Sensitivity of coherent dual-comb spectroscopy

Coherent dual comb spectroscopy can provide high-resolution, high-accuracy measurements of a sample response in both magnitude and phase. We discuss the achievable signal-to-noise ratio (SNR) due to both additive white noise and multiplicative noise, and the corresponding sensitivity limit for trace gas detection. We show that sequential acquisition of the overall spectrum through a tunable filter, or parallel acquisition of the overall spectrum through a detector array, can significantly improve the SNR under some circumstances. We identify a useful figure of merit as the quality factor, equal to the product of the SNR, normalized by the square root of the acquisition time, and the number of resolved frequency elements. For a single detector and fiber-laser based system, this quality factor is 10 10 Hz. Work of the U.S. government, not subject to copyright. OCIS codes: (300.6320) Spectroscopy, heterodyne; (300.6300) Spectroscopy, Fourier transforms References and links 1. F. Keilmann, C. Gohle, and R. Holzwarth, “Time-domain mid-infrared frequency-comb spectrometer”, Opt. Lett. 29, 1542–1544 (2004). 2. A. Schliesser, M. Brehm, F. Keilmann, and D. van der Weide, “Frequency-comb infrared spectrometer for rapid, remote chemical sensing”, Opt. Express 13, 9029–9038 (2005). 3. T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, and T. Araki, “Terahertz frequency comb by multifrequencyheterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy”, Appl. Phys. Lett. 88, 241104 (2006). 4. P. Giaccari, J. D. Deschenes, P. Saucier, J. Genest, and P. Tremblay, “Active fourier-transform spectroscopy combining the direct rf beating of two fiber-based mode-locked lasers with a novel referencing method”, Opt. Express 16, 4347–4365 (2008). 5. B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hansch, and N. Picque, “Cavity-enhanced dual-comb spectroscopy”, Nat. Photon. 4, 55–57 (2009). 6. I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs”, Phys. Rev. Lett. 100, 013902 (2008). 7. I. Coddington, W. Swann, and N. Newbury, “Time-domain spectroscopy of molecular free-induction decay in the infrared”, Opt. Lett., accepted (2010). 8. I. Coddington, W. Swann, and N. Newbury, “Coherent dual-comb spectroscopy at high signal to noise”, http://arxiv.org/abs/1001.3865 (2010). 9. R. J. Bell, Introductory Fourier transform spectroscopy (Academic Press, 1972). 10. J. Chamberlain, The Principles of Interferometric Spectroscopy (John Wiley and Sons, Inc, 1979). 11. J. R. Birch, “Dispersive fourier-transform spectroscopy”, Mikrochimica Acta 3, 105–122 (1987). 12. N. Almoayed and M. Afsar, “High-resolution absorption coefficient and refractive index spectra of carbon monoxide gas at millimeter and submillimeter wave-lengths”, IEEE T. Instrum. Meas. 55, 1033–1037 (2006). 13. S. Schiller, “Spectrometry with frequency combs”, Opt. Lett. 27, 766–768 (2002). 14. J. W. Brault, High Resolution in Astronomy (Geneva Observatory, 1985), Fourier transform spectrometry, pp. 1– 65. 15. L. A. Sromovsky, “Radiometric errors in complex fourier transform spectrometry”, Appl. Opt. 42, 1779–1787 (2003). 16. S. P. Davis, M. C. Abrams, and J. W. Brault, Fourier Transform Spectrometry (Academic Press, 2001). 17. V. V. Protopopov, Laser Heterodyning (Springer Berlin / Heidelberg, 2009). 18. W. Demtroder, Laser Spectroscopy (Springer, 1996), 2nd ed. 19. I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent linear optical sampling at 15 bits of resolution”, Opt. Lett. 34, 2153–2155 (2009). #123518 $15.00 USD Received 29 Jan 2010; revised 12 Mar 2010; accepted 15 Mar 2010; published 31 Mar 2010 (C) 2010 OSA 12 April 2010 / Vol. 18, No. 8 / OPTICS EXPRESS 7929 20. N. R. Newbury and W. C. Swann, “Low-noise fiber-laser frequency combs (invited)”, J. Opt. Soc. Am. B 24, 1756–1770 (2007). 21. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber”, Rev. Mod. Phys. 78, 1135–1184 (2006). 22. N. R. Newbury, B. R. Washburn, K. L. Corwin, and R. S. Windeler, “Noise amplification during supercontinuum generation in microstructure fiber”, Opt. Lett. 28, 944–946 (2002). 23. K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, “Fundamental noise limitations to supercontinuum generation in microstructure fiber”, Phys. Rev. Lett. 90, 113904 (2003). 24. T. W. Hänsch, “Nobel lecture: Passion for precision”, Rev. Mod. Phys. 78, 1297–1309 (2006). 25. J. L. Hall, “Nobel lecture: Defining and measuring optical frequencies”, Rev. Mod. Phys. 78, 1279–1295 (2006).


