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Dynamic nonlinearity errors in laser Doppler vibrometer measurements induced by environmental vibration and error correction

Lu Cheng, Xu Ziyi, Liu Guodong, Liu Bingguo, Chen Fengdong, Gan Yu, and Lu Binghui

Open Access Open Access

Abstract

Laser Doppler vibrometers (LDVs) are widely used for vibration testing in various fields. Nonlinearity errors are the key factor affecting the measurement accuracy of LDVs. The conventional Heydemann method cannot correct nonlinearity errors produced by noisy environments. Thus, we establish a novel model to describe dynamic nonlinearity errors produced in noisy environments and propose a compensation method to mitigate signal distortion. The performance of the proposed method is assessed by performing both simulations and experiments. The results of experiments carried out in a noisy environment indicate that the proposed method suppresses the nonlinearity to 30 nm compared to 737 nm using the conventional Heydemann correction. The proposed method can improve the accuracy of LDV measurements in industrial environments.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Laser Doppler vibrometers (LDVs) can be used to make high-precision noncontact optical measurements of the vibration velocity and amplitude of a surface without mass effects. LDV has been widely used in applications such as aerospace, precision manufacturing, structural health monitoring, and the life sciences [1–6]. The overall accuracy of an LDV measurement depends on a variety of factors. Unlike interferometers used for dimensional metrology, the laser wavelength instability and variations in the refractive index of air are not the main sources of error for measuring vibrations with relatively low displacement amplitudes. Undesirable reflection of the optical path, which is called multipath interference, distorts a measurement signal and introduces nonlinearity errors. Therefore, it is necessary to correct nonlinearity errors to improve the accuracy of the measurement results.

Nonlinearity errors can be corrected by well-chosen postprocessing algorithms. A commonly used correction method based on ellipse fitting was proposed by Heydemann [7]. The Heydemann correction method has been applied in many studies to compensate for nonlinearity errors caused by polarization mixing, unequal gain of detectors, and lack of quadrature [8–11]. However, nonlinearity errors have been reduced in some studies by adjusting the gain of the quadrature detector of the homodyne interferometer [12,13], realigning the axes of waveplates to specific angles in the homodyne interferometer [14], and adding an optical shutter to the homodyne laser vibrometer to pre-extract nonlinear parameters before measurement [15]. Although good results have been obtained in these studies, the nonlinear residuals introduced by the ghost reflection of the waveplates, lens, and other optical devices in the optical system were neglected, and the compensated signal retained nonlinearity errors. A ghost reflection is multiorder and therefore introduces complex nonlinearity errors. Many models for these ghost reflections have been developed, but no specific correction methods have been proposed [16–19]. Some attempts have been made to reduce ghost reflection by improving the optical system, such as by coating the optics [20], adjusting the ghost reflection angle, and using spatial filtering [21,22]. Although these approaches have improved correction results, the production of nonlinear residuals prevents elimination of the nonlinearity error caused by ghost reflection. Li et al. developed a compensation algorithm [23] for strong second-order ghost reflections caused by the lens in an optical system. This algorithm can eliminate the nonlinearity error to some extent, but the result is deteriorated by a very strong or weak second-order ghost reflection.

The aforementioned methods were proposed under the preconditions that the multipath interference is stable and the Lissajous curve forms an ellipse whose parameters can be treated as constant. Constant ellipse parameters can be used to correct nonlinearity errors. However, multipath interference is usually time-varying in practical measurements, especially those performed in noisy environments. For example, when the target to be measured is located in a vacuum tank, the multipath interference introduced by the vacuum window changes drastically with vibrations of the vacuum tank. In this case, the nonlinearity error appears dynamic and cannot be completely eliminated by the Heydemann correction. A piecewise Heydemann correction method is typically used to overcome this problem, which however is still based on ellipse fitting [24]. However, for severe environmental vibration, this method produces poor results and is computationally intensive.

In this study, we theoretically analyze the influence of dynamic multipath interference to find a helical trajectory for the Lissajous curves corresponding to a noisy environment. Thus, we propose a model to describe the Lissajous curves of the quadrature signals. This model is in turn used to develop a method based on a helix-fitting method to correct the nonlinearity error. We use simulations and experiments to verify the effectiveness of the proposed method. Experimental results show that the residual error obtained using the proposed method is an order of magnitude lower than that obtained using existing methods.

