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Abstract
To make dual-comb interferometry usable in a wide range of applications, it is important to achieve reproducible measurement results even in non-ideal environments that affect the repetition-rate stability. Here, we consider dual-comb interferometry based on a pair of fully referenced optical frequency combs (OFCs) and investigate the impact of fluctuations in the OFC repetition frequencies on the peak position of the center burst in the interferogram. We identify a phase-locking scheme that minimizes the impact of these fluctuations through choosing a special combination of phase-locked frequencies, and the resulting type of operating condition is termed integer-locking condition. Under the integer-locking condition, the number of sampling points in each interferogram remains constant regardless of repetition-rate variations, and this enables more stable phase-resolved measurements in non-ideal environments. We demonstrate the application of this approach using absolute path-length measurements and discuss the accuracy limit imposed by the integer-locking condition. Our findings offer a strategy for robust dual-comb interferometry outside metrology laboratories.
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1. Introduction
Interferometry using dual-comb sources, which provide a pair of frequency-stabilized optical frequency combs (OFCs), has attracted considerable attention in the past two decades. One area of application of such light sources is dual-comb spectroscopy (DCS) [1], which allows us to perform ultra-precise phase-resolved spectroscopic measurements of molecules [2–14] and solid materials [15–20]. Another area of application concerns absolute distance measurements with a resolution on the order of sub-nanometers to picometers [21–32]. Recently, this technology has also been applied to asynchronous optical sampling (ASOPS) [33,34] for time-resolved measurements [35,36] and terahertz time-domain spectroscopy measurements [37,38], which benefit from an extremely low jitter in the (time-dependent) emission-time difference between OFCs with tight phase locking.
When performing phase-resolved dual-comb interferometry measurements, it is important to maintain mutual coherence between the OFCs through tight phase-locking [1], and to achieve this, a narrow-linewidth continuous wave (cw) laser with frequency $f_{\mathrm {cw}}$ can be used [2]. For each OFC, the beat note between $f_{\mathrm {cw}}$ and one of the comb lines that is contained in both OFCs is tightly phase-locked as illustrated in Fig. 1. Here, the repetition frequencies of the two OFCs, referred to as Comb 1 and Comb 2, are $f_\mathrm {rep1}$ and $f_\mathrm {rep2}$, respectively. The frequency of the beat note due to Comb 1 is $f_{\mathrm {beat1}}$, and that due to Comb 2 is $f_{\mathrm {beat2}}$. In addition, a tight phase-locking of the carrier-envelope offset (CEO) frequencies, denoted by $f_{\mathrm {ceo1}}$ and $f_{\mathrm {ceo2}}$, is also important.
Fig. 1. The phase-locked frequencies in dual-comb interferometry. For each OFC (Comb 1 and Comb 2), the beat note between the cw-laser frequency $f_{\mathrm {cw}}$ and one of the comb lines that is contained in both OFCs is tightly phase-locked to a radio-frequency synthesizer. The repetition frequencies of the two OFCs are $f_{\mathrm {rep1}}$ and $f_{\mathrm {rep2}}$, and the beat note frequencies are denoted by $f_{\mathrm {beat1}}$ and $f_{\mathrm {beat2}}$. Additionally, the CEO frequencies of the two OFCs, $f_{\mathrm {ceo1}}$ and $f_{\mathrm {ceo2}}$, are also phase-locked. Together with the CEO and beat frequencies, the mode numbers $n$ and $n+\Delta n$ associated with the comb lines in the two OFCs used for the beat notes determine $f_{\mathrm {cw}}$. The integer-locking condition is established if the relationships $n/\Delta n \in \mathbb {Z}$, $f_{\mathrm {ceo1}}=f_{\mathrm {ceo2}}$, and $f_{\mathrm {beat1}}=f_{\mathrm {beat2}}$ are satisfied.
