Dual-comb spectroscopy has enabled new approaches for optical precision measurements. Although Doppler-limited resolution can be achieved over long-time scales across a large bandwidth, the development of dual-comb spectroscopy is hindered by strict demands for light source stability. Typically, expensive and complex self-reference systems are required to lock the carrier-envelope offset frequency (*f*_{ceo}) of the laser. Additionally, simply locking the repetition frequency (*f*_{rep}) to a radio frequency reference source still results in residual relative timing jitter between light sources. Here we extracted the relative *f*_{ceo} fluctuation between the *f*_{rep}-locked lasers from the high-precision passive notch filtering characteristics of the phase-shifted fiber Bragg grating and then eliminated it through online phase calibration. By introducing a passive broadband Fabry–Perot cavity with excellent thermal wavelength stability, we subsequently corrected residual relative timing jitter with online wavelength calibration, and the standard deviation of the relative wavelength drift was reduced to less than 0.4 pm within the full operating range. The spectral profile can also be extracted and removed by the Fabry–Perot cavity through intensity calibration. By calibrating these three dimensions, we built a reference-free post-calibration dual-comb spectroscopy and used this powerful tool to measure the Fabry–Perot cavity resonance peaks, the notch filtering narrow band of phase-shifted fiber Bragg gratings, and the absorption characteristics of hydrogen cyanide gas. The system achieves a spectral resolution of 0.8 pm over a bandwidth of more than 100 nm. This low-cost and convenient scheme provides new ideas for the application of dual-comb spectroscopy systems.

## I. INTRODUCTION

Since its birth two decades ago,^{1} the emerging dual-comb spectroscopy (DCS) system implemented on various optical platforms^{2,3} has provided new ideas for precision measurement and has been widely used in applications ranging from molecule metrology,^{4,5} greenhouse gas monitoring,^{6} laser lidar,^{7,8} and photothermal spectroscopy.^{9} Inspired by the Fourier transform spectrometer (FTS),^{10} where two optical frequency combs (OFCs) with slightly different repetition frequencies (*f*_{rep}) are used as the measurement signal and the local signal, respectively. Differences in *f*_{rep} form inter-sampling in time, which eliminates the mechanical movement of the interference arm in the FTS and maps the spectral information to the radio frequency (RF) domain. The comb-like structure with equal frequency spacing due to mutual coherence between two OFCs makes DCS a powerful tool for high-accuracy spectral precision measurements. The frequency of each comb line can be described as *ν*_{n} = *n* · *f*_{rep} + *f*_{ceo}, where *n* denotes the number of comb modes, *f*_{rep} is determined by the cavity length, and *f*_{ceo} is the offset frequency between carrier and envelope, which is basically affected by various noise sources, such as the intracavity amplified spontaneous emission (ASE) and the pump noise.^{11} In the free-running state, unconstrained two frequency degrees of freedom will greatly degrade the measurement resolution (e.g., to the order of GHz),^{2} leading to distortion of the measurement results over long time scales.

The most effective way of eliminating the fluctuation of *f*_{ceo} is to tightly lock it through a complex feedback system using a self-referential approach known as the *f*–2*f* interferometer.^{12} This method needs the participation of nonlinear processes to achieve the spectral octave of the OFCs. In addition, the signal-to-noise ratio (SNR) of the *f*_{ceo} signal is affected by a series of factors such as supercontinuum spectrum broadening, the efficiency of nonlinear crystals, the design of the detection optical path, etc.^{13} Besides feedback loops, feedforward schemes based on an acoustic-optic frequency shifter (AOFS) have also been proposed and verified in the near-infrared^{14} and mid-infrared^{15} regions. In addition, a stable atomic clock source or narrow linewidth CW laser with a hertz level is required as a reference, which greatly increases cost and system complexity.

