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Abstract
Passive demodulation scheme using 3 × 3 coupler has been widely used in phase-sensitive optical time-domain reflectometry (φ-OTDR), interrogation of fiber Bragg gratings or fiber optic interferometric sensors, and sensor multiplexing. However, the asymmetry of the 3 × 3 coupler in real applications affects the demodulation performance seriously. We proposed an ameliorated 3 × 3 coupler-based demodulation algorithm using iteratively reweighted ellipse specific fitting (IRESF) to overcome the drawback. IRESF combines iterative reweight technology with ellipse specific fitting, which decreases the weights of high noise points and always outputs ellipse solutions. Any two output signals from the 3 × 3 coupler-based interferometer are fitted by the IRESF and then corrected as a pair of quadrature signals. The stability of the fitting parameters is utilized to resolve the failures of IRESF under small signals. A real-time 1/4 ellipse arc judging module is designed, if the Lissajous figure is larger than 1/4 ellipse arc, IRESF is executed to offer ellipse correction parameters. Otherwise, the fixed parameters preset in the algorithm are used. The fixed parameters are mean values of the fitting parameters of IRESF under a large stimulus. The desired phase signal is finally extracted from the corrected quadrature signals. Experimental results show that the ameliorated algorithm does not require strict symmetry of the 3 × 3 coupler and can work under small signals. The noise floor of the proposed algorithm is −112 dB re rad/√Hz and the demodulated amplitude is 23.15 dB (14.37 rad) at 1 kHz when THD is 0.0488%. Moreover, the response linearity is as high as 99.999%. Compared to the algorithm using direct least squares, the proposed demodulation algorithm is more robust and precise, which has broad application prospects.
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1. Introduction
The phase demodulation scheme using a 3 × 3 fiber coupler has the advantages of high sensitivity, passive detection and low cost as it does not require phase or frequency modulation. It has been widely used in phase-sensitive optical time-domain reflectometry (φ-OTDR) [1–3], interrogation of fiber Bragg gratings or fiber optic interferometric sensors [4–6], and sensor multiplexing [7–9].
There are two conventional 3 × 3 coupler-based demodulation schemes for fiber interferometers: the first one was proposed by the Naval Research Laboratory (NRL) in 1982, two of the three outputs of the 3 × 3 coupler are used to obtain quadrature signals from their sum and difference [10]. Then the desired phase shift signal is demodulated by the differential-cross-multiplying (DCM) approach. The second one was reported by the Naval Postgraduate School (NPS) in 1991, all three outputs are used symmetrically in the proposed algorithm to extract the desired phase shift signal [11]. The NRL scheme is simple, but it requires a symmetric 3 × 3 coupler and a direct current (DC) offset in the processing electronics to eliminate the DC component of the sum of the two inputs, therefore the DC component will not be eliminated completely when the light power in the interferometer changes. The NPS scheme can eliminate the signal DC components by subtracting the mean value of the three outputs and compensate for the signal fading induced by the variation of the polarization states in the interferometer. However, if the 3 × 3 coupler is asymmetric, it will rise a rather serious problem: the demodulator may fail. In real applications, the 3 × 3 coupler is difficult to be strictly symmetric, the asymmetry of the 3 × 3 coupler affects the demodulation performance seriously.
To overcome the drawback of the conventional demodulation schemes, different ameliorated demodulation algorithms are reported. An improved NPS scheme for fiber optic interferometers using an asymmetric 3 × 3 coupler is demonstrated by introducing a modified DC components removing circuit and an automatic gain control [12]. A method to eliminate the influence of frequency-modulating-induced auxiliary amplitude modulation on the calibration of 3 × 3 coupler asymmetric parameters is proposed, it is realized by dividing the three interference signals by the laser intensity [13]. However, the method is complex and additional hardware overhead is required. Normalization technology is proposed for the NRL scheme in the φ-OTDR, but the phase difference asymmetry of the 3 × 3 coupler cannot be eliminated [14]. A new demodulation technique using ellipse fitting algorithm (EFA) is proposed for a 3 × 3 coupler based Mach-Zehnder interferometer, which eliminates the dependence on the symmetry of the 3 × 3 coupler, provides enhanced tolerance to the variance of photoelectric converters, and is anti-polarization in a certain extent [15]. A phase demodulation technology using 3 × 3 coupler and three EFAs is demonstrated to reduce the influence of light source intensity noise [16]. There is a drawback that the EFA fails to work when the signal to be detected is small. An ameliorated 3 × 3 coupler-based demodulation scheme using the EFA and an external active piezoelectric transducer modulator is proposed to guarantee the effectiveness of the EFA under small signals [17,18]. However, the external active modulator violates the motivation and merits inherent in using fiber optic sensors.