Introduction
There have been a number of demonstrations of coherent dual comb spectroscopy [1][2][3][4][5][6][7][8] providing broadband spectroscopic data often with very high frequency accuracy and resolution. In the configuration considered here, and demonstrated fully in Ref. [6][7][8], the output of one comb is transmitted through a sample and then combined with a second "local oscillator" (LO) comb source to measure the full complex response of a sample (i.e., phase and amplitude) with the frequency resolution and accuracy inherent to the comb source. In the time domain, the approach mirrors time-domain spectroscopy [7] or dispersive Fourier transform spectroscopy (FTS) [9][10][11][12]. In the frequency domain, the approach resembles a massively parallel laser heterodyne spectrometer, where each pair of comb teeth contributes to the heterodyne signal at a specific rf frequency [13]. Although this technique can provide the complex response (phase and amplitude) with high frequency accuracy and resolution, the achievable signal-to-noise ratio (SNR) is limited by the usual factors, namely detector noise, shot-noise, technical laser noise and detection dynamic range. A low SNR can, of course, diminish many of the benefits of the approach. For example, the ability to find the line center of broad spectral lines will have a statistical limit set by the SNR that can exceed the inherent comb accuracy. Here we derive scaling laws and quantitative SNR expressions and examine the sensitivity of this technique. These expressions will be useful in system design, in evaluating the system performance, and in comparisons to more conventional approaches.
We derive three main equations. First, we give the limiting SNR due to detector noise, shot noise, excess laser relative intensity noise (RIN), and detector dynamic range. We point out the potential advantages of sequential or parallel acquisition across the full spectrum. We also compare with a tunable laser spectrometer. Second, we give the corresponding sensitivity limit for trace gas detection. Third, we give the SNR limitations due to multiplicative noise, in particular due to residual carrier phase noise between the combs. Under the appropriate operating parameters and with either active phase-locking [6][7][8] or active monitoring [4] to cancel relative phase and timing jitter between the combs, this multiplicative noise is shown to be negligible. The derivations are straightforward and the final scaling laws follow closely related expressions in conventional FTS [14][15][16] and laser spectroscopy [17,18]. Therefore, in Section 2, we present and discuss the basic results and only later in Section 3 do we give a detailed derivation.