2. Principle

2.1 Basic LDV

A schematic of a basic LDV is shown in Fig. 1. The output of the single-frequency laser (SFL) is split using a 99:1 optical coupler, and both the resulting outputs are frequency-shifted by the acousto-optic modulators (AOM1 and AOM2). The two AOMs have different frequency shifts. Ninety-nine percent of the AOM1 output is used as a measurement light source, which passes through the fiber collimator, polarizing beam splitter, focusing lens, and quarter waveplate and converges on the target surface. The reflected light from the target is frequency-shifted by the target vibration. As the returned light passes through the quarter-wave plate twice, this light enters a second fiber collimator after being reflected by the PBS. Then, this measurement light source and one output of AOM2 enter the PD1 and interfere. The other outputs of AOM1 and AOM2 enter an auxiliary interferometer.

 figure: Fig. 1.

Fig. 1. Schematic of the heterodyne laser Doppler vibration measurement system. (SFL: single-frequency laser; AOM: acousto-optic modulator; CO: fiber coupler, where the numbers represent the splitting rate; FC: fiber collimator; PBS: polarizing beam splitter; QWP: quarter-wave plate; PD: photodetector).

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The measurement signal detected by PD1 can be expressed as

$${I_m}(t) = {A_m}\cos \left[ {2\pi {f_{AOM}}t + \frac{{4\pi \Delta d(t )}}{\lambda }} \right] = {A_m}\cos [{2\pi {f_{AOM}}t + \theta (t)} ],$$
and the reference signal detected by PD2 can be expressed as
$${I_a}(t) = {A_a}\cos (2\pi {f_{AOM}}t),$$
where Am represents the amplitude of the measurement signal, Aa represents the amplitude of the reference signal, and fAOM represents the difference in the frequency shifts of the two AOMs. The term Δd is the change in the distance caused by the target vibration, and λ is the SFL wavelength. The term θ(t) is the Doppler phase shift caused by the target motion. The phase θ(t) needs to be calculated to obtain Δd. The calculation process is shown in Fig. 2.

 figure: Fig. 2.

Fig. 2. Flow chart of quadrature demodulation (LPF: low-pass filter).

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2.2 Influence of multipath interference

Figure 3 shows the undesirable reflection that typically occurs due to interference during practical measurements, such as lens return light or quarter wave plate (QWP).

 figure: Fig. 3.

Fig. 3. Schematic of multipath interference.

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In this case, the measurement signal detected by PD1 can be expressed as

$${I_m}(t) = {A_m}\cos [{2\pi {f_{AOM}}t + \theta (t)} ]+ \sum\limits_{k = 1}^K {{A_{ek}}\cos [{2\pi {f_{AOM}}t + {\theta_{ek}}(t)} ]} ,$$
where K represents the number of the multipath interference, Aek and θek(t) represent the amplitude and phase of each multipath interference, respectively. For I\Q demodulation, the quadrature signals I and Q shown in Fig. 2 can be expressed as
$$I(t) = \frac{{{A_m}{A_a}}}{2}\cos [{\theta (t)} ]+ \sum\limits_{k = 1}^K {\frac{{{A_{ek}}{A_a}}}{2}\cos [{{\theta_{ek}}(t)} ]} = A\cos [{\theta (t)} ]+ \sum\limits_{k = 1}^K {{B_k}\cos [{{\theta_{ek}}(t)} ]} ,$$
$$Q(t) = \frac{{{A_m}{A_a}}}{2}\sin [{\theta (t)} ]+ \sum\limits_{k = 1}^K {\frac{{{A_{ek}}{A_a}}}{2}\sin [{{\theta_{ek}}(t)} ]} = A\sin [{\theta (t)} ]+ \sum\limits_{k = 1}^K {{B_k}\sin [{{\theta_{ek}}(t)} ]} ,$$
where for simplicity, AmAa/2 is denoted by A, and AekAa/2 is denoted by Bk. Then, the demodulation phase θcal is obtained as
$${\theta _{cal}}(t )= \arctan \left[ {\frac{{Q(t )}}{{I(t )}}} \right] = \arctan \left\{ {\frac{{A\sin [{\theta (t)} ]+ \sum\limits_{k = 1}^K {B\sin [{{\theta_{ek}}(t)} ]} }}{{A\cos [{\theta (t)} ]+ \sum\limits_{k = 1}^K {{B_k}\cos [{{\theta_{ek}}(t)} ]} }}} \right\}.$$
θcal is not equal to θ(t), and the deviation between these two terms is usually called the nonlinearity error. Figure 4 shows the simulation and demodulation results for different ratios of the main reflection Am to the multipath reflection Ae (MMRs), with only one multipath interference. The vibration frequency and amplitude of the target are 300 Hz and 1 µm, respectively. An ideal Lissajous curve is a circle centered at the origin of the coordinate system. If the multipath inference is stable, the Lissajous curve will be a circle for which the center deviates from the origin.

 figure: Fig. 4.

