Abstract

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

1. Introduction

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 [8]. 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 [9]. 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. [9] 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. [10] 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 [11] 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. Principles

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:

I=I1+I2+2I1I2cos(ϕ1ϕ2),
where I1 and I2 are the intensities of the two combined beams, and ϕ1 and ϕ2 are the phases. While the optical frequency v varies from v1 to v2, the phase ϕ of the interference fringes varies continuously from ϕ1 to ϕ2. The measured length L is half of the optical path difference between the reference and measurement paths, and is determined as:
L=12nΔϕ2πcΔv,
where c is the velocity of light in vacuum, n is the refractive index of air, Δv is the range of the optical frequency sweep (i.e., v2v1), and Δϕ is the phase change ϕ2ϕ1 that corresponds to Δv.

 

Fig. 1 Schematic illustration of the principles of FSI.

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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 [12]. 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.

 

Fig. 2 The FSI system. In the laboratory environment, the setup was constructed on a floating optical table to reduce interference caused by external noise.

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2.2 Amplification effect of the measurement error in FSI

If the length L varies Δεduring 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:

Δϕ'=4nπc[(L+Δε)v2Lv1].

Thus, the measured length LM can be obtained by:

LM=cΔϕ'4nπΔv=Ltrue+v2v2ν1Δε=Ltrue+ΩΔε,
where Ω=v2/Δv is the amplification factor, as shown in Eq. (4), and the length variation Δε is amplified by the factor Ω. In our established FSI method, the center wavelength λ=780 nm, the optical frequency sweep range Δv=150 GHz, and the corresponding amplification factor v2/Δv is about 2670. That is, any small variation Δε in the length result in an error of Δε×2670 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 [11] 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:

Lk+1=Lk+sT+12aT2,
where Lk and Lk+1 are the instantaneous lengths of the moving target at time steps k and k + 1, s and a are the first and second derivatives of L, respectively, and refer to the velocity and acceleration at time step k, and T is the sampling time interval from time step k to time step k + 1.

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.

 

Fig. 3 The state estimation for dynamic FSI.

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By combining Eqs. (4) and (5), we can write:

LkM=Lk+Ωksktk+12Ωkaktk2,
where the measured LkM contains two parts. The first is the true length Lk of the target at the start moment of the kth measurement period, while the second corresponds to the OPL variation Δlk caused by movement of the target during the kth optical frequency sweep.

We utilized a Kalman filter to model the dynamic measurement as follows:

xk+1=Φkxk+wk,
yk=Hkxk+υk,
where Φk is the state transition matrix, Hk is the measurement matrix at time step k, and Tk is the time interval between xk and xk+1, which is related to the sampling period of the Kalman filter. The term Ωk is the amplification factor of the kth measurement, and wkRn and υkRm are mutually independent Gaussian noise vectors with zero mean and covariance matrices Qk>0 and Rk>0, respectively. We consider the first derivative of the acceleration as the process noise in the state transition. The Qk and Rk can be written as:
Qk=σw2[Tk636Tk512Tk46Tk512Tk44Tk32Tk46Tk32Tk2],Rk=[συ2],
Where the parameter συ2 depends on the precision of the FSI measurement, so it can be determined by the variance deduced from the raw FSI measurements. However, the value of σw2 is difficult to be measured directly. The value of σw2 can be determined through the numerical simulations of KF aiming at different test environments.

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).

Tk={N(Tup,σTup)N(Tdown,σTdown),tk={N(tup,σtup)N(tdown,σtdown),
where up and down indicate the optical frequency increasing and decreasing sections, respectively.

 

Fig. 4 Statistical distribution of T and t. Part a (blue) represents the sampling period T of the Kalman filter, and Part b (red) represents the FSI period t. These plots are based on 6,000 FSI measurement samples.

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That is, the corresponding Φk and Hk are not time-invariant under real conditions. Therefore, Φk and Hk should be refreshed according to Tk and tk at each measurement period. The dynamic measurement model of the FSI method should be time-varying [17], and the state transition matrices Φk and measurement matrices Hk should be expressed as:

Φk=[1TkTk2/201Tk001]={ΦupΦdown,Hk=[1ΩktkΩktk22]={HupHdown.
The amplification factor Ωk can be rewritten as:

Ωk={Ωup=v2v2v1Ωdown=v1v1v2.

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:

x^k|k=x^k|k1+Kk(ykHkx^k|k1),
Pk|k=Pk|k1KkHkPk|k1,
and
x^k+1|k=Φk+1x^k|k,Pk+1|k=Φk+1Pk|kΦk+1T+Qk,
Kk=Pk|k1Hk1T(HkPk|k1HkT+R)1,x^0|1=x0,P0|1=P0,
where x^k|k is the a posteriori state estimate of the measured target; Pk|k denote the a posteriori estimate error covariance; Kk is the Kalman gain matrix; and x^0|1 and P0|1 are the initial state estimate and initial error covariance matrix, respectively. In the FSI-KF system, the initial length L0 of the x0 is set to the first coarse length measurement using FSI without filtering, and the initial velocity v0 and acceleration a0 are set to be zero. The initial error covariance matrix P0 is set to a three-order identity matrix.

