## Sensitivity of coherent dual-comb spectroscopy

Optics Express, Vol. 18, Issue 8, pp. 7929-7945 (2010)

http://dx.doi.org/10.1364/OE.18.007929

Acrobat PDF (1513 KB)

### Abstract

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^{6} – 10^{7} Hz^{1/2}.

© 2010 Optical Society of America

## 1. Introduction

1. F. Keilmann, C. Gohle, and R. Holzwarth, “Time-domain mid-infrared frequency-comb spectrometer”, Opt. Lett. **29**, 1542–1544 (2004). [CrossRef] [PubMed]

6. I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs”, Phys. Rev. Lett. **100**, 013902 (2008). [CrossRef] [PubMed]

7. I. Coddington, W. Swann, and N. Newbury, “Time-domain spectroscopy of molecular free-induction decay in the infrared”, Opt. Lett., accepted (2010). [CrossRef] [PubMed]

13. S. Schiller, “Spectrometry with frequency combs”, Opt. Lett. **27**, 766–768 (2002). [CrossRef]

## 2. Sensitivity of dual-comb spectrometers

### 2.1 Configuration of dual-comb spectrometer

*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

*V*(

_{ref}*t*) =

*S*(

*t*) ⊗

*E*(

_{s}*t*) 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

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). [CrossRef] [PubMed]

*H*(

*t*), the digitized voltage is

*V*(

*t*) =

*S*(

*t*) ⊗

*E*(

_{s}*t*) ⊗

*H*(

*t*). The goal is to isolate the sample response, which is easily done in the frequency domain from the ratio

*v*is the optical frequency.

*H*

_{0}(

*ν*) is the true sample response and

*σ*is the inevitable added noise, which is the subject of the remainder of this paper. For a weakly absorbing sample,

_{H}*H*̃

_{0}(ν) = 1 + 4

*π*

^{2}

*ic*

^{-1}

*νχ*(ν)

*L*≈ 1-

*α*(ν)

*L*/2+

*i*Δ

*k*(ν)

*L*, where

*χ*is the linear susceptibility,

*L*is the sample length,

*α*(ν) is the attenuation coefficient and Δ

*k*(ν) the phase shift.

*ν*) into sub-bands that can be acquired sequentially or in parallel. Parallel acquisition is accomplished through a fixed filter that multiplexes

*N*spectrally filtered bands to an array of

_{d}*N*detectors. Sequential acquisition is accomplished through a tunable optical bandpass filter in front of each detector that steps through

_{d}*F*different filter bands. In all, the total spectral bandwidth is divided into

*F*×

*N*sub-bands with filtered bandwidth Δ

_{d}*ν*= Δν/(

_{A}*FN*). Each sub-band is measured for a period

_{d}*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, Δ

*f*, (while still satisfying Nyquist constraints). Since this difference, Δ

_{r}*f*, is exactly the separation between the down-converted rf heterodyne beat signals, a larger value of Δ

_{r}*f*translates to a relaxed requirement on the relative comb linewidths. Filtering also helps normalize strong spectral variations in the comb spectra [8

_{r}8. I. Coddington, W. Swann, and N. Newbury, “Coherent dual-comb spectroscopy at high signal to noise”, http://arxiv.org/abs/1001.3865 (2010).

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). [CrossRef] [PubMed]

*σ*

^{-1}

_{H}) across the central FWHM. Effectively, then, for no filtering (

*FN*= 1), we assume a Gaussian source spectrum while for filtering (

_{d}*F*> 1), 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

_{Nd}*f*/2. Fourth, we assume that the difference in comb repetition rates, Δ

_{r}*f*, and spectral width of the sub-bands are chosen to satisfy the Nyquist constraint [1–8

_{r}1. F. Keilmann, C. Gohle, and R. Holzwarth, “Time-domain mid-infrared frequency-comb spectrometer”, Opt. Lett. **29**, 1542–1544 (2004). [CrossRef] [PubMed]

*f*(i.e. we assume negligible “1/

_{r}*f*” noise in relative phase and timing jitter between the combs [20

20. N. R. Newbury and W. C. Swann, “Low-noise fiber-laser frequency combs (invited)”, J. Opt. Soc. Am. B **24**, 1756–1770 (2007). [CrossRef]

### 2.2 Effects of additive noise: detector noise, shot noise, laser RIN & detector dynamic range

*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 low-frequency component can lead to multiplicative noise but we ignore it under the assumption that the interferogram update rate, equal to Δ

*f*, 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.