Fig. 4. Cases of different MMRs with dynamic nonlinear errors: (a) the Lissajous curves, (b) the simulated demodulation results.

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However, multipath interference in industrial environments is dynamic. In this case, the Lissajous curve is not a circle. Figure 4(a) shows that the Lissajous curve is a helix for a vibration frequency and amplitude of multipath interference of 15 Hz and 0.25 µm, respectively. The Lissajous curve moves increasingly away from the origin as the MMR decreases. The calculated target vibration is shown in Fig. 4(b), and the dynamic nonlinearity error increases as the MMR decreases. That is, the dynamic nonlinearity error has a larger effect when the return light of the target is weaker than the multipath interference, which is very common in rough noncooperative target measurement.

2.3 Correction method for dynamic nonlinearity error based on helix-fitting

Figure 4(a) shows that the effect of the dynamic nonlinearity error is that the Lissajous curve is a helix. The existing methods based on circle- or ellipse-fitting are no longer applicable in this case. Moreover, θe(t) is related to the environmental vibration and is impossible to determine by precalibration.

The multipath interference term can be defined as

$${I_e}(t) = \sum\limits_{k = 1}^K {{I_{ek}}(t)} = \sum\limits_{k = 1}^K {{B_k}\cos [{{\theta_{ek}}(t)} ]} ,$$
$${Q_e}(t) = \sum\limits_{k = 1}^K {{Q_{ek}}(t)} = \sum\limits_{k = 1}^K {{B_k}\sin [{{\theta_{ek}}(t)} ]} .$$

Taylor expansion of Iek(t) and Qek(t) in Eq. (7) and (8) around (t0 = 0, |t-t0|<1) yields the following results:

$$\begin{aligned} {I_{ek}}(t) &= {B_k} \ast \cos [{{\theta_{ek}}(0)} ]- {B_k} \ast {\theta _{ek}}^{\prime}(0) \ast \sin [{{\theta_{ek}}(0)} ]\ast t\\ &- 0.5 \ast {B_k} \ast \{{{{[{{\theta_{ek}}^{\prime}(0)} ]}^2} \ast \cos [{{\theta_{ek}}(0)} ]+ {\theta_{ek}}^{\prime\prime}(0) \ast \sin [{{\theta_{ek}}(0)} ]} \}\ast {t^2} + {R_{Ienk}}(t), \end{aligned}$$
$$\begin{aligned} {Q_{ek}}(t) &= {B_k} \ast \sin [{{\theta_{ek}}(0)} ]+ {B_k} \ast {\theta _{ek}}^{\prime}(0) \ast \cos [{{\theta_{ek}}(0)} ]\ast t\\ &- 0.5 \ast {B_k} \ast \{{{{[{{\theta_{ek}}^{\prime}(0)} ]}^2} \ast \sin [{{\theta_{ek}}(0)} ]- {\theta_{ek}}^{\prime\prime}(0) \ast \cos [{{\theta_{ek}}(0)} ]} \}\ast {t^2} + {R_{Qenk}}(t), \end{aligned}$$
where RIenk(t) is the residual of the Iek(t) Taylor expansion and RQenk(t) is the residual of the Qek(t) Taylor expansion and are given below:
$$\begin{aligned} {R_{Ienk}}(t) &\approx \frac{{{B_k}}}{6}\{{\sin [{{\theta_{ek}}(0)} ]\ast {{[{{\theta_{ek}}^{\prime}(0)} ]}^3} - 3 \ast \cos [{{\theta_{ek}}(0)} ]\ast {\theta_{ek}}^{\prime}(0) \ast {\theta_{ek}}^{\prime\prime}(0)} \\& { - \sin [{{\theta_{ek}}(0)} ]\ast \theta_{ek}^{(3)}(0)} \}\ast {t^3}, \end{aligned}$$
$$\begin{aligned} {R_{Qenk}}(t) &\approx \frac{{{B_k}}}{6}\{{ - \cos [{{\theta_{ek}}(0)} ]\ast {{[{{\theta_{ek}}^{\prime}(0)} ]}^3} - 3 \ast \sin [{{\theta_{ek}}(0)} ]\ast {\theta_{ek}}^{\prime}(0) \ast {\theta_{ek}}^{\prime\prime}(0)} \\& { + \cos [{{\theta_{ek}}(0)} ]\ast \theta_{ek}^{(3)}(0)} \}\ast {t^3}. \end{aligned}$$

