Quantitative modeling of complex molecular response in coherent cavity-enhanced dual-comb spectroscopy

Abstract

We present a complex-valued electric field model for experimentally observed cavity transmission in coherent cavity-enhanced (CE) multiplexed spectroscopy (i.e., dual-comb spectroscopy, DCS). The transmission model for CE-DCS differs from that previously derived for Fourier-transform CE direct frequency comb spectroscopy [Foltynowicz et al., Appl. Phys. B 110, 163–175 (2013)] by the treatment of the local oscillator which, in the case of CE-DCS, does not interact with the enhancement cavity. Validation is performed by measurements of complex-valued near-infrared spectra of CO and CO₂ by an electro-optic frequency comb coherently coupled to an enhancement cavity of finesse F = 19600. Following validation, we measure the 30012 ← 00001 ¹²C¹⁶O₂ vibrational band origin with a combined standard uncertainty of 770 kHz (fractional uncertainty of 4 × 10⁻⁹).

1. Introduction

Optical frequency combs (OFCs) [1–4] enable increasingly precise atomic and molecular spectroscopies [5], sub-Doppler and Doppler-free spectroscopy [6–17], ultrafast and multidimensional spectroscopy [18–21], survey spectroscopy of molecular ions and cold molecules [22–24], and time-resolved spectroscopy for fundamental chemical kinetics [25–28]. Applied spectroscopies [29] also make novel use of optical frequency combs for actively monitoring greenhouse gas fluxes [30–32], methane for open-path detection [33] and unambiguous source attribution [34] and reactive atmospheric species [35], elucidating the chemical composition of combustion and open flames [36, 37], and for developing nonlinear spectroscopies for medical imaging and diagnosis [38, 39]. In additional, laboratory trace gas sensing is also aided by several fundamental properties of optical frequency combs, naming their high temporal and spatial coherence and high spectral resolution, which enable efficient coupling to an enhancement cavity [40, 41].

Despite the burgeoning list of applications utilizing OFCs, there still exists certain areas of spectroscopy which have proven challenging for combs. One example is the merger of dual-comb spectroscopy [42] with cavity-enhanced direct frequency comb spectroscopy [40, 41], and thus the pairing of high acquisition rates and a high duty cycle with the highest possible sensitivities. This combination has the potential to enable time-resolved spectroscopy with truly zero dead-time. In practice, cavity-enhanced dual-comb spectroscopy (CE-DCS) requires high mutual coherence between three entities: a probe comb, the enhancement cavity, and a second comb serving as a local oscillator (LO). On timescales routinely required to achieve a sufficient signal-to-noise ratio for trace gas detection (≥1 s), this remains technically challenging. Complexity associated with phase locking these three entities has limited CE-DCS using mode-locked lasers to a single and seminal proof-of-principle demonstration [43]. In principle, adaptive sampling and/or real-time phase correction approaches to DCS [44–49] should be amenable to CE-DCS, thus mitigating inherent technical complexity associated with maintaining several high-bandwidth phase-locked servo loops.

Recently, an alternative approach to CE-DCS using electro-optic (EO) frequency combs with a high degree of frequency agility and robustness was demonstrated [50, 51]. Electro-optic frequency combs originating from a single continuous-wave (CW) laser have inherently high mutual coherence, and therefore were readily used to record the CE-DCS spectrum of CO₂ [50], as well as the simultaneous spectrum of CO₂, CO, H₂O, and HDO in synthetic air via the demonstration of coherent CE-DCS, capable of averaging time-domain CE-DCS waveforms for >7200 s [51]. Importantly, CE-DCS with EO frequency combs can be readily extended to a much broader, potentially power-leveled optical bandwidth via any combination of intensity, acousto-optic (AO) and/or EO modulator comb-generation schemes [52–64].

Here we present a model for the complex-valued (i.e., amplitude and phase) electric field components of an OFC which are transmitted through a high-finesse optical resonator containing an absorbing gas sample. Using a heterodyne detection scheme, the transmitted electric field of the probe OFC is interrogated by and down-converted to radiofrequencies by a second local oscillator (LO) OFC. Unlike other well-established Fourier transform methods in CE comb spectroscopy [65], a full complex-valued electric field model for cavity transmission in CE-DCS has not yet been reported. To validate the model derived herein, we performed CE-DCS on samples of either dilute CO or pure CO₂ with EO frequency combs in the near-infrared. The EO frequency combs were generated using dual-drive Mach-Zehnder modulators (DMZs) chosen for their relatively flat power spectrum [66, 67] and sufficient bandwidth for precision molecular spectroscopy [50, 52]. Following model validation, we performed quantitative CE-DCS measurements of the 30012 ← 00001 ¹²C¹⁶O₂ vibrational band (38 rovibrational transitions) near λ = 1575 nm, determining the vibrational band origin frequency with a combined standard uncertainty of 770 kHz (fractional uncertainty of 4 × 10⁻⁹).

2. Model for complex cavity transmission

2.1. General cavity transmission expressions

Many of the concepts introduced in this section are illustrated in Fig. 1. For a two-mirror optical resonator of length L, the generalized complex expression for the transmitted electric field ${\tilde{E}}_t$ as a function of angular optical frequency 𝜔 = 2π𝜈 is defined by Siegman [68] in terms of the incident electric field ${\tilde{E}}_i$, the effective mirror intensity reflectivity 𝑅_𝑒, the effective mirror intensity transmittivity 𝑇_𝑒, the intracavity single-pass absorption intensity coefficient α, and the round-trip phase-shift coefficient φ (which includes both molecular and mirror dispersion) as:

$${\tilde{E}}_t(\omega )={\tilde{E}}_i(\omega )\frac{T_e(\omega )\exp [-\alpha (\omega )L/2-i\phi (\omega )/2]}{1-R_e(\omega )\exp [-\alpha (\omega )L-i\phi (\omega )]}.$$ (1)

Fig. 1.