Configuration of dual-comb spectrometer
Coherent dual comb spectroscopy can be viewed as a massively parallel multi-heterodyne measurement between a source comb that passes through a sample and an LO comb with a different repetition rate; each pair of comb teeth from the source and LO combs forms its own RF beat signal, thereby down-converting the optical response to the rf domain. In the time domain, the method is essentially Fourier spectroscopy and involves the acquisition of an interferogram (or cross correlation) between the individual pulses of the source, E s (t), and the LO, E LO (t); the overlap between successive pairs of pulses occurs with an effective time step equal to the difference in pulse periods so that the LO pulses slowly walk through the signal pulses. In this picture, the digitized voltage with no sample present is ( ) as a function of effective time t where ⊗ denotes a convolution and S(t) ~ E LO * (-t) is the sampling function. (A more accurate expression for S(t) that preserves the multi-heterodyne frequency-domain picture is given in Eq. (12) and Ref. [19].) In the presence of a sample with time-domain response H(t), the digitized voltage is ( ) The goal is to isolate the sample response, which is easily done in the frequency domain from the ratio where the tilde represents a Fourier transform and ν is the optical frequency.   Fig. 1. (a) Schematic of dual-comb layout including the possibility of sequential or parallel data acquisition. In the figure, the signal and reference are time-multiplexed [7,8], but the reference could also be acquired in a separate interferometer [6] or sequentially with an empty cell. (b) Simulated time-domain signal versus the effective down-converted time (and not laboratory time). (c) Corresponding frequency-domain sample response over a 10 THz window at 100 MHz resolution for 100,000 M = resolved elements. For illustrative purposes, we assume a sample with 20 equally spaced, Lorentzian transitions 2 GHz wide and a spectral domain SNR of σ H -1 = 100. In the time domain, the corresponding signal is a one-sided interferogram with two components: a centerburst and a trailing free induction decay signal. In the frequency domain, each transition is highly resolved in magnitude and phase. We consider the general configuration shown in Fig. 1, which is motivated by the following consideration. While Fourier spectroscopy can significantly reduce the effects of detector noise (Felgett's advantage), it magnifies the effects of excess source noise and dynamic range limitations [9,10,14,16]. These noise terms are mitigated by breaking up the measured spectrum (Δν) into sub-bands that can be acquired sequentially or in parallel. Parallel acquisition is accomplished through a fixed filter that multiplexes N d spectrally filtered bands to an array of N d detectors. Sequential acquisition is accomplished through a tunable optical bandpass filter in front of each detector that steps through F different filter bands. In all, the total spectral bandwidth is divided into F×N d sub-bands with filtered bandwidth Each sub-band is measured for a period T/F, where T is the total acquisition period. Within each sub-band, the response is calculated according to Eq. (1). The stability of the combs allows one to coherently stitch together the full bandwidth signature from these sub-bands while preserving the frequency resolution and accuracy. In addition to potentially improving the SNR, this spectral division allows for a larger difference in comb repetition rates, Δf r , (while still satisfying Nyquist constraints). Since this difference, Δf r , is exactly the separation between the down-converted rf heterodyne beat signals, a larger value of Δf r translates to a relaxed requirement on the relative comb linewidths. Filtering also helps normalize strong spectral variations in the comb spectra [8]. However, these benefits must be balanced with the fact that spectral division also magnifies multiplicative noise (Section 2.4) and can come with added overhead in time.
We make several assumptions in the analysis. First, we assume negligible differential chirp between the filtered source and LO pulses. Differential chirp between the pulses has the benefit of reducing the dynamic range limitation [4,16] and furthermore would be cancelled in Eq. (1). However, it can also introduce correlations in the phase and timing noise which are not treated here. Second, we assume the optical filter is Gaussian in shape and calculate the average SNR (equal to , we assume a roughly uniform source spectrum. Third, we assume the detection is followed by a hardware or software low-pass filter with bandwidth equal to the Nyquist frequency of f r /2. Fourth, we assume that the difference in comb repetition rates, Δf r , and spectral width of the sub-bands are chosen to satisfy the Nyquist constraint [1][2][3][4][5][6][7][8]. Fifth, we assume that the relative linewidth of the two comb sources is sufficiently controlled to be below Δf r (i.e. we assume negligible "1/f" noise in relative phase and timing jitter between the combs [20]).

