FMCW LiDAR with an FM nonlinear kernel function for dynamic-distance measurement

Yu Zehao; Yu Zehao; Lu Cheng; Lu Cheng; Lu Cheng; Liu Guodong; Liu Guodong

doi:10.1364/OE.458235

1. Introduction

Among modern industrial measurements, high-fidelity three-dimensional (3D) imaging of scan surfaces and measuring the vibrations of large machine parts vibration measurement with micron-grade precision can prove to be challenging tasks. During the manufacture and assembly of spacecraft, gas turbines, and satellite solar panels, the precise 3D shape and vibration mode often need to be measured simultaneously. For example, to ensure the working stability of a gas turbine blade, it is always necessary to test the dynamic mechanical properties and high-precision 3D shape simultaneously under impact. Measurements of topography and mode shapes are currently performed using different instruments and are therefore not synchronized in time. Generally, laser trackers and FMCW LiDAR [1–3] are used to measure 3D shapes, and laser Doppler vibrometers are used to measure the vibration shape of targets. The measured absolute distance and dynamic measurement indicators have an important impact on the measurement system.

The FMCW LiDAR technique is suitable for high-speed, high-precision, long-distance measurement systems that can measure distance and speed simultaneously [1–6]. However, the accuracy of this technique is affected by frequency-modulated (FM) nonlinearity, which deteriorates the distance spectrum. Various linearization methods have been proposed to achieve a linear modulated frequency [7].

Three main approaches have been used to solve this problem: (1) linearization of the modulated laser frequency, (2) frequency sampling, and (3) the phase-ratio method. The first approach involves designing and manufacturing a tunable laser source with a tuning curve that is linear in time [8–11], which is difficult and less convenient than the other two methods. In 2015, Mateo et al. obtained two-dimensional (2D) surface profiles of different aluminum-plated shapes at a standoff of 1.5 m by using trilateration and high-resolution FMCW LiDAR [12]. A fast loop and a slow loop were used to control the PZT and the injection current, respectively, thereby linearizing the sweep frequency. The coordinate precision in all dimensions was below 200 µm. In 2017, Jia et al. used a time-varying Kalman filter to track the instantaneous motion of a target [13]. The time-varying Kalman filter was proposed to model the instantaneous movement of a target for different durations of optical-frequency tuning of a single external cavity diode laser (ECDL). The second method employs an auxiliary interferometer to sample measured interferometer signals at equal frequency intervals using an external clock hardware or resampling algorithm. The first and second methods have been reported to successfully address nonlinear frequency modulation but cannot be used for velocity detection because uneven sampling in the time domain obscures the Doppler signal. In addition, the optical path distance (OPD) of the auxiliary interferometer used is more than twice the measured value, which introduces severe residual nonlinearity and necessitates the use of a high sampling rate [11,14]. Baumann et al. proposed a frequency-comb-calibrated FMCW system in 2014. The femtosecond optical comb is used as a frequency reference to calibrate and therefore correct the nonlinearity in the frequency sweeping process. The system can obtain 3D images of a diffusely scattering surface at a distance of 10.5 m with a precision below 10 µm [15]. However, optical combs are an expensive means of correction, and the proposed method cannot be used to perform dynamic distance measurements. Masayuki et al. proposed a LiDAR scheme in 2020 called “swept source LiDAR” to perform FMCW ranging and nonmechanical beam steering simultaneously [16]. The third method is based on the principle that the phase ratio between measurement and auxiliary interferometers is proportional to the distance ratio of these instruments, does not require the use of a long auxiliary interferometer and is compatible with velocity detection; however, a high SNR is required to determine the instantaneous phase of the measurement signal, making the method only suitable for measuring cooperative targets, such as a spherical retroreflecting target or smooth mirror. In 2014, Matthew Warden et al. developed a setup using FSI and gas absorption cells for distance measurement and obtained sub-ppm accuracy in the 20-m range for a retroreflector target. [17,18]. This method involves using the phase-ratio technique to correct the nonlinearity of the sweep frequency, which requires a high signal-to-noise ratio for the return light from a target and cannot be used for noncooperative targets. In 2018, Fumin et al. proposed an FMCW LiDAR autofocusing system with a precision below 126 µm over a range of up to 60 m [19]. In 2020, Bin et al. proposed using frequency-fixed interferometry to eliminate the error associated with the Doppler effect [20]. Most of the setups used in the abovementioned methods have complex structures and cannot be used for dynamic measurements. Nonideal measurement results are obtained for noncooperative targets. Therefore, it is necessary to develop a method that can be used to measure both noncooperative targets and dynamic distances.

