1. Introduction

Frequency combs have revolutionized the fields of spectroscopy and optical frequency metrology over the past few decades due to their long coherence times, high precision, and ability to create a direct link between optical and microwave frequencies [1]. This impact has been felt in a diversity of areas from attosecond science to tests of fundamental physics to even exoplanetary science [2]. For spectroscopy in particular, the unique combination of high precision and fast acquisition time enabled by dual-comb spectroscopy (DCS) has challenged Fourier Transform Infrared Spectroscopy (FTIR) - the longtime gold standard of high-resolution broadband spectroscopy [3]. DCS has proven to be highly proficient at identifying and measuring the concentrations of molecules, which is desirable for a variety of applications such as greenhouse gas sensing, toxin detection, and biomedical tissue analysis [4]. To succeed in these endeavors, it is imperative to be able to accurately and precisely measure the properties (i.e. line center, width, and amplitude) of some spectral feature. Of course the speed and accuracy of this task benefit from a higher signal-to-noise ratio (SNR); the understanding and improvement of which have been the focus of many previous works [5–9]. However, the SNR is not the only significant parameter – as we show in this work, the repetition rates of the two combs have a tremendous impact on the error and uncertainty of the ascertained line properties assuming all else is equal. This impact is greatest when allowing for the selection of arbitrary-harmonic related repetition rates.

In the conventional implementation of DCS, one comb (referred to as the signal comb with a repetition rate of f_Sig) is transmitted through a medium of interest while the second comb (referred to as the local oscillator (LO) or “read-out” comb with a repetition rate of f_LO) bypasses the medium and interferes with the signal comb on a detector. The repetition rates of the two combs are slightly different so that their ratio (ρ_r ≡ f_LO : f_Sig) is very close to unity. In the frequency-domain, this ratio means that the beat notes between any two signal and LO comb lines seen on the detector occur at unique, evenly spaced frequencies thereby forming a radio-frequency (rf) comb. This one-to-one mapping links the measured rf comb lines and the optical comb lines that generated them. Because the amplitude and phase of each individual signal comb line can be measured, the spectral line is sampled with a spectral point spacing, δν, equal to f_Sig. In the time-domain, the use of nearly identical repetition rates implies that the time delay between sequential pairs of signal and LO pulses increases by a small, very precise amount. This configuration is analogous to FTIR in the sense that the relative time-delay between two pulses is scanned so that they sample one another at discrete points and indeed the recorded interferograms appear very similar. However, by taking advantage of the high precision of combs and avoiding the use of a mechanical delay stage, DCS has significantly better spectral resolution, compactness, robustness, and acquisition times.

Although the repetition rates of the two combs used in the vast majority of conventional DCS setups possess a near-unity ratio, in principle any near-harmonic (i.e. near-multiple) relationship between the repetition rates is acceptable. The concept of heterodyning two periodic optical waveforms with arbitrary near-harmonic repetition rates has been used to characterize the envelope [10] and more recently the envelope and spectral phase [11, 12] of an unknown waveform. The latter two techniques are quite similar to DCS in the way that they retrieve optical information from the rf signal by leveraging the one-to-one mapping between a pair of signal and LO comb lines and their beat note on a detector. The use of near-harmonic repetition rate combs has also been applied to DCS – but rather than directly using the rf-to-optical mapping, the authors viewed the measured rf signal as a temporal-multiplexing of independent traditional DCS interferograms then demultiplexed and combined them [13]. Another method that utilizes two pulse trains actually allows for any ρ_r although its sub-picosecond time accuracy is insufficient for DCS [14].

An example of how the optical information is mixed into the rf domain on a detector for ρ_r near 2:1 is shown in Fig. 1. From Fig. 1 it is clear that the maximum optical bandwidth, Δν_max, that can be used without ambiguity is

 

Fig. 1 (a) Optical spectra of a signal comb (showing an absorption dip) and LO comb with equal average powers and a repetition rate ratio ρ_r ≈ 2 : 1. Every other signal comb line is dotted to illustrate where the rf comb lines in (b) originate. (b) The rf comb seen on a detector in an electrical bandwidth of f_LO/2. A pair of gray lines illustrates the optical-to-rf mapping for a solid (dotted) rf comb line which in this specific example occur at even (odd) multiples of δf (k and k′ are integers). Note that each “interferogram” in the time-domain will contain two unique bursts (corresponding to the solid and dotted rf combs) because the LO pulse train sweeps through a given signal pulse twice during each τ_min ≡ δf⁻¹.

