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Precision and repeatability improvement in frequency-modulated continuous-wave velocity measurement based on the splitting of beat frequency signals

Xingrui Cheng, Junchen Liu, Linhua Jia, Fumin Zhang, and Xinghua Qu

Open Access Open Access

Abstract

The basic principle of frequency-modulated continuous-wave lidars is to measure the velocity of a moving object through the Doppler frequency shift phenomenon. However, the vibration generated by the moving object will cause the spectrum to broaden and the precision and repeatability of speed measurement to decrease. In this paper, we propose a speed measurement method based on H13C14N gas cell absorption peak splitting the sweep signal of a large bandwidth triangular wave modulated frequency laser. This method obtains the speed of a continuously moving target by re-splicing an accurately-split frequency sweep signal, which effectively solves the problem of simultaneous processing of excessive amounts of data when measuring the speed of a continuously moving target. At the same time, the H13C14N gas cell absorbs the spectra of specific wavelengths, which reduces the phase delay of the beat signal corresponding to the up- and down-scanning, thus reducing the signal spectrum broadening caused by frequency deviation, and improving the speed measurement resolution and range effectively. The experimental results show that for speeds of up to 30mm/s, the mean error was less than 23µm/s and the mean standard deviation was less than 61µm/s.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

In recent years, frequency-modulated continuous-wave (FMCW) lidar systems have been widely used in large-size measurement, speed measurement, automatic driving systems, etc. [1,2]. The common frequency modulation methods of FMCW lidars include triangle wave, sawtooth wave, coded frequency modulation and so on. FMCW lidars can measure distance and velocity directly and the measurement accuracy is established through a calibration process [3–6]. FMCW lidars have the advantages of large bandwidth, high resolution, and large time-bandwidth product, so high peak power is not a requirement for measurement [7]. FMCW lidar systems emit a continuous wave with a continuously changing frequency within the set sweep frequency range. The emitted continuous wave is called the local wave signal, which is reflected back by a corner prism; the continuous wave reflected back is called the echo signal, which is received by the collimator. Therefore, the local wave and the echo signals pass through a coupler to generate a beat frequency signal, the frequency of which is related to the speed and distance measured. Triangular-wave frequency-modulated lidars can not only measure distance, but also generate Doppler beat signals that can be measured directly.

Based on the above characteristics, many researchers have used FMCW lidars to measure the vibration or velocity of objects [8,9]. As early as 2001, Richard Schneider et al. [10] produced two opposite frequency-scanning lasers to measure the length of an object in industrial environments. In 2017, Seiichi Kakuma [11] used a servo control system based on a phase-locked loop to stably control a pair of vertical cavity surface emitting lasers with opposite sweep directions. Finally, the sum and difference frequency waveforms were obtained, and the radial movement speed and absolute length of the moving block gauge were measured. The experimental results showed that the method can measure radial velocities of 70.83µm/s, with a velocity resolution of 4.7µm/s, and a distance resolution of 208µm. After that, many speed measurement and frequency modulation lidar systems based on the principle of opposite frequency scanning directions were proposed.

However, the optical paths of the above methods are complicated and the simultaneity of the two sets of laser signals cannot be guaranteed. Therefore, many researchers have used a single laser source to achieve geometric measurement. In 2015, Juan Jose Martinez et al. [12] proposed to use the semiconductor optical amplifier medium to produce the Kerr effect and generate a second set of frequency sweep sources that are in a reverse mirror image relationship with the original light source. Using this approach, authors maintained simultaneity with the original frequency sweep light source, and obtained a lidar system with two opposite frequency-scanning directions. In 2020, L Yi [13] used a single triangular wave sweep laser to obtain two opposite frequency-scanning directions and measured the distance and radial velocity of the corner prism. The average error for velocity measurements of 10mm/s and below did not exceed 52µm, while the relative standard deviation did not exceed 2.5%. However, due to the influence of the vibration of the moving object itself, the frequency spectrum was severely broadened and so velocities above 10mm/s were not measured.

The above measurement methods mentioned all used the Doppler frequency shift phenomenon produced by moving objects to measure speed, but faced the issues that the measurement speed was too low or the repeatability was not good. In this paper, we use a single FMCW laser source and a triangular wave frequency modulation method, with a sweep period of 20nm and the focus is on measuring the speed of the corner prism. In addition, in order to reduce the impact of the frequency deviation caused by the movement of the object on the measurement accuracy and repeatability, we use the absorption peak signal of an H13C14N gas cell to split the beat signal generated by the up scanning and the down scanning. According to the absorption characteristics of the H13C14N gas cell at a specific wavelength, 18 sets of precisely matched two-way beat signals were generated. The experimental results show that this method reduces the impact of frequency deviation and significantly improves the measurement repeatability. Finally, we introduce a section of a Mach-Zehnder Interferometer (MZI) optical path to perform equal frequency resampling on the beat frequency signal of the measurement path, which further eliminates the nonlinearity of the FMCW lidar source. The FMCW lidar system designed in this paper is based on the characteristics of the absorption peak of the H13C14N gas cell. The optical path of the measurement system is established using optical fiber, which can reduce the impact of environmental factors on the measurement.

