Dynamic Distance Measurement Based on a Fast Frequency-Swept Interferometry

To improve the precision of dynamic distance measurement based on the frequency-swept interferometry (FSI) system, a Doppler-induced error compensation model based on a scheme increasing the frequency sweeping rate is proposed. A distance demodulation method based on a Fourier transformation is investigated when the defined quasi-stationary coefficient approaches a constant. Simulations and experiments based on dynamic distance with a sinusoidal change demonstrate that the proposed method has a standard deviation of 0.09 μm within a distance range of 4 μm at a sweeping rate of 60 KHz.


Introduction
Frequency-swept interferometry (FSI) technology has many advantages, such as high precision, high response speed, etc., when applied in absolute distance measurement for a static target. However, it will introduce the Doppler-induced error for a dynamic target [1][2][3]. The common way to solve this problem is to introduce additional hardware or components to increase a known quantity. For example, Richard Schneider et al. [4] devised a dual-laser sweeping system. The Doppler-induced error is eliminated by averaging the two-phase shifts produced by the frequency-sweeping of two lasers in opposite directions. Warden et al. [5,6] presented a dual-FSI system with a gas absorption cell to achieve dynamic OPD measurement at any sampling point. Pollinger and Liu et al. [7,8] used an additional heterodyne interferometry to directly measure the target movement and used an FSI system to calculate the distance. Shao Bin et al. [9] reported a fixed-frequency laser to measure the target velocity and used a FSI system to calculate the distance. Although these methods all successfully reduced the Doppler-induced error, they also increased the complexity, both for the demodulation algorithm and the measurement system.
To simplify hardware configuration, based on the hypothesis that the target is drifting at a constant speed or acceleration during the laser sweeping, Swinkels et al. [10] proposed an algorithm to combine four consecutive phase measurements instead of the normal two to reduce the Doppler-induced error, without utilizing auxiliary laser. Z. Liu et al. [2,11] proposed real-time models using the Kalman filter with one frequency-sweeping laser and in-phase and quadrature detection, which also assumed the target varied by a constant velocity or acceleration during the laser sweeps.
Aiming at measuring the dynamic distance where the target velocity is an arbitrary variable, in this paper, a Doppler-induced error compensation model based on a scheme increasing the frequency sweeping rate is proposed with only one frequency-sweeping laser. A distance demodulation method combining a Fourier transformation with a correlationlike algorithm is investigated when the defined quasi-stationary coefficient approaches a Figure 1a shows the schematic of a static distance measurement with an FSI system. One part of the light beam is reflected at the end face of the fiber, its frequency is recorded as u a (t). The rest of light is transmitted out of the fiber and reflected by the target, then coupled back into the fiber, its frequency is recorded as u b (t). The beat frequency can be written as [12]:

Principle of Reducing the Doppler-Induced Error
where n is the refractive index between the fiber end-face and the target, n = 1 when in the air; u is the optical frequency; ∆u is the optical frequency range; c is the speed of light; f b is the beat frequency of the FSI signal; t is the sweeping time; and T is the sweeping cycle.
Sensors 2022, 22, x FOR PEER REVIEW 2 of 12 laser. A distance demodulation method combining a Fourier transformation with a correlation-like algorithm is investigated when the defined quasi-stationary coefficient approaches a constant. Without utilizing auxiliary laser and complex algorithms, this method can greatly simplify the distance measurement based on the fast FSI. Figure 1a shows the schematic of a static distance measurement with an FSI system. One part of the light beam is reflected at the end face of the fiber, its frequency is recorded as ua(t). The rest of light is transmitted out of the fiber and reflected by the target, then coupled back into the fiber, its frequency is recorded as ub(t). The beat frequency can be written as [12]:

