We performed broadband dual-frequency-comb spectroscopy in the near-infrared region with a much higher resolution than the Fourier limit by using discrete Fourier transforms and spectral interleaving. We observed the resonant spectrum of a Fabry-Perot cavity over a spectral range of 187 to 218 THz using this technique, and measured its free spectral ranges and finesses. The recorded spectrum includes cavity resonance lines with widths of about 2 MHz, which is much narrower than the resolution of 48 MHz determined by the observation time window.
© 2015 Optical Society of America
Fourier transform spectroscopy (FTS) is a ubiquitous technology for recording the broadband spectra of various types of radiation and it is used in the fields of scientific research, industry, and medicine. In comparison with conventional dispersive spectrometers, FTS is advantageous because it has a high signal-to-noise ratio and the simultaneous acquisition of signals in a broadband spectrum. The recorded spectra are useful for identifying spectral carriers and evaluating optical components such as optical filters. Conventional FTS with high spectral resolution requires a long-range mechanical delay scan because the spectral resolution is determined by the observation time window (Fourier limit). Recently, precisely periodic pulses from optical combs have been combined with FTS. This combination has produced new types of FTS such as dual-frequency-comb (dual-comb) spectroscopy  and has been demonstrated in the visible , infrared [3–6 ], and terahertz  regions. In dual-comb spectroscopy, a high spectral resolution is available without any mechanical delay scan, and the spectral resolution was believed to be limited by the repetition frequency of the sequential pulses. Mode-resolved dual-comb spectroscopy, which was realized by extending the time-window width and employing spectral interleaving, has been demonstrated to overcome this limit [8–11 ]. In this case, a series of long-time window data including several or several tens of pulses is treated as a temporally observed signal. However, the spectral resolution is still limited by the time-window width, and the data volume increases greatly. Some results of dual-comb spectroscopy using coherent averaging  and spectral interleaving techniques have indicated the possibility of achieving a spectral resolution beyond the Fourier limit [4, 5 ]. However, a concrete method for improving the spectral resolution has not been clearly described, and such high-resolution spectroscopy has not been reported.
Recently, we showed that discrete Fourier transform spectroscopy (dFTS) has a spectral resolution beyond the Fourier limit when the interferogram is precisely periodic including the carrier-phase relation . This scheme was implemented with the asynchronous optical sampling method [14, 15 ] in the terahertz region and a resolution was achieved beyond the Fourier limit. The combination of dFTS and spectral interleaving enhances the spectral resolution and accuracy, and enables us to assign absorption lines with linewidths narrower than the resolution limited by the time-window width. Furthermore, dFTS reduces the data volume needed to obtain a linewidth as narrow as that of mode-resolved dual-THz-comb spectroscopy, and consequently the required measurement time is also reduced. On the other hand, the timing and phase jitter in the comb pulse is large in the visible and infrared regions because the carrier frequency is higher than the terahertz region. Therefore, to satisfy the conditions required for dFTS, the pulse-to-pulse relative timing and phase jitter of the two combs should be small, which corresponds to the relative linewidth of the two combs being narrow (see Sec. 2 for details).
In this paper, we describe the characteristic evaluation of a Fabry-Perot cavity in the near-infrared region that we performed by combining our dual-comb spectrometer , dFTS and spectral interleaving. As a result, we observed the resonant spectra of a Fabry-Perot cavity with a few MHz linewidth across 187-218 THz (1380-1600 nm) region and demonstrated that the spectral resolution is not limited by the time-window width. We measured the free spectral ranges (FSRs) and finesses of the cavity and observed the optical frequency dependences.
2. Operating principle of dual comb spectroscopy using dFTS
Figure 1 shows the operating principle of dual-comb spectroscopy . Dual-comb spectroscopy is a type of FTS that uses two combs with repetition rates of f rep,S (signal comb) and f rep,L = f rep,S − Δf rep (local comb, f rep,S >> Δf rep). In this situation, since the time interval between two optical pulses from the local comb and the signal comb gradually increases by Δf rep/(f rep,S f rep,L) every optical pulse, the two optical pulses overlap every 1/Δf rep ( = T) as shown in the top part of Fig. 1. Therefore, the periodic overlaps of the two pulse trains from the two frequency combs repeatedly generate interferogram signals with a waveform h(t) every period of time T in the time domain as shown in the bottom part of Fig. 1. The Fourier transform (spectrum) of the interferogram is recorded as a product of the signal and local comb spectra in the frequency domain, and its frequency scale is compressed from the optical to the RF region by multiplying Δf rep/f rep,S. The spectral resolution of the waveform h(t) is limited up to the inverse of the T by Fourier limit, which corresponds to the spectral resolution of f rep,S in the original optical frequency domain.