In practical applications, the environmentally induced vibration frequency is often on the order of a hundred Hz. When t is on the millisecond scale, the first three terms of the Taylor expansion in Eq. (9) and (10) are considerably larger than the residual. Using this approximation, Eq. (9) and (10) can be simplified as

$$\begin{aligned} {I_e}(t) &\approx \underbrace{{\sum\limits_{k = 1}^K {{B_k} \ast \cos [{{\theta_{ek}}(0)} ]} }}_{{{c_1}}}\underbrace{{ - \sum\limits_{k = 1}^K {{B_k} \ast {\theta _{ek}}^{\prime}(0) \ast \sin [{{\theta_{ek}}(0)} ]} }}_{{{b_1}}} \ast t\\ &\underbrace{{ - \sum\limits_{k = 1}^K {0.5 \ast {B_k} \ast \{{{{[{{\theta_{ek}}^{\prime}(0)} ]}^2} \ast \cos [{{\theta_{ek}}(0)} ]+ {\theta_{ek}}^{\prime\prime}(0) \ast \sin [{{\theta_{ek}}(0)} ]} \}} }}_{{{a_1}}} \ast {t^2}\\ &\textrm{ = }{a_1}{t^2} + {b_1}t + {c_1}, \end{aligned}$$
$$\begin{aligned} {Q_e}(t) &\approx \underbrace{{\sum\limits_{k = 1}^K {{B_k} \ast \sin [{{\theta_{ek}}(0)} ]} }}_{{{c_2}}} + \underbrace{{\sum\limits_{k = 1}^K {{B_k} \ast {\theta _{ek}}^{\prime}(0) \ast \cos [{{\theta_{ek}}(0)} ]} }}_{{{b_2}}} \ast t\\ &\underbrace{{ - \sum\limits_{k = 1}^K {0.5 \ast {B_k} \ast \{{{{[{{\theta_{ek}}^{\prime}(0)} ]}^2} \ast \sin [{{\theta_{ek}}(0)} ]- {\theta_{ek}}^{\prime\prime}(0) \ast \cos [{{\theta_{ek}}(0)} ]} \}} }}_{{{a_2}}} \ast {t^2}\\ &\textrm{ = }{a_2}{t^2} + {b_2}t + {c_2}. \end{aligned}$$

Equation (13) and (14) can be used to rewrite Eq. (4) and Eq. (5) as

$$I(t) = A\cos [{\theta (t)} ]+ {a_1}{t^2} + {b_1}t + {c_1},$$
$$Q(t) = A\sin [{\theta (t)} ]+ {a_2}{t^2} + {b_2}t + {c_2}.$$

Equation (15) and Eq. (16) can be used to show that the Lissajous curve satisfies Eq. (17).

$${[{I(t) - {a_1}{t^2} - {b_1}t - {c_1}} ]^2} + {[{Q(t) - {a_2}{t^2} - {b_2}t - {c_2}} ]^2} - {A^2} = 0.$$

Least-squares fitting of Eq. (17) can be used to calculate the six parameters (a1, a2, b1, b2, c1, c2). As the Lissajous curve [I, Q] is a helix, so the process of this least squares fitting is similar to finding the parameters of a helix. With these six parameters, the measured phase change can be expressed as

$${\theta _{cal}}(t) = \arctan \left\{ {\frac{{A\sin [{\theta (t)} ]}}{{A\cos [{\theta (t)} ]}}} \right\} = \arctan \left[ {\frac{{Q(t) - {a_2}{t^2} - {b_2}t - {c_2}}}{{I(t) - {a_1}{t^2} - {b_1}t - {c_1}}}} \right].$$

A method based on the aforementioned principles is proposed for correcting the dynamic nonlinearity error by fitting the helix of the quadrature signal. This method is used in conjunction with the quadrature demodulation algorithm. The overall process is shown in Fig. 5.

 figure: Fig. 5.

Fig. 5. Flow chart of quadrature demodulation based on helix-fitting correction.

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The vibration producing the dynamic multipath interference determines the angular velocity of the circular motion of the Lissajous curves. For a single-frequency vibration, the circular angular velocity changes direction twice in one period. A second-order approximation can only fit a single change in the angular velocity direction. Thus, the length of time for a single fit should not exceed the half-period of the vibration. The long fitting time leads to the model in Eq. (17) not satisfying the conditions under which the approximation is valid. However, if the length of the fitting time is too short, the few data available for fitting result in the circular movement being covered by the noise. Hence, the length of the fitting time needs to be chosen to trade off the aforementioned considerations.

