Frequency scanning interferometry (FSI) with a single external cavity diode laser (ECDL) and time-invariant Kalman filtering is an effective technique for measuring the distance of a dynamic target. However, due to the hysteresis of the piezoelectric ceramic transducer (PZT) actuator in the ECDL, the optical frequency sweeps of the ECDL exhibit different behaviors, depending on whether the frequency is increasing or decreasing. Consequently, the model parameters of Kalman filter appear time varying in each iteration, which produces state estimation errors with time-invariant filtering. To address this, in this paper, a time-varying Kalman filter is proposed to model the instantaneous movement of a target relative to the different optical frequency tuning durations of the ECDL. The combination of the FSI method with the time-varying Kalman filter was theoretically analyzed, and the simulation and experimental results show the proposed method greatly improves the performance of dynamic FSI measurements.
© 2017 Optical Society of America
The frequency scanning interferometry (FSI) method for absolute distance measurement is very precise, relatively low-cost, and easy to implement; thus, it has been widely used in many applications [1–11]. Fox-Murphy et al. proposed and constructed an FSI system for precise alignment and deformation monitoring of the ATLAS Detector [4,5], and Yang et al. applied an FSI system to the alignment of the tracker associated with the international linear collider (ILC) [6,7]. Alexandre Cabral et al. proposed an FSI system to measure the distances between spacecraft in the ESA-PROBA3 space mission . More recently, Hughes proposed a traceable large-scale 3D coordinate measurement using the FSI method in which the 3D coordinates are calculated according to the multilateration distances provided by FSI . However, in the conventional FSI method, the target mirror is assumed to be static for the duration of one measurement period while the optical frequency of the ECDL is scanned. In other words, the conventional FSI method is only suitable for measuring the absolute distance of a static target. In practice, the FSI method is highly sensitive to variations in the optical length (OPL) during optical frequency sweeps, and even small variations are amplified [4–11], which introduces large errors into the measurement results. These variations may be due to vibration, movements in the target mirror, or air turbulence over the finite measurement period. To address this issue, Coe et al. proposed a dual FSI system to realize dynamic measurements based on the interferometer phase ratio between two lasers tuned in opposite directions [5,8]. However, this method requires two differently tuned laser sources and a complicated technique to synchronize both sweeps. With a single tuned laser source and a frequency-stabilized He-Ne laser, Martinez and Campbell, et al.  used four-wave mixing (FWM) as a means to create a second swept light source suitable for dual FSI systems. The FWM sweep is based on a non-linear effect that generates additional optical frequencies when two beams of light pass through certain optical media. The generated light is mirror copy of the original swept laser, and the two light beams are synchronized and tuned in opposite directions. A disadvantage of this method is that it requires an additional frequency-stabilized laser source and complex non-linear optical setup. In situations requiring the movement of target mirrors at constant velocities, Cabral et al.  proposed a movement error compensation algorithm. Instead of using two swept lasers, this method used a single swept laser and performed two consecutive measurements while increasing and decreasing optical frequency sweep durations at the same sweep rate. Compensation for the measurement errors could be partially achieved by using two consecutive fringe countings. Previous experience suggests that the tuning linearity of the ECDL changes with sweep direction. For this reason, the two consecutive measurements, in general, are performed with different sweep durations.
In our previous work, Tao  used a single ECDL to construct an FSI system for dynamic absolute length measurements. In the current paper, a Kalman filter is used to model the movement of the target mirror, for which the length, velocity, and acceleration are parameters that are estimated by using multiple sequential FSI measurements based on the discrete Kalman filter algorithm. The proposed method effectively eliminates the amplification effect of the OPL variation during optical frequency sweeps, and realizes the distance measurement of a dynamic target mirror. In the method, two consecutive measurements are considered to have the same optical frequency sweep durations; therefore, the movement of the target mirror can be described by a time-invariant Kalman filter model. However, in practice, the increasing and decreasing optical frequency sweeps of the ECDL exhibit different behaviors due to hysteresis in the piezoelectric ceramic transducer (PZT) actuator in the ECDL and this will introduce additional errors into the parameter estimation process of the Kalman filter. The consequence is that the accumulated errors slow down the convergence of the movement estimation. Divergence trends also appear over a long period of continuous measurements while tracking the target mirror.
To address these issues, we propose a new FSI method with a single ECDL for use in dynamic measurements. In the proposed method, the optical frequency sweep of the ECDL is divided into two different stages, and a time-varying Kalman filter is used to describe the instantaneous movement of the target, which corresponds to different optical frequency tuning durations. The convergence of the movement estimation process using a time-varying Kalman filter is analyzed in detail later in this paper. The simulation and experimental results show that our proposed Kalman filter algorithm enhances the convergence of the movement estimation process, and effectively addresses the error divergence trends in measurements taken over a long period of time.