Effects of additive noise: detector noise, shot noise, laser RIN & detector dynamic range
In terms of noise, we consider detector noise, shot noise, and excess Relative Intensity Noise (RIN). The detector noise is characterized by an effective Noise Equivalent Power (NEP) that includes any and all additive noise terms up to the digitization of the signal such as dark current, Johnson noise, amplifier noise figure etc. This NEP will vary significantly depending on wavelength band and type of detector. We do assume the heterodyne signal is at high enough frequency to be well away from any "1/f" detection so that the NEP is white with frequency. The shot noise is easily calculated from the comb power, evaluated at the photodetector. The laser RIN can have a component at lower frequencies from pulse-to-pulse variations and a component at all frequencies from additive amplified spontaneous emission (ASE) emitted from the laser or from any subsequent optical amplification. The lowfrequency component can lead to multiplicative noise but we ignore it under the assumption that the interferogram update rate, equal to Δf r , is faster than this low frequency noise. Alternatively, this RIN component could be experimentally suppressed through active feedback or cancelled through a separate measurement. Therefore, we only consider the white, broadband RIN due to additive ASE. Furthermore, we assume the ASE has the same uniform spectral shape as the comb output. Finally, we also include a dynamic range limit to the detection set by the digitizer, amplifier, or, ultimately, the photodetector itself.
By considering only the white, broadband noise sources above, we essentially assume "perfect" source and LO combs, other than the effect of additive ASE. However, there will also be residual timing and carrier phase noise between the combs that can lead to multiplicative noise. In Section 2.4 and 3.4, we consider the multiplicative noise from carrier phase jitter since it will likely dominate over timing jitter. We show there that the multiplicative noise is negligible provided the optical filter bandwidth is sufficiently large and that the phase jitter is sufficiently suppressed through either active feedback or monitoring.
The measurement of ( ) H ν of Eq. (1) yields a set of complex points covering a bandwidth Δν with M resolved spectral elements. Including only the broadband noise contributions, the averaged uncertainty across the spectrum at each spectral element for the magnitude or phase (in radians) is, with the variables given in Table 1. (Eq. (2) is derived in Section 3.2.) The SNR is simply 1 H σ . The first two terms in the sum are detector noise and shot noise. The last term reflects the limits to measuring a pulse set by either the laser RIN or by detection dynamic range. ε accounts for the mismatch between the required resolution and the resolution set by the comb repetition rate. It can be viewed as a duty cycle correction in the time domain. It is theoretically absent in FTS because the scan length would be matched to the desired resolution (although even then, conventional FTS would still have a similar term due to the turn-around time of the mirror.) Note that modulation of the difference in repetition frequencies [2] can in principle reduce this duty cycle factor.   . SNR and quality factor for a dual comb spectrometer versus the total comb power, P c , for a single detector with no optical filtering (F = N d = 1) from Eq. (2), normalized to a 1 sec total acquisition time. The quality factor is the product of the number of resolved spectral elements and the SNR for either amplitude or phase. The SNR for 100, 000 M = resolved elements is also shown. The SNR (or quality factor) (solid blue line) is limited by detector noise (long dashed brown line) at low comb powers, by shot noise (short dashed grey line) at medium comb powers, and by either laser RIN (dash-dot light green line) or detector dynamic range (dash-dotted dark green line) at high comb powers. With our assumption of a uniform spectrum and Gaussian filter shape, the average power per comb tooth is 0.8 Values used: NEP = 2 pW/Hz 1/2 , D = 7000, From Eq. (2) regardless of the dominant noise, the uncertainty scales linearly with M. An M 1/2 dependence is expected in Fourier spectroscopy because there are ~M time steps per interferogram. In dual-comb spectroscopy, an additional M 1/2 arises because we assume a constant laser power, therefore doubling the spectral width reduces the source spectral power density. In conventional FTS it is not normally present since the source power spectral density is fixed. From both this scaling, and from a practical point of view, a useful figure of merit is the normalized quality factor, H M σ at T =1 s [9], or the product of the SNR per unit time and number of resolved spectral elements. Figure 2 plots the quality factor and SNR versus total comb power. In the next subsections, we discuss other consequences of Eq. (2). Figure 3 illustrates the scaling of the quality factor and SNR with the degree of sequential or parallel spectral acquisition, as quantified by F and N d . The quantities are all normalized to the same 1 sec total acquisition time. In the detector noise limit (low P c ), there is a distinct disadvantage to sequential acquisition ( 1 F > ), as expected from Fellgett's advantage and no benefit to multiple detectors. In the shot noise limit (medium P c ), the same SNR is reached whether the spectrum is measured sequentially or at once, but there is benefit to multiple detectors. Finally, in either the RIN or dynamic range limit (high P c ), there is an advantage to sequential acquisition and an even stronger advantage to multiple detectors.