Figure 2 shows the concept of dFTS. We assume that a series of signals h(t) is precisely periodic and repeatedly overlaps with an integer multiple delay of T as shown in Fig. 2(a). In this situation, the data acquired in a finite time-window width contain multiple signals h(t + NT) generated at different timings, where N is an integer. If the repetitive time-window widths are precisely and always constant (T in this case), sequential signals h(t + NT) and h(t + (N + 1)T) in the time window can be temporally connected, and the total signal can be considered equivalent to the continuous signal shown in Fig. 2(b). Consequently, the discrete Fourier-transform of the recorded interferogram of which the time window is precisely matched with repetition period T provides a discrete spectrum composed of data plots with an infinitesimal spectral resolution and a finite frequency space of 1/T.
Figure 3 shows combined schematics of the dFTS and the spectral interleaving. The spectral interleaving is a method for overlapping multiple and slightly frequency-shifted spectra as shown in Fig. 3(a). In dual-comb spectroscopy, we sweep T incrementally by appropriately changing f rep,S, f rep,L, f ceo,S, and f ceo,L to overlay the resultant spectra and fill the frequency gap (f rep,S) between the data plots. f ceo,S and f ceo,L are the carrier-envelope-offset frequencies of the signal and local combs, respectively. We expect to obtain a spectrum with a spectral resolution beyond 1/T by using spectral interleaving. However, with non-dFTS, we are not able to resolve the narrow lines in the spectrum because each data point has a resolution of 1/T, and the narrow lines are spectrally smoothed (Fig. 3(b)). In contrast, as mentioned above, we are able to obtain a sufficiently high spectral resolution when we employ both dFTS and spectral interleaving (Fig. 3(c)).
Next, we describe the conditions needed to execute dFTS in dual-comb spectroscopy. As we mentioned above, the interferogram should be precisely periodic. Executing dFTS is equivalent to executing coherent averaging  in dual-comb spectroscopy. If two combs satisfy the following two conditions, the dual-comb spectroscopy interferogram becomes precisely periodic with a period of 1/Δf rep and the spectroscopy can be executed as dFTS. First, the pulse-to-pulse delay between the two combs should be periodic with a period of 1/Δf rep. This is given byEqs. (2) means that the phase difference returns exactly to its original value l times in the period of time taken to generate the interferogram (1/Δf rep). When Eqs. (1) and (2) are precisely satisfied, dual-comb spectroscopy based on dFTS is executed. When we apply the principle in the THz region, Eq. (2) is satisfied because THz combs are usually generated by the difference frequency generation process, and such combs have an f ceo of zero . In contrast, in the near-infrared case, we need to adjust f ceo to satisfy Eq. (2). Moreover, the pulse-to-pulse relative timing and phase jitter in the time domain (mode-to-mode relative linewidth on the frequency domain) between the two combs induce significant phase noise in the interferogram. To obtain an interferogram with a sufficient signal-to-noise ratio, the timing and phase jitter should be sufficiently reduced. In other words, the relative linewidth of the combs should be sufficiently narrower than Δf rep.
3. Experimental setup
Figure 4 shows a schematic of the experimental setup. The measurement itself is similar to that described in our previous work . Briefly, we employ a pair of mode-locked lasers based on an erbium-doped fiber as the oscillator of the signal and local combs. It has an electro-optic modulator (EOM) in the laser cavity for fast servo control [17, 18 ]. The repetition rate of the signal comb (f rep,S) is phase-locked to the reference frequency of 48 MHz obtained from an RF synthesizer. The local comb is phase-locked to the signal comb through a CW buffer-laser using the following procedure. First, f ceo,S and f ceo,L are phase-locked to RF reference frequencies by controlling the injection current of the pump lasers for the comb oscillators. Second, the beat frequency (f beat,S) between the CW laser and one of the signal comb modes is phase-locked to another RF reference frequency. The error signal is fed back to the injection current of the CW laser, thus phase-locking the CW laser to the signal comb. Then the beat frequency (f beat,L) between the CW laser and one of the local comb modes is also phase-locked to another reference frequency. The error signal is fed back to the EOM in the local comb oscillator, and in this way the local comb is phase-locked to the signal comb through the CW laser. All of the reference frequencies are linked to coordinated universal time (UTC). We set Δf rep at 21 Hz (k at 2,323,818), and l at 475, which correspond to the span (56 THz) and the center (197 THz) of the dual-comb spectrum.
