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Dynamic cascade-model-based frequency-scanning interferometry for real-time and rapid absolute optical ranging

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Abstract

A new frequency-scanning interferometry (FSI) scheme using in-phase and quadrature (IQ) detection for real-time and rapid absolute optical ranging is presented. Dynamic measurement with FSI modulates interference signal frequency by both target movement and time-varying optical-frequency scanning rate; hence, a dynamic model is proposed to decouple dynamic absolute distance from the instantaneous frequency of interference signals. The unscented Kalman filter and particle filter algorithms are implemented for the nonlinear first-layer and non-Gaussian second-layer models, respectively. The proposed FSI scheme eliminates nonlinear optical-frequency scanning effects in dynamic measurements and realizes real-time measurement only current observed data are used. Experimental results verify high tracking performance for a vibrating target with approximately 10 μm amplitude and 50–500 Hz frequency.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

As one of the most promising techniques for absolute distance measurement (ADM), FSI has found a wide range of applications in science and industry in recent decades, such as autonomous driving and navigation [1], optical coherence tomography (OCT) [2,3], depth-mapping or 3D imaging [4–6], and high-precision configuration of satellites in space missions [7]. Compared to other techniques for ADM, including time of flight, multi-wavelength interferometry, and frequency comb interferometry, FSI is more suitable for implementation in practical applications owing to its compactness, low instrumental complexity, and cost-effectiveness while providing sufficient accuracy [8–14]. Conventional single FSI is always used for static target measurement owing to its high sensitivity to variations in the optical path length. Any variation of the optical path length can be considerably amplified by a factor given by the ratio of the laser optical frequency to the mode-hop-free (MHF) frequency scanning range υ/Δυ [15,16]. To address the dynamic measurement problem, some studies proposed dual-FSI systems using two different tunable lasers [17,18] or using four-wave mixing [19]. However, dual-FSI systems require an additional laser source, which implies an increase in instrumental complexity and costs. Additionally, several compensation algorithms based on two consecutive measurements [7,20] or on a motion state estimation model have been developed for single FSI systems using Kalman filters [21,22]. However, with respect to dynamic measurement, the frequency of the interference signal is always modulated by both target movement and time-varying optical-frequency scanning rate, which means that the performance of all the aforementioned investigations are also limited by the nonlinearities of optical-frequency scanning with tunable lasers during measurement [23,24]. Moreover, these works required a certain length of the interference signal in the time domain for offline processing, either to obtain a distance measurement average or for distance measurements at any sampling point over this period of time. This not only reduces the measurement bandwidth but also hinders real-time measurement. In other words, precise measurement is obtained at the expense of longer acquisition times and poor real-time performance.

To address the challenges of nonlinear optical-frequency scanning and poor real-time performance in high-speed measurements, herein we show that a simple and inexpensive, yet precise FSI technique provides rapid and real-time measurement of a dynamic target. Considering the nonlinear optical-frequency scanning of an external cavity diode laser (ECDL), the instantaneous frequency of the measured interference signal contains information on not only the instantaneous absolute distance of the dynamic target but also the time-varying optical-frequency scanning rate. Hence, we propose a novel dynamic model with a cascade architecture to decouple the dynamic absolute distance from the instantaneous frequency of the interference signal layerwise. Using the detected IQ signals from measured and auxiliary interferometers as observed data, the instantaneous frequency of the interference signal from the interferometer and the time-varying optical-frequency scanning rate are simultaneously estimated as states based on the first-layer model. Then, the estimated states of the last layer model are regarded as inputs to the second-layer model, and the absolute distance of the dynamic target is finally estimated based on the second-layer model. Based on the dynamic cascade model and the corresponding signal processing, the effects of nonlinear optical-frequency scanning in dynamic measurements can be removed. By applying the unscented Kalman filter (UKF) and particle filter in the first and second layers respectively, our proposed system can handle the nonlinear first-layer model and non-Gaussian second-layer model, respectively. Since both Bayesian estimation approaches do not need a complete history of the observed data, the cascade-model-based FSI can guarantee real-time measurement. Meanwhile, the measurement bandwidth is only determined by the sampling rate of the data acquisition (DAQ) card, which overcomes the measurement bandwidth limit of the conventional FSI. We experimentally demonstrated that based on the dynamic cascade model, our proposed FSI system achieves high tracking performance for dynamic targets at a relatively long distance compared to the conventional models. The proposed FSI method is promising for a wealth of applications requiring rapid and real-time dynamic ranging systems.

The remainder of this paper is structured as follows: the principle of FSI for dynamic measurement is described in Section 2. Section 3 provides details on the development of the dynamic cascade model. In Section 4, a detailed description of our experimental demonstration is provided.

2. Principle of FSI for dynamic measurement

To account for arbitrary nonlinear scanning of the ECDL, the electric field at the Michelson interferometer input takes the form:

E(t)=E0eiϕ(t),
where E0 is the field amplitude and ϕ(t) is a time-varying phase. By differentiation of the phase, the instantaneous optical frequency of the laser is given by

υ(t)=12πdϕdt.

Within a basic FSI system, the time delay between the reference and measurement paths is defined as τ=2nL/c, where L is the optical path difference, n is air refractive index and c is the speed of light in vacuum. The optical fringe pattern generated by the interferometer produces a voltage at a square-law detector given by

U(t)=ε|E(t)+E(t+τ)|2=U0{1+cos[ϕ(t+τ)ϕ(t)]},
where ε is the permittivity of free space that depends on the detector sensitivity and U0=2ε|E0|2. Here, we focus on the alternating-current (AC) component within Eq. (3) and Taylor expand the function ϕ(t+τ) for t:

ϕ(t+τ)=i=0τii!ϕ(i)(t).

Considering the optical path difference and laser scanning rate in our case, τ2dυ/dt1. Hence, neglecting the second- and higher-order terms of the sum in Eq. (4) and using Eq. (2) yields an expression for the phase φ of the fringe pattern:

φ=ϕ(t+τ)ϕ(t)=2πυ(t)τ+2πi=2τii!υ(i1)(t)2πυ(t)τ.

For dynamic object measurement, the time delay τ is no longer constant owing to the time-varying L(t). Hence,τ(t)=2nL(t)/c. Then, the interference signal is given as cos[2πυ(t)τ(t)+φ0] and the frequency of the fringe pattern can be expressed as:

fInterf(t)=12πdφdt=2nc[υ(t)L(t)+υ(t)L(t)].

It can be clearly seen in Eq. (6) that, with the assumption of linear frequency scanning, the frequency of the fringe pattern is only modulated by the movement of the measured target, which means that this movement can be extracted from the interference signal by estimating its frequency in real time. However, in practice, the nonlinearity of the frequency scanning of the laser source is coupled into the modulation in addition to the movement of the target.

Figure 1(a) shows a schematic diagram of the proposed measurement system, which comprises an ECDL, a measuring interferometer, and an auxiliary interferometer module consisting of a Mach–Zehnder (M-Z) interferometer and a Fabry–Pérot (F-P) cavity. The measuring interferometer is used for obtaining the IQ interference signals caused by the movement of the dynamic target and nonlinear optical frequency scanning, whereas the auxiliary interferometer module is used to synchronously monitor the time-varying optical frequency scanning rate. As shown in Fig. 1(b), owing to the nonlinear optical frequency sweeping caused by hysteresis of the PZT [23], the frequency of the measured interference signal is modulated simultaneously by both nonlinear optical frequency scanning and the target movement. Therefore, to determine the absolute distance of the moving target in real time, the target movement must be decoupled from the multi-modulated interferometric pattern. To address this challenge, we propose a novel cascade dynamic model consisting of two layers of the state-space model. To satisfy the high-speed dynamic measurement in real time, we use a specific recursive Bayesian filter to estimate the state of each layer of the state-space model, which does not require complete history of observation data.

 figure: Fig. 1

Fig. 1 (a) Conceptual optical layout of cascade dynamic model based FSI for rapid measurement. BS: beam splitter, PBS: polarizing beam splitter, RR: retroreflector, PD: photodetector, QWP: quarter wave plate, PZT: piezoelectric translator (b) Relationships between the frequency of interference signals and optical frequency or target movement.

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3. Cascade model

3.1 First-layer model design for frequency tracking of interference signal

The first layer model is designed to track the instantaneous frequency of fringe pattern in real time based on the quadrature detection using the UKF. As shown in Fig. 1(a), the measured interferometric setup uses a PBS cube. The incoming light enters a PBS that is linearly polarized through linear polarization oriented at 45° with respect to horizontal/vertical axes. The light beams return and are recombined in the beam-splitter cube. A QWP placed in each arm of the measured interferometer is used to direct the recombined horizontal and vertical components of the incoming polarization. After two recombination copies of the light are generated using a beam splitter. As the light from each arm has orthogonal polarizations, it must be sent through a polarizer oriented at 45° to the polarization axis of each light beam in order to observe interference. One of the copies of the light beam has a phase difference between the light from each arm added with the use of a quarter wave plate. The offsets and amplitude of the two interference signals can be first calibrated. After offset subtraction and normalization of the amplitude of the detected interference signals, the in-phase and quadrature signals were obtained from PDs

I(t)=cos[2πυ(t)τ(t)]Q(t)=sin[2πυ(t)τ(t)].

