A phase-stable dual-comb interferometer measures materials’ broadband optical response functions, including amplitude, frequency, and phase, making it a powerful tool for optical metrology. Normally, the phase-stable dual-comb interferometer is realized via tight phase-locking methods. This paper presents a post-correction algorithm that can compensate for carrier wave phase noise and interferogram timing jitter. The compensating signal is a beat between two combs using a free-running continuous wave laser as an optical intermediary. In our experiment, sub-hertz relative linewidth, ~1 ns relative timing jitter, and 0.2 rad precision in the carrier phase is demonstrated.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
An optical frequency comb (OFC) is a broadband coherent light source composed of many discrete longitudinal modes. Each mode can be described by repetition frequency (frep) and carrier-envelope offset frequency (fceo) as f(n) = nfrep + fceo [1, 2]. To fully detect and utilize these comb lines, researchers initially proposed a dual-comb interferometer using two combs with repetitive frequency difference Δfrep where the optical comb lines are down-converted to the RF domain [3, 4]. Multi-heterodyne spectroscopy and a radio-frequency (RF) comb are generated. Recently, dual-comb interferometry has been recognized as a powerful technique that provides high accuracy and fast measurement. Practical applications include gas analysis [5–13], absolute distance measurement [14–17], ellipsometry  and material characterization , hyperspectral imaging [20, 21], vibrometry , and strain sensors . Among these, the key point of an accurate dual-comb interferometer is high mutual coherence and resolved phase information. Such an accurate dual-comb interferometer is also called phase-stable dual-comb interferometer. Two approaches are available to realize such a system.
The conventional approach uses two ultra-stable continuous wave (CW) lasers as optical references to stabilize the fceo and frep [5, 10, 14]. When the fceo is stabilized by the f-2f scheme, only one ultra-stable CW laser is needed . Because mutual coherence is required in a dual-comb interferometer (more so than the stability of each comb), an electro-optic modulator-based synchronous locking approach [11, 19, 24] and acousto-optic frequency shifter-based feed-forward relative stabilization  are proposed as alternatives. These tight-locking approaches provide narrow linewidth and a stable RF comb phase; however, the complexity and cost of tight-locking two combs renders extended challenging applications.
The other approach involves continuously monitoring parameter fluctuations between the two combs during the post-process. Two effective post-process methods realized via analog signal processing [26, 27] and digital signal computation [28–30] share the same principle: both use two free-running CW lasers as optical intermediaries and obtain two relative beat signals between two combs. One corrects the RF comb’s fceo noise, and the difference signal between two beats is used to correct the Δfrep noise by reconstructing an interferogram’s sampling sequence; the difference signal is irrelevant to fceo noise. This process is simpler if post-correction is applied to a high-repetition-rate dual-comb system  because the two beat signals are obtained directly from the RF comb. Even self-corrected spectroscopy can be realized when two combs initially having a high-level mutual coherence . The post-process greatly suppresses the relative frequency noise between the two combs, which is sufficient for obtaining mode-resolved spectroscopy.
Some researchers have generated two combs by modulating a single-source CW laser [33–35] or using a single bidirectional ring cavity [36, 37]. Novel light sources can enhance mutual coherence to a certain extent, but tight-locking approaches or post-processes remain necessary for applications requiring high accuracy.
In summary, the post-process approach provides promising mutual coherence and is low-cost, easily operable, and readily available. But post-process capabilities should be further demonstrated via comparison with phase-stable dual-comb interferometers realized through tight-locking approaches. In this paper, we present a clear and effective post-correction algorithm based on a digital process. The process also demonstrates high mutual coherence (narrow relative linewidth), low timing jitter, and a stable carrier phase. To the best of our knowledge, this study marks the first time a digital post-correction method realizes such a phase-stable dual-comb interferometer without any tight locking. The corrected dual-comb interferometer’s performance is nearly equal to that of the tight-locking method [5, 11, 14], indicating that the proposed system alleviates dependence on an ultra-stable CW laser as an optical reference and a fast optical modulator as a feedback actuator.