In this letter, we propose a novel dual-laser FMCW-based dynamic-distance measurement method based on a preliminary study [21–23]. An FM nonlinear kernel function is used to determine the absolute distance, which may reduce the influence of dispersion mismatch. The use of a short auxiliary interferometer considerably reduces the detection and sampling bandwidth of the measurement system. As time information is retained, the velocity can be measured simultaneously and used to determine the dynamic distance. Simulation and experimental results show that the proposed method can be used to measure the dynamic distance of a noncooperative target with high precision. Finally, the profile measurement of a plaster cast is presented.

2. Principles

A schematic of the basic FMCW LiDAR setup is shown in Fig. 1; the electric field of the output laser can be defined as

(1)$$E(t )= {E_0}\exp [{j\varphi (t )} ]$$

where φ(t) is the phase of the FM laser and E₀ is a constant amplitude. The laser angular frequency ω(t) is the derivative of the phase with respect to time.

(2)$$\omega (t )= \frac{{d\varphi }}{{dt}}$$

Fig. 1. Diagram of FMCW ranging. TLS, tunable laser source; DAQ, data acquisition; PD, photodetector; PBS, polarizing beam splitter.

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The output of the FM laser source is split into a reference arm and a measuring arm. The interference signal of the auxiliary interferometer can be expressed as

(3)$$I(t )\textrm{ = }{|{{\sigma_r}E(t )+ {\sigma_m}E({t + \tau } )} |^2} = B + A\cos [{\varphi ({t + \tau } )- \varphi (t )} ]$$

where A and B represents the amplitude and DC bias of the signal, σ_r and σ_m are proportionality constants that depend on the detector sensitivity and optical power. τ is the group delay difference between the two paths. Then, the function φ(t +τ) can be Taylor expanded about t as given below.

(4)$$\begin{aligned} I(t ) &= A\cos [{\varphi ({t + \tau } )- \varphi (t )} ]\\ &= A\cos \left[ {\sum\limits_{n = 0}^\infty {\frac{{{\tau^n}}}{{n!}}{\varphi^{(n )}}(t )} - \varphi (t )} \right]\\ &= A\cos \left[ {\varphi (t )- \varphi (t )+ \tau \varphi^{\prime}(t )+ \frac{{{\tau^2}}}{2}{\varphi^{(2 )}}(t )+ \frac{{{\tau^3}}}{6}{\varphi^{(3 )}}(t )+ \ldots } \right]\\ &= A\cos \left[ {\tau \omega (t )+ \sum\limits_{n = 2}^\infty {\frac{{{\tau^n}}}{{n!}}{\omega^{({n - 1} )}}(t )} } \right]\\ &\approx A\cos [{\tau \omega (t )} ]\end{aligned}$$

In general, the higher-order terms in Eq. (4) are neglected, whereby the phase of the interference signal is linear to the laser frequency. If the laser frequency is linearly modulated in time, the interference signal is a standard cosine signal that changes with time, and a Fourier transform can be used to determine the group delay. However, as it is difficult to ensure that the FM characteristics of a laser source are completely linear, a nonlinearity correction method is required. Generally, an auxiliary interferometer should be used to correct the FM nonlinearity in the measurement interferometer signal.

2.1 Principle of the dual-laser FMCW

In industrial environments characterized by vibration and air turbulence, the influence of the Doppler effect on the signal of a measurement interferometer can be expressed as

(5)$$\begin{aligned} {I_m}(t ) &\approx {A_m}\cos [{{\tau_m}(t )\omega (t )} ]\\ &= {A_m}\cos \{{[{{\tau_{m0}} + \Delta {\tau_m}(t )} ][{{\omega_0} + \Delta \omega (t )} ]} \}\\ &= {A_m}\cos [{{\tau_{m0}}\Delta \omega (t )+ \Delta {\tau_m}(t ){\omega_\textrm{0}}\textrm{ + }{\omega_\textrm{0}}{\tau_{m\textrm{0}}}\textrm{ + }\Delta \omega (t )\Delta {\tau_m}(t )} ]\end{aligned}$$

where A_m is the measured signal amplitude. Δω(t) is the time-varying laser angular frequency variation during FM, and ω₀ is the FM starting angular frequency. Δτ_m(t) is the variation in the group delay during FM, and τ_m₀ is the group delay for the first FM. The first term of Eq. (5) contains information about the target distance. The second term results from the Doppler effect. The angular frequency of the laser ω₀ is approximately 2π×200 THz, which is much larger than the variation in the angular frequency Δω. Therefore, the second term of Eq. (5) cannot be neglected even if the variation in the group delay Δτ_m is very small, showing that the Doppler effect significantly influences the measurement precision.