The state vector of the first layer state-space model can be defined as x(k)=[I(k),Q(k),f(k)]T=[x1(k),x2(k),x3(k)]T, where I(k) and Q(k) are the in-phase and quadrature components at time step k, and f(k) is the unknown time-varying instantaneous frequency of the interference signal. Frequency change between the time step k and k −1 can be ignored since the sampling rate is much higher than the frequency of the interference signal in our case. Hence, the following set equations holds:

x1(k)=x1(k1)cos[2πΔTx3(k1)]x2(k1)sin[2πΔTx3(k1)]x2(k)=x1(k1)sin[2πΔTx3(k1)]x2(k1)cos[2πΔTx3(k1)]x3(k)=x3(k1),y1(k)=x1(k)y2(k)=x2(k),
where ΔT is the sampling interval of the DAQ between time step k and k1, and the observation vectors y(k)=[y1(k),y2(k)] are the detected in-phase and quadrature signals. Accordingly, the state-space dynamic model of the first layer can be designed as follows:

x(k)=f(x(k1),v(k1))=Fx(k1)+v(k1)y(k)=h(x(k),u(k))=Hx(k)+u(k),

Here, the state transition matrix and measurement matrix are

F=[cos[2πΔTx3(k1)]sin[2πΔTx3(k1)]0sin[2πΔTx3(k1)]cos[2πΔTx3(k1)]0001]H=[100010],
where v(k) and u(k) denote the uncorrelated zero-mean Gaussian process noise and measurement noise, respectively. It should be noted that the transition matrix contains the state component, which means that the model of this layer is nonlinear. Hence, the classical Bayesian state estimation only for the linear Gaussian filtering problem, including the basic Kalman filter, becomes invalid in this case and an improved method is required to handle the nonlinear model.

With the above state space model of first layer, the instantaneous frequency of the interference signal can be estimated by the UKF. Therefore, we implement the UKF to handle the nonlinear Gaussian model of this layer. The UKF is based on the unscented transformation, and it is widely used to estimate the state vector of nonlinear processes [25,26]. Unlike the basic Kalman filter, the key idea behind the UKF is to apply a set of deterministic sampling points to the nonlinear model, enabling the capturing of the posterior mean and covariance accurately to the 3rd order. Given 2m+1 sigma points χ(k1) at time step k, where m is the dimension of the state vector,

χ(k1)=[x^0(k1)x^i(k1)+(γP(k1))ix^i+m(k1)(γP(k1))i+m]T(i=1,....,m),
where x^0(k1) is the estimate of the state mean and Pk1 denotes the evaluated covariance matrix. γ weights the covariance matrix and is defined as γ=L+ψ, where ψ=α2(m+κ)m is a scaling parameter. The parameter α determines the spread of sigma points around x^k1 and κ is a secondary scaling parameter. Each column of the sigma point matrix is propagated at a step through the state transition matrix. Then the implementation of UKF follows the steps mentioned below [27]

A priori estimation

χi(k)=F(χi(k1))x^(k)=i=12m+1Wimeanχi(k)P(k)=i=12m+1Wicov[χi(k)x^(k)][χi(k)x^(k)]T+Qχ(k)=[x^0(k)x^i(k)+(γP(k))ix^i+m(k)(γP(k))i+m]Y(k)=H(χ(k))y(k)=i=12m+1WimeanYi(k).

A posteriori estimation

Py(k)y(k)=i=12m+1Wicov[Yi(k)y(k)][Yi(k)y(k)]T+RPx(k)y(k)=i=12m+1Wicov[χi(k)x^(k)][Yi(k)y(k)]Tχ(k)=[x^0(k)x^i(k)+(γP(k))ix^i+m(k)(P(k))i+m]K(k)=Py(k)y(k)Px(k)y(k)1P(k)=P(k)Py(k)y(k)KT(k)x^(k)=x^(k)+K(k)[y(k)y(k)].

In Eq. (13), the weights Wimean and Wicov are defined as:

W1mean=ψm+ψ,Wimean=12(m+ψ)W1cov=ψm+ψ+(1α2+β),Wicov=Wimean,
where β is a parameter used to incorporate prior knowledge of the higher order moments of the state distribution. As shown in the Eq. (13), the final state estimation steps are still performed using the basic Kalman filter. By implementing this algorithm recursively, we can obtain the estimated state x^(k) and the third component x^3(k) is the frequency of the interference signal, which is used in the second-layer model. It should be noted that unlike many other methods of obtaining the instantaneous frequency, such as empirical mode decomposition (EMD) and wavelet transformation, the UKF based on the first-layer model does not use complete history of observed data, ensuring real-time estimation.

3.2 Second-layer model design for absolute distance ranging

After obtaining the frequency of the interference signal using UKF based on the first layer model, the second layer model is designed for estimating the absolute distance of the target in real time using the frequency of the interference signal. This signal is estimated from the first-layer model and the auxiliary interferometer, as shown in Fig. 1(a). The state vector of the second layer state-space model is defined as

x(k)=[L(k),L(k),L(k)]T=[x1(k),x2(k),x3(k)]T,
where L(k) is the instantaneous distance at time step k, and L(k) and L(k) denote the first and second derivatives of time-varying L(k). Based on the second-order Taylor expansion and Eq. (6), the following equations hold:
L(k+1)=L(k)+L(k)ΔT+L(k)ΔT2/2L(k+1)=L(k)+L(k)ΔT,L(k+1)=L(k)fInterf(k)=2nc[υ(k)L(k)+υ(k)L(k)].
Here, the observation vector y(k)=[fInterf(k)] is the estimated frequency of the interference signals in the first layer. Accordingly, the state-space dynamic model of the second layer is given as follows:
x(k)=Fx(k1)+v(k1),y(k)=Hx(k)+u(k),
where v(k) and u(k) denote the process and measurement noises, respectively. The state transition matrix and measurement matrix is

F=[1ΔTΔT2/201ΔT001]H=[2ncυ(k)2ncυ(k)0][2ncυ(k)2ncυ00],

Note that the optical frequency is tuned from υ0 to υ0+Δυ during the sweep. In our case, the tuning range Δυ(≈150 GHz) is extremely small compared with the nominal optical frequency (≈384 THz). Thus, the instantaneous optical frequency υ(k) can be replaced with a fixed nominal value, which means that the instantaneous value need not be measured. The only parameter required in real-time is the instantaneous optical frequency scanning rate υ(k). Here, we propose the use of the auxiliary interferometer module shown in Fig. 1(a) to simultaneously calculate the time-varying optical frequency scanning rate υ(k).

Conventionally, the optical frequency can be roughly described by processing the F-P interference signal, as shown in Fig. 2(a). Because the free spectral range (FSR) of the F-P cavity corresponds to the time interval ΔTFP(j) between two adjacent peaks of the F-P signal in the time domain, the optical frequency scanning υ(k) can be expressed indirectly from the F-P signal as

 figure: Fig. 2

Fig. 2 (a) Principle of optical frequency fitting method with F-P signal. The green dashed line is the true optical frequency curve. (b) Relationship between instantaneous frequency of the interferometric signal of auxiliary interferometer and optical frequency scanning rate.

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υ(j)=υ0+jFSR.

Then, υ(k) can be achieved by deriving the curve of υ(k) fitted with optical frequency sampling points υ(j). However, we could acquire only up to 100 sampling points to υ(k) because ΔTFP(j) is much larger than the sampling interval ΔT of DAQ card and the FSR in our case is 1.5 GHz, which is much lower than the tuning range of the ECDL (≈150 GHz). In other words, some important features of the υ(k) might be missed if only the sampling points generated by F-P signal are used to fit the υ(k) curve. In addition, the υ(k) calculation method requires the complete history of F-P signal data to fit the υ(k) curve, which means that this method does not satisfy the real-time requirement.

Here, we exploit the auxiliary interferometer module to estimate the instantaneous υ(k) in real time. According to Eq. (6), the frequency of interferometric signals is only linearly dependent on the optical tuning rate when the time delay is fixed.

faux(k)=ncLauxυ(k),
where Laux is optical path difference of the auxiliary interferometer.

Hence, the same method mentioned in section 2.1 is implemented to determine the frequency of the interferometric signal of the auxiliary interferometer. Then the scanning rate can be synchronously obtained as

υ(k)=cnLauxfaux(k),
where the fixed Laux can be pre-calibrated with the F-P cavity and M-Z interferometer using the conventional FSI detection method [15] for a static target .
Laux=cΔφFP2πnΔυFP,
where ΔφFP is phase difference corresponding to the optical frequency scanning rangeΔυFP=Δυ(j), as shown in Fig. 2 (b). By combining Eq. (21) with Eq. (22), υ(k) is obtained without loss of detailed features as υ(k)=2πΔυFPfaux(k)/ΔφFP.