Scaling with filtering and number of detectors
Increasing the overall degree of filtering, N d F, reduces the optical power on a given detector and shifts the limiting noise from RIN or dynamic range to either shot noise or detector noise. Ideally, one would like to increase the number of detectors, N d , until the noise is limited by the detector noise. In practice, the use of multiple detectors must be balanced by the corresponding increase in system complexity and for practical reasons N d may be limited to one. It is still advantageous to then increase the sequential filtering, F, until the RIN or dynamic range no longer dominates. Based on Eq. (2), the optimal value of spectral filtering is ( ) , corresponding to ~ 1 nm bandwidth filtering for a 10 THz source at 1550 nm. This bandwidth is probably too narrow given the need to suppress the effects of multiplicative noise (see Section 2.5) and one would likely operate the system at a slightly broader filtering level.

Quantitative values and comparison with existing demonstrations
In general, for systems operating near the gain bandwidth of the laser or amplifier, the comb power can be high enough that we expect the limiting noise term in Eq.
(2) to be the RIN/dynamic range term, at least for a low F and N d . For supercontinuum sources, it may well dominate as well due to the degradation in RIN [21][22][23]. In this limit, Eq. (2) can be rewritten as the quality factor in the simplified expression, We measure fiber-laser RINs as low as 10 -15 , or -150 dBc/Hz. However, with amplification, values of -145 dBc/Hz are perhaps more realistic, and even higher RIN may be reached for significant spectral broadening. The dynamic range contribution, is -148 dBc/Hz at 12 bits and -124 dBc/Hz at 8 bits [16]. Therefore, depending on the system, either laser RIN or dynamic range can dominate Eq. (3). As a useful benchmark, the SNR is resolved elements and a single detector with no filtering at RIN = -145 dBc/Hz.
In our recent work [7,8], the SNR was limited by the poor dynamic range of ~500 D at the center of the spectrum and by detector noise at the edges. For the corresponding value of , or ~3 times worse than the calculated value, which is attributed mainly to the rolloff in the SNR on the spectral edges due to reduced comb power (the peak SNR was ~2× better) and to unequal source and LO powers.
We can also compare Eq. (3) to several other demonstrations of fiber-based dual-comb spectroscopy [4,5]. In the initial cavity-enhancement demonstration of Ref. [5], there were M = 1500 resolved elements. Including a reference scan, the peak SNR 100 2 = and the acquisition time 4 ms. (In [5] the quoted acquisition period is the apodization window, but in Eq. (2) it is the time required for a full scan or Δf r -1 ). With these values, the experimental quality factor was 6 1 /2

10 Hz
. In Ref. [4], both sources passed through the sample but the SNR should still apply to within a factor of two. We estimate 3100 M ≈ , ε = 115, an SNR of √2×100 for magnitude, and an acquisition time of 2 T = s, to calculate 6 1 /2 0.25 10 Hz H M σ ≈ × (with higher values possible for Δf r closer to the Nyquist limit).
In general for a single detector, these results, as well as Eq. (3) and Fig. 2, point to a quality factor of H M σ < 10 7 Hz 1/2 for coherent dual comb spectrometry, at least with fiberbased combs; solid-state combs have lower RIN and could reach higher quality factors.