Principle of Reducing the Doppler-Induced Error
where n is the refractive index between the fiber end-face and the target, n = 1 when in the air; u is the optical frequency; Δu is the optical frequency range; c is the speed of light; fb is the beat frequency of the FSI signal; t is the sweeping time; and T is the sweeping cycle. Specifically, the light frequency ua(t) and ub(t) vary linearly in a sweeping cycle and the time delay between two beams is constant, as shown in Figure 1b. So, the beat frequency fb is a constant in a sweeping cycle. According to Equation (1), the static distance is: However, for a dynamic target shown in Figure 2, the distance L changes during a sweeping cycle, so the beat frequency fb is no longer a constant. To analysis the instantaneous beat frequency, the light frequency reflected at fiber end-face is still assumed as ua(t), the light frequency arriving at the target is uc(t), the light travelling time from the fiber end-face to the target is τ, then, uc(t) can be written as:  Specifically, the light frequency u a (t) and u b (t) vary linearly in a sweeping cycle and the time delay between two beams is constant, as shown in Figure 1b. So, the beat frequency f b is a constant in a sweeping cycle. According to Equation (1), the static distance is: However, for a dynamic target shown in Figure 2, the distance L changes during a sweeping cycle, so the beat frequency f b is no longer a constant. To analysis the instantaneous beat frequency, the light frequency reflected at fiber end-face is still assumed as u a (t), the light frequency arriving at the target is u c (t), the light travelling time from the fiber end-face to the target is τ, then, u c (t) can be written as: Sensors 2022, 22, x FOR PEER REVIEW 2 of 12 laser. A distance demodulation method combining a Fourier transformation with a correlation-like algorithm is investigated when the defined quasi-stationary coefficient approaches a constant. Without utilizing auxiliary laser and complex algorithms, this method can greatly simplify the distance measurement based on the fast FSI. Figure 1a shows the schematic of a static distance measurement with an FSI system. One part of the light beam is reflected at the end face of the fiber, its frequency is recorded as ua(t). The rest of light is transmitted out of the fiber and reflected by the target, then coupled back into the fiber, its frequency is recorded as ub(t). The beat frequency can be written as [12]:

