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Abstract
For scan-free high-resolution direct-comb spectroscopy, mode spacing of an optical frequency comb is reduced down to 260 kHz by phase modulation. It turns out that time-domain signal is degraded by averaging because of slow optical path length fluctuations and fast optical pulse timing jitter. In this study, compensation of these effects is introduced, and signal degradation by averaging is avoided. With demonstrations of direct-comb spectroscopy with the narrow-mode-spacing optical frequency comb, Doppler-limited absorption spectrum of methane and reflection spectrum from an optical ring cavity are observed. As a result, detailed resonance spectral line profile of 8 MHz linewidth for the optical ring cavity is obtained in 50 ms measurement time.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Invention of an optical frequency comb (OFC) introduced drastic progress in optical frequency measurement, and it is an indispensable tool for optical frequency atomic clocks [1,2]. Applications of the OFCs have been expanded not only to simple optical frequency measurement but also to distance measurements [3], optical communications [4], astronomy [5], and direct-comb spectroscopy. Direct-comb spectroscopy, in which an OFC is utilized as a broad bandwidth spectroscopic light source, is studied broadly since the invention of OFCs. Heterodyne technique is employed to convert optical mode of an OFC to radio-frequency (rf) signal to measure each comb mode intensity. In dual-comb spectroscopy [6–8], another OFC is used as a local oscillator of heterodyne detection (multiheterodyne), and in comb-continuous-wave-heterodyne (CCH) spectroscopy [9–11], a single-mode continuous-wave (cw) laser is used as a local oscillator. In dual-comb spectroscopy, broad spectral range is available (from 1.0 $\mu$m to 1.9 $\mu$m [7]), and for example, application to environmental observation is studied [12]. In CCH spectroscopy, the bandwidth of spectroscopy is limited to a few gigahertz, which is determined by electric bandwidth of photodetectors. Instead, CCH spectroscopy is advantageous in detection of weak spectral lines, because intense spectral power of a local oscillator (a cw laser) enables us to obtain heterodyne signal of high signal-to-noise (S/N) ratio [11]. In CCH spectroscopy, a single set of heterodyne signal (interferogram) can be acquired within the inverse of mode spacing of the OFC ($\sim$4 $\mu$s in this study), which is much faster than in dual-comb spectroscopy ($\sim$1 ms [7] determined by the repetition rate difference between two OFCs). Such fast measurement time in CCH spectroscopy is useful for time-resolved spectroscopy, and in addition, a large number of averaging for high S/N ratio becomes practical.
In general, resolution of direct-comb spectroscopy is determined by pulse repetition rate of the OFC ($f_r$). To achieve higher resolution than $f_r$, scanning OFC mode frequency is the simplest scheme, and Doppler-free spectroscopy such as double-resonance spectroscopy of rubidium [13] and two-photon spectroscopy [14,15] has been carried out with dual-comb spectroscopy. As another approach, OFC mode density multiplication by phase modulation is demonstrated [16–18]. With this technique, scan-free direct-comb spectroscopy becomes possible.
In this study, mode spacing of 260 kHz of a phase-modulated OFC is achieved to make scan-free high-resolution CCH spectroscopy feasible. In order to take advantage of fast measurement of CCH spectroscopy, averaging before data transfer to a computer is favored to reduce data transfer time and data processing time, because in CCH spectroscopy, data transfer time to the computer can determine total measurement time of spectrum, which includes not only sampling time of the interferogram but also data transfer time to the computer and data processing time in the computer. Since the sampling time is short in CCH spectroscopy, data transfer time and data processing time can be dominant. Then it is preferred to average time-domain signal on a digitizer before transferring the data to the computer to suppress data size to be transferred. The time-domain signal, however, is not always the same because of slow phase shift caused by optical fiber fluctuations and pulse repetition jitter of OFCs, resulting degradation of the averaged signal. In the case of dual-comb spectroscopy, software phase compensation, with which the phase fluctuations are compensated in the computer after the data transfer, is carried out [19]. In this study, in order to average before the data transfer, hardware phase compensation technique and data acquisition trigger synchronous to the actual pulse repetition of the OFC are introduced, and as a result, averaging can be processed without the signal degradation. As demonstrations of CCH spectroscopy with 260 kHz mode-spacing OFC, spectra of methane absorption and reflection from a ring cavity are observed at 1645 nm.