In phase-resolved dual-comb interferometry, it is usually essential to obtain the same number of sampling points for each interferogram (IGM) recorded throughout the data accumulation process. The number of sampling points is determined by the so-called compression factor $M \equiv f_{\mathrm {rep1}} /(f_{\mathrm {rep1}}-f_{\mathrm {rep2}})$ [1], where $f_{\mathrm {rep1}}$ is the repetition frequency of the OFC used as the sampling clock, and $f_{\mathrm {rep2}}$ is the repetition frequency of the other OFC. Since fluctuations in $f_{\mathrm {rep1}}$ lead to a temporal variation in $M$, it is difficult to record consistent waveforms in successive IGMs, and this issue is particularly important for the coherent averaging process [21]. In metrology laboratories, this issue is usually addressed by tightly stabilizing $f_{\mathrm {cw}}$ to a resonance frequency of a high-finesse cavity [39,40], which ensures that the accuracies of $f_{\mathrm {rep1}}$ and $f_{\mathrm {rep2}}$ are high enough to keep the value of $M$ constant. In applications of dual-comb interferometry where a high-finesse cavity is difficult to use, for example in certain outdoor applications [41], $f_{\mathrm {cw}}$ is instead determined by a frequency of a narrow-linewidth external cavity laser (ECL) which is stabilized to one of the two OFCs, and the chosen OFC serves as a frequency standard by phase-locking $f_{\mathrm {ceo1}}$ and $f_{\mathrm {rep1}}$. In particular, in the situation where one of the OFCs lacks the capability for high-bandwidth cavity length control using an electro-optic modulator (EOM) [42] and a specially designed small piezo-electric transducer (PZT) attached to an optical fiber [41] that can be costly and technically challenging, only four channels can be used for tight frequency locking of the dual OFC systems: two for $f_{\mathrm {ceo1}}$ and $f_{\mathrm {ceo2}}$ via feedback to the currents of the laser diodes, one for $f_{\mathrm {beat1}}$ via feedback to the current of the external cavity laser, and one for $f_{\mathrm {beat2}}$ via feedback to the cavity length of Comb 2 using a high-bandwidth cavity length control. In this experimental scene, the stabilization of $f_{\mathrm {rep1}}$ is achieved through feedback control of the OFC’s cavity length [43], which is relatively slow, and the frequency fluctuation or drift of $f_{\mathrm {rep1}}$ directly causes the frequency fluctuation and drift of $f_{\mathrm {cw}}$. Consequently, the stability of $M$ depends on the stability of $f_{\mathrm {rep1}}$. The stabilization of $M$ occurs on a timescale corresponding to the inverse of the PZT bandwidth. If this timescale exceeds the measurement time required to obtain a single IGM ($t_\mathrm {IGM}=1/|f_{\mathrm {rep1}}-f_{\mathrm {rep2}}|$), variations in $M$ are likely to occur during the acquisition of successive IGMs, affecting the reproducibility of the signal. Although many laboratories use this method to improve mutual coherence in dual comb systems [15,18,35,36,38,44,45], the impact of fluctuations in $f_{\mathrm {rep1}}$ on $M$ (and the quality of the IGM) has not yet been thoroughly investigated. Such investigations may reveal an effective strategy to minimize the detrimental influence of fluctuations in $f_{\mathrm {rep1}}$.
There are some existing technologies for achieving long-term coherent averaging in dual-comb interferometry. Numerical error correction routines provide a powerful solution for reconstructing the phase information of the dual-comb interferometric signal using two free-running OFCs [46–48]. However, numerical phase correction can sometimes lead to artifacts in the signal, as pointed out by Chen et al.[12]. The proposal of a stable, fieldable dual-comb spectroscopy system using an ECL was investigated by Truong et al. [49]. They achieved long-term frequency stability by phase-locking one of the repetition rates of the OFCs against a stable quartz oscillator. However, their algorithm considers an experimental scenario with five tightly frequency-locked channels, which is not applicable to the system described above, which has only four frequency-locked channels. Moreover, Chen et al. [12] proposed a phase-stable dual-comb interferometer with mutual coherence between the two OFCs over 2000 seconds, but they did not explicitly discuss the effect of repetition rate fluctuations on the phase stability of the interferometer.
In this study, we investigate how temporal variations in $f_{\mathrm {rep1}}$ affect the compression factor $M$. We identify a phase-locking scheme that minimizes the impact of repetition-rate fluctuations in a dual-comb source, and the type of operating condition obtained by applying this scheme is termed integer-locking condition. This condition consists of three criteria: (i) The mode numbers $n$ and $n+\Delta n$ are integer multiples of $\Delta n$ (hence the name integer-locking), (ii) the two CEO frequencies are equal, and (iii) the two beat-note frequencies are equal. Under the integer-locking condition, the relationship $M=(n+\Delta n)/\Delta n$, where $M$ is an integer, is maintained even if $f_{\mathrm {rep1}}$ varies. This enables phase-resolved dual-comb interferometry under conditions where $f_{\mathrm {rep1}}$ fluctuates. We demonstrate the effectiveness of this approach by modulating $f_{\mathrm {rep1}}$ and recording successive IGMs. A stable IGM peak position and phase can be obtained even when $f_{\mathrm {rep1}}$ changes by 100 Hz. Furthermore, we discuss how repetition-rate fluctuations affect absolute distance measurements under the integer-locking condition and propose a method to correct the measurement error related to the used phase-locking scheme. The proposed locking scheme is beneficial for phase-resolved dual-comb interferometry under non-ideal environmental conditions that affect the repetition-rate stability.