The output beam from the signal comb is led to a Fabry-Perot cavity and the reflected beam from the cavity is extracted by an optical circulator. The extracted beam overlaps the output beam from the local comb at a polarization beam splitter (PBS). These beams are divided into two parts at another PBS after the polarization has been rotated with a half-wave plate to adjust the dividing ratio, and are then detected with two InGaAs photodiodes for balanced detection. The detected signal is an interferogram of the signal and local beams, which is guided to a 14-bit digitizer through a low-pass filter and an amplifier, and sampled at the repetition rate of the local comb (f rep,L). The sampled data are transferred to a PC, and each precisely repetitive interferogram is coherently averaged in 95 seconds under a condition satisfying Eqs. (1) and (2) . The averaged interferogram is Fourier-transformed into a spectrum, and then stored in the PC. The horizontal axis of the spectrum is scaled with the absolute frequency calculated from the repetition rates, carrier-envelope-offset frequencies, and the index numbers of the comb modes used to observe the beat frequencies with the CW laser. The index numbers are determined by the CW laser frequency measured with a wavelength meter. To interleave the frequency gap of the horizontal axis of the spectrum, we repeat the set of measurements while increasing f rep,S incrementally using the RF synthesizer. The step frequency is set at 0.12 Hz, which corresponds to interleaving the frequency gap (f rep,S) with 100 points. We adjust the f ceo,L and f beat,L by controlling the phase-locking reference frequencies in real time to satisfy Eqs. (1) and (2) . Therefore, k and l are always constant whenever f rep,S is incrementally increased by 0.12 Hz.
We observed the reflection spectrum of a Fabry-Perot cavity with a length of 27 cm and a mirror reflectance of 99% at 1542 nm. These parameters give a free spectral range (FSR) of 560 MHz, a finesse of 300, and a full-width at half-maximum (FWHM) of the linewidth of 1.9 MHz. To prevent long-term drift as regards the cavity length, the frequency of one of the cavity modes is stabilized to that of an acetylene-stabilized laser by using the Pound-Drever-Hall technique . A 5-MHz phase modulation is applied to the output beam from the acetylene-stabilized laser by an EOM. The beam overlaps the signal comb at a PBS and enters the cavity. The reflected beam from the cavity is extracted by an optical circulator and detected by a photodiode. The detected signal is demodulated at 5 MHz to obtain an error signal. It is fed back to a piezoelectric transducer through a servo circuit to stabilize the cavity length precisely.
Figure 5(a) shows that the observed spectrum of the reflected beam from the cavity spread from 187 to 218 THz. The envelope of the spectrum indicates the product of the spectra of the two combs. The frequency step of the horizontal scale is f rep,S/100 = 0.48 MHz. Figures 5(b)-5(d), respectively, show enlargements of Fig. 5(a) in the 189, 201, and 213 THz regions with a frequency span of 30 GHz. Absorption lines generated by the cavity can be clearly observed for every FSR. Figures 5(e)-5(g) show enlargements of Figs. 5(b)-5(d) focusing on a single absorption line with a frequency span of 70 MHz. Resonant lines with FWHMs of a few MHz are observed, which is narrower than the resolution limited by the time-window width ( = f rep,S = 48 MHz).
The shapes of the resonant spectra are somewhat asymmetric as shown in Figs. 5(e)-5(g), and there are some sidebands caused by the higher order transverse modes of the cavity as shown in Figs. 5(b)-5(d). This is particularly prominent at around 189 THz and considered to be due to the mismatch of the mode coupling of the incident beam and the cavity. Since we adjust the mode coupling by using a CW laser, the coupling efficiency could become worse at frequency regions far from that of the CW laser because of chromatic aberration in the mode matching lenses. Such an asymmetric spectral shape is caused by a mismatch of the profile of the beam reflected at an input mirror of the cavity and that exiting the inside of the cavity .