2. Mode-density multiplication by phase modulation
To obtain an OFC of narrow mode spacing, we employ mode-density multiplication by phase modulation [18], and here we briefly review how to multiply the mode density. Suppose that the $j$-th pulse of the OFC is subjected to the phase modulation of $\phi _j$, and this phase modulation is supposed to be applied to each pulse by an electro-optic phase modulator. The modulation sequence ${\phi _j}$ is periodic with $P$ (integer), namely,
Then the value of $P$ is the factor of mode-density multiplication, namely, the mode spacing of the OFC is reduced to $f_r/P$. In [11], the value of $P$ is chosen to be $N^2$ ($N$ is an integer), and the phase modulation sequence is given as where $j'$ and $j''$ are integers between 0 and $N-1$, and (mod $N$) means remainder after division by $N$. Then each optical mode of the mode-density multiplied OFC becomes to have equal intensity. With this phase sequence, $\phi _j$ is $N$-valued and has the period of $N^2$.3. Experimental setup
In this study, the multiplication factor $P$ is $16^2=256$. The experimental setup is essentially identical to that in [18] as shown in Fig. 1 (details of colored area in light yellow and the trigger generator are given in the following sections). The OFC is an erbium-doped fiber laser with $f_r=$ 66.87 MHz, which is stabilized with respect to a local oscillator (LO1). The spectral bandwidth of the OFC is broadened by a highly-nonlinear fiber after optical amplification by a fiber amplifier, and the output is divided into two beams. One beam is detected by balanced photodetectors (BPD1 in Fig. 1, Thorlabs, PDB415C-AC) after mixed with a single-mode cw laser beam (a grating-feedback external-cavity laser diode at 1645 nm). The beat-note signal between a single mode of the OFC and the cw laser is used for phase locking of the cw laser [20]. The other is for CCH spectroscopy. The beam for CCH spectroscopy passes through an electro-optic modulator (Thorlabs, LN65S-FC) for the mode-density multiplication and a device under test (DUT in Fig. 1), which is a glass cell filled with methane gas or an optical ring cavity in this study. Then this beam is mixed with the cw laser beam by a 50:50 beam splitter, and the beat-note signal between the OFC and the cw laser (interferogram) is detected by fast balanced photodetectors (BPD2 in Fig. 1, Thorlabs, PDB480C-AC). An arbitrary function generator (NF Corp., WF1968) generates the modulation sequence for mode-density multiplication of $P=256$, resulting the mode spacing of 261 kHz. The instruments in Fig. 1 are synchronized with a rubidium clock (not shown in the figure). The signal of BPD2 in Fig. 1 has been analyzed by a rf spectrum analyzer in [18]. The signal acquisition by a digitizer makes total measurement time much shorter than in [18]. For this purpose, we employ a fast digitizer (Spectrum, M4i.2230-x8, $5\times 10^9$ samples/s). It should be mentioned that the carrier-envelope-offset (CEO) frequency of the OFC is not stabilized in the present setup, although it must be stabilized for high-resolution spectroscopy.
Fig. 1. Schematic representation of experimental setup. Dotted lines imply transistor-transistor-logic signals. Colored area in light yellow is for phase compensation described in Section 4. Details of the trigger generator are given in Section 5. Abbreviations: ECLD for an external-cavity laser diode, AFG for an arbitrary function generator, EOM for an electro-optical modulator, FC for fiber couplers, DUT for a device under test, BS for beam splitters, FA for a fiber amplifier, HNLF for a highly nonlinear fiber, PID for proportional-integral-derivative controllers, M for rf mixers, PD for a photodetector, and PZT for a piezo-electric transducer. The arbitrary function generator, the two local oscillators (LO1 and LO2), and the digitizer are synchronized with the 10 MHz standard signal provided by a rubidium atomic clock (not shown in the figure).
By mode-density multiplication, each optical mode intensity is reduced by a factor of $P$ because of conservation of the total optical intensity. Therefore, averaging the signal is important to achieve high S/N ratio for large mode-density multiplication factor. For averaging, there are two methods: one is that each data set is Fourier-transformed and then averaged in frequency domain (frequency-domain averaging), and the other is that each data set is averaged in time domain and then Fourier-transformed (time-domain averaging). Calculation time for the frequency-domain averaging is much longer than that for the time-domain averaging because of the number of the Fourier transformation. In addition, data transfer time from the digitizer to the computer should be taken into account. In the case of time-domain averaging, the averaging can be carried out before or after the data transfer. If data size for a single interferogram is less than memory size on the digitizer, data transfer from the digitizer to the computer can be once for time-domain averaging, whereas it is $N_{\mathrm {ave}}$ times for frequency-domain averaging ($N_{\mathrm {ave}}$ is the average number). For CCH spectroscopy, the data acquisition rate is generally high, and data number to be transferred per a certain time tends to be large. Therefore, data transfer time and data processing time can determine the total measurement time. Specifically, the digitizer used in this study requires $\sim 26 \mu$s for $2^{17}$ samples for a single data set acquisition (it takes $\sim 15 \mu$s for a single trace of interferogram). For time-domain averaging of 100 times, it requires 150 ms, which is an actual measured value for data acquisition (1.5 ms), data transfer to the computer, and data processing in the computer, whereas for frequency-domain averaging of 100 times, it requires 17 seconds. Consequently, time-domain averaging is preferred for CCH spectroscopy. It is noted that in the case of dual-comb spectroscopy, the sampling rate of data acquisition does not have to be so fast compared to the case of CCH spectroscopy, and therefore frequency-domain averaging or time-domain averaging on the computer is favored because of software phase compensation technique.