2. Background theory

2.1 Dual-path FMCW lidar systems

The frequency modulation mode of FMCW laser sources is triangular wave modulation. When the laser modulation remains linear, its instantaneous frequency is expressed as:

$$f(t) = {f_0} + \alpha t \qquad 0 < t < \frac{T}{2}$$
where ${f_0}$ is the initial frequency of laser modulation, $\alpha$ is the frequency modulation rate of the laser, and $T$ is the laser modulation period. In the measurement optical path, 10% of the laser light is taken as the original wave, which is expressed as:
$$E(t) = {E_0}\exp [{j\varphi (t)} ]$$
After the remaining 90% of the laser light passes through a circulator, it is emitted from a collimator and is reflected by a corner prism to form an echo signal. The echo signal is expressed as:
$$E(t + {\tau _m}) = {E_0}\exp [{j\varphi (t + {\tau_m})} ]$$
where ${E_0}$ and $\varphi (t)$ are the laser’s amplitude and instantaneous phase, and ${\tau _m}$ is the time delay related to measured distance. The two lasers pass through the 50/50 laser coupler and produce mixing interference on the photodetector. $\sigma$ represents the sensitivity coefficient of the photodetector. The electrical signal measured by channel 1 of the oscilloscope is expressed as:
$$U(t) = \sigma {|{E(t + {\tau_m}) + E(t)} |^2} = 2\sigma {|{{E_0}} |^2}\{{1 + \cos [{\varphi (t + {\tau_m}) - \varphi (t)} ]} \}$$
The instantaneous phase of the laser light source is the ${n^{th}}$ derivative with respect to time:
$$\varphi (t + \tau ) = \sum\limits_{n = 0}^\infty {\frac{{{\tau ^n}}}{{n!}}{\varphi ^{(n)}}(t)}$$
Since the photodetector is insensitive to very high frequencies, Eq. (5) only needs to be expanded to the first order, that is as $\varphi (t + \tau ) \approx \varphi (t) + 2\pi f(t){\tau _m}$. Expressing $2\sigma {|{{E_0}} |^2}$ as ${U_{m0}}$, the electrical signal after removing the DC component becomes:
$${U_m}(t) = {U_{m0}}\cos [{2\pi f(t){\tau_m}} ]= {U_{m0}}\cos [{2\pi \alpha {\tau_m}t + 2\pi {\tau_m}{f_0}} ]$$
Using Eq. (6), the beat frequency ${f_b}$ is equal to $\alpha {\tau _m}$, so the distance to be measured can be expressed as:
$$R = \frac{{c{f_b}}}{{2\alpha n}}$$
In actual measurements, the continuous laser light generated by the FMCW laser source is nonlinear. The nonlinear term in the laser modulation rate is ${v_\varepsilon }(t)$ and can be expressed as ${v_\varepsilon }(t) = \sum\nolimits_{i = 2}^M {({{{\alpha _i}} / \alpha }) \cdot {t^i}}$, where ${\alpha _i}$ is the non-linear frequency modulation rate. In practical applications, $f(t )= {f_0} + \alpha \varphi ^{\prime}(t )$, with $\varphi ^{\prime}(t) = t + {v_\varepsilon }(t )$. The beat signal is then expressed as:
$${U_m}(t )= {U_{m0}}\cos [{2\pi \alpha {\tau_m}\varphi^{\prime}(t) + 2\pi {\tau_m}{f_0}} ]$$
In order to eliminate the error caused by the nonlinearity of the laser light source, we adopt the method of equal frequency interval resampling. As shown in Fig. 1, a section of an MZI optical path is added as the reference optical path. The reference signal optical path uses a long optical fiber with a known length to produce a time delay ${\tau _r}$. The actual correction signal is expressed as:
$${U_r}(t) = {U_{r0}}\cos [{2\pi \alpha {\tau_r}\varphi^{\prime}(t) + 2\pi {\tau_r}{f_0}} ]$$
The same light source illuminates both the correction and the measurement signal path, so the initial phases of the two paths are the same. We then extract the position points corresponding to the peak and trough of the reference signal to resample the measurement signal. The resampling frequency is expressed as:
$$F_s^{\prime} = \frac{1}{{\Delta \varphi }} = \frac{1}{{{\varphi _{k + 1}}(t) - {\varphi _k}(t)}} = 2\alpha {\tau _r} \quad k = 1,2,3\ldots \ldots $$

 figure: Fig. 1.

Fig. 1. Schematic of the experimental setup. The triangular wave-modulated continuous laser light is divided into three parts by the optical splitter. From top to bottom, they are the measurement optical path, the H13C14N gas cell optical path and the reference optical path. The corner prism is fixed on a CMM, relative to which the radial velocity of the collimator lens is measured. CIR: circulator; CM: collimating mirror; CMM: Coordinate measuring machine; PD: photodetector. (a) Local wave signal after the power amplifier. (b) Local wave signal entering the non-cooperative space through the collimator, and reflected back to the collimator by the corner prism to form an echo signal. (c) Beat signal of the measurement optical path. (d) Absorption peak signal of gas cell. (e) Beat signal of reference optical path.

