Interferogram-based determination of the absolute mode numbers of optical frequency combs in dual-comb spectroscopy

Tatsuhiro Fukuda; Makoto Okano; Shinichi Watanabe

doi:10.1364/OE.431104

1. Introduction

The invention of the optical frequency comb (OFC) technique [1,2] has opened a new era in high-precision optical metrology. To date, OFCs have been shown to be useful in many different fields of science, such as molecular spectroscopy [3,4], astronomy [5], and particle physics [6]. One important application scheme of OFCs is dual-comb spectroscopy (DCS). In this coherent multi-heterodyne detection technique, a signal in the optical frequency range is down-converted to the radio frequency (RF) range by using two OFCs with slightly different repetition frequencies [7,8]. Thus, the information of the amplitude and phase of each frequency-comb component in the optical frequency range can be directly obtained from the resulting signal in the RF range. Various spectroscopic measurements can be implemented using DCS, for instance, molecular absorption spectroscopy [9–18], spectroscopy for chemical detection [19–21], and solid-state spectroscopy [22–26]. Recently, various types of OFC sources including chip-based devices have been developed [27,28]. These emerging OFC sources promise the realization of compact DCS systems [29,30]. Although several DCS systems without full stabilization of the OFC frequency have been proposed [31–37], appropriate frequency stabilization of the two OFCs is still essential for practical operation of a DCS system.

In a typical DCS system [8], the carrier envelope offset frequency (f_ceo) of each OFC is stabilized to an RF synthesizer. Furthermore, each OFC has a comb line that is located in the vicinity of a reference continuous-wave (cw) laser in the frequency domain. The two optical beat frequencies that are obtained between the reference cw laser and each OFC’s comb line next to the cw laser frequency f_cw, are also stabilized to the RF source. By stabilizing each beat frequency f_beat with respect to the same cw laser, the two OFCs are mutually phase-locked, which enables the DCS measurement. In order to perform a DCS measurement with high accuracy and high resolution (in terms of optical frequency), the absolute mode numbers of the comb lines of both OFCs have to be determined precisely.

In the case where the two OFCs are linked via the cw laser as mentioned above, the absolute mode numbers of all comb lines of the two OFCs can be sequentially calculated, if we can determine the absolute mode numbers of the two comb lines (one in each OFC) that are located immediately below f_cw. This mode number in one of the two OFCs is hereafter referred to as n_cw (the mode number of the corresponding comb line in the other OFC is denoted by n_cw+Δn). Conventionally, n_cw is determined from the precise knowledge of f_cw. For instance, f_cw can be determined by stabilizing the cw laser to an ultra-stable cavity [10,38–40] or by the direct measurement with a wavemeter that has a better frequency resolution than the DCS system [33,41,42]. However, these techniques require additional instruments that are expensive and complex. On the other hand, the determination of n_cw without additional instruments has been also demonstrated [43]. In this technique, which is based on a dual-comb Vernier technique [44–48], n_cw is determined by calculating the ratio of the repetition frequency of one OFC, f_rep, to the difference between the f_rep values of the two OFCs, Δf_rep. Here, the uncertainty of the calculated value of n_cw is proportional to the tracking instability of the repetition frequency at a given integration time [44]. Therefore, according to this technique, it is difficult to uniquely determine n_cw within a measurement time of 1 s if Δf_rep < 100 Hz and f_rep ∼ 100 MHz. Note that Δf_rep is inversely proportional to the measurable frequency range, and Δf_rep is set to values much smaller than 100 Hz for broadband DCS [40]. Since broadband DCS has advantages for applications, a method that enables a fast determination of n_cw at relatively small Δf_rep values, would be beneficial.

The aim of the present study is to demonstrate a measurement protocol that enables a fast determination of the absolute mode numbers of both OFCs in a DCS system. If the previously proposed method is used, the direct measurement of a small values of Δf_rep (several Hz) may sometimes be necessary to determine n_cw. However, such a measurement requires a relatively long measurement time. Here, instead of measuring Δf_rep, we measured the ratio of f_rep to Δf_rep. This is relatively easy and fast if we use the sampling point interval between the peak positions in the interferogram data in the time domain, i.e., if we measure the periodicity in the interferogram data. By using this method, n_cw can be precisely determined within about a second even at Δf_rep < 10 Hz. This measurement time is shorter than that of the previously reported method [44]. Then, by using the gained knowledge of n_cw, we adjusted the stabilized frequencies of the two OFCs and measured the absorption of acetylene gas by DCS. By comparing our results with data from the HITRAN database, we clarified the validity of our method.