Owing to the nonlinear optical frequency sweeping mentioned above, the optical frequency scanning rate of the laser source υ(k) is not constant but varies with time. Furthermore, in interferometry using tunable laser diodes, the phase and optical-frequency noise of laser source comprises Gaussian white noise part owing to the spontaneous emission and flicker noise part [28]. Besides, the external effects such as current, vibrational, and thermal variations of the laser source also contribute to the non-Gaussian noise part. Moreover, the non-Gaussian noise component becomes more dominant with increasing distance. Considering Eq. (17), this means that the measurement noise is no longer Gaussian. Hence, the conventional Bayesian state estimation methods for linear Gaussian filtering problems, including basic Kalman filter, become invalid in this case. Therefore, we use the more robust approach of the particle filter method to estimate the state of this time-varying non-Gaussian model.

From a Bayesian point of view, our goal is to finally estimate the posterior distribution p(x0:k|y1:k), where y1:k={y1,y2,yk}. The key point of a Monte-Carlo based particle filter is to represent the posterior density function by a set of random samples with associated weights ω [29–31]. With the Monte-Carlo random sampling, the particle filter assumes no function, but a series of particles is used instead to estimate the posterior distribution. In this manner, the time-varying non-Gaussian system can be handled.

Firstly, the particle filter starts with a set of equally weighted random samples {xk1i;i=1,...,N}, also known as particles, thus obtaining an approximation for p(xk1|y1:k1),

p(xk1|y1:k1)p^(xk1|y1:k1)=i=1Nwk1(xk1i)δxk1i(dxk1),
where δ is the Dirac delta function. Each particle is then propagated through the dynamic model to update both the particles and their corresponding importance weights {x˜k1i;wk1i}, thus yielding an approximation ofp(xk1|y1:k1). Given that the states follow a hidden Markov process, the recursive estimate for the importance weights can be expressed as [31]
wki=wk1ip(yk|xki)p(xki|xk1i)q(xki|x0:k1i,y1:k),w˜ki=wki[j=1Nwkj]1,
where the transition prior p(xki|xk1i) and observation likelihood p(yk|xki) are known, q(xki|x0:k1i,y1:k) is the proposal distribution, and w˜ki are normalized importance weights. To achieve better filtering performance, we employed a UKF to generate the proposal distribution, such that the most recent observed data can be taken into consideration in the proposal distribution [31]. Based on the importance weights calculated through Eq. (24), the weighted measure {x˜ki;w˜ki} is mapped to the unweighted measure {x˜ki;N1} by the random resampling algorithm [30]. Using Eq. (23), the absolute distance of the target can be finally obtained as
x˜k=i=1Nwkix˜kiN,
where x˜k is a state estimate of the measured target at time step k and it can be readily used as the tracking result of the proposed FSI system.

Accordingly, the conceptual illustration of the cascade model based FSI detection architecture is shown in Fig. 3.

 figure: Fig. 3

Fig. 3 Structure of the cascade dynamic model and signal processing scheme. Frequency of Aux.: instantaneous frequency of auxiliary interference signal. Frequency of Meas.: instantaneous frequency of measured interference signal. The first layer model is used not only to estimate the instantaneous frequency of the measured interference signal but also simultaneously estimate the instantaneous frequency of the auxiliary interference signal. The former is taken as the observation value in the second-layer model and the latter is used to compute optical frequency scanning rate υ(k), which is the time-varying parameter of the measurement matrix in the second-layer model. The second layer model is used to estimate the absolute distance of the moving target.

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Based on the two-layer model, the target movement is decoupled from the multi-modulated frequency of interference signal from measured interferometer, and we can recursively estimate the instantaneous frequency of interference signal f(k) and the absolute distance of the moving target L(k) using only the observation data at time step k1 without the complete history of the observation data. In this manner, rapid and real-time measurements for dynamic absolute distance of the target mirror can be achieved using the FSI system. Because ΔT between two adjacent time steps in the cascade model is the sampling interval of the DAQ system, the measurement rate is increased to the sampling frequency of DAQ

4. Experiment

We verified the viability of the proposed cascade dynamic model based FSI through conducting proof-of-concept experiments. The experimental setup of the proposed FSI system is illustrated in Fig. 4. The tunable laser source was an ECDL (Newport TLB6812) with a center wavelength of 780 nm and an MHF tuning range of 150 GHz. The spectral bandwidth was less than 200 kHz, corresponding to a coherence length much larger than the optical path length in the setup. The auxiliary interferometer module consisting of a high-finesse F-P device (Thorlabs SA210-5B with an FSR of 1.5 GHz) and a fiber M-Z interferometer was surrounded by a shielding chamber on a floating optical table for thermal stabilization and vibration isolation. It should be noted that a long fiber is not necessary for the M-Z interferometer because our method does not require long optical path difference. The output power was 35 mW, which is sufficiently high to allow the detection of both the interference and transmitted F-P signals. Five photodetectors (Newport Model 1801) were used to detect the auxiliary and measured IQ interference signals and transmitted F-P signals. An eight-channel DAQ card (NI-PXle-5105) was used to synchronously acquire the F-P and interference signals at a maximum rate of 60 MS/s, and an environmental sensor (Agilent E1738A) was used to measure the refractive index. A PZT actuator (Thorlabs, NF15AP25 with a resolution of 25 nm) was used to generate vibration that was applied to the target mirror in measured interferometer. A capacitive displacement sensor (Micro-Epsilon, capaNCDT DL6220-CS02, with a dynamic resolution of 4 nm) was used as an external witness to monitor the vibration of the target mirror.

 figure: Fig. 4

Fig. 4 Experimental setup of the measurement system. COL: collimator, PA: PZT actuator, ES: environmental sensor, POL: polarizer, CDS: captive displacement sensor.

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First, the vibration frequency and amplitude of the PZT actuator were set to 500 Hz and approximately 10 µm. Based on the designed quadrature detection scheme mentioned above, a set of IQ signals can be obtained. For display convenience, we selected only a segment of the raw data of IQ signals from the measured interferometer, as shown in Fig. 5(a). Obvious frequency modulation of the IQ interference signals caused by the movement of target can be observed, which is consistent with the qualitative analysis in Fig. 1(b). To verify the phase shift π/4 between IQ signals, a Lissajous circle was obtained, as shown in Fig. 5(c), by plotting the PD outputs against each other in Fig. 5(b). One revolution of the rotating vector path corresponds to a phase change of 2π. A circle was fitted with IQ data and the circle center and radium of the fitted circle with IQ data is shown in Fig. 5(c). With detected IQ signals from the auxiliary and measured interferometers, the proposed signal processing scheme can be implemented to obtain the absolute distance values.

 figure: Fig. 5

Fig. 5 (a) Raw signals: In-phase and quadrature signals from measured interferometer (b) Enlarged view of normalized part of raw data corresponding to 4π phase change. (c) Lissajous figure of the corresponding IQ data.

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Figure 6 shows the absolute distance values measured with the cascade dynamic model based FSI and the displacement values measured with the captive displacement sensor in 500 Hz vibration at a standoff distance of approximately 3 m. To evaluate the results of measurement, the amplitude spectra of measurement by the FSI and displacement sensor are shown in Fig. 6 (c), and the spectra were calculated using the chirp-z transformation (CZT) method. The primary frequency components of the measurements of the FSI and sensor were 499.926 and 500.006, corresponding to 9.727 µm and 9.355 µm, respectively. The relative error (RE) between the vibration frequencies and amplitudes measured using the proposed FSI and displacement sensor were 0.016% and 3.82%.

 figure: Fig. 6

Fig. 6 (a) FSI measurements at a frequency of 500 Hz. (b) Displacement sensor measurements at a frequency of 500 Hz. (c) Amplitude spectra of the measured data

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To further test the frequency resolution of the FSI based on the cascade dynamic model, we also measured the displacement with a 501 Hz vibration with the same amplitude of PZT actuator driving signal setting. The obtained results are shown in Fig. 7(a)-(c), which show good agreement between the results of the cascade dynamic model based FSI and the displacement sensor. The Fourier analysis of the measured data for the proposed FSI at frequencies of 500 Hz and 501 Hz are shown in Fig. 7 (d). Using the proposed method, 1 Hz difference between the two different vibration frequencies could be clearly distinguished, with high precision in both amplitude and frequency.

 figure: Fig. 7

Fig. 7 (a) FSI measurements at a frequency of 500 Hz. (b) Displacement sensor measurements at a frequency of 500 Hz. (c) Amplitude spectra of measured data (d) Fourier analysis of the sequence of measured results at 500 Hz and 501 Hz vibrations.