In general, a reference source is also required to realize the locking of *f*_{rep} by fast electro-optical modulation or precise cavity length control.^{11} Practically, a commercial arbitrary waveform generator (AWG) can provide an RF reference signal with a frequency stability of 2 ppm, which is considered a comb-line frequency deviation after locking the *f*_{rep} when ignoring the *f*_{ceo} jitter. This RF domain deviation will be drastically amplified by *n* times through the frequency transfer characteristics of OFC when upconverting to optical frequency. For instance, at a wavelength of 1500 nm (200 THz), *n* equals 2 × 10^{6}, so the calculated wavelength deviation is about 3 pm. When it comes to DCS systems, the deviation between the two OFCs, i.e., residual relative timing jitter, will also be mapped to the RF domain and cause the broadening of the RF comb line over a long time scale.^{16} For high-accuracy spectroscopic measurement applications, performing effective calibration algorithms in the time^{17} or frequency domain is necessary.

Besides locking a single OFC, another solution is to extract the relative jitter of two independent OFCs, and afterward, the electrical adaptive scheme^{18} or algorithm processing^{19,20} is implemented for locking or real-time calibration and compensation, even in free-running DCS systems.^{18,19} Although each laser still drifts, the relative deviation is fixed, which is sufficient for most DCS systems.^{2} This approach has attracted widespread attention and led to a series of impressive works. Typically, auxiliary CW lasers are required as the medium to extract the relative jitter between two OFCs.^{21} Usually, fiber Bragg gratings (FBGs) are used as filters, and the beat notes of the two most adjacent modes of the two OFCs are directly selected as calibration signals.^{22} However, due to the large filtering bandwidth (>10 GHz), the valid window of the calibration signal is limited (<100 ps), so only a small part of the interference signal can be calibrated, resulting in the degradation of the spectral resolution and sensitivity. Using electrical filtering, the filter bandwidth can be extremely narrow, for example, on the order of several MHz, so the time window of the calibration signal can completely cover the delay range of the interference signal. However, due to the technological limitations of the current electrical filter, the adaptive solution is not ideal for filtering edges, putting forward higher requirements for electrical components, and a complex frequency mixing process is required to achieve perfect band-pass filtering. Moreover, the delay introduced by electrical devices cannot be ignored, and as *a priori*, it needs to be carefully measured, which undoubtedly increases the complexity of the experiment.^{18}

With the development of hardware and data processing technology, free-running DCS systems based on calibration algorithms to correct residual phase fluctuations have been realized, including a cross-ambiguity function borrowed from radar technology,^{23} real-time frequency estimation,^{24} and Kalman filtering algorithms.^{25} In this way, the proper design can easily realize the digital filter bandwidth of orders of MHz. However, such approaches were usually applied for OFCs with large *f*_{rep}, i.e., a few hundred MHz or even GHz, provided by passively mode-locked waveguide lasers (WGLs), quantum cascade lasers (QCLs), and silicon microresonators. This can be interpreted as a larger *f*_{rep} (small period), which helps to avoid 2*π* ambiguity during phase unwrapping.

In this paper, we propose and demonstrate a reference-free post-calibration DCS system assisted by two common passive fiber devices, including a phase-shifted fiber Bragg grating (PS-FBG) and a Fabry–Perot cavity (F–P cavity). The notch filtering bandwidth of a PS-FBG, typically several hundreds of MHz, corresponds to a calibration window >10 ns, which is two orders of magnitude higher than the calibration window of the traditional FBG-based scheme. Compared to the FBG scheme, our PS-FBG scheme has two main advantages. First, the calibration signal has a larger time window because the filtering bandwidth and time window satisfy the relationship between Fourier transform pairs, thereby obtaining more time-domain information to support phase calibration. Second, for a *f*_{rep} of 100 MHz, our scheme can ensure that only one comb tooth is continuously filtered out, resulting in more singular frequency components in the calibration signal, manifested as sine waves covering the entire interference cycle. An F–P cavity was also introduced in the system as the wavelength reference to solve the issue of residual relative timing jitter. Benefiting from the excellent thermal wavelength stability of the F–P cavity, the standard deviation of relative wavelength drift was decreased by about 11 and 518 times through an online wavelength calibration algorithm at the calibration center wavelength and away from it when multi-frames of the interferogram are overlayed. Meanwhile, the comb-like structure of the fiber F–P cavity makes it have sufficient peak points, which can be used to fit the spectral profile instead of introducing another reference arm. By performing intensity calibration, the spectral envelope can be eliminated to generate a normalized sample absorption spectrum. Therefore, this scheme can extract the real-time calibration signal in the full delay range and retrieve the absorption characters of the sample that cover the operation wavelength without degrading the resolution of the system. After such three dimensions of calibration, the coherent averaging process can be carried out to obtain the spectral characteristics of samples with high resolution and high SNR. We used this powerful spectral measurement system for F–P cavity resonance peaks, high-precision PS-FBG notch filtering bandwidth, and gas molecular absorption measurements. A high resolution of 100 MHz (0.8 pm) in the more than 100 nm range was achieved.

