Dual-comb mid-infrared spectroscopy with free-running oscillators and absolute optical calibration from a radio-frequency reference

By using free-running independent femtosecond OPOs with a repetition-rate difference of 500 Hz we demonstrate methane absorption spectroscopy with spectral coverage simultaneously spanning the methane P, Q and R branches and with a resolution of 0.5 cm-1. Absolute optical frequency calibration with an accuracy of 0.25 cm-1 (0.27 nm) is achieved from simultaneous repetition-rate and carrier-envelope-offset frequency measurements, without the need for any optical reference. The calibration technique allows registration and averaging of consecutively acquired dual-comb spectra, leading to a high quality and low-noise absorbance measurement in good agreement with the HITRAN database. c © 2017 Optical Society of America OCIS codes: (300.6300) Spectroscopy, Fourier transforms; (300.6390) Spectroscopy, molecular; (120.6200) Spectrometers and spectroscopic instrumentation; (190.4970) Parametric oscillators and amplifiers. References and links 1. F. Keilmann, C. Gohle, and R. 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Notwithstanding the success of near-IR DCS [9][10][11][12], the principal spectral region of interest for molecular spectroscopy is at wavelengths above 3 µm, where molecular absorption cross-sections are highest. While in the near-IR fiber frequency combs have proven themselves as the platform of choice for DCS, the situation concerning mid-IR combs [13,14]  , which has the advantage of inherently high coherence and-when the pump and seed are derived from the same laser-of zero-offset frequency combs, which provide a simple and direct mapping from the measured radio-frequency spectrum to the corresponding optical spectrum [18]. In all other implementations of mid-IR DCS there has to our knowledge been no absolute wavelength calibration reported which did not rely on a pre-existing optical reference such as a Fourier-transform spectrometer [20,25,26], an auxiliary reference laser [16,27,28] or spectral registration with a known gas spectrum [21,24]. In this paper we introduce a mid-IR DCS system in which the wavelength scale is entirely derived from radio-frequency references, achieving an absolute wavelength accuracy of 0.27 nm at 3150 nm. An OPO-based dual-comb spectrometer was first demonstrated in a mode where two femtosecond lasers pumped a single OPO [24]. While this approach offers advantages in terms of stability [25,26], it places a limit on the repetition-frequency difference between the two combs because spectrally different idler pulses are produced when the pump repetition rates differ significantly [24]. Using a single OPO cavity also makes it complex to independently measure and control the carrier-envelope offset frequency (fCEO) for each OPO [29], limiting the ability to utilize this signal for stabilization or calibration. Here, by simultaneously recording the DCS interferogram, repetition-rate and carrier-envelope offset frequencies, we now demonstrate absolute optical wavelength calibration in a measurement of the mid-infrared dual-comb spectroscopy of methane gas using two free-running independent oscillators with a repetition-rate difference of 500 Hz. A comparison with the HITRAN database [30] shows that a wavelength accuracy of 0.27 nm is achieved after averaging 23 calibrated spectra spanning a large portion of the P, Q and R branches of the methane mid-IR absorption spectrum.

Optical parametric oscillators and performance characterization
The pump lasers for the OPOs were two all-normal dispersion Yb:fiber oscillators [31], identical in their design but operated with slightly different repetition rates. The highly chirped pulses, which are naturally output-coupled from such lasers, are suitable for direct amplification without further dispersive broadening and were used to seed two identical cladding-pumped Yb:fiber amplifiers. Each amplifier channel produced chirped 3-ps pulses with average powers of 2.3 W, center wavelengths of 1055 nm and bandwidths >20 nm. One fiber collimator in each cavity was mounted on a piezo-electric transducer (PZT) to allow the cavity length to be finely adjusted.
Each OPO incorporated a 20-mm-long MgO:PPLN crystal with a grating period of 30.49 µm and followed a previously described design which achieved broadband parametric transfer of the pump bandwidth to the idler pulses [24]. The experimentally measured signal spectra from each OPO (Fig. 1a) showed a 4-nm spectral bandwidth and the idler spectra (Fig. 1b) had a bandwidth of 160 nm centered around 3.35 µm, corresponding to a frequency bandwidth of 4.27 THz, close to the original pump bandwidth as is expected for the phasematching geometry used [24]. When pumped with up to 1.8 W of 1055-nm pulses the OPO exhibited slope efficiencies of 39.5% and 6.7% (Fig. 1c) for the signal and idler channels respectively.