For time-domain averaging, relative phase between the OFC carrier and the cw laser must always be the same (otherwise, the interferogram may be averaged out). Therefore, the lowest beat-note frequency between the OFC and the cw laser must be in the simple ratio to the mode spacing frequency $f_r/P$ (reduced repetition rate) to record the interferogram repeatedly in phase. In this study, the lowest beat-note frequency is set at $1/4$ of the mode spacing frequency by choosing the offset frequency of the cw-laser phase locking (LO2) at $309/1024$ of $f_r$ ($\sim 20.18$ MHz).
It seems that increasing the mode-density multiplication factor is just tailoring the modulation sequence for higher value of $N$ according to Eq. (2), but some improvement in experimental setup is required. We describe details of the improvement in the following.
4. Phase compensation
The relative phase between the modulated OFC carrier and the cw laser drifts slowly even with the phase locking of the cw laser with respect to the OFC, because of slow fluctuations in the out-of-loop beam path [19–21]. In the presence of this phase shift, the single-shot interferogram becomes different every time as shown in Fig. 2(a), in which four independent traces of a part of interferogram are shown. In Fig. 2(a), the signals of interferogram are different each other, and time-domain averaging may degrade the averaged interferogram (after the data transfer to the computer, software phase compensation can be carried out to prevent the signal intensity from degrading). Therefore, phase compensation process is necessary to achieve high S/N ratio by time-domain averaging on the digitizer.
Fig. 2. A part of the interferogram observed without (a) and with (b) the hardware phase compensation. Period of the interferogram is $\sim 15 \mu$s. The observation is repeated four times. The plots are vertically shifted for clarity.
We employ hardware phase compensation as shown in the colored area in light yellow in Fig. 1. In this scheme, the lowest frequency component in the interferogram is measured by the use of a lock-in amplifier with the reference that is provided by the arbitrary function generator for phase modulation sequence. Because the lowest beat-note frequency is $1/4$ of the phase sequence frequency ($f_r/1024$), the reference signal is produced by frequency-dividing the synchronized transistor-transistor-logic (TTL) output (Sync out in Fig. 1) of the arbitrary function generator by 4. The output of the lock-in amplifier is fed through a PID (proportional-integral-differential) control circuit back to a piezo-electric transducer to compensate the phase drift by adjusting the optical path length.
Figure 3 shows the result of the phase measurement without and with the hardware phase compensation. In this measurement, the beat-note frequency between the OFC and the cw laser is phase-locked at the frequency of 20.179 263 105 469 MHz, which is $309/1024$ of $f_r=$ 66.872 380 MHz. Without the phase compensation, the phase fluctuates between $-\pi$ and $\pi$, and after the hardware phase compensation loop is closed, the phase is stabilized. For repeated interferogram measurement with this feedback loop, the signals are almost identical each other as shown in Fig. 2(b), and therefore time-domain averaging is expected to improve the S/N ratio of the interferogram.
Fig. 3. Out-of-loop phase ($\varphi$) of the OFC carrier with respect to the cw laser. Although the cw laser is phase-locked to the OFC carrier, out-of-loop beam path fluctuations cause slow phase drift. The phase stabilization feedback loop is closed at the moment indicated by a red dotted line.
In Fig. 4(a) and (b), Fourier spectra of the interferogram in Fig. 2 are shown (to record the interferogram, the number of data is $2^{19}$ samples for higher resolution). Figure 4(a) and (b) are without and with the hardware phase compensation, respectively, and black (red) curves show single-shot (100-time averaged) spectra. Without the hardware phase compensation, each OFC mode intensity becomes small by averaging ($\sim 2$ dB at 100 MHz and 10 dB at 700 MHz), whereas with the hardware phase compensation, deterioration of the OFC mode intensity by averaging is prevented at lower frequency region ($\sim 100$ MHz). Meanwhile, the averaging reduces the noise level by 20 dB, improving the S/N ratio. The 20 dB noise reduction by 100-time averaging agrees with expectation (note that the detected optical power is averaged, and the rf power in Fig. 4 is square of the optical power). At higher frequency ($\sim 700$ MHz), although improved, the signal reduction by averaging is still observed ($\sim 3$ dB). We think that this is due to timing jitter of the OFC pulse repetition. Next, we discuss how to suppress the effect of the timing jitter.