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Additionally, we calculated the RE between the vibration amplitudes and frequencies measured using the proposed system and the displacement sensor, and the measurement was carryout 10 times at frequencies of 50, 100, 250, 300, and 500 Hz. Table 1 shows the experimental results including the measurement RE and STD (standard deviation) of frequency and amplitude at different frequencies. As shown in Table 1, the RE of frequency was less than 0.04% in all the measurements, which indicates the high temporal accuracy of the cascade dynamic model based FSI. For amplitude, the RE was less than 7.2%; we believe that part of the amplitude discrepancy was caused by both Abbe and cosine errors, which could be further suppressed by improving the alignment between the FSI and the displacement sensor.

Tables Icon

Table 1. Characteristics of vibration measurements at different frequencies.

5. Discussion

The experiments above were performed at a standoff distance of about 3 m. For some applications requiring dynamic measurements, the measurement system needs to have longer measurement standoff distance. We hence also attempted to track high frequency vibrations using our proposed method at longer standoff distances. For a frequency of 500 Hz vibration at about 15 m standoff distance, the experimental results are shown in Fig. 8. The blue line in Fig. 8(a) represents the fitted line based on the Fourier fitting method. As shown in Fig. 8(b), the residual error in frequency remained small (0.115 Hz). However, in terms of amplitude, the proposed system could no longer track the target movement effectively at such a long standoff distance. We believe that this is caused by the significant decrease of signal to noise ratio (SNR) of the interference signal as the standoff distance increased to about 15 m, which influenced the performance of the first-layer model based algorithm. Moreover, at the long standoff distance, more significant absolute optical distance drift caused by the thermal drift and air turbulence also affected the effectiveness and robustness of the second-layer model. Therefore, the algorithm of the signal processing based on the cascade model still has room to improve to enhance the robustness of the measurement in the case of longer measurement standoff distances.

 figure: Fig. 8

Fig. 8 (a) FSI measurements at a frequency of 500 Hz for a standoff distance of about 15 m. (b) Amplitude spectra of the data measured by FSI and sensor

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6. Summary and conclusion

In summary, we have successfully demonstrated a simple FSI system capable of real-time and rapid dynamic absolute ranging. A novel dynamic model with cascade architecture is proposed to rapidly decouple the dynamic absolute distance from the instantaneous frequency of interference signal and the influence of nonlinear optical frequency scanning in dynamic measurement is thus eliminated. Based on the cascade model, the UKF and particle filter algorithms play an important role in handling the nonlinear first-layer model and non-Gaussian second-layer model, respectively, and ensuring real-time measurement by the proposed FSI system. The effectiveness of the proposed cascade model based FSI was confirmed through experiments. The proposed system paves the way for technically straightforward, rapid and real-time dynamic absolute ranging systems. There is still room for improvement of the proposed FSI system. Our future work will focus on improving the accuracy of the measurement system at longer standoff distance by increasing the SNR of the interference signals, and realizing the conceptual FSI proposed in this paper through real time processing in field-programmable gate array applications to further improve the measurement bandwidth.

Funding

National Natural Science Foundation of China (NSFC) (51875447, 51635010).

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1. R. Komissarov, V. Kozlov, D. Filonov, and P. Ginzburg, “Partially coherent radar unties range resolution from bandwidth limitations,” Nat. Commun. 10(1), 1423 (2019). [CrossRef]   [PubMed]  

2. T. Bonin, G. Franke, M. Hagen-Eggert, P. Koch, and G. Hüttmann, “In vivo Fourier-domain full-field OCT of the human retina with 1.5 million A-lines/s,” Opt. Lett. 35(20), 3432–3434 (2010). [CrossRef]   [PubMed]  

3. E. D. Moore and R. R. McLeod, “Correction of sampling errors due to laser tuning rate fluctuations in swept-wavelength interferometry,” Opt. Express 16(17), 13139–13149 (2008). [CrossRef]   [PubMed]  

4. W. Zhang, H. Wei, H. Yang, X. Wu, and Y. Li, “Comb-referenced frequency-sweeping interferometry for precisely measuring large stepped structures,” Appl. Opt. 57(5), 1247–1253 (2018). [CrossRef]   [PubMed]  

5. D. J. Lum, S. H. Knarr, and J. C. Howell, “Frequency-modulated continuous-wave LiDAR compressive depth-mapping,” Opt. Express 26(12), 15420–15435 (2018). [CrossRef]   [PubMed]  

6. E. Baumann, F. R. Giorgetta, J.-D. Deschênes, W. C. Swann, I. Coddington, and N. R. Newbury, “Comb-calibrated laser ranging for three-dimensional surface profiling with micrometer-level precision at a distance,” Opt. Express 22(21), 24914–24928 (2014). [CrossRef]   [PubMed]  

7. A. Cabral, M. Abreu, and J. M. Rebordão, “Dual-frequency sweeping interferometry for absolute metrology of long distances,” Opt. Eng. 49, 085601 (2010).

8. K. Meiners-Hagen, T. Meyer, J. Mildner, and F. Pollinger, “SI-traceable absolute distance measurement over more than 800 meters with sub-nanometer interferometry by two-color inline refractivity compensation,” Appl. Phys. Lett. 111(19), 191104 (2017). [CrossRef]  

9. R. Yang, F. Pollinger, K. Meiners-Hagen, J. Tan, and H. Bosse, “Heterodyne multi-wavelength absolute interferometry based on a cavity-enhanced electro-optic frequency comb pair,” Opt. Lett. 39(20), 5834–5837 (2014). [CrossRef]   [PubMed]  

10. H. Pan, X. Qu, and F. Zhang, “Micron-precision measurement using a combined frequency-modulated continuous wave ladar autofocusing system at 60 meters standoff distance,” Opt. Express 26(12), 15186–15198 (2018). [CrossRef]   [PubMed]  

11. G. Shi, F. Zhang, X. Qu, and X. Meng, “High-resolution frequency-modulated continuous-wave laser ranging for precision distance metrology applications,” Opt. Eng. 53(12), 122402 (2014). [CrossRef]  

12. C. Lu, G. Liu, B. Liu, F. Chen, and Y. Gan, “Absolute distance measurement system with micron-grade measurement uncertainty and 24 m range using frequency scanning interferometry with compensation of environmental vibration,” Opt. Express 24(26), 30215–30224 (2016). [CrossRef]   [PubMed]  

13. M. Cui, M. G. Zeitouny, N. Bhattacharya, S. A. van den Berg, H. P. Urbach, and J. J. M. Braat, “High-accuracy long-distance measurements in air with a frequency comb laser,” Opt. Lett. 34(13), 1982–1984 (2009). [CrossRef]   [PubMed]  

14. X. Jia, Z. Liu, Z. Deng, W. Deng, Z. Wang, and Z. Zhen, “Dynamic absolute distance measurement by frequency sweeping interferometry based Doppler beat frequency tracking model,” Opt. Commun. 430, 163–169 (2019). [CrossRef]  

15. H.-J. Yang, J. Deibel, S. Nyberg, and K. Riles, “High-precision absolute distance and vibration measurement with frequency scanned interferometry,” Appl. Opt. 44(19), 3937–3944 (2005). [CrossRef]   [PubMed]  

16. C. Lu, Y. Xiang, Y. Gan, B. Liu, F. Chen, X. Liu, and G. Liu, “FSI-based non-cooperative target absolute distance measurement method using PLL correction for the influence of a nonlinear clock,” Opt. Lett. 43(9), 2098–2101 (2018). [CrossRef]   [PubMed]  

17. R. Schneider, P. Thuermel, and M. Stockmann, “Distance measurement of moving objects by frequency modulated laser radar,” Opt. Eng. 40, 5 (2001).

18. J. Dale, B. Hughes, A. J. Lancaster, A. J. Lewis, A. J. 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). [CrossRef]   [PubMed]  

19. J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. J. Copner, and A. J. Lewis, “Dual-Sweep Frequency Scanning Interferometry Using Four Wave Mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015). [CrossRef]  

20. B. L. Swinkels, N. Bhattacharya, and J. J. Braat, “Correcting movement errors in frequency-sweeping interferometry,” Opt. Lett. 30(17), 2242–2244 (2005). [CrossRef]   [PubMed]  

21. X. Jia, Z. Liu, L. Tao, and Z. Deng, “Frequency-scanning interferometry using a time-varying Kalman filter for dynamic tracking measurements,” Opt. Express 25(21), 25782–25796 (2017). [CrossRef]   [PubMed]  

22. 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). [CrossRef]   [PubMed]  

23. Z. Deng, Z. Liu, B. Li, and Z. Liu, “Precision improvement in frequency-scanning interferometry based on suppressing nonlinear optical frequency sweeping,” Opt. Rev. 22(5), 724–730 (2015). [CrossRef]  

24. S. Kakuma and Y. Katase, “Resolution improvement in vertical-cavity-surface-emitting-laser diode interferometry based on linear least-squares estimation of phase gradients of phase-locked fringes,” Opt. Rev. 17(5), 481–485 (2010). [CrossRef]  

25. S. J. Julier and J. K. Uhlmann, “New extension of the Kalman filter to nonlinear systems,” in Signal Processing, Ssensor Fusion, and Target Recognition VI (International Society for Optics and Photonics, 1997), 182–194.