Frequency-domain illustration of comb-cavity coupling in the presence of intracavity absorption. An electro-optic (EO) frequency comb (blue/red lines) is phase locked to a unique cavity mode (black curves) at the points labeled PDH (Pound-Drever-Hall). Each comb tooth has a unique detuning frequency of ±βΔ_c (see the main text). In the presence of absorption (orange), the EO comb complex transmission spectrum can unambiguously differentiate between +Δ_c (red) and −Δ_c (blue dashed) cases. Above, two orange arrows emphasize the local cavity dispersion by dramatically moving the cavity modes from their expected vacuum frequencies (dashed black curves). Finally, the absorption enables the theoretical model to fit any detuning of the PDH locking point δ_PDH (green double-headed arrow).

Note that Eq. (1) assumes lossless mirrors and equal electric field reflectivity and transmittivity coefficients 𝑟₁ = 𝑟₂ and 𝑡₁ = 𝑡₂. Therefore, the effective cavity mirror conservation of energy expression is 1 = 𝑅_𝑒 + 𝑇_e. In CE-DCS, the frequency of the transmitted optical electric field is down-converted into the radiofrequency (RF) domain by combination with an LO electric field ${\tilde{E}}_{LO}$ which produces a suite of heterodyne beat signals which are measured with a fast photodetector. The resulting interferogram contains both the amplitude and phase of the multiheterodyne signal ${\tilde{E}}_t{\tilde{E}}_{LO}^{*}$, and is in practice limited by the photoreceiver/digitizer electronic bandwidth to include only the nearest-neighbor heterodyne beats. Therefore, the normalized intensity (𝐼_𝑡/𝐼₀) and phase (Φ) spectrum of the cavity transmission, summed over all cavity modes, is defined in Eqs. (2) and (3).

$$\frac{I_t}{I_0}=\frac{\left|{\tilde{E}}_t{\tilde{E}}_{LO}^{*}\right|}{\left|{\tilde{E}}_0{\tilde{E}}_{LO}^{*}\right|}$$ (2)

$$\mathrm{\Phi }={\tan }^{-1}\left[\frac{\mathfrak{J}\left({\tilde{E}}_t{\tilde{E}}_{LO}^{*}\right)}{\Re \left({\tilde{E}}_t{\tilde{E}}_{LO}^{*}\right)}\right]-{\tan }^{-1}\left[\frac{\mathfrak{J}\left({\tilde{E}}_0{\tilde{E}}_{LO}^{*}\right)}{\Re \left({\tilde{E}}_0{\tilde{E}}_{LO}^{*}\right)}\right]$$ (3)

Above, ${\tilde{E}}_0$ (defined in Eq. 4) is the transmitted electric field absent intracavity absorbers (i.e., intracavity vacuum) and other residual phase shifts.

$${\tilde{E}}_0(\omega )={\tilde{E}}_i(\omega )\frac{T_e(\omega )}{1-R_e(\omega )}$$ (4)

Finally, the general expression for the normalized, complex-valued electric field transmission is:

$$\frac{{\tilde{E}}_t}{{\tilde{E}}_0}=\frac{\left[1-R_e(\omega )\right]\exp [-\alpha (\omega )L/2-i\phi (\omega )/2]}{1-R_e(\omega )\exp [-\alpha (\omega )L-i\phi (\omega )]}.$$ (5)

Equation (4) suggests that ${\tilde{E}}_0(\omega )$ can be approximated by a scaled version of ${\tilde{E}}_i(\omega )$. Importantly, the definition of ${\tilde{E}}_0(\omega )$ in Eq. (4) assumes no phase shifts related to cavity-comb coupling (including mirror dispersion). Put another way, Eq. (4) assumes that all comb teeth are perfectly matched to the center of their unique optical cavity resonances in the absence of molecular absorption. As we choose to bypass the optical cavity and scale ${\tilde{E}}_i(\omega )$, our complex transmission spectra will include contributions from mirror dispersion (Section 2.3.2).

In practice, spectral normalization can be performed in one of several ways. The spectrum of ${\tilde{E}}_0(\omega )$ can be recorded with the enhancement cavity at vacuum pressures, or by diverting the incident comb ${\tilde{E}}_i(\omega )$ around the optical cavity and through a variable attenuator (as was done here). We note that the choice of an alternative normalization procedure will potentially alter the definition of ${\tilde{E}}_0(\omega )$.

2.2. Limiting cases – weak/strong resonant intracavity absorption

For the high-reflectivity case considered here, Eq. (5) exhibits a series of widely spaced, narrow resonances that can be treated individually (i.e., 𝜈_FSR ≫ 𝛿_cav, where 𝜈_FSR is the cavity free spectral range and 𝛿_cav is the cavity mode line width). Neglecting the single-pass term in the numerator, assuming that both α𝐿 and φ are much less than unity and linearizing the exponential term in the denominator, then the transmitted signal in the vicinity of an individual cavity resonance reduces to the complex-valued Lorentzian function

$$\frac{{\tilde{E}}_t(\omega )}{{\tilde{E}}_0}=\frac{1}{\left(1+\frac{F}{\pi }\alpha (\omega )L\right)+i\frac{F}{\pi }\phi (\omega )},$$ (6)

where $F=\frac{\pi }{1-R_e}$ is the empty-cavity finesse. Equation (6) has a normalized magnitude that is closely approximated by

$$\frac{I_t(\omega )}{I_0}=1-\frac{F}{\pi }\alpha (\omega )L,$$ (7)

at the on-resonance condition corresponding to φ = 0. This result is the familiar expression for the cavity-enhanced transmission spectrum of a heterodyne signal in the low-absorption limit. More generally, for off-resonance excitation of a given cavity mode, the normalized magnitude, $\frac{I_t(\omega )}{I_0}$ and phase, Φ(ω), of the complex-valued heterodyne transmission signal are given by

$$\frac{I_t(\omega )}{I_0}=\frac{1}{\sqrt{{\left(1+\frac{F}{\pi }\alpha (\omega )L\right)}^2+{\left(\frac{F}{\pi }\phi (\omega )\right)}^2}},$$ (8)

and

$$\mathrm{\Phi }(\omega )=-{\tan }^{-1}\left(\frac{\frac{F}{\pi }\phi (\omega )}{1+\frac{F}{\pi }\alpha (\omega )L}\right).$$ (9)

Note that the on-resonance cavity enhancement factor 𝐹/π for CE-DCS appearing is half that found in Fourier transform (FT) frequency comb spectroscopy [65]. Also, inspection of Eqs. (7) and (9) illustrates that this on-resonance factor amplifies both the absorbance and phase-shift coefficients equally.

