## Abstract

Long-term mutual coherence is a key factor that affects the signal-to-noise ratio and resolution of a dual-comb interferometer. To realize a phase-stable dual-comb interferometer configuration, tightly phase-locked loop systems or digital error correction methods with external optical reference are commonly used. This paper presents a self-referencing digital error correction method based on the short-term spectral characteristics of interferograms to reduce the cost and complexity of the phase-stable dual-comb interferometer configuration. In our experiment, fully mutual coherence of a dual-comb interferometer is reconstructed and 1 Hz theoretical linewidth in 1 s acquisition time is achieved by digitally compensating for time jitter, center frequency jitter, and carrier-envelope-phase jitter, offering an effective technique for advanced dual-comb applications.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

An optical frequency comb (OFC) is a coherent laser source with wide spectral range and narrow pulse width simultaneously [1], consisting of a series of longitudinal modes with equal spacing in frequency domain [2]. The interval between adjacent longitudinal modes is referred to as repetition frequency (${f_r}$). If carrier-envelope-offset (CEO) frequency (${f_{ceo}}$) can be identified at the same time, then all the continuous wave (CW) laser components in the OFC can be described by the mathematical formula ${\nu _n} = {f_{ceo}} + n{f_r}$ [3]. When a pair of OFCs with slight repetitive frequency differences are combined [4,5], the optical longitudinal modes of dual OFCs originally over THz band beat reciprocally and down-convert to the MHz frequency band. The original frequency, amplitude, and phase information are preserved in the corresponding radio frequency (RF) comb, whose adjacent interval decreases from ${f_r}$ to $\Delta {f_r}$. The original information encoded in OFCs can be reconstructed by using analog and digital schemes prevalently conducted in RF analysis [6]. This dual-comb interferometer (DCI) configuration has been widely used in applications, such as gas molecular absorption spectroscopy [7–9], absolute distance measurement [10–12], coherent Raman spectroscopy [13–15], and microscopy [16–18] because of its high resolution, high sensitivity, broad spectral range, and fast refresh rate [19,20].

Despite its impressive strengths, the DCI configuration requires an extremely high relative phase stability of OFCs. Coherence between dual OFCs suffers from instability of repetition frequency and offset frequency induced by resonant cavity length vibrations, temperature drifts, and quantum fluctuations [21,22]. Thus, RF interferograms (IGMs) generated by asynchronous optical sampling would be severely distorted and spectral information would be submerged in jitter noise due to the relative frequency variation. Even if conventional closed-loop frequency-locked electronic systems are implemented and the repetition frequency and CEO frequency are traced to an RF frequency reference, the linewidth of the RF components generated by DCI is still larger than the repetitive frequency difference due to the limited bandwidth of the actuator and the short-term stability of the RF frequency Ref. [23]. This aliasing phenomenon between adjacent RF longitudinal modes causes the resolution and sensitivity of the actual DCI configuration to become worse than the theoretical parameters. Furthermore, the performance cannot be directly improved by coherent averaging on account of the lack of long-term mutual coherence, which limits the application of DCI in various fields [19,20].

To improve the mutual coherence of DCI configuration, a common laboratory method involves ultra-stable CW laser as optical frequency reference. Ultra-high bandwidth actuators and f-2f nonlinear interferometers are also required to achieve a tightly locked phase-stable DCI configuration [24–26]. Tightly locked DCI can obtain sub-hertz level relative linewidth and substantial mutual coherence at the cost of large size, complex experimental setup, and high cost, thereby making it difficult to replicate outside of the laboratory scene. Another common method is to measure optical phase jitter and time jitter using narrowband FBGs [27] or narrow linewidth CW lasers as intermediaries [28]; this method is prevalently referred to as optical referencing technique. Analogous adaptive sampling [29] or digital error correction [30] steps are then imposed upon IGMs to improve the mutual coherence of a DCI. The corrected IGMs are capable of performing long-term coherent averaging with acquisition time of over 24 hours [31], reaching comparable performance to the tightly locked DCI configuration. In general, digital error correction scheme based on the optical referencing technique simplifies the DCI stabilization setup and reduces the practical threshold outside the lab. However, CW laser intermediaries (at least kHz level linewidth) and extra sampling channels are still necessary to provide optical referencing error, which limits the minimum complexity and cost of the DCI configuration. In addition, the difference between reference path and measurement path would lead to deviation of the digital error correction [28].