The relationship between the length of the fitting time and the root mean square (RMS) of the deviation of the demodulated signal with different SNRs is shown in Fig. 6: for an MMR of 1, the frequency and the amplitude are 300 Hz and 1 µm, respectively, for the target, and 15 Hz and 0.25 µm, respectively, for the multipath interference.

 figure: Fig. 6.

Fig. 6. Simulated RMS amplitude deviations.

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Although the minimum RMS and the corresponding length of the fitting time vary for different SNRs, the trends of the curves are similar. These curves prove that the fitting time length could be reduced further.

The aforementioned algorithm is validated by simulation. The vibration frequency of the target to be measured is 300 Hz, the amplitude is 1 µm, and fAOM is 5 MHz. The MMR is set to 0.5, the vibration frequency of multipath interference is 15 Hz, and the amplitude is 0.25 µm. The results are shown in Fig. 7.

 figure: Fig. 7.

Fig. 7. (a) Lissajous curve corrected by different correction methods. (b) Results without correction. (c) Results with piecewise Heydemann correction. (d) Results with helix fitting correction.

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Figure 7(a) shows the Lissajous curve before and after correction using the two methods. Although most of the nonlinearity errors have been eliminated using the piecewise Heydemann correction method, the Lissajous curve exhibits residual nonlinearity. The helix-fitting correction method corrects the Lissajous curve to a circle centered on the origin of the coordinate system. Figure 7(b) shows that the uncorrected amplitude and phase of the demodulated signal are severely distorted by the dynamic multipath interference. Figures 7(c) and 7(d) present the demodulated signal corrected by the two methods. Although the conventional method compensates for most of the nonlinearity errors, distortion of some segments makes the compensated demodulated signal inconsistent with the original signal. Application of the proposed method makes the amplitudes of the demodulated and original signals consistent. In simulations with dynamic multipath interference, the proposed method exhibits excellent performance both in terms of correcting the residual nonlinearity of the Lissajous curves and compensating for the nonlinearity errors of the demodulated signal compared to the performance of the conventional method.

Figure 8 shows the deviation of the signal compensated by the two methods from the original signal.

 figure: Fig. 8.

Fig. 8. Deviation of simulated compensated demodulated signal from the original signal for different correction methods.

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The maximum deviation of the compensated demodulated signal from the original signal is 19.69 nm using the conventional method and 0.21 nm using the proposed method. Compensation by the conventional method produces displacement jumps in the demodulated signal and therefore a large deviation from the original signal.

To illustrate the effect of multipath interference parameters (reflection intensity, amplitude and frequency of the vibration) on the algorithm, the following transformations were made. The number of multipath disturbances is considered here as one.

$$\left\{ \begin{array}{l} {k_a} = \frac{B}{A}\\ {k_d} = \frac{{{d_e}}}{d}\\ {k_f} = \frac{{{f_e}}}{f} \end{array} \right.,$$
where ka represents the ratio of multipath reflection to main reflection intensity, kd represents the ratio of multipath reflection vibration amplitude to main reflection vibration amplitude, and kf represents the ratio of multipath reflection vibration amplitude to main reflection vibration frequency. Figure 9 shows the RMS curve of the error with respect to the three parameters. The vibration frequency of the target to be measured is 300 Hz, the amplitude is 1 µm.

 figure: Fig. 9.

Fig. 9. (a) RMS curve of ka. (b) RMS curve of kd. (c) RMS curve of kf.

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The error after compensated by the proposed method is negatively correlated with the values of the three parameters (ka, kd, kf).

Figure 10 shows the deviation of the compensated demodulated signal from the original signal using the two methods with different MMRs.

 figure: Fig. 10.

Fig. 10. (a) The deviation of the demodulated signal compensated by piecewise Heydemann correction. (b) The deviation of the demodulated signal compensated by helix fitting correction.

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The following conclusions can be drawn from Fig. 10: the smaller the MMR is, the larger the deviation between the compensated and original signals is, and the compensation by the proposed method is better than that by the conventional method for the same fitting length.

3. Experiment

Figure 11 shows the experimental setup used to validate the proposed method. The measured target is driven by a piezoelectric element (PZT) (PI P-753.21C, controller PI E-501.00), for which the displacement can be fed back by a capacitive sensor. The laser source is an SFL operating at 1520 nm. The PZT vibrates in the frequency range of 250 Hz to 500 Hz at 50-Hz intervals.

 figure: Fig. 11.