In the limit of high absorption where 𝐹α(𝜔)𝐿 ≫ 1, changes in the round-trip phase shift are dominated by absorption-induced dispersion in the cavity medium. For cases where absorption corresponds to discrete quantum transitions or a linear combination thereof, there is a simple relationship between the phase of the heterodyne signal and that of the complex-valued line shape $\tilde{g}(\omega )$ [69]. Assuming perfect matching between the interrogating comb and empty-cavity modes, the phase of the transmission signal becomes

$$\mathrm{\Phi }(\omega )=-{\tan }^{-1}\left(\frac{\phi (\omega )}{\alpha (\omega )L}\right)={\tan }^{-1}\left(\frac{\Im [\tilde{g}(\omega )]}{\Re [\tilde{g}(\omega )]}\right),$$ (10)

where $\Re [\tilde{g}(\omega )]$ and $\Im [\tilde{g}(\omega )]$ represent the real (absorptive part) and imaginary components (dispersive part), respectively, of $\tilde{g}(\omega )$. (For a complete derivation which relates φ(𝜔) to $\Im [\tilde{g}(\omega )]$, see Section 2.3.) In this limit, the phase of the transmitted heterodyne signal equals the phase of the complex-valued line shape. Moreover, for the more general case corresponding to an arbitrary amount of absorption and to non-zero mismatch between the comb and empty-cavity mode spacing, the complete solution (discussed below) remains sensitive to the complex-valued line shape. The ability of CE-DCS to interrogate both the real and imaginary components of the line shape contrasts with conventional cavity-enhanced spectroscopy methods which are only sensitive to intensities. Applications of such complex-valued line profile measurements include the study of advanced line shapes [70], as well as broadband effects like non-Lorentzian behavior in the far-wings, line mixing and collisional induced absorption [71, 72].

2.3. The round-trip phase shift coefficient

The round-trip phase shift coefficient φ(𝜔) is treated here as the sum of two parts: resonant molecular dispersion [φ_m(𝜔)] and cavity-comb dispersion [φ_c(𝜔)], each defined relative to the grid of optical frequencies established by the laser comb. The derivation of φ(𝜔) presented here makes use of the prior works of Thorpe et al. [73] and Foltynowicz et al. [65] as well as the following textbooks and book chapters: Siegman [68], Yariv [74], Lehmann [75], Boyd [76], and Nagourney [77].

2.3.1. Resonant molecular dispersion

The round-trip phase-shift coefficient for resonant molecular dispersion is defined as ${\phi }_{\text{m}}(\omega )={\phi }_{\text{m}}^{\prime }(\omega )-{\phi }_{\text{m}}^0(\omega )$, where ${\phi }_{\text{m}}^{\prime }$ and ${\phi }_{\text{m}}^0$ are the perturbed and unperturbed optical phases at each cavity mode. Absent mirror dispersion (Section 2.3.2), the unperturbed optical phases are defined as ${\phi }_{\text{m}}^0(\omega )=\omega t_{\text{rt}}$, where 𝑡_rt = 1/𝜈_FSR is the unperturbed cavity round-trip time (equal to the inverse of the cavity free spectral range 𝜈_FSR). Ignoring for now the mode offset due to mirror dispersion, we define ${\nu }_{FSR}={\nu }_q^0/q$ for the lowest order transverse modes, where ${\nu }_q^0$ is the nominal optical frequency of longitudinal mode order 𝑞. Returning to angular optical frequency, we write ${\phi }_{\text{m}}^0(\omega )=2\pi q\omega /{\omega }_q^0$.

The angular frequencies of the perturbed cavity modes are ${\omega }_q={\omega }_q^0\sqrt{1-{\chi }^{\prime }(\omega )}$, where χʹ(ω) is the real part of the complex linear susceptibility $\tilde{\chi }(\omega )$. If |χʹ(ω)| ≪ 1, the perturbed cavity mode angular frequencies are well-approximated by ${\omega }_q={\omega }_q^0\left[1-{\chi }^{\prime }(\omega )/2\right]$. This expression for 𝜔_𝑞 yields ${\phi }_{\text{m}}^{\prime }(\omega )=2\pi q\omega /\left\{{\omega }_q^0\left[1-{\chi }^{\prime }(\omega )/2\right]\right\}$ for the perturbed optical phases. Using the above expressions for ${\phi }_{\text{m}}^0(\omega )$ and ${\phi }_{\text{m}}^{\prime }(\omega )$, we arrive at an expression for the round- trip phase-shift coefficient for resonant molecular dispersion in Eq. (11).

$${\phi }_{\text{m}}(\omega )=2\pi \frac{\omega {\chi }_s^{\prime }(\omega )/2}{{\omega }_{\text{FSR}}}=2\pi \frac{\mathrm{\Delta }{\omega }_m}{{\omega }_{\text{FSR}}}$$ (11)

In Eq. (11), $\mathrm{\Delta }{\omega }_m=\frac{\omega {\chi }_s^{\prime }(\omega )}{2}$ and the cavity free spectral range in angular units is 𝜔_FSR = 2π𝜈_FSR. Note that $\tilde{\chi }(\omega )$ could be a linear combination of linear susceptibilities from any number of discrete molecular (or atomic) transitions.