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2. Simulation

The impact of various experimental parameters on the determination of the properties (i.e. line center, width, and amplitude) of some spectral feature can be quantified using a Monte Carlo method. Many randomly generated data sets with appropriate SNRs are each fit using a regression approach to extract the properties of the spectral feature. The errors between these measured values and the ones used to generate the data are then analyzed as a function of the experimental parameter to reveal its influence.

A number of approximations and assumptions have been made to simplify our analyses. First, we only consider a sample medium that possesses a single Lorentzian resonance and only analyze the absorption due to this resonance i.e. phase information is neglected. Second, we approximate the spectral envelopes of the signal and LO pulse trains as “top hats” whose width, Δν, is less than Δν_max. In Sec. 2.1 the amplitude of the top hat is such that the average power of each beam is fixed while in Sec. 2.2 it is irrelevant. Third, the response function of the detector is assumed to be flat and the rf comb (which we assume lines up perfectly with the measurement frequency grid) is sampled at a rate of f_LO. Fourth, we assume that the offset frequency and repetition rate noise of the combs is negligible and that the dominant noise in the measurement may be modeled as additive complex white Gaussian noise, N(ν). Fifth, we assume that all systematic errors (e.g. baseline issues from etalon effects) are negligible.

These assumptions allow us to write the normalized, dimensionless, noise-free, optical-domain signal as

2.1. Varying the repetition rates

For a fair comparison of repetition rates to be used in DCS, all other experimental parameters are fixed. In this case, Eq. (6) implies that the SNR scales with the repetition rates as

 

Fig. 2 SNR and corresponding standard deviations of the errors in the fitted center frequency (σ_c), HWHM (σ_γ), and strength (σ_s) as functions of the repetition rates of the signal and LO combs (which have units of γ, the HWHM of the resonance). Conventional (ρ_r ≈ 1) DCS is the middle diagonal line on each plot while the first line above (below) the diagonal corresponds to ${\rho }_{\mathrm{r}}\approx 2({\rho }_{\mathrm{r}}^{-1}\approx 2)$ and so forth. The space between neighboring lines shrinks exponentially due to the logarithmic axes.

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In DCS the spectral point spacing, δν, is usually equal to the signal comb repetition rate, f_Sig. However, it is possible to achieve the same spectral point spacing with a signal comb repetition rate that is m times greater (where m is an integer) by using the spectral interleaving technique in which a series of m separate spectra are collected with each one having shifted signal comb lines [17,18]. According to Eqs. (6) and (7) the SNR in the final interleaved spectrum is identical to one acquired without interleaving because the increase in the signal comb repetition rate by a factor of m (increasing the SNR by a factor $\sqrt{m}$) is counteracted by the reduction in acquisition time allowed for each measurement by a factor of m (decreasing the SNR by a factor $1/\sqrt{m}$) assuming all other parameters are the same. Thus, the results in Fig. 2 also apply to systems using the spectral interleaving technique but with the understanding that the horizontal axis should be interpreted as the final spectral point spacing, δν, which is f_Sig/m.

The core result of this work is that the repetition rates of the signal and LO combs greatly impact the error and uncertainty in the measured parameters of a Lorentzian resonance which is clearly shown in Fig. 2. When all other experimental parameters are fixed, it is ideal to have f_Sig ≈ f_LO ≈ γ when using the traditional implementation of DCS in which ρ_r ≈ 1. However, by allowing the repetition rates to have a near-harmonic relationship, the uncertainty in the fit parameters can be orders of magnitude lower (assuming f_Sig ⪅ γ) resulting from an SNR that scales with $\sqrt{{\rho }_{\mathrm{r}}}$ (see Eq. 6 and the following paragraph). In this case, it is ideal to have any f_Sig ⪅ γ in addition to the highest near-harmonic LO repetition rate that is feasible. Of course there is a lower limit set on the signal comb repetition rate that corresponds to the SNR approaching 1. Note that it is never beneficial to use a near-subharmonic LO comb as can be seen in Fig. 2 – this would imply that many signal pulses are never sampled, wasting potential data. Although any signal comb repetition rate below the HWHM of the resonance should produce nearly the same fitting error for a given f_LO (see Fig. 2), there may be practical reasons to choose one f_Sig over another. For example, the higher the signal comb repetition rate, the less congested the rf comb will be because there are fewer comb teeth in a given electrical bandwidth. On the other hand, the higher the signal comb repetition rate, the less versatile that system will be in the sense that fewer samples will meet the f_Sig ⪅ γ requirement.

