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Abstract
In this paper, we propose a method aiming to measure the absolute distance via the slope of the inter-mode beat phase by sweeping the repetition frequency of the frequency comb. The presented approach breaks the inertial thinking of the extremely stable comb spacing, and the bulky phase-locking circuit of the repetition frequency is not required. In particular, the non-ambiguity range can be expanded to be infinite. To verify the performance of presented method, a series of distance experiments have been devised in different scenarios. Compared with the reference values, the experimental results show the differences within 25 µm at 65 m range in the laboratory, and within 100 µm at 219 m range out of the lab.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
During the past decades, optical frequency combs (OFCs), honored by the Nobel Prize, have attracted the attention of the whole world [1–3], and have seen a variety of applications as a versatile tool, e.g., frequency metrology [4,5], precision spectroscopy [6,7], time/frequency transfer [8,9], astrocombs [10,11], and absolute distance measurement [12–14], etc. In the optical domain, OFC is composed of a series of equally-spaced frequency modes covering across a board spectral band, and these coherent lines can be expressed as n×frep+fceo clearly, where n is an integer, frep is the comb spacing, and fceo is the carrier-envelope-offset frequency. As long as frep and fceo are tightly locked to a RF frequency reference by a phase-locking loop, the bridge is constructed between the optical frequency and the RF frequency, and all the spectral lines will share the same frequency stability as the external reference. This inherent advantage has given rise to the substantial progress in the field of absolute distance measurement, which is of fundamental importance in both science and technology.
In the time domain, OFC is in the form of a pulse train with stable time interval, and the pulse period can be expressed simply as 1/frep. By means of the time-domain characteristics of frequency comb, researchers have made great progress during the past years. The way how to extract the accurate distance cues is the key point, and the measuring schemes including the pulse-to-pulse alignment [15], peak extraction [16], intensity detection [17] and the slope of the unwrapped phase [18] are well analyzed and conducted. However, the scanning of mechanical stage is indispensable to shift one pulse with regard to the other, to generate the interferograms. This long moveable component directly results in that the optical alignment in the free space is not easy, and the measuring speed is low. To overcome the above defects, researchers proposed an ingenious method called dual-comb ranging, which relies on two mode-locked lasers with slightly different repetition frequencies, and the pulses can automatically scan each other at a rapid pace to generate the interferograms. The limitation is that two well-stabilized mode-locked lasers are needed, making the system expensive and bulky [19–21]. In the frequency domain, methods of dispersive interferometry [22–24], chirped pulse interferometry, multi-heterodyne interferometry [25,26], and multi-wavelength interferometry [27,28] are proposed from diverse perspectives. Dispersive interferometry and chirped pulse interferometry can measure the distances via the appropriate processing of the spectral interferograms. Nevertheless, the measurement performances are limited by the resolution of the optical spectrum analyzer. The phases of the inter-mode beat can be used to determine the distances, and in general the high harmonic is needed to finely measure the distances with a smaller synthetic wavelength [29,30].
The frequency comb offers an intriguing solution to achieve the higher accuracy and the greater non-ambiguity range simultaneously, and of which the attraction of greater measuring range has prompted various efforts to improve the measuring scheme. In particular, the non-ambiguity range can be significantly extended to kilometer level by slightly changing the repetition frequency of the signal laser, for all the methods mentioned above (with one single comb or dual combs) in the time and frequency domains. However, in this case, the additional measurement step of changing the repetition frequency is required to unequally determine the distances. In this work, we propose a method enabling long distance measurement by using one single laser frequency comb. The distance can be measured by the phase slope of the inter-mode beats, while the repetition frequency is swept linearly. Compared with the general pulsed laser, the dynamic optical frequency comb can be achieved by linearly changing the cavity length, and the repetition rate can be controlled precisely through a high-precision linear stage, which greatly maintains the high coherence of frequency comb. Furthermore, the ultra-short pulse duration of optical frequency comb corresponds to a broadband spectrum, which can achieve the extension of the stable higher harmonics, and greatly improve the measurement accuracy. Surprisingly, the method can achieve arbitrary distance measurement in contrast to conventional time-of-flight methods, and the non-ambiguity range can be extended to be infinite in principle, which is very valuable to the large-scale measurement. The complex locking circuit for the stabilization of the repetition frequency is not needed any more, which can well satisfy the field-base experiments. The experimental results show that our method can measure the distances with high accuracy and precision.
