Comb-locked Lamb-dip spectrometer

Overcoming the Doppler broadening limit is a cornerstone of precision spectroscopy. Nevertheless, the achievement of a Doppler-free regime is severely hampered by the need of high field intensities to saturate absorption transitions and of a high signal-to-noise ratio to detect tiny Lamb-dip features. Here we present a novel comb-assisted spectrometer ensuring over a broad range from 1.5 to 1.63 μm intra-cavity field enhancement up to 1.5 kW/cm2, which is suitable for saturation of transitions with extremely weak electric dipole moments. Referencing to an optical frequency comb allows the spectrometer to operate with kHz-level frequency accuracy, while an extremely tight locking of the probe laser to the enhancement cavity enables a 10−11 cm−1 absorption sensitivity to be reached over 200 s in a purely dc direct-detection-mode at the cavity output. The particularly simple and robust detection and operating scheme, together with the wide tunability available, makes the system suitable to explore thousands of lines of several molecules never observed so far in a Doppler-free regime. As a demonstration, Lamb-dip spectroscopy is performed on the P(15) line of the 01120-00000 band of acetylene, featuring a line-strength below 10−23 cm/mol and an Einstein coefficient of 5 mHz, among the weakest ever observed.

the difficulty to achieve at once the intensity level required to saturate the transition and the absorption level needed for the dip, which is a fraction of the absorption, to emerge from the noise. In the near-infrared, where the line-strengths are rather small and typically below 10 −22 cm/mol, the adoption of a high-finesse cavity may serve both purposes, since the finesse effectively multiplies both intra-cavity power and absorption length.
Among cavity-enhanced techniques, cavity-ring-down-spectroscopy (CRDS) 26,27 has the merit of a substantially simple setup and of a strong immunity against the laser frequency noise. Such noise decreases the cavity throughput but does not necessarily lead to a strong degradation of the signal-to-noise ratio on the exponentially decaying waveform, at least as long as the cavity throughput favourably compares with the detector noise. This was brilliantly demonstrated by Campargue and Kassi 28 , with a noise limit at a 6•10 −11 cm −1 level on the single ring-down event in spite of 250-Hz cavity resonances and of a 2-MHz laser linewidth. Lamb dips by CRDS have been observed on NO 2 lines near 0.8 μ m 29 , on water lines near 1.4 μ m 30 and on 17 O 12 C 16 O at natural abundance near 2.3 μ m 31 . Recently, by adoption of a particularly evolved CRDS apparatus seeded by a an ultra-stable laser, CO 2 lines as weak as 5·10 −25 cm/mol with Einstein coefficients down to 7 mHz were saturated and exploited to revisit the spectroscopic constants of the CO 2 30013 ← 00001 band 32 . Interestingly, the noise level of 1.7·10 −11 cm −1 over a 7-minute-long measurement remained nearly 2 orders of magnitude above the limit of detection of the same spectrometer in a linear absorption regime, equal to 5•10 −13 cm −1 over 1 s 33 . This derives from two main issues of Doppler-free CRDS spectroscopy: i) the shorter ring-down time that occurs at the centre of the absorption line, where the dip takes place, and ii) the failure of the exponential function for the fit of ring-down decays in a saturated regime. This makes the fit more susceptible to fluctuating conditions such as the optical power coupled into the cavity at the beginning of the decay. Introducing a second decay constant in the fit function improves the fit quality and also helps in normalizing the absorption against the empty-cavity losses, as demonstrated by Galli et al. 34 , yet it may increase the error bar due to the correlation between the two decay constants.