Dual-comb spectroscopy configuration
The integration of the pump lasers, nonlinear interferometry and OPOs into a complete dualcomb spectroscopy system is shown in Fig. 2. Firstly, the idler fCEO for each channel was obtained from separate nonlinear interferometers which heterodyned the pump supercontinuum (pSC) with sum-frequency mixing (SFM) light between the pump and idler (p + i) [11]. These were implemented using a beamsplitter (BS1) to direct 30% of the pump light for each OPO to a grating compressor and into a photonic crystal fiber (PCF) to produce supercontinuum light at 800 nm. The spatially overlapped idler and undepleted pump beams exiting the OPO were focused by a concave silver mirror (Ag) into a 2-mm-long PPLN crystal with a grating period of 22 µm to generate SFM light with an average power of 1 mW. A dichroic mirror was used to separate the idler from the SFM light at 800 nm. The SFM light was combined with the pump supercontinuum and the heterodyne mixing signal detected by an avalanche photodiode (APD). A 10-nm bandwidth interference filter (IF) was used before the APD to improve the signal:noise level of the heterodyne frequency. By adjusting the time delay between the SFM and supercontinuum pulses, two signals were obtained containing the idler CEO frequencies of both channels independently.
The laser repetition frequencies fREP1 and fREP2 were sampled from the other output of the beamsplitter cube in the nonlinear interferometers. In order to obtain high resolution, the tenth harmonics of the repetition frequencies were detected following a 50-MHz-bandwidth RF filter centered at 1 GHz. Mixing these signals provided a third signal at the tenth harmonic of the repetition-frequency difference (10ΔfREP), necessary for accurate calibration. The absolute repetition frequency of Channel 1 was acquired with a precision frequency counter (Hameg, HM8123).
The DCS configuration was a heterodyne scheme, in which idler pulses from one OPO passed through the gas cell before being combined with those from the other OPO at a 50:50 CaF2 beamsplitter. The gas cell was 20 cm long and contained a 1 atm mixture of N2 with <1% of CH4. A half-wave plate in one idler channel was used to adjust the power balance between the channels before detection. The idler pulses were detected by two identical cryogenically cooled HgCdZnTe detectors with a bandwidth of 100 MHz (Kolmar KMPV11-0.25-J1). Using two detectors allowed us to record the DCS interferogram signal (SDCS) and its antiphase replica (S'DCS). By differencing these signals we significantly reduced common-mode noise present on the two channels. The typical idler power needed was <10 mW per channel. After Nyquist filtering to < 50 MHz, the DCS interferogram signals were sampled at 12-bit resolution at a rate of 200 MSa/s, together with the signals 10ΔfREP (= 10fREP1 -10fREP2), fCEO1 and fCEO2.

Signal handling and wavelength calibration
An overview of the signal handling and wavelength calibration process is presented in Fig.  3. Data acquisition constraints limited the number of dual-comb interferograms which could be consecutively acquired to twenty-three. These were individually windowed, along with the co-acquired fREP, fCEO and ΔfREP calibration signals, from which a unique look-up table was constructed for each spectrum. As observed in [15], residual instability in ΔfREP at the level of a few hundred mHz leads to nm-scale fluctuations in the center wavelength of each spectrum (see Appendix A), which if left uncorrected prevents spectral averaging. A full-spectrum crosscorrelation approach (see Appendix B) was used to co-align the spectra with better than 0.1-nm precision before averaging was implemented to yield the final spectrum.