In this paper, an ameliorated 3 × 3 coupler-based demodulation algorithm using iteratively reweighted ellipse specific fitting (IRESF) is proposed, which does not require strict symmetry of the coupler. IRESF is robust as ellipse specific fitting algorithm introduces a 6 × 6 constraint matrix to prevent hyperbola solutions, furthermore, the iterative reweight technology gives a high accuracy fitting result. Because the 3 × 3 coupler is a passive device and the laser intensity is stable, mean values of the fitting parameters of IRESF under a large stimulus are used as fixed parameters and preset in the algorithm. IRESF is used to offer the ellipse correction parameters when the Lissajous figure is larger than 1/4 ellipse arc. Otherwise, fixed parameters are used to correct the outputs of the interferometer. The proposed ameliorated 3 × 3 coupler-based demodulation algorithm is experimentally demonstrated and shows a good performance.
2. Principle
The schematic of the 3 × 3 coupler-based demodulation algorithm for the interferometer is shown in Fig. 1. A laser is launched into a fiber Michelson interferometer (FMI) consists of a symmetric 3 × 3 fiber coupler and two Faraday rotation mirrors (FRMs). The output power from each leg of the 3 × 3 coupler is nominally 120° out of phase with either of its neighbors and can be written as
The output optical powers from any two legs of the 3 × 3 coupler are detected by a balanced photodetector (BPD) and converted into electrical signals. The signals are sampled by a data acquisition (DAQ) and then processed by the proposed demodulation algorithm. In practical applications, due to asymmetry of the 3 × 3 coupler, time delay caused by transmission optical fibers or electronic devices and influence of the ambient temperature [18], the detected interference signals of the BPD are not ideal and written as
The Lissajous figure of V1 and V2 can be fitted as an ellipse that is expressed as
where A, B, C, D, E and F are ellipse fitting parameters.As the radian of the Lissajous figure affects the accuracy of ellipse fitting results and the ellipse fitting even fails when the radian is too small, a Lissajous figure radian judging module based on quadrant distribution is designed. If the Lissajous figure is equal or larger than 1/4 ellipse arc, IRESF is used to process the data to offer the fitting parameters for ellipse correction, otherwise, fixed parameters are used to correct the two signals. V1 and V2 are sampled by two analog-to-digital converters (ADCs) in the DAQ and the corresponding digital signals at time k are xk and yk. Translation and rotation operations are firstly executed for the digital signals, and the transformed signals can be written as
Let us represent Eq. (3) by an implicit second order polynomial [19], it can be re-written as
where u = [A, B, C, D, E, F]T and X = [x2, xy, y2, x, y, 1]T. u and X are six-dimensional vectors representing the ellipse parameters and the position, respectively.In real application, we obtain Xk by replacing x and y in the position vector X by xk and yk, as Xk inevitably contains noise, Eq. (5) can be written as F(u, Xk) that is usually not equal to zero. Consequently, F(u, Xk) is called the “algebraic distance” of a point Xk to the conic F(u, X) = 0. If we denote the inner product of vectors Xk and u by (Xk, u), the algebraic distance F(u, Xk) = (Xk, u). Ellipse fitting is to minimize the sum of squared algebraic distances over the set of N data points in the least squares, that is
- 1. let u0 = 0 and Wk = 1, k = 1, …, N.
- 2. Minimization Eq. (6) is realized by Lagrange multiplier method that is famous in mathematical optimization problems and the Lagrange multiplier λ is introduced. where Wk are the weights and Ccon is a 6 × 6 constraint matrix preventing hyperbola solutions. Ccon can be written as
This system is readily solved by considering the eigenvalue problem, and the eigenvector u for the positive eigenvalue is computed.