Sensitivity to a trace gas
The reconstructed response provides high-frequency resolution and accuracy for characterizing a sample and therefore comes with the potential to detect a particular gas in the presence of large background clutter. The best possible sensitivity (with no background clutter) can be calculated based on Eq. (2). To compare with laser spectrometers, it is useful to express the sensitivity in terms of the minimum absorption sensitivity at a discrete line where the sum is over the different rovibrational lines, each with Lorentzian FWHM , Lor j ν Δ and peak absorption j α . The square root term represents the enhancement over the sensitivity calculated at a single frequency element. For the mock signal of Fig 1, the sum is over 20 equally strong lines with widths ~20×ν res , yielding an enhancement of ~20. More realistically, for example, for the HCN data of Refs. [6][7][8], the enhancement factor relative to the strongest line is ~11 for res ν =200 MHz. Eq. (4) also assumes the gas response is known.
While this enhancement helps, the overall sensitivity is low by the standards of laser spectrometers. As with conventional FTS, this lower sensitivity is a tradeoff with the fact one has access to broadband spectrum.

Multiplicative noise from residual phase noise between the combs
An important noise contribution in any Fourier spectroscopy is multiplicative noise. It can occur here due to relative amplitude noise, timing jitter or phase noise between the two comb sources. (Shot noise is technically multiplicative but treated as additive here for reasons explained in Section 3.2.) We assume most of the timing and phase noise between the combs is suppressed either through active phase-locking or monitoring approaches [4,[6][7][8]. (It would be possible to suppress pulse-to-pulse amplitude noise in this way as well, in principle.) Nevertheless, there will remain residual noise between the combs and it is useful to calculate the effect of this noise on the measured sample response. Carrier phase noise is particularly troublesome even for highly stabilized combs [20], so we concentrate on carrier phase noise rather than timing jitter. We can divide the phase noise into slow and fast components. Phase noise slow compared to the interferogram update rate  [8,16]. With these considerations, slow phase noise is not generally a problem.
Residual fast phase noise that occurs over the timescale of the interferogram (and particularly over the centerburst) cannot be similarly removed. Assuming the noise is small (< 1 radian), it adds a multiplicative noise term that will average down with the square root of the number of interferograms, but can still easily exceed the additive white noise. For example, a 0.1 radian variation leads to ~ 10 % noise over the centerburst, which will completely mask any weak underlying signal. To retain sensitivity in the time domain to the sample response (i.e., FID signal for a trace gas), the centerburst duration (~1/ A ν Δ ) must be shorter than the sample coherence time (~inverse width of the spectral features). In the frequency domain, this multiplicative noise results in phase and amplitude noise with a correlation width equal to the spectral width of the signal in a given sub-band , e.g., Δ > Δ , the multiplicative noise was negligible. Therefore, through a combination of suppressing residual noise between the combs and the built-in common-mode rejection of Fourier spectroscopy, multiplicative noise can be suppressed in dual comb spectroscopy to a level where it is unimportant. Of course, with greater sequential or parallel filtering, A ν Δ is reduced and there is a corresponding reduction in the effective common mode rejection so that one must be careful to always ensure that this noise contribution remains below Eq. (2).

Comparison with tunable laser spectrometer & grating spectrometer
It is also useful to compare the SNR of a dual-comb spectrometer to a tunable laser spectrometer (TLS) [17,18] covering the same bandwidth at the same resolution. We first compare to the dual-comb spectrometer with a single point detector and no filtering ( 1 d F N = = ), ignoring geometry-dependent numerical factors and assuming spectrally flat RIN and comb spectrum. In the detector noise limit, the dual-comb spectrometer is equivalent to a TLS with a power teeth tooth N P where tooth P is the power per tooth and teeth r N f ν = Δ is the number of participating comb teeth (equal to M for 1 ε = ). Of course, the TLS is unlikely to be detector noise limited. In the shot-noise limit, the dual-comb spectrometer is equivalent to a TLS at a power equal to tooth P . In the RIN limit, the dual comb spectrometer is equivalent to a TLS with a RIN that is N teeth greater. The comparison improves if sequential or parallel acquisition is used. For example, in the shot-noise limit the dual-comb spectrometer is equivalent to a TLS with power d tooth N P and, in the RIN limit, to a TLS with a RIN of ( ) There are several additional points in favor of dual-comb FTS over TLS. First, the comb spectrum can, in principle, cover spectral regions where a TLS is unavailable. Second, a conventional TLS measures the intensity (amplitude) and does not recover the phase of the signal. It is possible to configure a TLS to measure phase as well as amplitude, but it is difficult to control the phase noise down to the sub-mradian level possible with dual comb spectroscopy. Third, high frequency accuracy and stability is challenging in a fast swept laser system. Fourth, the comb spectrometer can acquire spectra more rapidly than many (but not all) swept lasers and allow for spectral measurements in dynamic systems. This potential for high speed spectral measurements is a strong attribute of dual comb spectroscopy.