2. Integer-locking condition
In this section, we explain the theoretical basis of our proposed phase-locking scheme. As shown in Fig. 1, we consider a pair of fully stabilized OFCs, where the CEO frequencies are phase-locked to a radio-frequency (RF) synthesizer referenced to an RF standard. Moreover, one of the comb lines of Comb 1 is phase-locked to the corresponding comb line of Comb 2 via a narrow-linewidth cw laser with frequency $f_\mathrm {cw}$. If we denote the optical beat frequency between the chosen comb line of Comb 1 (Comb 2) and $f_\mathrm {cw}$ by $f_{\mathrm {beat1}}$ ($f_{\mathrm {beat2}}$), $f_\mathrm {cw}$ can be expressed as follows:
Each IGM contains a specific waveform (until Section 4.2 we consider a single center burst), and to maintain the stability of the peak position of this signal during the measurement of successive IGMs, $M$ needs to be an integer. There are two methods to ensure this: In the first method, $n/ \Delta n$ is set to an integer value and $\Delta f$ is set to zero, which is a prerequisite of the integer-locking condition. In the second method, $n/ \Delta n$ is set to a non-integer value and $\Delta f$ is set to an appropriate value to make $M$ an integer, and the type of operating condition achieved by this method is hereafter referred to as the non-integer-locking condition.
In this work, we consider experimental conditions where $f_\mathrm {rep1}$ is not perfectly stabilized and thus is subject to fluctuations. According to the elastic-tape model for OFCs, a variation in $f_\mathrm {rep1}$ results in an accordion-like expansion of the entire frequency comb [43,50]. Consequently, a change in $f_\mathrm {rep1}$ also affects the frequency of the cw laser. When $f_{\mathrm {rep1}}$ changes to $f_{\mathrm {rep1}}+\delta f_{\mathrm {rep1}}$, the (modified) frequency of the cw laser becomes
Furthermore, because $f_{\mathrm {ceo2}}$ and $f_{\mathrm {beat2}}$ are phase-stabilized, $f_{\mathrm {rep2}}$ also changes in response to the change in $f_{\mathrm {rep1}}$. When $f_{\mathrm {rep1}}$ changes, the (modified) repetition frequency of Comb 2 becomes
Thus, the compression factor after the change in $f_{\mathrm {rep1}}$ is
By using Eqs. (3), (4), and (5), we can rewrite $M'$ as
From this equation, we find that $M'=M$ holds irrespective of $\delta f_{\mathrm {rep1}}$, if $\Delta f=0$. Furthermore, if $n/\Delta n$ is an integer, the envelope of the center burst in successive IGMs remains at the same position. Therefore, under the integer-locking condition, which contains these two requirements, the envelope of the center burst remains at the same position even if $f_\mathrm {rep1}$ fluctuates. This is the key idea presented in this work, and the experimental verification of this idea is shown later.
In addition, we also explore the impact of the above idea on carrier-phase measurements using DCS. In such measurements, the phase of the IGM signal needs to remain zero, which is achieved if $f_{\mathrm {ceo1}}-f_{\mathrm {ceo2}}=l\cdot (f_{\mathrm {rep1}}+\delta f_{\mathrm {rep1}}-f_{\mathrm {rep2}}')$ [6], where $l$ is an integer. This condition is satisfied for any value of $\delta f_\mathrm {rep1}$ if $l=0$, i.e. if $f_{\mathrm {ceo1}}-f_{\mathrm {ceo2}}=0$. Therefore, the integer-locking condition for stable DCS measurements is defined by the following three equations:
where $\mathbb {Z}$ denotes the set of all integers.3. Experimental setup
Figure 2 shows the experimental setup used in this work. The setup is divided into three parts: the dual OFC system, Setup A, and the IGM measurement system. For the experiments in Section 4.3, Setup A is replaced with Setup B. The dual OFC system is indicated by the green broken rectangle in Fig. 2. Here, two Er-doped fiber-based OFCs, labelled Comb 1 and Comb 2, are used as the light sources. The CEO frequencies $f_{\mathrm {ceo1}}$ and $f_{\mathrm {ceo2}}$ are detected by the $f$–$2f$ self-referencing technique [51,52] and phase-locked to an RF signal from a function generator (WF1968, NF Corp.) via a feedback signal to the pump-LD current. An ECL with a wavelength of $\approx$ 1550.11 nm (PLANEX, Redfern Integrated Optics) is used to detect the two beat frequencies $f_{\mathrm {beat1}}$ and $f_{\mathrm {beat2}}$. $f_{\mathrm {beat1}}$ is phase-locked to a second RF signal via a feedback signal to the ECL current, and $f_{\mathrm {beat2}}$ is phase-locked to a third RF signal via feedback signals to the intra-cavity EOM, the PZT, and the thermoelectric cooler (TEC) for Comb 2. To demonstrate the robustness of DCS measurements against fluctuations of $f_{\mathrm {rep1}}$, we phase-locked $f_{\mathrm {rep1}}$ to a fourth RF signal using another feedback loop that controls the PZT and the TEC for Comb 1; this allows us to alter $f_{\mathrm {rep1}}$ by adjusting the RF frequency. Finally, the output light is amplified using erbium-doped fiber amplifiers (EDFAs). To correlate the data and the fluctuations, we monitored $f_{\mathrm {rep1}}$ and $f_{\mathrm {rep2}}$ continuously by a 2-channel universal counter (SC7215A, Iwatsu). All function generators and the universal counter were referenced to a 10-MHz signal from a global-positioning-system (GPS)-controlled rubidium (Rb) clock with a relative uncertainty on the order of $10^{-12}$.
