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An optical frequency comb interferometer with a 342-m-long fiber-based optical reference path was developed. The long fiber-based reference path was stabilized to 10−12-order stability by using a fiber noise cancellation technique, and small temperature changes on the millikelvin order were detected by measuring an interferometric phase signal. Pulse number differences of 30 and 61 between the measurement and reference paths were determined precisely, with slight tuning of the 53.4 MHz repetition frequency. Moreover, with pulse number difference of 61, a 6.4-m-wide scanning for the relative pulse position is possible only by 1 MHz repetition frequency tuning, which makes pulses overlapped for arbitrary distance. Such wide-range high-precision delay length scanning can be used to measure arbitrary distances by using a highly stabilized long fiber-based reference path.
© 2015 Optical Society of America
1. Introduction
An optical frequency comb is an essential tool for high-accuracy, long-range absolute distance measurements [1–12]. An optical frequency comb interferometer (OFCI), which uses interference between pulses in a mode-locked laser as a light source [13], is one such technique. In a general case of interferometry, optical path length scanning is necessary in order to measure the phase and to determine the absolute distance, i.e., integer multiples of the phase. Conventionally, a mechanical moving stage is used to scan the optical path, and a long-range mechanical moving stage is required for a wide scanning range. In this case, mechanical uncertainty in the displacement owing to various factors, such as yawing and pitching of the stage and motion-induced instability, causes interference signal deterioration. With an OFCI, the optical path difference between the two pulses can be scanned precisely by tuning the repetition frequency (frep) of the frequency comb [11, 14]. On the other hand, meter-wide scanning of the relative optical path length is essential for an OFCI, because pulse-to-pulse interference signals are observed only when the optical path difference between the measurement and reference paths corresponds to a multiple of the pulse separation, ngL, which is equal to c/frep, where c is the speed of light in vacuum, L is the geometrical pulse separation, and ng is the effective group refractive index of the interferometer medium, such as air or glass. However, it is difficult to realize a wide scanning range corresponding to ngL, because the tunable range of frep (Δfrep) is limited by the laser cavity configuration.
Because of the narrow linewidth and long coherence length of each mode of the comb, an interference signal is observed between separated pulses in a mode-locked pulse train. Therefore, an unbalanced interferometer, utilizing interference between the measurement and the reference pulses with a large optical path length difference, can be realized. Such a pulse-to-pulse interferometer with an unbalanced path has a so-called “multiplication effect” [15–20], as indicated in Fig. 1.
Fig. 1 Multiplication effect with slight frep tuning. Since each pulse-to-pulse separation is changed by the same amount by repetition frequency tuning Δfrep, the peak position of the m-th pulse (delay) is changed by m times the interval change.
When the optical path difference ngl, between the measurement and reference paths, is m × ngL (m: an integer for the pulse number difference between the measurement and reference pulses), interference fringes between the measurement and reference pulses are observed. In this case, when frep is changed by Δfrep by changing the laser cavity optical length, the relative position between the two pulses, i.e., the optical delay length, is changed by m × ngΔL. In this way, the range of optical delay length scanning, i.e., the effective optical path length scanning (nΔl) to generate pulse-to-pulse interferometric signals for both the envelope and the internal fringe, is multiplied by m; thus, wide-range scanning can be realized even with a slight change in Δfrep, as follows:
where n is the refractive index of the interferometer medium, which can be either group- or phase-refractive index (i.e., ng and np) for envelope or fringe scanning, respectively. In this study, the refractive indices were determined at the center of gravity of the ultrashort pulse spectrum with the spectral width of 10 nm, and the uncertainty in the determination of the center wavelength was in the range of 0.5–1 nm. For the simplicity of discussion, the right hand side of Eq. (1) was approximated because the change in frep is at most few % (i.e., frep >> Δfrep). However, this effect is useful for very long-distance measurements only, e.g., in the case of a mode-locked laser with frep = 53.4 MHz, ngL = c/frep = 5.6 m, where a scanning range of ngΔl = 2.8 m is required to realize a pulse overlap in the worst case when one of the pulses, such as a reference pulse, is in the middle of a pair of others, such as signal pulses. When the maximum Δfrep is 1 MHz (1.9% of frep), a minimum of m = 28 is required to yield a sufficient multiplication effect to achieve the full scanning range through Δfrep tuning only. Thus, an optical path difference above 157 m (5.6 m × 28 = 156.8 m) is required, and this effect is applicable to measure absolute distance above 157 m. Thus, it is difficult to utilize this effect in distance measurements that range from a few meters to even 100 m, which is important for many scientific and industrial applications.The multiplication effect has been utilized for autocorrelation measurements [15, 19], refractive index measurements [14, 16], three-dimensional measurements [17], THz spectroscopy [18], and Fourier-transform spectroscopy [20]; however, a precise demonstration of the multiplication effect for interferometric phase measurements of arbitrary distance has not been reported.
