Hz. Importantly, the uncertainties reported here for the P10e and R16e transitions are the lowest reported to our knowledge for Doppler-broadened molecular transition frequencies [38,40,41,44,61] and smaller than state-of-the-art near-IR comb-linked saturation molecular spectroscopy [22–26].

4. MONTE CARLO SIMULATIONS

To further investigate statistical uncertainties and potential bias in the fitted line positions, we performed a Monte Carlo numerical study simulating our experimental spectra with Gaussian noise sources along both the detuning and absorption axes. Our calculations indicate that uncertainty in the fitted line center for a single spectrum is dominated by noise in the absorption axis. We find that the uncertainty is inversely proportional to the SNR and equals $\approx {13.5}\;{\rm{kHz}}$ for a SNR of 10,000:1. As can be seen in Fig. 7, these calculations are in good agreement with our measurements of the SNR for the various lines studied and the associated uncertainties in the fitted line position. For statistically stationary behavior and after averaging results, the standard uncertainty for the line position is given by $u({f_0}) \approx {{135}}\;{\rm{MHz}}\;({\rm{1/SNR}}) \times ({\rm{1/}}{n^{1/2}})$, where $n$ is the number of fitted spectra and independent determinations of the line center. At the specified SNR, we estimate the statistical uncertainty to be less than 1 kHz after averaging 200 spectra (roughly 4 h for our measurement acquisition rate). In general, we note that for a purely Doppler-broadened profile that is limited by Gaussian noise along the absorption axis, provided that $\Delta {f_s} \lt {\sigma _D}$, our simulations show that uncertainty in the fitted line center for a single spectrum scales as $u({f_0})\sim({\sigma _D}/{\rm{SNR}})\;{(\Delta {\nu _s}/{\sigma _D})^{1/2}}$. Here, ${\sigma _D} = {\Gamma _D}/{[{\rm{2 \; ln}}({{2}})]^{1/2}}$, where ${\Gamma _D}$ is the HWHM of the Doppler profile, and $\Delta {f_s}$ is the experimental spectral step size. This result shows that there is no inherent advantage to oversampling the spectrum for a purely isolated Doppler profile.

 

Fig. 7. Comparison of line center uncertainty, $u({f_0})$, versus signal-to-noise ratio (SNR), determined experimentally (open circles) and by Monte Carlo modeling (red line).

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We also performed Monte Carlo simulations of the transmitted signal to estimate how much measured temperature drifts (${\rm{nominally}} \pm {{10}}\;{\rm{mK}}$) during the acquisition of individual spectra can affect the measurement statistics. Linear variation in the sample temperature of this magnitude introduces a small amount of asymmetry in the measured profile, which leads to positive- and negative-going biases in the fitted line center ${\lt}{{150}}\;{\rm{Hz}}$ in magnitude. However, averaging over many spectra with uncorrelated temperature excursions will tend to randomize the observed variation in line centers. Most importantly, we find that the principal factor limiting the statistical component of the line position uncertainty was the noise in the measured spectrum losses, as shown in Fig. 6.

The Monte Carlo simulations also reveal that interfering transitions, which are below the noise level, can bias the fitted line center, depending on the number of spectra averaged, their relative line positions, and absorption strength. A potentially insidious case involves weakly perturbing transitions, located near the target line, but which are too weak to be observed in a single spectrum. Our calculations reveal that the fractional bias, $R$, (equal to the change in fitted mean value divided by the standard error of the ensemble mean) associated with neglecting the perturbing line scales with the number of spectra, $n$, the ${{\rm{SNR}}_p}$ (SNR of the perturbing line), and its detuning relative to the target line, $\Delta {f_p} = {f_{p\:}}\! -\! \;{f_0}$, shown as

As indicated in Table 1, the combined uncertainty (212 Hz for the R16e transition) is driven by statistical effects. In our experiments, these variations are manifested as a distribution of measured transition frequencies with a standard error of the mean equal to 208 Hz. Based on our Monte Carlo simulations, the distribution in measured transition frequencies is dominated by the effects of a finite spectrum SNR and temperature drift (combined contributions of 189 Hz) and to a much lesser extent the statistical variability (60 s timescale) in the frequency chain discussed above (contribution of 87 Hz). Our analysis allows us to account for those physical effects that dominate the observed distribution of measured line positions.

5. COMPARISON TO SATURATION SPECTROSCOPY

The frequency agility and coupling efficiency of CL-CRDS also allow for nonlinear operation in the saturated spectroscopy regime. This allows for a direct comparison between Doppler-broadened and sub-Doppler features. We have observed saturation dips in these $({{30012}}) \leftarrow ({{00001}})$ $^{12}{{\rm{C}}^{16}}{{\rm{O}}_2}$ vibrational band transitions (to the best of our knowledge, the first reported saturated measurements in this band). The spectra were recorded within a 6 MHz bandwidth of the expected line center from Doppler-broadened measurements, at a spectral step size of 50 kHz and corresponding to 120 spectral points per spectrum. The spectra were measured sequentially in frequency, shifting ${f_{\rm{offset}}}$, and subsequently the cavity PZT position. Because we were not attempting to determine the profile of the saturation spectrum, we fit the non-exponential decays (caused by the time-dependent absorption loss during the decay [80]) purely as an exponential function. Any distortion of the saturation spectrum is expected to be symmetrical and would not bias the fitted center line position. The saturation spectra are fit with a Lorentzian function and a linear baseline term.