Fig. 2. Experimental setup for dual-comb interferometry. The dual-comb source is indicated by the green broken rectangle. Setup A, which combines the two OFCs, is indicated by the blue broken rectangle at the center. In Section 4.3, Setup A is replaced with Setup B. The IGM measurement system is indicated by the red broken rectangle. The solid black lines represent optical fibers, and the thick broken lines indicate electric cables. EDF: erbium-doped fiber; WDM: wavelength division multiplexing coupler (980 nm/1550 nm); EOM: electro-optic modulator; PZT: piezoelectric element; TEC: thermoelectric cooler; Q: quarter-wave plate; H: half-wave plate; P: polarizer; ISO: isolator; LD: laser diode; ECL: external cavity laser; $f$–$2f$: $f$–$2f$ interferometer; EDFA: erbium-doped fiber amplifier; VOA: variable optical attenuator; CL: collimator lens; BPF: bandpass filter; PC: personal computer.
Regarding the two different setups indicated by the blue broken rectangles, in Setup A, Comb 1 and Comb 2 are combined by a broadband coupler, and in Setup B, Comb 2 is first split into a reference path and a sample path: The light in the reference path propagates through a single-mode fiber with a length of about 5 meters and a variable optical attenuator (VOA) in the middle of the fiber. The light in the sample path propagates through an optical fiber with a total length of 1.32 km (a 1.19-km-long single-mode fiber and a 0.13-km-long dispersion-compensating fiber). The light propagating through the sample and reference paths is combined by a broadband coupler, before it is combined with Comb 1 by another broadband coupler.
The IGM measurement system is indicated by the red broken rectangle in Fig. 2. Here, the combined beam from Setup A or Setup B is first detected by an amplified photodetector (PDB415C, Thorlabs). The detected electrical signal is passed through an electrical bandpass filter (BPF) with a frequency bandwidth of 3–30 MHz to remove the DC and repetition-frequency components in the signal, and then it is recorded with a digitizer (M2p.5962-x4, Spectrum) with a sampling rate equal to $f_{\mathrm {rep1}}$.
The integer-locking condition is achieved by using the following procedure: To satisfy the relations given by Eqs. (9) and (10), we first set $f_{\mathrm {ceo1}}= f_{\mathrm {ceo2}}= f_{\mathrm {beat1}}= f_{\mathrm {beat2}} =$ 22 MHz. To satisfy Eq. (8), the phases of $f_{\mathrm {ceo1}}, f_{\mathrm {ceo2}}, f_{\mathrm {beat1}}, f_{\mathrm {beat2}}$, and $f_{\mathrm {rep1}}$ are first locked and the values of $n$ and $\Delta n$ are roughly estimated [53]. Then, the locking mechanism for $f_{\mathrm {ceo2}}, f_{\mathrm {beat1}}$ and $f_{\mathrm {beat2}}$ is deactivated, and the current applied to the ECL is either increased or decreased to let $f_{\mathrm {cw}}$ shift to a value where $n$ equals a multiple of 10. After phase locking $f_{\mathrm {beat1}}$ again, the resonator length of Comb 2 is adjusted while monitoring $f_{\mathrm {rep2}}$ with a frequency counter. Thereby, the condition $\Delta n = -10$ can be reached. After phase locking $f_{\mathrm {ceo2}}$ and $f_{\mathrm {beat2}}$ again, the values of $n$ and $\Delta n$ are measured to verify that $n/\Delta n$ is an integer.
Table 1 summarizes the frequencies of the RF signals (for the phase stabilization of the discussed frequency parameters) used in the experiments in Section 4.1, as well as the corresponding values of $n$, $\Delta n$, and $M$. The experiments were performed under four different conditions, which are hereafter referred to as measurement conditions (A), (B), (C), and (D). In (A), the integer-locking condition is fulfilled, because $n/\Delta n$ is an integer and $\Delta f=0$. In (B), $f_{\mathrm {rep1}}$ is 100 Hz larger than in (A), while the other parameters remain the same. In (C), which corresponds to the non-integer-locking condition, $n/\Delta n$ is not an integer, but the $M$ is an integer, because $f_{\mathrm {beat1}}$ and $f_{\mathrm {beat2}}$ are appropriately set. In (D), $f_{\mathrm {rep1}}$ is 100 Hz larger than in (C), while $f_{\mathrm {ceo1}}$, $f_{\mathrm {ceo2}}$, $\Delta f$, $n$, and $\Delta n$ remain the same. Therefore, the compression factor is not an integer. Note that $|f_{\mathrm {rep1}}-f_{\mathrm {rep2}}|$ was approximately $200$ Hz in each condition, so $t_\mathrm {IGM}\approx 0.005$ s. The parameters for the experiments described in Sections 4.2 and 4.3 are provided in the Appendix.