26. P. Regulski and V. Terzija, “Estimation of Frequency and Fundamental Power Components Using an Unscented Kalman Filter,” Trans. Instrum. Meas. 61(4), 952–962 (2012). [CrossRef]  

27. E. A. Wan and R. Van Der Merwe, “The unscented Kalman filter for nonlinear estimation,” in Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (Cat. No. 00EX373), (IEEE, 2000), 153–158. [CrossRef]  

28. Y. Salvadé and R. Dändliker, “Limitations of interferometry due to the flicker noise of laser diodes,” J. Opt. Soc. Am. A 17(5), 927–932 (2000). [CrossRef]   [PubMed]  

29. N. J. Gordon, D. J. Salmond, and A. F. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” in IEEE Proceedings F-radar and Signal Processing, (IET, 1993), 107–113. [CrossRef]  

30. A. Doucet, S. Godsill, and C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Stat. Comput. 10(3), 197–208 (2000). [CrossRef]  

31. R. Yong and C. Yunqiang, “Better proposal distributions: object tracking using unscented particle filter,” in Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001 (IEEE, 2001), 786–793. [CrossRef]  

References

  • View by:

  1. R. Komissarov, V. Kozlov, D. Filonov, and P. Ginzburg, “Partially coherent radar unties range resolution from bandwidth limitations,” Nat. Commun. 10(1), 1423 (2019).
    [Crossref] [PubMed]
  2. T. Bonin, G. Franke, M. Hagen-Eggert, P. Koch, and G. Hüttmann, “In vivo Fourier-domain full-field OCT of the human retina with 1.5 million A-lines/s,” Opt. Lett. 35(20), 3432–3434 (2010).
    [Crossref] [PubMed]
  3. E. D. Moore and R. R. McLeod, “Correction of sampling errors due to laser tuning rate fluctuations in swept-wavelength interferometry,” Opt. Express 16(17), 13139–13149 (2008).
    [Crossref] [PubMed]
  4. W. Zhang, H. Wei, H. Yang, X. Wu, and Y. Li, “Comb-referenced frequency-sweeping interferometry for precisely measuring large stepped structures,” Appl. Opt. 57(5), 1247–1253 (2018).
    [Crossref] [PubMed]
  5. D. J. Lum, S. H. Knarr, and J. C. Howell, “Frequency-modulated continuous-wave LiDAR compressive depth-mapping,” Opt. Express 26(12), 15420–15435 (2018).
    [Crossref] [PubMed]
  6. E. Baumann, F. R. Giorgetta, J.-D. Deschênes, W. C. Swann, I. Coddington, and N. R. Newbury, “Comb-calibrated laser ranging for three-dimensional surface profiling with micrometer-level precision at a distance,” Opt. Express 22(21), 24914–24928 (2014).
    [Crossref] [PubMed]
  7. A. Cabral, M. Abreu, and J. M. Rebordão, “Dual-frequency sweeping interferometry for absolute metrology of long distances,” Opt. Eng. 49, 085601 (2010).
  8. K. Meiners-Hagen, T. Meyer, J. Mildner, and F. Pollinger, “SI-traceable absolute distance measurement over more than 800 meters with sub-nanometer interferometry by two-color inline refractivity compensation,” Appl. Phys. Lett. 111(19), 191104 (2017).
    [Crossref]
  9. R. Yang, F. Pollinger, K. Meiners-Hagen, J. Tan, and H. Bosse, “Heterodyne multi-wavelength absolute interferometry based on a cavity-enhanced electro-optic frequency comb pair,” Opt. Lett. 39(20), 5834–5837 (2014).
    [Crossref] [PubMed]
  10. H. Pan, X. Qu, and F. Zhang, “Micron-precision measurement using a combined frequency-modulated continuous wave ladar autofocusing system at 60 meters standoff distance,” Opt. Express 26(12), 15186–15198 (2018).
    [Crossref] [PubMed]
  11. G. Shi, F. Zhang, X. Qu, and X. Meng, “High-resolution frequency-modulated continuous-wave laser ranging for precision distance metrology applications,” Opt. Eng. 53(12), 122402 (2014).
    [Crossref]
  12. C. Lu, G. Liu, B. Liu, F. Chen, and Y. Gan, “Absolute distance measurement system with micron-grade measurement uncertainty and 24 m range using frequency scanning interferometry with compensation of environmental vibration,” Opt. Express 24(26), 30215–30224 (2016).
    [Crossref] [PubMed]
  13. M. Cui, M. G. Zeitouny, N. Bhattacharya, S. A. van den Berg, H. P. Urbach, and J. J. M. Braat, “High-accuracy long-distance measurements in air with a frequency comb laser,” Opt. Lett. 34(13), 1982–1984 (2009).
    [Crossref] [PubMed]
  14. X. Jia, Z. Liu, Z. Deng, W. Deng, Z. Wang, and Z. Zhen, “Dynamic absolute distance measurement by frequency sweeping interferometry based Doppler beat frequency tracking model,” Opt. Commun. 430, 163–169 (2019).
    [Crossref]
  15. H.-J. Yang, J. Deibel, S. Nyberg, and K. Riles, “High-precision absolute distance and vibration measurement with frequency scanned interferometry,” Appl. Opt. 44(19), 3937–3944 (2005).
    [Crossref] [PubMed]
  16. C. Lu, Y. Xiang, Y. Gan, B. Liu, F. Chen, X. Liu, and G. Liu, “FSI-based non-cooperative target absolute distance measurement method using PLL correction for the influence of a nonlinear clock,” Opt. Lett. 43(9), 2098–2101 (2018).
    [Crossref] [PubMed]
  17. R. Schneider, P. Thuermel, and M. Stockmann, “Distance measurement of moving objects by frequency modulated laser radar,” Opt. Eng. 40, 5 (2001).
  18. J. Dale, B. Hughes, A. J. Lancaster, A. J. Lewis, A. J. 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).
    [Crossref] [PubMed]
  19. J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. J. Copner, and A. J. Lewis, “Dual-Sweep Frequency Scanning Interferometry Using Four Wave Mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).
    [Crossref]
  20. B. L. Swinkels, N. Bhattacharya, and J. J. Braat, “Correcting movement errors in frequency-sweeping interferometry,” Opt. Lett. 30(17), 2242–2244 (2005).
    [Crossref] [PubMed]
  21. X. Jia, Z. Liu, L. Tao, and Z. Deng, “Frequency-scanning interferometry using a time-varying Kalman filter for dynamic tracking measurements,” Opt. Express 25(21), 25782–25796 (2017).
    [Crossref] [PubMed]
  22. 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).
    [Crossref] [PubMed]
  23. Z. Deng, Z. Liu, B. Li, and Z. Liu, “Precision improvement in frequency-scanning interferometry based on suppressing nonlinear optical frequency sweeping,” Opt. Rev. 22(5), 724–730 (2015).
    [Crossref]
  24. S. Kakuma and Y. Katase, “Resolution improvement in vertical-cavity-surface-emitting-laser diode interferometry based on linear least-squares estimation of phase gradients of phase-locked fringes,” Opt. Rev. 17(5), 481–485 (2010).
    [Crossref]
  25. S. J. Julier and J. K. Uhlmann, “New extension of the Kalman filter to nonlinear systems,” in Signal Processing, Ssensor Fusion, and Target Recognition VI (International Society for Optics and Photonics, 1997), 182–194.
  26. P. Regulski and V. Terzija, “Estimation of Frequency and Fundamental Power Components Using an Unscented Kalman Filter,” Trans. Instrum. Meas. 61(4), 952–962 (2012).
    [Crossref]
  27. E. A. Wan and R. Van Der Merwe, “The unscented Kalman filter for nonlinear estimation,” in Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (Cat. No. 00EX373), (IEEE, 2000), 153–158.
    [Crossref]
  28. Y. Salvadé and R. Dändliker, “Limitations of interferometry due to the flicker noise of laser diodes,” J. Opt. Soc. Am. A 17(5), 927–932 (2000).
    [Crossref] [PubMed]
  29. N. J. Gordon, D. J. Salmond, and A. F. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” in IEEE Proceedings F-radar and Signal Processing, (IET, 1993), 107–113.
    [Crossref]
  30. A. Doucet, S. Godsill, and C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Stat. Comput. 10(3), 197–208 (2000).
    [Crossref]
  31. R. Yong and C. Yunqiang, “Better proposal distributions: object tracking using unscented particle filter,” in Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001 (IEEE, 2001), 786–793.
    [Crossref]

2019 (2)