Dual-laser FMCW is commonly used to correct the error resulting from the Doppler effect [18,24], and the setup is shown in Fig. 2. Two tunable lasers are utilized with similar start frequencies and frequency modulations with opposite directions and different rates. The outputs of TSL 1 and TSL 2 are combined using a 50/50 coupler and then split between the measurement and auxiliary interferometers.

Fig. 2. Experimental layout of the dual-modulation experiment. TLS: tunable laser source; PD, photodetector; PBS, polarizing beam splitter; HPF: high-pass filter; LPF: low-pass filter.

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As the two lasers have different FM rates, the signals of both the auxiliary and measurement interferometers are separated using a high-pass filter (HPF) and a low-pass filter (LPF). These signals can be expressed as

(6)$${I_{a1}}(t )\textrm{ = }{A_{a1}}\cos [{{\tau_a}{\omega_1}(t )} ]$$

(7)$${I_{a2}}(t )\textrm{ = }{A_{a2}}\cos [{{\tau_a}{\omega_2}(t )} ]$$

(8)$${I_{m1}}(t )= {A_{m1}}\cos [{{\tau_{m0}}\Delta {\omega_1}(t )+ \Delta {\tau_m}(t ){\omega_\textrm{0}}\textrm{ + }{\omega_\textrm{0}}{\tau_{m\textrm{0}}}\textrm{ + }\Delta {\omega_1}(t )\Delta {\tau_m}(t )} ]$$

(9)$${I_{m2}}(t )= {A_{m2}}\cos [{{\tau_{m0}}\Delta {\omega_2}(t )+ \Delta {\tau_m}(t ){\omega_\textrm{0}} + {\omega_\textrm{0}}{\tau_{m\textrm{0}}} + \Delta {\omega_2}(t )\Delta {\tau_m}(t )} ]$$

where I_a₁ and I_a₂ are the auxiliary interferometer signals corresponding to the two lasers and have amplitudes A_a₁ and A_a₂, respectively, and Δω₁ and Δω₂ are the angular frequency variations corresponding to the two lasers, where Δω₂ is negative. τ_a is the group delay of the auxiliary interferometer. I_m₁ and I_m₂ are the measurement interferometer signals corresponding to the two lasers and have amplitudes A_m₁ and A_m₂, respectively.

The measurement interferometer signals pass through multipliers and HPFs. The output is given below.

(10)$$\begin{aligned} {I_{mc}}(t ) &= HPF[{{I_{m1}}(t ){I_{m2}}(t )} ]\\ &= \frac{{{A_{m1}}{A_{m2}}}}{2}\cos \{{{\tau_{m0}}[{\Delta {\omega_1}(t )- \Delta {\omega_2}(t )} ]+ \Delta {\tau_m}(t )[{\Delta {\omega_1}(t )- \Delta {\omega_2}(t )} ]} \}\\ &\approx \frac{{{A_{m1}}{A_{m2}}}}{2}\cos \{{{\tau_{m0}}[{\Delta {\omega_1}(t )- \Delta {\omega_2}(t )} ]} \}\end{aligned}$$

As Δτ_m is considerably smaller than τ_m₀, the second term in Eq. (10) can be neglected. The influence of the Doppler effect is eliminated, and τ_m₀ is the measured absolute distance. As Δω₁ and Δω₂ are unknown and nonlinear, the signal of the auxiliary interferometer is required to correct the FM nonlinearity.