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 [18], and Bruno Sinopoli et al provided the convergence conditions for a Kalman filter with intermittent observations [19]. It is known that if Φk,Hk are bounded and [Φk,Hk] is uniformly detectable with [Φk,Qk] uniformly stable, the associated optimal filter error covariance is bounded. The convergence of the error covariance is shown in Eq. (15).

P0,s.t.supEPk=0ppc
where E is the conditioned expected value operator.

In FSI system, with the data Tk, tk and Ωk 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 Φk and Hk are bounded, as shown in Eq. (16):

supkΦk21.01799,supkHk267.45363

Next, it should be verified that the time-varying FSI-Kalman filter is detectable and stable. As Anderson and Moore proved [18], the pair [Φk,Hk] is uniformly detectable and [Φk,Qk] is uniformly stable if there exist some integers N0 and constants ξ1,ξ2 that satisfy the conditions in Eq. (17):

0<ξ1Ii=kN+1kΦk,iQi,i1Qi,i1TΦk,iT0<ξ2Ij=kN+1kΦj,kTHjTHjΦj,k.

In the case of our constructed FSI, when N3, Eq. (17) satisfies these conditions. As a result, Eq. (17) can be expressed as follows:

i=kN+1kΦk,iQi,i1Qi,i1TΦk,iT=σw2[f11(N,T)f12(N,T)f13(N,T)f21(N,T)f22(N,T)f23(N,T)f31(N,T)f32(N,T)f33(N,T)]3.85×1011σw2I>0.
j=kN+1kΦj,kTHjTHjΦj,k=[Nf12(Ω,N,T,t)f13(Ω,N,T,t)f21(Ω,N,T,t)f22(Ω,N,T,t)f23(Ω,N,T,t)f31(Ω,N,T,t)f32(Ω,N,T,t)f33(Ω,N,T,t)]1.15×109I>0

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:

tk={tup,k(2n1)tdown,k(2n),Tk={Tup,k(2n1)Tdown,k(2n),nN+.

The corresponding amplification factor Ω is:

Ωk={Ωup=v2v2v1,k(2n1)Ωdown=v1v1v2,k(2n),nN+.

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.

Tables Icon

Table 1. Simulation Parameters

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.

 

Fig. 5 Length errors between the time-invariant and time-varying Kalman filters.

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Fig. 6 Velocity errors between the time-invariant and time-varying Kalman filters.

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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.

 

Fig. 7 The simulation results of vibration tests.

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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.

 

Fig. 8 The RMSE of the vibration in variable amplitude.

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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.

 

Fig. 9 The simulations of vibration test in variable frequencies.

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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.

 

Fig. 10 Conditions of vibration measurement.

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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.

 

Fig. 11 Results of the 1D dynamic velocity measurement. a, the measured absolute length of the moving target. b, Estimated velocity using the FSI-Kalman filter. c, Estimated acceleration of the target. d, Deviation between the estimated velocity and the velocity of the moving stage.

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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.

 

Fig. 12 The results of the vibration measurement. a, measured length results of standard FSI (black) and the FSI-Kalman filter (red); b, the measured velocity (deep blue). c, the estimated acceleration of FSI-Kalman filter; d, the measurements results of vibration amplitude by using eddy3010 and FSI-Kalman filter.

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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.

5.3 Discussion

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.

6. Conclusions

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.

Funding

National Natural Science Foundation of China (NSFC) (Grant Nos. 51375376 and 51635010).

References and links

1. J. Dale, B. Hughes, A. J. Lancaster, A. J. Lewis, A. J. H. Reichold, and M. S. Warden, “Multi-channel absolute distance measurement system with sub ppm-accuracy and 20 m range using frequency scanning interferometry and gas absorption cells,” Opt. Express 22(20), 24869–24893 (2014). [PubMed]  

2. B. L. Swinkels, N. Bhattacharya, A. A. Wielders, and J. J. M. Braat, “Absolute distance metrology for space interferometers,” Proc. SPIE 554, 559–561 (2005).

3. G. Shi, F. M. Zhang, X. H. Xu, and X. S. Meng, “High-resolution frequency-modulated continuous-wave laser ranging for precision distance metrology applications,” Opt. Eng. 53(12), 122402 (2014).

4. A. F. Fox-Murphy, D. F. Howell, R. B. Nickerson, and A. R. Weidberg, “Frequency scanned interferometry (FSI): the basis of a survey system for ATLAS using fast automated remote interferometry,” Nucl. Instrum. Methods Phys. Res. 383(1), 229–237 (1999).

5. P. A. Coe, D. F. Howell, and R. B. Nickerson, “Frequency scanning interferometry in ATLAS: remote, multiple, simultaneous and precise distance measurements in a hostile environment,” Meas. Sci. Technol. 15(11), 2175–2187 (2004).

6. H. J. Yang, T. Chen, and K. Riles, “Frequency scanned interferometry for ILC tracker alignment,” https://arxiv.org/abs/1109.2582 (2011).

7. H. J. Yang, S. Nyberg, and K. Riles, “High-precision absolute distance measurement using dual-laser frequency scanned interferometry under realistic conditions,” Nucl. Instrum. Methods Phys. Res. 575(3), 395–401 (2007).

8. P. A. Coe, A. Mitra, S. M. Gibson, D. F. Howell, and R. B. Nickerson, ” Frequency scanning interferometry - a versatile, high precision, multiple distance measurement technique,” Seventh International Workshop on Accelerator Alignment (2002).