2. Measurement principle

In this section, we explain the method that we used to determine the absolute mode numbers of the comb lines of both OFCs in our DCS system. Our method relies on the determination of the absolute mode numbers of the two comb lines immediately below f_cw: n_cw in one OFC and n_cw+Δn in the other OFC. Once the absolute number n_cw and the offset Δn are known, the mode numbers of all comb lines observed in the DCS measurement can be calculated sequentially.

Hereafter, we explain the measurement principle for a DCS system that consists of two OFCs called “signal comb (S-comb)” and “local comb (L-comb).” f_ceo, f_beat, and f_rep of the L-comb are denoted by f_ceoL, f_beatL, and f_repL, respectively. Similarly, we use f_ceoS, f_beatS, and f_repS for the S-comb. The following phase-lock condition is considered; f_ceoL, f_beatL, f_repL, f_ceoS, and f_beatS are phase-locked to the RF synthesizer referenced to a frequency standard. This means that these five frequencies are known parameters. On the other hand, it is assumed that f_cw is roughly known, for example from the manufacture specifications of the cw laser or by using an optical bandpass filter that filters the cw laser to detect f_beat.

In this case, f_cw can be described with the frequency set of each OFC as follows:

(1)$${f_{\textrm{cw}}} = {f_{\textrm{ceoL}}} + {n_{\textrm{cw}}}{f_{\textrm{repL}}} + {f_{\textrm{beatL}}}, $$

(2)$${f_{\textrm{cw}}} = {f_{\textrm{ceoS}}} + ({n_{\textrm{cw}}} + \Delta n){f_{\textrm{repS}}} + {f_{\textrm{beatS}}}, $$

where n_cw and Δn are integers. n_cw corresponds to the absolute mode number of the comb line of the L-comb that is located immediately below f_cw in the frequency domain. From Eq. (1), it is found that f_cw is indirectly stabilized by phase-locking of f_ceoL, f_beatL, and f_repL. In addition, although f_repS is not directly phase-locked to the RF synthesizer in our scheme, it is actually stabilized through the frequency stabilization of f_cw, f_ceoS and f_beatS. We emphasize that precise determination of both Δn and n_cw is required for performing DCS under such a condition.

First, to determine Δn, we adopted the method proposed by Peng and Shu [46]. In this method, Δn is evaluated by

(3)$$\Delta n = \frac{{{f_{\textrm{repL}}} - {f_{\textrm{repS}}}}}{{{f_{\textrm{repS}}}/{n_{\textrm{cw}}}}} - \frac{{{f_{\textrm{ceoS}}} - {f_{\textrm{ceoL}}} + {f_{\textrm{beatS}}} - {f_{\textrm{beatL}}}}}{{{f_{\textrm{repS}}}}}. $$

This procedure requires a rough estimation of n_cw. Under typical DCS conditions, i.e., f_repL and f_repS are ∼100 MHz and Δf_rep ≡ f_repS - f_repL$\; $ is ∼100 Hz, Δn can be uniquely determined by measuring f_repS using a frequency counter with an uncertainty of ∼10 Hz if n_cw has been determined with an uncertainty of 0.05%. This determination of Δn is relatively easy and can be achieved within 1 s under typical conditions.

From Eqs. (1) and (2), as well as Δn, the value of n_cw can be determined by the following equation [46]:

(4)$${n_{\textrm{cw}}} ={-} \frac{{\Delta n({f_{\textrm{repL}}} + \Delta {f_{\textrm{rep}}}) + {f_{\textrm{ceoS}}} - {f_{\textrm{ceoL}}} + {f_{\textrm{beatS}}} - {f_{\textrm{beatL}}}}}{{\Delta {f_{\textrm{rep}}}}}. $$

According to Eq. (4), Δf_rep needs to be determined precisely in order to uniquely determine n_cw. Because the number of digits of n_cw is about six (its magnitude is ∼10⁶) under typical conditions for a near-infrared DCS system, Δf_rep has to be determined with at least seven significant digits. However, since Δf_rep is typically ∼100 Hz, the required measurement time of Δf_rep is long compared to that of f_repS (which is needed to assess Δn). As a result, it is possible that the determination of n_cw cannot be carried out fast enough if we only consider Eq. (4).