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After equispaced-phase resampling, the measurement beat signal is given by:

$$\begin{aligned} {U_r}(t) &= {U_{r0}}\cos [2\pi \alpha {\tau _m}k\Delta \varphi + 2\pi {\tau _m}{f_0}]\\ &= {U_{r0}}\cos [2\pi \alpha {\tau _m}\frac{k}{{2\alpha {\tau _r}}} + 2\pi {\tau _m}{f_0}] \quad k = 1,2,3\ldots \ldots \\ &= {U_{r0}}\cos [\pi \frac{{{\tau _m}}}{{{\tau _r}}}k + 2\pi {\tau _m}{f_0}] \end{aligned}$$
Finally, we apply a fast Fourier transform (FFT) on the resampled measurement signal to calculate the absolute distance to the target. if ${\tau _r}$ is the delay of the correction signal, $N^{\prime}$ is the number of sampling points after re-sampling, and $k^{\prime}$ is the number of re-sampling times corresponding to the peak after the FFT, the frequency corresponding to the peak of the spectrum signal is expressed as:
$${f_b} = {F_s}^{\prime}{{k^{\prime}} / {N^{\prime}}} = 2\alpha {\tau _r}{{k^{\prime}} / {N^{\prime}}}$$
Due to the characteristics of the triangular wave modulation, the distance to the target to be measured is finally expressed as:
$$R = {{c{\tau _m}} / 2} = k^{\prime}{{c{\tau _r}} / {N^{\prime}}}$$

2.2 Principle of velocity measurement

When measuring the distance to a stationary object, the phase expression of the beat signal generated by the interference between the measuring light and the local oscillator light is:

$$\varphi (t) = 2\pi ({{f_0}\tau - 0.5{\alpha_0}{\tau^2} + {\alpha_0}t\tau } )\quad \tau \le t \le \frac{{{T_m}}}{2}$$
where ${T_m}$ represents the modulation period of the triangular wave. When measuring the speed of an object moving at a speed $v$, then the expression of the time delay corresponding to the distance to be measured is expressed as:
$$\tau (t) = \frac{{2({{R_0} - vt} )}}{c} = {\tau _0} - \frac{{2vt}}{c}$$
where ${R_0}$ is the distance to the target to be measured at the initial moment of a single frequency sweep period, and ${\tau _0}$ is the time delay corresponding to that distance. Then, the phase of the measured beat signal is expressed as:
$${\varphi _v}(t) = 2\pi ({f_0}{\tau _0} - \frac{{2vt{f_0}}}{c} + {\alpha _0}t\tau - \frac{{2v{\alpha _0}{t^2}}}{c}) \quad \tau \le t \le \frac{{{T_m}}}{2}$$
According to Eq. (16), the frequency change of the measured beat signal in a single scan period can be expressed as:
$$\Delta f = \frac{{4v{\alpha _0}}}{c}(\frac{{{T_m}}}{2} - \tau ) = \frac{{vB}}{c}(4 - \frac{{16R}}{{c{T_m}}})$$
where $R$ is the instantaneous distance value of the target to be measured at the initial moment of a single frequency sweep cycle, and $B$ is the frequency modulation range. Frequency deviation is the main factor that affects speed and distance measurement resolution. According to Eq. (17), in order to reduce the spectrum broadening, the frequency modulation range of the tunable laser needs to be small. To achieve this, we used a gas cell to split the beat signal and obtain the smallest possible frequency modulation range, which reduced the influence of frequency deviation on speed and distance resolution.

We perform low-pass filtering on the gas cell signal, and then initially locate the absorption peaks of the gas cell. Then, a part of the absorption peak signal near the absorption peak is selected for quadratic function fitting according to the preliminary positioning result. Finally, an accurate absorption peak position is obtained. In the comparison of the two absorption peak positions in Fig. 2, the deviation of the absorption peak position relative to an entire absorption signal is not more than $0.025\%$. Since the beat signals and H13C14N gas cell absorption peak signal were generated by the optical splitter at the same time. There is no phase shift between the two signals, which ensures the accuracy of signal splitting. Therefore, the beat signal can be accurately separated in the frequency domain according to the position of the absorption peak of the gas cell.

 figure: Fig. 2.

Fig. 2. Schematic diagram of the positioning of the absorption peak of the gas cell signal. Purple is the gas cell signal collected by the PD, green is the signal after low-pass filtering, and blue is the signal fitted by the quadratic function. The black and red asterisks indicate the position of the absorption peak before and after filtering, respectively.

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The time delay $\tau ^{\prime}$ and center wavelength $\lambda $ between each group of two-way scanning signals can be accurately obtained using the wavelength corresponding to the absorption peak of the gas cell. When the scanning direction of the FMCW is changed, the beat frequency signal generated is unstable. Therefore, in this study we exclude the front and back parts of the scanning range, and finally 18 sets of two-way scanning beat frequency signals are obtained.

When the object is moving at a constant speed, the generated beat signal is shown in Fig. 3(c), and is related to the speed of the object and the scanning direction of the FMCW signal. Objects moving at a constant speed will cause a Doppler frequency shift given by:

$${f_d} = \frac{{2v}}{\lambda }$$
where $v$ represents the speed of the object to be measured and $\lambda $ represents the laser wavelength. In our experiment, the total frequency sweep range was 20nm, and the center wavelength was determined by the absorption peak of the gas cell. The sweep frequency range is much smaller than the center wavelength, so $\lambda $ is the center wavelength. Due to the single-laser frequency sweep measurement, the up- and down-scanning beat frequency signals have a time delay $\tau ^{\prime}$.

 figure: Fig. 3.

Fig. 3. Schematic diagram of gas cell splitting beat signal. (b) Absorption peaks of the gas cell signal corresponding to the up- and down-sweep frequencies are symmetrical to each other. (c) Absorption peak of the gas cell splitting the FMCW signals of different frequency ranges in the frequency domain. (d) According to the wavelength corresponding to the absorption peak of the gas cell, the beat signals corresponding to the up- and down-scanning during a frequency sweep period are divided into 18 groups, and $\Delta f$ represents the frequency corresponding to the adjacent absorption peak of the gas cell.