2. Experimental setup and principle
2.1 Experimental setup
Figure 1 shows the experimental setup and principle of the dual-comb interferometer. As Fig. 1(a) shows, a pulse train from Comb 1 (frep1 = 56.090 MHz) passed through a Michelson interferometer and then interfered with Comb 2 (frep2 = 56.092 MHz), generating interferograms (IR and IM) with a certain update time Tupdate = 1/Δfrep. The interferograms (IGMs) were sampled and computed along with the relative beat signal fb, which was generated using a free-running CW laser as an optical intermediary. The generated relative beat signal is described in Fig. 1(b). The two beats (fCW1,1 and fCW1,2) between the two modes (f1 and f2) and the CW1 laser were extracted by filters and electronically mixed, generating a beat signal (Beat 1) whose frequency fb1 equaled f1 –f3 and was immune to noise from the CW1 laser. Similarly, another beat signal (Beat 2) with frequency fb2 = f4 –f2 was generated using another laser, CW2. The detail schematic of beat generation process is given in our previous work . The two relative beat signals demonstrated how the frequency noise or phase noise transfers among different modes; but only one beat (Beat 1) signal was used for IGM correction when we use full-stabilized two combs, and the Beat 2 generated from CW 2 was only used for verification of the noise model. Figure 1(c) shows the multiheterodyne process of dual-comb interferometry in the frequency domain. Two combs with different repetition frequencies converted the optical comb lines into the RF domain. The RF comb’s frequency components can be described as f(k) = fc ± k·Δfrep (k = 0, 1, 2···), where fc is the comb’s center frequency. In the time domain, IGMs can be described as envelopes with a period of 1/Δfrep, multiplied by the carrier wave cos(2πfct) as shown in Fig. 1(d). Thus, the frequency noise of fc and Δfrep (or the phase noise of the carrier wave and timing jitter of envelopes) were each incorporated into our correction algorithm.
2.2 Principle of the correction algorithm
As we have introduced by Fig. 1(d) in Section 2.1, the IGMs can be described as envelopes multiplied by the carrier wave. The timing jitter of the envelopes depends on the noise of Δfrep and the RF carrier phase noise depends on the frequency noise of the corresponding optical carrier wave. Actually, the RF carrier frequency is a beat signal between the optical carrier waves of the two combs as fc = nc1fr1 − nc2fr2 + fceo1− fceo2, where nc1 and nc2 is the order number of such two comb lines. According to the comb model, the frequency modes’ value and corresponding noise are linear versus the modes’ order, thus the instantaneous phase noise of carrier wave f(nc1) is expressed asFig. 1(b). Thus corresponding timing jitter can be expressed as [28, 30]
In fact, such dual-comb noise model is not strictly linear with respect to Δfrep, where the nc1 does not equal to nc2 for generating RF comb lines because of the repetition rate difference. But compared with the value s of nc1 and nc2 (both ~3.4 × 106), we ignore their difference between nc1 and nc2 (~60) when computing the relative noise between two combs.
Our algorithm focuses on compensating the carrier phase noise and timing jitter of IGMs, no matter the dual-comb system is free-running or fully-stabilized. Here, fully-stabilize means ordinary locking frep and fceo to a RF standard, not tight locking. To obtain available phase information of a certain frequency mode, we need a fully-stabilized system to fix fceo and frep.
The above model would be simplified when the both fceo are locked by f-2f interferometer to avoid drift (both frep are also locked in fact), which is necessary because we aim to recover available phase information of carrier wave f(nc1) = nc1fr1 + fceo1. We assume the noise of the locked fceo can be ignored compared with the noise of (nc1fr1 − nc2fr2) in carrier frequency, even both fr1 and fr2 are locked in our system. Therefore, the fc ≈nc1fr1 − nc2fr2. The value of the fc can be determined by a fitting method or estimated directly from the center of the RF comb. It is not very accurate but the uncertainty of fc has little effect on the phase correction performance according our results shown in the next section. Then we can calculate the value of the optical carrier wave f(nc). The solution of nc1 and nc2 is unique because it is limited by the optical bandpass filter.