To circumvent this problem, we propose to use the so-called compression factor. The compression factor, M, is defined as [8]

(5)$$M \equiv \frac{{{f_{\textrm{repL}}}}}{{\Delta {f_{\textrm{rep}}}}}. $$

By using the definition given in Eq. (5), Eq. (4) becomes

(6)$${n_{\textrm{cw}}} = \frac{{{f_{\textrm{ceoL}}} - {f_{\textrm{ceoS}}} + {f_{\textrm{beatL}}} - {f_{\textrm{beatS}}} - \Delta n{f_{\textrm{repL}}}}}{{{f_{\textrm{repL}}}}}M - \Delta n. $$

In this equation, Δf_rep is not included explicitly. The most essential point of our method is the replacement of the direct measurement of Δf_rep with the direct measurement of M. Because f_repL is much larger than Δf_rep under typical conditions, the measurement of M with a large number of significant digits can be achieved much faster than that of Δf_rep.

We employ the following method for the direct measurement of M: In a DCS system, the peaks of the interferogram in the time domain appear with a period of 1/Δf_rep. When the RF signal for the interferogram is recorded with a sampling frequency of f_repL, the number of sampling points between the periodically appearing peaks in the interferogram data corresponds to M. We stress that M does not have to be an integer value, because we can evaluate M including the decimal part by recording an interferogram with a sufficiently long duration as demonstrated in Section 4. Thus, by extending the measurement time to record many peaks, we can precisely determine the value of M, and this value allows us to precisely determine the absolute number of n_cw. Once the absolute number of n_cw has been determined, we can determine the absolute mode numbers of all comb lines of both OFCs by counting from the comb line with the mode number n_cw.

3. Experimental setup

In this section, we explain the experimental setup used to demonstrate and verify our method. Figure 1 shows the three experimental setups (Setup A–C) used in this study to perform different measurements. In these experiments, the same DCS system was used as a light source.

Fig. 1. Schematics of the three DCS setups. ECL: external cavity laser, LD: laser diode, PZT: piezoelectric transducer, TEC: Peltier element, EOM: electro-optic phase modulator, FG: function generator, Atten.: attenuator, CL: collimator lens, QWP: quarter-wave plate, POL: polarizer, BS: beam splitter, EDFA: erbium-doped fiber amplifier, ¹²C₂H₂: acetylene gas cell, BPF: bandpass filter, and PC: personal computer.

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The DCS system contained two Er-doped-fiber-based OFCs (the L-comb and the S-comb) and a cw external cavity laser (ECL) for reference with a wavelength of ≈1550.11 nm and a linewidth of 1.6 kHz (Redfern Integrated Optics). f_repL was phase-locked to the RF reference signal generated by a function generator (FG) (WF1968, NF Corp.) via a feedback to a piezoelectric transducer (PZT) and a Peltier element (TEC). f_ceoL and f_ceoS were detected by f−2f self-referencing interferometers [1,2], and were also phase-locked to RF reference signal generated by another FG (WF1948, NF Corp.) via feedbacks to the currents of the pump laser diodes (LD). f_beatL was phase-locked to the RF reference signal via a feedback to the current of the ECL, whereas f_beatS was phase-locked to the RF reference signals via a feedback loop consisting of an electro-optic phase modulator (EOM), a PZT, and a TEC [24,49]. The FGs were referenced to a global-positioning-system-controlled rubidium (Rb) clock with a relative uncertainty on the order of 10⁻¹². The output beams from the L-comb and the S-comb were introduced either into Setup A, B, or C and then combined. Finally, the combined beam was detected by an amplified photodetector (PDB415C, Thorlabs) in air. The detected signal passed through an electrical bandpass filter (3–30 MHz) and was sampled by a digitizer (M2p.5962-x4, Spectrum) with a sampling frequency of f_repL. For the RF frequency measurement, we used a universal counter (SC7215A, Iwatsu), which was also referenced to the same Rb clock as the FGs.

In Setup A, the electrical signal of f_repS measured via another amplified photodetector (PDA05CF2, Thorlabs) was also sampled by the digitizer. The two data sets (the electrical signal of f_repS and the interferogram in the time domain) were measured simultaneously for comparison. In Setup B, the output beams from both combs propagated in free space. In Setup C, the output beam from the S-comb passed through a fiber-coupled acetylene gas cell with a pressure of 5-Torr (C2H2-12-H(15)-5-FCAPC, Wavelength References) before combining it with the output beam from the L-comb.

4. Experimental results

4.1 Determination of n_cw based on the direct measurement of M

In this subsection, we demonstrate the determination of n_cw using the protocol described in Section 2. For this experiment, we stabilized f_repL, f_ceoL, f_beatL, f_ceoS and f_beatS to the frequencies described in Table 1. f_repL was stabilized to a repetition frequency that is almost the same as that of the L-comb in the free-running condition. Regarding f_beatL, we stabilized f_repL-f_beatL to an RF reference signal with a frequency of 22,000,000 Hz as an example. f_repS was evaluated to be ≈61,530,655.4593 Hz by using a frequency counter with a measurement time of 1 s. Since the uncertainty of f_repS for this measurement time is approximately 0.01 Hz, this value of f_repS includes an uncertainty and is only used to determine M₀ as described below.