An alternative and particularly elegant approach to Doppler-free spectroscopy that does not suffer from the above trade-offs is represented by Noise-Immune Cavity-Enhanced Optical-Heterodyne Molecular Spectroscopy (NICE-OHMS). NICE-OHMS is a smart combination of tight laser-to-the-cavity locking, achieved through the Pound-Drever-Hall (PDH) technique 35 , with a frequency-modulation-spectroscopy approach where sidebands are placed coincident with the neighbouring cavity modes. The immunity of NICE-OHMS against laser-cavity frequency noise was demonstrated by Ma et al. 36 almost two decades ago with the saturation of a very weak overtone line of CH 2 D at 1064 nm, with a nearly shot-noise-limited sensitivity of 10 −14 cm −1 Hz −0.5 . A similar value has been recently approached even in a Doppler-broadening regime on the P(11) line of C 2 H 2 37 , yet recurring, as in the first demonstration, to an ultra-narrow seed laser with very poor frequency tunability to achieve a tight lock between laser and cavity. With more versatile and tunable laser sources such lock and thus the sensitivity limit was found to be degraded by one or even two or three orders of magnitude 38,39 , which explains why NICE-OHMS failed to be applied to broad line surveys in a saturated regime. This paper describes a versatile cavity-enhanced optical spectrometer for Lamb-dip spectroscopy over the range from 1.5 to 1.63 μ m, featuring frequency accuracy at the kHz-level and a limit of detection below 2.5•10 −11 cm −1 over a typical 200-s acquisition time. The accuracy of the spectrometer is based on the referencing to an Er:fiber frequency comb whereas the sensitivity on an extremely tight laser-cavity PDH lock that brings the relative amplitude noise at the cavity output below 10 −3 in a direct detection mode, without recurring to any wavelength-or frequency-modulation-spectroscopy approach and thus with a very simple scheme. A cavity finesse in excess of 100000 allows Lamb-dip spectroscopy to be performed also on weak transitions, whereas the possibility to vary the intra-cavity field intensity by more than two orders of magnitude up to the kW/cm 2 level enables saturation also of highly symmetric vibration modes with poor electric dipole moment. This is experimentally demonstrated on the P(15) line of the 01120-00000 band of C 2 H 2 , which features a line-strength below 10 −23 cm/mol and an Einstein coefficient of only 5 mHz. At the opposite side of the application range, saturation is also demonstrated for the P(30), P(28) and P(26) lines of the well characterized 10100-00000 band of C 2 H 2 , with line strengths of 10 −22 cm/mol and Einstein coefficients 1000 times higher. Thousands of lines in different bands for several molecules may be found within these wide boundary conditions, which makes the system an unprecedented tool for Doppler-free spectroscopy in a spectral region of extreme spectroscopic interest.

Experimental Setup
The layout of the spectrometer is sketched in Fig. 1. An extended-cavity-diode-laser (ECDL) with broad tunability from 1.5 to 1.63 μ m (Toptica DL pro) is locked with a Pound-Drever-Hall (PDH) approach to a stainless-steel cavity with a finesse > 100000. PDH locking is achieved through a single proportional-integral-derivative (PID) servo that provides external correction to the laser frequency through a voltage-controlled-oscillator (VCO) driven single-sideband modulator (SSM), that acts as an ultrafast frequency actuator, as described in ref. 40. An output voltage dynamics of ± 3.5 V for the PID and a frequency-to-voltage conversion of 22 MHz/V for the VCO allow the laser to remain steadily locked upon tuning the cavity mode by up to ± 75 MHz, thus far in excess of a typical Lamb-dip spectral width. The control bandwidth, as high as 5 MHz, squeezes the residual laser-cavity linewidth down to 4 mHz, strongly suppressing intensity noise at the cavity output caused by frequency-to-amplitude conversion of cavity and laser frequency noise.