2. Measuring principle based on dynamic comb spacing
Theoretically, the measuring system is devised as shown in Fig. 1. The output of the laser source is divided into three channels. One is the monitoring channel (green), which is used to monitor the real-time repetition frequency. One is the reference channel (blue), and the third one is the measurement channel (black). The repetition frequency of the laser source is linearly swept, and accordingly the phase difference between the measurement and reference channels would change linearly. The different distance L will correspond to a different phase slope, which means that the distance can be measured.
2.1 Dynamic comb spacing based on fundamental inter-mode beat
The phases of the fundamental inter-mode beat for the reference and measurement channels (i.e., ϕref and ϕmeas) can be expressed respectively as:
where λ is the pulse-to-pulse interval, λ=c/(Ng·frep), c is the light speed in vacuum, Ng is the group refractive index, r is the length of the reference channel, and L is the measured distance. Equation (2) can be rewritten as: Hence, the phase difference ϕ can be calculated via Eq. (1) and Eq. (3) as: With the sweeping of frep, the difference of phase ϕ will generate the linear modulation. Therefore, the measured distance L can be calculated from the slope of phase change as: The phase difference can be greater than 2π as the measured distance increased, and it will cause the nominal ambiguous range for distance measurement. Hence, the non-ambiguity range LNAR needs to satisfy the following condition as: where frep1 and frep2 are adjacent values in the process of repetition frequency variation, and LNAR can be collated as: where Δfrep is the difference of contiguous change for the comb spacing, it also can be considered as a resolution of variation. We can observe that Δfrep is the decisive factor of non-ambiguity range from the above formula. Fortunately, the Δfrep could be tuned by external trigger of the signal generator. In theory, the Δfrep can be adjusted to infinitesimal if the trigger frequency ft is high enough.2.2 Dynamic comb spacing based on higher harmonic
From the Eq. (5), with the sweeping of repetition frequency, the small phase variation will be the challenge we may encounter. In response to this problem, we utilize the higher harmonic as an “amplification factor” acting on the phase slope. Then, the phase difference ϕm between the reference and measurement channels can be redefined by:
where m is an integer (i.e., the order of the higher harmonic). The slope of phase change can be derived as: Consequently, the measured distance L can be calculated as:3. Demonstration of the Yb-doped mode-locked laser
The mode-locked laser is generated based on the mechanism of nonlinear polarization rotation (NPR), using Yb-doped gain fiber. The high-gain efficient of the Yb-doped fiber makes the system easy to achieve the mode-locked state, and can maintain the high output power. The laser configuration is shown in Fig. 2. The fiber-ring cavity consists of the Yb-doped fiber (25 cm), single mode fiber, wavelength division multiplexing WDM, two collimators, optical isolator (Thorlabs IO-5-1030-VLP), and the NPR module. One collimator is installed at a mechanical stage (50 mm moving range), which is used to tune the repetition frequency slowly and largely. Part of the single mode fiber is stuck on a piezoelectric transducer (Thorlabs PK2FSP2), so that the repetition frequency can be fast and finely adjusted. In our experiments, the repetition frequency is about 100 MHz, with tuning range of about 1 MHz. The output power of the laser source is about 100 mW.
Fig. 2. Schematic of homemade frequency comb. LD, laser diode; WDM, wavelength division multiplexer; PZT, piezoelectric transducer; QWP, quarter-wave plate; HWP, half-wave plate; PBS, polarization beam splitter; ISO, isolator.
Figure 3(a) shows the generated pulse train, with the period of 9.65 ns. The spectrum of the laser source is indicated in Fig. 3(b). We find that, the center wavelength is about 1030 nm, with about 15 nm spectral width (FWHM). The repetition frequency can be locked to the Rb clock by using the phase-locking loop [14]. Figure 4(a) shows the scatters of the stabilized repetition frequency for 5 hours, and the standard deviation is 0.2 mHz. The Allan deviation, shown in Fig. 4(b), can achieve 1.8 × 10−12 at 1 s, and 5.9 × 10−14 at 1000-s averaging time.