In this study, we developed an OFCI for measuring arbitrary distances, not limited only to very long distances. Here, we utilize a long fiber-based reference path (Fig. 2) instead of relying on a long distance to be measured, i.e., a long measurement path. The measurement path can be both space- and fiber-based. In this scheme, although a fiber-based reference path has advantages with respect to the system compactness, unwanted issues such as optical length fluctuation and large phase noise in long fibers cause interference signal instability. In order to overcome this problem, a correction of the interferogram instability, e.g., by using a digital signal processing algorithm, was demonstrated [20]. In contrast, we applied fiber noise cancellation techniques [21–26] to the OFCI with a long fiber-based optical reference path. As a result, the fiber-based reference path was stabilized at nm-level stability in the optical path up to a distance of 342 m, which corresponds to a fluctuation on the order of 10−12.
Moreover, a pulse number difference of m = 61 was precisely determined with slight repetition frequency tuning (Δfrep = 2 Hz) because of high stability and accuracy in the phase measurements. Large pulse number difference of m = 61 yields a wide scanning range of nΔl = 6.4 m only by the repetition frequency change of Δfrep = 1 MHz; thus, the measurement and reference pulses can be made to overlap at arbitrary distances without requiring a mechanical moving stage. The developed interferometer can be applied to absolute distance measurement without prior measurements, such as in separate time-of-flight measurements [2, 10], and can be also applied to a high-sensitivity millikelvin-order temperature sensing.
2. Optical frequency comb interferometer with fiber noise cancellation technique
Figure 3 shows a schematic of the OFCI with a fiber noise cancellation technique. The setup consists of two Mach-Zehnder type interferometers, i.e., “main” and “monitor” interferometers, with an Er:fiber-based comb and a single frequency CW laser as the light sources, respectively. The main interferometer (red dashed line in Fig. 3) is used to measure the interference signal. The monitor interferometer (black dashed line in Fig. 3) is used to stabilize the reference path length fluctuation that is caused by environmental instability.
Fig. 3 Optical frequency comb interferometer with a fiber noise cancellation technique. CW: Continuous wave; BPF: Band pass filter; AOM: Acousto-optic modulator; PMF: Polarization-maintaining fiber; DCF: Dispersion compensation fiber; PD1 and PD2: Photodetectors 1 and 2, respectively. Optical path difference ngl between the measurement and reference paths is preset to 168 m and 342 m, which corresponds to 30 and 61 times the pulse separation of 5.6 m. The indicated fiber length is the optical length including the fiber refractive index.
In the main interferometer, we used a home-built diode-laser-pumped nonlinear-polarization-rotation mode-locked Er:fiber ring laser with a central wavelength of 1574 nm as a light source. The laser system was similar to the one described in [27, 28], and the frep and the carrier-envelope-offset frequency (fceo) were fully stabilized to the microwave frequency reference synthesized from an Rb frequency standard. The frep was 53.4 MHz, and it could be varied precisely by using a piezoelectric transducer, and coarsely up to 1 MHz by using a fiber delay line integrated into the laser cavity. The fiber delay line consisted of a fiber collimator, a corner reflector, and a moving stage (Santec ODL-330, delay: 300 ps, insertion loss variation: <0.5 dB). During frep tuning, the mode-locking operation was maintained and the output power variation was below 1%.
The fiber comb output was spectrally filtered by using a band pass filter (BPF) (central wavelength: 1574 nm, bandwidth: 10 nm), to avoid nonlinear and dispersion effects in the optical fiber, and its output was divided into two optical fiber paths by an optical fiber coupler. The optical path difference (ngl) was preset to a multiple of ngL = 5.6 m, i.e., ngl = 168 m and 342 m, which corresponded to m = 30 and 61. In the reference path, an acousto-optic modulator (AOM) was introduced for two purposes: 1) to stabilize the reference path as is described later, and 2) to shift the comb mode to 77 MHz for heterodyne interferometric detection [29]. To minimize the effect of pulse deformation, in addition to the mentioned BPF that was used to shape and narrow the spectrum, the reference path consisted of a polarization-maintaining fiber (PMF) and a dispersion compensation fiber (DCF) for net dispersion compensation. In future applications for distance measurements, further consideration of pulse deformation, which could arise for the measurement path, might be needed. The interference signal between the measurement and reference pulses was detected by PD1.