The above conditions make a simple cavity-enhanced direct-detection approach of great attractiveness. The power transmitted by the cavity is measured by a commercial detector with variable gain from 10 3 to 10 7 V/W (Thorlabs PDB150C). This ensures an electrical signal of few Volts, well above the quantization noise of the 16-bit FPGA-based board used for data acquisition, whatever is the optical power from below 1 μ W to the maximum achieved value of about 100 μ W. The latter case, with a cavity length of 50 cm, mirrors with losses of 1.1•10 −5 and 1-m radius of curvature, corresponds to an intra-cavity field intensity in excess of 1.5 kW/cm 2 , which is sufficient to provide saturation parameters of few % for transitions with Einstein coefficients down to few mHz. Absolute calibration of the frequency scans is obtained through the referencing of the cavity-locked probe laser to a frequency comb synthesized by a 100-MHz Er:fibre oscillator (Toptica FFS) with repetition-and carrier-envelope-offset-frequencies stabilized to a primary clock through a GPS-referred Rubidium oscillator. The comb is made to beat on an avalanche photodetector with a fraction of the ECDL and their beat note is stabilized by slow feedback (700 Hz bandwidth) to a piezoelectric transducer acting on the cavity length and thus on the cavity mode that the ECDL is locked to. In this way the cavity mode (which the ECDL is locked to) is forced to lie at a given offset from the nearest comb mode. In order to perform comb-calibrated scans of the laser frequency, the comb undergoes a double pass in an acousto-optic frequency shifter (AOFS) before superimposing the ECDL: the AOFS provides a single-pass diffraction efficiency higher than 75% in the 75-95 MHz RF range, giving rise to an optical tuning range of 40 MHz for the comb and thus for the ECDL. Frequency scans are thus accomplished by means of a RF synthesizer that drives in a controlled way the input signal of the AOFS. In order to maintain a stable frequency offset between comb and ECDL over multiple scans, the beat note signal at the input of the feedback loop must exhibit a good signal-to-noise ratio (SNR): experimentally, by counting the beat note during the scans and looking at its deviations, we found a 30-dB SNR at a resolution bandwidth of 10 kHz to be necessary to push rms deviations below 0.2 kHz. In our case, due to the relatively low repetition frequency and average optical power of the comb, such threshold was not reached over the entire tuning range of the ECDL using the comb supercontinuum output. We then exploited a second 1.55-μ m comb output and got it spectrally broadened in a simple dispersion-shifted telecom fiber over a region much narrower than an octave, roughly from 1.4 to 1.7 μ m, in order to enhance the average power per comb mode over the rather large tuning range of the ECDL (1.5-1.63 μ m).

Results and Discussion
In a cavity-enhanced direct-detection regime the minimum detectable absorption α min is related to the laser intensity noise at the cavity output through the simple formula: which derives from a first-order approximation of the Lambert-Beer law. In Eq. (1) Δ I/I is the root-mean-square value of the relative intensity noise over a given measurement time and L eff is the effective interaction length. Under weak absorption conditions this is in turn proportional to the geometrical length L and to the finesse F of the cavity, by the relation L eff = 2•L•F/π 41 . Increasing F and L is thus beneficial for the sensitivity of the spectrometer provided that the intensity noise term is preserved. This is in practice a challenge since the higher is L eff the steeper is the slope of the frequency-to-amplitude response given by the cavity modes, with the consequence that any residual frequency noise between laser and cavity may severely impact on Δ I/I and thus on α min . In our case, thanks to a very tight laser-cavity locking, such a detrimental noise conversion is strongly inhibited. This is shown in Fig. 2(a), where an Allan plot (blue curve) is reported for Δ I/I as a function of the averaging time for times longer than 1 ms. Below 1 ms the data are not relevant since they correspond to measurement bandwidths that can't be exploited experimentally since they are in excess of the 700 Hz-bandwidth of the cavity-comb feedback loop. Moreover, below 1 ms, it is more