Fig. 4. (a) Repetition rate variation. (b) Allan deviation of the relative repetition rate. The counter (53230A, Keysight) is synchronized to the Rb clock, whose gate time is 0.1 s.
As shown in Fig. 5, the input voltage to the PZT (piezoelectric transducer) has a minor periodic fluctuation, which leads to a nonlinear phase error. The actual level of fluctuation of input voltage is 8.9 mV, and the standard deviation is 1.8 mV. The displacement of PZT (Thorlabs, PK4GA7P2) is lower than 0.67 µm/V at 25 °C. Hence, the impact on the cavity length is less than 5.9 nm, and the destabilization of repetition rate Dfrep is 0.15 Hz with regard to the input voltage of PZT. When the measured distance L is 65 m, the nonlinear phase error ϕerror can be calculated as:
where c is the light speed in vacuum, Ng is the air group refractive index. The level of periodic nonlinear phase error is below 4.0 × 10−7 rad, the standard deviation is 8.1 × 10−8 rad and the resulting error for the distance measurement is 9.4 × 10−8 m. We could find that the repetition rate fluctuation in Fig. 4 (1 mHz) is much less than the predicted value and consistent with the frequency stability of Rb clock. Due to the excessively marginal impact, the nonlinear phase error with respect to the input voltage can be ignored.Fig. 5. Nonlinear phase error with the input voltage to the PZT. The left Y axis is the input voltage to the PZT, and the right Y axis is the nonlinear phase error.
4. Long distance measurement using phase slope of the inter-mode beat
The experimental setup is shown in Fig. 6. The output of the laser source is split into three channels, as described in Sec. 2. To meet the requirements of ultralong distance experiments, the laser is amplified by Yb-doped fiber amplifier YDFA. Corresponding to the principle of the method, three photodetectors (MenloSystems FPD310-F) are used to measure the repetition frequency, the phase of the reference arm, and the phase of the measurement arm, respectively. As depicted in Fig. 6, two frequency counters (Keysight 53230A), which are synchronized by a signal generator (Tektronix AFG31000), are used as a tool to record repetition rates and phase signals. In the process of data acquisition, the measuring system shares the common time base of the rubidium clock (Microsemi 8040) as an external reference.
As shown in Fig. 6, the measurement and reference signals pass through a low-pass filter with a cut-off frequency of 100 MHz. After being amplified by the amplifier, they are connected to the frequency counter, and the fundamental frequency is electronically obtained. For the way of determining high inter-mode beats, the measurement and reference signals pass through a band-pass filter with a desired frequency band. After being amplified by the amplifier, they are connected to a mixer meeting the bandwidth requirement, and are mixed with the reference signal to obtain a low-frequency signal that retains the original phase difference information. Considering the change of the mixing signal caused by dynamic higher harmonics, a low-pass filter with a cut-off frequency of 5 MHz is subsequently connected, and finally they are input into the frequency counter after the secondary amplification of the amplifier.
To acquire the sweeping repetition rate accurately, both the frequency counter and the sampling signals output by the signal generator use the Rb clock as an external reference. At the same time, the frequency counter, which detects the phase change, is also locked to the same external reference.
4.1 Absolute distance measurements on the long optical rail
Based on the principle of approach, we performed a series of experiments on the 80 m long optical rail, at National Institute of Metrology, which has the stable environment conditions (T: 24.08 °C, P: 991.93 hPa, RH: 43.5%), and the group refractive index is 1.0002624 based on the Ciddor formula [31]. The experimental photograph is shown in the Fig. 7.
The incremental distance meter (Agilent 5519B) is used as the reference, and a series of experiments have been devised for different measured distances with 5 m step. In preparation for the measurement, the measurement arm is ensured to be parallel to the optical rail. By means of the long optical rail, the distance measurements have been performed, covering 65 m. With the dynamic comb spacing of fundamental frequency, the original data of phase changes are illustrated in Fig. 8.
