1. Introduction

Direct-comb spectroscopy, in which optical frequency combs are used as light sources, has enabled us to carry out broad-band spectroscopy with a high spectral resolution and accuracy in optical frequency determination. To utilize the excellent performance of direct-comb spectroscopy, every comb mode is to be resolved and the mode intensity must be measured. To this end, dual-comb spectroscopy [1,2] employs a second comb, and two-dimensional imaging comb spectroscopy [3] uses a combination of a diffraction grating and a virtually-imaged phased array. These techniques have been widely employed for broadband linear spectroscopy and have recently been extended to nonlinear spectroscopy such as saturation spectroscopy [4, 5], coherent anti-Stokes Raman spectroscopy [6], and two-photon absorption spectroscopy [7].

Another technique to resolve comb modes is the heterodyne measurement between the outputs from a continuous-wave (cw) laser and frequency comb. This cw-comb heterodyne (CCH) technique was demonstrated for a near-infrared electronic transition of metastable argon using a Ti:sapphire laser comb and an extended-cavity laser diode [8–10]. Subsequently, the CCH technique was applied to molecular spectroscopy of acetylene at 1550 nm [11] and sub-Doppler resolution spectroscopy of potassium [12]. Since the spectral bandwidth of CCH spectroscopy is limited by the rf bandwidth of the photodetector used, the broad bandwidth of the optical frequency comb is not fully utilized [13]. However, the use of cw laser as a local source enhances the sensitivity, and the narrow bandwidth reduces the data acquisition time. Therefore, phase-modulated cw laser were used as the frequency comb in CCH spectroscopy [13, 14]. The high resolution and sensitivity of CCH spectroscopy are more beneficial to molecular spectroscopy than atomic spectroscopy, because molecules have more crowded spectral lines than atoms because of the freedom of vibration and rotation, and the vibration-rotation transitions are considerably weaker than the electronic transitions [11]. The bandwidth of CCH spectroscopy can be extended by using several cw lasers with different frequencies at the same time. This scheme develops novel applications of CCH spectroscopy to real-time measurements of optical thermometry [15], environmental monitoring, and isotope ratio measurements [16].

In this study, we have applied the CCH technique to molecular spectroscopy of the 2ν₃ overtone vibration band of methane at 1651 nm using a spectrally broadened Er-fiber frequency comb. The optical power level of the single comb mode is usually low and further reduced when the frequency comb is spectrally broadened by a highly-nonlinear fiber (HNLF) for extending the spectral coverage. In the present study, the optical power density and the absorption of molecular transitions are about 2000 photons per mode per pulse (18 nW/mode) and 20 % at most, respectively. They are smaller than the previous work, e.g., more than 300 nW/mode and 75 % in [11]. The noise source of CCH spectroscopy is dominated by the frequency noise of the heterodyne beat note, which was reduced by the phase correction algorithm in [11].

To reduce the frequency noise, in the present study, the cw laser frequency is stabilized with respect to one of the comb modes. This enables us to accumulate the signal for a long measurement time. We also introduce balanced-detection of the heterodyne signal to remove the intensity noise of the cw laser and the irrelevant rf signal of the fundamental and harmonics of the repetition rate of the optical comb. The latter prevents the relevant rf signal from saturation due to the nonlinear response of the rf electronics. These remarkably improve the signal-to-noise ratio (SNR) in the present study.

2. Principle

Figure 1 presents schematics of the CCH technique. The principle of CCH spectroscopy is described in detail in [8]. Briefly, the comb output passes through a gas sample [Fig. 1(a)], and the spectral information is stored in the individual-comb modes [Fig. 1(b)]. Then the comb beam overlaps with a beam from the single-mode cw laser, and they are detected together with a fast photodetector. The detected signal contains rf beat notes between the comb modes and cw laser.