The real part of the molecular susceptibility ${\chi }^{\prime }(\omega )=c^2{n}_{\text{a}}{S}_{\text{int}}\Im [\tilde{g}(\omega )]/\omega =c\varphi (\omega )/\omega$, where $\varphi (\omega )=c{n}_{\text{a}}{S}_{\text{int}}\Im [\tilde{g}(\omega )]$ is the familiar molecular dispersion coefficient in dimensions of inverse length, n_a = p_a/(k_BT) is the number density of absorbers, 𝑝_a is the partial pressure of the absorbers, 𝑘_𝐵 is the Boltzmann constant, 𝑇 is the sample temperature, and 𝑆_int is the transition line intensity in dimensions of area/(wave number × molecule). Therefore, Eq. (11) can be written in terms of ϕ(𝜔) as

$${\phi }_{\text{m}}(\omega )=\varphi (\omega )L.$$ (12)

2.3.2. Cavity-comb dispersion

Dispersion in an optical cavity is conveniently expressed as an expansion of the propagation constant (𝜔) about some angular frequency 𝜔₀ as

$$k(\omega )=k_0+{\left(\frac{\partial k}{\partial \omega }\right)}_{{\omega }_0}\left(\omega -{\omega }_0\right)+\frac{1}{2}{\left(\frac{{\partial }^{2}k}{\partial {\omega }^2}\right)}_{{\omega }_0}{\left(\omega -{\omega }_0\right)}^2+\cdots ,$$ (13)

where here the expansion is about the Pound-Drever-Hall (PDH, [78]) lock point (see Fig. 1). In this section, we define propagation constant expansions for both the cavity [𝑘_cav(ω)] and the comb [𝑘_comb(ω)], ultimately revealing the cavity-comb coupling dispersion term φ_c(ω) =2L[𝑘_cav(ω)–𝑘_comb(ω)]. The zeroth-order expansion coefficients, 𝑘₀, for the cavity and the comb are 𝑘₀,_{cav =}(ω₀ – δω_PDH)/c and 𝑘₀,_{comb =} ω_0/c, respectively, where 𝜔₀ is the angular optical frequency of the cavity mode to which the CW comb seed laser is locked, and δω_PDH is any offset in the angular frequency of the PDH locked comb tooth from the center of the reference cavity mode.

The first-order expansion coefficient 𝜕𝑘/𝜕𝜔 in Eq. (13) is the inverse group velocity (1/𝑣_𝑔). When local changes in n(𝜔) are small (i.e., 𝜔𝜕(𝜔)/𝜕𝜔 ≪ 1, assuming 𝑛(𝜔) = 1), 𝑣_𝑔 ≈ 𝑐/𝑛. This assumption is consistent with the general assumption that the propagation constant depends weakly on the angular frequency, and therefore allows for the expansion in Eq. (13). In this limiting case we can express the group velocity of the cavity (𝑣_𝑔,cav) and the group velocity of the comb (𝑣_𝑔,comb) in terms of the cavity free spectral range (𝑣_𝑔,cav = 2𝐿𝜈_FSR) and the best-match comb repetition rate (𝑣_𝑔,comb = 2𝐿𝜈_rep).

The second-order expansion coefficient 𝜕²𝑘/𝜕ω² is the group velocity dispersion (GVD), equal to 𝜕(1/𝑣_𝑔)/𝜕ω Note that the GVD is equal to the group delay dispersion (GDD) 𝐷_ω divided by the round-trip path length 2L. Gathered together, these terms yield two unique expressions for the cavity and the comb wave numbers, respectively.

$$k_{\text{cav}}(\omega )\approx \frac{{\omega }_0-\delta {\omega }_{\text{PDH}}}{c}+\frac{1}{2L{\nu }_{\text{FSR}}}\left(\omega -{\omega }_0\right)+\frac{D_{\omega }}{4L}{\left(\omega -{\omega }_0\right)}^2$$ (14)

$$k_{\text{comb}}(\omega )\approx \frac{{\omega }_0}{c}+\frac{1}{2L{\nu }_{\text{rep}}}\left(\omega -{\omega }_0\right)$$ (15)

The round-trip phase shift coefficient for cavity-comb dispersion is φ_c(ω) = 2𝐿[𝑘_cav(ω) − 𝑘_comb(ω)]. Defining the frequency axis reference frame so that 𝜔₀ is the angular frequency of the PDH locked comb tooth, we write ω − ω₀ = 𝛽ω_rep, where ω_𝑟𝑒𝑝 = 2π𝜈_𝑟𝑒𝑝 and β is a signed integer identifying each unique comb tooth relative to the PDH lock point (see also Section 2.4). Note that, at the PDH lock point, β = 0. Using the above expressions, and neglecting the GDD term in (ω) over the EO comb bandwidth (an assumption justified in the following paragraph) we arrive at the following equation:

$${\phi }_{\text{c}}(\omega )=2\pi (\frac{\beta {\mathrm{\Delta }}_{\text{c}}-\delta {\omega }_{\text{PDH}}}{{\omega }_{\text{FSR}}})=2\pi \frac{\mathrm{\Delta }\omega }{{\omega }_{\text{FSR}}},$$ (16)

where Δ_c = ω_FSR − ω_rep, and Δω = 𝛽Δ_c − δω_PDH is the apparent shift in angular frequency at each cavity mode relative to its unique comb tooth. Note that Eq. (16) and Eq. (11) are identical, and reveal an intuitive result: the phase-shift coefficient is simply described as being proportional to the frequency shift at each cavity mode divided by the cavity free spectral range.