To solve these problems, computational algorithms, such as Kalman filter [32] or cross-ambiguity function [33], have been utilized to calculate the jitter noise information contained in the IGMs themselves and recover the mutual coherence of the DCI configuration. However, Kalman filter method is complex with high numerical difficulty and long calculation time, moreover, it is more suitable to process approximately continuous IGMs based on QCL combs and electro-optic combs rather than mode-locked fiber lasers. Cross-ambiguity function method demands two-dimensional integration and image search, costing considerable computing resources, and the accuracy would decrease in some cases as the absence of extracting phase jitter. This paper presents a self-referencing digital error correction method that applies simple short-time Fourier transform on apodized IGM across the region around each single centerburst, extracting time jitter, center frequency jitter, and carrier-envelope phase jitter of IGMs from the real-time amplitude spectrum and phase spectrum with a refresh rate equivalent to the value of $\Delta {f_r}$. Digital compensation steps are then imposed on IGMs to obtain a long-term phase-stable DCI configuration where the calculated jitters are corrected substantially. This method also alleviates the requirement on the absolute optical frequency stability of DCI, completely dispensing with the detection and active control of ${f_{ceo}}$, which guarantees the low cost and easy realization of a DCI configuration with remarkable long-term mutual coherence.

The rest of the paper is organized as follows. First, the principle and mathematical model of DCI are described in detail, while theoretical support and specific steps of the post-processing algorithm based on self-referencing digital error correction method are illustrated. Second, a numerical simulation is conducted according to the mathematical model of DCI, thereby proving the effectiveness of the algorithm in error extraction and correction. Third, an experiment based on a pair of erbium-doped fiber OFCs is demonstrated, providing solid evidence that the self-referencing digital error correction method achieves the performance of the optical referencing technique and realizes a phase-stable DCI configuration. Finally, the unique properties and potential extensions of this method are discussed.

## 2. Principle

#### 2.1 Principle of DCI and mathematical model of IGMs

The schematic of a basic DCI setup is shown in Fig. 1. A pair of OFCs with small repetition frequency differences are injected into the same photodetector, down-converting the information from optical domain to RF domain within the electrical bandwidth. A pair of photodetectors separately extract the repetition frequency of dual OFCs, whose unbiased reference values can be determined by two low-bandwidth phase-locked loops. The loose locking system can suppress the random wandering of the optical modes, enabling time repetition frequency optimization and longer measurement time compared with the totally free-running DCI configuration. An optical band-pass filter should be exploited before the detector to satisfy the requirement of the band-pass sampling rule. In the electrical path, an electronic low-pass filter is used to obtain the pure IGMs by suppressing the independent detector response of dual OFCs. IGMs are digitized by a high-speed ADC and further post-processing algorithms are exploited to retrieve the origin optical information.

The principle of DCI can be revealed from two aspects: time domain and frequency domain. Figure 2 demonstrates the down-converting process from optical scale to RF scale in terms of the time domain and frequency domain, which corresponds to asynchronous optical sampling and multi-heterodyne interference respectively. The principle and mathematical model of DCI in the time domain have been described in [34]. The cross-correlation function of dual-carrier envelope optical pulse sequences is obtained by asynchronous optical sampling, with the time scale stretched factor $k = {f_r}/\Delta {f_r}$. For the frequency domain interpretation, DCI arises from heterodyne beating located on different RF frequencies between pairs of optical components, thereby generating an RF comb structure. In this study, the mathematical model is built in detail based on the frequency domain principle. The origin and ultimate forms of jitter noise are considered in the model to simulate the actual experimental setup.

From the frequency domain characteristics shown in Fig. 2(c), the electric field equations of two OFCs can be expressed as follows:

*p*

^{th}component of OFC1 and the

*q*

^{th}component of OFC2 are limited to ${{\upsilon }_{1p}} = {f_{ceo1}} + p{f_{r1}}$ and ${{\upsilon }_{2q}} = {f_{ceo2}} + q{f_{r2}}$ according to the nature of mode-lock lasers, where ${f_{r1}}$, ${f_{r2}}$, ${f_{ceo1}}$ and ${f_{ceo2}}$ represent the repetition frequencies and CEO frequencies of two separate OFCs. Neglecting the effects of nonlinear dispersion in DCI, we can write the corresponding phases of longitudinal modes in optical frequency domain as ${\phi _{1p}} = {\varphi _{01}} + 2{\pi }{{\upsilon }_{1p}}{\tau _{01}}$ and ${\phi _{2q}} = {\varphi _{02}} + 2{\pi }{{\upsilon }_{2q}}{\tau _{02}}$, where $-{\tau _{01}}$, $-{\tau _{02}}$ are the initial pulse center positions and ${\varphi _{01}}$, ${\varphi _{02}}$ are carrier envelope phases of two OFCs. Thus, continuous time-domain carrier envelope pulse sequences are uniquely determined.