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According to Fig. 3(c), as the scanning frequency rises (i.e., up-scanning), the frequency of measurement beat signal changes to:

$${f_{up}} = \alpha {\tau _m} + \frac{{2v}}{\lambda }$$
When the scanning frequency drops (i.e., down-scanning), the scanning rate is $- \alpha $.The time difference between the two scans is $\Delta t$. The frequency of the measurement beat signal changes to:
$${f_{down}} ={-} \alpha ({\tau _m}\textrm{ + }\tau ^{\prime}) + \frac{{2v}}{\lambda } ={-} \alpha ({\tau _m}\textrm{ + }\frac{{2v\Delta t}}{c}) + \frac{{2v}}{\lambda }$$
After resampling, the up- and down-scanning measurement beat signals can be written as:
$$f{^{\prime}_{up}} = \alpha ({\tau _m} + \frac{{2v}}{\lambda }) \times \frac{1}{{2\alpha {\tau _r}}} = \frac{{{\tau _m}}}{{2{\tau _r}}} + \frac{v}{{\lambda \alpha {\tau _r}}}$$
$$f{^{\prime}_{down}} = [ - \alpha ({\tau _m} + \frac{{2v\Delta t}}{c}) + \frac{{2v}}{\lambda }] \times \frac{1}{{2( - \alpha ){\tau _r}}} = \frac{{{\tau _m}}}{{2{\tau _r}}} + \frac{{v\Delta t}}{{c{\tau _r}}} - \frac{v}{{\lambda \alpha {\tau _r}}}$$
Combining the above two equations, the speed and distance of the object to be measured moving in a uniform radial direction can be expressed as:
$$v = \frac{{(f{^{\prime}_{up}} - f{^{\prime}_{down}})\lambda \alpha {\tau _r}}}{{2 - \frac{{\Delta t}}{c}\lambda \alpha }} = \frac{{\frac{{K{^{\prime}_{up}} - K{^{\prime}_{down}}}}{{N^{\prime}}}\lambda \alpha {\tau _r}}}{{2 - \frac{{\Delta t}}{c}\lambda \alpha }}$$
$$R = \frac{{2c}}{{{n_r}}}{\tau _m} = \frac{{2c}}{{{n_r}}}[(f{^{\prime}_{up}} + f{^{\prime}_{down}}){\tau _r} - \frac{{v\Delta t}}{c}] = \frac{{2c}}{{{n_r}}}[(\frac{{K{^{\prime}_{up}} + K{^{\prime}_{down}}}}{{N^{\prime}}}){\tau _r} - \frac{{v\Delta t}}{c}]$$
$f{^{\prime}_{up}}$ and $f{^{\prime}_{down}}$ are obtained by performing FFT on the measurement beat signal, where $K{^{\prime}_{up}}$ and $K{^{\prime}_{down}}$ are the peak positions of the up-scanning beat signal and down-scanning signals after the FFT, respectively, and $N^{\prime}$ is the total number of sampling points.

3. Experiments and results

The experiment was carried out under constant temperature, humidity and pressure conditions to reduce the impact of dispersion. As shown in Fig. 4, the movement speed of the object to be measured was controlled using a coordinate-measuring machine (CMM) (Brown & Sharpe CHAMELEON 9159). The CMM in this experiment moved along the Y-axis direction, and the radial movement was relative to the collimator in the measurement system. The corner prism was fixed to the CMM, and the radial movement speed and instantaneous distance of the three-coordinate measurement machine were measured.

 figure: Fig. 4.

Fig. 4. Lab environment. For the experimental measurements, the corner prism was fixed on the probe of the CMM to obtain a stable movement speed. A collimating lens integrated with the transceiver was used. Within the radial range of the corner prism movement, the collimator lens and the corner prism were always kept in a horizontal straight line to ensure that the local wave signal and the echo signal could be coupled to form a beat signal with strong energy.

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The measurement system mainly included a tunable laser (Luna PHOENIX 1400) with a scanning range of 1515-1565 nm and a frequency modulation rate of 1–2000 nm/s, and an H13C14N gas cell (HCN-13-H(16.5)-25-FC/APC) to absorbs light of a specific wavelength and form a spectrum of absorption peaks. The wavelength absorption characteristics were certified by the National Institute of Standards Technology (NIST). Using this device as a reference source for splitting the frequency modulation range, we matched the up- and down-scanning signals with precision.

When the sweep direction changes, the beat signal has no practical meaning. Combined with the wavelength interval of the H13C14N’s absorption peak, the scanning bandwidth and modulation rate were set to 1545-1565nm and 100.04nm/s, respectively. Using these parameter settings, a group of triangular wave modulation measurement signals can be split and combined into 18 groups of measurement signals, with the frequency modulation range of each group being about 100GHz.

As shown in Fig. 1, the source laser light passed through the 90/10 optical splitter, and the light from the stronger path passed through a 90/10 optical splitter again and entered the measurement optical path and the H13C14N gas cell. One strand in the measurement optical path passed through the circulator and the collimator to enter the corner prism fixed on the CMM, and then was received by the collimator to obtain the echo signal. The other strand was used as the local wave signal. The echo signal and the local wave signal generated the measurement beat signal in the coupler. The measurement beat signal was detected by the PD. Since the measurement optical path signal and H13C14N gas cell absorption peak signal were generated by the optical splitter at the same time, the two signals had perfect simultaneity, ensuring the accuracy of signal splitting.