## II. PRINCIPLE AND SYSTEM EXPERIMENTAL SETUP

*f*

_{rep}of 100 MHz. The direct output powers are 14 and 13 dBm, respectively. By optimizing the intra-cavity dispersion and pump power, the 10 dB bandwidths reach 156 and 130 nm, respectively. The optical fields of two OFCs can be approximately described as

*a*(

*t*) =

*a*

_{0}(

*t*) · exp(

*i*2

*πν*

_{0}

*t*), and

*a*

_{0}(

*t*) is the envelope in the time domain,

*ν*

_{0}is the carrier frequency, and

*A*(

*ν*) is the Fourier transform of

*a*

_{0}(

*t*). A 99:1 coupler (not shown in the figure) divides the laser output power into two parts: 99% of the optical power is used for measurement, while 1% is for actively locking the

*f*

_{rep}, which is completed by a two-channel digital phase lock loop (DPLL) module designed based on a field programable gate array (FPGA) platform. We have introduced an electrically controlled optical delay line (EODL) with a free-space delay range of 300 ps and a step of 2 fs in the fiber cavity, which increases the

*f*

_{rep}tuning to the MHz level, far exceeding the kHz level that piezoelectric transducers (PZTs) can provide, while also bringing more controllability. Therefore, a combined cavity length adjustment mechanism is formed by embedding the PZT onto the mirror of the EODL. The cavity length can be controlled precisely by adjusting the EODL and PZT simultaneously, thereby locking the

*f*

_{rep}to the RF reference provided by the vector signal generator (VSG). With proper global control algorithm design, the phase-locked function can be self-activated and

*f*

_{rep}stabilization for up to several days. The entire module is packaged in a metal box with a volume of 30 × 22 × 13 cm

^{3}, which is compact, flexible, and convenient to use in dual-comb interferometer structure applications.

^{26}In this work, the frequencies of the two VSG channels were set to 100 and 100.0001 MHz, respectively. 99% of the optical power is split again by a 30:70 coupler, where 70% is used to measure the sample’s absorption characteristics, and a polarization controller (PC) inserted here is used to maximize the intensity of the interference signal.

*I*

_{s}(

*t*) generated by two OFCs is then photoelectrically detected. The approximate expression of

*I*

_{s}(

*t*) is

*f*

_{rep}/2. The corresponding comb frequency can be approximately expressed as

*p*is constant within a certain range, and

*A*(

*f*) can be regarded as a slowly changing function relative to

*f*, that is,

*A*

_{1}[

*ν*(

*n*

_{i})] ≈

*A*

_{2}[

*ν*(

*n*

_{i}+

*p*)], then Eq. (3) can be further expressed as

_{1}) centered at 1550.5 nm and a filtering bandwidth of 0.5 nm for pre-filtering, power amplification via a modular erbium-doped fiber amplifier (EDFA), then take the transmission port of PS-FBG for narrowband notch filtering, and finally, place FBG

_{2}with the same parameters as FBG

_{1}to filter out redundant sidebands and ASE noise introduced by the EDFA module. Therefore, only the spectral component within the notch filtering narrowband is preserved. Take it as the calibration signal

*I*

_{c}(

*t*).