Fig. 8. Experimental results for the fundamental frequency. (a) Sweeping of repetition rate for the experiment. (b) Measured data of unwrapped phase corresponding to 5 m, 35 m, 65 m.
Figure 8 reveals that, the experimental results show a good agreement with theoretical derivation. We could observe the phase slope increases continuously with the increase of measured distance. It implies the measured distance has certain correspondence with the phase slope as the theoretical derivation. As shown in Fig. 8, the phase change is more distinct with the increase of measured distance. Nevertheless, the phase jitter is the problem we must solve for the calculation of phase slope. Due to the environmental disturbances, photoelectric conversion and the interference from electronics, etc., the acquired phase signal accompanying phase jitter is unavoidable. By means of the verification of original phase data from repeated experiments, the actual level of phase jitter is 8.6 mrad, and the standard deviation attains 6.7 mrad.
Based on the above factors, the acquired phase signals will generate phase jitter caused by random errors and non-linear periodic errors. The phase jitter has the characteristics of high frequency, hence, the frequency of phase jitter will be much higher than the frequency of phase change generated by the dynamic comb spacing. Compared with traditional Fourier transform, The Empirical Mode Decomposition (EMD) can analyze the local time-frequency characteristics of the signal [32], instead of performing a global transformation on the signal to obtain the overall frequency spectrum. Based on the time scale characteristics of the data itself, the method can decompose the signal into several Intrinsic Mode Functions (IMFs), and reconstruct the phase signals from component IMFs (C1, C2, C3…), as shown in Fig. 9. Because of time-frequency localization characteristics of EMD, the method is suitable for non-stable signals. For the situation of phase signal instability, phase jitter can be solved by taking advantage of EMD. Due to the local characteristics of the phase change, the IMF (C12) that matches the phase change is approximately linear. In the subsequent processing, the linear change of phase is obtained by linear fitting. We fit phase slope at different regions of the phase variation and average them to measure the distance, thereby eliminating the effect of phase jitter.
Fig. 9. Data processing by using EMD. It is clear that the phase jitter is greatly suppressed after EMD.
Therefore, based on the phase slope of the inter-mode beat, the distances can be measured. In our experiments, each distance is fast measured for seven times. The gate time of the frequency counter is 1 ms, i.e., 1 kHz sampling rate. Compared with the reference distances, the residuals are shown in Fig. 10. Among that the cross marks represent the scatters of the measurements, and the solid mid points indicate the average of seven measurements, and the error bars show the standard deviation. We find, the residuals can be below 3 mm at 65 m range.
Next, we use the 9th harmonic of repetition frequency (about 932.4 MHz) to finely measure the distances. Correspondingly, the gate time of the frequency counter is updated to 0.11 ms, i.e., 9 kHz sampling rate. The phase changes that are caused by the dynamic comb spacing have been depicted in Fig. 11.
Fig. 11. Experimental results of the 9th harmonic. (a) 9th harmonic of repetition rate sweeping. (b) Experimental data of unwrapped phase change for 5 m, 35 m, 65 m based on the 9th harmonic.
In contrast to Fig. 8, Fig. 11 clearly indicates the fine step using the 9th harmonic of the repetition frequency shows more sensitive phase changes, which is actually magnified by 9 times. Each distance is measured for 21 times. The experimental results are depicted in Fig. 12. In the figure, the cross marks represent the scatters of the measurements, the solid mid points indicate the average of 21 measurements, and the error bars show the standard deviation. The residuals can be well below 25 µm at 65 m range, which is 3.8×10−7 in relative.
4.2 Long distance measurement on the base line out of the lab
To further validate the performance of the presented approach, and verify the practical application of frequency comb outside the lab, open-air experiments are performed on the base line. The experimental scheme is shown in Fig. 13.
Fig. 13. (a) Experimental photograph on the base line at 1:00 am. (b) Top view of the measurement path.