We discuss the asymmetric line shape of the cavity reflection spectrum to measure the FSR and finesse of the cavity precisely. The cavity reflection spectrum is observed as an interference signal of the beam reflected at the input mirror and that exiting the inside of the cavity. The amplitude of the outgoing beam has a complex Lorentzian lineshape (see ref , for example). If the wavefront of the incident beam completely matches that of the cavity spatial mode, the interference signal reflects the real part of the complex Lorentzian lineshape function. Otherwise the interference signal reflects a superposition of the real and imaginary parts of the complex Lorentzian function because the beam reflected at the input mirror has phase-mismatched components with that exiting the cavity. Therefore, the reflection lineshape of the cavity can be written asEquation (3) is fitted to the observed spectrum with ν 0, δν, α, and β as adjustable parameters after the baseline of the spectrum has been eliminated. The red curves in Figs. 5(e)-5(g) show that the fitting results agree well with the asymmetric lineshape of the observed spectrum.
We measured the center frequencies and FWHMs of the resonant lines of the Fabry-Perot cavity, and Eq. (3) is fitted to the plots of the resonant lines to determine the FSRs and finesses of the cavity in several frequency regions. Figure 6 shows the obtained frequency characteristics of the FSR and finesse. Each FSR value is derived by fitting a line to ten successive center frequencies of the resonant lines. The results are around 567.4 MHz, and the deviations indicate that the cavity length is affected by the group velocity dispersion of the cavity mirrors. On the other hand, the finesse values are derived from the weighted mean of the FSR-to-FWHM ratios. The resultant finesses decrease in the frequency region higher than 205 THz, which indicates that the reflectance of the cavity mirrors decreases in the higher frequency region. The interleaved spectrum with a wide spectral bandwidth and a high spectral resolution enables us to evaluate the reflectance and dispersion of the cavity mirrors across the 30-THz frequency region.
One of the resonant lines of the cavity is considered to be the frequency of the acetylene-stabilized laser used as a reference. We measured the frequency by fitting Eq. (3) to the observed spectrum. The resultant frequency is 194 369 569.353(37) MHz, which agrees with the value of 194 369 569.384(5) MHz recommended by the International Committee for Weights and Measures (CIPM) who supervise the Metre Convention . The discrepancy of −31 kHz is within the uncertainty of the fitting. This result suggests that the interleaved spectrum has an accurate frequency axis for determining the center frequency of resonant lines narrower than the resolution limited by the time-window width.
In addition, we describe the resolution limit of our near-infrared dFTS measurement. We consider the absolute linewidth of the signal comb modes and the relative linewidth of the signal to local comb modes to be possible factors limiting the resolution. Since f rep,S is stabilized at the reference frequency generated by the RF synthesizer with a large time constant to allow us to ignore the fast frequency noise components of the synthesizer, the absolute linewidth of the signal comb is comparable to that of the comb under a free-running condition and is 10 kHz at most . On the other hand, the relative linewidth of the two combs is calculated from the measured phase noise, and is 0.15 Hz, which is limited by the fluctuations of the optical path length in the optical fibers. The relative linewidth is converted to the resolution by multiplying it with the k value and the result is approximately 380 kHz. Therefore, the relative linewidth of the combs limits the resolution in this study and is comparable to the frequency step of the observed spectrum of 480 kHz.
We performed near-infrared dual-comb spectroscopy with a spectral resolution beyond the limitation of the finite time-window width by using dFTS. We observed the reflection spectrum of the Fabry-Perot cavity across 187-218 THz and resolved the resonant lines with a linewidth of 2 MHz. This linewidth is considerably narrower than the resolution determined by the time-window width ( = f rep,S = 48 MHz). In addition, the frequency characteristics of the FSR and the finesse of the cavity were measured from the observed spectrum. These results indicate that the dFTS technique is valid in the near-infrared region. The dFTS technique will allow us to perform dual-comb spectroscopy with sub-Doppler resolution, and could be employed to evaluate narrow-band optical filters and/or cavities as in the demonstration reported here.
This work was supported by Collaborative Research Based on Industrial Demand from the Japan Science and Technology Agency, and Grants-in-Aid for Scientific Research No. 23244084 and 26246031 from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. We also gratefully acknowledge financial support from The Canon Foundation. In addition, we are grateful to Prof. M. Katsuragawa of the University of Electro-Communications for a fruitful discussion as regards asymmetric spectral lineshape. We are grateful to T. Suzuyama and M. Amemiya for maintaining UTC at NMIJ.
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