For such a stabilized dual-comb system, the respective noise of fc and Δfrep were obtained from either Beat 1 or Beat 2. Because δφb1(t)/fCW1 = δφb2(t)/fCW2 once we ignore the effect of fceo. In other word, only one CW laser is enough. We use Beat 1 as an example. The instantaneous phase φb1(t) of Beat 1 was computed using the Hilbert transform , with the instantaneous frequency denoted as fb1(t) = dφb1(t)/dt. The exact frequency value (without any noise) of Beat 1 could be calculated because the rough value of fCW1 and the locked value of the two combs’ fceo and frep were known . In practice, fb1 is the integer multiple of Δfrep because the values of all comb parameters are integer multiples of Δfrep, which can be computed as the round number of [fb1(t)/Δfrep]·Δfrep. Thus, we can obtain the instantaneous frequency noise of Beat 1 δfb1(t) and its corresponding phase noise δφb1(t) asEq. (1) as:
After the carrier phase noise and timing jitter are obtained, we have sufficient information to correct the raw IGMs (denoted as I0). The correction algorithm is realized in the time domain and divided into two steps:
First, we correct the carrier phase noise, which is as simple as
Then, we compensate the IGMs’ timing jitter:Eq. (7) is equivalent to shifting the envelopes with Tjitter, shown in Fig. 1(d). In practice, however, we can only manipulate the whole phase-corrected IGMs; thus, an additional phase shift exp(−i2πfcTjitter) should be taken into account.
The two correction steps are simple and comprehensive linear processes because we divide the noise into two parts: the phase noise of the carrier wave and the timing jitter of the envelopes (noise of Δfrep). We have made some assumptions and approximations in correction algorithm, experimental results are provided in the next section to demonstrate the algorithm’s validity.
The experimental results are organized as follows: Section 3.1 presents the phase relation between Beat 1 and Beat 2 to demonstrate the phase (frequency) noise transmission model expressed in Eqs. (3) and (4). Section 3.2 outlines the IGMs’ timing jitter and correction signal expressed in Eq. (5). Section 3.3 shows the detailed process and results of the post-correction in Eqs. (6) and (7). We also present the linewidth, timing jitter, and carrier-phase stability of phase-timing corrected IGMs.
3.1 Phase noise of different modes
In our algorithm, the compensating signal for the carrier wave is computed from the noise of Beat 1 as indicated in Eqs. (3) and (4). However, this method’s performance is difficult to demonstrate because it is hard to extract the carrier wave individually. Alternatively, we generated another beat signal (Beat 2) as a comparison to obtain the phase noise of both beat signals: δφb1(t) and δφb2(t). Curve ‘i’ in Fig. 2 represents the phase noise δφb1(t). Curve ‘ii’ is computed from δφb2(t)·fCW1 /fCW2. Both curves fluctuate from within ± 104 rad, mainly due to the frequency noise of repetition frequency. Comparatively, the phase noise difference between curve ‘i’ and curve ‘ii’ only has ± 1 rad fluctuations as indicated by the gray curve, which comprises only 0.01% of the phase noise of either beat signal.
The high correlation between the two beats’ phase noise suggests they can compensate noise mutually. Using Beat 1 as an example, the compensated signal equals the raw signal multiplied by exp[−i δφb2(t)·fCW1 /fCW2]. Figure 3(a) shows the spectrum of raw Beat 1, and Fig. 3(c) shows the spectrum of compensated Beat 1 as introduced above. About 1 s data length was used, so the ~1Hz linewidth of compensated Beat 1 reached the theoretical limitation. Figure 3(b) also depicts the spectrum of the compensated signal, but the compensating phase signal is exp[−i δφb2(t)]. This performance decline indicates that only part of the noise of the repetition frequency was compensated; thus, phase noise conversion is necessary. From the experimental results in this section, it is clear that IGMs’ carrier-phase noise can be compensated.
3.2 Timing jitter of IGMs
Another key parameter is the phase noise of Δfrep in our correction algorithm. It was computed using Eq. (5), and the validity must be demonstrated. Thus, we computed the timing jitter of IGMs as a comparison. The time delay of IGMs can be calculated through the phase-frequency slope of the spectrum by Fourier transform [14, 15]. The time information of IGMs is calculated every Tupdate. The first IGM is considered an ‘initial point’ in time axis; thus, the timing jitter of all IGMs can be obtained and shown as the ‘i’ curve in Fig. 4. The timing jitter computed from the beat signal in Eq. (5) is shown as the ‘ii’ curve, which shares the same ‘initial point’ in time axis. The timing jitter varied within ± 200 ns with ~80 ns standard deviation. The difference between the two curves shows a much higher stability with ~1 ns standard deviation. This result indicates that our algorithm can greatly compensate for timing jitter.