Table 1. Experimental Condition in Section 4.1

View Table

Firstly, because we filtered the cw laser by a dense wavelength division multiplexing filter with an international telecommunication union grid (channel #34), f_cw can be roughly estimated to be 193.4 ± 0.1 THz. Using this value, we can estimate a value of 3,143,200 ± 1,600 for n_cw (≈f_cw⁄f_repL) from Eq. (1). Since this uncertainty of n_cw is small enough to uniquely determine Δn using Eq. (3), Δn is evaluated to be –5.

Next, because the sampling frequency was set to f_repL, the recorded electrical signal of f_repS repeats itself with Δf_rep. Thus, it is possible to evaluate M from the repetition period of the recorded electrical signal of f_repS or also from that of the interferogram.

Figure 2(a) shows the interferogram and the electrical signal of f_repS for a measurement time of 10⁄Δf_rep (∼0.1 s) using Setup A. Both data curves contain 10-cycles of a signal with the same period. Note that M is not an integer value under this condition. To evaluate M, we divide both data curves into 10 segments with a length of M₀ (an integer), which is calculated by the following equation:

(7)$${M_0} = \left|{\left[ {\frac{{{f_{\textrm{repL}}}}}{{f_{\textrm{repS}}^{\textrm{meas}} - {f_{\textrm{repL}}}}}} \right]} \right|, $$

where $f_{\textrm{repS}}^{\textrm{meas}}$ is the f_repS measured by the frequency counter and the square brackets [] are used to indicate the operation of rounding off to the nearest integer value. From Table 1, we find M₀=594,732. The 10 segments of the interferogram and 10 segments of the electrical signal of f_repS are shown in Figs. 2(b) and 2(d), respectively. The data curves are offset for clarity and the signal at the bottom is the first segment in Fig. 2(a). From these figures, it is found that both signals are roughly repeated with the period M₀. The data in the vicinities of the peaks in Figs. 2(b) and 2(d) are plotted in Figs. 2(c) and 2(e), respectively. The evaluated peak position of each data curve is indicated by a black triangle. We find that the peak positions in the segmented interferogram data [Fig. 2(c)] gradually move to the left as the segment number increases. This behavior indicates that |M| < M₀. In contrast to Fig. 2(c), no obvious peak shift is observed in Fig. 2(d). This different behavior for the two data types originates from the difference in the sharpness of the peak signal. The broad peak in the electrical signal of f_repS is partly caused by the small bandwidth of the fast photodiode (150 MHz).

Fig. 2. (a) Interferogram (red) and electrical signal of f_repS (blue) for a measurement duration of 10/Δf_rep. (b) Ten segments of the interferogram data and (d) ten segments of the electrical signal of f_repS. Each segment has a length of M₀= 594,732 sampling points. Magnified views of the regions around peaks in (b) and (d) are plotted in (c) and (e), respectively. Black triangles indicate the signal peak position of each data segment. (f) Signal peak position of the interferogram (red) and that of the electrical signal (blue) as a function of the number of segments for a longer measurement time. The inset of (f) shows a magnified view of the data obtained from the interferogram.

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The red data in Fig. 2(f) shows the peak positions in the data segments of the interferogram for a long measurement time plotted as a function of the segment number. The corresponding data of the electrical signal of f_repS is shown as the blue data. We can clearly see that the peak positions of both data types decrease linearly with almost the same slope, although the data from the electrical signal of f_repS has a larger fluctuation than that from the interferogram. This slope (α) is caused by the difference between |M| and M₀, and we can determine M from the following equation:

(8)$$M = \textrm{sign(}\Delta {f_{\textrm{rep}}}\textrm{)} \cdot ({M_0} + \alpha ). $$

From the interferogram data in Fig. 2(f), α is evaluated to be −2.45 ± 0.04 within approximately 0.2 s, and therefore, M = 594,729.55 ± 0.04. By using this value of M, n_cw can be uniquely determined from Eq. (6); n_cw = 3,143,096.