 

Fig. 1 (a) Schematic representation of CCH spectroscopy, (b) optical spectrum of the comb transmitting through the absorption cell and cw laser, and (c) heterodyne rf spectrum observed by a photodetector. The absorption information is contained within the rf signal. In (b), the beat frequency between the cw laser and the nearest comb mode (mode number is 0) is expressed as f_b.

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The beat frequencies distribute from |f_b| up to |nf_r + f_b| below the bandwidth of the detector, where f_r is the repetition rate of the comb, f_b is the beat frequency between the cw laser and nearest comb mode (− f_r/2 < f_b ≤ f_r/2), and n is the signed mode number from the comb mode giving the lowest beat frequency. Figure 1(c) presents the rf spectrum beated down from Fig. 1(b). It is noted that the rf spectral lines labeled with negative n are folded back in Fig. 1(c).

CCH spectroscopy is not broadband because the spectral bandwidth is limited to twice that of the photodetector. However, the data acquisition time is as short as 1/ |f_b| in principle, which is much shorter than 1/δ of dual-comb spectroscopy, where δ is the repetition-rate difference between the signal and local combs. The spectral resolution of CCH spectroscopy is determined by the repetition rate of the comb in a similar manner as other types of direct-comb spectroscopy. Any cw laser with a spectral linewidth narrower than half the repetition rate is applicable to CCH spectroscopy. For high sensitivities, however, a narrower linewidth is better because a higher signal intensity is expected.

3. Signal and noise intensity of CCH spectroscopy

In the heterodyne technique, the signal intensity is generally proportional to the square root of the local oscillator power. Therefore, higher local oscillator (cw laser) power is desired in CCH spectroscopy, when the noise is due to the photodetector noise. When the noise is mainly due to the local-oscillator shot noise, the SNR does not depend on the local oscillator power. Therefore, for high SNR, the best local oscillator power is the highest possible while avoiding saturation of the photodetector. Accordingly, CCH spectroscopy is sensitive because power of the local oscillator per mode is much higher than in dual-comb spectroscopy.

The optical intensity detected with the photodetector is expressed by

The noise intensity in CCH spectroscopy is usually determined by the photodetector noise and the photon shot noise. As long as the measurement time is identical, the photodetector noise intensity of CCH spectroscopy may be identical to that of dual-comb spectroscopy. The photon shot noise is determined by the total photon number, being proportional to $\sqrt{I_{\text{comb}}+I_{\text{cw}}}$ and $\sqrt{I_{\text{comb}}+I_{\text{lc}}}$ for CCH spectroscopy and dual-comb spectroscopy, respectively (I_lc is the optical intensity of the local-oscillator comb). Assuming a 1-mW cw laser and 10-mW combs again, the shot noise of CCH spectroscopy is estimated to be 0.75 times that of dual-comb spectroscopy (when I_lc = I_comb).

In CCH spectroscopy, the comb modes that are far from the cw laser frequency beyond the photodetector bandwidth contribute not to the signal but to the photon shot noise. Therefore, the bandwidth of the comb is matched to that of the photodetector for the best sensitivity. The matching of the bandwidth also works for dual-comb spectroscopy, but the advantage of dual-comb spectroscopy is lost by the narrowing. In the case that the total comb power is much less than the cw laser power, the shot noise is mainly due to the cw laser, and the narrowing of the optical bandwidth does not drastically reduce the shot noise.

4. Experimental setup and results

Figure 2(a) illustrates the experimental setup. The frequency comb is a hand-made mode-locked Er-doped fiber laser with a repetition rate of 66.8 MHz and central wavelength of 1550 nm. This value of f_r is small enough to observe the 600-MHz-wide (full-width at half-maximum, FWHM) Doppler-broadened lines of the 2ν₃ band of CH₄ at 1650 nm. The comb mode frequency, which drifts at a speed less than f_r per 1 hour, is sufficiently stable to observe these spectral lines during a data acquisition time of a few seconds even in the free-running operation. Hence, neither the repetition rate nor carrier-envelope offset frequency is stabilized in this study.