In the case of the monitor interferometer, we used a single frequency CW laser with a central wavelength of 1560 nm (PLANEX, RIO) to stabilize the reference path fluctuation. The optical frequency of the CW laser was also stabilized to the Rb clock by using another frequency comb (not shown in the figure) and shifted to 77 MHz by the AOM, and the phase fluctuation in the reference path was detected as intensity change in the signal of the monitor interferometer by PD2. The detected signal was used as the error signal and was fed back to the AOM via a servo-loop, and the AOM shifted the optical phase by using diffraction, following which the reference path was stabilized [21–26]. The CW laser interference signal followed the phase refractive index np of the medium. In the main interferometer, the detected pulse-to-pulse interference signals also followed np. Thus, the stabilization method with the CW laser is effective even for the long-term fluctuation of the interference signal. The signals of two interferometers from PD1 and PD2 were sent to the signal port of the lock-in amplifier 1 and 2 (SR844, Stanford Research Systems Inc.), and they were converted into the phase signals, ϕmea and ϕref.
3. Evaluation of fiber noise cancellation
Figure 4 shows the error signal of the servo control for the fiber noise cancellation in the monitor interferometer. When the fiber noise cancellation was conducted at the time of 10 s, the observed error signal was stabilized to a constant with a standard deviation of 0.01 V, indicating that the servo control was successful.
Fig. 4 The error signal of the servo control for the fiber noise cancellation. The error signal is the variation in the intensity of the monitor interference signal generated by the CW laser. The transitions between −11 V and 11 V correspond to a half wavelength change in the optical path.
In order to evaluate the performance of the fiber noise cancellation, we tested the short-term stability of the system with the reference path lengths set to 168 m and 342 m, respectively. Figure 5 shows the measured stability for the two interference signals, i.e., those from the monitor interferometer (black curve, ϕref) and those from the main interferometer (red curve, ϕmea), when the fiber noise cancellation was conducted to the monitor interferometer. In the case of the monitor interferometer, ϕref indicates the residual instability of the interference signal, which is mainly owing to the reference path fluctuation, because the fiber length for the other path is much shorter (0.56 m). The observed result is very stable, with a standard deviation of the residual fluctuation on the 1.0 nm level, corresponding to the order of 10−12 for each reference path length, e.g., 168 m and 342 m, respectively. Because the residual stability does not depend on the reference path length, it is suggested that the residual fluctuations reflect the stability of the fiber noise cancellation system, which is small enough to realize nm-precision length measurements. On the other hand, in the case of the main interferometer, the observed ϕmea exhibits fluctuations with standard deviations of 15.7nm [Fig. 5(a)] and 11.0 nm [Fig. 5(b)], respectively. Although the signals contain the residual fluctuation, the interference signals were precisely detected regardless of the long fiber-based reference path.
Fig. 5 Optical path length variation with fiber noise cancellation performed with a 168 m (a) and with a 342 m (b) reference optical path. ϕref is the error signal corresponding to the reference path stability (black), and ϕmea is the interferometric signal corresponding to the residual optical length fluctuation in the measurement path (red). The vertical origins of plots ϕref and ϕmea are shifted for clarity.
In the case of the main interferometer, the residual variation in the ϕmea could be caused by the variations in the optical path, which were attributed to the ambient temperature variation. In order to estimate the effect of temperature change, we measured the ambient temperature in the experimental setup and interference signal simultaneously, when the temperature change was rather large. In this experiment, the entire setup, except the measurement fiber path, was covered by a thermal insulator. Figure 6(a) shows the measured optical path length variation corresponding to the residual variation of ϕmea when the reference path of 342 m was stabilized. The data shows an average drift of −1.7 nm/s, similar to the temperature drift of −0.09 mK/s around the experimental setup [Fig. 6(b)]. In this case, by setting the optical length of the measurement fiber to 0.62 m, and the temperature coefficient of refractive index dn/dT of silica fiber to 10−5, we estimated the corresponding temperature drift as −0.40 mK/s. This result is on the same order of magnitude as the observed temperature drift. Though the two values are not same, the result suggests the applicability of the developed technique to detect small temperature variation with millikelvin-order resolution.