Table 1. Experimental parameters used in Section 4.1: the five phase-locked frequencies $f_\mathrm {rep1}$, $f_\mathrm {ceo1}$, $f_\mathrm {ceo2}$, $f_\mathrm {beat1}$, and $f_\mathrm {beat2}$, the mode numbers $n$ and $\Delta n$, and the compression factor $M$ or $M'$ are shown for four different measurement conditions
4. Results and discussion
In this section, we demonstrate the effectiveness of the integer-locking condition for dual-comb interferometry. The first two subsections investigate how variations in $f_{\mathrm {rep1}}$ affect IGMs in the cases of integer-locking and non-integer-locking conditions. The experiments were performed with two OFCs that are fully stabilized, while the repetition frequencies were subject to variations: Section 4.1 considers the situation where a step-like change in $f_{\mathrm {rep1}}$ occurs. In Section 4.2, we added a sinusoidal offset to $f_{\mathrm {rep1}}$ to mimic environmental fluctuations. Section 4.3 focuses on the suitability of the integer-locking condition for absolute distance measurements. We also discuss the accuracy limit of such distance measurements under the integer-locking condition, which is particularly evident for long distances, and suggest a method to correct this error.
4.1 Impact of a step-like change in the repetition frequency
Here, we compare our experimental results obtained using four different measurement conditions to evaluate the effect of the integer-locking condition. For each measurement condition, we continuously measured IGMs using Setup A for approximately 50 s. Each collected data set contains $10000 \times |M|$ sampling points, which corresponds to 10000 IGMs with a segment length equal to $|M|$ in the cases of measurement conditions (A) to (C). In the case of measurement condition (D), the data set was divided into segments with a length equal to the integer nearest to $|M'|$. Figures 3(a)–3(d) display representative IGM data near the center-burst peak measured under conditions (A) to (D), respectively. The sampling index $m$ is used to specify the temporal position of each data point in an IGM. Each panel shows ten IGMs, where the IGM at the bottom is data of the first segment in the collected data, and the other IGMs are the data of later segments at intervals of 1000 IGMs, corresponding to a time interval of about 5 s. Figures 3(e)–3(h) show the peak position of the center-burst signal, $m_{\mathrm {peak}}$ as a function of the segment number for each measurement condition.
Fig. 3. (a) Representative results of a continuous measurement of 10000 IGMs under condition (A), where the integer-locking (IL) scheme is applied. Each IGM is identified by a so-called segment number ranging from 1 to 10000. The ten IGMs with segment numbers $1+ i \times 1000$ ($i$ = 0, 1, 2,…, 9) are shown, and the data are offset for clarity. The origin of the sampling index $m$ is set to the peak position of the center burst in the first IGM shown at the bottom of the figure. To focus on the center burst, only the data in the range $m=0 \pm 100$ points are shown. (b) The corresponding data measured under condition (B). (c), (d) The representative results for measurement conditions (C) and (D), respectively, where the non-integer-locking (NIL) scheme is applied. (e)–(h) The peak position of the center-burst signal, $m_{\mathrm {peak}}$, as a function of the segment number for each of the four measurement conditions.
Figures 3(e) and 3(g) evidence that the peak position remains constant under measurement conditions (A) and (C), where we applied the integer-locking (IL) and non-integer-locking (NIL) schemes, respectively. This result is attributed to the fact that $M$ is an integer in these measurements. When the $f_{\mathrm {rep1}}$ in these measurements is increased by 100 Hz, which corresponds to measurement conditions (B) and (D), a clear difference can be observed between the measurement results: In the case of the integer-locking condition, $m_{\mathrm {peak}}$ remains the same when $f_{\mathrm {rep1}}$ is changed [Fig. 3(f)], while $m_{\mathrm {peak}}$ starts to continuously shift in the case of the non-integer-locking condition [Fig. 3(h)]. In the former case, $m_{\mathrm {peak}}$ is unaffected by the change in $f_{\mathrm {rep1}}$, because $M$ remains the same as a result of $\Delta f=0$ (Eq. (7)). The continuous shift in the latter case is a result of the change in $M'$ to a non-integer value when $f_{\mathrm {rep1}}$ is increased: The slope of the curve in Fig. 3(h) is 0.0137, which is equal to the difference between the compression factors of measurement conditions (C) and (D).