_{r}*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,

*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.

_{H}*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

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). [CrossRef] [PubMed]

*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,

*M*/

*σ*at

_{H}*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).

#### 2.2.1 Scaling with filtering and number of detectors

*F*and

*N*. The quantities are all normalized to the same 1 sec total acquisition time. In the detector noise limit (low

_{d}*P*), there is a distinct disadvantage to sequential acquisition (

_{c}*F*> 1), as expected from Fellgett’s advantage and no benefit to multiple detectors. In the shot noise limit (medium

*P*), 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

_{c}*P*), there is an advantage to sequential acquisition and an even stronger advantage to multiple detectors.

_{c}*N*, 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,

_{d}F*N*, 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

_{d}*N*may be limited to one. It is still advantageous to then increase the sequential filtering,

_{d}*F*, until the RIN or dynamic range no longer dominates. Based on Eq. (2), the optimal value of spectral filtering is

*D*and

*ε*=

*b*=

*γ*=

*c*

_{γ2}=1). For the values in Fig. 1 and

*P*=2 mW, one finds

_{c}*F*≈100

_{opt}*N*

^{-1}

_{d}, 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.

#### 2.2.2 Quantitative values and comparison with existing demonstrations

*F*and

*N*. For supercontinuum sources, it may well dominate as well due to the degradation in RIN [21–23

_{d}21. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber”, Rev. Mod. Phys. **78**, 1135–1184 (2006). [CrossRef]

*b*=

*γ*=

*c*

_{γ2}= 1. 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, 8

*D*

^{-2}

*f*

^{-1}

_{r}, 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 1/

*σ*~ 100

_{H}*T*

^{1/2}for

*M*= 100,000 resolved elements and a single detector with no filtering at

*RIN*= -145 dBc/Hz.

7. I. Coddington, W. Swann, and N. Newbury, “Time-domain spectroscopy of molecular free-induction decay in the infrared”, Opt. Lett., accepted (2010). [CrossRef] [PubMed]

8. I. Coddington, W. Swann, and N. Newbury, “Coherent dual-comb spectroscopy at high signal to noise”, http://arxiv.org/abs/1001.3865 (2010).

*D*~ 500 at the center of the spectrum and by detector noise at the edges. For the corresponding value of 8

*D*-2

*f*

^{-1}

_{r}=-125 dBc/Hz and the experimental parameters

*F*~45 and

*ε*=2, Eq. (3) yields

*M*/

*σ*= 6.7 × 10

_{H}^{6}Hz

^{1/2}. At the measured

*M*= 41000,

*σ*= 4 × 10

_{H}^{-4}and

*T*= 2700 sec the measured quality factor was

*M*/

*σ*= 2 × 10

_{H}^{6}Hz

^{1/2}, 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.

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). [CrossRef] [PubMed]

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). [CrossRef]

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). [CrossRef]

*M*= 1500 resolved elements. Including a reference scan, the peak SNR=100/√2 and the acquisition time

*T*= 2 Δ

*f*

^{-1}

_{r}/1 ~ 4 ms. (In [5

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). [CrossRef]

*f*

^{-1}

_{r}). With these values, the experimental quality factor was

*M*/

*σ*=1.7×10

_{H}^{6}Hz

^{1/2}. In Ref. [4

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). [CrossRef] [PubMed]

*M*≈ 3100,

*ε*= 115, an SNR of √2× 100 for magnitude, and an acquisition time of

*T*= 2 s, to calculate

*M*/

*σ*≈ 0.25×10

_{H}^{6}Hz

^{1/2}(with higher values possible for Δ

*f*closer to the Nyquist limit).