9. J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. Copner, and A. J. Lewis, “Dual-sweep frequency scanning interferometry using four wave mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).

10. A. Cabral and J. Rebordao, “Absolute distance metrology with frequency sweeping interferometry,” Proc. SPIE 5879, 195–204 (2005).

11. L. Tao, Z. Liu, W. Zhang, and Y. Zhou, “Frequency-scanning interferometry for dynamic absolute distance measurement using Kalman filter,” Opt. Lett. 39(24), 6997–7000 (2014). [PubMed]  

12. M. Zhang, Z. Liu, Y. Zhu, M. Bu, and J. Hong, “Integral force feedback control with input shaping: Application to piezo-based scanning systems in ECDLs,” Rev. Sci. Instrum. 88(7), 075006 (2017). [PubMed]  

13. R. E. Kalman, “Contribution to the theory of optimal control,” Bol. soc. mat. mexicana 5(63), 102–119 (1960).

14. B. Ristic, S. Arulampalam, and N. Gordon, “Beyond the Kalman filter: Particle filters for tracking applications,” IEEE Trans. Aerosp. Electron. Syst. 19(7), 37–38 (2003).

15. Z. W. Deng and Z. G. Liu, “Precision improvement in frequency-scanning interferometry based on suppressing nonlinear optical frequency sweeping,” Opt. Rev. 22(5), 724–730 (2015).

16. Z. Liu, Z. Liu, Z. Deng, and L. Tao, “Interference signal frequency tracking for extracting phase in frequency scanning interferometry using an extended Kalman filter,” Appl. Opt. 55(11), 2985–2992 (2016). [PubMed]  

17. A. Jain, P. K. Krishnamurthy, P. Landais, and P. M. Anandarajah, “EKF for Joint Mitigation of Phase Noise, Frequency Offset and Nonlinearity in 400 Gb/s PM-16-QAM and 200 Gb/s PM-QPSK Systems,” IEEE Photonics J. 9(1), 1–10 (2017).

18. B. D. O. Anderson and J. B. Moore, “Detectability and stabilizability of time-varying discrete-time linear systems,” SIAM J. Control Optim. 19(1), 20–32 (1981).

19. Y. L. Mo and B. Sinopoli, “Kalman filtering with intermittent observations: critical value for second order system,” 2003 Proc. IEEE Conf. Decision Control, 1453–1464 (2004).

References

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  1. J. Dale, B. Hughes, A. J. Lancaster, A. J. Lewis, A. J. H. Reichold, and M. S. Warden, “Multi-channel absolute distance measurement system with sub ppm-accuracy and 20 m range using frequency scanning interferometry and gas absorption cells,” Opt. Express 22(20), 24869–24893 (2014).
    [PubMed]
  2. B. L. Swinkels, N. Bhattacharya, A. A. Wielders, and J. J. M. Braat, “Absolute distance metrology for space interferometers,” Proc. SPIE 554, 559–561 (2005).
  3. G. Shi, F. M. Zhang, X. H. Xu, and X. S. Meng, “High-resolution frequency-modulated continuous-wave laser ranging for precision distance metrology applications,” Opt. Eng. 53(12), 122402 (2014).
  4. A. F. Fox-Murphy, D. F. Howell, R. B. Nickerson, and A. R. Weidberg, “Frequency scanned interferometry (FSI): the basis of a survey system for ATLAS using fast automated remote interferometry,” Nucl. Instrum. Methods Phys. Res. 383(1), 229–237 (1999).
  5. P. A. Coe, D. F. Howell, and R. B. Nickerson, “Frequency scanning interferometry in ATLAS: remote, multiple, simultaneous and precise distance measurements in a hostile environment,” Meas. Sci. Technol. 15(11), 2175–2187 (2004).
  6. H. J. Yang, T. Chen, and K. Riles, “Frequency scanned interferometry for ILC tracker alignment,” https://arxiv.org/abs/1109.2582 (2011).
  7. H. J. Yang, S. Nyberg, and K. Riles, “High-precision absolute distance measurement using dual-laser frequency scanned interferometry under realistic conditions,” Nucl. Instrum. Methods Phys. Res. 575(3), 395–401 (2007).
  8. P. A. Coe, A. Mitra, S. M. Gibson, D. F. Howell, and R. B. Nickerson, ” Frequency scanning interferometry - a versatile, high precision, multiple distance measurement technique,” Seventh International Workshop on Accelerator Alignment (2002).
  9. J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. Copner, and A. J. Lewis, “Dual-sweep frequency scanning interferometry using four wave mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).
  10. A. Cabral and J. Rebordao, “Absolute distance metrology with frequency sweeping interferometry,” Proc. SPIE 5879, 195–204 (2005).
  11. L. Tao, Z. Liu, W. Zhang, and Y. Zhou, “Frequency-scanning interferometry for dynamic absolute distance measurement using Kalman filter,” Opt. Lett. 39(24), 6997–7000 (2014).
    [PubMed]
  12. M. Zhang, Z. Liu, Y. Zhu, M. Bu, and J. Hong, “Integral force feedback control with input shaping: Application to piezo-based scanning systems in ECDLs,” Rev. Sci. Instrum. 88(7), 075006 (2017).
    [PubMed]
  13. R. E. Kalman, “Contribution to the theory of optimal control,” Bol. soc. mat. mexicana 5(63), 102–119 (1960).
  14. B. Ristic, S. Arulampalam, and N. Gordon, “Beyond the Kalman filter: Particle filters for tracking applications,” IEEE Trans. Aerosp. Electron. Syst. 19(7), 37–38 (2003).
  15. Z. W. Deng and Z. G. Liu, “Precision improvement in frequency-scanning interferometry based on suppressing nonlinear optical frequency sweeping,” Opt. Rev. 22(5), 724–730 (2015).
  16. Z. Liu, Z. Liu, Z. Deng, and L. Tao, “Interference signal frequency tracking for extracting phase in frequency scanning interferometry using an extended Kalman filter,” Appl. Opt. 55(11), 2985–2992 (2016).
    [PubMed]
  17. A. Jain, P. K. Krishnamurthy, P. Landais, and P. M. Anandarajah, “EKF for Joint Mitigation of Phase Noise, Frequency Offset and Nonlinearity in 400 Gb/s PM-16-QAM and 200 Gb/s PM-QPSK Systems,” IEEE Photonics J. 9(1), 1–10 (2017).
  18. B. D. O. Anderson and J. B. Moore, “Detectability and stabilizability of time-varying discrete-time linear systems,” SIAM J. Control Optim. 19(1), 20–32 (1981).
  19. Y. L. Mo and B. Sinopoli, “Kalman filtering with intermittent observations: critical value for second order system,” 2003 Proc. IEEE Conf. Decision Control, 1453–1464 (2004).