*I*

_{c}(

*t*) can be approximately expressed as

*n*

_{c}is the number of comb modes at the corresponding wavelength of the notch band of the PS-FBG. It can be found in Eqs. (5) and (6) that the phases of both

*I*

_{s}(

*t*) and

*I*

_{c}(

*t*) contain the common term Δ

*f*

_{ceo}(

*t*). Therefore, the Hilbert transform can extract the phase separately and then perform a subtraction to remove the Δ

*f*

_{ceo}(

*t*). It is worth noting that not only the influence of Δ

*f*

_{ceo}(

*t*) can be eliminated during the phase calibration process, but the carrier frequency of the measurement signal is also greatly reduced. It can be approximated as being reduced from

*n*· Δ

*f*

_{rep}to (

*n*−

*n*

_{c}) · Δ

*f*

_{rep}, close to zero frequency. Moreover, when the notch filtering band wavelength of PS-FBG shifts, only the absolute frequency of the filtered comb line

*ν*(

*n*

_{c}) is changed. Since the relative jitter is maintained, it will not affect the results of phase calibration. From the frequency domain, as viewed in Fig. 1(b), the phase calibration process can be regarded as taking the notch position of the PS-FBG as a zero-frequency reference, as confirmed by the fact that the filtering bandwidth is much smaller than the effective working bandwidth of dual combs. The severely distorted spectrum directly obtained from

*I*

_{s}(

*t*) is then shifted and reshaped with the reference as the origin to eliminate common components. After the phase calibration process, the spectral absorption characteristics of the sample can be retrieved undistorted. As for the second branch, we select an F–P cavity with a free spectral range (FSR) of about 100 GHz to perform both wavelength and intensity calibration. Practically, it can offer an average wavelength thermal stability of ±0.73 GHz in the temperature range of −5–70 °C, corresponding to a wavelength deviation of about 12 pm around 1550 nm. This excellent wavelength thermal stability makes it a meritorious reference for wavelength calibration, as viewed in Fig. 1(c). In addition, its transmission spectrum also reflects the spectral profile of dual combs and can easily cover the operating wavelength range. Therefore, instead of using another common spectral reference signal, we also use the F–P cavity for intensity calibration, simplifying the system as viewed in Fig. 1(d). Finally, all three optical signals were detected by a 200 MHz PD array, filtered by three 48 MHz low-pass filters (LPFs) to satisfy the Nyquist sampling law, and the data were recorded by a three-channel analog-to-digital converter (ADC) card with a sample rate of 100 MSa/s. The optical path part of the whole system is encapsulated in a metal sealing box and placed at a constant temperature of 25 °C to guarantee the stability of the performance of the PS-FBG and F–P cavities.

Figure 2 shows the characterization results of key modules and devices. Figure 2(a) illustrates the performance of the *f*_{rep} locking of the fiber laser. Taking comb 1 as an example, the noise characteristics of 1.2 GHz (corresponding to the 12th harmonic of the *f*_{rep}) were measured by directly injecting the fiber laser into a fast photodetector (200 MHz) and recorded by an electrical spectrum analyzer (ESA) operated in free-running and digital phase-locked modes, respectively. The noise characteristics of higher harmonics can be used to approximately describe the phase noise of a mode-locked laser.^{27} The phase noise PSD was then scaled to 100 MHz, as shown by the blue and red curves in Fig. 2(a). It exhibits about 43 dB of noise suppression within the locking bandwidth of about 250 Hz. As shown by the black dotted line in Fig. 2(a), the phase noise PSD shows a characteristic 1/*f*^{4} slope for higher Fourier frequency in the free-running state, which is confirmed to be the random walking noise introduced by the timing jitter.^{28} The vibration in the low Fourier frequency region may be caused by mechanical resonance, which can be suppressed by further vibration isolation packages.^{11} In addition, the phase noise PSD of the 100 MHz reference signal provided by the RF source is plotted. It can be seen that the noise level after digital phase locking within the locked bandwidth is comparable to that provided by RF sources, which proves the effective locking performance.