difficult to evaluate the impact of residual laser-cavity frequency noise since the cavity acts as a low pass filter (with a pole at half the 3-kHz-wide cavity mode). The Allan deviation amounts to 8•10 −4 at the 10-ms averaging time used in the measurements, it decreases till 4·10 −4 at 20 ms while remaining almost constant up to 10 s. This is the signature of a noise dominated by a Flicker contribution, which is mostly due to residual frequency noise between cavity and laser. This is confirmed by the green trace reported in the figure, which refers to a measurement performed before the cavity where Δ I/I is mostly affected by laser intensity noise: this remains a factor greater than 4 below the frequency noise contribution over the typical 2-s-long measurement time taken by each frequency scan. On the other hand the detector noise remains more than an order of magnitude below whatever is the gain setting (black line). To improve the limit of detection in a system dominated by Flicker noise it is beneficial to speed up the acquisition of a single spectrum while averaging over many subsequent spectra: in our case, due to the relatively slow cavity-comb feedback loop, optimum conditions were found by stepping the laser frequency at every 20 ms, by letting the cavity-locked laser to stabilize against the comb within the first 10 ms, then acquiring and averaging the cavity output power over the last 10 ms. This implies a 100-spectral-points spectrum to be acquired in 2 s. Figure 2(b) reports the normalized rms deviation of residuals for N-times averaged blank spectra acquired in those conditions and fitted with a linear baseline. At N = 1 the deviation amounts to 8·10 −4 , in perfect agreement with the Allan plot over the corresponding time window; at increasing N, it decreases with the square root of N as expected for a normal noise distribution, attaining a value of 2.5·10 −5 at N = 1000, i.e. 2000 s. This corresponds to a detection limit of 7.8•10 −12 cm −1 , which is, to the best of our knowledge, the lowest ever demonstrated in a direct-detection cavity-enhanced regime starting from a widely tunable laser with linewidth in excess of 100 kHz. The inverse square-root law shown in Fig. 2(b) was found to hold for frequency scans below 30 MHz that are typical of the Lamb-dip regime thanks to the absence in the setup of parasitic cavities with comparable free-spectral-range. Precision and accuracy of the spectrometer were tested on three well-characterized 17,18 sub-Doppler lines of the 10100-00000 band of acetylene, namely P(30), P(28) and P (26). Due to their high intensity, up to 3•10 −22 cm/mol, and to an empty-cavity finesse of 100000, low pressures were adopted, from 0.2 to 0.4 Pa depending on the line, to keep the linear absorption below 60%.
On the other hand, the optical power was reduced to 0.5 μ W at the cavity output to minimize power broadening of the lines while maximizing the precision of the transition frequency determination. In such conditions the saturation parameter remains around unity (calculation according to ref. 42). Lower power values have not been adopted to prevent the signal-to-noise ratio from being degraded, whereas higher power values would have turned the regime into strong saturation. This regime offers the chance to accurately determine line saturation intensities, yet its exploration remained outside the scope of the paper. The trade-off between SNR degradation and strong saturation effects explains why the investigation has been here limited to the less intense transitions of the 10100-00000 band of acetylene. Actually, the operating conditions used for these lines represent a sort of boundary for the best performance of the spectrometer, which better lends itself to the saturation of weaker transitions, also associated with more symmetric vibration modes and lower electric dipole moments.  collisional contribution of 220 kHz calculated from a 0.094 cm −1 /atm broadening coefficient (HITRAN source 4 ) and a power-broadening contribution of 180 kHz from a saturation parameter of 0.94. The latter comes as the ratio of an intra-cavity field intensity of 12.7 W/cm 2 , which was calculated dividing the power transmitted from the cavity by the experimental mirror transmission (1.1•10 −5 ), and a saturation intensity of 13.5 W/cm 2 , the latter calculated according to Eq. (22) from ref. 42.