2017 (2)

M. Zhang, Z. Liu, Y. Zhu, M. Bu, and J. Hong, “Integral force feedback control with input shaping: Application to piezo-based scanning systems in ECDLs,” Rev. Sci. Instrum. 88(7), 075006 (2017).
[PubMed]

A. Jain, P. K. Krishnamurthy, P. Landais, and P. M. Anandarajah, “EKF for Joint Mitigation of Phase Noise, Frequency Offset and Nonlinearity in 400 Gb/s PM-16-QAM and 200 Gb/s PM-QPSK Systems,” IEEE Photonics J. 9(1), 1–10 (2017).

2016 (1)

2015 (2)

Z. W. Deng and Z. G. Liu, “Precision improvement in frequency-scanning interferometry based on suppressing nonlinear optical frequency sweeping,” Opt. Rev. 22(5), 724–730 (2015).

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. Copner, and A. J. Lewis, “Dual-sweep frequency scanning interferometry using four wave mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).

2014 (3)

2007 (1)

H. J. Yang, S. Nyberg, and K. Riles, “High-precision absolute distance measurement using dual-laser frequency scanned interferometry under realistic conditions,” Nucl. Instrum. Methods Phys. Res. 575(3), 395–401 (2007).

2005 (2)

A. Cabral and J. Rebordao, “Absolute distance metrology with frequency sweeping interferometry,” Proc. SPIE 5879, 195–204 (2005).

B. L. Swinkels, N. Bhattacharya, A. A. Wielders, and J. J. M. Braat, “Absolute distance metrology for space interferometers,” Proc. SPIE 554, 559–561 (2005).

2004 (1)

P. A. Coe, D. F. Howell, and R. B. Nickerson, “Frequency scanning interferometry in ATLAS: remote, multiple, simultaneous and precise distance measurements in a hostile environment,” Meas. Sci. Technol. 15(11), 2175–2187 (2004).

2003 (1)

B. Ristic, S. Arulampalam, and N. Gordon, “Beyond the Kalman filter: Particle filters for tracking applications,” IEEE Trans. Aerosp. Electron. Syst. 19(7), 37–38 (2003).

1999 (1)

A. F. Fox-Murphy, D. F. Howell, R. B. Nickerson, and A. R. Weidberg, “Frequency scanned interferometry (FSI): the basis of a survey system for ATLAS using fast automated remote interferometry,” Nucl. Instrum. Methods Phys. Res. 383(1), 229–237 (1999).

1981 (1)

B. D. O. Anderson and J. B. Moore, “Detectability and stabilizability of time-varying discrete-time linear systems,” SIAM J. Control Optim. 19(1), 20–32 (1981).

1960 (1)

R. E. Kalman, “Contribution to the theory of optimal control,” Bol. soc. mat. mexicana 5(63), 102–119 (1960).

Anandarajah, P. M.

A. Jain, P. K. Krishnamurthy, P. Landais, and P. M. Anandarajah, “EKF for Joint Mitigation of Phase Noise, Frequency Offset and Nonlinearity in 400 Gb/s PM-16-QAM and 200 Gb/s PM-QPSK Systems,” IEEE Photonics J. 9(1), 1–10 (2017).

Anderson, B. D. O.

B. D. O. Anderson and J. B. Moore, “Detectability and stabilizability of time-varying discrete-time linear systems,” SIAM J. Control Optim. 19(1), 20–32 (1981).

Arulampalam, S.

B. Ristic, S. Arulampalam, and N. Gordon, “Beyond the Kalman filter: Particle filters for tracking applications,” IEEE Trans. Aerosp. Electron. Syst. 19(7), 37–38 (2003).

Bhattacharya, N.