4.2 Demonstration of robustness to oscillatory fluctuations
In this subsection, we evaluate the impact of a continuous change in $f_\mathrm {rep1}$ on the stability of $m_{\mathrm {peak}}$. To simulate variations in the repetition frequency due to environmental fluctuations, we modulated $f_\mathrm {rep1}$ by adding a sinusoidal offset with an amplitude of 5 Hz and a frequency of 0.02 Hz. The details of the parameters used for this experiment are provided in the Appendix [see measurement conditions (E) and (F)]. The temporal evolution of $f_\mathrm {rep1}$ measured under condition (E), where the integer-locking (IL) scheme is applied, is shown in Fig. 4(a), and that measured under condition (F), where the non-integer-locking (NIL) scheme is applied, is shown in Fig. 4(b). Figures 4(c) and 4(d) plot the corresponding data of $m_{\mathrm {peak}}$. We find that, in the case of the integer-locking condition, $m_{\mathrm {peak}}$ remains almost the same despite the modulation of $f_\mathrm {rep1}$. In contrast, in the case of the non-integer-locking condition, an oscillation of $m_{\mathrm {peak}}$ is observed, which can be attributed to a combination of the change in the $M$ and the change in the carrier phase of IGM. In fact, the maximum value of the modulated compression factor $M'$ is calculated from Eq. (7) as $M'= -305581.00069$ when the repetition frequency exceeds its average value by 5 Hz. However, this change in the compression factor is too small to account for the observed change in $m_{\mathrm {peak}}$. Therefore, a change in the carrier phase of IGM during the modulation of the repetition frequency is a plausible explanation for the variation in $m_{\mathrm {peak}}$ in Fig. 4(d).
Figures 5(a) and (b) show the IGMs with segment number 1 measured under conditions (E) and (F), respectively, and Figs. 5(c) and (d) show the IGMs with segment number 101. In the case of the integer-locking (IL) condition [Figs. 5(a) and (c)], the phase is relatively stable within the time span between the first and hundred-and-first segment ($\approx$ 0.5 s) despite the continuous change in $f_\mathrm {rep1}$. The good degree of stability is related to the fact that we set $f_{\mathrm {ceo1}}=f_{\mathrm {ceo2}}$ and $M\in \mathbb {Z}$, which ensures a stable phase as long as mutual coherence between the two OFCs is maintained. This implies that frequency locking of $f_\mathrm {rep1}$ is even not necessary, and four phase locked loops are enough to achieve the stable measurement of the dual-comb interferometry in the integer-locking condition. On the other hand, in the case of the non-integer-locking (NIL) condition [Figs. 5(b) and (d)], the phase undergoes significant changes within $\approx$ 0.5 s, although we set $f_{\mathrm {ceo1}}=f_{\mathrm {ceo2}}$. This result indicates that a correct adjustment of the CEO frequencies to satisfy the relation $f_{\mathrm {ceo1}} - f_{\mathrm {ceo2}} = l \cdot (f_{\mathrm {rep1}} - f_{\mathrm {rep2}})$, where $l$ is an integer, as discussed in Refs. [1,6,12,47], to preserve the coherent averaging condition is not sufficient for coherent averaging if $f_{\mathrm {rep1}}$ is subject to changes, because the condition $M \in \mathbb {Z}$ is not satisfied. Hence, the integer-locking condition is important in the case of repetition-rate fluctuations. A complete visualization of the time evolution of these IGMs over a time span of 50 s is presented in Visualization 1.
Fig. 4. (a) The temporal variation of $f_\mathrm {rep1}$ during data collection under condition (E), where the integer-locking (IL) scheme is applied. (b) The temporal variation of $f_\mathrm {rep1}$ during data collection under condition (F), where the non-integer-locking (NIL) scheme is applied. (c), (d) The results of $m_{\mathrm {peak}}$ extracted from the data collected under conditions (E) and (F), respectively.
Fig. 5. Comparison of the IGMs with segment numbers 1 and 101 for the integer-locking (IL) and non-integer-locking (NIL) conditions. (a), (b) IGM data of the first segment for conditions (E) and (F), respectively, where $f_\mathrm {rep1}$ is modulated by adding a sinusoidal offset. (c), (d) IGM data of the 101st segment for conditions (E) and (F), respectively. A complete visualization of the waveform changes over a time span of 50 s is provided in Visualization 1.
4.3 Absolute distance measurements under the integer-locking condition
In this subsection, we address the accuracy limit imposed by the integer-locking condition if repetition-rate fluctuations are present: these fluctuations lead to minor changes in the repetition frequencies of OFCs, which can have a significant impact on the measurement accuracy. To illustrate this effect, we apply DCS to absolute distance measurements [21–32]. DCS-based absolute distance measurements can combine time-of-flight (ToF) and interferometric techniques, and here we consider the error in ToF measurements under the integer-locking condition. For this experiment, we used Setup B and either measurement condition (G), where $f_\mathrm {rep1}$ is not modulated, or measurement condition (H), where the frequency modulation was applied to $f_\mathrm {rep1}$ with a peak deviation of 5 Hz and a modulation frequency of 0.02 Hz. The details of the parameters used for this experiment are provided in the Appendix [see measurement conditions (G) and (H)]. For each measurement condition, we continuously measured IGMs for 400 s, resulting in approximately 80000 IGMs. Figure 6(a) shows the IGM with segment number 1 measured under condition (G). Two features can be identified: the larger signal is a result of interference between Comb 1 and the light from the reference path, and the smaller signal is a result of interference between Comb 1 and the light from the sample path, which includes an optical fiber with a length of 1.32 km.