The analysis of the line centre-frequencies retrieved from the fit of the individual dip profiles acquired consecutively within a set of 100 scans, reveals a scattering of about 6-7 kHz rms, depending on the set. This is consistent with a signal-to-noise ratio of nearly 100 that derives from a noise at the 10 −3 level (see Fig. 2) on a dip with 11% contrast with respect to quite a strong linear absorption greater than 50%. The corresponding precision is 3 parts over 10 11 over 2 s, which is worse than the 4 parts over 10 12 that can be attained with an optimized dither-based laser-to-the-dip locking 18 on the same time window, but more than a decade better than what can be obtained in a comb-assisted Doppler broadening regime 43,44 . As compared to dip-locked spectrometers, the main advantage of our approach is the full access to the dip profile, enabling a quantitative study of effects related to collisions, transit-time, saturation, sub-Doppler line-shapes. Figure 3(b) reports the line-centres retrieved from the fit of 25 fully independent sets of measurements, each one encompassing 100 scans, as performed over few days taking care before each set of resetting all locking servos, emptying and refilling the cavity. The rms deviation of data amounts to 1.1 kHz only, which accounts for an impressive Type A uncertainty of 5 parts over 10 12 . The error bars in Fig. 3(b) provide the rms scatter of centre frequencies within a given set, corresponding to the 6-7 kHz value discussed above. Table 1

Sources of systematic errors Estimated shift [kHz]
Frequency  Table 1. Summary of estimated systematic uncertainties associated with the absolute measurement of line centre frequencies.
The influence of electronic offsets in the PDH lock is negligible since with a PDH discriminator of 310 V/kHz 40 , typical offsets drifts by few mV are not appreciable. A systematic shift was found to originate in our case by a drifting offset between comb and cavity-locked laser during the scan (both having a linewidth of about 10 kHz over 1 ms 40,46 ). Due to the relatively loose lock between these two signals, a mutual excursion by nearly 200 Hz may happen during the scans, thus adding a further 0.2 kHz term to the uncertainty budget. A 20% uncertainty on the intra-cavity power, as weighted with a power shift coefficient of − 11 Hz/mW 47 , gives an almost negligible contribution < 0.1 kHz. An uncertainty contribution less than 1 kHz is in the end ascribed to asymmetric residuals in the dip fit, that cannot be accounted for with a purely symmetric Lorentzian function. The quadrature sum of the various systematic shifts leads to a type B standard uncertainty of 1.09 kHz and to an overall uncertainty of ∼ 1.5 kHz with the quadrature addition of the Type A contribution (1.1 kHz). Table 2 compares the line-centre frequencies retrieved for the P(30), P(28) and P(26) lines with data determined by Madej et al. 18 at the Canadian metrological institute, which exhibit, amongst the published values, the lowest uncertainties. These have been determined with a cavity of much lower finesse (< 400) yet with similar intra-cavity field intensities at a pressure of 2.7 Pa through the typical top-of-the-dip locking approach adopted for frequency standards. The comparison requires quite a severe correction by more than 6 kHz for the pressure shift, due to the use in our case of nearly 10-times lower pressure values (0.4 Pa for the P(30), 0.2 Pa for the P(28) and P(26) lines): this correction has been reported with a + sign in Table 2 to make frequencies from Madej et al.
(higher pressure) comparable with ours. A second correction term with opposite sign has been introduced to account for the effect of a rather large 1.8-MHz peak-to-peak modulation depth adopted in ref. 18 for dip locking: the same authors report for this effect a shift coefficient of − 4.7 kHz/MHz pp 47 , which translates into a final correction by − 8.5 kHz due to the absence, in our case, of any modulation. Interestingly, despite the very different working conditions and the strong impact of both pressure and amplitude modulation effects, the offset between the two frequency sets stays within the respective uncertainties, in both cases reported at the 2-kHz level only. The offset amounts to roughly 2.6 kHz with a sub-kHz dispersion, thus attesting a very high precision for both approaches. The comparison is unfortunately limited to three lines due to the difficulty, in our setup, to preserve the same level of accuracy for the much more intense lines that occur with lower rotational quantum numbers of the upper state. On the other hand, it emerges from Table 2 that one of the main hurdles in the traceability and eventually in the accuracy of spectroscopic determinations derives from the impact that different working conditions may have on the systematic errors: with synchronous detection schemes the impact given by the modulation depth, if not properly accounted for, may significantly impair the comparison between different determinations. Moreover, if the dip-profile is not measured but simply used as a marker for a top-of-fringe locking, one may miss the impact of profile asymmetries on the final line center value 48 . With our approach, instead, the cavity is not modulated and the frequency axis is fully defined by the comb at every point of the dip profile, thus making possible to catch rather subtle effects that do not emerge at the output of a lock-in amplifier. A further strength of the approach is given by the possibility to work at very low pressures, thus with smaller linewidths and lesser impact from pressure-induced shifts that cannot be easily accounted for in the absence of high-quality spectroscopic data. This is for example the case of HITRAN, which reports for the whole band a single pressure shift value of − 1 kHz/Pa.