To assess the performance of our method in terms of speed, we evaluated the time required for a unique determination of n_cw. Figure 3(a) shows the deviation of the calculated value of n_cw from the correct value (3,143,096) as a function of the measurement time for the interferogram-based evaluation. Figure 3(b) shows the corresponding deviation for the data extracted from the electrical signal of f_repS. We define the uncertainty of n_cw, δn_cw, as 1.96 times the standard deviation σ of the evaluated values of n_cw. The error bars in Figs. 3(a) and 3(b) correspond to δn_cw. These graphs reveal the following important characteristics: (i) the length of the error bar shows a monotonic decrease with the measurement time, and (ii) the converged values of both signals are the identical.

Fig. 3. Values of n_cw determined from (a) the interferogram and (b) the electrical signal of f_repS as a function of the measurement time. The error bars correspond to δn_cw. (c) Measurement time dependence of δn_cw for the determination based on the interferogram (red line) and that based on the electrical signal of f_repS (blue line). The gray horizontal line corresponds to δn_cw = 0.5. The dotted line is proportional to T^−1.5.

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As evidenced in Figs. 3(a) and 3(b), both types of signals can be used to determine M. To compare the degrees of convergence, we plotted δn_cw for each signal type as a function of the measurement time, T, in Fig. 3(c). As T increases, the δn_cw values of both signal types decrease with almost the same slope. The condition δn_cw < 0.5 is reached for T larger than ≈0.2 and larger than ≈50 s for the interferogram data and the electrical signal of f_repS, respectively. These times correspond to the minimum time required for unique determination of n_cw (this time is hereafter referred to as T₀). It is clearly found that the convergence speed achieved by using the interferogram data is two orders of magnitude faster than that achieved by using the electrical signal of f_repS. The reason for this difference is the different sharpness in the peak signal. Thus, hereafter we only focus on the absolute mode number determination based on the measurement of the interferogram. In addition, Fig. 3(c) shows that the curves for both data are proportional to T^−1.5. The reason why these data are proportional to T^−1.5 is discussed in Section 5.

In order to clarify the validity of our method for various values of Δf_rep, we performed similar experiments using Setup B. The measurement time dependences of δn_cw for Δf_rep values in the range from 5.58 to 738 Hz are plotted in Fig. 4(a). All measurement time dependences exhibit almost the same slope, and the absolute value of δn_cw for a given measurement time tends to be smaller for a larger absolute value of Δf_rep. To quantitatively evaluate the Δf_rep dependence of the minimum required measurement time, the values of T₀ evaluated from Fig. 4(a) are plotted as a function of Δf_rep in Fig. 4(b). The smaller T₀ for larger |Δf_rep| is caused by an increase in the number of signal repetitions (smaller segment length) in an interferogram for a certain measurement time as |Δf_rep| increases. Note that n_cw is uniquely determined within about 1 s even at |Δf_rep| < 10 Hz. Thus, our protocol is useful for situations where Δf_rep should be set to <10 Hz, for instance, in ultra-broadband DCS [40].

Fig. 4. (a) Measurement time dependence of δn_cw determined from the peak position in the interferogram for different values of Δf_rep. The gray horizontal line corresponds to δn_cw= 0.5. The dotted lines are proportional to T^−1.5. (b) |Δf_rep| dependence of T₀ evaluated from the crossing between the data curves and the gray horizontal line in (a).

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4.2 Absorption measurement of acetylene gas

To clarify the validity of our method of mode number determination, we measured the absorption of acetylene gas by DCS. In this experiment, we utilized the experimental Setup C. First, we stabilized f_repL to 61,531,201 Hz, and f_ceoL, f_beatL, f_ceoS and f_beatS to 22,000,000 Hz, respectively. Next, by measuring the interferogram, we obtained Δn = −11 and n_cw = 3,143,086. Finally, to achieve a coherent averaging condition (see Appendix), we changed f_beatS to 22,002,153.44 Hz with M^INT = 285,735 according to Eq. (22) in the Appendix. Thus, Δf_rep = f_repL⁄M^INT = 215.3436 Hz.