Principle of Reducing the Doppler-Induced Error
where n is the refractive index between the fiber end-face and the target, n = 1 when in the air; u is the optical frequency; Δu is the optical frequency range; c is the speed of light; fb is the beat frequency of the FSI signal; t is the sweeping time; and T is the sweeping cycle. Specifically, the light frequency ua(t) and ub(t) vary linearly in a sweeping cycle and the time delay between two beams is constant, as shown in Figure 1b. So, the beat frequency fb is a constant in a sweeping cycle. According to Equation (1), the static distance is: However, for a dynamic target shown in Figure 2, the distance L changes during a sweeping cycle, so the beat frequency fb is no longer a constant. To analysis the instantaneous beat frequency, the light frequency reflected at fiber end-face is still assumed as ua(t), the light frequency arriving at the target is uc(t), the light travelling time from the fiber end-face to the target is τ, then, uc(t) can be written as:  Consider the Doppler effect, the light frequency reflected at the target u d (t) is [13]: where v(t) is the velocity of the target. Assume that the light traveling time from the fiber end-face to the target is equal to the time from the target to the fiber end-face, the light frequency going back to the fiber end-face u b (t) is: Then the instantaneous beat frequency is: According to Equation (2), the dynamic instantaneous distance in a sweeping cycle is calculated to be: where f sweep = 1/T, f sweep is the sweeping rate. The first term in Equation (7) is the real distance L, the second term is the Dopplerinduced error L error . We can see, L error is related to the velocity of the target v(t) and the sweeping rate of the light f sweep . As f sweep increases, L error rapidly decreases and then tends to be constant. When v(t) and light frequency are fixed, the greater the f sweep is, the smaller the L doppler is, as shown in Figure 3.
where v(t) is the velocity of the target.
Assume that the light traveling time from the fiber end-face to the target is eq the time from the target to the fiber end-face, the light frequency going back to th end-face ub(t) is: Then the instantaneous beat frequency is: According to Equation (2), the dynamic instantaneous distance in a sweeping is calculated to be: The first term in Equation (7) is the real distance L, the second term is the Do induced error Lerror. We can see, Lerror is related to the velocity of the target v(t) a sweeping rate of the light fsweep. As fsweep increases, Lerror rapidly decreases and then to be constant. When v(t) and light frequency are fixed, the greater the fsweep is, the s the Ldoppler is, as shown in Figure 3. Although increasing the sweeping rate helps to reduce the Doppler-induced Ldoppler, it is still 12 μm at a very high sweeping rate (300 KHz), which is too large compared to the distance variation of 0.3 μm.
The real instantaneous distance, which is the first term in Equation (7), can be f described to be [2]: Although increasing the sweeping rate helps to reduce the Doppler-induced error L doppler , it is still 12 µm at a very high sweeping rate (300 KHz), which is too large when compared to the distance variation of 0.3 µm.
The real instantaneous distance, which is the first term in Equation (7), can be further described to be [2]: where L(0) is the initial distance in one sweeping cycle and a is the acceleration. From Equation (7), we can see that when the sweeping cycle becomes small, the distance change in one sweeping cycle will also be small. At this time, take Equation (8) into Equation (7), let L dyn (T), which is the distance at the end moment T of a sweeping cycle, represent the distance after the whole sweeping cycle. It is: In Equation (9), the sweeping cycle T is assumed to be sufficiently small. In this case, the distance change in one cycle is considered to be very small, the velocity or acceleration is regarded as a constant.
Define a Doppler coefficient to be: Let both sides of Equation (9) be divided by Equation (10) after moving L(0) in Equation (9) from right to the left. The distance variation is: Then after further suppressing the Doppler-induced error, the measured distance is: The principle residual error is expressed as: From Equation (13) we can see that L error is related to a and T. When T is getting smaller, which means the sweeping rate f sweep is higher, L error decreases. The larger a is, the larger L error becomes. Further simulation based on Equation (13) is given in Figure 4. When f sweep is 60 KHz, the residual error L error has been reduced to 0.13 nm and L error is only 1.3 nm even when the acceleration a is 10 m/s 2 . That is, we can suppress the Doppler-induced error to the order of a nanometer, by increasing the sweeping rate of the light frequency to some extent.

Applicable Conditions of Frequency Demodulation
For a static distance measurement, the interference signal in one sweeping cycle is [14]: It is a standard cosine function with a single beat frequency fb, as shown in Figure 5a. The fast Fourier transformation (FFT) can be used to get its interference frequency in the frequency domain shown in Figure 5b, then calculate the distance according to the relationship between beat frequency and distance using Equation (2).

Applicable Conditions of Frequency Demodulation
For a static distance measurement, the interference signal in one sweeping cycle is [14]: It is a standard cosine function with a single beat frequency f b , as shown in Figure 5a. The fast Fourier transformation (FFT) can be used to get its interference frequency in the frequency domain shown in Figure 5b, then calculate the distance according to the relationship between beat frequency and distance using Equation (2). 5.1 × 10 12 Hz, ua = 1.9365 × 10 14 Hz, v = 0.1 m/s, a = 0.1 m/s 2 , fsweep = 60 KHz).