Fig. 6. (a) The IGM with segment number 1 obtained using experimental setup B and measurement condition (G). The peak positions of the signals due to the reference path and the sample path are indicated by arrows. The origin of the sampling index $m$ is set to the peak position of the reference signal. (b), (c) The peak position of the reference signal as a function of the measurement time for conditions (G) and (H), respectively. The axis on the right-hand side indicates the effective time. (d), (e) The peak position of the sample signal as a function of the measurement time for conditions (G) and (H), respectively.
Here, we determined the peak positions of the reference and sample signals using a weighted average of the absolute IGM intensity $I$:
Each sum extends over 475 points around the corresponding IGM peak. Moreover, $m_{\mathrm {peak}}$ can be converted into a variation of the effective time $\tau _{\mathrm {eff}}$, which is defined below and reflects the effective relative time delay between Comb 1 and Comb 2 at a given $m_{\mathrm {peak}}$:
Here, to define $\tau _{\mathrm {eff}}$, $f_{\mathrm {rep1}}$ is treated as a constant and is taken as its averaged value, because the impact on $\tau _{\mathrm {eff}}$ is relatively small if the change in $f_{\mathrm {rep1}}$ is only on the order of several hertz. Figures 6(b) and (c) show the temporal variations of $m_{\mathrm {peak}}$ of the reference signals measured under conditions (G) and (H), respectively (the effective time is indicated by the axis on right-hand side). We find that the peak position of the reference signal is also nearly constant and the standard deviation of $m_{\mathrm {peak}}$ in terms of $\tau _{\mathrm {eff}}$ is only 85 fs under measurement condition (H), although here $f_\mathrm {rep1}$ is subject to modulation. This result proves the robustness of DCS measurements under the integer-locking condition.
Figure 6(d) shows the temporal variation of $m_{\mathrm {peak}}$ of the sample signal for measurement condition (G). We can confirm a shift in $m_{\mathrm {peak}}$ of approximately $-54$ points within 400 s, and this shift is equivalent to an effective time difference of about $-2.8$ ps. Because $f_\mathrm {rep1}$ is not modulated in this measurement, the observed shift is not attributed to repetition-rate fluctuations. This shift is most likely caused by a variation in the optical path length as a result of environmental temperature fluctuations during the experiment: The effective time difference of $-2.8$ ps corresponds to an optical path difference of roughly $-0.84$ mm and a refractive-index change of $-9.4 \times 10^{-7}$ in the optical fiber. Based on prior studies on the temperature dependence of the refractive index of optical fibers [54,55], this refractive-index change corresponds to a temperature change of about $-0.13$ K, which is a plausible value for our experimental setup.
Figure 6(e) shows the temporal variation of $m_{\mathrm {peak}}$ of the sample signal for measurement condition (H). Besides the above-mentioned gradual shift in $m_{\mathrm {peak}}$ due to the refractive-index change of the optical fiber, we can also confirm a distinct oscillation with an amplitude of around 20 points and a frequency of 0.02 Hz. This oscillation frequency is equal to the modulation frequency of $f_\mathrm {rep1}$, indicating that this feature is a direct consequence of the repetition-rate fluctuation. The observed oscillation is a result of oscillations in the repetition frequencies of OFCs and the relatively long optical path length (exceeding 1 km) that is to be determined by the ToF method.
If the integer-locking condition is used, even a small variation in $f_\mathrm {rep1}$ can have a significant impact on ToF results. Here, the impact becomes clearly visible when the time delay between the pulses from the reference path and the sample path that result in the IGM peak greatly exceeds the pulse interval of Comb 1 ($1/f_{\mathrm {rep1}}$). In Figs. 6(d) and (e), we observe an effective time delay of approximately 268 ps. However, the actual time delay $\tau _{\mathrm {actual}}$ is much longer:
where $N_\mathrm {pulse}$ is an integer. Based on the total length and the refractive index of the optical fiber, we estimate that $\tau _{\mathrm {actual}}$ is approximately 6484 ns, which suggests that $N_\mathrm {pulse}$ is around 400. When $f_{\mathrm {rep1}}$ ($f_{\mathrm {rep2}}$) shifts to $f_{\mathrm {rep1}}+\delta f_{\mathrm {rep1}}$ ($f'_{\mathrm {rep2}}$) during the measurement, while $\tau _{\mathrm {actual}}$ remains the same, the equation becomes where $\tau '_{\mathrm {eff}}$ is the effective time where the IGM peak appears after the change in $f_{\mathrm {rep1}}$. From Eqs. (13) and 14, we obtainFor a frequency shift of $f'_{\mathrm {rep2}} = f_{\mathrm {rep2}} + 5$ Hz, the second term in this equation amounts to 0.53 ps, consistent with the oscillation amplitude seen in Fig. 6(e).