2.2 Influence of dispersion

In most systems, the auxiliary interferometer is built from fiber-optic devices. Unfortunately, the fiber has dispersion, and the group delay τ_a varies with time. Thus, Eqs. (6) and (7) should be rewritten as

(11)$$\begin{aligned} {I_{a1}}(t ) &= {A_{a1}}\cos [{{\tau_a}(t ){\omega_1}(t )} ]\\ &= {A_{a1}}\cos \left[ {\frac{{{n_g}(t ){R_a}}}{c}{\omega_1}(t )} \right]\\ &= {A_{a1}}\cos [{{\beta_{l1}}(t ){R_a}} ]\end{aligned}$$

(12)$${I_{a2}}(t )= {A_{a2}}\cos [{{\beta_{l2}}(t ){R_a}} ]$$

where R_a is the difference in the lengths of the two paths of the auxiliary interferometer, n_g is the group refraction index of the fiber, and β_l₁ and β_l₂ are the propagation constants for the two lasers. To approximate dispersion effects, the propagation constants can be written as a Taylor expansion about the starting frequency ω₀:

(13)$${\beta _{l1}}(t )\approx {\beta _0} + \Delta {\omega _1}(t ){\beta _1} + \frac{1}{2}\Delta {\omega _1}{(t )^2}{\beta _2}$$

(14)$${\beta _{l2}}(t )\approx {\beta _0} + \Delta {\omega _2}(t ){\beta _1} + \frac{1}{2}\Delta {\omega _2}{(t )^2}{\beta _2}$$

where the propagation constants are approximated to second-order. β₀ = n_g₀ω₀/c, and n_g₀ is the group refraction index at the tuning start frequency ω₀. β₁ is the inverse group velocity. β₂ is the group velocity dispersion or dispersion coefficient of a single-mode optical fiber. Substituting Eqs. (13) and (14) into Eqs. (11) and (12) yields

(15)$$\begin{aligned} {I_{a1}}(t ) &= {A_{a1}}\cos [{{\beta_{l1}}(t ){R_a}} ]\\ &= {A_{a1}}\cos \left[ {{\beta_0}{R_a} + \Delta {\omega_1}(t ){\beta_1}{R_a} + \frac{1}{2}\Delta {\omega_1}{{(t )}^2}{\beta_2}{R_a}} \right] \end{aligned}$$

(16)$${I_{a2}}(t )= {A_{a2}}\cos \left[ {{\beta_0}{R_a} + \Delta {\omega_2}(t ){\beta_1}{R_a} + \frac{1}{2}\Delta {\omega_2}{{(t )}^2}{\beta_2}{R_a}} \right]$$

Equations (15) and (16) show that dispersion makes the phase of the signal nonlinear in the optical frequency. Using this signal to correct the FM nonlinearity will introduce a measurement error.

3. Dynamic-distance measurement based on an FM nonlinear kernel function

3.1 Measurement of the initial distance

To eliminate the influence of FM nonlinearity and dispersion, a nonuniform Fourier-transform-based range extraction method is proposed in this study. This method consists of applying a Hilbert transform to determine the phase increments (φ_a₁ and φ_a₂) of the auxiliary interferometer signals (Eqs. (11) and (12)).

(17)$$\Delta {\omega _1}(t ){\beta _1}{R_a} + \frac{1}{2}\Delta {\omega _1}{(t )^2}{\beta _2}{R_a}\textrm{ = }{\varphi _{a1}}(t )$$

(18)$$\Delta {\omega _2}(t ){\beta _1}{R_a} + \frac{1}{2}\Delta {\omega _2}{(t )^2}{\beta _2}{R_a}\textrm{ = }{\varphi _{a2}}(t )$$

The equations presented above can be solved for the frequency variations of the lasers.

(19)$$\Delta {\omega _1}(t)\textrm{ = }\frac{{ - {\beta _1}{R_a}\textrm{ + }\sqrt {{\beta _1}^2{R_a}^2 + 2{\beta _2}{R_a}{\varphi _{a1}}(t )} }}{{{\beta _2}{R_a}}}$$

(20)$$\Delta {\omega _2}(t)\textrm{ = }\frac{{ - {\beta _1}{R_a}\textrm{ + }\sqrt {{\beta _1}^2{R_a}^2 + 2{\beta _2}{R_a}{\varphi _{a2}}(t )} }}{{{\beta _2}{R_a}}}$$

The kernel function is used as a representation of the frequency sweep characteristics of the laser. Different distance information corresponds to different interference signals, and the actual distance information can be obtained after inner product with the measurement. The kernel function can be used to decompose the measurement interferometer signal as given below. We assume that the phase of the measurement interferometer is φ_m(t), and the phase of the auxiliary interferometer is φ_a(t), and the initial phase of the measuring signal is expressed as φ₀. The cross-correlation operation of the above two signals can be expressed as