Fig. 11. (a) Schematic of the measurement setup. (b) Experimental device photo.

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Figure 12 shows the measurement results when the PZT vibrates at a frequency of 300 Hz and an amplitude of 1.15 µm.

 figure: Fig. 12.

Fig. 12. (a) Lissajous curve corrected by different methods. (b) Different demodulated signals and PZT output values. (c) Detailed view of the signal shown within the black dashed rectangular frame in (b). (d) Deviation between the compensated and PZT signals for different methods.

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Figure 12(a) shows the Lissajous curves before and after correction using the two methods. Figure 12(b) shows the different demodulated signals and output values of the PZT. The maximum deviation between the uncompensated demodulated signal and the original signal is 1156.55 nm. A detailed view of the signals within the black rectangular frame in Fig. 12(b) is provided in Fig. 12(c). Figure 12(c) shows the correction effect of the two methods more clearly. Figure 12(c) shows that the conventional method compensates for the demodulated signal to some extent but significant distortions remain. By comparison, the proposed method provides better compensation of the demodulated signal.

Figure 12(d) shows the deviation of the demodulated signal from the PZT output value after compensation by the two correction methods. This deviation is a visual measure of the efficacy of compensation by the two methods. The maximum deviation between the original and compensated demodulated signals is 737 nm using the piecewise Heydemann correction method and 30 nm using the helix-fitting method. Thus, there is a large difference between the simulation results for the two methods, where the 30 nm deviation can be considered the better result considering the acquisition noise, environmental vibration, and AOM frequency instability.

Table 1 presents the deviations between the demodulated and original signals for different frequencies. These results show that the smallest deviation between the compensated and original signals using the helix-fitting correction is 37 nm. The deviation between the compensated and original signals obtained using the helix-fitting correction is at least one order of magnitude lower than that obtained using the piecewise Heydemann correction. Environmental vibration during the experiment, such as the vibration of the vacuum tank, can affect the vibration of the target. Consequently, the output of the capacitive sensor becomes inconsistent with the true vibration state of the target. The magnitude of the deviation depends on the environment. The deviation result obtained using the helix-fitting correction is acceptable for the noisy environment considered.

Tables Icon

Table 1. Maximum deviation between demodulated and original signals at different frequencies

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Many noncooperative targets need to be measured in industrial environments. In this case, multipath interference from the lens, waveplates, and fiber pigtail inside the LDV system cannot be neglected. Moreover, multipath interference can be coupled with vibration due to the noisy environment. Corrections must be made to improve the accuracy of the signal measurement. Therefore, to verify the generalizability of our method, the experimental device was placed in a noisy environment, and the glass window between the LDV and PZT was removed.

Figure 13 shows that the multipath interference return power is weaker than the measured target return bit; nevertheless, a large deviation is observed between the measured and original results, with a maximum value of 134 nm. The maximum deviation between the compensated demodulated signal and original signal is 79 nm using the conventional method and 28 nm using the proposed method.

 figure: Fig. 13.

Fig. 13. (a) Lissajous curve corrected by different correction methods. (b) Different demodulated signals and the output values of PZT. (c) Details of the signal in the black dashed rectangle in (b). (d) The deviation of demodulated signals compensated by different correction methods from the PZT signals.

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The deviations between the compensated demodulated signals and original signals at different frequencies are given in Table 2. These results show that the smallest deviation obtained using the helix-fitting correction is 42 nm. These compensation results demonstrate the universality and effectiveness of the helix-fitting correction method.

Tables Icon

Table 2. Maximum deviation of the compensated demodulated signal from the original signal at different frequencies

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4. Summary and discussion

In this study, we have analyzed the nonlinearity errors introduced by dynamic multipath interference in LDVs. Multipath interference for a measurement system in a noisy environment results in helical Lissajous curves of the quadrature signals. A mathematical model is established to describe this case, and a compensation method is proposed to mitigate signal distortion.

Nonlinearity errors introduced by multipath interference are conventionally corrected using the Heydemann correction method. However, a significant residual error is incurred by applying this method to dynamic multipath interference. This residual error results from the shift of the center of the Lissajous curve of the quadrature signal. The Heydemann correction method based on ellipse fitting cannot address this phenomenon. Therefore, in this study, a model that is applicable to this case is developed and used to propose a compensation method. The accuracy of the model and the efficiency of the compensation method are validated by the results of simulations and experiments. The residual errors obtained using the proposed method are an order of magnitude lower than those obtained using the conventional method. The results of the experiment show that the conventional Heydemann correction method reduces the dynamic nonlinearity to 737 nm, while the present method can reduce it to 30 nm.