2.1 Principles of the FSI method
An FSI system consists of an ECDL, Fabry–Pérot (F-P) cavity, and Michelson interferometer, as depicted in Fig. 1. The light output of the tunable laser is split by the first beam splitter (BS1). One part is transmitted into the F-P cavity, which measures the optical frequency sweep range, while another part is transmitted into the Michelson interferometer. In the Michelson interferometer, the beam is split by BS2 into two parts: one beam is directed along a path of fixed length, while the other is directed along the length to be measured. The laser is continuously tuned in the rising and falling directions, and the phase changes continuously. The order k is the sequence of the laser tuning. The PD1 records the phase shift induced in the interferometer during the optical frequency sweep, which allows the distance between the reference and measurement mirrors to be determined. The intensity I recorded by PD1 can be expressed as:
The FSI experimental setup is shown in Fig. 2. The FSI utilized an ECDL (Newport TLB6800) as the light source with a mode-hop-free sweep range of 150 GHz and a center wavelength of 780 nm. A signal generator (Tektronix, AFG3052C) produced a modified triangular waveform with smoothed turnaround region to provide optical frequency modulation for the ECDL . A high-finesse Fabry–Pérot (Thorlabs SA210-5B with an FSR of 1.5 GHz, made of invar with high thermal stability) was employed to determine the optical frequency range. A thermal shield box made of polystyrene walls is used to cover the F-P cavity for the purpose of reducing the fluctuation of the refractive index of air. Two photodetectors PD1 and PD2 (Newport Model 1801) were used to detect the interference signals of the Michelson interferometer and transmitted F-P signal, respectively. A data acquisition card (DAQ) with two channels (NI-PXle-5105) was used to acquire the F-P signal and interference signals, and an environmental sensor (Agilent E1738A) was located near the beam to measure the refractive index n of air using the Edlen equation.
2.2 Amplification effect of the measurement error in FSI
If the length L varies during optical frequency scanning, it is not possible to distinguish whether the phase change was induced by scanning the optical frequency or by the change in the length L. The total phase difference can be written as:
Thus, the measured length can be obtained by:Eq. (4), and the length variation is amplified by the factor . In our established FSI method, the center wavelength nm, the optical frequency sweep range GHz, and the corresponding amplification factor is about 2670. That is, any small variation in the length result in an error of in the measured length. Therefore, it is very important to eliminate the influence of the additional error on the measured length LM caused by length changes during the optical frequency sweep.
3. Kalman filter for dynamic FSI measurements
3.1 Time-varying Kalman filter for dynamic FSI measurements
In our previous work, Tao  used a Kalman filter [13,14] to realize the dynamic measurement of a moving target using the FSI method with a single ECDL. In this method, the absolute length L was regarded as a continuous function of time during one period of the optical frequency sweep. If L is first and second order differentiable at the sampling time, and the sampling time interval T is very short, L can be expressed by a second-order Taylor expansion in discrete time as:
The state vector of the state-space dynamic model is defined as x = [L, s, a]T, and the measurement function LkM = f(x, s, a). Figure. 3 shows the state transition process from time step k to time step k + 1 in consecutive measurements. A 20 Hz triangle wave was used to drive the ECDL. The optical frequency tuning period of the ECDL is divided into optical frequency rising and falling stages, and in each stage, one measurement update is performed. In practice, only the linear parts of the optical frequency tuning are adapted. In Fig. 3, the blue points denote the starting moments of the respective measurement periods, the violet points denote the ending moments, and time tk refers to the measurement period of the FSI at time step k. For a moving target, the OPL varies continuously during the measurement period tk.
By combining Eqs. (4) and (5), we can write:
We utilized a Kalman filter to model the dynamic measurement as follows:
Since the optical frequency output of the ECDL exhibits nonlinear behavior due to the hysteresis of the PZT actuator [15,16], the optical sweeps in the increasing and decreasing sections are different, and Tk and tk are no longer constant. In our FSI system, a 20 Hz modified triangular wave is used to drive the ECDL, the corresponding sampling period Tk and measurement tk exhibit the statistical properties of the bimodal distribution, which is a mixture of two normal distributions, as shown in Fig. 4 and it can be described by Eq. (10).