(21)$$\begin{aligned} &\sum {\cos ({\varphi _m}(t) + {\varphi _0})\exp [j({\varphi _a}(t))]} \\ \textrm{ = }&\sum {\cos ({\varphi _m}(t) + {\varphi _0})\cos ({\varphi _a}(t))} + j\sum {\cos ({\varphi _m}(t) + {\varphi _0})\sin ({\varphi _a}(t))} \\ \textrm{ = }&\frac{{\sum {\cos ({\varphi _m}(t) + {\varphi _0} - {\varphi _a}(t))} }}{2} + \frac{{\sum {\cos ({\varphi _m}(t) + {\varphi _0} + {\varphi _a}(t))} }}{2}\\ \textrm{ + }&j\frac{{\sum {\sin ({\varphi _m}(t) + {\varphi _0} - {\varphi _a}(t))} }}{2} + j\frac{{\sum {\sin ({\varphi _m}(t) + {\varphi _0} + {\varphi _a}(t))} }}{2} \end{aligned}$$

By calculating the maximum value of the absolute value of the above formula, we can obtain φ_a(t) such that φ_m(t) φ_a(t), and the distance to be measured of the target can be accurately calculated at this time.

To extract the range of the measurement interferometer, a series of kernel functions are established to simplify the above process as shown below:

(22)$$W({{R_p},t} )= \exp \left[ {j{R_p}\frac{{\Delta {\omega_1}(t) - \Delta {\omega_2}(t)}}{c}} \right]$$

where R_p is a series of distances within the possible range of the target. The kernel functions contain the FM nonlinearity and is not affected by the dispersion of the auxiliary interferometer. This orthogonal basis can be used to decompose the measurement interferometer signal as given below.

(23)$$X({{R_p}} )= \sum\limits_{}^{} {{I_{mc}}(t )W({{R_p},t} )} \textrm{ = }\sum\limits_{}^{} {{I_{mc}}(t )\exp \left[ {j{R_p}\frac{{\Delta {\omega_1}(t) - \Delta {\omega_2}(t)}}{c}} \right]}$$

This equation shows that the maximum X(R_p) is obtained at R_p = τ_m₀c. Therefore, the initial distance can be determined from a plot of X(R_p).

As a long auxiliary interferometer is not required to implement the proposed method based on the FM nonlinear kernel function, the residual nonlinear error is relatively small and a high sampling rate is not required. Furthermore, the proposed method can solve the problem of dispersion mismatch.

3.2 Dynamic distance measurement

Measuring the dynamic distance is useful for analyzing the vibration characteristics of the target. Thus, two reference signals should be generated as given below.

(24)$${R_I}(t )= \sin [{{\tau_{m0}}\Delta {\omega_1}(t )} ]$$

(25)$${R_Q}(t )= \cos [{{\tau_{m0}}\Delta {\omega_1}(t )} ]$$

Figure 3 shows how the dynamic distance can be determined. The FMCW signal shown in Eq. (8) is multiplied by the reference signals (Eqs. (24) and (25)) and sent to the low-pass filters. The output is given below.

(26)$$\begin{aligned} {I_I}(t ) &= \frac{{{A_{m1}}}}{2}\sin [{\Delta {\tau_m}(t ){\omega_0} + {\omega_\textrm{0}}{\tau_{m\textrm{0}}}\textrm{ + }\Delta {\omega_1}(t )\Delta {\tau_m}(t )} ]\\ &\approx \frac{{{A_{m1}}}}{2}\sin [{\Delta {\tau_m}(t ){\omega_0}} ]\end{aligned}$$

(27)$${I_Q}(t )= \frac{{{A_{m1}}}}{2}\cos [{\Delta {\tau_m}(t ){\omega_0}} ]$$

Fig. 3. The processing flow chart for dynamic-distance measurement.

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Then, the dynamic distance is

(28)$$\Delta {\tau _m}(t )\textrm{ = }\frac{1}{{{\omega _0}}}\textrm{ta}{\textrm{n}^{ - 1}}\left[ {\frac{{{I_I}(t )}}{{{I_Q}(t )}}} \right]$$

4. Experiment and analysis

4.1 Absolute-distance measurement

4.1.1 Parameter calibration

To verify the effectiveness of the proposed method, a laser-ranging system is built based on the device diagram shown in Fig. 2. Two external-cavity lasers (NewFocus.corp, TLM-8700, up to 110 nm mode-hop-free tuning, coherence length > 5km in swept mode.) are used, one of which has a tuning range of 1,520-1,580 nm and a tuning rate of 1000 nm/s or 100 THz/s. The second laser has a tuning range of 1,580-1,600 nm, and a tuning rate of 500 nm/s or 50 THz/s. Different frequency sweeping rates are used to enable the corresponding laser signals to be separated by filtering. The optical power of the external-cavity lasers can reach 10mW. The frequency-swept light from the external-cavity lasers is split using a 95:5 optical coupler, and 5% of the optical power is sent to an auxiliary Mach–Zehnder-type fiber interferometer.