We measured and fit saturated spectra of both the R16e and P10e transitions. We obtained an SNR of roughly 500:1 for each saturation spectrum, yielding single-spectrum line center uncertainties of about 10 kHz for each transition and ensemble standard errors of 1.9 kHz and 2.6 kHz for R16e and P10e, respectively. The average fitted line positions were 190,667,021.4113(19) MHz and 190,059,681.7231(18) MHz for R16e and P10e, representing deviations of 1.9 kHz and 1.4 kHz from the value obtained from the linear absorption Doppler regime measurements. These deviations are within the combined uncertainty of the two measurements. The saturation dip half-width is 270 kHz, roughly that expected from the cavity geometry and laser fluence. It should be noted that each saturation spectrum contained nearly twice as many spectral points as the linear CL-CRDS Doppler spectra and produced a fitted line center uncertainty that was approximately ${n^{- 1/2}}$ lower than the Doppler-broadened single-spectrum line center uncertainty of nominally 14 kHz. However, because it is necessary to change the cavity length and reestablish the transmission lock at each point, the measurement time for the saturation spectra is 15 times greater than that of the linear CL-CRDS case. These saturation spectra have lower SNR and are also significantly more complicated, with observed asymmetric baseline features likely caused by the linearly varying cavity length and various degrees of saturation caused by changes in intracavity fluence. Our comparison of linear and nonlinear absorption techniques provides a rigorous confirmation of the linear method at the kHz level of uncertainty. Indeed, saturation spectroscopy remains a powerful technique for resolving blended transitions, which cannot be readily modeled as symmetric line profiles. However, for largely isolated transitions, such as the ${{\rm{CO}}_2}$ transitions reported here, equivalently low single-spectrum uncertainties with significantly superior long-term averaging may be achieved through linear CL-CRDS of Doppler-broadened spectra.

6. COMPARISON TO LITERATURE

Several previous comb-linked studies have investigated various transitions in the $({{30012}}) \leftarrow ({{00001}})$ band. A comparison of these previous results to the current work may be found in Table S1. We note that the transition frequencies reported here agree well within 2σ to those reported in both Long et al. [61] and Truong et al. [41], both determined by CRDS with the comb-referenced PDH-FARS method. The results are slightly outside the combined uncertainties reported in two different implementations of comb-locked CRDS by Gotti et al. [37] and Gatti et al. [38]. The R16e transition frequency determined using OFC-FTS by Rutkowski et al. [81] deviates from that determined here by nearly 3 MHz, likely caused by the significantly higher pressure in the OFC-FTS measurement and the correspondingly larger pressure shift. Our measurements have the lowest reported combined uncertainties for near-IR molecular transition frequencies, including those determined for numerous transitions in the $({{30012}}) \leftarrow ({{00001}})$ $^{12}{{\rm{C}}^{16}}{{\rm{O}}_2}$ vibrational band.

7. CONCLUSIONS

We emphasize that our technique may be readily applied to unblended molecular transitions without having to satisfy saturation power requirements or fitting non-exponential decays or secondary saturation dip features. Additionally, because the molecular line intensity was employed to determine the sample pressure, the present determination was not subject to biases caused by zero-pressure offset errors common in capacitance manometers, which may affect line center determinations that employ a linear extrapolation from a series of higher-pressure line center measurements to determine ${f _0}$. We also note that the cavity transmission modes are actively brought into overlap with the OFCL probe beam, which makes our spectrum frequency axis insensitive to frequency shifts in the cavity modes caused by absorption-induced dispersion.

We have demonstrated a linear Doppler-limited absorption technique based on OFCL-CRDS that achieves combined relative uncertainties in transition frequencies as low as ${1.1} \times {{10}^{- 12}}$: a figure of merit superior to state-of-the-art Doppler-free optical molecular spectroscopy methods in the near-IR spectral region. Our method exploits the accurate frequency axis achieved using a Cs-clock-referenced OFCL-CRDS system and the high degree of symmetry in the experimental observations of isolated absorption lines. We have demonstrated the wide suitability of this method by making extensive ensemble measurements of five $^{12}{{\rm{C}}^{16}}{{\rm{O}}_2}$ molecular transitions in the $({{30012}}) \leftarrow ({{00001}})$ vibrational band and have compared these results to previously determined values. Our uncertainties are significantly smaller than those of existing experiments and mitigate several potential sources of systematic bias.

This study illustrates the wide applicability of this technique for the determination of unperturbed molecular transition frequencies, allowing for more extensive ultra-high accuracy and precision determinations than have been previously achieved. This approach will allow for new secondary frequency standards and refinements of calculated potential energy surfaces in molecular systems [34]. Importantly, this technique may be applied to molecular transitions that are not suitable for Doppler-free spectroscopy and may also be employed to determine collisional-shift coefficients with improved accuracy.

Future refinements in SNR, spectrum acquisition rate, and temperature stability should allow for relative uncertainties in transition frequency of $10^{-13}$ or below, ultimately limited by the quality of the reference.