_{r}### 2.3 Sensitivity to a trace gas

*α*

_{0}

*L*)

_{min}. One might simply equate this to the uncertainty in spectral intensity, 2

*σ*. However, the actual sensitivity is considerably better because there are many measured points across a single absorption line and many absorption lines. We derive

_{H}*ν*and peak absorption

_{Lor,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×

_{j}*ν*, yielding an enhancement of ~20. More realistically, for example, for the HCN data of Refs. [6–8

_{res}6. I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs”, Phys. Rev. Lett. **100**, 013902 (2008). [CrossRef] [PubMed]

*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.

_{res}### 2.4 Multiplicative noise from residual phase noise between the combs

**16**, 4347–4365 (2008). [CrossRef] [PubMed]

6. I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs”, Phys. Rev. Lett. **100**, 013902 (2008). [CrossRef] [PubMed]

20. N. R. Newbury and W. C. Swann, “Low-noise fiber-laser frequency combs (invited)”, J. Opt. Soc. Am. B **24**, 1756–1770 (2007). [CrossRef]

*f*

^{-1}

_{r}will cause a reduction in the summed interferogram signal of (1 -

*σ*

^{2}

_{φ,slow}/2) where

*σ*

^{2}

_{φ,slow}slow is the (slow) component of the phase variation. For

*σ*

^{2}

_{φ,slow}= 0.4rad, the net effect is a negligible reduction of 8 % [19

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). [CrossRef] [PubMed]

*σ*

^{2}

_{φ,slow}> 1 rad, phase correction should be applied in the usual way [8

8. I. Coddington, W. Swann, and N. Newbury, “Coherent dual-comb spectroscopy at high signal to noise”, http://arxiv.org/abs/1001.3865 (2010).

*ν*) 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

_{A}*ν*yielding a slow baseline wander. To retain sensitivity in the frequency domain to a sample response, the correlation width Δ

_{A}*ν*must be larger than the width of the spectral features (exactly the same condition as in the time domain). The slow wander can be removed by a simple polynomial fit [6–8

_{A}**100**, 013902 (2008). [CrossRef] [PubMed]

*σ*

^{2}

_{φ,fast}and a spacing between filter centers equal to the FWHM, the noise on the magnitude or phase is approximately

### 2.5 Comparison with tunable laser spectrometer & grating spectrometer

17. V. V. Protopopov, *Laser Heterodyning* (Springer Berlin / Heidelberg, 2009). [CrossRef]

*F*=

*N*= 1), 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

_{d}*P*is the power per tooth and

_{tooth}*N*= Δ

_{teeth}*ν*/

*f*is the number of participating comb teeth (equal to

_{r}*M*for

*ε*= 1). 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

*P*. In the RIN limit, the dual comb spectrometer is equivalent to a TLS with a RIN that is

_{tooth}*N*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

_{teeth}*N*and, in the RIN limit, to a TLS with a RIN of

_{d}P_{tooth}*N*/(

_{teeth}RIN*N*

^{2}

_{d}

*F*).

### 2.6 Summary

*f*noise between the combs. This latter assumption requires some combination of a sufficiently high scan rate (Δ

*f*) 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.

_{r}^{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.

## 3. Derivations

*ν*= Δ

_{A}*ν*/(

*FN*) for an acquisition period

_{d}*T*=

_{A}*T*/

*F*. The responses within each spectral sub-band are concatenated in the frequency domain to cover the full source bandwidth, Δ

*ν*, (and then inverse Fourier transformed to yield the time-domain response). Assuming roughly uniform spectral power, the spectral SNR within a subband Δ

*ν*is identical to that over the full band Δ

_{A}*ν*, 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.