To determine the absolute distance, the group delay and dispersion coefficient parameters (β₁R_a, β₂R_a) need to be calibrated. Figure 4 shows the calibration device constructed for this purpose. The measurement object is a corner cube on a platform 10 m from the device. The fiber collimator, two mirrors and corner cube constitute the optical path of our measurement system (marked by the yellow dashed box); the reference cube and the measurement cube constitute the measurement optical path of the laser interferometer (Renishaw.corp ML10, marked by the blue dashed box). Additionally, the cube mirror and the measuring mirror of the laser interferometer are fixed on the slider of the same guide rail, and after linear calibration, they can slide along the guide rail. Obviously, the sum of the measured value of our measurement system and the measured value of the laser interferometer is a constant value. The measured distance is compared with the known position of the laser interferometer Through the above process, we have completed the calibration of our system by the laser interferometer. The frequency modulation ranges of the two lasers are set to 40 nm and 20 nm. The device is placed on a vibration isolation platform, whereby the Doppler effect can be neglected.

Fig. 4. Device used for the comparative experiment.

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Equations (15) and (8) shows that in the absence of the Doppler effect, the phase increments of the signals of the auxiliary and measurement interferometers satisfy the following formula:

(29)$$\begin{aligned} \Delta {\varphi _a} &= {\varphi _{astart}} - {\varphi _{aend}}\\ \Delta {\varphi _m} &= {\varphi _{mstart}} - {\varphi _{mend}} \end{aligned}$$

(30)$$\frac{{\Delta {\varphi _{ai}}}}{{\Delta {\varphi _{mi}}}}\textrm{ = }\frac{{c[{\beta _1} + ({\omega _1} - {\omega _0}){\beta _2}]{R_a}}}{{{R_{mcal}}}}$$

Where c is the speed of light in vacuum, R_mcal is the distance measured with a laser interferometer (Renishaw ML10) and the entire FM period (1,520-1,580 nm) is divided into 14 segments. ω₀ is the initial angular frequency, which is approximately 2π×190 THz. i denotes the FM segment under consideration. ω_i is the center angular frequency of the ith FM segment. Different FM segments are used to calibrate the parameters as follows:

(31)$${\beta _2}{R_a} = \left( {\frac{{\Delta {\varphi_{ai}}}}{{\Delta {\varphi_{mi}}}} - \frac{{\Delta {\varphi_{ak}}}}{{\Delta {\varphi_{mk}}}}} \right)\frac{{{R_{mcal}}}}{{c({{\omega_i} - {\omega_k}} )}}$$

(32)$${\beta _1}{R_a} = \frac{{\Delta {\varphi _{ai}}}}{{\Delta {\varphi _{mi}}}}{R_{mcal}} - c({{\omega_i} - {\omega_0}} ){\beta _2}{R_a}$$

A target located at approximately 6 m from the interferometer is measured in this study.

The sampling number N is 20 million, which corresponds to a wavelength swept from 1,520 to 1,580 nm or a frequency swept from 198 to 190 THz. Figure 5 shows the measurement results obtained for different sampling sections, which can be approximately considered as functions of the laser frequency. We use Eq. (29) to determine β₂≈−2.3× 10⁻²⁶s²/(m·rad).

Fig. 5. Measurement results obtained using different center frequencies.

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4.1.2 Correction of the dispersion mismatch

An experiment was carried out to verify the calibration results. During the experiment, the target moves on a rail, and the true value of the movement is measured by the laser interferometer. A comparative analysis of the measurement results and those obtained using the laser interferometer is presented below.

The measurement result is verified, and a typical spectrum is shown in Fig. 6(a). A comparison of the measurement target spectrum before and after compensation following the Eq. (19) and Eq. (20) shows that the effect of dispersion has been completely eliminated. The results are both obtained under the condition of using nonlinear kernel function method. Compensation has eliminated the spectrum broadening and offset caused by fiber dispersion. As the length of the optical guide is limited to ∼2 m, we investigated 20 different displacements from the fiber collimator. Figure 6(b) shows the error between the laser interferometer readout position and the FMCW measurement versus the laser interferometer position. Before dispersion compensation, the standard deviation of 20 measurements is 16 µm. After compensation, the standard deviations are less than 1.2 µm for all measurements.