The heterodyne interference output voltage generated by the square effect of the detector is given by

*p*

^{th}component in OFC1 and the

*q*

^{th}component in OFC2.

The RF frequency components after multi-heterodyne interference are determined by

*p*a common index between the optical comb and RF comb.

Electronic filter is used to map each optical frequency in OFC1 to one single RF frequency. Then, the ordinal numbers *p* and *q* match each other and are limited to satisfy the following inequality:

*p*is defined by the center wavelength and bandwidth of the optical band-pass filter, and only one possible

*q*responds to a specific value of

*p*under the limitation of Eq. (5). However, Eq. (5) cannot guarantee the inverse mapping from RF frequency to optical frequency unless the location of RF spectrum keeps away from “dead zones” to avoid aliasing [35].

Under the preceding necessary requirement, ${\Delta }p = p - q$ is constant across the entire observed spectrum, and corresponding phases of the RF frequency components are still linear and encoded by

Due to the high similarity between Eq. (9) and the OFC’s electric field equation (Eq. (1)), except for the difference in scale, the IGMs can be expressed analogically in the form of carrier-envelope pulse sequences as

#### 2.2 Algorithm of self-referencing digital error correction

Equation (10) is based on an ideal DCI system without any jitter noise. However, the existence of typical jitter noise in two OFCs is intrinsic. To add jitter noise to our mathematical model, we introduce a basic assumption that parameters of IGMs are invariable within the apodized IGM around each centerburst. This assumption can be simply explained by the remarkable intrinsic short-term stability of the free running mode-locked lasers [5,36,37] and was also proved in our experimental results. Then, short-time Fourier transform could be implemented on each apodized IGM, extracting transient carrier frequency, pulse envelope center position, and carrier-envelope phase simultaneously. Due to the highly discrete property of IGMs, these three time-variant parameter should be used to describe the relative instability between separate apodized IGMs, deriving center frequency jitter, time jitter and carrier envelope phase jitter for digital error correction respectively.

In our DCI configuration, as the application of two loose PLL feedback circuits shown in Fig. 1, repetition frequency accuracy is maintained over a long time scale and classic phase noise model can be used to characterize the instability [38]. In this case, ${f_{r1}}$ and ${f_{r2}}$ are regarded as constants with the reference value measured by a frequency counter. The real-time instability of two OFCs’ repetition frequency introduces the time jitter ${\delta }{\tau _{01}}(t )$, ${\delta }{\tau _{02}}(t )$, along with the carrier envelope phase jitter ${\delta }{\varphi _{01}}(t )$, ${\delta }{\varphi _{02}}(t )$ in optical scale, which ultimately induces the time jitter $\mathop {\delta \tau }\nolimits_0^{RF} (N )$ and the carrier envelope phase jitter $\delta \mathop \varphi \nolimits_0^{RF} (N )$ in IGMs according to Eqs. (7) and (8). However, considering the absence of absolute frequency stabilization in our experimental system, CEO frequencies of two OFCs are time variable, whose contributions to carrier envelop phase jitter are ${\delta }{\varphi _{01}}(t )= \int_0^t {[{{f_{ceo1}}(\tau )- {f_{ceo1}}(t )} ]d\tau } $ and ${\delta }{\varphi _{02}}(t )= \int_0^t {[{{f_{ceo2}}(\tau )- {f_{ceo2}}(t )} ]d\tau } $ in optical scale. As a result, extra carrier envelope phase jitter $\delta \mathop \varphi \nolimits_0^{RF} (N )$ is added into IGMs, as well as the periodically updated $\mathop f\nolimits_c^{RF} (N )$. Therefore, IGMs with jitter noise caused by frequency instability can be amended as