To test the ability of the spectrometer to saturate much weaker transitions we moved to the P(15) line of the 01120-00000 band. The line strength is 9.468•10 −24 cm/mol and gives rise to a linear absorption of about 10% at the adopted pressure of 0.18 Pa. The Einstein coefficient of the transition is 5.632 mHz, i.e. three orders of magnitude lower than those of the 10100-00000 band and close to the smallest ever explored in a saturated regime 36 . To saturate the transition the power level at the cavity output has been raised to a level of ∼ 50 μ W, corresponding to an intra-cavity field-intensity of ∼ 1.5 kW/cm 2 . In such conditions the dip was clearly visible, as shown by the 100-fold-averaged Lamb-dip spectrum reported in Fig. 4. The dip contrast is 0.3%, thus in rather good agreement with an expected saturation parameter of 0.08. In such conditions the linewidth is negligibly affected by power broadening and amounts to 359 kHz only. The residuals of a Lorentzian fit were found to be limited by a non perfect line shape model and by a non perfect baseline to a rms normalized value of 3.55•10 −5 . Remarkably, this is nearly an order of magnitude below that given by the Allan variance plot of Fig. 2 and resulted from an extremely careful optimization of the setup and suppression of ground loops that was found hardly reproducible, which is the reason why the discussion above has been set on the normal performance regime of the spectrometer. However, those residuals are a demonstration that a sensitivity limit of 3.1•10 −12 cm −1 over 200 s can be achieved with a direct detection approach. In such conditions the statistics on the line centre frequencies retrieved from the fit of 100 consecutive scans is characterized by a scattering of 18-24 kHz, thus only a factor of 3-4 worse than

Conclusions
This paper describes a versatile comb-assisted cavity-enhanced system for Lamb-dip spectroscopy at a variety of wavelengths in a rather large spectral range from 1.5 to 1.63 μ m. It relies on two servo loops only, one for the laser-to-the-cavity locking, the other for the cavity-to-the-comb locking. Detection is performed directly at the cavity output with high signal-to-noise ratio thanks to an extremely tight Pound-Drever-Hall (PDH) locking based on an integrated single-sideband modulator. Thanks to the high finesse of the cavity and to an intra-cavity field intensity that can be varied over two decades overcoming the kW/cm 2 threshold, the system lends itself to the saturation of a variety of lines never saturated so far, also including weak ones. From a spectroscopic point of view it may become a workhorse for highly accurate surveys of line centre-frequencies of thousands of transitions, over several absorption bands of a variety of molecules, as well as for the investigation of line-shape models and saturation effects in a Doppler-free regime. From a technological point of view, the current sensitivity at the 10 −11 cm −1 level may be further improved by minimizing the cavity leakage, which currently bounds the averaging time to few hundreds of seconds. A leap forward would be possible by arranging the system in a NICE-OHMS configuration. A vast deployment of NICE-OHMS has been prevented so far by the difficulty in achieving tight PDH locking with conventional broadly tunable cw lasers. The single-sideband modulation scheme demonstrated here removes this stumbling block and paves the way to shot-noise limited spectroscopy at the output of high-finesse cavities.