Applicable Conditions of Frequency Demodulation
For a static distance measurement, the interference signal in one sweeping cycle is [14]: It is a standard cosine function with a single beat frequency fb, as shown in Figure 5a. The fast Fourier transformation (FFT) can be used to get its interference frequency in the frequency domain shown in Figure 5b, then calculate the distance according to the relationship between beat frequency and distance using Equation (2). For a dynamic target, the interference signal can be derived by the instantaneous beat frequency fbd in Equation (6) as: Equation (15) indicates that the beat frequency fbd varies as the sweeping time t. The interference signal is not a uniform distribution, as shown in Figure 5a. To understand the signal from the frequency domain, the instantaneous beat frequency fbd is obtained after applying the FFT to Equation (15), as shown in Figure 6. When fsweep is low, the FFT spectrum contains multiple instantaneous frequencies (black curve), which correspond to the instantaneous distance change because of the movement of the target in a sweeping cycle. When fsweep is gradually rising, the frequencies gradually merge (red curve). For a dynamic target, the interference signal can be derived by the instantaneous beat frequency f bd in Equation (6) as: Equation (15) indicates that the beat frequency f bd varies as the sweeping time t. The interference signal is not a uniform distribution, as shown in Figure 5a. To understand the signal from the frequency domain, the instantaneous beat frequency f bd is obtained after applying the FFT to Equation (15), as shown in Figure 6. When f sweep is low, the FFT spectrum contains multiple instantaneous frequencies (black curve), which correspond to the instantaneous distance change because of the movement of the target in a sweeping cycle. When f sweep is gradually rising, the frequencies gradually merge (red curve). When the ratio of fsweep to the movement frequency of the target reaches "a certain value", the frequencies merge into one peak (blue curve). That means, there is only one beat frequency, which is the same as the static distance situation (shown in Figure 5b). It can be considered that the movement of the target compared to the light frequency sweeping is quasi-static. Therefore, the FFT frequency algorithm, instead of the complex demodulation methods [15][16][17][18], can be used to demodulate the beat frequency then calculate the distance at the end moment of one sweeping cycle using Equation (12).
To analyze the quantitative condition of the quasi-static state, define ρ as the quasistationary coefficient to describe the ratio of the static beat frequency to the dynamic beat frequency, which is: Equation (16) shows that ρ is related to v(t) and fsweep. Suppose v(t) is a cosine function When the ratio of f sweep to the movement frequency of the target reaches "a certain value", the frequencies merge into one peak (blue curve). That means, there is only one beat frequency, which is the same as the static distance situation (shown in Figure 5b). It can be considered that the movement of the target compared to the light frequency sweeping is quasi-static. Therefore, the FFT frequency algorithm, instead of the complex demodulation methods [15][16][17][18], can be used to demodulate the beat frequency then calculate the distance at the end moment of one sweeping cycle using Equation (12).
To analyze the quantitative condition of the quasi-static state, define ρ as the quasistationary coefficient to describe the ratio of the static beat frequency to the dynamic beat frequency, which is: Equation (16) shows that ρ is related to v(t) and f sweep . Suppose v (t) is a cosine function to imitate the target moving at an arbitrary speed. That is: where A is the range of distance change and ω is the frequency. The value ρ is simulated according to Equations (16) and (17), and plotted in Figure 7. The value of ρ is related to ω, A and f sweep , as shown in Figure 7a. To understand the relationship between ρ and f sweep more clearly, extract some specific curves when A is ±10 µm, ±50 µm, and ±100 µm and plot in Figure 7b-d. It shows that, as the sweeping rate f sweep of the laser increases, ρ gradually approaches a constant, which indicates the movement of the target compared to the light frequency-sweeping is quasi-static. However, the quasi-static state is affected by ω and A. Comparing the f sweep corresponding to the points with ρ = 0.9 and ω = 1000 Hz in Figure 7b-d, it can be seen that ρ rises slower when A is larger. That is, the larger the ω or A, the longer ρ takes to reach a constant, the larger the corresponding f sweep . Therefore, before using the FFT frequency algorithm, the highest frequency ω max and the range of distance change A need to be calculated. For a simple calculation, when f sweep is around 20~50 times larger than the highest frequency of distance, the FFT algorithm would work, according to Figure 7 and is demonstrated by our practice.

Demodulation Algorithm
The FSI signal is actually a discrete sampled signal. The sampling points are equal to the swept points of the light frequency. To increase the resolution of FFT, a zero-padding extension to the discrete signal is often applied. Then, according to the FFT algorithm: where s[n] is the discrete form of Equation (15), NFFT is the length of s[n] after applying the zero-padding extension.