Figure 5(a) shows 6 interferogram segments in the vicinity of the peak signal, which appears at about every 50 ms. By choosing appropriately stabilized frequencies, the measured interferogram segments have almost the same profile (including the carrier phase). In addition, to average the interferogram segments over a long measurement time (several tens of minutes), we corrected the long-term phase change by the method proposed in [40,50]. Figure 5(b) shows the amplitude spectrum obtained by the Fourier transform of the interferogram data averaged for 76 minutes (984,960 segments). Several sharp absorption peaks corresponding to the optical transitions in the ν₁+ν₃ vibration band of acetylene are clearly observed [40]. Figure 5(c) shows the amplitude spectrum around 197.221 THz, corresponding to the R(9) line. To extract the influence of the background of the signal on the spectrum, we fitted the amplitude spectrum in Fig. 5(c) by a combination of a Gaussian function and a linear function, which correspond to the components of the absorption peak and the background, respectively. We determined the peak frequency of each absorption line by such fits, and in Fig. 5(d), the differences between the obtained absorption peak frequencies and the data from the HITRAN database are plotted with respect to the peak frequency [51]. Our data shows a good agreement with the HITRAN database; except for one point, our data is within the range of ± f_repS/2, corresponding to half of the frequency resolution of the spectrum. We consider that the reason for the deviations of the observed peak frequencies at the low and high frequency regions in Fig. 5(d) might be the small amplitude of the signal in these regions as shown in Fig. 5(b). Nevertheless, this result confirms that our determination method of the absolute mode numbers of both OFCs is valid.

Fig. 5. (a) Measured interferogram segments for a segment interval of 11/Δf_rep (approximately 50 ms). The interferogram segments are offset for clarity. (b) Amplitude spectrum obtained by averaging 984,960 segments (76 minutes). (c) Magnified view of the spectrum in (b) around 197.221 THz. The blue curve is the result of fitting the data to Gaussian function and a linear background. (d) Frequency difference between the observed peak frequencies in (b) and those from the HITRAN database. The interval ± f_rep ⁄ 2 is shown as the red region.

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5. Discussion

Finally, we discuss the reason why δn_cw is proportional to T^−1.5 as evidenced in Figs. 3(c) and 4(a). As shown below, this discussion enables us to predict T₀, the minimum required time to uniquely determine n_cw, for arbitrary Δf_rep values.

Prior to the discussion of the measurement time dependence of δn_cw, we discuss the uncertainty of Δn, δ(Δn), in our method. Based on error propagation and Eq. (3), δ(Δn) is given by

(9)$$\mathrm{\delta }(\Delta n)\sim \frac{1}{{{f_{\textrm{repL}}}{f_{\textrm{repS}}}}}\sqrt {{{({\Delta {f_{\textrm{rep}}}} )}^2}{{({\mathrm{\delta }{f_{\textrm{cw}}}} )}^2} + {{({{f_{\textrm{CW}}}} )}^2}{{({\mathrm{\delta }{f_{\textrm{repS}}}} )}^2}}, $$

where δf_cw and δf_repS are the uncertainties of f_cw and f_repS, respectively. Here, we approximated n_cw as f_cw/f_repL according to Eq. (1) (f_cw>>f_beatL, f_ceoL), and ignore the uncertainties of the stabilized frequencies f_ceoS, f_ceoL, f_beatS, f_beatL, and f_repL. δf_cw is ∼0.1 THz according to the specification of the optical bandpass filter. δf_repS was estimated to be ∼ 0.01 Hz by using a frequency counter with a measurement time of 1 s under the experimental condition of f_repS ∼$\; $60 MHz. By using these values and Δf_rep ~ 100 Hz, we obtain δ(Δn) = 3×10⁻³ from Eq. (9), which is sufficiently small to uniquely determine Δn.

Next, we discuss δn_cw. According to Eqs. (3) and (6) and the above experimental conditions, δn_cw can be written as

(10)$$\mathrm{\delta }{n_{\textrm{cw}}}\sim \frac{{{n_{\textrm{cw}}}}}{{{f_{\textrm{repL}}}}}|{\Delta {f_{\textrm{rep}}}} |\mathrm{\delta }M, $$

where δM is the uncertainty of M. As for Eq. (9), here we ignored the uncertainties of the stabilized frequencies. From Eq. (10), we can understand that the measurement time (T) dependence of δn_cw is equal to that of δM. From Eq. (8), it is found that δM is equal to the uncertainty of the slope α, δα, because M₀ has no uncertainty. Thus, we consider the uncertainty of the slope in the linear fitting procedure.

Generally, δα is determined by the fluctuations of the peak positions of the periodic signals as shown in Fig. 2(f). For a quantitative discussion of the fluctuation of the peak positions, we introduce a new parameter σ_fit, which represents the standard deviation of the difference between the actual peak position and the linear fitting line. σ_fit is thus defined as [52],

(11)$${\sigma _{\textrm{fit}}} = \sqrt {\frac{{\sum\nolimits_{i = 1}^N {{{({y_\textrm{i}} - {{\hat{y}}_\textrm{i}})}^2}} }}{{N - 2}}}, $$

where N is the number of segments in the considered data set, such as the total recorded interferogram or the electrical signal of f_repS. Therefore, N corresponds to the number of data points used in the fitting procedure. y_i is the index of the peak in the i-th data segment, and ${\hat{y}_\textrm{i}}$=αx_i + β is the predicted value, where x_i is the index of the data segment and β is the intercept of the linear fitting line with the y-axis.