Look-up table construction
The digitized signals were processed in Matlab and a low-noise DCS interferogram obtained by differencing SDCS and S'DCS, after first balancing their powers. Power balancing was implemented numerically by using the Hilbert transform to extract the envelope of each interferogram. The interferograms were normalized using their respective envelopes before subtracting one from the other. By rejecting common-mode noise, this procedure improved the signal:noise ratio of the data by approximately a factor of three and enhanced the vertical symmetry of the resulting interferogram. The resulting interferogram was Fourier transformed and calibrated on an absolute wavelength scale in the following way.
Noting that the beat frequencies from the nonlinear interferometers may either be fCEO or fREP -fCEO, the true offset frequencies for each comb were first determined. The wavelength range of interest was defined and then used to obtain the corresponding range of mode numbers Fig. 3. Signal handling and wavelength calibration process. Five signals are acquired simultaneously: two carrier-envelope offset frequencies (fCEO1, fCEO2), a signal providing the repetition-rate difference (10DfREP) and two anti-phase dual-comb interferograms (SDCS and S'DCS). The first three signals are used to construct a look-up table providing a direct mapping from interferogram carrier frequency to wavelength, used to initially align the dual-comb spectra, following which precise co-alignment is applied, followed by averaging to produce the final spectrum.
for Comb 1: A wavelength range from 3.2 μm (λMIN) to 3.4 μm (λMAX) was selected, corresponding to mode numbers from 864437 to 918464. Except for very small values of ΔfREP, interference occurs between comb teeth of different mode numbers. In practice, we chose ΔfREP to be sufficiently high to provide a wide range of radio frequencies while still being low enough to avoid aliasing [2].
The mode-number difference was calculated as: with the bracket notation representing rounding down to the nearest integer. Thus, for mode n 1 in Comb 1, the nearest mode in Comb 2 is n 2 = n 1 + Δn. Modes n 1 in Comb 1 and n 2 in Comb 2 interfere to produce a unique frequency in the DCS interferogram: where f 1 = n 1 f R EP1 + f CEO1 and f 2 = n 2 f R EP2 + f CEO2 . Only the correct assignment for each channel of f CEO to either the measured signal or f R EP − f CEO (amounting to testing four different options) yields a value for f DC which matches the experimental data. Once the correct assignments have been identified these are applicable to the entire set of interferograms and persist indefinitely.
The instantaneous frequencies of the 10Δ f R EP , f CEO1 and f CEO2 signals were obtained directly in the time domain using a zero-crossings algorithm to produce an average for each frequency within the time windows corresponding to the individual interferograms (~100 μs).
After Fourier transforming, these frequencies were used along with f R EP1 to create a look-up table which uniquely mapped f DC to the wavelength of Comb 1, λ = c/ f 1 . Using this lookup table, each spectrum was individually calibrated in wavelength and stored for subsequent co-alignment to the other spectra prior to averaging the entire dataset.

Co-alignment and averaging of calibrated spectra
Using the tenth harmonic of the repetition frequency difference allows Δ f R EP to be determined with better accuracy, however short-term excursions of Δ f R EP introduce wavelength shifts in the calibrated spectra of up to a few nm. The total recording time for the dual-comb data was 50 ms, corresponding to a Nyquist-limited resolution of 20 Hz, but providing Δ f R EP to a nominal 2-Hz resolution. This is the origin of the uncertainty which is responsible for the deviations in the wavelength calibrations of the different spectra which can be seen in Fig. 4, in which the spectra of 23 consecutively acquired interferograms are shown as an intensity map.
Dealing with this problem by using co-alignment to an independently measured narrow spectral feature was proposed in the original demonstration of mid-IR DCS [15] and was recently implemented in an OPO DCS system [32]. In fact, an independent reference is not necessary for accurate co-alignment. Instead, we achieved this by computing the misalignment between each spectrum and every other spectrum from their cross-correlations, amounting to 23 × 23 comparisons, when self-comparisons are counted. Such a correlation method [33] is general, very reliable and automatically returns the mean location around which the spectra are evenly shifted. The shifts identified in this way are shown in Fig. 5 for the entire dataset and exhibit a standard deviation in wavelength of 2.4 nm, or 359 mHz in the repetition-frequency difference. The mapping from the RF to optical domain involves a scaling factor of f R EP /Δ f R EP , so uncertainties in Δ f R EP dominate the calibration, since f R EP is passively stable to Hz levels (10 −8 instability) on the timescale of the DCS acquisition. The error in calculating the relative shifts is estimated to be less than 0.1 nm; for clarity no corresponding error bars have been shown in Fig. 5.