- 3. As the variance is σ2(u,V0[Xk]u), if u≈u0 up to sign, return u and stop. Else, update and go back to step 2. V0[Xk] is the covariance matrix and σ is the standard deviation.
Initially, we let Wk = 1, so the initial solution is the ellipse specific fitting (ESF) solution, from which the iterations start. If the solution u of Eq. (7) is not close to u0, the weights Wk and u0 are updated. The weights Wk are small for uncertain terms and large for certain terms. Thus, IRESF decreases the weights of high noise points and always outputs ellipse solutions.
Then the parameters h, k, a, b and δ can be calculated by the solved u [21] and written as
Consequently, correction is performed and the two signals of V1 and V2 are corrected as a pair of quadrature signals [22].
Finally, the desired phase signal φ(t) is extracted from the corrected quadrature signals by the DCM approach and the low frequency noise is filtered out by a high passed filter (HPF).
3. Simulation
Assuming that h = 1, k = 0.8, a = 1.5, b = 1.5, δ=2π/3 and φ(t)= π/4*cos(2π*1000*t), the signals V1 and V2 in Eq. (2) can be simulated. Moreover, σ is the standard deviation (STD) of Gaussian noise that is added in φ(t). The Lissajous figures of the input data, the true ellipse and the fitted ellipses are plotted in Fig. 2. The Lissajous figure of the input data is a 1/4 ellipse arc. Ellipse fitting based on direct least squares (DLS) is mentioned in Ref. [23] and compared with the proposed IRESF in our work. DLS is realized by minimizing the sum of squared algebraic distances in Eq. (6) and the unit eigenvector for the smallest eigenvalue is computed by solving the eigenvalue problem of $\frac{1}{N}\sum\limits_{k = 1}^N {{{\boldsymbol X}_k}{\boldsymbol X}_k^T{\boldsymbol u}} = \boldsymbol {\lambda {I}u}$, where I is the identity matrix. Figure 2(a) shows that the fitted ellipse of DLS deviates seriously from the true shape while the fitted ellipse of IRESF overlaps with the true shape when there is no Gaussian noise (σ=0). Figures 2(b)–2(d) show that the two fitted ellipses deviate more from the true shape as the STD increases. However, the fitted ellipse of IRESF is always better than that of DLS. We also observed that convergence is realized when the number of iterations is about 4 for IRESF.
Fig. 2. Input data of 1/4 ellipse arc and the ellipse fitting results as (a) σ=0, (b) σ=0.001, (c) σ=0.005 and (d) σ=0.01.
Assuming that σ=0.01, the amplitude of φ(t) is changed from π/4 to π/2 and 3π/4, respectively. The fitted ellipses are plotted in Fig. 3, the Lissajous figures of the input data are a 1/2 ellipse arc and a 3/4 ellipse arc, respectively. The fitted ellipses of IRESF and DLS are close to the true shape when the Lissajous figure is a 1/2 ellipse arc and the results of IRESF is slightly better than that of DLS. The both fitted ellipses overlap with the true shape when the Lissajous figure is a 3/4 ellipse arc, which means the accuracy of the two fitting methods are high enough when the Lissajous figure is larger than 3/4 ellipse arc. It is consistent with Ref. [24]. However, in real applications, the Lissajous figure is varying timely due to the influence of ambient temperature, IRESF is obviously superior to DLS when the amplitude of φ(t) is small. Consequently, we set 1/4 ellipse arc as the threshold in the Lissajous figure judging module.
Fig. 3. The ellipse fitting results of the input data of (a) 1/2 ellipse arc and (b) 3/4 ellipse arc.