R. Komissarov, V. Kozlov, D. Filonov, and P. Ginzburg, “Partially coherent radar unties range resolution from bandwidth limitations,” Nat. Commun. 10(1), 1423 (2019).
[Crossref] [PubMed]

X. Jia, Z. Liu, Z. Deng, W. Deng, Z. Wang, and Z. Zhen, “Dynamic absolute distance measurement by frequency sweeping interferometry based Doppler beat frequency tracking model,” Opt. Commun. 430, 163–169 (2019).
[Crossref]

2018 (4)

2017 (2)

K. Meiners-Hagen, T. Meyer, J. Mildner, and F. Pollinger, “SI-traceable absolute distance measurement over more than 800 meters with sub-nanometer interferometry by two-color inline refractivity compensation,” Appl. Phys. Lett. 111(19), 191104 (2017).
[Crossref]

X. Jia, Z. Liu, L. Tao, and Z. Deng, “Frequency-scanning interferometry using a time-varying Kalman filter for dynamic tracking measurements,” Opt. Express 25(21), 25782–25796 (2017).
[Crossref] [PubMed]

2016 (1)

2015 (2)

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. J. Copner, and A. J. Lewis, “Dual-Sweep Frequency Scanning Interferometry Using Four Wave Mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).
[Crossref]

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

2014 (5)

2012 (1)

P. Regulski and V. Terzija, “Estimation of Frequency and Fundamental Power Components Using an Unscented Kalman Filter,” Trans. Instrum. Meas. 61(4), 952–962 (2012).
[Crossref]

2010 (3)

S. Kakuma and Y. Katase, “Resolution improvement in vertical-cavity-surface-emitting-laser diode interferometry based on linear least-squares estimation of phase gradients of phase-locked fringes,” Opt. Rev. 17(5), 481–485 (2010).
[Crossref]

A. Cabral, M. Abreu, and J. M. Rebordão, “Dual-frequency sweeping interferometry for absolute metrology of long distances,” Opt. Eng. 49, 085601 (2010).

T. Bonin, G. Franke, M. Hagen-Eggert, P. Koch, and G. Hüttmann, “In vivo Fourier-domain full-field OCT of the human retina with 1.5 million A-lines/s,” Opt. Lett. 35(20), 3432–3434 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (1)

2005 (2)

2001 (1)

R. Schneider, P. Thuermel, and M. Stockmann, “Distance measurement of moving objects by frequency modulated laser radar,” Opt. Eng. 40, 5 (2001).

2000 (2)

Y. Salvadé and R. Dändliker, “Limitations of interferometry due to the flicker noise of laser diodes,” J. Opt. Soc. Am. A 17(5), 927–932 (2000).
[Crossref] [PubMed]

A. Doucet, S. Godsill, and C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Stat. Comput. 10(3), 197–208 (2000).
[Crossref]

Abreu, M.

A. Cabral, M. Abreu, and J. M. Rebordão, “Dual-frequency sweeping interferometry for absolute metrology of long distances,” Opt. Eng. 49, 085601 (2010).

Andrieu, C.

A. Doucet, S. Godsill, and C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Stat. Comput. 10(3), 197–208 (2000).
[Crossref]

Baumann, E.

Bhattacharya, N.

Bonin, T.

Bosse, H.

Braat, J. J.

Braat, J. J. M.

Cabral, A.

A. Cabral, M. Abreu, and J. M. Rebordão, “Dual-frequency sweeping interferometry for absolute metrology of long distances,” Opt. Eng. 49, 085601 (2010).

Campbell, M. A.

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. J. Copner, and A. J. Lewis, “Dual-Sweep Frequency Scanning Interferometry Using Four Wave Mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).
[Crossref]

Chen, F.

Coddington, I.

Copner, N. J.

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. J. Copner, and A. J. Lewis, “Dual-Sweep Frequency Scanning Interferometry Using Four Wave Mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).
[Crossref]

Cui, M.

Dale, J.

Dändliker, R.

Deibel, J.

Deng, W.

X. Jia, Z. Liu, Z. Deng, W. Deng, Z. Wang, and Z. Zhen, “Dynamic absolute distance measurement by frequency sweeping interferometry based Doppler beat frequency tracking model,” Opt. Commun. 430, 163–169 (2019).
[Crossref]

Deng, Z.

X. Jia, Z. Liu, Z. Deng, W. Deng, Z. Wang, and Z. Zhen, “Dynamic absolute distance measurement by frequency sweeping interferometry based Doppler beat frequency tracking model,” Opt. Commun. 430, 163–169 (2019).
[Crossref]

X. Jia, Z. Liu, L. Tao, and Z. Deng, “Frequency-scanning interferometry using a time-varying Kalman filter for dynamic tracking measurements,” Opt. Express 25(21), 25782–25796 (2017).
[Crossref] [PubMed]

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

Deschênes, J.-D.

Doucet, A.

A. Doucet, S. Godsill, and C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Stat. Comput. 10(3), 197–208 (2000).
[Crossref]

Filonov, D.

R. Komissarov, V. Kozlov, D. Filonov, and P. Ginzburg, “Partially coherent radar unties range resolution from bandwidth limitations,” Nat. Commun. 10(1), 1423 (2019).
[Crossref] [PubMed]

Franke, G.

Gan, Y.

Ginzburg, P.

R. Komissarov, V. Kozlov, D. Filonov, and P. Ginzburg, “Partially coherent radar unties range resolution from bandwidth limitations,” Nat. Commun. 10(1), 1423 (2019).
[Crossref] [PubMed]

Giorgetta, F. R.

Godsill, S.

A. Doucet, S. Godsill, and C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Stat. Comput. 10(3), 197–208 (2000).
[Crossref]

Gordon, N. J.

N. J. Gordon, D. J. Salmond, and A. F. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” in IEEE Proceedings F-radar and Signal Processing, (IET, 1993), 107–113.
[Crossref]

Hagen-Eggert, M.

Howell, J. C.

Hughes, B.

Hughes, E. B.

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. J. Copner, and A. J. Lewis, “Dual-Sweep Frequency Scanning Interferometry Using Four Wave Mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).
[Crossref]

Hüttmann, G.

Jia, X.

X. Jia, Z. Liu, Z. Deng, W. Deng, Z. Wang, and Z. Zhen, “Dynamic absolute distance measurement by frequency sweeping interferometry based Doppler beat frequency tracking model,” Opt. Commun. 430, 163–169 (2019).
[Crossref]

X. Jia, Z. Liu, L. Tao, and Z. Deng, “Frequency-scanning interferometry using a time-varying Kalman filter for dynamic tracking measurements,” Opt. Express 25(21), 25782–25796 (2017).
[Crossref] [PubMed]

Kakuma, S.

S. Kakuma and Y. Katase, “Resolution improvement in vertical-cavity-surface-emitting-laser diode interferometry based on linear least-squares estimation of phase gradients of phase-locked fringes,” Opt. Rev. 17(5), 481–485 (2010).
[Crossref]

Katase, Y.

S. Kakuma and Y. Katase, “Resolution improvement in vertical-cavity-surface-emitting-laser diode interferometry based on linear least-squares estimation of phase gradients of phase-locked fringes,” Opt. Rev. 17(5), 481–485 (2010).
[Crossref]

Knarr, S. H.

Koch, P.

Komissarov, R.

R. Komissarov, V. Kozlov, D. Filonov, and P. Ginzburg, “Partially coherent radar unties range resolution from bandwidth limitations,” Nat. Commun. 10(1), 1423 (2019).
[Crossref] [PubMed]

Kozlov, V.

R. Komissarov, V. Kozlov, D. Filonov, and P. Ginzburg, “Partially coherent radar unties range resolution from bandwidth limitations,” Nat. Commun. 10(1), 1423 (2019).
[Crossref] [PubMed]

Lancaster, A. J.

Lewis, A. J.

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. J. Copner, and A. J. Lewis, “Dual-Sweep Frequency Scanning Interferometry Using Four Wave Mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).
[Crossref]

J. Dale, B. Hughes, A. J. Lancaster, A. J. Lewis, A. J. 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).
[Crossref] [PubMed]

Li, B.

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

Li, Y.

Liu, B.

Liu, G.

Liu, X.

Liu, Z.

X. Jia, Z. Liu, Z. Deng, W. Deng, Z. Wang, and Z. Zhen, “Dynamic absolute distance measurement by frequency sweeping interferometry based Doppler beat frequency tracking model,” Opt. Commun. 430, 163–169 (2019).
[Crossref]

X. Jia, Z. Liu, L. Tao, and Z. Deng, “Frequency-scanning interferometry using a time-varying Kalman filter for dynamic tracking measurements,” Opt. Express 25(21), 25782–25796 (2017).
[Crossref] [PubMed]

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

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

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).
[Crossref] [PubMed]

Lu, C.

Lum, D. J.

Martinez, J. J.