Fig. 4. Fourier spectra of the interferogram. (a) and (b) are results without and with the hardware phase compensation, respectively, and the signal acquisition is carried out with the FG trigger source. (c) is with the hardware phase compensation and with the OFC trigger source. Black and red plots are single-shot and 100-time averaged results, respectively.
5. Signal acquisition trigger
In the observation of Fig. 4(a) and (b), we use the TTL output of the arbitrary function generator synchronized with the mode-density multiplication phase modulation sequence (Sync out in Fig. 1) as a trigger source of the data acquisition by the digitizer. Ideally, the output of Sync out is supposed to be synchronized with the pulse timing. Practically, however, timing jitter of the pulse train is anticipated because of time constant of the repetition rate stabilization feedback loop. To observe the timing jitter, the interferogram in short time scale is shown in Fig. 5(a), in which the hardware phase compensation is carried out (black and red curves are single-shot and 100-time averaged results, respectively). In the averaged interferogram, fast oscillation disappears, meaning that high frequency component is filtered out.
Fig. 5. Interferogram observed with the FG trigger source (a) and with the OFC trigger source (b). Black and red curves are single-shot and 100-time averaged results. The plots are vertically shifted for clarity.
In order to reduce the effect of the timing jitter, we try another trigger source for the signal acquisition, which is provided by “trigger generator" in Fig. 1. Details of the trigger generator are given in Fig. 6. This circuit can be replaced with a 10-bit ($2^{10}=1024$) counter with the clock input from the repetition rate detection, but the circuit in Fig. 6 can be used for any mode-density multiplication factor. For convenience, the trigger sources of Sync out of the arbitrary function generator and of the trigger generator are referred as “FG trigger source" and “OFC trigger source", respectively. The circuit of Fig. 6 generates the TTL signal that is synchronous with not AFG in Fig. 1 but with actual pulse timing of the OFC. The logic circuit used in Fig. 6 is for reducing the trigger frequency down to $f_r/1024$. With the OFC trigger source, it is expected that the signal acquisition timing is always synchronous with the OFC pulse train regardless of the pulse timing jitter.
Fig. 6. The circuit to generate a TTL trigger source synchronized with the actual OFC pulse timing. Dotted lines are for TTL logic signals. Terminal S is connected to the TTL output of the arbitrary function generator for the phase modulation sequence (Sync out in Fig. 1), R is to the photodetector to monitor the repetition rate of the OFC (PD in Fig. 1), and T is the signal acquisition trigger to be connected to the digitizer. HPF is a high-pass filter (cut-off frequency is 1 MHz), C is a comparator (Texas Instruments, TLV3501), and D-FF is a delay flip flop (Texas Instruments, SN74AS74A). Even when the transition from 0 level to 1 comes to the terminal S, the transition at the terminal T is delayed until the TTL pulse triggers the D-FF at the clock input (CLK).
Figure 5(b) is the result of interferogram observed with the OFC trigger source. Black curves and red curves are single-shot and 100-time averaged interferogram, respectively. From this result, the fast oscillation in the averaged interferogram is observed similarly to the single-shot interferogram. Hence the signal intensity is kept unchanged by the averaging, whereas the noise level is expected to be reduced.
In Fig. 4(c), the Fourier spectrum of Fig. 5(b) is shown. As expected, the averaged Fourier signal intensity is unchanged even in higher frequency region ($\sim 700$ MHz), whereas the noise level is suppressed substantially by averaging. Even with the OFC trigger source, the timing jitter that occurs during the signal acquisition time cannot be compensated, and this residual timing jitter causes a small amount (less than $\sim 1$ dB at 700 MHz) of signal intensity reduction in Fig. 4(c) by time-domain averaging.