*f*

_{rep}of the two OFCs was locked, the noise characteristics of the Δ

*f*

_{rep}between the two OFCs were investigated. In free-running and digital phase locking states, a real-time oscilloscope with a sample rate of 2 MSa/s was used to record the Δ

*f*

_{rep}signal for 5 s, respectively, and the Hilbert transform was then performed to extract the Δ

*f*

_{rep}as a function of time, that is, Δ

*f*

_{rep}(

*t*). Hence, its phase noise PSD was calculated by

^{29}

^{,}

*f*

_{rep}(

*t*)〉 is the expected value of Δ

*f*

_{rep}(

*t*), and

*f*in the denominator is the Fourier frequency variable. Factor 2 at the front of the formula indicates that the single sideband PSD is considered. The calculated results are shown in Fig. 2(b). After digital phase locking, the phase noise PSD of Δ

*f*

_{rep}drops by about 73.4 dB, and the spike frequency appearing in the given curve represents Δ

*f*

_{rep}and its higher harmonics. It is worth noting that during free-running, since the PZT in the cavity is not driven by the outer voltage and returns to the original length, the Δ

*f*

_{rep}no longer maintains the set value, that is, 100 Hz, but about 162.5 Hz, which can be confirmed from Fig. 2(b) by the first spike frequency of the blue curve. The 1/

*f*

^{4}slope line is also referenced here to identify the contribution of the residual relative timing jitter, as mentioned before. The noise floor of the two curves in Fig. 2(b) is about −100 dBc/Hz, which is dominated by the shot noise introduced in the photodetection and electrical mixing processes. In this work, though, we only focus on the change of phase noise at low Fourier frequencies after locking Δ

*f*

_{rep}through two DPLLs, which means that the detection sensitivity is sufficient. Higher detection sensitivity for Δ

*f*

_{rep}phase noise measurement can be achieved by optical heterodyne detection

^{29}or spectral interferometry combined with dispersive Fourier transform.

^{30}We also recorded the frequency of the electrical signal of the phase calibration branch path for 8 h at a constant temperature of 25 °C with a frequency counter, and the results are shown in Fig. 2(c). As mentioned before, its fluctuation reflects the relative

*f*

_{ceo}jitter between two OFCs. It drifted between 25 and 40 MHz over 8 h; the inset on the right is the frequency distribution histogram.

As the core device of the system, the notch filtering bandwidth of the PS-FBG needs to be strictly measured. Since the bandwidth is only a few hundred MHz, which cannot be measured by a commercial optical spectrum analyzer (OSA), the notch filtering bandwidth is measured using fast frequency sweeping. The frequency sweep CW laser can be used to scan the frequency near the filtering narrowband. When the frequency falls within the notch band, the output power will change sharply. Therefore, the real-time oscilloscope records the power change during the frequency sweep, and the duration of the change can be used to infer the bandwidth of the notch. The measurement results of the transmission port of the PS-FBG are shown in Fig. 2(d). An arbitrary waveform generator (AWG) emits a 200 Hz triangular wave to drive the CW source with a swept bandwidth of 18 GHz, which is much larger than the notch bandwidth of the PS-FBG. In Fig. 2(d), we captured the voltage change during the CW sweeping of the PS-FBG and the corresponding driving signal within 10 ms. Figure 2(e) is a zoomed-in view of the third peak in Fig. 2(d). When the frequency sweeps over the notch band, the voltage change occurs within 5 *µ*s, which is much smaller than the time scale of the notch position and bandwidth change. From this, it can be inferred that the full width at half maximum (FWHM) is about 118.3 MHz, and under the experimental setting, only one comb of each OFC can always be filtered out, which can be verified by the frequency range shown in Fig. 2(c).