That is, the corresponding and are not time-invariant under real conditions. Therefore, and should be refreshed according to Tk and tk at each measurement period. The dynamic measurement model of the FSI method should be time-varying , and the state transition matrices and measurement matrices should be expressed as:
With the time-varying state-space model, the Kalman filter time update equations and measurement update equations are developed to iteratively estimate the absolute length and velocity of the target, as follows:
3.2 Convergence analysis of the time-varying Kalman filter
In this section, we analyze the convergence of our proposed FSI time-varying Kalman filter. Anderson and Moore proved the convergence of a time-varying discrete-time linear system , and Bruno Sinopoli et al provided the convergence conditions for a Kalman filter with intermittent observations . It is known that if are bounded and is uniformly detectable with uniformly stable, the associated optimal filter error covariance is bounded. The convergence of the error covariance is shown in Eq. (15).
In FSI system, with the data Tk, tk and from the experiments under the condition of using a modified triangle waveform of 20 Hz to driving the ECDL, according to Eq. (10), we can see that and are bounded, as shown in Eq. (16):
Next, it should be verified that the time-varying FSI-Kalman filter is detectable and stable. As Anderson and Moore proved , the pair is uniformly detectable and is uniformly stable if there exist some integers and constants that satisfy the conditions in Eq. (17):
Based on this analysis, the proposed time-varying FSI-Kalman filter is theoretically convergent.
4. Simulation and results
4.1 Time-varying FSI-Kalman filter simulations
To verify the performance of our proposed time-varying Kalman filter for FSI with a single ECDL, a series of simulations were performed. In the time-varying Kalman filter measurement model, the periods of the optical frequency sweeps at time step k are subdivided into forward scanning tup and backward scanning tdown, which can be expressed as:
The corresponding amplification factor is:
The simulation parameters are listed in Table 1. In order to compare the time-varying Kalman filter algorithm with the time-invariant version, the sampling time Tk and measurement time tk for the time-invariant Kalman filter are set to 0.025 s and 0.0225 s, respectively.
4.2 Simulation of 1D uniform motion
Uniform linear motion is the most common scenario for moving target measurements. In this section, we perform an invariable velocity motion simulation.
The initial length was set to 1.5 m, and the velocities of the targets were set to 1 mm/s and 10 mm/s, respectively. The simulation results are shown in Figs. 5 and 6, where it can be seen that the errors in the time-invariant Kalman filter increased with time, while the errors in the time-varying Kalman filter quickly converged to zero. As the velocity increased to 10 times, the error in the time-invariant Kalman filter increased by a factor of 10. In terms of velocity estimation, the time-invariant Kalman filter performed similar to that of the length estimation. The error of the time-varying Kalman filter was observed to converge toward zero.
4.3 Simulation of periodic vibration
The following simulations illustrate the motion of a target in periodic synthesis vibrations, which generally manifest as periodic OPL drift during continuous measurements.
The parameters in the vibration function are as shown in Table 1, and represent a synthetic vibration with frequencies of 1 and 3 Hz and amplitudes of 10 and 9 μm. The sampling period T of the Kalman filter was approximately 0.025 s.
The simulation results without the Kalman filter are shown in Fig. 7. From these figures, it is evident that the basic FSI method using a single ECDL cannot track the periodic vibration. Figure. 7(b) and 7(c) show the measurement results with time-invariant and time-varying Kalman filters, respectively. Both filters were able to accurately track the trajectories of the vibrating target. To compare the performance of the two methods, we calculated the tracking errors shown in Fig. 7(d). From the figure, it can be seen that the errors in tracking the vibrating target with the two methods appear to converge, although the proposed FSI method with the time-varying Kalman filter exhibited much better performance than that with the time-invariant Kalman filter. The relative error with the time-varying Kalman filter was less than 2 ppm, while that with the time-invariant Kalman filter was less than 35 ppm. The root mean square error (RMSE) of the time-varying filter was approximately 0.2 μm, while that of the time-invariant filter was approximately 0.35 μm.
Furthermore, in order to analyze the influence of the vibration amplitude on the accuracy of tracking the target, a series of simulations with amplitudes from 1 μm to 100 mm was performed, as shown in Fig. 8. It can be seen that the RMSEs of the two types of FSIs increase as the vibration amplitude increased. Nevertheless, the FSI method with the time-varying Kalman filter had higher accuracy than that with the time-invariant Kalman filter.
To further validate the performance of measuring the vibrations with the proposed method and time-varying Kalman filter, a random vibration was synthesized that contained frequency components of 1, 1.5, 3, 6, 7, and 8.5 times the fundamental frequency, and the fundamental frequency was set to 0.5, 1, 1.5, and 2 Hz during the tests. A 20 Hz triangle wave was used to drive the ECDL. With this setup, approximately 40 measurements were carried out per second. The measurement results are shown in Fig. 9. As the frequency of vibration increased, the accuracy of our proposed FSI method gradually decreased until it became invalid.