Demodulation Algorithm
The FSI signal is actually a discrete sampled signal. The sampling points are equal to the swept points of the light frequency. To increase the resolution of FFT, a zero-padding extension to the discrete signal is often applied. Then, according to the FFT algorithm: where s[n] is the discrete form of Equation (15), N FFT is the length of s[n] after applying the zero-padding extension. The distance is calculated to be: where n max is the location of the peak frequency. The demodulated distance L corr contains the real distance L and the Doppler-induced error L doppler . According to Equations (11) and (12), the measured distance L mear after suppressing the Doppler-induced error is: The data processing flow is shown in Figure 8. The first process of the raw spectrum is the mean value removing, normalization and zero-padding extension. Then, the distance is obtained using the FFT algorithm. After the Doppler-induced error compensation according to Equation (20), the dynamic distance can be demodulated.

Simulation
In order to verify the analysis above, a dynamic distance L(t) varies as a sinusoidal function which is given as: Figure 9a shows the distance L(t) varies as a sinusoidal function according to Equation (21). When the sweeping cycle T = 0.05 ms and the sweeping rate fsweep = 1/T = 20 KHz, the variation of the distance in one sweeping cycle is shown as Figure 9b, which is corresponding to the red part in Figure 9a. According to the quasi-static state condition, when the frequency of the dynamic dis-

Simulation
In order to verify the analysis above, a dynamic distance L(t) varies as a sinusoidal function which is given as: (21) Figure 9a shows the distance L(t) varies as a sinusoidal function according to Equation (21). When the sweeping cycle T = 0.05 ms and the sweeping rate f sweep = 1/T = 20 KHz, the variation of the distance in one sweeping cycle is shown as Figure 9b, which is corresponding to the red part in Figure 9a.

Simulation
In order to verify the analysis above, a dynamic distance L(t) varies as a sinusoidal function which is given as: Figure 9a shows the distance L(t) varies as a sinusoidal function according to Equation (21). When the sweeping cycle T = 0.05 ms and the sweeping rate fsweep = 1/T = 20 KHz, the variation of the distance in one sweeping cycle is shown as Figure 9b, which is corresponding to the red part in Figure 9a. According to the quasi-static state condition, when the frequency of the dynamic distance is 1 KHz, the sweeping rates of the laser fsweep is set 20 KHz, 40 KHz or 60 KHz. The demodulation results are calculated by our proposed method and shown in Figure 10a-c, respectively. The corresponding demodulation errors for each sweeping rate are shown in Figure 10d-f. The mean value of the absolute demodulation error is 0-0.019 μm and the  According to the quasi-static state condition, when the frequency of the dynamic distance is 1 KHz, the sweeping rates of the laser f sweep is set 20 KHz, 40 KHz or 60 KHz. The demodulation results are calculated by our proposed method and shown in Figure 10a-c, respectively. The corresponding demodulation errors for each sweeping rate are shown in Figure 10d-f. The mean value of the absolute demodulation error is 0-0.019 µm and the maximum value is 0.13-0.16 µm, accounting for 0.3% of the demodulation range (60 µm). Results show our proposed method is able to achieve high demodulation accuracy even when the dynamic distance is in the form of a sinusoidal function, which means both velocity and acceleration are changing during sweeping. According to the quasi-static state condition, when the frequency of the dynamic distance is 1 KHz, the sweeping rates of the laser fsweep is set 20 KHz, 40 KHz or 60 KHz. The demodulation results are calculated by our proposed method and shown in Figure 10a-c, respectively. The corresponding demodulation errors for each sweeping rate are shown in Figure 10d-f. The mean value of the absolute demodulation error is 0-0.019 μm and the maximum value is 0.13-0.16 μm, accounting for 0.3% of the demodulation range (60 μm). Results show our proposed method is able to achieve high demodulation accuracy even when the dynamic distance is in the form of a sinusoidal function, which means both velocity and acceleration are changing during sweeping.