For coupling femtosecond laser pulses to enhancement cavities, cavity mirrors with near-zero GDD are desirable. As an example, we see from Fig. 3 of [73] that zero-GDD mirrors can be engineered to have excursions in |D_v| = |D_𝜔|/2π < 10 fs² over an optical bandwidth of 40 nm. For the purposes of estimating the influence of mirror GDD on Δ𝜔, we assume 𝐷_𝑣 = 10 fs² (normal dispersion) and calculate ${\phi }_{\text{c}}^{(2)}(\omega )$ (where the superscript (2) indicates the use of only the second-order expansion coefficient) at the largest optical detuning from the PDH lock point used herein: 15𝜈_rep ≈ 3 GHz. The resulting phase shift is ${\phi }_{\text{c}}^{(2)}(\omega )=1.15\times {10}^{-8}$ rad. Rearranging Eq. (16), we estimate Δ𝜈 = Δω/(2π) = 0.37 Hz, a frequency shift due to GDD which is much less than the cavity line width of Δ_cav ≈ 10 kHz. At a mode order of 𝑚 = 2500 (optical detuning of 0.5 THz with 𝜈_rep = 203 MHz), the frequency shift due to a GDD of 10 fs² is virtually equivalent to Δ_cav. Therefore, quantitative modeling of CE-DCS performed with broadband frequency combs will require the inclusion of GDD. (The prerequisite inclusion of GDD was already alluded to [79] and ultimately exquisitely controlled [80] in the early works on CE spectroscopy using mode-locked lasers and non-DCS methods.) Broadband EO combs [55, 58, 63] provide a unique opportunity to interrogate GDD in a mode-resolved fashion because the CW laser which seeds the probe EO comb generator can be tightly phased-locked to a unique mode of the optical cavity (as was done here and in the first demonstrations of CE-DCS [50] and coherent CE-DCS [51] using EO combs).

Fig. 3.

A single time-domain interferogram (period of 2 μs), upper panel, and its corresponding Fourier transform, bottom panel, are shown in blue. In the bottom panel, the Fourier transform of 100 consecutive interferograms is also show in orange.

Finally, we note that for the general case of locking a broadband comb to a cavity with near-zero GDD at the lock-point wavelength, the next higher-order expansion term, proportional to 𝜕𝐷_ω/𝜕ω [which scales as (ω − ω₀)³], may not be negligible compared to the GDD term (second-order expansion coefficient).

2.4. Model expression for spectral fitting

The CE-DCS demonstrated here using EO frequency combs achieves cavity-comb coupling via a PDH phase lock of a single comb mode to a single resonator mode. For spectral analysis, it is therefore useful to shift the reference frame from absolute angular frequency to a relative frequency axis with the locking point as the reference frequency (as was introduced in Section 2.3.2.). The individual comb modes transmitted by the resonator are then identified by their unique signed mode number β, where β = 0 at the PDH locking point.

By substitution of φ = φ_m + φ_c derived in Section 2.3 into Eq. (1), we arrive at the following expression for the normalized transmitted electric field.

$$\frac{{\tilde{E}}_t}{{\tilde{E}}_0}=\frac{\left[1-R_e(\beta )\right]\exp \left\{-\frac{\alpha (\beta )L}{2}-\frac{i[\varphi (\beta )-\varphi (0)]L}{2}-\frac{i\pi \left(\beta {\mathrm{\Delta }}_{C,\nu }-{\delta }_{\mathrm{PDH},\nu }\right)}{{\nu }_{\mathrm{FSR}}}\right\}}{1-R_e(\beta )\exp \left\{-\alpha (\beta )L-i[\varphi (\beta )-\varphi (0)]L-\frac{i2\pi \left(\beta {\mathrm{\Delta }}_{c,\nu }-{\delta }_{\mathrm{PDH},\nu }\right)}{{\nu }_{\mathrm{FSR}}}\right\}}$$ (17)

The full model is obtained by substitution of Eq. (17) into Eqs. (2–3) with the addition of a complex-valued linear baseline. For the data analysis reported below, we fit this model to the measured multi-heterodyne transmission spectrum.

3. Experimental methods

An illustration of the experimental setup is shown in Fig. 2. The experiment utilized principles of frequency-stabilized cavity ring-down spectroscopy (FS-CRDS) [81], frequency agile, rapid-scanning (FARS) CRDS [82], and electro-optic comb generation and cavity coupling [50, 51] taught elsewhere in the literature. Therefore, only a brief description of the components illustrated in Fig. 2 is provided here.

Fig. 2.

Optical layout for coherent CE-DCS using EO frequency combs. Fiber components are in black, the free-space HeNe laser beam path and optics are in red, the probe comb free-space path is in blue, and the free-space continuous-wave Pound-Drever-Hall locking laser path is in green. Acronyms are defined as follows: AOM, acousto-optic modulator; ECDL, external cavity diode laser; FA, fiber amplifier; Iso, optical isolator; FS, fiber splitter; EOM, electro-optic phase modulator; WM, wavelength meter; OFC, optical frequency comb; PD, photodetector; DMZ, dual-drive Mach-Zehnder modulator; Sw, fiber optic switch; Att, variable optical attenuator; FC, fiber combiner; C, near-infrared camera.

Beginning in Fig. 2 with the external-cavity diode laser (ECDL) and working clockwise in the direction indicated by the solid black arrows, the optical output of a fiber amplifier (FA) passes through a fiber isolator (Iso) and into a fiber splitter (FS). The outer path is further split at a second FS to generate both the probe/reference and local oscillator (LO) combs using dual-drive Mach-Zehnder modulators (DMZ). An acousto-optic modulator (AOM) was added to the LO comb to shift the resulting dual-comb interferograms away from DC. Using a fiber-optic switch (Sw), the probe/reference comb was either coupled to the optical cavity containing a gas sample (probe) or directed to bypass the gas sample (reference) and through a variable fiber attenuator (Att) in order to generator a reference interferogram. Fiber combiners (FC) are used to combine the optical signals onto a fiber-coupled photodetector (PD) for digitization and analysis.

A representative interferogram signal, along with representative Fourier transforms of individual and averaged interferograms, are plotted in Fig. 3. Individual interferograms were coherently averaged in the time domain with the aid of a low-bandwidth (1 kHz) dual-comb servo loop described in [51].