 

Fig. 2 (a) Experimental setup, and (b) optical spectra after HNLF (black) and after the band elimination by diffraction grating and pinhole (red). The blue curve indicates the background noise level recorded by the optical spectrum analyzer. In (a), an alternative setup (purple) following the beamsplitter for the balanced detection is shown (see Fig. 4). FA: fiber amplifier; HNLF: highly-nonlinear fiber; BS: 50 : 50 beamsplitter; PD: fast photodetector; DA: differential rf amplifier; SA: rf spectrum analyzer.

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To cover the spectral range around 1650 nm, the spectral width must be extended by the HNLF. For this purpose, the output of the comb is amplified with a hand-made fiber amplifier, followed by the HNLF (10 cm-long, zero-dispersion wavelength of 1445 nm, Sumitomo Electric). Figure 2(b) depicts the optical spectral distribution recorded by an optical spectrum analyzer (Anritsu, MS9710C) immediately after the HNLF in a black curve. The output of the HNLF passes through a 40-cm-long absorption cell filled with methane gas at a pressure of about 500 Pa.

Subsequently, the comb beam is dispersed by a diffraction grating (600 lines/mm) and is then spatially filtered by a pinhole to eliminate the comb modes that are far from the cw laser frequency to suppress the photon shot noise level. Figure 2(b) represents the spectral distribution of the filtered comb in a red curve, while the blue curve is the background noise level of the optical spectrum analyzer. The filtered comb is centered at 1651 nm with a FWHM of 2 nm (220 GHz), and the total power is about 50 μW. The corresponding photon number per mode per pulse is approximately 2000. The comb beam is combined with the beam from a cw distributed-feedback (DFB) laser (Toptica, LD-1665-0010-DFB-1, 2.5 mW at the photodetector) and is detected using a fast photodetector with a bandwidth of 1 GHz (New Focus, 1611FC-AC). The detected rf signal is divided into two: one is for spectroscopy, and the other is for the frequency stabilization of the cw laser [not shown in Fig. 2(a)]. The cw laser frequency, which is close to the R(4) transition absorption line of CH₄ at 1650.96 nm, is stabilized so that the lowest beat frequency with the comb is a quarter of the repetition rate using the optical phase lock loop technique [17]. Hence, the n-th comb mode has an offset frequency of (n − 1/4) f_r apart from the cw laser frequency (the sign of 1/4 is determined by the feedback loop polarity). The frequency stabilization of the cw laser is not required for CCH spectroscopy [8], but preferred for data accumulation and averaging. The linewidth of the DFB laser is about 1 MHz (determined from the beat-frequency measurement with the comb). As this linewidth is limited by the Schawlow-Townes linewidth, the linewidth cannot be reduced by the frequency stabilization. Therefore, the stabilization is performed to prevent the laser frequency drift, and the feedback time constant is as slow as 0.1 s.

Figure 3(a) displays the rf spectrum observed with an rf spectrum analyzer (Rohde and Schwarz, FSC-3) with a resolution bandwidth of 300 kHz, a video bandwidth of 1 MHz, and averaging over 100 measurements. The rf spectral lines of the positive and negative mode numbers appear in an alternating fashion as shown in Fig. 1(c). Figure 3(a) presents the beat signal lines of −7 ≤ n ≤ 7 (from −500 to 500 MHz), and those of −19 ≤ n ≤ 20 (from −1.3 to 1.3 GHz) are indeed observed. Without the frequency stabilization, the beat notes are washed out during the averaging.

 

Fig. 3 (a) Observed rf spectrum (averaged over 100 measurements). The corresponding mode numbers are also shown. The line with the mode number of zero corresponds to that nearest to the cw laser in frequency. (b) Normalized mode intensity as a function of laser frequency detuning with respect to the cw laser frequency. The corresponding mode numbers are displayed on the upper horizontal axis. The red trace is the fitted curve with the sum of four Gaussian functions, and the residuals are shown in (c). The green trace is the laser absorption spectrum recorded by sweeping the cw laser frequency (vertically shifted for clarity).