B. L. Swinkels, N. Bhattacharya, A. A. Wielders, and J. J. M. Braat, “Absolute distance metrology for space interferometers,” Proc. SPIE 554, 559–561 (2005).

Braat, J. J. M.

B. L. Swinkels, N. Bhattacharya, A. A. Wielders, and J. J. M. Braat, “Absolute distance metrology for space interferometers,” Proc. SPIE 554, 559–561 (2005).

Bu, M.

M. Zhang, Z. Liu, Y. Zhu, M. Bu, and J. Hong, “Integral force feedback control with input shaping: Application to piezo-based scanning systems in ECDLs,” Rev. Sci. Instrum. 88(7), 075006 (2017).
[PubMed]

Cabral, A.

A. Cabral and J. Rebordao, “Absolute distance metrology with frequency sweeping interferometry,” Proc. SPIE 5879, 195–204 (2005).

Campbell, M. A.

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. Copner, and A. J. Lewis, “Dual-sweep frequency scanning interferometry using four wave mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).

Coe, P. A.

P. A. Coe, D. F. Howell, and R. B. Nickerson, “Frequency scanning interferometry in ATLAS: remote, multiple, simultaneous and precise distance measurements in a hostile environment,” Meas. Sci. Technol. 15(11), 2175–2187 (2004).

P. A. Coe, A. Mitra, S. M. Gibson, D. F. Howell, and R. B. Nickerson, ” Frequency scanning interferometry - a versatile, high precision, multiple distance measurement technique,” Seventh International Workshop on Accelerator Alignment (2002).

Copner, N.

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. Copner, and A. J. Lewis, “Dual-sweep frequency scanning interferometry using four wave mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).

Dale, J.

Deng, Z.

Deng, Z. W.

Z. W. Deng and Z. G. Liu, “Precision improvement in frequency-scanning interferometry based on suppressing nonlinear optical frequency sweeping,” Opt. Rev. 22(5), 724–730 (2015).

Fox-Murphy, A. F.

A. F. Fox-Murphy, D. F. Howell, R. B. Nickerson, and A. R. Weidberg, “Frequency scanned interferometry (FSI): the basis of a survey system for ATLAS using fast automated remote interferometry,” Nucl. Instrum. Methods Phys. Res. 383(1), 229–237 (1999).

Gibson, S. M.

P. A. Coe, A. Mitra, S. M. Gibson, D. F. Howell, and R. B. Nickerson, ” Frequency scanning interferometry - a versatile, high precision, multiple distance measurement technique,” Seventh International Workshop on Accelerator Alignment (2002).

Gordon, N.

B. Ristic, S. Arulampalam, and N. Gordon, “Beyond the Kalman filter: Particle filters for tracking applications,” IEEE Trans. Aerosp. Electron. Syst. 19(7), 37–38 (2003).

Hong, J.

M. Zhang, Z. Liu, Y. Zhu, M. Bu, and J. Hong, “Integral force feedback control with input shaping: Application to piezo-based scanning systems in ECDLs,” Rev. Sci. Instrum. 88(7), 075006 (2017).
[PubMed]

Howell, D. F.

P. A. Coe, D. F. Howell, and R. B. Nickerson, “Frequency scanning interferometry in ATLAS: remote, multiple, simultaneous and precise distance measurements in a hostile environment,” Meas. Sci. Technol. 15(11), 2175–2187 (2004).

A. F. Fox-Murphy, D. F. Howell, R. B. Nickerson, and A. R. Weidberg, “Frequency scanned interferometry (FSI): the basis of a survey system for ATLAS using fast automated remote interferometry,” Nucl. Instrum. Methods Phys. Res. 383(1), 229–237 (1999).

P. A. Coe, A. Mitra, S. M. Gibson, D. F. Howell, and R. B. Nickerson, ” Frequency scanning interferometry - a versatile, high precision, multiple distance measurement technique,” Seventh International Workshop on Accelerator Alignment (2002).

Hughes, B.

Hughes, E. B.

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. Copner, and A. J. Lewis, “Dual-sweep frequency scanning interferometry using four wave mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).

Jain, A.

A. Jain, P. K. Krishnamurthy, P. Landais, and P. M. Anandarajah, “EKF for Joint Mitigation of Phase Noise, Frequency Offset and Nonlinearity in 400 Gb/s PM-16-QAM and 200 Gb/s PM-QPSK Systems,” IEEE Photonics J. 9(1), 1–10 (2017).

Kalman, R. E.

R. E. Kalman, “Contribution to the theory of optimal control,” Bol. soc. mat. mexicana 5(63), 102–119 (1960).

Krishnamurthy, P. K.

A. Jain, P. K. Krishnamurthy, P. Landais, and P. M. Anandarajah, “EKF for Joint Mitigation of Phase Noise, Frequency Offset and Nonlinearity in 400 Gb/s PM-16-QAM and 200 Gb/s PM-QPSK Systems,” IEEE Photonics J. 9(1), 1–10 (2017).

Lancaster, A. J.

Landais, P.

A. Jain, P. K. Krishnamurthy, P. Landais, and P. M. Anandarajah, “EKF for Joint Mitigation of Phase Noise, Frequency Offset and Nonlinearity in 400 Gb/s PM-16-QAM and 200 Gb/s PM-QPSK Systems,” IEEE Photonics J. 9(1), 1–10 (2017).