### 3.1 System response

*T*and

_{S}*T*, so that the LO pulses advance through the source pulses by Δ

_{L}*T*=

*T*-

_{L}*T*every LO pulse, yielding a down-conversion factor in time of

_{S}*T*> 0.) A single pass of the LO pulse through the source pulses generates a single interferogram, which will contain

*K*points and take a time Δ

*f*

^{-1}

_{r}. Similarly, in the frequency domain, the repetition rates

*f*=

_{rS}*T*

^{-1}

_{S}and

*f*=

_{rL}*T*

^{-1}

_{L}differ by Δ

*f*=

_{r}*f*-

_{rS}*f*so that their heterodyne signal is an rf comb with repetition rate Δ

_{rL}*f*yielding a down-conversion factor in frequency of [1–8

_{r}1. F. Keilmann, C. Gohle, and R. Holzwarth, “Time-domain mid-infrared frequency-comb spectrometer”, Opt. Lett. **29**, 1542–1544 (2004). [CrossRef] [PubMed]

13. S. Schiller, “Spectrometry with frequency combs”, Opt. Lett. **27**, 766–768 (2002). [CrossRef]

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). [CrossRef] [PubMed]

**16**, 4347–4365 (2008). [CrossRef] [PubMed]

**34**, 2153–2155 (2009). [CrossRef] [PubMed]

*K*is an integer. This condition ensures that the overlap of the*K*LO pulse with (_{th}*K*+1)_{th}source pulse has exactly the same time offset as the overlap of the 0th LO and source pulse.- (
*K*+ 1)Δ*θ*=_{ceo,S}*K*Δ*θ*+2_{ceo,L}*πq*, where Δ*θ*_{ceo,S(L)}is the carrier-envelope offset (ceo) phase shift per pulse [24, 2524. T. W. Hänsch, “Nobel lecture: Passion for precision”, Rev. Mod. Phys.

**78**, 1297–1309 (2006). [CrossRef]] and25. J. L. Hall, “Nobel lecture: Defining and measuring optical frequencies”, Rev. Mod. Phys.

**78**, 1279–1295 (2006). [CrossRef]*q*is an integer. This insures the relative phase of the*K*LO pulse and (_{th}*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,*ν*_{0n}=*ν*_{0}+*nKf*sit at identical frequencies for both source and LO combs. For simplicity, we select_{rL}*ν*_{0}to correspond to the particular “shared” comb tooth,*ν*_{0n}, 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 Δ*f*), part of the bookkeeping is to track jumps in_{r}*ν*_{0}. - The entire instantaneous bandwidth Δ
*ν*falls between_{A}*ν*_{0}and*ν*_{0}±*Kf*/2, i.e., 0 < ∣_{rS}*ν*-*ν*_{0}∣ <*Kf*/2. This condition insure there are no rf beats that fall at zero or Nyquist and avoids aliasing effects. Clearly, in an experiment, it is necessary to first shift_{rs}*ν*_{0}if a portion of the optical spectrum covering a shared tooth, or equidistant between two shared teeth, is to be measured (see Fig. 4). - All spectral filtering is Gaussian with FWHM Δ
*ν*and a spacing between filtered sub-bands equal to the FWHM. (The formulae then apply to_{A}*N*_{d}=*F*= 1 for a Gaussian source.) Figure 4 clarifies the different frequencies involved.

*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,

*k*, where

_{TL}*k*is an integer. Substituting (8) into (9) yields

*t*=

_{eff}*k*Δ

*T*, noting

*T*=

_{L}*T*+ Δ

_{S}*T*, and with the change of variables

*r*=

*n*-

*m*,

*p*≡

*k*-

*m*,

*τ*=

*τ*′-

*kT*+

_{L}*pT*, after some rearranging,

_{S}*S*(

*t*) =

*a*∑

_{p}

*R*(

*pT*+

_{S}*t*)

*A*

^{*}

_{L}(

*p*Δ

*T*-

*t*).