Summary
We derive the basic SNR achievable with a dual comb spectrometer, Eq. (2), considering the effects of additive white noise from detection, shot noise, or RIN due to additive ASE under the assumption of negligible differential chirp, Nyquist filtering, and negligible 1/f noise between the combs. This latter assumption requires some combination of a sufficiently high scan rate (Δf r ) and effective phase-locking between the combs through active feedback or monitoring of error signals. We show that under the appropriate conditions, multiplicative noise can be controlled to below the noise level of Eq. (2). We also derive the effective SNR for trace gas detection considering the additive noise of Eq. (2). We allow for the possibility of detecting the entire spectrum at once, sequentially through a tunable filter, or in parallel through a fixed filter and detector array.
The equations can be used to guide the design of dual-comb spectrometers, depending on the contribution of the different noise sources. For example, with appropriate care in treating multiplicative noise, the best SNR is attained through a dual comb spectrometer that uses a detector array. A useful figure of merit, both in terms of the scaling of the SNR and for practical reasons, is the quality factor given by the product of the number of resolved elements and the SNR. For fiber-based systems, it is on the order of 10 6 Hz 1/2 -10 7 Hz 1/2 ; however, the equations are general and apply as well to future dual comb spectroscopy into the mid-and long-wave infrared. The limit to the sensitivity for measuring a specific gas spectrum will be set by the SNR, the overall path length (which can be quite large for these systems through either cavity enhancement or multipass cells) and the total achievable integration period, which can also be quite large through coherent signal averaging. Although not as sensitive as tunable laser spectrometers, dual-comb spectrometers are promising for broadband detection of multiple gases and for high resolution spectroscopy, due to the excellent frequency resolution and accuracy, the high acquisition speed, and the possibility of accessing spectral regions inaccessible to tunable lasers.

Derivations
The organization of this section mirrors that of Section 2, i.e. Section 3.1 derives the basic response of the system, additive noise is considered in 3.2, trace gas sensitivity in 3.3, and multiplicative noise in 3.4. For added insight, the derivation is given in both the time domain, where the data are acquired, and the frequency domain, where they are often used.
As shown in Fig. 1, a single detector will see an "instantaneous" spectral subband of width The responses within each spectral sub-band are concatenated in the frequency domain to cover the full source bandwidth, Δν, (and then inverse Fourier transformed to yield the time-domain response). Assuming roughly uniform spectral power, the spectral SNR within a subband A ν Δ is identical to that over the full band ν Δ , and therefore we need calculate only the SNR in a sub-band. The subscript A is added to the variables in Table 1 to indicate filtered quantities.

System response
In dual comb spectroscopy, the source and the LO have different pulse repetition periods, T S and T L , so that the LO pulses advance through the source pulses by rS rL As noted in previous work [19], to achieve fully synchronous sampling, we would like each time-domain interferogram to be identical so that we can simply add them. In the frequency domain, the rf frequency comb then falls exactly on the measured frequency grid, which is not necessary [4] but does simplify the math. To achieve this, we assume [19]: (1) K is an integer. This condition ensures that the overlap of the K th LO pulse with (K+1) th source pulse has exactly the same time offset as the overlap of the 0th LO and source pulse.
is the carrier-envelope offset (ceo) phase shift per pulse [24,25] and q is an integer. This insures the relative phase of the K th LO pulse and (K+1) th source pulse is exactly the same as the 0th LO and source pulse. An equivalent condition is that a regular series of comb teeth, 0 0 n r L nKf ν ν = + sit at identical frequencies for both source and LO combs. For simplicity, we select 0 ν to correspond to the particular "shared" comb tooth, 0n ν , closest to the carrier frequency of the filtered light. As the filter is tuned, or for different point detectors, or if a comb offset frequency is shifted (in increments of Δf r ), part of the bookkeeping is to track jumps in 0 ν . ν if a portion of the optical spectrum covering a shared tooth, or equidistant between two shared teeth, is to be measured (see Fig. 4).