In the case of a linear fitting procedure, the standard deviation of the slope, σ_α, can be written as [52],

(12)$${\sigma _\alpha } = \frac{{{\sigma _{\textrm{fit}}}}}{{\sqrt {\sum\nolimits_{i = 1}^N {{{({x_i} - {{\bar{x}}_i})}^2}} } }}, $$

where ${\bar{x}_i}$ represents the average concentration of the number of segments, which is equal to (N+1)/2. Thus, δM is given by

(13)$$\mathrm{\delta }M = \frac{{1.96{\sigma _{\textrm{fit}}}}}{{\sqrt {\sum\nolimits_{i = 1}^N {{{({x_i} - {{\bar{x}}_i})}^2}} } }}. $$

Here, we multiply σ_α by 1.96 because δM is defined as the 95%-confidence interval of M. The denominator of the right-hand side of Eq. (13) is equal to

(14)$$\sqrt {\frac{{(N - 1)N(N + 1)}}{{12}}} \sim \sqrt {\frac{{{N^3}}}{{12}}}, $$

when N is sufficiently large. Because N is the integer calculated by rounding down the product of |Δf_rep| and T, Eq. (10) can be rewritten with Eqs. (13) and (14) as follows:

(15)$$\mathrm{\delta }{n_{\textrm{cw}}} = 2\sqrt 3 \cdot 1.96{\sigma _{\textrm{fit}}} \cdot \frac{{{n_{\textrm{cw}}}}}{{{f_{\textrm{repL}}}}} \cdot {|{\Delta {f_{\textrm{rep}}}} |^{ - \frac{1}{2}}} \cdot {T^{ - \frac{3}{2}}}. $$

This equation clearly proves that δn_cw is proportional to T^−1.5, in agreement with the experiment.

According to Eq. (15), T₀ can be approximated by

(16)$${T_0}\sim {\left( {4\sqrt 3 \cdot 1.96{\sigma_{\textrm{fit}}} \cdot \frac{{{n_{\textrm{cw}}}}}{{{f_{\textrm{repL}}}}}} \right)^{\frac{2}{3}}}{|{\Delta {f_{\textrm{rep}}}} |^{ - \frac{1}{3}}}. $$

By using this equation, we can easily determine the minimum required time for unique determination of n_cw for arbitrary Δf_rep values. In addition, based on Eq. (16), we can better understand the reason for the large discrepancy between the T₀ values obtained from the interferogram and the electrical signal of f_repS: The interferogram has a relatively small σ_fit (due to the sharp peaks), which is in stark contrast to the large σ_fit for the electrical signal of f_repS as shown in Fig. 2(f). Because T₀ strongly depends on σ_fit, T₀ is smaller if it is determined from the interferogram. Therefore, for a fast determination of n_cw, it is important to minimize σ_fit, for instance, by measuring the interferogram or the electrical signal with a high-speed photodiode.

6. Conclusions

In summary, a fast evaluation method of the absolute mode numbers of the two OFCs used in DCS has been demonstrated. This method is based on the measurement of M instead of the measurement of Δf_rep. By measuring the sampling point interval between the peaks of the interferogram (or the electrical signal of the repetition frequency of the S-comb), we precisely determined M. Then we determined the absolute mode number n_cw, which is the mode number of the L-comb line that is located immediately below the oscillation frequency of the reference cw laser linking the two OFCs. By using this information, the absolute mode numbers of both OFCs can be calculated correctly.

With this method, we were able to determine the absolute mode number n_cw within about 1 s even at relatively small values of Δf_rep (∼several Hz.) To verify the validity of our method, we measured the absorption of acetylene gas by DCS after adjusting the stabilized frequencies with our method. Because the measured frequencies of the absorption lines are consistent with those from the HITRAN database, we confirmed that the determined mode numbers are correct. In addition, we derived the measurement time dependence of the uncertainty of n_cw and the equation of the minimum required measurement time for a unique determination of n_cw. Our method is considered useful especially for researchers who do not possess an ultra-stable cavity or an expensive wavemeter with very high frequency resolution.

Appendix: Procedure to determine the frequencies for the coherent averaging

Here, we explain the practical procedure used to determine the frequencies that are required for the “coherent averaging condition” [53].