Experiment
The dynamic distance measurement experiment was carried out to verify the proposed method, as shown in Figure 11. The distance was formed by the gap between the fiber end-face and the PZT (Pk4FA2H3P2, Thorlabs, Newton, NJ, USA). The PZT was driven by a signal generator (DG4102, RIGOL) and the amplifier (Has 4011, NF Corporation, Yokohama, Japan). The optical fiber was connected to a frequency-swept laser (Arcadia Optronix, Zhuhai, China, GC-760001c-01) and the photodetector module (Xilinx Artix-7, Conquer, Beijing, China). The entire setup was placed on a vibration-isolated optical stage.

Experiment
The dynamic distance measurement experiment was carried out to verify the proposed method, as shown in Figure 11. The distance was formed by the gap between the fiber end-face and the PZT (Pk4FA2H3P2, Thorlabs, Newton, NJ, USA). The PZT was driven by a signal generator (DG4102, RIGOL) and the amplifier (Has 4011, NF Corporation, Yokohama, Japan). The optical fiber was connected to a frequency-swept laser (Arcadia Optronix, Zhuhai, China, GC-760001c-01) and the photodetector module (Xilinx Artix-7, Conquer, Beijing, China). The entire setup was placed on a vibration-isolated optical stage. The sinusoidal vibration frequency of the PZT was set to 100 Hz, 500 Hz, and 1 KHz, and the laser was scanned at different frequency intervals. The dynamic distance was expressed as: The sinusoidal vibration frequency of the PZT was set to 100 Hz, 500 Hz, and 1 KHz, and the laser was scanned at different frequency intervals. The dynamic distance was expressed as: where L(0) is 250 µm, ω PZT is the PZT vibration frequency, and the vibration amplitude The interference spectra of each sweeping cycle were collected by the laser at the sweeping rates of 20 KHz, 40 KHz, and 60 KHz, respectively, and the demodulation results were calculated according to the data processing flow shown in Figure 8 and given in Figures 12-14. Figure 12 shows the demodulation distances with f sweep = 20 KHz, and the frequency of the distance is ω PZT of 100 Hz, 500 Hz, and 1 KHz, respectively. Figure 13 shows f sweep = 40 KHz and Figure 14 shows f sweep = 60 KHz. The red dots indicate the demodulated distance, and the black delineated lines are the ideal distance according to the parameters of the signal generator. The sinusoidal vibration frequency of the PZT was set to 100 Hz, 500 Hz, and 1 KHz, and the laser was scanned at different frequency intervals. The dynamic distance was expressed as: where L(0) is 250 μm, ωPZT is the PZT vibration frequency, and the vibration amplitude APZT = ±2 μm.
The interference spectra of each sweeping cycle were collected by the laser at the sweeping rates of 20 KHz, 40 KHz, and 60 KHz, respectively, and the demodulation results were calculated according to the data processing flow shown in Figure 8 and given in Figure 12-14. Figure 12 shows the demodulation distances with fsweep = 20 KHz, and the frequency of the distance is ωPZT of 100 Hz, 500 Hz, and 1 KHz, respectively. Figure 13 shows fsweep = 40 KHz and Figure 14 shows fsweep = 60 KHz. The red dots indicate the demodulated distance, and the black delineated lines are the ideal distance according to the parameters of the signal generator.   The demodulation errors between the demodulated distances and ideal distances in Figures 12-14 are calculated and given in Figure 15. The average standard deviation is 0.14 μm, 0.11 μm, and 0.09 μm when fsweep is 20 KHz, 40 KHz, and 60 KHz, respectively. The larger fsweep, the smaller the fluctuation of the demodulated distance, which means the smaller the error, the mean value of the demodulation errors in each condition is plotted in Figure 16. It can be seen that when fsweep is fixed, Lerror increases with the increase in the ωPZT; when ωPZT is fixed, Lerror decreases with the increase in fsweep. The maximum error is 0.41 μm and the minimum error is 0.001 μm.  