Lewis, A. J.

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. Copner, and A. J. Lewis, “Dual-sweep frequency scanning interferometry using four wave mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).

J. Dale, B. Hughes, A. J. Lancaster, A. J. Lewis, A. J. H. Reichold, and M. S. Warden, “Multi-channel absolute distance measurement system with sub ppm-accuracy and 20 m range using frequency scanning interferometry and gas absorption cells,” Opt. Express 22(20), 24869–24893 (2014).
[PubMed]

Liu, Z.

Liu, Z. G.

Z. W. Deng and Z. G. Liu, “Precision improvement in frequency-scanning interferometry based on suppressing nonlinear optical frequency sweeping,” Opt. Rev. 22(5), 724–730 (2015).

Martinez, J. J.

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. Copner, and A. J. Lewis, “Dual-sweep frequency scanning interferometry using four wave mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).

Meng, X. S.

G. Shi, F. M. Zhang, X. H. Xu, and X. S. Meng, “High-resolution frequency-modulated continuous-wave laser ranging for precision distance metrology applications,” Opt. Eng. 53(12), 122402 (2014).

Mitra, A.

P. A. Coe, A. Mitra, S. M. Gibson, D. F. Howell, and R. B. Nickerson, ” Frequency scanning interferometry - a versatile, high precision, multiple distance measurement technique,” Seventh International Workshop on Accelerator Alignment (2002).

Mo, Y. L.

Y. L. Mo and B. Sinopoli, “Kalman filtering with intermittent observations: critical value for second order system,” 2003 Proc. IEEE Conf. Decision Control, 1453–1464 (2004).

Moore, J. B.

B. D. O. Anderson and J. B. Moore, “Detectability and stabilizability of time-varying discrete-time linear systems,” SIAM J. Control Optim. 19(1), 20–32 (1981).

Nickerson, R. B.

P. A. Coe, D. F. Howell, and R. B. Nickerson, “Frequency scanning interferometry in ATLAS: remote, multiple, simultaneous and precise distance measurements in a hostile environment,” Meas. Sci. Technol. 15(11), 2175–2187 (2004).

A. F. Fox-Murphy, D. F. Howell, R. B. Nickerson, and A. R. Weidberg, “Frequency scanned interferometry (FSI): the basis of a survey system for ATLAS using fast automated remote interferometry,” Nucl. Instrum. Methods Phys. Res. 383(1), 229–237 (1999).

P. A. Coe, A. Mitra, S. M. Gibson, D. F. Howell, and R. B. Nickerson, ” Frequency scanning interferometry - a versatile, high precision, multiple distance measurement technique,” Seventh International Workshop on Accelerator Alignment (2002).

Nyberg, S.

H. J. Yang, S. Nyberg, and K. Riles, “High-precision absolute distance measurement using dual-laser frequency scanned interferometry under realistic conditions,” Nucl. Instrum. Methods Phys. Res. 575(3), 395–401 (2007).

Rebordao, J.

A. Cabral and J. Rebordao, “Absolute distance metrology with frequency sweeping interferometry,” Proc. SPIE 5879, 195–204 (2005).

Reichold, A. J. H.

Riles, K.

H. J. Yang, S. Nyberg, and K. Riles, “High-precision absolute distance measurement using dual-laser frequency scanned interferometry under realistic conditions,” Nucl. Instrum. Methods Phys. Res. 575(3), 395–401 (2007).

Ristic, B.

B. Ristic, S. Arulampalam, and N. Gordon, “Beyond the Kalman filter: Particle filters for tracking applications,” IEEE Trans. Aerosp. Electron. Syst. 19(7), 37–38 (2003).

Shi, G.

G. Shi, F. M. Zhang, X. H. Xu, and X. S. Meng, “High-resolution frequency-modulated continuous-wave laser ranging for precision distance metrology applications,” Opt. Eng. 53(12), 122402 (2014).

Sinopoli, B.

Y. L. Mo and B. Sinopoli, “Kalman filtering with intermittent observations: critical value for second order system,” 2003 Proc. IEEE Conf. Decision Control, 1453–1464 (2004).

Swinkels, B. L.

B. L. Swinkels, N. Bhattacharya, A. A. Wielders, and J. J. M. Braat, “Absolute distance metrology for space interferometers,” Proc. SPIE 554, 559–561 (2005).

Tao, L.

Warden, M. S.

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. Copner, and A. J. Lewis, “Dual-sweep frequency scanning interferometry using four wave mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).

J. Dale, B. Hughes, A. J. Lancaster, A. J. Lewis, A. J. H. Reichold, and M. S. Warden, “Multi-channel absolute distance measurement system with sub ppm-accuracy and 20 m range using frequency scanning interferometry and gas absorption cells,” Opt. Express 22(20), 24869–24893 (2014).
[PubMed]

Weidberg, A. R.

A. F. Fox-Murphy, D. F. Howell, R. B. Nickerson, and A. R. Weidberg, “Frequency scanned interferometry (FSI): the basis of a survey system for ATLAS using fast automated remote interferometry,” Nucl. Instrum. Methods Phys. Res. 383(1), 229–237 (1999).

Wielders, A. A.