## III. RESULTS

### A. Verification of the calibration algorithm

As a prerequisite step, we use the F–P cavity in the system (which itself can also be regarded as the sample to be measured) to verify the validity of the calibration algorithm mentioned before. The retrieved results from the Fourier transform of the single-frame interferogram are shown in Fig. 3. Due to the presence of phase fluctuations, the spectrum of the interference signal appears very chaotic, as shown in the gray area of Fig. 3(a). Then the phase calibration is performed in the range of 1500–1600 nm, eliminating the relative *f*_{ceo} jitter between the two OFCs and thus recovering the spectral information of the F–P cavity, showing a clear comb-like structure with an FSR of 100 GHz (0.8 nm near 1550 nm) in Fig. 3(b). Record the intensity of each resonant peak of the F–P cavity to fit the spectral profile, as shown in the red curve in Fig. 3(b). It is worth noting that the obvious gap between the fitted profile and the retrieved result is the 1.8 dB insertion loss of the F–P cavity itself. The spectrum profile shows a 10 dB bandwidth of about 100 nm, which can support intensity calibration over the entire operating range. The intensity calibration is implemented by removing the envelope, as shown in Fig. 3(c).

We use an OSA (Yokogawa AQ6319) to measure the absorption spectrum of the F–P cavity multiple times in the range of 1500–1600 nm, with a resolution of 0.01 nm, and then average it to obtain a reference spectrum to support wavelength calibration. Next, record the wavelength position of each resonant peak of the F–P cavity named as {*λ*_{1}, *λ*_{2}, …, *λ*_{n}}, where *n* is the number of resonant peaks supported by the F–P cavity within the operating range. Compared with the standard resonant wavelength positions {*λ*_{01}, *λ*_{02}, …, *λ*_{0n}}, there exist some wavelength deviations *δλ*_{i} between *λ*_{i} and *λ*_{0i}, *i* ∈ [1,*n*]. By cubic spline interpolation, the *λ*_{i} is interpolated on the grid of the fixed resonant peak reference *λ*_{0i}, and the wavelength axis is reconstructed. This process is called wavelength calibration, which is reliable owing to the superior thermal wavelength stability of the F–P cavity. We consecutively acquired 50 groups of single-frame interferograms and stacked the retrieved results together to characterize the wavelength calibration, as illustrated in Fig. 4. In the stacked results, record the peak wavelength position of each resonant peak in the F–P cavity in 50 interferograms, calculate its standard deviation, and define it as the wavelength standard deviation *σ*_{λ}, reflecting the relative drift of the wavelength. In Fig. 4(a), the decline in *σ*_{λ} before and after wavelength calibration is intuitively shown, corresponding to black circles and pink diamonds, respectively. Since the notch filtering band position of PS-FBG is taken as the wavelength reference *λ*_{ref} in the phase calibration process, in this work, *λ*_{ref} is about 1547 nm. Therefore, the absolute wavelength deviation *δλ*_{i} presents a distribution of gradual accumulation and diffusion to both sides away from *λ*_{ref}, with *λ*_{ref} as the center. Therefore, a V-shaped distribution of *σ*_{λ} is formed before wavelength calibration, as shown in Fig. 4(a). Figure 4(b) corresponds to the blue-shaded area of the results after wavelength calibration, where the *σ′*_{λ} significantly decreased. The *σ′*_{λ} after wavelength calibration is less than 0.4 pm within the entire operating range, which is lower than the theoretical optical resolution (0.8 pm), providing the possibility for precise spectral measurement. In cubic spline interpolation, a new wavelength axis is established by fitting the F–P resonant peak wavelength point, which fluctuates between the fitting point, inevitably leading to the difference around different wavelengths, as shown in Fig. 4(b).

To demonstrate the results of wavelength calibration further, resonant peaks around 1523, 1551, and 1595 nm were selected from top to bottom in Fig. 5, corresponding to one central wavelength and two sidebands. Among them, 1523 and 1595 nm have the same noise floor level before wavelength calibration. The deviation between frames makes the results aliased, and it is difficult to distinguish the shape of the resonant peak of the F-P cavity. As for the case near *λ*_{ref}, although there is wavelength deviation, frames can still be aligned to show clear spectral information. Select one of the two F–P cavity resonant peaks in three different ranges in Fig. 5(a) and record its peak wavelength position called *λ*_{a}, *λ*_{b}, and *λ*_{c}, as marked by red triangles, green squares, and blue circles. Record the peak wavelength positions of F–P cavity resonance peaks recovered at these three wavelengths before wavelength calibration; the calculated *σ*_{λ} at *λ*_{a}, *λ*_{b}, and *λ*_{c} are 53.4, 3.9, and 88 pm, respectively, as plotted in Fig. 5(c). It can be found that the farther away from the central wavelength, the greater the *σ*_{λ}, which verifies the V-shape distribution mentioned earlier.