Fig. 6. (a) Dispersion-corrected results, where the blue and red curves show the uncalibrated and calibrated results, respectively; (b) measurement results compared to those obtained using a single-frequency laser interferometer.

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4.1.3 Measured absolute distance

When the measurement target is taken off the isolation platform, the Doppler effect cannot be neglected. The target measurement process is shown in Fig. 7.

Fig. 7. Left: photograph of the process of measuring the prototype. Right: the plaster head scanned in this study on which a red indicator light is being shone.

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The measurement signal acquired by the balanced photodetector is shown in Fig. 8. The original signal spectrum obtained by different detectors is shown in Fig. 8(a) and 8(b), and the quantities required for the calculation presented above can be determined using different bandwidth filters. The time domain signal before and after filtering is shown in Fig. 8(c) and 8(d).

Fig. 8. (a) Measurement signals acquired by PD1, including those for the fiber end face and the target at ∼10 m; (b) signals of auxiliary interferometers at a distance of ∼20 m; (c) and (d) show magnified regions of the signals shown in (a) and (b), respectively.

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Before using the auxiliary signal to perform the nonlinear correction on the measurement signal, the measurement signal is filtered to obtain the time-domain waveform and frequency spectrum, which are shown in Fig. 9(a).

Fig. 9. (a) (b)Measurement signal (a) before and (b) after correction. (c) Distance spectrum measured using the proposed method. (d) Measurement results

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Using the proposed method to change the frequency spectrum of the signal of the measurement path from that of the FFT affects the frequency subdivision, and the resulting signal of the measurement path is shown in Fig. 9(b). Figure 9(c) shows the distance spectrum measured by the proposed method. The standard deviation of 20 measurements is 7.8 µm as shown in Fig. 9(d) and mainly results from the undulation angle and vibration of the target surface.

4.2. Dynamic distance measurement

We performed a dynamic measurement of a single point on a plaster head; Fig. 10 shows the results obtained by processing the measured signals using the method presented in Fig. 3 and the phase-ratio method.

Fig. 10. Dynamic distance measured using (a) the proposed method and (b) the phase-ratio method.

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As the experiment described above could not be used to verify the accuracy of dynamic distance measurement, we designed the following experiment. A measurement experiment for vibration targets was designed to demonstrate the advantages offered by dynamic measurement. The measurement device is shown in Fig. 11. The focusing lens focuses the measurement beam onto a white paper pasted on the surface of a piezoelectric ceramic (PI corp, P-753, 38-µm travel range and 0.1-nm resolution), thereby obtaining a measurement signal with a low signal-to-noise ratio (SNR) and simulating actual test conditions. We measured the distance of the paper from the device as 3.136845 m (including the length of the optical fiber). The excitation signal of the piezoelectric ceramic is the sum of a 50-Hz signal and a 66.6-Hz signal and has an amplitude of ∼1 µm.

Fig. 11. Device used to measure the dynamic distance

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The measured results are shown in Fig. 12. The phase-ratio method can produce the correct measurement result for a target signal with a high SNR but does not yield an accurate result in this experiment because the signal has a low SNR. However, if the phase of the measured signal does not need to be determined, the phase-ratio method can be used under low SNR conditions.

Fig. 12. Comparison of the dynamic distance measured using (a) and (b) the phase-ratio method and the proposed method. (c) Measurement error in results obtained using the proposed method and a capacitive sensor. (d) Spectra of the capacitive sensor signal and measured signal.

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Figure 12(a) and 12(b) show the dynamic measurement results obtained using the phase-ratio method and the proposed measurement method. For signals with a low SNR, the measurement result obtained using the proposed method is significantly more accurate than that obtained using the phase-ratio method. For the device proposed for measuring the dynamic distance, the wavelength of the measuring light used is 1550 nm, the signal-to-noise ratio of the acquisition system is approximately 68 dB (Alazar.corp ATS9350), and the optimal measurement resolution of vibration is calculated by the formula SNR = −20lg(Δφ_n) to be approximately 130 pm. Figure 12(c) shows that the standard deviation of the error is less than 0.1 µm for all measurements. Figure 12(d) shows the spectra of the capacitive sensor signal and measured signal, where 49.8-Hz and 66.6-Hz sine wave signals generated by the function generator are used as excitations. The spectral analysis results are 49.7949 Hz and 66.6113 Hz for the feedback signal of the capacitance sensor and 49.7656 Hz and 66.6182 Hz for the measured vibration. There are some low-frequency components in the actual measurement signal, which are mainly caused by environmental vibrations impinging on the optical platform.