In conclusion, the current compensation method can effectively reduce the nonlinearity errors introduced by dynamic multipath interference. The proposed method has wide applicability because noncooperative targets and noisy environments are encountered in most cases.

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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Data availability

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

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Figures (13)
Fig. 1.
Fig. 1. Schematic of the heterodyne laser Doppler vibration measurement system. (SFL: single-frequency laser; AOM: acousto-optic modulator; CO: fiber coupler, where the numbers represent the splitting rate; FC: fiber collimator; PBS: polarizing beam splitter; QWP: quarter-wave plate; PD: photodetector).

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Fig. 2.
Fig. 2. Flow chart of quadrature demodulation (LPF: low-pass filter).

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Fig. 3.
Fig. 3. Schematic of multipath interference.

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Fig. 4.
Fig. 4. Cases of different MMRs with dynamic nonlinear errors: (a) the Lissajous curves, (b) the simulated demodulation results.

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Fig. 5.
Fig. 5. Flow chart of quadrature demodulation based on helix-fitting correction.

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Fig. 6.
Fig. 6. Simulated RMS amplitude deviations.

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Fig. 7.
Fig. 7. (a) Lissajous curve corrected by different correction methods. (b) Results without correction. (c) Results with piecewise Heydemann correction. (d) Results with helix fitting correction.

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Fig. 8.
Fig. 8. Deviation of simulated compensated demodulated signal from the original signal for different correction methods.

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Fig. 9.
Fig. 9. (a) RMS curve of ka. (b) RMS curve of kd. (c) RMS curve of kf.

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Fig. 10.
Fig. 10. (a) The deviation of the demodulated signal compensated by piecewise Heydemann correction. (b) The deviation of the demodulated signal compensated by helix fitting correction.

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Fig. 11.
Fig. 11. (a) Schematic of the measurement setup. (b) Experimental device photo.

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Fig. 12.
Fig. 12. (a) Lissajous curve corrected by different methods. (b) Different demodulated signals and PZT output values. (c) Detailed view of the signal shown within the black dashed rectangular frame in (b). (d) Deviation between the compensated and PZT signals for different methods.

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Fig. 13.
Fig. 13. (a) Lissajous curve corrected by different correction methods. (b) Different demodulated signals and the output values of PZT. (c) Details of the signal in the black dashed rectangle in (b). (d) The deviation of demodulated signals compensated by different correction methods from the PZT signals.

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Tables (2)

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Table 1. Maximum deviation between demodulated and original signals at different frequencies

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Table 2. Maximum deviation of the compensated demodulated signal from the original signal at different frequencies

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Equations (19)