Returning to the first FS, the inner optical fiber path was split for both absolute optical frequency referencing using either a wavelength meter (WM) or a self-referenced optical frequency comb (OFC), and for locking of the EO comb seed laser to the optical cavity as described in [82]. For laser locking, an electro-optic modulator was used to generate PDH sidebands. Upon reflection by the optical cavity, the PDH error signal was detected using another PD and the transmitted optical mode was imaged onto a near-infrared camera (C). Finally, the optical cavity length was actively stabilized to a frequency-stabilized HeNe laser using another AOM, a piezoelectric transducer fixed to one of the cavity mirrors, and a low-bandwidth (10 Hz) feedback loop. The HeNe transmission signal was detector using a third PD.

4. Experimental validation of the transmission model

The complex transmission model presented in Section 2 contains the phase shift parameter φ which describes the mismatch between the comb mode spacing and the cavity free spectral range (Δ_c,ν = Δ_c/2π = 𝜈_rep − 𝜈_FSR) and the PDH locking offset parameter δ_PDH,ν = δ_PDH/2π. The value of Δ_c,ν is non-zero if the cavity length drifts 1) from the time that 𝜈_FSR was last measured, 2) between successive spectral acquisitions, or 3) if 𝜈_rep of the probe comb is manually detuned from 𝜈_FSR. The value of δ_PDH,ν is non-zero if the PDH locking electronics introduce an apparent DC offset (potentially when the molecular dispersion at the PDH reference mode is significant).

4.1. Influence of Δ_c,ν on the spectrum of carbon monoxide (CO)

Figures 4 and 5 show the transmission and phase spectrum of 10 kPa (75 Torr) of NIST Standard Reference Material (SRM^®) 2637a, CO in N₂ (CO concentration of 2472.8 μmol/mol ± 4.2 μmol/mol), recorded with intentional detunings of Δ_c,ν = −500 Hz (Fig. 4, approximately −δ_cav/20) and Δ_c,ν = −2 kHz (Fig. 5, approximately −δ_cav/5), respectively, where δ_cav = 𝜈_FSR/𝐹 is the cavity line width. The line center and line strength of the CO transition are 𝜈₀ = 6364.767 625 cm⁻¹ and 𝑆_int = 1.551 × 10⁻²³ cm/molecule [69]. In Figs. 4 and 5, normalized transmission and phase spectra are plotted in the top and bottom panels, respectively. For a given complex cavity transmission spectrum, we fitted both the transmission and phase spectra simultaneously using Eq. (17) to retrieve the fitted detuning Δ_c,ν. The fitted models (black lines) reproduced the experimental data points (black circles) well, as evidenced by the absence of systematic structure in the fitted residuals. The retrieved values of Δ_c,ν = −456 Hz and Δ_c,ν = −1.92 kHz are in good agreement with the intended detunings of Δ_c,ν = −500 Hz and Δ_c,ν = −2 kHz. We also observed a fitted value of δ_PDH,ν ≠ 0, most likely from an unintended consequence of optimizing the cavity throughput via adjustment of the PDH locking conditions while maintaining a constant comb mode spacing (see Section 4.2).

Fig. 4.

Transmission (top) and phase (bottom) spectrum of 10 kPa of 2472.8 μmol/mol CO in a balance of N₂ recorded with an intentional detuning of 𝑓_rep from 𝑓_FSR of Δ_c,ν = −500 Hz. Experimental data points are plotted in the black circles and the fitted model in the solid black lines. Along with Δ_c,ν, the PDH offset parameter δ_PDH,ν was also floated in the fitting routine, resulting in fitted values of Δ_c,ν = −456(5) Hz and δ_PDH,ν = −630(50) Hz for the black trace. The blue dashed lines show the model calculated at Δ_c,ν = +456 kHz, clearly revealing our ability to unambiguously determine the sign of Δ_c,ν. The red dotted lines are calculated with Δ_c,ν = δ_PDH,ν = 0, representative of only the cavity-enhanced complex molecular response.

Fig. 5.

Transmission (top) and phase (bottom) spectrum of 10 kPa of 2472.8 μmol/mol CO in a balance of N₂ recorded with an intentional detuning of 𝑓_rep from 𝑓_FSR of Δ_c,ν = −2 kHz. Experimental data points are plotted in the black circles and the fitted model in the solid black lines. Along with Δ_c,ν, the PDH offset parameter δ_PDH,ν was also floated in the fitting routine, resulting in fitted values of Δ_c,ν = −1.91(6) kHz and δ_PDH,ν = 510(90) Hz for the black trace. The blue dashed lines show the model calculated at Δ_c,ν = +1.92 kHz, again clearly revealing our ability to unambiguously determine the sign of Δ_c,ν. The red dotted lines are calculated with Δ_c,ν = δ_PDH,ν = 0, representative of only the cavity-enhanced complex molecular response.

Also plotted in Figs. 4 and 5 are additional simulations of the transmission and phase spectra. For the blue dashed lines, we reverse the sign of the fitted Δ_c,ν to be −Δ_c,ν; for the red dotted lines we fix Δ_c,ν = δ_PDH,ν = 0. The red dotted lines, with Δ_c,ν = δ_PDH,ν = 0, correspond to the hypothetical transmission and phase spectrum absent round-trip phase shifts associated with the cavity-comb coupling. The additional simulations illustrate the sensitivity of our multiplexed fit to the round-trip phase shift parameter φ, which we uniquely interrogate in this experiment with full comb tooth (cavity mode) resolution. Importantly, we are also measuring these round-trip phase parameters relative to a single cavity mode with fixed phase, i.e. the PDH locking point with no optical detuning.