4.3 3D imaging

3D imaging was performed by performing the FMCW range measurement using a two-dimensional turntable. An adjustable focus lens is used to focus light onto various targets. A red beam is coaxially coupled into the collimator as an indicator light for convenience and is unnecessary in practical applications. The emitted optical power is 10 mW. To prevent error in recognition of the FFT spectral peak, we introduce a threshold value for data processing. The beat frequency is only recorded for peak intensities above the threshold and used to calculate the distance; otherwise, the distance is null.

A piece of an aluminum workpiece with a fixed surface spacing (2 mm, 4 mm, 6 mm, 8 mm, and 10 mm) was customized, as shown in Fig. 13(a), and calibrated by a coordinate machine. The piece was placed at approximately 20 meters (with reflection by a plane mirror to increase the distance) from the measurement system and scanned in three dimensions: Fig. 13(b) shows the obtained point cloud. Figure 13(c) shows the result of matching the point cloud with the standard plane. The measured value at each point is subtracted from the true value to obtain the coordinate error of each point, which is shown in Fig. 13(d). These results show that the measurement error for most points is less than 40 µm. We then imaged a plaster head from a distance of 7 m. A photograph and the measurement results for the plaster head are shown in Fig. 13(e) and 13(f). The images contain 150×150 measurement points. The rendered 3D image is shown in Fig. 13(f). The 3D image is a good reproduction of the real scene, proving the reliability of the proposed LiDAR system in 3D imaging. The proposed LiDAR system has potential applications in many fields, such as 3D imaging and automatic vehicles.

Fig. 13. 3D imaging result obtained using the proposed FMCW-based LiDAR system. (a) Metal steps and (b) corresponding measured point cloud. (c) Measured height of a column of points. (d) Measured coordinate error compared with the true value. (e) Plaster head and (f) corresponding measured point cloud.

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5. Conclusion

A high-precision dynamic-distance extraction method is proposed in this study to improve the accuracy of a FMCW laser-ranging system. First, the principle of FMCW laser-ranging is used to establish a double-modulation system to improve the ranging precision for a moving target. A fiber FMCW laser-ranging system with an auxiliary interferometer is designed and built. The design also realizes the nonlinear correction of FM signal dispersion. The theoretical measurement range of the system can reach approximately 2.5 km (determined by a coherence length of 5 km). For an FM period of approximately 10 ms, the sampling bandwidth of the traditional method reaches approximately GHz, compared to only approximately several dozens of MHz for the proposed system. Experimental results obtained using the measurement system show that the system is not affected by target movement and dispersion mismatch and does not require a high sampling rate. For a target placed 10 m from the system, the ranging accuracy is better than 1.2 µm, which is considerably superior to that for the existing resampling range extraction method under the same conditions; thus, the proposed technique can be used to measure absolute distance with high accuracy and speed. Unlike the phase-ratio method, the proposed method can extract real-time vibration information of a target with a low signal-to-noise ratio, with a measurement error below 100 nm and an optimal measurement resolution of vibration of approximately 130 pm: thus, the proposed method is practical for measuring vibrations in industrial applications. Our system can simultaneously measure the absolute distance and vibration with high precision. Therefore, the proposed system has significant application value in the fields of precision manufacturing and space technology.

Funding

National Natural Science Foundation of China (61805059); China Postdoctoral Science Foundation (2018M641820).

Acknowledgments

Lu Cheng thanks the National Natural Science Foundation of China for funding.

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed to obtain the presented results.

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FMCW LiDAR with an FM nonlinear kernel function for dynamic-distance measurement

Abstract

1. Introduction

2. Principles

2.1 Principle of the dual-laser FMCW

2.2 Influence of dispersion

3. Dynamic-distance measurement based on an FM nonlinear kernel function

3.1 Measurement of the initial distance

3.2 Dynamic distance measurement

4. Experiment and analysis

4.1 Absolute-distance measurement

4.1.1 Parameter calibration

4.1.2 Correction of the dispersion mismatch

4.1.3 Measured absolute distance

4.2. Dynamic distance measurement

4.3 3D imaging

5. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (13)

Equations (32)

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