In actual measurements, due to the nonlinear wavelength-current tuning relationship of the semiconductor laser and the temperature change of the gain medium, the triangular wave modulation signal of the semiconductor laser is nonlinear. The nonlinearity of the triangle wave modulation will cause the beat frequency signal to manifest a spectrum-broadening phenomenon after FFT, as shown in Fig. 1. In order to reduce the non-linearity of the laser light source, another laser with lower energy was drawn through the MZI system to generate a reference beat signal. In this paper, the method of extreme value resampling is used to extract the peak and trough of the reference beat signal through the algorithm, and the measurement signal is resampled according to the position corresponding to the extreme value. The comparison of the experimental results in Fig. 5(b) shows that extreme value resampling suppresses spectrum broadening effectively.

 figure: Fig. 5.

Fig. 5. Signal processing. (a) The three signals measured by the oscilloscope, namely the beat signal of the measurement optical path, beat signal of the reference optical path and the gas cell absorption peak signal. The position corresponding to the absorption peak of the H13C14N signal is marked with an asterisk. Blue signal in (b): due to the non-linearity of the laser frequency modulation, the frequency spectrum after the FFT of the beat signal appears severely broadened and the position of the spectral peak cannot be determined. Red signal in (b): after re-sampling the measurement signal in the frequency domain, the peak energy of the spectrum is extremely large, and the amplitude of the sidebands is low, so the frequency of the beat signal can be calculated accurately to improve the precision of the measurement. Purple signal (c): signal directly collected by the oscilloscope. The larger (smaller) amplitude is the beat signal of the measurement (reference) optical path. Both signals have obvious glitches. The measurement signal was band-pass filtered, while the reference signal was low-pass filtered. The two signals after filtering are relatively smooth, which improved the re-sampling accuracy.

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The length of the delay fiber in the experiment was 20m. Based on the experimental setup, the distance of the object to be measured was about 4m, while the length of the fiber of the circulator was about 3.2m. The length of the delay fiber was about 5 times the distance to be measured, which satisfies the Nyquist sampling theorem.

The three signals shown in Fig. 5(a) were all collected by the PD. The oscilloscope (TELEDYNE LECROY WAVERUNNER 610Zi) received the signal collected by the photodetector and stored the data through the USB interface. When the acquisition frequency of the oscilloscope was 25M/s, the glitches in the collected data were fewer, which was convenient for subsequent data processing.

According to Fig. 6, when the frequency modulation range was 20nm, the spectrum broadening was significant, which affected the accuracy and repeatability of the speed measurement. The absorption peak interval of the H13C14N gas cell was about 0.7nm, and the beat frequency signal was split by the absorption peak signal, so the modulation range of each group after splitting reached 0.7nm. We obtained a smaller frequency modulation range and the beat signal showing good linearity. As shown in Fig. 6, the sidelobe energy of the spectrum signal obtained after splitting was low, and there was almost no spectrum broadening.

 figure: Fig. 6.

Fig. 6. Comparison of frequency spectrum of beat signal splitting effect at a speed of 8 mm/s, (a) is the frequency spectrum of the beat signal obtained by sweeping over a wide range, and (b) is corresponding spectrum after splitting the wide range sweep.

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In this experiment, the probe of the CMM to move at a constant speed controlled by software program. The movement speed of the probe was increasingly set to 1mm/s, 5mm/s, 8mm/s, 10mm/s, 15mm/s, 20mm/s, 25mm/s and 30mm/s. In order to verify that the method proposed in this paper reduces the impact of frequency deviation caused by wide frequency modulation range and increases precision and repeatability of speed measurement, we designed a set of comparative experiments. According to theoretical analysis, the frequency modulation range needs to be as small as possible without signal splitting, otherwise serious spectrum broadening will occur. Since the minimum frequency modulation range of the laser is 1nm, the frequency modulation range of the laser was set to 1548.9-1550.1nm. The frequency modulation rate to 99.95nm/s and we did not use the gas cell optical path in the Fig. 1. The speed was measured in the same experimental environment.

The full width at half maximum of the two sets of signals show that the spectrum broadening of blue signals is more serious than the red signals in the Fig. 7. This verifies that our proposed method reduces the influence of the frequency modulation range on the spectrum signal, and makes the speed measurement resolution significantly improved.

 figure: Fig. 7.

Fig. 7. Spectrum comparison chart. Red signal: spectrogram after the large-range sweep frequency is split, with a frequency modulation range of about 0.7 nm; blue signal: spectrogram using the narrow linewidth sweep frequency directly, with a frequency modulation range of 1.2 nm.

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 figure: Fig. 8.

Fig. 8. Frequency spectrum of the two scanning directions. Red line: up-scanning laser; blue line: down-scanning laser.

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As shown in Fig. 8, the peak positions (${f_{up}}$ and ${f_{down}}$) representing the corner prism echo signal at up- and down-scanning respectively were obtained. A radial velocity of 1.006 mm/s and instantaneous distance 3.921285 m were obtained via Eq. (23) and Eq. (24). Since the instantaneous distance cannot be calibrated using laser trackers, etc., we only analyze results of speed measurements.

Next, we analyzed the accuracy and repeatability of the two sets of experimental data. According to Fig. 9(a) and Fig. 9(b) and Table 1, when the movement speed was 1mm/s, 5mm/s and 8mm/s, the errors of the two sets of experiments are all less than 92µm/s. When the speed was greater than 10mm/s, the maximum error of the average velocity of the experiment without a H13C14N gas cell was 152µm/s, and the maximum standard deviation of the average velocity was 348µm/s. The maximum mean error of the method proposed in this paper was 83µm/s, and the maximum standard deviation of the mean velocity was 61µm/s. According to Fig. 9(c), our method achieved good measurement stability when the measured speed was 15mm/s, 20mm/s, 25mm/s, and 30mm/s.

 figure: Fig. 9.