*N*covers negligible duration time compared with the repetition period, overlapping between separate single IGM could be ignored. Therefore, apodized IGM around the

*N*

^{th}centerburst is reasonable to be regarded as the independent part of IGMs corresponding to

*N*. Thus, FFT is conducted to calculate the center of frequency gravity based on amplitude spectra as $\mathop f\nolimits_c^{RF} (N )$, while the slope and intercept of phase spectra are $2{\pi }({ - N\mathop T\nolimits_r^{RF} + \mathop \tau \nolimits_{_0}^{RF} + \delta \mathop \tau \nolimits_{_0}^{RF} (N )} )$ and $N\Delta \mathop \varphi \nolimits_{ce}^{RF} + \mathop \varphi \nolimits_0^{RF} + \delta \mathop \varphi \nolimits_0^{RF} (N )$, respectively.

For a practical IGM sampled by fast ADC, linearity of phase spectra can be fairly distorted on account of optical nonlinear chirp and electronic phase delay, and considerable artifacts in the linear fitting process can be generated. Fortunately, nonlinear process has a negligible impact on the difference between two adjacent phase spectra, and artifacts can be effectively suppressed by linear fitting the differential phase spectra. After this differential process, an accurate relative slope and intercept of the fitted curve can be obtained, which correspond to $\textrm{2}\pi ( - {T_{RFr}} + \delta {\tau _{RF0}}({N + 1} )- \delta {\tau _{RF0}}(N ))$ and $\Delta \mathop \varphi \nolimits_{ce}^{RF} + \delta \mathop \varphi \nolimits_0^{RF} ({N + 1} )- \delta \mathop \varphi \nolimits_0^{RF} (N )$. The first apodized IGM is set as the reference ($N = 0$), where $\delta \mathop \tau \nolimits_{_0}^{RF} (0 )= 0$ and $\delta \mathop \varphi \nolimits_0^{RF} (N )= 0$ are identified as the initial conditions for the following accumulation process. As ${T_{RFr}} = 1/{\Delta }{f_r}$ can be acquired due to our repetition frequency locking system, $\delta \mathop \tau \nolimits_{_0}^{RF} (N )$, $\mathop f\nolimits_{_c}^{RF} (N )- \mathop f\nolimits_{_c}^{RF} (0 )$, and $\delta \mathop \varphi \nolimits_0^{RF} (N )$ are calculated as time jitter, center frequency jitter, and carrier envelope phase jitter, which are three fundamental forms of jitter in separate IGMs.

Correction algorithm based on the calculated jitter consists of three steps to compensate for three fundamental forms of jitter respectively, which are listed as follows:

Step 1: Compensating for time jitter by digital interpolation and resampling. Timing-corrected IGMs can be mathematically expressed as

Step 2: A Hilbert transform process is conducted to generate the analytical signal ${S_{cor1}}({{t_N}} )$ from ${U_{cor1}}({{t_N}} )$. Thereafter, a frequency compensation operator is multiplied by apodized IGM as## 3. Results

Actual parameters of the DCI include 57.199 MHz and 57.200 MHz repetition frequency of two OFCs in the simulation and experimental systems. The time scale stretched factor $k = {f_r}/\Delta {f_r}$ is set to an integer to obtain theoretically identical IGMs, which is beneficial to verify the error compensation effect and enable potential coherent average for better SNR. Center wavelength of the band-pass filter was 1560 nm with 2 nm full-width half maximum (FWHM). Sampling rate was set to 57.200 MHz and acquisition time lasted for one second.

For the random parameters of simulation, additional time jitter $\delta {\tau _{10}}(t )$, $\delta {\tau _{20}}(t )$ was simulated as a Weiner stochastic process with 3 Hz repetition frequency Allan deviation. ${\Delta }{f_{ceo}}$ was set to linearly shift with 20 kHz/ms drift velocity, which was sufficient to simulate the slow variation of the CEO frequency. Considering the background noise induced by the detector and ADC, we added white noise with Gauss distribution, whose standard deviation was 0.05 times that of the pulse amplitude. Time-domain simulated IGMs are presented at different time scales in Figs. 3(a)–(c), which show a strong similarity with the experimental sampled signal in Figs. 3(d)–(f).