After an effective wavelength calibration process, the results of the corresponding range are illustrated in Fig. 5(b). It is evident from the horizontal comparison of Figs. 5(a) and 5(b) that all results are well aligned after performing the wavelength calibration process. Similarly, record the peak wavelength positions of 50 groups of interferograms at three wavelengths after wavelength calibration, which are marked by five-pointed stars of corresponding colors in Fig. 5(b). The point distribution maps are plotted in Fig. 5(c), and the calculated wavelength standard deviations *σ′*_{λ} after wavelength calibration at *λ*_{a}, *λ*_{b}, and *λ*_{c} are 0.23, 0.36, and 0.17 pm, respectively. For the same wavelength position, *σ′*_{λ} is significantly reduced after wavelength calibration. At the central wavelength and the wavelength away from the central wavelength, the wavelength standard deviation decreased by about 11 times and 518 times, respectively, which increased by more than one order of magnitude. Moreover, from Figs. 4 and 5, it can be confirmed that the wavelength standard deviation supported by the F–P cavity for wavelength calibration is better than that of DPLL (corresponding to before wavelength calibration). This further allows a free-running DCS system, where F–P replaces DPLL to lock Δ*f*_{rep}.

### B. System characterization

Another PS-FBG was selected as the sample to demonstrate the high-resolution capability of the system and used its reflection port as the output. As mentioned before, it has a notch filtering bandwidth on the order of 100 MHz, which is close to the theoretical resolution of the system. An interferogram is recorded in 1.4 s, corresponding to 140 frames, which is limited by the memory depth of the ADC card. By applying the post-calibration process frame by frame, the mode-resolved spectrum over 100 nm was obtained. The spectrum near the reflection spectrum of the PS-FBG is shown in Fig. 6. Figure 6(a) shows that individual comb lines are obvious with 100 MHz spacing. The figure inserted on the right plots the spectrum near 1550.42 nm, showing the details of the transmission notch of the PS-FBG with a transmittance of about 5 dB. In addition, some side lobes are also observed near the transmission notch. After intensity calibration, the spectral profile is fitted, as shown in Fig. 6(b). It can be confirmed from the inserted figure that the FWHM of the transmission notch is about 187.5 MHz (1.5 pm), which shows the high-resolution measurement capability of the system. The measurement capability of up to 100 MHz spectral resolution is sufficient for spectral characterization of most samples such as solids, liquids, gases, or optical communication devices within the operating range.^{3}

To verify the reliability of the system in the application of gas molecular spectroscopy, we measured hydrogen cyanide (H^{13}C^{14}N, 25 Torr, 16.5 cm) at a room temperature of 25 °C. In the experiment, a C-band wavelength division multiplexer (WDM) is inserted before the gas cell to improve SNR. We coherently averaged 168 frames of the interferogram (corresponding to 1.68 s of coherent time), and the measurement results are shown in Fig. 7. The orange curve in Fig. 7(a) is the retrieved result of the frequency in the range of 3.7 THz (the corresponding wavelength is 1530–1560 nm), and the black curve is the spectral profile. After intensity calibration, the absorption characteristics of R brunch and P brunch of H^{13}C^{14}N can be obtained, referenced with the standard absorption characteristics provided by the HITRAN database^{31} as the blue curve. The insufficient absorption intensity of the absorption peak at R brunch is caused by the low average power of DCS near the short wavelength. While in P brunch, higher optical power makes the recovery consistent with HITRAN results. The illustration on the right side of Fig. 7(b) shows the detailed results of the P9 line centered around 1548.96 nm, showing the good consistency between the measurement results and the HITRAN database.