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I m ( t ) = A m cos ⁡ [ 2 π f A O M t + 4 π Δ d ( t ) λ ] = A m cos ⁡ [ 2 π f A O M t + θ ( t ) ] ,
I a ( t ) = A a cos ⁡ ( 2 π f A O M t ) ,
I m ( t ) = A m cos ⁡ [ 2 π f A O M t + θ ( t ) ] + ∑ k = 1 K A e k cos ⁡ [ 2 π f A O M t + θ e k ( t ) ] ,
I ( t ) = A m A a 2 cos ⁡ [ θ ( t ) ] + ∑ k = 1 K A e k A a 2 cos ⁡ [ θ e k ( t ) ] = A cos ⁡ [ θ ( t ) ] + ∑ k = 1 K B k cos ⁡ [ θ e k ( t ) ] ,
Q ( t ) = A m A a 2 sin ⁡ [ θ ( t ) ] + ∑ k = 1 K A e k A a 2 sin ⁡ [ θ e k ( t ) ] = A sin ⁡ [ θ ( t ) ] + ∑ k = 1 K B k sin ⁡ [ θ e k ( t ) ] ,
θ c a l ( t ) = arctan ⁡ [ Q ( t ) I ( t ) ] = arctan ⁡ { A sin ⁡ [ θ ( t ) ] + ∑ k = 1 K B sin ⁡ [ θ e k ( t ) ] A cos ⁡ [ θ ( t ) ] + ∑ k = 1 K B k cos ⁡ [ θ e k ( t ) ] } .
I e ( t ) = ∑ k = 1 K I e k ( t ) = ∑ k = 1 K B k cos ⁡ [ θ e k ( t ) ] ,
Q e ( t ) = ∑ k = 1 K Q e k ( t ) = ∑ k = 1 K B k sin ⁡ [ θ e k ( t ) ] .
I e k ( t ) = B k ∗ cos ⁡ [ θ e k ( 0 ) ] − B k ∗ θ e k ′ ( 0 ) ∗ sin ⁡ [ θ e k ( 0 ) ] ∗ t − 0.5 ∗ B k ∗ { [ θ e k ′ ( 0 ) ] 2 ∗ cos ⁡ [ θ e k ( 0 ) ] + θ e k ′ ′ ( 0 ) ∗ sin ⁡ [ θ e k ( 0 ) ] } ∗ t 2 + R I e n k ( t ) ,
Q e k ( t ) = B k ∗ sin ⁡ [ θ e k ( 0 ) ] + B k ∗ θ e k ′ ( 0 ) ∗ cos ⁡ [ θ e k ( 0 ) ] ∗ t − 0.5 ∗ B k ∗ { [ θ e k ′ ( 0 ) ] 2 ∗ sin ⁡ [ θ e k ( 0 ) ] − θ e k ′ ′ ( 0 ) ∗ cos ⁡ [ θ e k ( 0 ) ] } ∗ t 2 + R Q e n k ( t ) ,
R I e n k ( t ) ≈ B k 6 { sin ⁡ [ θ e k ( 0 ) ] ∗ [ θ e k ′ ( 0 ) ] 3 − 3 ∗ cos ⁡ [ θ e k ( 0 ) ] ∗ θ e k ′ ( 0 ) ∗ θ e k ′ ′ ( 0 ) − sin ⁡ [ θ e k ( 0 ) ] ∗ θ e k ( 3 ) ( 0 ) } ∗ t 3 ,
R Q e n k ( t ) ≈ B k 6 { − cos ⁡ [ θ e k ( 0 ) ] ∗ [ θ e k ′ ( 0 ) ] 3 − 3 ∗ sin ⁡ [ θ e k ( 0 ) ] ∗ θ e k ′ ( 0 ) ∗ θ e k ′ ′ ( 0 ) + cos ⁡ [ θ e k ( 0 ) ] ∗ θ e k ( 3 ) ( 0 ) } ∗ t 3 .
I e ( t ) ≈ ∑ k = 1 K B k ∗ cos ⁡ [ θ e k ( 0 ) ] ⏟ c 1 − ∑ k = 1 K B k ∗ θ e k ′ ( 0 ) ∗ sin ⁡ [ θ e k ( 0 ) ] ⏟ b 1 ∗ t − ∑ k = 1 K 0.5 ∗ B k ∗ { [ θ e k ′ ( 0 ) ] 2 ∗ cos ⁡ [ θ e k ( 0 ) ] + θ e k ′ ′ ( 0 ) ∗ sin ⁡ [ θ e k ( 0 ) ] } ⏟ a 1 ∗ t 2  =  a 1 t 2 + b 1 t + c 1 ,
Q e ( t ) ≈ ∑ k = 1 K B k ∗ sin ⁡ [ θ e k ( 0 ) ] ⏟ c 2 + ∑ k = 1 K B k ∗ θ e k ′ ( 0 ) ∗ cos ⁡ [ θ e k ( 0 ) ] ⏟ b 2 ∗ t − ∑ k = 1 K 0.5 ∗ B k ∗ { [ θ e k ′ ( 0 ) ] 2 ∗ sin ⁡ [ θ e k ( 0 ) ] − θ e k ′ ′ ( 0 ) ∗ cos ⁡ [ θ e k ( 0 ) ] } ⏟ a 2 ∗ t 2  =  a 2 t 2 + b 2 t + c 2 .
I ( t ) = A cos ⁡ [ θ ( t ) ] + a 1 t 2 + b 1 t + c 1 ,
Q ( t ) = A sin ⁡ [ θ ( t ) ] + a 2 t 2 + b 2 t + c 2 .
[ I ( t ) − a 1 t 2 − b 1 t − c 1 ] 2 + [ Q ( t ) − a 2 t 2 − b 2 t − c 2 ] 2 − A 2 = 0.
θ c a l ( t ) = arctan ⁡ { A sin ⁡ [ θ ( t ) ] A cos ⁡ [ θ ( t ) ] } = arctan ⁡ [ Q ( t ) − a 2 t 2 − b 2 t − c 2 I ( t ) − a 1 t 2 − b 1 t − c 1 ] .
{ k a = B A k d = d e d k f = f e f ,
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