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. J. Copner, and A. J. Lewis, “Dual-Sweep Frequency Scanning Interferometry Using Four Wave Mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).
[Crossref]

McLeod, R. R.

Meiners-Hagen, K.

K. Meiners-Hagen, T. Meyer, J. Mildner, and F. Pollinger, “SI-traceable absolute distance measurement over more than 800 meters with sub-nanometer interferometry by two-color inline refractivity compensation,” Appl. Phys. Lett. 111(19), 191104 (2017).
[Crossref]

R. Yang, F. Pollinger, K. Meiners-Hagen, J. Tan, and H. Bosse, “Heterodyne multi-wavelength absolute interferometry based on a cavity-enhanced electro-optic frequency comb pair,” Opt. Lett. 39(20), 5834–5837 (2014).
[Crossref] [PubMed]

Meng, X.

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

Meyer, T.

K. Meiners-Hagen, T. Meyer, J. Mildner, and F. Pollinger, “SI-traceable absolute distance measurement over more than 800 meters with sub-nanometer interferometry by two-color inline refractivity compensation,” Appl. Phys. Lett. 111(19), 191104 (2017).
[Crossref]

Mildner, J.

K. Meiners-Hagen, T. Meyer, J. Mildner, and F. Pollinger, “SI-traceable absolute distance measurement over more than 800 meters with sub-nanometer interferometry by two-color inline refractivity compensation,” Appl. Phys. Lett. 111(19), 191104 (2017).
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Moore, E. D.

Newbury, N. R.

Nyberg, S.

Pan, H.

Pollinger, F.

K. Meiners-Hagen, T. Meyer, J. Mildner, and F. Pollinger, “SI-traceable absolute distance measurement over more than 800 meters with sub-nanometer interferometry by two-color inline refractivity compensation,” Appl. Phys. Lett. 111(19), 191104 (2017).
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R. Yang, F. Pollinger, K. Meiners-Hagen, J. Tan, and H. Bosse, “Heterodyne multi-wavelength absolute interferometry based on a cavity-enhanced electro-optic frequency comb pair,” Opt. Lett. 39(20), 5834–5837 (2014).
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Qu, X.

H. Pan, X. Qu, and F. Zhang, “Micron-precision measurement using a combined frequency-modulated continuous wave ladar autofocusing system at 60 meters standoff distance,” Opt. Express 26(12), 15186–15198 (2018).
[Crossref] [PubMed]

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

Rebordão, J. M.

A. Cabral, M. Abreu, and J. M. Rebordão, “Dual-frequency sweeping interferometry for absolute metrology of long distances,” Opt. Eng. 49, 085601 (2010).

Regulski, P.

P. Regulski and V. Terzija, “Estimation of Frequency and Fundamental Power Components Using an Unscented Kalman Filter,” Trans. Instrum. Meas. 61(4), 952–962 (2012).
[Crossref]

Reichold, A. J.

Riles, K.

Salmond, D. J.

N. J. Gordon, D. J. Salmond, and A. F. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” in IEEE Proceedings F-radar and Signal Processing, (IET, 1993), 107–113.
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Salvadé, Y.

Schneider, R.

R. Schneider, P. Thuermel, and M. Stockmann, “Distance measurement of moving objects by frequency modulated laser radar,” Opt. Eng. 40, 5 (2001).

Shi, G.

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

Smith, A. F.

N. J. Gordon, D. J. Salmond, and A. F. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” in IEEE Proceedings F-radar and Signal Processing, (IET, 1993), 107–113.
[Crossref]

Stockmann, M.

R. Schneider, P. Thuermel, and M. Stockmann, “Distance measurement of moving objects by frequency modulated laser radar,” Opt. Eng. 40, 5 (2001).

Swann, W. C.

Swinkels, B. L.

Tan, J.

Tao, L.

Terzija, V.

P. Regulski and V. Terzija, “Estimation of Frequency and Fundamental Power Components Using an Unscented Kalman Filter,” Trans. Instrum. Meas. 61(4), 952–962 (2012).
[Crossref]

Thuermel, P.

R. Schneider, P. Thuermel, and M. Stockmann, “Distance measurement of moving objects by frequency modulated laser radar,” Opt. Eng. 40, 5 (2001).

Urbach, H. P.

van den Berg, S. A.

Van Der Merwe, R.

E. A. Wan and R. Van Der Merwe, “The unscented Kalman filter for nonlinear estimation,” in Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (Cat. No. 00EX373), (IEEE, 2000), 153–158.
[Crossref]

Wan, E. A.

E. A. Wan and R. Van Der Merwe, “The unscented Kalman filter for nonlinear estimation,” in Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (Cat. No. 00EX373), (IEEE, 2000), 153–158.
[Crossref]

Wang, Z.

X. Jia, Z. Liu, Z. Deng, W. Deng, Z. Wang, and Z. Zhen, “Dynamic absolute distance measurement by frequency sweeping interferometry based Doppler beat frequency tracking model,” Opt. Commun. 430, 163–169 (2019).
[Crossref]

Warden, M. S.

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. J. Copner, and A. J. Lewis, “Dual-Sweep Frequency Scanning Interferometry Using Four Wave Mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).
[Crossref]

J. Dale, B. Hughes, A. J. Lancaster, A. J. Lewis, A. J. 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).
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Wei, H.

Wu, X.

Xiang, Y.

Yang, H.

Yang, H.-J.

Yang, R.

Yong, R.

R. Yong and C. Yunqiang, “Better proposal distributions: object tracking using unscented particle filter,” in Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001 (IEEE, 2001), 786–793.
[Crossref]

Yunqiang, C.

R. Yong and C. Yunqiang, “Better proposal distributions: object tracking using unscented particle filter,” in Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001 (IEEE, 2001), 786–793.
[Crossref]

Zeitouny, M. G.

Zhang, F.

H. Pan, X. Qu, and F. Zhang, “Micron-precision measurement using a combined frequency-modulated continuous wave ladar autofocusing system at 60 meters standoff distance,” Opt. Express 26(12), 15186–15198 (2018).
[Crossref] [PubMed]

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

Zhang, W.

Zhen, Z.

X. Jia, Z. Liu, Z. Deng, W. Deng, Z. Wang, and Z. Zhen, “Dynamic absolute distance measurement by frequency sweeping interferometry based Doppler beat frequency tracking model,” Opt. Commun. 430, 163–169 (2019).
[Crossref]

Zhou, Y.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

K. Meiners-Hagen, T. Meyer, J. Mildner, and F. Pollinger, “SI-traceable absolute distance measurement over more than 800 meters with sub-nanometer interferometry by two-color inline refractivity compensation,” Appl. Phys. Lett. 111(19), 191104 (2017).
[Crossref]

IEEE Photonics Technol. Lett. (1)

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. J. Copner, and A. J. Lewis, “Dual-Sweep Frequency Scanning Interferometry Using Four Wave Mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).
[Crossref]

J. Opt. Soc. Am. A (1)

Nat. Commun. (1)

R. Komissarov, V. Kozlov, D. Filonov, and P. Ginzburg, “Partially coherent radar unties range resolution from bandwidth limitations,” Nat. Commun. 10(1), 1423 (2019).
[Crossref] [PubMed]

Opt. Commun. (1)

X. Jia, Z. Liu, Z. Deng, W. Deng, Z. Wang, and Z. Zhen, “Dynamic absolute distance measurement by frequency sweeping interferometry based Doppler beat frequency tracking model,” Opt. Commun. 430, 163–169 (2019).
[Crossref]

Opt. Eng. (3)

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

R. Schneider, P. Thuermel, and M. Stockmann, “Distance measurement of moving objects by frequency modulated laser radar,” Opt. Eng. 40, 5 (2001).

A. Cabral, M. Abreu, and J. M. Rebordão, “Dual-frequency sweeping interferometry for absolute metrology of long distances,” Opt. Eng. 49, 085601 (2010).