3.3 IGM correction
We demonstrated the effectiveness of the correction algorithm above. Similarly, the algorithm shown in Eqs. (6) and (7) can be applied to compensate the carrier-phase noise and timing jitter of IGMs. In Fig. 5, only the results of reference IGMs (IR) are shown. The measured IGMs (IM) were compensated similarly: two series of IGMs used the same compensating signal but at different times. The IGMs with time interval Tupdate are displayed together with a 2-μs time window to present the detailed process because the Tupdate equals integer multiples of sampling interval 1/fs. The whole data length is about 1 s, corresponding to 2000 IGMs. The IGMs are shown every 50ms in the time domain (i.e., 20 IGMs are shown together). The spectrum was computed using 1 s samples of either reference or measurement results. The spectrum was located between 0 and frep/2 with ~13 MHz 3 dB bandwidth, resulted from ~56 MHz frep, 2 kHz Δfrep, and 3nm optical bandwidth. Only part of the spectrum is shown to display distinguishable frequency lines.
Figure 5(a) shows the raw IGMs with an obvious timing jitter and unstable carrier phase; Fig. 5(d) is the spectrum of raw IGMs between 14.637960 MHz and 14.639040 MHz. The expected frequency line at 14.638000 MHz is difficult to distinguish. Figure 5(b) shows the corrected IGMs in which carrier phase noise and timing jitter were both compensated. The phase-corrected IGMs are not shown in the time domain because they were nearly identical to the raw IGMs in Fig. 5(a). The spectra of phase-corrected IGMs and phase-timing corrected IGMs were almost the same as those shown in Fig. 5(e). Because the uncertainty caused by timing jitter is 80ns/Tupate = 1.6 × 10−4 in the time domain, which was smaller than relative resolution RBW/Δfrep = 5 × 10−4 in the frequency domain (the RBW was 1Hz, determined by the 1s data length). This result indicates that only the linewidth performance in the frequency domain was insufficient to evaluate the effectiveness of the post-correction method. Both spectra shown in Fig. 5(e) had an 18-Hz sideband, caused by carrier phase residuals after phase compensation. The sideband could be suppressed after we compensated the residuals manually as shown in Figs. 5(c) and 5(f). The phase information of aligned IGMs could not be utilized, but it did provide a better spectrum, because the IGMs aligned perfectly. Coherent averaging [16, 19] could also be applied to suppress the intensity noise. Fig. (g) shows a comparison of the spectrum (from 14.618 MHz to 14.658 MHz) between raw IGMs and corrected IGMs. The raw spectrum is shown by multiplying a −1 factor. No comb line can be distinguished in it because the raw linewidth (0.4 MHz) is much wider than the RF comb’s spacing (2 kHz). By contrast, our correction algorithm recovered the comb lines successfully.
According to these experimental results, we obtained not only the narrow linewidth of the RF comb but also a low timing jitter and stable carrier phase. The method to calculate the timing jitter was introduced in Section 3.2. The carrier phase was also computed simultaneously from Fourier transform of the IGMs. The carrier phase was computed from the IGMs shown in Fig. 5(b) instead of Fig. 5(c). For eliminating the slow phase drift of either IM or IR caused by the fiber paths in the system, actually we calculated the carrier phase difference between IM and IR. Figure 6 shows the precision (Allan deviation) of the carrier phase and timing jitter. About 1 ns timing jitter and 0.2 rad phase precision were obtained without averaging (~0.22 ns and ~0.43 rad, respectively, with 0.01 s averaging time; ~0.025 ns and ~0.0051 rad, respectively, with 1 s averaging time). The high carrier phase precision and low timing jitter provide strong evidence that our post-correction algorithm reached nearly the same performance compared to the tight-locking method [14, 16, 17].