The experiments are performed at early morning with relatively stable environment (T: 20.06 °C, P: 988.28 hPa, RH: 47.4%) corresponding to the group refractive index of 1.0002651. We measure the distances at the initial position, 9, 27, 75, and 219 m, respectively. The phase changes at 219 m position for the fundamental and the 9th harmonic are indicated in Fig. 14, and the distance can be measured via the phase slope.
Fig. 14. Experimental results corresponding to 219 m. (a) Phase changes based on the fundamental frequency. (b) Phase changes based on the 9th harmonic.
Figure 15 shows the experimental results of distance measurement. The reference values are provided by National Institute of Metrology. In the step of coarse measurement using fundamental frequency, the residuals can be within 4 mm. In the case of fine measurement using 9th harmonic, the residuals can be greatly improved to be below 100 µm.
Fig. 15. Distance measurement results over a travel of 219 m. (a) Experimental results based on the fundamental frequency. (b) Experimental results based on the 9th harmonic.
We performed long-term measurements at the distance of 219 m, based on the fundamental frequency and the higher harmonic respectively. The scatters of residuals and Allan deviation are depicted in Fig. 16, and of which the data rates are 1 Hz and 5 Hz for the fundamental and the higher harmonic respectively. The standard deviations, for the fundamental and the higher harmonic, are 2.2 mm and 51.6 µm, respectively. The Allan deviation for the fundamental beat can reach 2.2 mm at 1 s, and 103.7 µm at 100 s averaging time. In the case of higher harmonic, the Allan deviation can achieve 20.4 µm at 1 s, and 5.6 µm at 100 s averaging time.
Fig. 16. Long-term experiments of the distance measurement at 219 m. (a) Measured residuals for fundamental frequency. (b) Measured residuals based on the higher harmonic. (c) Allan deviation of the distance measurement versus averaging time.
4.3 Uncertainty evaluation
The uncertainty of measured distance is a tool that we appraise our method. Based on Eq. (10), the main influence factors of measurement uncertainty are related to the slope of phase S=∂ϕ/∂frep and the group refractive index Ng, hence it can be calculated as:
Considering the above-mentioned factors in the open-air environment, the uncertainties of temperature, air pressure and humidity are 39.3 mK, 16.5 Pa and 5.1%, corresponding to the uncertainty of air refractive index of 3.6 × 10−8, 4.5 × 10−8, and 4.4 × 10−8, respectively. The combined uncertainty is 7.5 × 10−8. The uncertainty related to the phase slope is 1.68 × 10−8 rad/Hz (standard deviation) based on fundamental frequency, and the combined uncertainty can be expressed as [(4 mm)2 + (7.5 × 10−8·L)2]1/2. With the help of higher harmonic, the uncertainty is 3.7 × 10−11 rad/Hz (standard deviation) for the first term, and the combined uncertainty is [(99 µm)2 + (7.5 × 10−8·L)2]1/2.
5. Summary
Since the advent of optical frequency comb, researchers in many fields have been attracted by its charm. This paper proposes a simple method, and the distance can be measured via the phase slope of the inter-mode beats. It makes the method break the constraint of periodic ambiguity, and benefit simplification of measuring facilities. The introduction of higher harmonic can also ensure the measuring accuracy effectively. A series of long distance experiments are designed to verify the performances. Besides, the dynamic comb spacing makes an attempt to explore the application potential of frequency comb. The open-air experimental results demonstrate the method has the better practicability and the commercial value. For the further research, our work will inspire the superiority of frequency comb in the fields of ultralong absolute distance measurement and dynamic parameters detection, such as underwater object positioning, vibration measurement and ultra-fast dynamic object measurement.
Funding
National Natural Science Foundation of China (61505140); Beijing Aerospace Xinfeng Machinery Equipment Co., Ltd. Aerospace Testing Technology and Application Laboratory Innovation Fund (QT-2018-004); Natural Science Foundation of Tianjin City (18JCYBJC17100); National Basic Research Program of China (973 Program) (2016YFC1401203); Tianjin Municipal Education Commission (JWK1616).
Acknowledgments
We thank National Institute of Metrology for the technical support.
Disclosures
The authors declare no conflicts of interest.
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