Opt. Express (7)

E. D. Moore and R. R. McLeod, “Correction of sampling errors due to laser tuning rate fluctuations in swept-wavelength interferometry,” Opt. Express 16(17), 13139–13149 (2008).
[Crossref] [PubMed]

D. J. Lum, S. H. Knarr, and J. C. Howell, “Frequency-modulated continuous-wave LiDAR compressive depth-mapping,” Opt. Express 26(12), 15420–15435 (2018).
[Crossref] [PubMed]

E. Baumann, F. R. Giorgetta, J.-D. Deschênes, W. C. Swann, I. Coddington, and N. R. Newbury, “Comb-calibrated laser ranging for three-dimensional surface profiling with micrometer-level precision at a distance,” Opt. Express 22(21), 24914–24928 (2014).
[Crossref] [PubMed]

J. Dale, B. Hughes, A. J. Lancaster, A. J. Lewis, A. J. 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).
[Crossref] [PubMed]

C. Lu, G. Liu, B. Liu, F. Chen, and Y. Gan, “Absolute distance measurement system with micron-grade measurement uncertainty and 24 m range using frequency scanning interferometry with compensation of environmental vibration,” Opt. Express 24(26), 30215–30224 (2016).
[Crossref] [PubMed]

H. Pan, X. Qu, and F. Zhang, “Micron-precision measurement using a combined frequency-modulated continuous wave ladar autofocusing system at 60 meters standoff distance,” Opt. Express 26(12), 15186–15198 (2018).
[Crossref] [PubMed]

X. Jia, Z. Liu, L. Tao, and Z. Deng, “Frequency-scanning interferometry using a time-varying Kalman filter for dynamic tracking measurements,” Opt. Express 25(21), 25782–25796 (2017).
[Crossref] [PubMed]

Opt. Lett. (6)

Opt. Rev. (2)

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

S. Kakuma and Y. Katase, “Resolution improvement in vertical-cavity-surface-emitting-laser diode interferometry based on linear least-squares estimation of phase gradients of phase-locked fringes,” Opt. Rev. 17(5), 481–485 (2010).
[Crossref]

Stat. Comput. (1)

A. Doucet, S. Godsill, and C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Stat. Comput. 10(3), 197–208 (2000).
[Crossref]

Trans. Instrum. Meas. (1)

P. Regulski and V. Terzija, “Estimation of Frequency and Fundamental Power Components Using an Unscented Kalman Filter,” Trans. Instrum. Meas. 61(4), 952–962 (2012).
[Crossref]

Other (4)

E. A. Wan and R. Van Der Merwe, “The unscented Kalman filter for nonlinear estimation,” in Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (Cat. No. 00EX373), (IEEE, 2000), 153–158.
[Crossref]

S. J. Julier and J. K. Uhlmann, “New extension of the Kalman filter to nonlinear systems,” in Signal Processing, Ssensor Fusion, and Target Recognition VI (International Society for Optics and Photonics, 1997), 182–194.

R. Yong and C. Yunqiang, “Better proposal distributions: object tracking using unscented particle filter,” in Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001 (IEEE, 2001), 786–793.
[Crossref]

N. J. Gordon, D. J. Salmond, and A. F. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” in IEEE Proceedings F-radar and Signal Processing, (IET, 1993), 107–113.
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 (a) Conceptual optical layout of cascade dynamic model based FSI for rapid measurement. BS: beam splitter, PBS: polarizing beam splitter, RR: retroreflector, PD: photodetector, QWP: quarter wave plate, PZT: piezoelectric translator (b) Relationships between the frequency of interference signals and optical frequency or target movement.
Fig. 2
Fig. 2 (a) Principle of optical frequency fitting method with F-P signal. The green dashed line is the true optical frequency curve. (b) Relationship between instantaneous frequency of the interferometric signal of auxiliary interferometer and optical frequency scanning rate.
Fig. 3
Fig. 3 Structure of the cascade dynamic model and signal processing scheme. Frequency of Aux.: instantaneous frequency of auxiliary interference signal. Frequency of Meas.: instantaneous frequency of measured interference signal. The first layer model is used not only to estimate the instantaneous frequency of the measured interference signal but also simultaneously estimate the instantaneous frequency of the auxiliary interference signal. The former is taken as the observation value in the second-layer model and the latter is used to compute optical frequency scanning rate υ (k), which is the time-varying parameter of the measurement matrix in the second-layer model. The second layer model is used to estimate the absolute distance of the moving target.
Fig. 4
Fig. 4 Experimental setup of the measurement system. COL: collimator, PA: PZT actuator, ES: environmental sensor, POL: polarizer, CDS: captive displacement sensor.
Fig. 5
Fig. 5 (a) Raw signals: In-phase and quadrature signals from measured interferometer (b) Enlarged view of normalized part of raw data corresponding to 4π phase change. (c) Lissajous figure of the corresponding IQ data.
Fig. 6
Fig. 6 (a) FSI measurements at a frequency of 500 Hz. (b) Displacement sensor measurements at a frequency of 500 Hz. (c) Amplitude spectra of the measured data
Fig. 7
Fig. 7 (a) FSI measurements at a frequency of 500 Hz. (b) Displacement sensor measurements at a frequency of 500 Hz. (c) Amplitude spectra of measured data (d) Fourier analysis of the sequence of measured results at 500 Hz and 501 Hz vibrations.
Fig. 8
Fig. 8 (a) FSI measurements at a frequency of 500 Hz for a standoff distance of about 15 m. (b) Amplitude spectra of the data measured by FSI and sensor

Tables (1)

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Table 1 Characteristics of vibration measurements at different frequencies.

Equations (25)

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E(t)= E 0 e iϕ(t) ,
υ(t)= 1 2π dϕ dt .
U(t)=ε | E(t)+E(t+τ) | 2 = U 0 { 1+cos[ ϕ(t+τ)ϕ(t) ] },
ϕ(t+τ)= i=0 τ i i! ϕ (i) (t).
φ=ϕ(t+τ)ϕ(t)=2πυ(t)τ+2π i=2 τ i i! υ (i1) (t)2πυ(t)τ.
f Interf (t)= 1 2π dφ dt = 2n c [ υ (t)L(t)+υ(t) L (t) ].
I(t)=cos[ 2πυ( t )τ( t ) ] Q(t)=sin[ 2πυ( t )τ( t ) ].
x 1 (k)= x 1 (k1)cos[ 2πΔT x 3 (k1) ] x 2 (k1)sin[ 2πΔT x 3 (k1) ] x 2 (k)= x 1 (k1)sin[ 2πΔT x 3 (k1) ] x 2 (k1)cos[ 2πΔT x 3 (k1) ] x 3 (k)= x 3 (k1), y 1 (k)= x 1 (k) y 2 (k)= x 2 (k),
x(k)=f(x(k1),v(k1))=Fx(k1)+v(k1) y(k)=h(x(k),u(k))=Hx(k)+u(k),
F=[ cos[ 2πΔT x 3 (k1) ] sin[ 2πΔT x 3 (k1) ] 0 sin[ 2πΔT x 3 (k1) ] cos[ 2πΔT x 3 (k1) ] 0 0 0 1 ] H=[ 1 0 0 0 1 0 ],
χ(k1)= [ x ^ 0 (k1) x ^ i (k1)+ (γ P(k1) ) i x ^ i+m (k1) (γ P(k1) ) i+m ] T (i=1,....,m),
χ i (k)=F( χ i (k1)) x ^ (k)= i=1 2m+1 W i mean χ i (k) P (k)= i=1 2m+1 W i cov [ χ i (k) x ^ (k)] [ χ i (k) x ^ (k)] T +Q χ(k)=[ x ^ 0 (k) x ^ i (k)+ (γ P (k) ) i x ^ i+m (k) (γ P (k) ) i+m ] Y(k)=H(χ(k)) y (k)= i=1 2m+1 W i mean Y i (k).
P y(k)y(k) = i=1 2m+1 W i cov [ Y i (k) y (k)] [ Y i (k) y (k)] T +R P x(k)y(k) = i=1 2m+1 W i cov [ χ i (k) x ^ (k)] [ Y i (k) y (k)] T χ(k)=[ x ^ 0 (k) x ^ i (k)+ (γ P (k) ) i x ^ i+m (k) ( P (k) ) i+m ] K(k)= P y(k)y(k) P x(k)y(k) 1 P(k)= P (k) P y(k)y(k) K T (k) x ^ (k)= x ^ (k)+K(k)[y(k) y (k)].
W 1 mean = ψ m+ψ , W i mean = 1 2(m+ψ) W 1 cov = ψ m+ψ +(1 α 2 +β), W i cov = W i mean ,
x(k)= [L(k), L (k), L (k)] T = [ x 1 (k), x 2 (k), x 3 (k)] T ,
L(k+1)=L(k)+ L (k)ΔT+ L (k)Δ T 2 /2 L (k+1)= L (k)+ L (k)ΔT, L (k+1)= L (k) f Interf (k)= 2n c [ υ (k)L(k)+υ(k) L (k)].
x(k)=Fx(k1)+v(k1),y(k)=Hx(k)+u(k),
F=[ 1 ΔT Δ T 2 /2 0 1 ΔT 0 0 1 ] H=[ 2n c υ (k) 2n c υ(k) 0 ][ 2n c υ (k) 2n c υ 0 0 ],
υ(j)= υ 0 +jFSR.
f aux (k)= n c L aux υ (k),
υ (k)= c n L aux f aux (k),
L aux = cΔ φ FP 2πnΔ υ FP ,
p( x k1 | y 1:k1 ) p ^ ( x k1 | y 1:k1 )= i=1 N w k1 ( x k1 i ) δ x k1 i (d x k1 ) ,
w k i = w k1 i p( y k | x k i )p( x k i | x k1 i ) q( x k i | x 0:k1 i , y 1:k ) , w ˜ k i = w k i [ j=1 N w k j ] 1 ,
x ˜ k = i=1 N w k i x ˜ k i N ,

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