4.1 Comparison between different post-correction methods
We outlined three different post-correction methods [26, 28, 31] for spectroscopy in the introduction section. Essentially, the idea behind the three methods is similar in that two beat signals between two combs are used. On one hand, the phase noise caused by fceo fluctuation can be eliminated by using either beat signal because the fceo fluctuation makes the same contributions to all comb lines. Meanwhile, the phase noise caused by frep can be partly eliminated this way. However, the phase noise is mainly caused by the noise frep, which is amplified by n (the order of the comb line). Thus, we presented the algorithm to compensate the full phase noise caused by frep; in other words, the phase noise of the carrier wave is compensated. A comparison between the conditions that noise of frep is sufficiently considered and not considered when compensating phase noise is shown in Figs. 3(c) and 3(b), respectively. Even without considering the timing jitter, a spectrum with narrow linewidth can be obtained as shown in Fig. 5(e), which would be especially useful in spectroscopy applications. Findings also indicate we should evaluate the full performance rather than focusing solely on the spectrum linewidth in the post-correction method. On the other hand, timing jitter presents a more substantial challenge because it stretches the time axis nonlinearly. In the aforementioned three methods, a nonlinear Fourier transform or reconstructed time axis were applied to obtain the narrow-linewidth spectrum. The algorithm proposed in this paper provides a solution that focuses directly on the timing jitter of every single IGM (or the noise of Δfrep). The fully corrected IGMs show promising results for recovering the stable and available carrier phase, which can be used in some applications requiring high accuracy and precision (e.g., nanometer precision in dual-comb ranging and dual-comb ellipsometry). Our post-correction method and results also indicate that the self-corrected dual-comb system could be realized, because the carrier phase noise can be obtained directly from the timing jitter of IGMs as Eq. (5) shows. But the performance would be limited by the compensating rate (Δfrep) and large initial phase noise, unless using two high-repetition-rate combs with high mutual coherence .
Another difference between our method and some previous approaches is that the two combs in our system are fully-stabilized instead of free-running. RF-stabilized combs cannot provide a narrow linewidth and stable phase; they simply provide locked values of frep and fceo. Thus, accurate frequency information and a stable phase could be utilized. Another convenience of the proposed method is that only one CW laser is needed to generate a compensating signal. Our algorithm can certainly be used for a free-running dual-comb system. In this case, two CW lasers are needed to provide noise information for the carrier wave and repetition frequency difference as we present in Section 2.2. However, the absolute frequency of the phase information would be lost unless the accurate initial values of frep and fceo were known during post-correction. Essentially, whether the two combs are stabilized in our post-correction method depends on applications’ accuracy demand. The correction algorithm we proposed separates the carrier phase noise and timing jitter, which is a simple approach with demonstrated effectiveness for phase-required applications.
The performance also varies between our method and previous post-correction methods. For example, the experimental results in Refs [28, 30, 31]. only show the mode-resolved spectrum that is enough only for spectroscopy application. Although the coherent averaging is successfully applied in Ref , it is resulted from a tight-locking method (a cavity-stabilized CW laser is employed as a reference) combined with a post correction method as presented in its Method section. In our method, not only mode-resolved spectrum is obtained, phase-stable IGMs are also recovered so that coherent averaging can be applied without any tight locking. We lock frep and fceo to a RF standard to make sure we can obtain accurate frequency value of comb modes, and makes the stable phase available for further applications. Generally speaking, our post-correction method reaches the performance by the tight-locking method .
4.2 Comparison between stabilization and post-correction methods
Basically, the post-correction methods are equivalent to the tight-locking method that uses an ultra-stable CW laser as an optical reference or a mutually locking scheme that uses a free-running CW laser as an optical intermediary between two combs. The beat signals, whether locked or compensating, are exactly the same; the only difference is the way the signals are used. When stabilizing the combs, fast actuators are needed in electronics for feedback control. However, the servo bandwidth is limited, which normally has a ~1MHz feedback rate. The post-correction method surmounts bandwidth feedback limitations. In fact, every sampled IGM point in our experiment is corrected because the IGMs and compensating beat signals use the same sampling clock. A slight drawback is that the compensating phase signal is the integral of the frequency noise from the initial time; it would consume too much memory if long-length IGMs needed to be corrected continuously. Fortunately, a several-seconds data length is sufficient for most applications.
We have presented a computational post-correction method to compensate the noise of a dual-comb interferometer. The algorithm is highly clear and focused on the carrier phase noise and timing jitter of IGMs. It effectively realized narrow linewidth, low timing jitter, and stable carrier phase of IGMs. This post-correction technique decreases the requirements of complex and tight dual-comb system stabilization, especially in phase-required applications.
National Natural Science Foundations of China (61575105, 61611140125); Shenzhen fundamental research funding (Grant No. JCYJ20170412171535171); Tsinghua University Initiative Scientific Research Program.
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