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The absorption spectrum can be reproduced by rearranging the rf spectral lines in order. However, the line intensity variation of the rf spectrum in Fig. 3(a) is not caused by the methane absorption (approximately 20 %) but the rf-dependent response of the photodetector and the following electronics. In fact, even though the methane absorption line of R(4) F₂ should lie around the rf spectral line of n = 7, any appreciable decrease in the rf intensity is not recognized in Fig. 3(a). Here note that the vertical axis is a logarithmic scale. In the case of weak optical lines, the signal is hardly observed because of the substantial background distribution in the rf spectrum. To overcome this difficulty, two sets of rf signals are recorded in the experiment. One (A_n,s) is with the cw laser frequency close to the absorption line center (within 1 GHz) and the other (A_n,b) is with the cw laser frequency far-off-resonance from any appreciable methane absorption lines (detuned by about 4 GHz). The normalized signal is then A_n,s/A_n,b. This normalization is imperfect, because the spectral distribution of the comb is not compensated and the cw laser power is not identical in the two cases. Nevertheless, the normalization greatly improves the sensitivity.

Figure 3(b) displays the normalized spectrum by rearranging according to mode number order in Fig. 3(a). The horizontal axis represents the frequency detuning with respect to the cw laser frequency, and the corresponding mode number, n, is represented by the upper axis. The F₂, F₁, A₁, and E tetrahedral components of the R(4) transition are observed with an absorption of approximately 20 %, whereas the F₁, A₁, and E components overlap considerably because of the Doppler broadening. The assignments are made in accordance with [18]. The absorption of 20 % is very weak compared to that in [8,9,11–14].

The spectrum in Fig. 3(b) is fitted to the sum of four Gaussian functions of

The rf spectral lines at the repetition rate and its harmonics in Fig. 3(a) are larger in the intensity than the beat notes but contain no spectroscopic information. They are suppressed by the balanced optical detection [19], in which each wave from the two output ports of the 50:50 beamsplitter is detected, and the difference between them is analyzed. This setup is identical to Fig. 2(a), but the photodetector part in Fig. 2(a) (green) is replaced with that for the balanced detection (purple). Figure 4 represents the rf spectrum of the balanced detector (PDB480C-AC, Thorlabs, rf bandwidth of 1 GHz). The rf signals at the repetition rate and its harmonics are suppressed down to the background noise level. The common noise caused by the intensity fluctuations of the cw laser and the comb decreases, and, in addition, the rf signal can be amplified without saturation in the readout electronics.

 

Fig. 4 Observed rf spectrum with the balanced detection.

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5. Discussion

The data acquisition time can be very short in CCH spectroscopy in principle. In the present study, however, the rf signal is transformed to the frequency spectrum using the rf spectrum analyzer, whose frequency sweep rate currently limits the data acquisition time (∼ 0.5 s). Therefore, the measurement carried out in this study is sequential, but the real-time measurement can be done by time-domain data acquisition of the rf signal and digital data processing in a computer. The real-time measurement may significantly reduce the data acquisition time, because of the slow sweep rate of the rf spectrum analyzer. The phase spectrum as well as the intensity spectrum is also obtained.