4.2. Influence of δ_PDH,ν on the spectrum of carbon dioxide (CO₂)

Examples of pure CO₂ spectra with near-zero values of Δ_c,ν and large non-zero values of δ_PDH,ν are shown in Fig. 6. For this series of complex-valued spectra, cavity throughput (i.e., transmitted optical power) was optimized while the seed laser for the probe comb was locked to an enhancement cavity mode experiencing significant molecular dispersion. By maximizing probe comb throughput, the optical frequency of the PDH-locked seed laser was detuned from the center of the cavity mode with = 0, resulting in δ_PDH,ν ≠ 0. This condition was achieved by adjusting the zero-crossing of the PDH error signal until a maximum in the down-converted probe comb interferogram was observed. Values of |δ_PDH,ν| nearly equal to half the empty-cavity mode line width are achieved, although we note that the local mode line width is significantly broadened by the presence of molecular absorption at the lock point. For the spectra in Fig. 6 (60 s acquisition time), the largest fitted value of |Δ_c,ν| was observed for a CO₂ pressure of 39 Pa (black circles), where Δ_c,ν = 221291(9) Hz, followed by Δ_c,ν = 80(6) Hz for the lowest pressure of 5.6 Pa CO₂ (red diamonds) and Δ_c,ν = 33(25) Hz for the intermediate pressure of 12 Pa CO₂ (blue squares).

Fig. 6.

Transmission (top) and phase (bottom) spectrum of pure CO₂ at pressures of 5.6 Pa (red), 12 Pa (blue) and 39 Pa (black). The approximate line center and line strength of the transition are ${\tilde{v}}_0=6336.242389{\text{cm}}^{-1}$ and 𝑆_int = 1.619 × 10⁻²³ cm/molecule [69]. Cavity-comb coupling conditions were optimized for maximum transmission of optical power through the cavity with Δ_c,ν ≈ 0. To achieve these conditions, δ_PDH,ν was forced to be large by adjusting the PDH locking conditions. Importantly, the model described in Section 2 (solid lines) is well fitted to the experimental data (open points) in all cases without a priori knowledge of either the magnitude or the sign of δ_PDH,ν, thus further validating the functionality of the complex transmission model.

5. Applications

5.1. Absolute transition frequencies of carbon dioxide (CO₂)

To apply the validated CE-DCS model, we measured absolute frequencies for 38 distinct rovibrational transitions within the 30012 ← 00001 ¹²C¹⁶O₂ band centered at λ = 1575 nm. Transition frequencies for this band have previously been measured by FARS CRDS with absolute uncertainties between 30 kHz and 800 kHz [30]. The resulting FARS-CRDS fitted accuracy for the 30012 ← 00001 band origin (G_v) was 59 kHz, or equivalently a fractional uncertainty of ${\sigma }_{G_{\nu }}/G_{\nu }=3\times {10}^{-10}$ [83].

Representative transmission (top left panel) and phase (bottom left panel) CE-DCS spectra of CO₂ are plotted in Fig. 7. The CE-DCS of a high-purity, isotopically enriched CO₂ gas sample (pressure 𝑝 = 13 Pa, temperature 𝑇 = 296 K, ¹²C isotopic purity 0.997 6), spanning nearly 2 THz, was measured in a line-by-line, multiplexed fashion. For each rovibrational transition, 60 multiplexed spectra (30 probe comb lines per spectrum, spaced by 𝜈_rep = 𝜈_FSR ≈ 203.085 MHz) were recorded in 2 s of integrated time per spectrum, and at an average duty cycle of 𝑡_int/𝑡_total = 60 %. Given lessons learned in Section 4, a slow integrator feeding back to the ECDL cavity length was added to the PDH servo to minimize DC offsets in the PDH phase locking servo loop.

Fig. 7.

Transmission (top panels) and phase (bottom panels) spectrum of the 30012 ← 00001 ¹²C¹⁶O₂ band. In the left panels, the band-wide spectrum is shown at two different detunings of the PDH locking point from the individual CO₂ transitions frequencies of ±1.2 GHz (black solid lines and blue dashed lines, respectively). The band-wide spectrum was recorded in a total integration time of 76 s (38 transitions, 2 s per spectrum). In the right panels, spectra of the R14e transition are shown. Again, the experimental data points recorded at two different optical detunings are plotted, this time as black dots and blue circles, while the fitted models are plotted as solid black lines and solid blue lines.

Individual interferograms were coherently averaged for 1 s using fast acquisition software and then analyzed in the time domain to retrieve the amplitude and phase at each comb mode [51]. The transmission and phase spectra were modeled using Eq. (17) and Eqs. (3–4), and fitted using a non-linear least-squared algorithm. Transition frequencies, fitted relative to the PDH locked carrier frequency, were ultimately referenced to a commercial self-referenced OFC as illustrated in Fig. 2. The 𝑓_ceo of the reference comb was stabilized via an f-2f interferometer, and the reference comb 𝑓_rep was stabilized to the same Cs clock signal used to stabilize both EO frequency comb generation and coherent DCS interferogram collection.

Following the consecutive acquisition of 60 multiplexed spectra, the ECDL was rapidly tuned to the next rovibrational transition and multiplexed spectral acquisition was resumed. Because molecular dispersion induces asymmetries in the observed transmission spectra, we recorded the entire 30012 ← 00001 band with two different constant frequency offsets between the PDH locked carrier comb mode and the expected molecular transition frequency of ±1.2 GHz, or approximately ±4 cavity modes. Despite the intentional frequency offset, which was intended to minimize the influence of molecular dispersion on the fitted parameter Δ_c,ν, phase offsets are still clearly observable in the bottom panels of Fig. 7. In the bottom right panel of Fig. 7, the black and the blue fitted R14e transitions exhibit nearly equal in magnitude but opposite in sign phase shifts relative to the empty-cavity mode phases (i.e., ϕ = 0). This is expected, and fully captured at every probe comb tooth by our complex cavity transmission model without a priori knowledge of the molecular or mirror dispersion.