B. L. Swinkels, N. Bhattacharya, A. A. Wielders, and J. J. M. Braat, “Absolute distance metrology for space interferometers,” Proc. SPIE 554, 559–561 (2005).

Xu, X. H.

G. Shi, F. M. Zhang, X. H. Xu, and X. S. Meng, “High-resolution frequency-modulated continuous-wave laser ranging for precision distance metrology applications,” Opt. Eng. 53(12), 122402 (2014).

Yang, H. J.

H. J. Yang, S. Nyberg, and K. Riles, “High-precision absolute distance measurement using dual-laser frequency scanned interferometry under realistic conditions,” Nucl. Instrum. Methods Phys. Res. 575(3), 395–401 (2007).

Zhang, F. M.

G. Shi, F. M. Zhang, X. H. Xu, and X. S. Meng, “High-resolution frequency-modulated continuous-wave laser ranging for precision distance metrology applications,” Opt. Eng. 53(12), 122402 (2014).

Zhang, M.

M. Zhang, Z. Liu, Y. Zhu, M. Bu, and J. Hong, “Integral force feedback control with input shaping: Application to piezo-based scanning systems in ECDLs,” Rev. Sci. Instrum. 88(7), 075006 (2017).
[PubMed]

Zhang, W.

Zhou, Y.

Zhu, Y.

M. Zhang, Z. Liu, Y. Zhu, M. Bu, and J. Hong, “Integral force feedback control with input shaping: Application to piezo-based scanning systems in ECDLs,” Rev. Sci. Instrum. 88(7), 075006 (2017).
[PubMed]

Appl. Opt. (1)

Bol. soc. mat. mexicana (1)

R. E. Kalman, “Contribution to the theory of optimal control,” Bol. soc. mat. mexicana 5(63), 102–119 (1960).

IEEE Photonics J. (1)

A. Jain, P. K. Krishnamurthy, P. Landais, and P. M. Anandarajah, “EKF for Joint Mitigation of Phase Noise, Frequency Offset and Nonlinearity in 400 Gb/s PM-16-QAM and 200 Gb/s PM-QPSK Systems,” IEEE Photonics J. 9(1), 1–10 (2017).

IEEE Photonics Technol. Lett. (1)

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. Copner, and A. J. Lewis, “Dual-sweep frequency scanning interferometry using four wave mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).

IEEE Trans. Aerosp. Electron. Syst. (1)

B. Ristic, S. Arulampalam, and N. Gordon, “Beyond the Kalman filter: Particle filters for tracking applications,” IEEE Trans. Aerosp. Electron. Syst. 19(7), 37–38 (2003).

Meas. Sci. Technol. (1)

P. A. Coe, D. F. Howell, and R. B. Nickerson, “Frequency scanning interferometry in ATLAS: remote, multiple, simultaneous and precise distance measurements in a hostile environment,” Meas. Sci. Technol. 15(11), 2175–2187 (2004).

Nucl. Instrum. Methods Phys. Res. (2)

H. J. Yang, S. Nyberg, and K. Riles, “High-precision absolute distance measurement using dual-laser frequency scanned interferometry under realistic conditions,” Nucl. Instrum. Methods Phys. Res. 575(3), 395–401 (2007).

A. F. Fox-Murphy, D. F. Howell, R. B. Nickerson, and A. R. Weidberg, “Frequency scanned interferometry (FSI): the basis of a survey system for ATLAS using fast automated remote interferometry,” Nucl. Instrum. Methods Phys. Res. 383(1), 229–237 (1999).

Opt. Eng. (1)

G. Shi, F. M. Zhang, X. H. Xu, and X. S. Meng, “High-resolution frequency-modulated continuous-wave laser ranging for precision distance metrology applications,” Opt. Eng. 53(12), 122402 (2014).

Opt. Express (1)

Opt. Lett. (1)

Opt. Rev. (1)

Z. W. Deng and Z. G. Liu, “Precision improvement in frequency-scanning interferometry based on suppressing nonlinear optical frequency sweeping,” Opt. Rev. 22(5), 724–730 (2015).

Proc. SPIE (2)

A. Cabral and J. Rebordao, “Absolute distance metrology with frequency sweeping interferometry,” Proc. SPIE 5879, 195–204 (2005).

B. L. Swinkels, N. Bhattacharya, A. A. Wielders, and J. J. M. Braat, “Absolute distance metrology for space interferometers,” Proc. SPIE 554, 559–561 (2005).

Rev. Sci. Instrum. (1)

M. Zhang, Z. Liu, Y. Zhu, M. Bu, and J. Hong, “Integral force feedback control with input shaping: Application to piezo-based scanning systems in ECDLs,” Rev. Sci. Instrum. 88(7), 075006 (2017).
[PubMed]

SIAM J. Control Optim. (1)

B. D. O. Anderson and J. B. Moore, “Detectability and stabilizability of time-varying discrete-time linear systems,” SIAM J. Control Optim. 19(1), 20–32 (1981).

Other (3)

Y. L. Mo and B. Sinopoli, “Kalman filtering with intermittent observations: critical value for second order system,” 2003 Proc. IEEE Conf. Decision Control, 1453–1464 (2004).

P. A. Coe, A. Mitra, S. M. Gibson, D. F. Howell, and R. B. Nickerson, ” Frequency scanning interferometry - a versatile, high precision, multiple distance measurement technique,” Seventh International Workshop on Accelerator Alignment (2002).