6. Spectroscopy
For demonstration of CCH spectroscopy with the OFC of mode spacing of 260 kHz, Doppler-broadened absorption spectrum of methane and reflection spectrum of an optical ring cavity are recorded. As mentioned before, the CEO frequency of the OFC must be stabilized for high-resolution spectroscopy such as double-resonance spectroscopy. In this demonstration, however, we leave the CEO frequency free-running, because the CEO frequency drift (a few megaheltz for 1 hour) is not so significant during the measurement time of CCH spectroscopy (0.1 s at most) compared to the Doppler-broadened linewidth of methane or resonance frequency drift of the optical ring cavity. Figure 7(a) shows absorption spectrum of methane recorded by CCH spectroscopy using the OFC of mode spacing of 260 kHz. For this purpose, a glass cell (20 cm long) filled with methane gas (100 Pa) is inserted as a device under test (DUT in Fig. 1). The cw laser wavelength is 1645.566 nm, around which R(6) absorption lines in 2$\nu _3$ band exist. For normalization of rf response, the spectrum around 1645.600 nm is also recorded as a reference. Each blue dot in Fig. 7(a) corresponds to individual OFC mode intensity (average of 100 data sets, and a single data set consists of $2^{19}$ samples). According to the HITRAN database [22], there are three absorption lines, and two of them are not resolved because of the Doppler broadening. It seems that the noise level increases compared to that in [18]. This noise increase is due to the reduction of each mode intensity for high mode density multiplication factor. In addition, the noise increase higher than 1 GHz and lower than $-1$ GHz is caused by response bandwidth of the fast photodiodes.
Fig. 7. (a) Absorption spectrum of the R(6) transitions of the $2\nu _3$ band of methane at 1645.566 nm recorded by CCH spectroscopy. The horizontal lower axis indicates the optical frequency of the phase-modulated OFC relative to the cw laser frequency. The upper axis shows the mode number of the OFC. The red dotted lines are at the center frequencies of the absorption, and their length implies relative intensity of the absorption lines. (b) Reflection spectrum of an optical ring cavity. The cavity length is 50 cm, resulting the free-spectral range of 600 MHz. The ring cavity is slightly misaligned, so that a number of lines of higher-order transverse modes are observed. Magnified spectrum is given in the inset.
As a demonstration for narrower linewidth, reflection spectrum of a slightly misaligned optical ring cavity is observed with the cw laser at 1645.604 nm. In this case, DUT in Fig. 1 is replaced with a ring cavity (spectrum of the reflection beam is observed). The ring cavity consists of four mirrors, and one of them is an input coupler of reflectivity of 98 %. The cavity length is 50 cm, resulting the free-spectral range (FSR) of 600 MHz. If no optical power loss except transmission at the input coupler exists, finesse of the cavity is expected to be $\sim$ 50. It should be noted that the optical components are originally designed for wavelength at 1550 nm, and for 1646 nm, the estimated value of finesse may be different from the actual value. The cavity mirrors are slightly misaligned, so that the input beam couples with a number of higher-order transverse modes. The cavity length is not stabilized, and therefore if the signal acquisition time is long, the spectrum may be washed out by fluctuation of the cavity length. In this demonstration, the total data acquisition time of CCH spectroscopy is set to be $\sim$ 54 ms ($2^{28}$ samples), which corresponds to averaging of $\sim 3500$ data sets of interferogram. The reference signal for rf normalization is obtained by inserting a beam stop in the ring cavity without changing wavelength of the cw laser.
The result is given in Fig. 7(b). Periodic resonance spectrum of the ring cavity is observed with resonance lines of higher-order transverse modes. The magnified spectrum is given in the inset, and linewidth is 8.6 MHz. The FSR is 600 MHz, and then the finesse is determined as 70, which is slightly larger than the expected value. This disagreement may be partly due to mismatch of the optical component design wavelength.
7. Conclusions
In conclusion, mode density multiplication of optical frequency combs by phase modulation in [18] is further extended for scan-free high-resolution direct-comb spectroscopy, and mode spacing of the OFC is reduced down to 260 kHz by mode-density multiplication factor of 256. This narrow-mode-spacing OFC is suitable for high-resolution direct-comb spectroscopy. It is found that slow out-of-loop phase noise and fast pulse timing jitter of the OFC degrade time-domain-averaged interferogram. To overcome, hardware phase compensation for slow out-of-loop phase noise and trigger source for data acquisition synchronized to actual OFC pulse timing to cancel the jitter are introduced. As a result, time-domain averaging before data transfer can be carried out without deterioration of interferogram, whereas noise level is reduced substantially by averaging. In order to demonstrate CCH spectroscopy with the 260 kHz resolution, Doppler-limited spectroscopy of methane is carried out. For demonstration of narrower resolution, reflection spectrum of an optical ring cavity is observed. Spectroscopy of narrow linewidth such as double-resonant spectroscopy and two-photon spectroscopy is expected without scanning the OFC mode frequency.
Funding
Precise Measurement Technology Promotion Foundation.
Disclosures
The author declares no conflicts of interest.
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