**100**, 013902 (2008). [CrossRef] [PubMed]

*t*=

_{eff}*k*Δ

*T*is aligned with the effective time offset

*T*, so that no interpolation is required in calculating the average interferogram, 〈

_{S}*V*(

*t*)〉. Henceforth, we drop the brackets and let

_{eff}*t*→

_{eff}*t*. Under assumption 3 above, Δ

*T*is sufficiently short that

*p*Δ

*T*is effectively a continuous variable (

*p*Δ

*T*→

*τ*), and the sum for

*S*(

*t*) can be rewritten as

*S*(

*t*)≈

*a*Δ

*T*

^{-1}∫

*R*(

*Kτ*+

*t*)

*A*

^{*}

_{L}(

*τ*-

*t*)

*dτ*with Eq. (6). With a change of variables,

*K*>> 1 and a broad detection bandwidth,

*S*(

*t*) ≈

*A*

^{*}

_{L}(-

*t*), as expected.

*q*,

*p*are integers. Substitution into Eq. (9) gives

*p*=

*w*+

*q*, and carrying out the integrations,

*w*= 0 term survives, and there is a finite range of

*q*, giving

*q*Δ

*f*is the beat between the neighboring

_{r}*q*

^{th}comb lines [1

**29**, 1542–1544 (2004). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

**100**, 013902 (2008). [CrossRef] [PubMed]

*ν*= (

*K*+1)

*f*as

*V*̃ (

*ν*) =

*e*

^{-iΔφ}

*a*(

*K*+1)

^{-1}

*R*̃(

*ν*/(

*K*+1))

*A*̃

^{*}

_{L}(

*νK*/(

*K*+ 1))

*A*̃

_{S}(

*ν*)∑

_{q}

*δ*(

*ν*~

*qf*). We observe for only a finite time so the delta function should be a sinc function; however, we sample on the grid of

_{rS}*ν*=

*qf*, and the values are unchanged and

_{rS}*ν*

_{0}, so that

*A*

_{S}_{(L)}→

*E*(

_{S}*L*), Fourier transforming, adding the effect of the sample response, and a noise term, gives

*K*time points separated by Δ

*T*and covering a time window

*T*, where

_{S}*n*(

*t*) is the noise in a single interferogram, which is reduced by the square root of the number of averaged interferograms equal to

*T*Δ

_{A}*f*. The interferogram has the appearance shown in Fig. 1 of a centerburst, oscillating at a frequency ∣

_{r}*ν*-

_{A}*ν*

_{0}∣ and with a width ~ Δ

*ν*

^{-1}

_{A}. If

*n*(

*t*) has a standard deviation

*σ*, the peak time-domain signal-to-noise ratio is

_{t}*K*/2+1) frequency points separated by

*f*and covering a bandwidth (2Δ

_{rS}*T*)

^{-1}with an optical frequency offset

*ν*

_{0}. The SNR in either quadrature is

*n*̃(

*ν*) has equal real and imaginary standard deviations,

*σ*.

_{ν}*R*(

*ν*) drops out. We will assume a bandwidth matched to Nyquist (thus avoiding aliased noise) but otherwise ignore

*R*(

*ν*) below.

### 3.2 Contributions of additive white noise sources

*σ*(in units of photon counts). The second and third term are shot noise, where

_{d}*n*

_{S(LO)}is the number of photons per source (LO) pulse within the filtered band Δ

*ν*. The shot-noise term is technically multiplicative, as it varies over the centerburst, but if the centerburst is short compared to the interferogram, we ignore this variation. The last two terms arise from the excess RIN of the lasers, which we assume here arises from an additive ASE field. This RIN is white so that variation in photons per pulse δ

_{A}*n*is (

_{s}*δn*/

_{S}*n*)

_{s}^{2}=

*RIN*(

_{S}*f*/2) since the RIN is single-sided. With balanced detection, this RIN from each individual laser is cancelled. However, the noise from the LO pulses beating against the source ASE component, and vice versa, is not cancelled and leads to the 4