Spectroscopy results and discussion
The average of 23 calibrated spectra from consecutively acquired DC interferograms is shown in Fig. 6 (top, black line) superposed on a representative spectrum acquired from a single interferogram (top, gray). The inset shows visually the improvement in the signal-to-noise ratio (SNR) provided by averaging multiple spectra. An estimate of the improvement in the SNR is illustrated by the data shown in the inset of Fig 6. Before averaging, the signal to noise (defined here as the inverse of the standard deviation of the spectral noise in a region without molecular absorption) is 10.7 dB. Incoherent averaging of 23 spectra acquired in 100 ms should provide a factor of 4.80 ( √ 23) improvement, and in the example presented here the SNR in the region between the absorption lines increases by 4.66 to 17.4 dB, close to the expected value. Decoherence between the OPO combs on a timescale of the interferogram time window is responsible for the line broadening that occurs as a result of the averaging procedure. This could be mitigated by operating the OPO at a higher repetition-rate difference, so that the interferograms were recorded in a shorter time. Aliasing considerations [2] constrain ∆ f R EP to ≤ 1 kHz in our system if the full spectral bandwidth of the OPOs is exploited, however the introduction of suitable optical bandpass filtering would allow this limit to be circumvented. Figure 6 (lower panel) shows the result of implementing a best fit using the HITRAN 2012 database to the measured transmittance data. The best fit was obtained for a resolution of 0.5 cm −1 , a methane concentration of 0.68% and a wavenumber (wavelength) shift of 0.25 cm −1 (0.27 nm). This shift reveals the accuracy limit of the calibration method as implemented, but which could be improved in a number of ways. Averaging more spectra would reduce the noise and improve the absolute accuracy further, however data acquisition constraints placed a limit on the number of interferograms which can be acquired in our current implementation. The standard deviation observed in the set of individually calibrated spectra would be reduced by recording a higher harmonic of ∆ f R EP . In this demonstration we acquired 10∆ f R EP , however this could be extended significantly further by heterodyning higher harmonics of the repetition frequency using high-speed detectors, or by directly recording a high harmonic of ∆ f R EP by mixing the pump combs on a high-speed two-photon-absorption photodiode.
The HITRAN comparison shows that the experimental transmittance spectrum clearly resolves the P and R branches of the ro-vibrational spectrum, as well as some of the structure associated with the Q branch (near 3.32 µm). Saturation effects in the Q-branch together with uncertainties in the OPO spectral profile before the gas cell may be responsible for the deviation between the experimental and HITRAN data in this region. Some spectra in the dataset also exhibited weaker Q-branch absorption features but their inclusion contributes to reducing the strength of the Q-branch in the averaged spectrum. In the analysis presented here, no such outliers were removed, however an automated process could be conceived that would allow such spectra to be identified and eliminated. Fig. 6. Top: Average of 23 calibrated spectra after re-centering on the mean wavelength (black) and comparison with a single representative spectrum (gray). The inset illustrates the 4.66× improvement in RMS noise between two adjacent absorption features, close to the expected factor of 4.80 ( √ 23). Bottom: Comparison of the measured transmittance (blue) with a HITRAN 2012 best fit for 0.68% methane in nitrogen for a 20-cm path length and a resolution of 0.5 cm −1 (red). An offset of 0.25 cm −1 (0.27 nm) was applied to the data to obtain the best fit, corresponding to the residual inaccuracy of the absolute wavelength calibration.