#### 3.1 Simulation results

Short-time Fourier transform process was used to extract time jitter, center frequency jitter, and carrier-envelope phase jitter from simulated IGMs, with a 1 kHz refresh rate according to ${\Delta }{f_r}$. Figure 4 demonstrates the performance of this error extraction method, which shows strong consistency between artificially added jitter and calculated jitter. The maximum discrepancies of the time jitter, center frequency jitter, and carrier envelope jitter in the RF frequency domain are 7.8 ns, 117 kHz, and 0.7 rad, respectively, with a standard deviation of 1.5 ns, 25 kHz, and 0.13 rad. The accuracy of relative optical time jitter measurement reaches the femtosecond order of magnitude, which is much less than the pulse width after optical filtering and chirp. In conclusion, these simulation results are characterized by a small statistical error, which proves the effectiveness of the error extraction algorithm.

Effects of the digital error correction steps are shown in Fig. 5. Step 1 is aimed to eliminate time jitter and align the pulse center with a constant period, which can be verified by the improvement from Figs. 5(a)–5(b). Figure 5(g) exhibits the resulting spectra after center frequency jitter correction without the visible location drifts. Fully phase-aligned IGMs are recovered in Fig. 5(d) and perform steady periodicity due to the integer multiple relation between ${f_r}$ and ${\Delta }{f_r}$, which indicates that our self-referencing digital error correction method is able to compensate all artificial relative jitters between separate apodized IGMs.

#### 3.2 Experimental results

Jitter noise in experimental DCI configuration has a wider range of source and more complex characteristics than that in simulation, with the linear phase distorted during the transmission in the optical and electric paths. Jitter within a single apodized IGM also exists besides the relative jitter between separate IGM periods, which increases uncertainty upon the availability of the self-referencing digital error correction method. Figure 6 demonstrates corrected IGMs and spectra after each process step, showing the similar compensating effects on three fundamental jitters to Fig. 5. In Fig. 6(h), spectra are highly consistent without distinct distortion caused by jitter within each single apodized IGM. Additionally, separate apodized IGMs after correction are completely coincident in Fig. 6(d). Short-term jitter within a single Fourier transform time window can be neglected as the previous assumption.

Various types of jitter noise in raw and fully phase-aligned IGM are shown in Fig. 7. After the correction, the variation ranges in the time, center frequency, and carrier-envelope phase jitters in the RF domain are reduced from 1 µs to 10 ns, from 2 MHz to 12 kHz, and from 130 rad to 100 µrad, respectively. Ultimate time jitter precision (Modified Allan deviation at 1 ms gate time) is ∼0.7 ns (∼12 fs in the optical domain), and center-frequency jitter precision is approximately 0.2 mHz. The high precision of fully corrected IGMs provides strong evidence that our self-referencing digital error correction method achieves nearly the same time-domain compensating performance compared with the optical referencing technique [30].

Spectra shown in Fig. 6(h) have outstanding spectral profile signal-to-noise ratio (SNR) at the expense of losing the comb structure completely; thus, the theoretical resolution cannot be realized. To reconstruct the frequency components of IGMs in the RF domain, FFT is needed for long-time IGMs with multiple interference periods [19]. Under the premise of fully mutual coherence, the SNR of the DCI system is proportional to the square root of measurement time, while the computational resolution is inversely proportional to the measurement time, showing the potential to improve the performance of DCI by extending acquisition time. However, 1 s coherence time is a common barrier for DCI stabilized by a tightly phase-locked loop or digital error correction [23]. In our experiment, 1 s raw IGMs were processed by FFT to calculate the RF spectrum in Fig. 8 (red). The profile shape is distorted with a fuzzy center frequency and fully overlapped frequency components. By contrast, a fully phase-aligned spectrum recovers the RF comb structure as the theoretical model, covering more than 10,000 individual components across a frequency range of 10 MHz in the RF domain. The SNR in the spectrum culmination is 60 dB, and each component has 1 Hz linewidth and 14 dB side lobe amplitude around 14 MHz. Experimental results prove the validity of our noise model and pre-processing algorithm, which reaches analogous long-term mutual coherence to tightly active stabilization [24] and conventional digital error correction [28]. Longest coherence time for 1 s was demonstrated in our experiment, but there is no sign that mutual coherence is about to break down after 1 s acquisition time. Furthermore, if the length of IGMs exceed the limitation of ADC memory or computing resource, time domain coherent average method [19,31] could be used to fulfill longer measurement time with improved DCI performance, removing the theoretical coherence time limitation in tightly active stabilization schemes. Considering the advantages of simple configuration, low cost, and easy implementation, it is fairly attractive to generalize our method to concrete practical application that require long-term phase-stable DCI in the future.