Fig. 9. Analysis of the accuracy and repeatability of two sets of experimental data

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Tables Icon

Table 1. Measurement results of the two experiments

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We then converted Eq. (23) into the following form to derive the expended uncertainty of speed:

$$v = \frac{{\frac{{K{^{\prime}_{up}} - K{^{\prime}_{down}}}}{{N^{\prime}}}\lambda \alpha {\tau _r}}}{{2 - \frac{{\Delta t}}{c}\lambda \alpha }} = \frac{{\lambda \alpha {R_r}}}{{2c - \alpha \Delta t\lambda }}F^{\prime}$$
$F^{\prime}$ represents $(K{^{\prime}_{up}} - K{^{\prime}_{down}})/N^{\prime}$ and ${R_r}$ represents $c{\tau _r}$. The synthetic uncertainty of the measured speed $v$ can be expressed as:
$${u_c}(v) = \sqrt {{{(\frac{{\partial v}}{{\partial \alpha }}u(\alpha ))}^2} + {{(\frac{{\partial v}}{{\partial {R_r}}}u({R_r}))}^2} + {{(\frac{{\partial v}}{{\partial \lambda }}u(\lambda ))}^2} + {{(\frac{{\partial v}}{{\partial F^{\prime}}}u(F^{\prime}))}^2}}$$
where $u(\alpha )$ represents the standard uncertainty of the laser modulation rate, $u({R_r})$ is the standard uncertainty of the auxiliary fiber length, $u(\lambda )$ is the standard uncertainty of the laser wavelength, $u(F^{\prime})$ represents the standard uncertainty of the signal spectrum extraction point after resampling. Here, $u(\alpha ) = 1.23\textrm{nm/s,}$ $u({R_r}) = 73\mathrm{\mu }\textrm{m,}$ $u(\lambda ) = 0.2\textrm{nm,}$ $u(F^{\prime}) = 58 \times {10^{ - 6}},\;\frac{{\partial v}}{{\partial \alpha }} \approx \frac{v}{\alpha }$, $\frac{{\partial v}}{{\partial {R_r}}} \approx \frac{v}{{{R_r}}}$, $\frac{{\partial v}}{{\partial \lambda }} \approx \frac{v}{\lambda }$, $\frac{{\partial v}}{{\partial F^{\prime}}} \approx \frac{{\alpha {R_r}\lambda }}{{2c - \alpha \varDelta t\lambda }}$.

In the actual measurements, $\alpha = 100.04n\textrm{m/s,}\;{R_r} = 20\textrm{m,}\;\lambda = 1550\textrm{nm,}\;\Delta t = 0.007547\textrm{s,}$ $c = 3 \times {10^8}\textrm{m/s}$, and the number of spectrum points after resampling was $2 \times {10^4}$.

$${U_c}(v) = \sqrt {{{(\frac{{\partial v}}{{\partial \alpha }}u(\alpha ))}^2} + {{(\frac{{\partial v}}{{\partial {R_r}}}u({R_r}))}^2} + {{(\frac{{\partial v}}{{\partial \lambda }}u(\lambda ))}^2} + {{(\frac{{\partial v}}{{\partial F^{\prime}}}u(F^{\prime}))}^2}} < 1.2 \times {10^{ - 4}}v + 2.2 \times {10^{ - 6}}$$
Taking the inclusion factor $k = 2$, and the expanded uncertainty ${U_c}(v)$ is less than $2.4 \times {10^{ - 4}}v + 4.4\mathrm{\mu }\textrm{m/s}$

The absorption peak wavelength of the gas cell is invariant, and this property can be traced to the NIST value. We used this feature to split the beat signal, not only because it ensured the phase correspondence of the beat signals at opposite scanning directions, but also because the spacing between the peaks was less than 1nm. This resulted in a significantly reduced amount of data processed simultaneously. The experimental comparison results show that this method improves the utilization of data, as when the scanning direction is opposite, resulting in a redundant spectrum. The total sweep range of this method is extremely large. The split spectrum signal will not be affected by the change of frequency-modulation direction, so the signal utilization rate is increased by about 10%. In addition, the modulation range of each group after splitting reached 0.7nm which resulted in a smaller frequency modulation range and the beat signal showing good linearity. Finally, comparing the spectrograms and experimental results of the two sets of experiments, it is evident that the method proposed in this paper reduces the influence of the frequency deviation caused by wide frequency modulation range and shows good accuracy and repeatability.

4. Conclusion and future work

In this paper, we proposed a novel FMCW velocity measurement method. This method splits the sweep signal of a large-bandwidth triangular wave-modulated laser based on the absorption peak of a H13C14N gas cell, which solves the problem of the excessive amount of data that needs to be processed in a single speed measurement of a continuously moving target effectively and reduces the complexity of the algorithm. In addition, the proposed method resulted in a smaller frequency modulation range and the beat signal showing good linearity, the resolution and range of speed measurement are improved significantly. In the experiment, the precision was below 61µm/s in a range up to 30 $\textrm{mm/s}$, and the relative error of the mean value of speed measurement did not exceed 0.42%, which shows good repeatability and accuracy. At the same time, this method improves the signal utilization rate effectively and reduces the influence of the frequency deviation on the quality of the speed measurement spectrum. In future work, we will use this FMCW speed measurement system and signal analysis method to measure the non-uniform instantaneous speed of moving targets with high precision.