(4) All spectral filtering is Gaussian with FWHM
A ν Δ and a spacing between filtered subbands equal to the FWHM. (The formulae then apply to d 1 N F = = for a Gaussian source.) Figure 4 clarifies the different frequencies involved.

Measured Bandwidth Δν
Filter Δν A K source comb teeth Intensity r Nyquist Nyquist Fig. 4. Schematic of the different optical frequencies involved in the calculation. The light blue (dark blue) lines are the source (LO) comb lines. Their repetition-rate difference is Δf r . The entire measured bandwidth is Δν, but for any given interferogram, the bandwidth is filtered to Δν Α either sequentially by tuning the filter across the spectrum or in parallel with a fixed filter bank to multiple detectors (or a combination of the two.) Note the "dead zones" where the rf beat notes are at either zero or Nyquist frequency; at these filter locations an overall frequency shift is applied to one source [7,8].
Under assumptions 1 and 2 above, the source and LO electric fields are The heterodyne portion of the detected voltage is where R(t) is the filtering from both hardware and software and a converts from squared electric field to measured photocounts. The voltage is digitized synchronously with the LO comb at a series of times, where the individual terms (interferograms) are with a sampling function ( ) ( ) ( ) Eq. (10) simply describes a series of concatenated interferograms. If Fourier transformed, one finds the rf comb of Ref. [6]. However, it is simpler to just add the individual interferograms, and just as effective in terms of noise suppression. With the assumption of synchronous sampling, there is no phase shift between interferograms, and the grid of time points, eff t k T = Δ is aligned with the effective time offset S T , so that no interpolation is required in calculating the average interferogram, In the limit K >> 1 and a broad detection bandwidth, ( ) ( )  ' .
Only the w = 0 term survives, and there is a finite range of q , giving  (16) or the standard "multi-heterodyne" result modified to account for the frequency response of the detector: an rf comb where each tooth at r q f Δ is the beat between the neighboring q th comb lines [1,2,6] . We rewrite this in terms of the optical frequencies ( ) We observe for only a finite time so the delta function should be a sinc function; however, we sample on the grid of rS qf ν = , and the values are unchanged and where ( ) In the frequency (spectral) domain, the Fourier transform of (19) yields evaluated at (K/2+1) frequency points separated by f rS and covering a bandwidth (2ΔT) -1 with an optical frequency offset ν 0 . The SNR in either quadrature is where the noise term ( ) n ν has equal real and imaginary standard deviations, v σ .
This SNR applies to both the reference and signal of Eq. (1). Therefore, the noise on the normalized transmittance, or sample response is where the numerator includes one factor of √2 from comparing reference and signal and one factor of √2 from the fact that equal periods must be spent measuring reference and signal. At this point, the detector response, R(ν) drops out. We will assume a bandwidth matched to Nyquist (thus avoiding aliased noise) but otherwise ignore R(ν) below.

Contributions of additive white noise sources
We now derive Eq. (2). The additive, white, Gaussian noise from detector noise, shot-noise and laser RIN to each interferogram has standard deviation

Multiplicative noise from residual phase noise between combs
The combs will exhibit residual amplitude, timing and phase jitter between pulses since no effort to remove the residual noise will be perfect. The carrier phase noise is typically dominant because it is the most challenging to control and, particularly outside the locking or monitoring bandwidth, the combs will have some residual high-frequency phase variations [20]. As a result, the first term of Eq.