Summary and conclusions
We have introduced here an f CEO -based wavelength calibration technique for mid-IR DCS, providing a 17-dB signal:noise ratio, broad bandwidth (>60 nm) and 0.5-cm −1 resolution sufficient to resolve gas-phase ro-vibrational absorption lines. Noise reduction using averaging of multiple interferograms and balanced detection provided a high quality DCS spectrum. The calibration approach benefits from operating at a high repetition frequency difference. Doing this distributes the wavelength scale across a wider range of frequencies, so fluctuations in f CEO , which only contribute additively to the interferogram frequency, have less impact. The absolute calibration accuracy demonstrated was 0.25 cm −1 (0.27 nm), and this was achieved without any optical references. It should be possible to further improve the calibration accuracy by recording a very high harmonic of ∆ f R EP , either by heterodyning high harmonics of f R EP1 and f R EP2 or by other techniques. One can also consider the potential to improve the calibration accuracy by recording and averaging a larger number of interferograms. The spectra used to construct Fig. 5 showed a standard deviation in the population of 2.4 nm, corresponding to a standard deviation in the mean of 0.51 nm (2.4 nm / √ 22), meaning that statistically the observed average deviation (i.e. 0.27 nm) should be less than this value. We can conclude that to improve the current calibration accuracy to better than 0.1 nm it would be necessary to record 573 spectra. With a suitable data acquisition system this would possible in a time of 1.15 seconds.
A major advantage of femtosecond OPOs over lasers for DCS is their intrinsically broad wavelength coverage. Recently we introduced a femtosecond OPO based on the new material orientation-patterned gallium phosphide (OP-GaP), which permits a 1-μm Yb-based femtosecond laser to be directly extended to the 5-13-μm molecular fingerprint spectroscopy region [34]. In such a DCS embodiment it would be possible to gain direct access to the idler (i) carrier-envelope offset frequency ( f i CEO ) by heterodyning the OPO signal pulses with spectrally broadened pump pulses. Signal (s) wavelengths from an Yb:fiber-pumped OP-GaP OPO are in the region of 1.15-1.35 μm, so only a moderate amount of nonlinear broadening of the pump (p) pulses would be needed to achieve sufficient spectral overlap. The beat signal obtained would contain a frequency of f p CEO − f s CEO , which is equivalent to f i CEO , allowing this frequency to be recorded without the need for access to pump-idler sum-frequency mixing light, as was necessary in the present implementation. In this way, fingerprint-region dual-comb spectroscopy could be implemented and traceably referenced to atomically-disciplined microwave oscillators without the need for auxiliary optical references in the long-wave infrared region. We note that difference-frequencygeneration routes to long-wave femtosecond pulses may be equally attractive [35].

Appendix A: Dependence of the wavelength calibration on ∆ f R EP
In §4.2 and Fig. 5 we presented data showing the distribution of the center wavelengths of a set of transmission spectra, each of which had been independently calibrated from one of 23 consecutively acquired interferograms and their corresponding f CEO and ∆ f R EP signals. The wavelength shifts were attributed to changes in ∆ f R EP which are unresolvable at the resolution of our current calibration system (see discussion in §5 for proposed improvements addressing this issue). The mapping between changes in wavelength λ and ∆ f R EP can be calculated in the following way.
In dual-comb spectroscopy a reduction factor of ∆ f R EP / f R EP relates an interval in optical frequency δ f OPT to an interval in the radio frequency δ f DC . This allows an interval in radio frequency to be expressed as an interval in wavelength: The comb-mode equations ( §4) show that an error in the repetition-frequency difference of δ(∆ f R EP ) leads to an error in the interferogram carrier frequency of δ f DC = nδ(∆ f R EP ). Thus for a mode number of n = 906, 300 corresponding to the Q branch (λ = 3.31 µm) we find that a wavelength error of δλ = 2.40 nm implies an error in the RF frequency of δ f DC = 0.359 Hz when ∆ f R EP = 500 Hz.

Appendix B: Co-alignment of spectra using cross-correlation
For two similar spectra, I 1 (ν) and I 2 (ν) the location of the maximum of their cross-correlation provides their relative shift [33]: δν = argma x ν [I 1 (ν) * I 2 (−ν)] (5) Figure 7 illustrates a set of mutually shifted spectra (red, green and blue). Cross-correlation between any pair of spectra yields the magnitude and direction of their relative shift, allowing a table of mutual shifts to be formed, which in the example amounts to 3 × 3 values, including self-comparisons. The illustration in Fig. 7 shows three spectra separated from each other by shifts of between two and ten elements. The shift of each spectrum relative to the average position can be computed by summing its mutual shifts and normalizing the resulting values by the number of spectra. Fig. 7. For a set of mutually shifted spectra (red, green and blue) a table of relative shifts can be formed from mutual cross-correlations, whose columns can be summed and normalized to yield the relative shifts of each spectrum from the mean position (gray dashed). Individual spectra are then co-aligned with the spectrum closest to the average position (green) and averaged. Finally, the average spectrum is shifted by the required amount (2) to align it with the average position.
In practice we used cross-correlations to compute a 23 × 23 table of shifts for the calibrated DCS spectra, allowing us to find the shift of each spectrum from the mean position. We then co-aligned all of the spectra to the member of the set which had the lowest deviation from the average position and performed an average. Finally, the average spectrum was shifted by the remaining amount so that it coincided with the exact average position. The result of this procedure led to a residual inaccuracy of 0.27 nm (0.25 cm −1 ) in the position of the spectrum when compared with the HITRAN 2012 database.