Funding

Key Research and Development Plan of Tianjin (2018YFB2003501, 2020YFB2010701); National Natural Science Foundation of China (51775379, 52035013).

Acknowledgments

We would like to thank the peer reviewers for their very helpful comments.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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3. L. Rovati, U. Minoni, and F. Docchio, “Dispersive white light combined with a frequency-modulated continuous-wave interferometer for high-resolution absolute measurements of distance,” Opt. Lett. 22(12), 850–852 (1997). [CrossRef]  

4. E. Baumann, F. R. Giorgetta, I. Coddington, L. C. Sinclair, K. Knabe, W. C. Swann, and N. R. Newbury, “Comb-calibrated frequency-modulated continuous-wave ladar for absolute distance measurements,” Opt. Lett. 38(12), 2026–2038 (2013). [CrossRef]  

5. M. U. Piracha, D. Nguyen, I. Ozdur, and P. J. Delfyett, “Simultaneous ranging and velocimetry of fast moving targets using oppositely chirped pulses from a mode-locked laser,” Opt. Express 19(12), 11213–11219 (2011). [CrossRef]  

6. E. Baumann, F. R. Giorgetta, J. D. Deschenes, W. C. Swann, I. Coddington, and N. R. Newbury, “Comb-calibrated laser ranging for three-dimensional surface profiling with micrometer-level precision at a distance,” Opt. Express 22(21), 24914–24928 (2014). [CrossRef]  

7. Q. Jie, Q. Zhou, W. Xie, X. Yan, and W. Hu, “Coherence enhancement of a chirped DFB laser for frequency-modulated continuous-wave reflectometry using a composite feedback loop,” Opt. Lett. 40(19), 4500–4503 (2015). [CrossRef]  

8. F.-M. Zhang, Y.-T. Li, H. Pan, C.-Z. Shi, and X.-H. Qu, “Vibration Compensation of the Frequency-Scanning-Interferometry-Based Absolute Ranging System,” Appl. Sci. 9(1), 147–157 (2019). [CrossRef]  

9. S. Kakuma and Y. Katase, “Frequency scanning interferometry immune to length drift using a pair of vertical-cavity surface-emitting laser diodes,” Opt. Rev. 19(6), 376–380 (2012). [CrossRef]  

10. R. Schneider, P. Thurmel, and M. Stockmann, “Distance measurement of moving objects by frequency modulated laser radar,” Opt. Eng. 40(1), 33–37 (2001). [CrossRef]  

11. S. Kakuma, “Frequency-modulated continuous-wave laser radar using dual vertical-cavity surface-emitting laser diodes for real-time measurements of distance and radial velocity,” Opt. Rev. 24(1), 39–46 (2017). [CrossRef]  

12. J. J. Martinez, A. M. Campbell, S. M. Warden, and B. E. Hughes, “Dual-Sweep Frequency Scanning Interferometry Using Four Wave Mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015). [CrossRef]  

13. F. Zhang, L. Yi, and X. Qu, “Simultaneous measurements of velocity and distance via a dual-path FMCW lidar system,” Opt. Commun. 474, 126066 (2020). [CrossRef]  

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (9)
Fig. 1.
Fig. 1. Schematic of the experimental setup. The triangular wave-modulated continuous laser light is divided into three parts by the optical splitter. From top to bottom, they are the measurement optical path, the H13C14N gas cell optical path and the reference optical path. The corner prism is fixed on a CMM, relative to which the radial velocity of the collimator lens is measured. CIR: circulator; CM: collimating mirror; CMM: Coordinate measuring machine; PD: photodetector. (a) Local wave signal after the power amplifier. (b) Local wave signal entering the non-cooperative space through the collimator, and reflected back to the collimator by the corner prism to form an echo signal. (c) Beat signal of the measurement optical path. (d) Absorption peak signal of gas cell. (e) Beat signal of reference optical path.

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Fig. 2.
Fig. 2. Schematic diagram of the positioning of the absorption peak of the gas cell signal. Purple is the gas cell signal collected by the PD, green is the signal after low-pass filtering, and blue is the signal fitted by the quadratic function. The black and red asterisks indicate the position of the absorption peak before and after filtering, respectively.

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Fig. 3.
Fig. 3. Schematic diagram of gas cell splitting beat signal. (b) Absorption peaks of the gas cell signal corresponding to the up- and down-sweep frequencies are symmetrical to each other. (c) Absorption peak of the gas cell splitting the FMCW signals of different frequency ranges in the frequency domain. (d) According to the wavelength corresponding to the absorption peak of the gas cell, the beat signals corresponding to the up- and down-scanning during a frequency sweep period are divided into 18 groups, and $\Delta f$ represents the frequency corresponding to the adjacent absorption peak of the gas cell.

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Fig. 4.
Fig. 4. Lab environment. For the experimental measurements, the corner prism was fixed on the probe of the CMM to obtain a stable movement speed. A collimating lens integrated with the transceiver was used. Within the radial range of the corner prism movement, the collimator lens and the corner prism were always kept in a horizontal straight line to ensure that the local wave signal and the echo signal could be coupled to form a beat signal with strong energy.