_{r}^{th}and 5

^{th}terms in Eq. (24). Without balanced detection, the RIN contribution is twice as big. The peak signal for the interferogram is

*c*= (1 +

_{γ}*γ*)/(2

*γ*),

*c*

_{γ2}= (1 +

*γ*

^{2})/(2

*γ*) for

*RIN*=

_{L}*RIN*=

_{S}*RIN*and we introduce the parameter

*b*=1(2) for balanced (unbalanced) optical detection. The last term in the denominator is added to include experimental dynamic range limitations in the detection that clamp the SNR of a single interferogram to

*D*. The fraction in brackets is the SNR of a single interferogram; for

*γ*= 1; it equals

*D*in the dynamic range limit, which can be set by the photodetection, amplification or digitizer.

_{ΔνA}, gives the ratio

*ν*<< (2Δ

_{A}*T*)

^{-1}(which is true to avoid aliasing). With

*P*≡

_{cA}*hνf*, in the frequency-domain Eq. (25) becomes,

_{r}n_{s}*f*~

_{rS}*f*~

_{rL}*f*,

_{r}*M*= Δ

_{A}*ν*/

_{A}*f*, and the equality

_{r}*T*=

_{A}*T*/

*F*,

*M*=

*M*(Δ

_{A}*ν*/Δ

*ν*),

_{A}*P*=

_{c}*P*(Δ

_{c,A}*ν*/Δ

*ν*) and Δ

_{A}*ν*= Δ

_{A}*ν*/(

*FN*) into Eq. (26), and the result into Eq. (23),

_{d}*ν*. In that case, we redefine

_{res}*M*= Δ

*ν*/

*ν*to retain its definition as the number of resolved elements and multiply by

_{res}*γ*= 1 so that the power per tooth,

*P*, is the same for the source and LO combs. The rms signal amplitude between two teeth is √2

_{tooth}*ηP*at an rf frequency corresponding to their optical frequency difference. The white noise power is [

_{tooth}*η*

^{2}

*NEP*

^{2}+4

*ηhν*

*P*

^{2}

_{cA}+2

*bη*

^{2}

*P*

^{2}

_{cA}(

*RIN*)]

*B*, where the first term is detector noise, the second term is the shot noise associated with the two sources, and the third term is the RIN of the two sources. Note that both the shot noise and RIN are given by the total comb power

*P*. The effective bandwidth is

_{cA}*B*= 1/(2

*T*). This random noise power contributes equally to the amplitude and phase noise, and

_{A}### 3.3 Sensitivity to a known gas

*ρ*and sample length

*L*will have a time-domain response equal to

*H*(

*t*) = sinc(

*πt*Δ

*ν*)⊗ [

*δ*(

*t*) +

*ρLF*(

*t*)] , where the sinc term reflects the instrument response over the bandwidth Δ

*ν*and the term in square brackets is the sample response characterized by the function

*F*(

*t*). We use a standard matched filter approach by defining a cutoff time

*t*

_{0}. We mask out points earlier than

*t*

_{0}to avoid the multiplicative noise across the centerburst. The uncertainty in the column density,

*ρL*, is

*σ*=

_{t,R}*σ*/√

_{H}*M*is the standard deviation of the spectrally reconstructed time-domain interferogram (See Fig. 1). For a Voigt profile, a single absorption line at

*ν*. with integrated line strength

_{j}*S*has a time-dependent response

_{j}*F*(

_{j}*t*) =

*S*(

_{j}θ*t*)

*e*

^{-2πiνjt}

*e*

^{-π2Δν2Dt2/(4ln2)-πΔνLor,jt}, where Δ

*ν*is the FWHM Doppler width, Δ

_{D}*ν*is the FWHM Lorentzian width and

_{Lor}*θ*(

*t*)is the Heavyside step function. (This time-dependent signal is exactly the free induction decay). We will assume a collisionally broadened line (Δ