The results shown in Fig. 9 are explained in Fig. 10. As the frequency of vibration increased, the frequency of the optical frequency tuning also increased to satisfy Shannon's sampling theorem. Thus, the frequency sweep of the tunable laser had to increase to accurately measure the vibrations that occurred at higher frequencies.
Up to this point, the results of the simulations verify that our proposed FSI with time-varying Kalman filter can effectively realize dynamic measurements. In next section, additional experiments are conducted to further test the performance of the proposed FSI method with a time-varying Kalman filter.
5. Experiments and discussion
5.1 1D uniform motion tracking
In the laboratory environment, the experimental FSI system shown in Fig. 4 was located on a floating optical table, and the target mirror was fixed on a stage attached to a linear guide rail (Newport, M-IMS400PP, minimum incremental motion = 1.25 μm). The moving speed of the stage was set to 1 mm/s. The stage remained stationary from 0 to 7.5 s and then moved to decrease the OPL at a speed of 1 mm/s.
Figure. 11 shows the results of the 1D motion. In Fig. 11(a), the black line represents the FSI method without the Kalman filter, while the red line represents the measurements with the time-varying Kalman filter. The results indicate the amplified motion errors were more effectively eliminated when the time-varying Kalman filter was used. From Fig. 11(b), we can see the velocity and acceleration of the target were quickly tracked, and Fig. 11(d) shows the deviation between the measured velocity and that of the moving stage. We observed the segment for 16 s, and found the upper and lower deviations to be ± 2.5 μm. Based on these results, our method was able to dynamically track and measure the 1D moving target.
5.2 Vibration measurement
To test the performance of the vibration measurement, we employed a PZT actuator (Thorlabs, NF15AP25 with a resolution of 25 nm) to produce a sinusoidal vibration that was applied to the target mirror. The amplitude of the vibration was set to 1 μm and the frequency was set to 1 Hz. An eddy current displacement sensor (eddyNCDT 3010 with a static resolution of 25 nm) was used to examine the vibration of the target mirror. The results of the measurement are shown in Fig. 12.
Figure. 12(a) shows that the measurements (black line) of the basic FSI method deviate significantly from the true values. Figure. 12(b) shows the velocity measurement of the FSI method with time-varying Kalman filter (deep blue line) quickly converges to a sinusoidal movement, and Fig. 12(c) shows the acceleration measurement of the moving target. From Fig. 12(d), we can see the measurements results of the eddy current displacement sensor are very close to that of our FSI with the time-varying Kalman filter. However, there was a systematic error of about 0.15 μm, which may be due to an assembly gap in the PZT actuator, which can be compensated for. Furthermore, since the two methods measured the vibration of the target retroreflector in opposite directions, the phases of the two measured signals are opposite.
As an optimal recursive Bayesian estimator, Kalman filter is used for somewhat restricted class of linear Gaussian problems. It is critical to determine the noise distribution before the FSI-KF is applied for dynamic measurement. If the FSI-KF were used in a hostile environment and the noises were not Gaussian, the Kalman filter would not be able to track the target accurately. In the case of non-Gaussian problems, some nonlinear filter such as particle filter could be an effective approach. In addition, the initial values of the state variables have an influence on the convergence speed of the FSI-KF. In our future work, some more efficient algorithms are worthy to be studied for initial values determination.
In the basic FSI system, the OPL of the measurement interferometer should be constant, but it is impossible for dynamic measurement. When the OPL changed during FSI measurements, the variations will be amplified in the measurements. To address this problem, a dynamic FSI system with time-varying Kalman filter was proposed to make high-precision dynamic measurement. A time-varying Kalman filter can effectively accommodate variances in the optical frequency sweeping durations. On this basis, a dynamic measurement model can be established to describe the instantaneous movement of the target mirror. The convergence of the movement estimation was analyzed, and indicated that the frequency of the optical frequency sweep of the ECDL is critical to dynamic measurement. The simulation and experimental result show that our proposed FSI method with a single ECDL and time-varying Kalman filter have improved the performance of tracking the target mirror. Under laboratory conditions, a velocity measurement precision of ± 2.5 μm was achieved for a target moving at 1 mm/s. During our investigation, we found that this method has the capability to measure vibration amplitudes as small as 1 μm without a priori knowledge.
National Natural Science Foundation of China (NSFC) (Grant Nos. 51375376 and 51635010).
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