The demodulation errors between the demodulated distances and ideal distances in Figures 12-14 are calculated and given in Figure 15. The average standard deviation is 0.14 μm, 0.11 μm, and 0.09 μm when fsweep is 20 KHz, 40 KHz, and 60 KHz, respectively. The larger fsweep, the smaller the fluctuation of the demodulated distance, which means the smaller the error, the mean value of the demodulation errors in each condition is plotted in Figure 16. It can be seen that when fsweep is fixed, Lerror increases with the increase in the ωPZT; when ωPZT is fixed, Lerror decreases with the increase in fsweep. The maximum error is 0.41 μm and the minimum error is 0.001 μm. The demodulation errors between the demodulated distances and ideal distances in Figures 12-14 are calculated and given in Figure 15. The average standard deviation is 0.14 µm, 0.11 µm, and 0.09 µm when f sweep is 20 KHz, 40 KHz, and 60 KHz, respectively. The larger f sweep , the smaller the fluctuation of the demodulated distance, which means the smaller the error, the mean value of the demodulation errors in each condition is plotted in Figure 16. It can be seen that when f sweep is fixed, L error increases with the increase in the ω PZT ; when ω PZT is fixed, L error decreases with the increase in f sweep . The maximum error is 0.41 µm and the minimum error is 0.001 µm. The demodulation errors between the demodulated distances and ideal distances in Figures 12-14 are calculated and given in Figure 15. The average standard deviation is 0.14 μm, 0.11 μm, and 0.09 μm when fsweep is 20 KHz, 40 KHz, and 60 KHz, respectively. The larger fsweep, the smaller the fluctuation of the demodulated distance, which means the smaller the error, the mean value of the demodulation errors in each condition is plotted in Figure 16. It can be seen that when fsweep is fixed, Lerror increases with the increase in the ωPZT; when ωPZT is fixed, Lerror decreases with the increase in fsweep. The maximum error is 0.41 μm and the minimum error is 0.001 μm.     Figure 15. The average standard dev 0.14 μm, 0.11 μm, and 0.09 μm when fsweep is 20 KHz, 40 KHz, and 60 KHz, resp The larger fsweep, the smaller the fluctuation of the demodulated distance, which m smaller the error, the mean value of the demodulation errors in each condition is in Figure 16. It can be seen that when fsweep is fixed, Lerror increases with the increa ωPZT; when ωPZT is fixed, Lerror decreases with the increase in fsweep. The maximum 0.41 μm and the minimum error is 0.001 μm.  μm. It can be seen from the comparison that the larger fsweep is, the smaller the error is when ωPZT is a constant.

Conclusions
A Doppler-induced error compensation model based on a scheme to increase the frequency sweeping rate is proposed. A distance demodulation method combining a Fourier transformation and a correlation-like algorithm is investigated when the defined quasistationary coefficient approaches a constant. Simulations and experiments based on dynamic distance with a sinusoidal change demonstrate that the proposed method has a standard deviation of 0.09 μm within a distance range of 4 μm at a sweeping rate of 60 KHz.
The proposed method is based only on the basic FSI system without any additional de-

Conclusions
A Doppler-induced error compensation model based on a scheme to increase the frequency sweeping rate is proposed. A distance demodulation method combining a Fourier transformation and a correlation-like algorithm is investigated when the defined quasi-stationary coefficient approaches a constant. Simulations and experiments based on dynamic distance with a sinusoidal change demonstrate that the proposed method has a standard deviation of 0.09 µm within a distance range of 4 µm at a sweeping rate of 60 KHz.
The proposed method is based only on the basic FSI system without any additional devices which can greatly simplify the hardware system; it is also based on the FFT and correlation-like algorithm instead of complex methods, which can simplify the calculation.

Data Availability Statement:
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest:
The authors declare no conflict of interest.