5.2. Accurate spectroscopic parameters for the 30012 ← 00001 ¹²C¹⁶O₂ band

Measured transition frequencies (Section 5.1) are plotted in Fig. 8 vs. m, where m = −J″ for the P-branch (ΔJ = J′ − J″ = −1) and m = J″ + 1 for the R-branch (ΔJ = +1), where J″ and J′ are the lower- and upper-rotational state quantum numbers. The absolute transition frequencies plotted in Fig. 8 are the weighted average of 60 individual multiplexed spectra. Fitted residuals from the spectroscopic model (black line, top panel) are plotted in the bottom panel of Fig. 8. The fitted spectroscopic model yielded values for the 30012 ← 00001 vibrational band origin frequency G_v along with the upper-state inertial parameters $B_{\nu }^{\prime },D_{\nu }^{\prime }$, and $H_{\nu }^{\prime }$ [84]. For the fitted model, the lower-state inertial parameters were fixed at the values reported in Ref. [83].

Fig. 8.

Top panel: Measured absolute transitions frequencies within the 30012 ← 00001 ¹²C¹⁶O₂ band (blue circles) plotted vs. m, along with the fitted spectroscopic model (black line). Bottom panel: Residuals from the fitted model, plotted along with experimental uncertainties (1σ, standard deviation of the fitted absolute transitions frequencies from 120 individual spectra, each recorded in 2 s of integration time).

Experimental error bars (±1σ uncertainty) plotted in the bottom panel of Fig. 8 are the average standard deviation from the two 60-spectra ensembles, with two exceptions (P10e and R12e, with m = −10 and m = 13) where absolute transition frequencies were only measured at a single detuning PDH locking point from the molecular transition frequency due to a loss in 𝑓_ceo stabilization of the self-referenced OFC during CE-DCS data acquisition.

Fitted values of G_v, $B_{\nu }^{\prime },D_{\nu }^{\prime }$, and $H_{\nu }^{\prime }$ are listed in Table 1. By fitting 38 unique transition frequencies across the 30012 ← 00001 band, we report G_v with a combined standard uncertainty of ${\sigma }_{G_{\nu }}=770$ kHz and a relative uncertainty of ${\sigma }_{G_{\nu }}/G_{\nu }=4\times {10}^{-9}$. The combined standard uncertainty was limited by the standard error of the fitted spectroscopic model, which in turn was dominated by the individual type-A uncertainties plotted in Fig. 8 and described in the preceding paragraph. Any residual systematic uncertainty associated with line asymmetries unaccounted for by our CE-DCS model was reduced by recording two sets of spectra with equal and opposite optical detunings of the PDH lock point with respect to the transition frequency. In addition, the uncertainty associated with the measurement of the comb seed laser optical frequency by beating with the self-referenced OFC is <20 kHz [83], and therefore also negligible compared to the standard error from the fitted spectroscopic model. All CE-DCS parameters measured here and listed in Table 1 exhibit ±2σ agreement with those measured by FARS-CRDS [83].

Table 1.

30012 ← 00001 ¹²C¹⁶O₂ band origin (G_v in MHz) and upper-state inertial parameters ($B_{\nu }^{\prime },D_{\nu }^{\prime }$ and $H_{\nu }^{\prime }$ in MHz) measured by CE-DCS (this work) compared with literature values measured by FARS-CRDS [83]. Experimental uncertainties (1σ) are in parentheses. Ground-state inertial parameters were fixed to those reported in Ref. [83].

	Gv	${\mathit{B}}_{\mathit{\nu }}^{\prime }$	${\mathit{D}}_{\mathit{\nu }}^{\prime }$ (10−3)	${\mathit{H}}_{\mathit{\nu }}^{\prime }$ (10−9)
Ref. [83]	190 303 782.258(59)	11 585.628 64(39)	−2.942 62(69)	15.81(32)
This work	190 303 781.28(77)	11 585.636 7(51)	−2.954 2(85)	20.6(3.6)

6. Conclusions

We derived, validated and applied a complex transmission model for coherent cavity-enhanced dual-comb spectroscopy (CE-DCS), using the model to measure the 30012 ← 00001 ¹²C¹⁶O₂ vibrational band origin with a combined standard uncertainty of 770 kHz. We also report low-uncertainty measurements of the upper-state inertial parameters of CO₂ which agree within ±2σ of the literature values. With a total integration time of 76 s, we recorded virtually an entire vibrational band of CO₂ covering an optical frequency range of nearly 2 THz.

Uniquely, coherent CE-DCS enables the multiplexed interrogation of enough enhancement cavity modes to retrieve the full complex (amplitude and phase) molecular response of individual rovibrational transitions without laser scanning, thus “freezing” potential sources of systematic uncertainty. With high acquisition rates (up to 2 × 1/Δ𝜈_rep) and a high-throughput data stream amenable to actively controlled coherent averaging [51], postprocessing [48], and/or adaptive sampling [49], deep averaging of fitted spectroscopic parameters is possible, and therefore so is high precision. Although intensity measurements of cavity transmission modes can also measure complex molecular response (via mode position and line width) [82, 85–87], those techniques each require laser scanning, and therefore are inherently susceptible to complications arising from temperature drifts, mechanical and acoustic vibrations, polarization and birefringence drifts, and parasitic etalons. The targeted, accurate multiplexed measurements reported herein are readily extended to continuous coverage over bandwidths of at least 2 THz by leveraging the rapid-scanning capabilities of modern continuous-wave (CW) lasers. For example, external cavity diode lasers are commercially available with tuning rates of up to 20 nm/s (2.4 THz/s at λ = 1575 nm). With the validated complex transmission model presented here, we envision routinely performing coherent CE-DCS measurements of individual rovibrational transitions without laser scanning, and then acquiring additional targeted rovibrational transitions with rapid and broadband laser agility, thus bridging the gap between precision molecular spectroscopy using CW lasers and broadband frequency combs while leveraging the high optical and data throughput advantages of both types of laser sources.