Fig. 6 (a) Variation of interferometric signal with stabilized 342-m-long reference path with fiber noise cancellation, showing the average drift of −1.7 nm/s. (b) Temperature change around the measurement path, showing the average drift of 0.09 mK/s.
4. Evaluation of multiplication effect
In order to evaluate the multiplication effect for the developed OFCI with a fiber-based reference, we determined the pulse number difference (m) by measuring the interferometric phase ϕmea changes when the frep change was applied. The evaluation was conducted for the optical path differences (ngl) of 168 m and 342 m, i.e., 30 and 61 times the value of ngL = 5.6 m, respectively. Figure 7 shows the effective optical path length variation (npΔl) for the interferometric fringe scanning, which was obtained by the measured interferometric phase ϕmea changes when Δfrep was swept to 2.0 Hz with a 0.2 Hz step by using a PZT actuator in the laser cavity. The fiber noise cancellation was applied during the measurements. By using linear regression on the data measured by changing ϕmea while sweeping Δfrep, we estimated m and the measurement uncertainty from the slope and its standard deviation. In this evaluation, m = 30.2 ± 0.3 (ngl = 168 m) and m = 60.8 ± 0.2 (ngl = 342 m) were obtained; thus, we determined m of 30 and 61 without ambiguity, within the measurement uncertainty. These obtained m values agreed with the multiples of ngL that were estimated from the actual path lengths. Although the Δfrep scanning range of 2.0 Hz was small compared with the frep of 53.4 MHz, m could be determined accurately owing to the high stability associated with the fiber noise cancellation technique. Therefore, the developed OFCI can be used to measure distances in terms of the m values. Moreover, we demonstrated 30 and 61 times multiplication effects precisely for long reference paths, which can be used to measure arbitrary distances, even to short measurement paths. Because the maximal Δfrep of our fiber-based frequency comb was 1 MHz, a wide scanning range of ngΔl = 6.4 m could be realized between the measurement and the reference pulses. Thus, at an arbitrary distance, pulses in the OFCI could be made to overlap only by using frep tuning without a mechanical stage, and the pulse-to-pulse interference signal could be measured precisely. For distances shorter than 157 m, the length of a fiber-based reference path was set to over 314 m in our case (frep = 53.4 MHz, Δfrep = 1 MHz). Thus, the optical path length difference ngl is always maintained above 157 m, and a multiplication effect for m above 28 is always achieved, and any pulses can be made to overlap only by using through frep tuning. On the other hand, for measurement paths longer than 157 m, the reference path is replaced with a short length fiber, and a multiplication effect for m above 28 is always achieved with a conventional long measurement-path interferometer. This implies that the developed OFCI with a long fiber-based reference path and fiber noise cancellation could demonstrate the efficacy of the developed technique for high-precision measurements of arbitrary distances.
Fig. 7 The effective optical path length variation npΔl for frep tuned with ngl = 168 m (black line) and ngl = 342 m (red line).
5. Conclusion
In conclusion, we have developed an optical frequency comb interferometer with a long fiber-based reference path and fiber noise cancellation technique, and we have demonstrated the efficacy of the interferometer for arbitrary distance measurement. The long fiber reference of 342 m was stabilized to a nm-level fluctuation by the fiber noise cancellation, and the interference signals were measured precisely. By employing this highly stabilized interferometer, temperature changes on the order of millikelvin were detected, and pulse number differences m of 30 and 61 between the measurement and reference pulses were determined without ambiguity within the measurement uncertainty. These obtained m values agree very well with the multiples of pulse separation that are estimated from the actual path lengths. Thus, the developed method is applicable to absolute distance measurement in terms of precise values of m. Such a good agreement was achieved owing to the high stability associated with the fiber noise cancellation. Because of a long fiber-based reference path with the fiber noise cancellation, multiplication effects of frep tuning can be used to measure arbitrary distances. Moreover, the multiplication effect of m = 61 with Δfrep = 1 MHz corresponds to a wide scanning range of 6.4 m, which is more than the pulse separation. Therefore, pulses can be overlapped without a mechanical moving stage, and high-accuracy interferometric measurements of arbitrary distances can be realized.
Acknowledgments
We would thank Drs. H. Inaba, K. Hosaka, and S. Okubo of AIST for their helpful discussion on developing the fiber noise cancellation system. We also thank K. Miyano of UEC for his help with construction of the experimental setup. This work was supported by the Japan Science and Technology Agency (JST) through the ERATO Intelligent Optical Synthesizer (IOS) Project and partially by JSPS KAKENHI, Grant Number 25286076.
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