## 4. Discussions

A vital paradigm shift is to replace active stabilization PLL circuits with computational post-correction algorithms, which significantly alleviate the bandwidth requirement of the actuator and improve the implementability of DCI. However, the error extraction configuration still inherits from the active stabilization scheme, which becomes the dominant limitation of the DCI integration and cost. For more simplification, we can divide all of the jitter noise into two categories in terms of the time scale of IGMs: jitter between separate apodized IGMs and jitter between different sample points within an apodized IGM. As the duty cycle (defined as the product of apodized IGM duration and $\Delta {f_r}$) is fairly small (approximately 1/200 in our experiment), the former jitter is dominant compared with the latter. Additionally, considering the ultra-short time scale, we can only compensate the latter jitter correctly if elusive bandwidth and accuracy are achieved, which has already been proved unnecessary in most applications [5,7,39]. In essence, the main purposes of DCI stabilization schemes are to eliminate the accumulated jitter as soon as a new centerburst appears. In other words, continuous high-speed active servo control or continuous computational correction is dispensable in this case. The self-referencing digital error correction method demonstrated in this paper completely removes the external reference path, executing a compensation process triggered by IGMs themselves. Although absolute stabilization bandwidth seems limited compared with the conventional stabilization methods, the correction rate of our method matches the IGMs perfectly, and the removed execution operation are proved unnecessary by our experimental results. Considering the tremendous simplification of the experimental setup, this reduction in compensation rate is quite cost-effective in most cases.

For a conventional digital error correction [28] or analog adaptive sampling [29] method with external optical reference, self-calibration ability is unavailable due to the intermediary nature of the CW laser. Similarly, our method is not capable of calibrating the absolute frequency axis. However, as full mutual coherence is achieved with our method, a certain linear relationship exists between the Fourier frequency axis and the absolute frequency axis. Priori knowledge, such as known molecule absorption line, can be utilized to recover the absolute frequency axis, achieving sufficient accuracy for dual-comb applications such as gas detection [33,40].

The self-referencing digital error correction method does not conflict with the active stabilization method and optical referencing digital error correction technique when they aim at noise in different time scales. These methods can be combined to improve the general performance of DCI according to specific demands. For example, a pair of loosely locked PLLs (approximately several tens of hertz) are exploited in our experiment to stabilize the conversion factor between optical frequency and RF frequency against temperature drifts. If a longer coherent average time is required, such as several hours, ultraslow digital servo control of cavity length and pump current as that in [31] can be utilized by FPGA to avoid spectral overlapping. For the situation where the jitter within an apodized IGM is deemed non-negligible [29], the optical referencing digital error correction method is suitable. For the applications with strict requirement on absolute optical stability, tightly phase-locked loop systems are indispensable. In other words, in view of tradeoffs between cost and performance, the combination of these methods would play an important role in expanding the application boundary of DCI.

## 5. Conclusions

This paper demonstrated a self-referencing digital correction method to retrieve a phase-stable DCI configuration. By extracting time jitter, center frequency jitter, and carrier-envelope phase jitter from the IGMs themselves, we can eliminate external optical reference (e.g., CW laser intermediary) to achieve a low-cost, easily operable DCI setup without losing its long-term mutual coherence. This method has broad prospects in numerous applications, which can significantly improve the performance of an elementary DCI at nearly negligible cost.

## Funding

Shenzhen Science and Technology Innovation Commission (JCYJ20170412171535171, JCYJ20170817160808432); National Natural Science Foundation of China (51835007); Ministry of Science and Technology of the People's Republic of China (2016YFF0100700).

## References

**1. **T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature **416**(6877), 233–237 (2002). [CrossRef]

**2. **J. Ye, S. T. Cundiff, F. X. Kärtner, E. P. Ippen, A. Bartels, A. L. Gaeta, R. S. Windeler, G. Steinmeyer, U. Keller, and T. Kobayashi, “Femtosecond Optical Frequency Comb: Principle, Operation, and Applications,” (2005).