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Fig. 5.
Fig. 5. Signal processing. (a) The three signals measured by the oscilloscope, namely the beat signal of the measurement optical path, beat signal of the reference optical path and the gas cell absorption peak signal. The position corresponding to the absorption peak of the H13C14N signal is marked with an asterisk. Blue signal in (b): due to the non-linearity of the laser frequency modulation, the frequency spectrum after the FFT of the beat signal appears severely broadened and the position of the spectral peak cannot be determined. Red signal in (b): after re-sampling the measurement signal in the frequency domain, the peak energy of the spectrum is extremely large, and the amplitude of the sidebands is low, so the frequency of the beat signal can be calculated accurately to improve the precision of the measurement. Purple signal (c): signal directly collected by the oscilloscope. The larger (smaller) amplitude is the beat signal of the measurement (reference) optical path. Both signals have obvious glitches. The measurement signal was band-pass filtered, while the reference signal was low-pass filtered. The two signals after filtering are relatively smooth, which improved the re-sampling accuracy.

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Fig. 6.
Fig. 6. Comparison of frequency spectrum of beat signal splitting effect at a speed of 8 mm/s, (a) is the frequency spectrum of the beat signal obtained by sweeping over a wide range, and (b) is corresponding spectrum after splitting the wide range sweep.

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Fig. 7.
Fig. 7. Spectrum comparison chart. Red signal: spectrogram after the large-range sweep frequency is split, with a frequency modulation range of about 0.7 nm; blue signal: spectrogram using the narrow linewidth sweep frequency directly, with a frequency modulation range of 1.2 nm.

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Fig. 8.
Fig. 8. Frequency spectrum of the two scanning directions. Red line: up-scanning laser; blue line: down-scanning laser.

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Fig. 9.
Fig. 9. Analysis of the accuracy and repeatability of two sets of experimental data

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Tables (1)

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Table 1. Measurement results of the two experiments

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Equations (27)

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f ( t ) = f 0 + α t 0 < t < T 2
E ( t ) = E 0 exp ⁡ [ j φ ( t ) ]
E ( t + τ m ) = E 0 exp ⁡ [ j φ ( t + τ m ) ]
U ( t ) = σ | E ( t + τ m ) + E ( t ) | 2 = 2 σ | E 0 | 2 { 1 + cos ⁡ [ φ ( t + τ m ) − φ ( t ) ] }
φ ( t + τ ) = ∑ n = 0 ∞ τ n n ! φ ( n ) ( t )
U m ( t ) = U m 0 cos ⁡ [ 2 π f ( t ) τ m ] = U m 0 cos ⁡ [ 2 π α τ m t + 2 π τ m f 0 ]
R = c f b 2 α n
U m ( t ) = U m 0 cos ⁡ [ 2 π α τ m φ ′ ( t ) + 2 π τ m f 0 ]
U r ( t ) = U r 0 cos ⁡ [ 2 π α τ r φ ′ ( t ) + 2 π τ r f 0 ]
F s ′ = 1 Δ φ = 1 φ k + 1 ( t ) − φ k ( t ) = 2 α τ r k = 1 , 2 , 3 … …
U r ( t ) = U r 0 cos ⁡ [ 2 π α τ m k Δ φ + 2 π τ m f 0 ] = U r 0 cos ⁡ [ 2 π α τ m k 2 α τ r + 2 π τ m f 0 ] k = 1 , 2 , 3 … … = U r 0 cos ⁡ [ π τ m τ r k + 2 π τ m f 0 ]
f b = F s ′ k ′ / N ′ = 2 α τ r k ′ / N ′
R = c τ m / 2 = k ′ c τ r / N ′
φ ( t ) = 2 π ( f 0 τ − 0.5 α 0 τ 2 + α 0 t τ ) τ ≤ t ≤ T m 2
τ ( t ) = 2 ( R 0 − v t ) c = τ 0 − 2 v t c
φ v ( t ) = 2 π ( f 0 τ 0 − 2 v t f 0 c + α 0 t τ − 2 v α 0 t 2 c ) τ ≤ t ≤ T m 2
Δ f = 4 v α 0 c ( T m 2 − τ ) = v B c ( 4 − 16 R c T m )
f d = 2 v λ
f u p = α τ m + 2 v λ
f d o w n = − α ( τ m  +  τ ′ ) + 2 v λ = − α ( τ m  +  2 v Δ t c ) + 2 v λ
f u p ′ = α ( τ m + 2 v λ ) × 1 2 α τ r = τ m 2 τ r + v λ α τ r
f d o w n ′ = [ − α ( τ m + 2 v Δ t c ) + 2 v λ ] × 1 2 ( − α ) τ r = τ m 2 τ r + v Δ t c τ r − v λ α τ r
v = ( f u p ′ − f d o w n ′ ) λ α τ r 2 − Δ t c λ α = K u p ′ − K d o w n ′ N ′ λ α τ r 2 − Δ t c λ α
R = 2 c n r τ m = 2 c n r [ ( f u p ′ + f d o w n ′ ) τ r − v Δ t c ] = 2 c n r [ ( K u p ′ + K d o w n ′ N ′ ) τ r − v Δ t c ]
v = K u p ′ − K d o w n ′ N ′ λ α τ r 2 − Δ t c λ α = λ α R r 2 c − α Δ t λ F ′
u c ( v ) = ( ∂ v ∂ α u ( α ) ) 2 + ( ∂ v ∂ R r u ( R r ) ) 2 + ( ∂ v ∂ λ u ( λ ) ) 2 + ( ∂ v ∂ F ′ u ( F ′ ) ) 2
U c ( v ) = ( ∂ v ∂ α u ( α ) ) 2 + ( ∂ v ∂ R r u ( R r ) ) 2 + ( ∂ v ∂ λ u ( λ ) ) 2 + ( ∂ v ∂ F ′ u ( F ′ ) ) 2 < 1.2 × 10 − 4 v + 2.2 × 10 − 6
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