Once the mode numbers of both OFCs have been determined precisely as described in Section 2, we can set the f_repL, f_ceoL, f_beatL, f_ceoS, and f_beatS values appropriate for coherent averaging. In this condition, all interferogram data segments have the same temporal profile including the carrier phase. Because we can accumulate the interferogram data under this condition, the coherent averaging technique is quite important to improve the signal-to-noise ratio. Here, we describe the process to determine the appropriate frequencies for the coherent averaging condition. f_repL, f_ceoL and f_beatL are not changed during this process in order to not change f_cw. On the other hand, f_ceoS and f_beatS are adjusted to satisfy the two following equations [54]:

(17)$${M^{\textrm{INT}}} \equiv \frac{{{f_{\textrm{repL}}}}}{{f_{\textrm{repS}}^u - {f_{\textrm{repL}}}}}, $$

(18)$${k^{\textrm{INT}}} \equiv \frac{{f_{\textrm{ceoS}}^u - {f_{\textrm{ceoL}}}}}{{f_{\textrm{repS}}^u - {f_{\textrm{repL}}}}}, $$

where M^INT and k^INT are integer values and $f_{\textrm{ceoS}}^u$ and $f_{\textrm{repS}}^u$ are the updated frequencies of f_ceoS and f_repS for coherent averaging, respectively. The initial values of M^INT and k^INT are calculated using Eq. (5) and k = (f_ceoS-f_ceoL)⁄(f_repS-f_repL), respectively. From Eqs. (1) and (2), we obtain the following relation;

(19)$${f_{\textrm{ceoL}}} + {n_{\textrm{cw}}}{f_{\textrm{repL}}} + {f_{\textrm{beatL}}} = f_{\textrm{ceoS}}^u + ({n_{\textrm{cw}}} + \Delta n)f_{\textrm{repS}}^u + f_{\textrm{beatS}}^u, $$

where $f_{\textrm{beatS}}^u$ is the updated frequency of f_beatS for coherent averaging. From Eqs. (17)–(19), we find

(20)$$f_{\textrm{ceoS}}^u = {f_{\textrm{ceoL}}} + \frac{{{k^{\textrm{INT}}}}}{{{M^{\textrm{INT}}}}}{f_{\textrm{repL}}}, $$

and

(21)$$f_{\textrm{beatS}}^u = {f_{\textrm{beatL}}} - \left( {\frac{{n_{\textrm{cw}} + \Delta n + {k^{\textrm{INT}}}}}{{{M^{\textrm{INT}}}}} + \Delta n} \right){f_{\textrm{repL}}}. $$

By setting f_ceoS and f_beatS to $f_{\textrm{ceoS}}^u$ and $f_{\textrm{beatS}}^u$ according to Eqs. (20) and (21), we can achieve the coherent averaging condition.

In this work, since f_ceoL and f_ceoS are set to the same frequency, k^INT is always zero. Thus, we actually only changed f_beatS to satisfy Eq. (21) and realize the coherent averaging condition.

Funding

JST CREST (JPMJCR19J4); JSPS KAKENHI (JP18H02040); MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) (JPMXS0118067246).

Acknowledgements

The authors acknowledge Dr. K. A. Sumihara, Mr. R. Tabuchi, Dr. S. Okubo, Dr. H. Inaba, and Prof. T. Hasegawa for technical support. S. W. acknowledges Dr. I. Coddington for discussion.

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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	Frequency (/Hz)
f_repL	61,530,552 (stabilized)
f_ceoL	22,000,000 (stabilized)
f_beatL	39,530,552 (stabilized)
f_ceoS	22,000,000 (stabilized)
f_beatS	22,000,000 (stabilized)
f_repS	61,530,655.4593 (measured)

Interferogram-based determination of the absolute mode numbers of optical frequency combs in dual-comb spectroscopy

Abstract

1. Introduction

2. Measurement principle

3. Experimental setup

4. Experimental results

4.1 Determination of n_cw based on the direct measurement of M

4.2 Absorption measurement of acetylene gas

5. Discussion

6. Conclusions

Appendix: Procedure to determine the frequencies for the coherent averaging

Funding

Acknowledgements

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Tables (1)

Equations (21)

Optics Express

Abstract

1. Introduction

2. Measurement principle

3. Experimental setup

4. Experimental results

4.1 Determination of ncw based on the direct measurement of M

4.2 Absorption measurement of acetylene gas

5. Discussion

6. Conclusions

Appendix: Procedure to determine the frequencies for the coherent averaging

Funding

Acknowledgements

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Tables (1)

Equations (21)

Optics Express

4.1 Determination of n_cw based on the direct measurement of M