*ν*> Δ

_{Lor}*ν*). If

_{D}*t*

_{0}<< Δ

*ν*

^{-1}

_{Lor,j}<<

*f*

^{-1}

_{rS}, the limits on the integral are 0 and ∞ and

*α*= 2

_{j}*ρS*/(

_{j}*π*Δ

*ν*) for a Lorentzian. To rewrite this in a more conventional form, we let

_{Lor, j}*σ*= (

_{ρL}*ρL*)

_{min}, the minimum detectable column density, and then scale it by a particular reference line (denoted

*j*= 0), to find the effective minimum absorption, normalized to a reference line, (

*α*

_{0}

*L*)

^{effective}

_{min}= 2

*S*

_{0}(

*π*Δ

*ν*)

_{Lor}^{-1}(

*ρL*)

_{min}, or Eq. (4).

### 3.4 Multiplicative noise from residual phase noise between combs

20. N. R. Newbury and W. C. Swann, “Low-noise fiber-laser frequency combs (invited)”, J. Opt. Soc. Am. B **24**, 1756–1770 (2007). [CrossRef]

*V*(

*t*) →

*e*

^{iδφ(Kt)}

*V*(

*t*), where the exponent describes the relative carrier phase jitter as a function of laboratory time. The fast phase variations are particularly problematic (see Section 2.5). If small (< 1 rad), they add a multiplicative noise

*n*(

*t*) =

*iV*(

*t*)

*δφ*(

*t*) to Eq. (19). From a generalized Eq. (22) and (23),

*n*̃(

*ν*) =

*φ*̃((

*ν*-

*ν*

_{0}) /

*K*)⊗

*iV*̃(

*ν*). Assuming a pulse-to-pulse phase variance of

*σ*

^{2}

*, 〈*

_{φ,fast}*φ*̃(

*f*)

*φ*̃

^{*}(

*f*′)〉 =

*σ*

^{2}

_{φ,fast}*f*

^{-1}

_{r}(

*f*-

*f*′) and a Gaussian filter profile,

*V*(

*ν*) = exp[-4ln(2)(

*ν*-

*ν*)

_{A}^{2}/Δ

*ν*

^{2}

_{A}],

## Acknowledgements

## References and links

1. | F. Keilmann, C. Gohle, and R. Holzwarth, “Time-domain mid-infrared frequency-comb spectrometer”, Opt. Lett. |

2. | A. Schliesser, M. Brehm, F. Keilmann, and D. van der Weide, “Frequency-comb infrared spectrometer for rapid, remote chemical sensing”, Opt. Express |

3. | T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, and T. Araki, “Terahertz frequency comb by multifrequency-heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy”, Appl. Phys. Lett. |

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 |

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. |

6. | I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs”, Phys. Rev. Lett. |

7. | I. Coddington, W. Swann, and N. Newbury, “Time-domain spectroscopy of molecular free-induction decay in the infrared”, Opt. Lett., accepted (2010). [CrossRef] [PubMed] |

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, |

10. | J. Chamberlain, |

11. | J. R. Birch, “Dispersive fourier-transform spectroscopy”, Mikrochimica Acta |

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. |

13. | S. Schiller, “Spectrometry with frequency combs”, Opt. Lett. |

14. | J. W. Brault, |

15. | L. A. Sromovsky, “Radiometric errors in complex fourier transform spectrometry”, Appl. Opt. |

16. | S. P. Davis, M. C. Abrams, and J. W. Brault, |

17. | V. V. Protopopov, |

18. | W. Demtroder, |

19. | I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent linear optical sampling at 15 bits of resolution”, Opt. Lett. |

20. | N. R. Newbury and W. C. Swann, “Low-noise fiber-laser frequency combs (invited)”, J. Opt. Soc. Am. B |

21. | J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber”, Rev. Mod. Phys. |

22. | N. R. Newbury, B. R. Washburn, K. L. Corwin, and R. S. Windeler, “Noise amplification during supercontinuum generation in microstructure fiber”, Opt. Lett. |