It is important to note that these results apply strictly to linear measurements because Eq. (3) does not accurately describe nonlinear signals in general. For example, nonlinear interactions typically generate new comb lines rather than attenuate existing ones [19–23]. Indeed one nonlinear experiment demonstrated that the SNR actually decreases with increasing f_Sig (fixed laser output powers), which would imply that lower signal comb repetition rates are always better for nonlinear measurements due to the higher energy per pulse [24]. Of course there are other techniques that can affect the SNR even for linear measurements. For instance, the GATOR technique can be used, which would favor a slower LO comb [25].

2.2. Varying the SNR and spectral point spacing

The results of Sec. 2.1 isolate the advantage stemming from the use of near-harmonically related repetition rates. However, most practical DCS systems have sufficiently different laser powers, detector efficiencies, types of dominant noise, etc., such that a more general simulation approach is necessary. For example, DCS has recently been performed with microresonator combs [26], but because these combs typically output only milliwatts of power [27], it is inappropriate to use the results from Fig. 2 to compare their ability to ascertain the properties of a spectral line to slower oscillators that have orders of magnitude more output power. To generalize our results, the simulations were performed again but instead of varying the repetition rates, the independent parameters selected were the SNR and the spectral point spacing, δν. In all other respects the simulation process was identical to that described in Sec. 2.1. The results are displayed in Fig. 3.

 

Fig. 3 Standard deviation of the errors in the fitted center frequency (σ_c), HWHM (σ_γ), and strength (σ_s) as functions of the SNR and spectral point spacing, δν. The dashed white lines are gradients to help illustrate the relationship in Eq. (9).

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Examining the left-hand side of each plot in Fig. 3 reveals that as long as δν ⪅ γ and the SNR is sufficiently high, then

All of the plots in Figs. 2 and 3 show that the error rapidly climbs for δν ⪆ 3γ. This observation corroborates one’s intuition because zero (or only one) signal comb lines are “hitting the resonance.” This condition means that the vast majority of the fitting involves the wings of the Lorentzian which are inherently insensitive to changes in ν_c, γ, and d (i.e. a large change in one of the parameters results in a minuscule change in data points far from line center) due to the exponential behavior of the wings. This lack of sensitive data points quickly leads to huge uncertainty in the fitted parameters despite the fact that the SNR monotonically increases with f_Sig (and therefore δν).

To verify the comprehensiveness of the results in Figs. 2 and 3 several series of simulations were performed. First, the overall patterns in the error plots remained the same as the SNR was scaled up and down over several orders of magnitude. Second, the number of data points over which the fitting was performed was varied between a fixed number of data points, a fixed frequency range of data points, and a hybrid approach. There was no significant effect on our results. Third, the strength of the resonance was varied. For sufficiently small strengths (s ⪅ 5) the general patterns in our results were unaffected. However, for larger strengths the absorption becomes saturated and the comb lines containing the most critical information (those near line center) have drastically reduced signal-to-noise ratios. This situation leads to significantly less accurate fits overall.

3. Conclusion

We have isolated the impact of the DCS comb repetition rates on the standard deviation of the error, σ, of the measured parameters of a single Lorentzian resonance (assuming all else is equal, i.e. lasers powers, acquisition time, etc.) and found that there is a maximum suitable spectral point spacing, δν ≈ γ, where γ is the HWHM of the resonance. Because of this, the repetition rates that yield the smallest errors in the fit parameters do not possess the highest SNR. For the traditional implementation of DCS the ideal repetition rates are f_Sig ≈ f_LO ≈ γ but even more accurate measurements can be made by allowing for arbitrary near-harmonic repetition rate ratios. In this case the ideal repetition rates are f_Sig ⪅ γ and f_LO → ∞. In general, $\sigma \propto 1/\sqrt{f_{\text{LO}}}$ (with zero f_Sig dependence) for f_Sig ⪅ γ. These are timely results given the recent advances in high-repetition-rate microresonator combs [26] and external repetition rate multiplying techniques [9, 29–31]. Since utilizing near-harmonic repetition rates requires only a slight modification to the rf-to-optical mapping process, any application of DCS where measuring the parameters of a spectral line is a top priority should readily receive higher accuracy and lower uncertainty by switching from similar to near-harmonic repetition rates assuming other experimental parameters remain more or less unchanged.

In practice, spectroscopic systems may differ wildly in everything from the type of dominant noise to the maximum allowable acquisition time. By running our simulations as a function of the SNR and spectral point spacing, δν, we identified a general fitting result, $\sigma \propto \sqrt{\delta \nu }/\text{SNR}$, which can be used to compare any two spectroscopic systems despite their differences. In fact, this relationship can be used to compare even non-comb-based systems as long as the noise is predominately white and Gaussian distributed. Such a metric shows that while the SNR is important, efforts to reduce the spectral point spacing should also be made.