## IV. CONCLUSIONS

To sum up, we used two common passive fiber optic devices to build a reference-free, post-calibrated DCS system. The notch filtering bandwidth was strictly measured to be about 118.3 MHz. By virtue of the ultra-narrow notch filtering width of PS-FBG, carrier-envelope offset frequency jitter between two OFCs can be extracted, real-time acquisition of calibration signals in the full delay range can be realized, and the corresponding time-domain calibration window can reach the order of 10 ns, which is two orders of magnitude higher than that of the FBG-based scheme. By removing the common component of the phase in both the measured and calibration signals, the relative *f*_{ceo} jitter can be eliminated so that the absorption spectrum information of the sample can be restored without distortion. However, the recovered spectrum will still fluctuate on the wavelength axis due to the residual relative timing jitter, which is manifested as the breath of the measured results. Therefore, a broadband F–P cavity with excellent wavelength and thermal stability was introduced as the wavelength and envelope reference. By implementing the online wavelength calibration algorithm, this wavelength deviation was significantly eliminated. After performing wavelength calibration, the wavelength standard deviation at the central wavelength and the sidebands away from the central wavelength decreased by about 11 and 518 times, respectively. The wavelength standard deviation within the operating range is less than 0.5 pm, which is smaller than the theoretical optical resolution, indicating the validity of wavelength calibration. Finally, the mode-resolved normalized absorption spectrum of the sample can be obtained by eliminating the spectral profile provided by the F–P cavity as well. With such three dimensions of calibration based on an online algorithm, a reference-free DCS system calibrated by passive fiber devices is established. With this powerful and dependable spectral measurement capability, we characterized the F–P cavity resonance peaks, the notch filtering bandwidth of PS-FBG, and the absorption characteristics of H^{13}C^{14}N gas. The system has achieved an ultra-high resolution of 100 MHz (0.8 pm) in the range of more than 100 nm and has great potential for precision spectral measurement.

In this work, we demonstrated the validity and spectral measurement ability of the scheme in the near-infrared band. The feasibility of this scheme can be further extended to other regions, such as mid-infrared, far infrared, and even terahertz, where abundant molecular fingerprint spectra and near-field optical measurement applications exist.^{2,3} At the same time, the post-calibration program can be deployed to the ADC card based on the FPGA platform so that the data acquisition and processing can be executed on the same FPGA platform, which is expected to realize the real-time DCS applications. Moreover, with the rapid development of on-chip and integrated OFCs,^{32,33} our scheme also provides new ideas for the system of free-running dual-comb interferometers on the chip. The improvement and maturity of on-chip optical filters based on waveguide gratings,^{34} cascaded Mach–Zehnder interferometers (MZIs),^{35} and high-Q microresonators^{36} make integration possible in the future.

## ACKNOWLEDGMENTS

This work was supported by the National Key Research and Development Program of China (Grant Nos. 2022YFF0705904 and 2019YFB2203102) and the National Natural Science Foundation of China (Grant Nos. 61735006, 61927817, and 62075072).

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflicts to disclose.

### Author Contributions

**Chen Liu**: Conceptualization (supporting); Data curation (lead); Formal analysis (lead); Methodology (equal); Software (equal); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). **Liang Xu**: Conceptualization (supporting); Data curation (supporting); Formal analysis (supporting); Methodology (equal); Software (supporting). **Lei Zhang**: Conceptualization (lead); Methodology (equal). **Danlu Wang**: Data curation (supporting); Formal analysis (supporting); Methodology (supporting). **Ziyu Cao**: Data curation (supporting). **Zheng Zhang**: Conceptualization (supporting); Funding acquisition (lead); Investigation (supporting). **Chi Zhang**: Conceptualization (lead); Methodology (lead); Writing – review & editing (lead). **Xinliang Zhang**: Conceptualization (supporting); Funding acquisition (lead); Investigation (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*ν*

_{3}band with an accurate midinfrared coherent dual-comb spectrometer

*μ*m