H. J. Yang, T. Chen, and K. Riles, “Frequency scanned interferometry for ILC tracker alignment,” https://arxiv.org/abs/1109.2582 (2011).

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Figures (12)

Fig. 1
Fig. 1 Schematic illustration of the principles of FSI.
Fig. 2
Fig. 2 The FSI system. In the laboratory environment, the setup was constructed on a floating optical table to reduce interference caused by external noise.
Fig. 3
Fig. 3 The state estimation for dynamic FSI.
Fig. 4
Fig. 4 Statistical distribution of T and t. Part a (blue) represents the sampling period T of the Kalman filter, and Part b (red) represents the FSI period t. These plots are based on 6,000 FSI measurement samples.
Fig. 5
Fig. 5 Length errors between the time-invariant and time-varying Kalman filters.
Fig. 6
Fig. 6 Velocity errors between the time-invariant and time-varying Kalman filters.
Fig. 7
Fig. 7 The simulation results of vibration tests.
Fig. 8
Fig. 8 The RMSE of the vibration in variable amplitude.
Fig. 9
Fig. 9 The simulations of vibration test in variable frequencies.
Fig. 10
Fig. 10 Conditions of vibration measurement.
Fig. 11
Fig. 11 Results of the 1D dynamic velocity measurement. a, the measured absolute length of the moving target. b, Estimated velocity using the FSI-Kalman filter. c, Estimated acceleration of the target. d, Deviation between the estimated velocity and the velocity of the moving stage.
Fig. 12
Fig. 12 The results of the vibration measurement. a, measured length results of standard FSI (black) and the FSI-Kalman filter (red); b, the measured velocity (deep blue). c, the estimated acceleration of FSI-Kalman filter; d, the measurements results of vibration amplitude by using eddy3010 and FSI-Kalman filter.

Tables (1)

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Table 1 Simulation Parameters

Equations (23)

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I= I 1 + I 2 +2 I 1 I 2 cos( ϕ 1 ϕ 2 ),
L= 1 2n Δϕ 2π c Δv ,
Δ ϕ ' = 4nπ c [ ( L+Δε ) v 2 L v 1 ].
L M = cΔ ϕ ' 4nπΔv = L true + v 2 v 2 ν 1 Δε= L true +ΩΔε ,
L k+1 = L k +sT+ 1 2 a T 2 ,
L kM = L k + Ω k s k t k + 1 2 Ω k a k t k 2 ,
x k+1 = Φ k x k + w k ,
y k = H k x k + υ k ,
Q k = σ w 2 [ T k 6 36 T k 5 12 T k 4 6 T k 5 12 T k 4 4 T k 3 2 T k 4 6 T k 3 2 T k 2 ], R k =[ σ υ 2 ],
T k ={ N( T up , σ Tup ) N( T down , σ Tdown ) , t k ={ N( t up , σ tup ) N( t down , σ tdown ) ,
Φ k =[ 1 T k T k 2 /2 0 1 T k 0 0 1 ]={ Φ up Φ down , H k =[ 1 Ω k t k Ω k t k 2 2 ]={ H up H down .
Ω k ={ Ω up = v 2 v 2 v 1 Ω down = v 1 v 1 v 2 .
x ^ k|k = x ^ k|k1 + K k ( y k H k x ^ k|k1 ),
P k|k = P k| k1 K k H k P k| k1 ,
x ^ k+1|k = Φ k+1 x ^ k|k , P k+1|k = Φ k+1 P k|k Φ k+1 T + Q k ,
K k = P k| k1 H k1 T ( H k P k| k1 H k T +R ) 1 , x ^ 0| 1 = x 0 , P 0| 1 = P 0 ,
P 0 , s.t. sup E P k =0p p c
sup k Φ k 2 1.01799, sup k H k 2 67.45363
0< ξ 1 I i=kN+1 k Φ k,i Q i,i1 Q i,i1 T Φ k,i T 0< ξ 2 I j=kN+1 k Φ j,k T H j T H j Φ j,k .
i=kN+1 k Φ k,i Q i,i1 Q i,i1 T Φ k,i T = σ w 2 [ f 11 ( N,T ) f 12 ( N,T ) f 13 ( N,T ) f 21 ( N,T ) f 22 ( N,T ) f 23 ( N,T ) f 31 ( N,T ) f 32 ( N,T ) f 33 ( N,T ) ]3.85× 10 11 σ w 2 I>0 .
j=kN+1 k Φ j,k T H j T H j Φ j,k = [ N f 12 ( Ω,N,T,t ) f 13 ( Ω,N,T,t ) f 21 ( Ω,N,T,t ) f 22 ( Ω,N,T,t ) f 23 ( Ω,N,T,t ) f 31 ( Ω,N,T,t ) f 32 ( Ω,N,T,t ) f 33 ( Ω,N,T,t ) ]1.15× 10 9 I>0
t k ={ t up ,k( 2n1 ) t down ,k( 2n ) , T k ={ T up ,k( 2n1 ) T down ,k( 2n ) ,n N + .
Ω k ={ Ω up = v 2 v 2 v 1 ,k( 2n1 ) Ω down = v 1 v 1 v 2 ,k( 2n ) ,n N + .

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