Note that $\tau _{\mathrm {actual}}$ can be determined from the IGM peak position of the sample signal ($\tau _{\mathrm {eff}}$ or $\tau '_{\mathrm {eff}}$) and the repetition frequency of Comb 2 ($f_{\mathrm {rep2}}$ or $f'_{\mathrm {rep2}}$) using the above equations. Figure 7 shows the result for $\tau _\mathrm {actual}$ in the sample path with $N_\mathrm {pulse} =$ 400. The calculated $\tau _\mathrm {actual}$ shows no oscillation related to the repetition-rate fluctuation, which suggests that the observed variation predominantly reflects the variation in the optical path length as a result of environmental temperature fluctuations during the experiment. In summary, while a fluctuation in $f_\mathrm {rep1}$ in the case of the integer-locking conditions can lead to a significant variation in $\tau '_{\mathrm {eff}}$, we can correct this error if the repetition frequencies of the OFCs have been monitored during the measurement.
Fig. 7. The actual time delay $\tau _{\mathrm {actual}}$ as a function of the measurement time for condition (H).
5. Conclusion
In this work, we explored the stability of dual-comb interferometry, focusing on the case where the two OFCs are fully stabilized, while the repetition frequencies are subject to variations. We have shown how variations in the repetition frequency of Comb 1 influence the peak position of the center-burst signal in the IGM, $m_{\mathrm {peak}}$, because changes in $m_{\mathrm {peak}}$ affect the analysis of successive IGMs. We identified of a phase-locking scheme that effectively minimizes peak-position variations through a special combination of phase-locked frequencies. We have demonstrated that the integer-locking condition is useful for dual-comb interferometry: In DCS [1], the integer-locking condition enables coherent averaging of IGMs even if the repetition frequencies are unstable. In the frequency-domain representation, a stable time-domain interferogram ensures the stability of both the amplitude and phase of each down-converted frequency component of the interferogram. Consequently, both the amplitude and phase of each frequency component can be effectively averaged. Furthermore, in absolute-distance measurements using ToF analysis [21], repetition-rate fluctuations lead to minor variations in the frequency standard if the integer-locking condition is used, which affects the ultimate accuracy of length measurements. However, it is possible to correct this error if the OFC repetition frequencies have been monitored during the measurement. The integer-locking condition is also beneficial for applying dual-comb interferometry to ASOPS measurements, as it ensures the stability of the time origin’s position in the repeated time-resolved signal. Consequently, this may facilitate the accumulation of the repeated time-resolved signal without requiring an external trigger.
The integer-locking condition may be used to relax the usually strict requirements regarding the stability of $f_{\mathrm {rep}}$, and thus may facilitate the use of high-repetition rate micro-resonator frequency combs [56] (in these devices, repetition-rate stabilization for dual-comb applications is a challenge). To make dual-comb interferometry usable in a wide range of applications, especially without strict frequency stabilization of the cw laser using a high-finesse cavity, it is usually considered that the stability of the OFC repetition rate is crucial. On the other hand, due to the limited bandwidth of the servo loops that stabilize the OFC repetition rate, there is a relatively large frequency noise, and this also affects the stability of all comb modes. Our findings suggest that this bottleneck can be overcome by adopting the integer-locking condition. Furthermore, the combination of the integer-locking condition and the feed-forward stabilization scheme [12] may significantly extend the time required to maintain the mutual coherence between the two OFCs, even when their repetition rates are fluctuating. The integer-locking condition offers a viable approach for achieving stable dual-comb interferometry outside metrology laboratories.
Appendix: Experimental conditions used in Sections 4.2 and 4.3
Table 2 summarizes the RF frequencies along with the values of $n$, $\Delta n$, and $M$ that define the four measurement conditions (E) to (H). $|f_{\mathrm {rep1}}-f_{\mathrm {rep2}}|$ was approximately $200$ Hz in each condition, so $t_\mathrm {IGM}\approx 0.005$ s. Measurement conditions (E) and (F) are applied in Section 4.2, and (G) and (H) are employed in Section 4.3. In (E), (G) and (H), the integer-locking (IL) scheme is applied, whereas in (F), the non-integer-locking (NIL) scheme is applied. The repetition rate is modulated in (E), (F) and (H). The only difference between (H) and (G) is the modulation of $f_{\mathrm {rep1}}$. Note that in (F), we phase-locked $f_{\mathrm {rep1}}-f_{\mathrm {beat1}}$ to 22 MHz, so $f_{\mathrm {beat1}}$ also undergoes a modulation in response to the modulation of $f_{\mathrm {rep1}}$. However, the resulting change in $M$ due to the variation in $f_\mathrm {beat1}$ is small.
Funding
Core Research for Evolutional Science and Technology (JPMJCR19J4); Japan Society for the Promotion of Science (JP22K14625, JP23H01471); Ministry of Education, Culture, Sports, Science and Technology (JPMXS0118067246); Precise Measurement Technology Promotion Foundation; Mizuho Foundation for the Promotion of Sciences; Murata Science Foundation.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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