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. |

24. | T. W. Hänsch, “Nobel lecture: Passion for precision”, Rev. Mod. Phys. |

25. | J. L. Hall, “Nobel lecture: Defining and measuring optical frequencies”, Rev. Mod. Phys. |

**OCIS Codes**

(300.6300) Spectroscopy : Spectroscopy, Fourier transforms

(300.6310) Spectroscopy : Spectroscopy, heterodyne

**ToC Category:**

Spectroscopy

**History**

Original Manuscript: January 29, 2010

Revised Manuscript: March 12, 2010

Manuscript Accepted: March 15, 2010

Published: March 31, 2010

**Citation**

Nathan R. Newbury, Ian Coddington, and William Swann, "Sensitivity of coherent dual-comb spectroscopy," Opt. Express **18**, 7929-7945 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-8-7929

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### References

- F. Keilmann, C. Gohle, and R. Holzwarth, "Time-domain mid-infrared frequency-comb spectrometer," Opt. Lett. 29, 1542-1544 (2004). [CrossRef] [PubMed]
- 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). [CrossRef] [PubMed]
- T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, and T. Araki, "Terahertz frequency comb by multifrequency heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy," Appl. Phys. Lett. 88, 241104 (2006). [CrossRef]
- 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). [CrossRef] [PubMed]
- 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). [CrossRef]
- I. Coddington, W. C. Swann, and N. R. Newbury, "Coherent multiheterodyne spectroscopy using stabilized optical frequency combs," Phys. Rev. Lett. 100, 013902 (2008). [CrossRef] [PubMed]
- I. Coddington, W. Swann, and N. Newbury, "Time-domain spectroscopy of molecular free-induction decay in the infrared," Opt. Lett., accepted (2010). [CrossRef] [PubMed]
- I. Coddington, W. Swann, and N. Newbury, "Coherent dual-comb spectroscopy at high signal to noise," http://arxiv.org/abs/1001.3865 (2010).
- R. J. Bell, Introductory Fourier transform spectroscopy (Academic Press, 1972).
- J. Chamberlain, The Principles of Interferometric Spectroscopy (John Wiley and Sons, Inc, 1979).
- J. R. Birch, "Dispersive Fourier-transform spectroscopy," Mikrochimica Acta 3, 105-122 (1987).
- 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). [CrossRef]
- S. Schiller, "Spectrometry with frequency combs," Opt. Lett. 27, 766-768 (2002). [CrossRef]
- J. W. Brault, High Resolution in Astronomy (Geneva Observatory, 1985), Fourier transform spectrometry, pp. 1-65.
- L. A. Sromovsky, "Radiometric errors in complex Fourier transform spectrometry," Appl. Opt. 42, 1779-1787 (2003). [CrossRef] [PubMed]
- S. P. Davis, M. C. Abrams, and J. W. Brault, Fourier Transform Spectrometry (Academic Press, 2001).
- V. V. Protopopov, Laser Heterodyning (Springer Berlin / Heidelberg, 2009). [CrossRef]
- W. Demtroder, Laser Spectroscopy (Springer, 1996), 2nd ed.
- I. Coddington, W. C. Swann, and N. R. Newbury, "Coherent linear optical sampling at 15 bits of resolution," Opt. Lett. 34, 2153-2155 (2009). [CrossRef] [PubMed]
- N. R. Newbury and W. C. Swann, "Low-noise fiber-laser frequency combs (invited), " J. Opt. Soc. Am. B 24, 1756-1770 (2007). [CrossRef]
- J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006). [CrossRef]
- 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). [CrossRef]
- 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). [CrossRef] [PubMed]
- T. W. Hänsch, "Nobel lecture: Passion for precision," Rev. Mod. Phys. 78, 1297-1309 (2006). [CrossRef]
- J. L. Hall, "Nobel lecture: Defining and measuring optical frequencies," Rev. Mod. Phys. 78, 1279-1295 (2006). [CrossRef]

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