**3. **S. T. Cundiff and J. Ye, “Colloquium: Femtosecond optical frequency combs,” Rev. Mod. Phys. **75**(1), 325–342 (2003). [CrossRef]

**4. **S. Schiller, “Spectrometry with frequency combs,” Opt. Lett. **27**(9), 766–768 (2002). [CrossRef]

**5. **F. Keilmann, C. Gohle, and R. Holzwarth, “Time-domain mid-infrared frequency-comb spectrometer,” Opt. Lett. **29**(13), 1542–1544 (2004). [CrossRef]

**6. **N. Picqué and T. W. Hänsch, “Frequency comb spectroscopy,” Nat. Photonics **13**(3), 146–157 (2019). [CrossRef]

**7. **A. Schliesser, M. Brehm, F. Keilmann, and D. W. van der Weide, “Frequency-comb infrared spectrometer for rapid, remote chemical sensing,” Opt. Express **13**(22), 9029–9038 (2005). [CrossRef]

**8. **M. Yan, P.-L. Luo, K. Iwakuni, G. Millot, T. W. Hänsch, and N. Picqué, “Mid-infrared dual-comb spectroscopy with electro-optic modulators,” Light: Sci. Appl. **6**(10), e17076 (2017). [CrossRef]

**9. **G. Ycas, F. R. Giorgetta, K. C. Cossel, E. M. Waxman, E. Baumann, N. R. Newbury, and I. Coddington, “Mid-infrared dual-comb spectroscopy of volatile organic compounds across long open-air paths,” Optica **6**(2), 165–168 (2019). [CrossRef]

**10. **I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics **3**(6), 351–356 (2009). [CrossRef]

**11. **Z. Zhu, G. Xu, K. Ni, Q. Zhou, and G. Wu, “Synthetic-wavelength-based dual-comb interferometry for fast and precise absolute distance measurement,” Opt. Express **26**(5), 5747–5757 (2018). [CrossRef]

**12. **P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science **359**(6378), 887–891 (2018). [CrossRef]

**13. **T. Ideguchi, S. Holzner, B. Bernhardt, G. Guelachvili, N. Picqué, and T. W. Hänsch, “Coherent Raman spectro-imaging with laser frequency combs,” Nature **502**(7471), 355–358 (2013). [CrossRef]

**14. **K. J. Mohler, B. J. Bohn, M. Yan, G. Mélen, T. W. Hänsch, and N. Picqué, “Dual-comb coherent Raman spectroscopy with lasers of 1-GHz pulse repetition frequency,” Opt. Lett. **42**(2), 318–321 (2017). [CrossRef]

**15. **M. Yan, L. Zhang, Q. Hao, X. Shen, X. Qian, H. Chen, X. Ren, and H. Zeng, “Surface-Enhanced Dual-Comb Coherent Raman Spectroscopy with Nanoporous Gold Films,” Laser Photonics Rev. **12**(9), 1800096 (2018). [CrossRef]

**16. **M. Brehm, A. Schliesser, and F. Keilmann, “Spectroscopic near-field microscopy using frequency combs in the mid-infrared,” Opt. Express **14**(23), 11222–11233 (2006). [CrossRef]

**17. **E. Hase, T. Minamikawa, T. Mizuno, S. Miyamoto, R. Ichikawa, Y.-D. Hsieh, K. Shibuya, K. Sato, Y. Nakajima, A. Asahara, K. Minoshima, Y. Mizutani, T. Iwata, H. Yamamoto, and T. Yasui, “Scan-less confocal phase imaging based on dual-comb microscopy,” Optica **5**(5), 634–643 (2018). [CrossRef]

**18. **H. Mikami, J. Harmon, H. Kobayashi, S. Hamad, Y. Wang, O. Iwata, K. Suzuki, T. Ito, Y. Aisaka, N. Kutsuna, K. Nagasawa, H. Watarai, Y. Ozeki, and K. Goda, “Ultrafast confocal fluorescence microscopy beyond the fluorescence lifetime limit,” Optica **5**(2), 117–126 (2018). [CrossRef]

**19. **I. Coddington, N. Newbury, and W. Swann, “Dual-comb spectroscopy,” Optica **3**(4), 414 (2016). [CrossRef]

**20. **T. Ideguchi, “Dual-Comb Spectroscopy,” Opt. Photonics News **28**(1), 32–39 (2017). [CrossRef]

**21. **R. Paschotta, A. Schlatter, S. C. Zeller, H. R. Telle, and U. Keller, “Optical phase noise and carrier-envelope offset noise of mode-locked lasers,” Appl. Phys. B: Lasers Opt. **82**(2), 265–273 (2006). [CrossRef]