Before the discussion about the SNR, we note that the SNR of the rf signal [Fig. 3(a)] should be distinguished from the SNR of the molecular spectrum [Fig. 3(b)]. For convenience, we refer the former as “rf SNR”, and the latter as “ spectrum SNR”. The rf SNR in Fig. 3(a) is approximately 20 dB and improves to about 25 dB by the balanced detection in Fig. 4. In this study, the detector noise and photon shot noise are responsible for the overall noise, and estimated as $20\hspace{0.17em}\text{pW}/\sqrt{\text{Hz}}$ and $25\hspace{0.17em}\text{pW}/\sqrt{\text{Hz}}$, respectively. The photon shot noise is mainly due to the cw laser power. To achieve the rf SNR of 25 dB in the present study, the 100-times average of rf spectrum with a resolution bandwidth of 300 kHz have been carried out. Therefore, the measurement time was approximately 300 μs (note that the rf spectrum analyzer spends much longer time in analyzing every frequency component). Ideally, a single measurement is completed in 1/ f_b (60 ns in this experiment), and therefore the rf SNR of 25 dB has been achieved by 5000 times data accumulation. In this study, the linewidth of the cw laser is quite large (1 MHz), and higher rf SNR may be expected if narrower-linewidth cw lasers, such as external-cavity lasers, are employed.

The spectrum SNR, which is estimated as 20 from the result of Fig. 3(c), depends not only on the rf SNR but also on the signal intensity fluctuations. The spectrum SNR is expected to be proportional to T^1/2 and P^1/2, where T is the measurement time of each data point and P is the optical power per mode of the comb, as long as the intensity fluctuations are random. By expressing the spectrum SNR as σ(TP)^1/2, the value of σ is independent of T and P, and estimated as $8.6\times {10}^6/\sqrt{\text{W}\cdot \text{s}}$. In the previous work [11], the spectrum SNR was 100 with the measurement time of 4400 μs and 300 nW/mode, and then σ is estimated as $2.8\times {10}^6/\sqrt{\text{W}\cdot \text{s}}$. The improvement of the spectrum SNR may be due to the phase locking of the cw laser to one of the comb modes. With the stabilization, the rf signal frequency is also stabilized, so that it is expected that the rf-frequency-dependent fluctuation in the rf signal caused by the rf electronics is reduced. In addition to the cw laser frequency stabilization, the phase correction algorithm in [11] can be employed after data acquisition for further improvement of the sensitivity.

The data acquisition time is limited by the lowest frequency of the beat notes. When the offset frequency of the phase lock is chosen as f_r/2, the data acquisition time will be the shortest. However, this setup is not appropriate because the rf beat components coincide with those caused by the neighboring modes in the frequency axis. They are separated when the offset frequency is lower than f_r/2 − Δν, where Δν is the linewidth of the cw laser (supposing the linewidth of each comb mode is infinitesimal). Hence, the narrow linewidth of the cw laser is desired again, especially in the case of high resolution spectroscopy with low-repetition-rate combs.

The resolution of CCH spectroscopy is f_r, as in other direct-comb spectroscopy. In this study, f_r was 66.8 MHz, and the resolution is high to observe Doppler-broadened spectral lines. However, this is insufficient to apply CCH spectroscopy to sub-Doppler resolution spectroscopy and optical thermometry. For higher resolution, low-repetition-rate combs, which can be prepared by repetition-rate downsampling [20] or phase-modulation technique [21, 22], are favored.

For wider bandwidth of CCH spectroscopy, fast photodetectors should be used, because the bandwidth of the CCH spectroscopy is twice the photodetector bandwidth. Currently, photodetectors as fast as 100 GHz are commercially available, and graphene photodetectors are expected to be faster than 500 GHz [23], promising a bandwidth of over 1 THz using this technique. These fast photodetectors allow us to carry out broadband and high-resolution spectroscopy.

6. Conclusions

We have carried out CCH spectroscopy of the R(4) transition in the 2ν₃ band of CH₄ at 1651 nm using the spectrally broadened Er-fiber comb and 1651-nm DFB laser. The power level of the comb modes and the sample absorption are weaker than those in the previous studies of the similar technique [8, 9, 11–14]. To enhance the sensitivity, the frequency of the cw laser is phase-locked to one of the comb modes to accumulate the signal for a long time, and the balanced detection of the heterodyne signal substantially suppresses irrelevant rf signals. The CCH technique has short data acquisition time compared to dual-comb spectroscopy and two-dimensional imaging comb spectroscopy and is demonstrated to have high sensitivity.