**22. **N. R. Newbury and W. C. Swann, “Low-noise fiber-laser frequency combs (Invited),” J. Opt. Soc. Am. B **24**(8), 1756–1770 (2007). [CrossRef]

**23. **Z. Chen, M. Yan, T. W. Hänsch, and N. Picqué, “A phase-stable dual-comb interferometer,” Nat. Commun. **9**(1), 3035 (2018). [CrossRef]

**24. **I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. **100**(1), 013902 (2008). [CrossRef]

**25. **I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent dual-comb spectroscopy at high signal-to-noise ratio,” Phys. Rev. A **82**(4), 043817 (2010). [CrossRef]

**26. **O. Sho, I. Kana, I. Hajime, H. Kazumoto, O. Atsushi, S. Hiroyuki, and H. Feng-Lei, “Ultra-broadband dual-comb spectroscopy across 1.0–1.9 µm,” Appl. Phys. Express **8**(8), 082402 (2015). [CrossRef]

**27. **P. Giaccari, J.-D. Deschênes, 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**(6), 4347–4365 (2008). [CrossRef]

**28. **J.-D. Deschênes, P. Giaccari, and J. Genest, “Optical referencing technique with CW lasers as intermediate oscillators for continuous full delay range frequency comb interferometry,” Opt. Express **18**(22), 23358–23370 (2010). [CrossRef]

**29. **T. Ideguchi, A. Poisson, G. Guelachvili, N. Picque, and T. W. Hansch, “Adaptive real-time dual-comb spectroscopy,” Nat. Commun. **5**(1), 3375 (2014). [CrossRef]

**30. **Z. Zhu, K. Ni, Q. Zhou, and G. Wu, “Digital correction method for realizing a phase-stable dual-comb interferometer,” Opt. Express **26**(13), 16813–16823 (2018). [CrossRef]

**31. **J. Roy, J.-D. Deschênes, S. Potvin, and J. Genest, “Continuous real-time correction and averaging for frequency comb interferometry,” Opt. Express **20**(20), 21932–21939 (2012). [CrossRef]

**32. **D. Burghoff, Y. Yang, and Q. Hu, “Computational multiheterodyne spectroscopy,” Sci. Adv. **2**(11), e1601227 (2016). [CrossRef]

**33. **N. B. Hébert, J. Genest, J.-D. Deschênes, H. Bergeron, G. Y. Chen, C. Khurmi, and D. G. Lancaster, “Self-corrected chip-based dual-comb spectrometer,” Opt. Express **25**(7), 8168–8179 (2017). [CrossRef]

**34. **H. Yu, K. Ni, Q. Zhou, X. Li, and G. Wu, “A simulation method for dual-comb spectroscopy with jitter noise,” in * Real-time Photonic Measurements, Data Management, and Processing III*, (International Society for Optics and Photonics, 2018), 108220Q.

**35. **N. R. Newbury, I. Coddington, and W. Swann, “Sensitivity of coherent dual-comb spectroscopy,” Opt. Express **18**(8), 7929–7945 (2010). [CrossRef]

**36. **L.-S. Ma, R. K. Shelton, H. C. Kapteyn, M. M. Murnane, and J. Ye, “Sub-10-femtosecond active synchronization of two passively mode-locked Ti:sapphire oscillators,” Phys. Rev. A **64**(2), 021802 (2001). [CrossRef]

**37. **H. Shi, Y. Song, F. Liang, L. Xu, M. Hu, and C. Wang, “Effect of timing jitter on time-of-flight distance measurements using dual femtosecond lasers,” Opt. Express **23**(11), 14057–14069 (2015). [CrossRef]

**38. **E. Rubiola, *Phase noise and frequency stability in oscillators* (Cambridge University Press, 2009).

**39. **B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hänsch, and N. Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics **4**(1), 55–57 (2010). [CrossRef]

**40. **T. Ideguchi, A. Poisson, G. Guelachvili, T. W. Hänsch, and N. Picqué, “Adaptive dual-comb spectroscopy in the green region,” Opt. Lett. **37**(23), 4847–4849 (2012). [CrossRef]