4. Experimental setup and results
4.1 Experimental setup
The experimental setup is shown in Fig. 4, a FMI is fabricated by a 3 × 3 coupler, a signal arm, a reference arm and two FRMs. The signal arm of the FMI is wound on a cylinder piezoelectric transducer (PZT) driven by a signal generator (Rigol, DG4202). The stimulating signal output from the signal generator is a sinusoidal signal. The length difference between the signal and reference arms is 1 m, and the third fiber is angled to prevent reflection from the fiber end face. A fiber laser (NKT, Koheras BASIK X15) with a center wavelength of 1550.12 nm is connected to port 1 of the 3 × 3 coupler and the light is launched into the FMI. The interference light output from port 2 and port 3 are detected by a BPD (THORLABS, PDB450C) and then sampled by a DAQ (NI USB-6356). Continuous sampling mode is set for the ADCs, the sampling rate is set as 500 kS/s and the samples to be read is set as 500 kS. The two sampled interference signals are sent to the ameliorated demodulation algorithm in the computer for real-time processing.
4.2 Experimental results
A 1 kHz stimulating signal with the amplitude of 1.5 V is set for the PZT and the PZT modulation induced radian amplitude is 2.15 rad. The two signals sampled from the BPD is shown in Fig. 5(a), it is found that the DC offsets and AC amplitudes are different. The original and corrected Lissajous figures are plotted in Fig. 5(b). The original Lissajous figure is a slanted ellipse, which shows an offset angle of approximately 45°, indicating the phase difference of V1 and V2 of about 2π/3 rad. The Lissajous figure is a full ellipse which means the stimulating signal induced phase shift is larger than 2π. Consequently, IRESF is utilized to fit the sampled signals of V1 and V2. The corrected Lissajous figure is a circle overlapping with a circle with the radius of 1 V. The result demonstrates that the signals of V1 and V2 are corrected as a pair of quadrature signals.
Fig. 5. (a) Time domain waveforms of the sampled signals of V1 and V2, and (b) original and corrected Lissajous figures.
The DCM approach and the HPF are then used to extract the desired phase shift signal from the corrected quadrature signals. Time domain waveform and frequency spectrum of the demodulated signal are shown in Fig. 6. The stimulating signal is well restored with a low harmonic distortion, it is found that the amplitude of the demodulated signal at 1 kHz is 2.15 rad (6.66 dB). The total harmonic distortion (THD) is as low as 0.0266% and the signal to noise and distortion ratio (SINAD) reaches 65.94 dB.
The parameters h, k, a, b and δ output from the IRESF are monitored for 10 minutes and the result is plotted in Fig. 7. It shows that the parameters are stable in 10 minutes, the reason is that the parameters are dominated by the 3 × 3 coupler and the laser intensity. The 3 × 3 coupler is a passive device with fixed parameters and the light intensity is fixed once the system is established. The mean values of h, k, a, b and δ are 1.2142 V, 0.9999 V, 1.0225 V, 0.8251 V and 2.1118 rad, and the corresponding STDs are 0.00084 V, 0.00126 V, 0.00100 V, 0.00083 V and 0.00110 rad, respectively. The results verify the stability of the parameters h, k, a, b and δ, and the mean values of h, k, a, b and δ are used as fixed parameters and preset in the proposed phase demodulation algorithm.
To compare the performance of the algorithms using DLS and IRESF, the frequency spectra of the demodulated signals under the stimulating amplitudes of 400 mV and 650 mV are plotted in Fig. 8. The Lissajous figures of V1 and V2 are about a 1/4 ellipse arc and a 1/2 ellipse arc when the stimulating amplitudes are 400 mV and 650 mV, respectively. The PZT modulation induced radian amplitudes at 400 mV and 650 mV are 0.57 rad and 0.93 rad, respectively. The THDs and SINADs of the demodulated signals of the two algorithms are 0.1207% (DLS), 0.0180% (IRESF), 55.17 dB (DLS) and 70.71 dB (IRESF) when the Lissajous figure is about a 1/4 ellipse arc. The THDs and SINADs of the demodulated signals of the two algorithms are 0.0345% (DLS), 0.0233% (IRESF), 65.53 dB (DLS) and 67.92 dB (IRESF) when the Lissajous figure is about a 1/2 ellipse arc. Compared to the algorithm using DLS, the THD and SINAD of the demodulated signal using IRESF are improved by 0.1027% and 15.54 dB when the Lissajous figure is about a 1/4 ellipse arc, and the THD and SINAD are improved by 0.0112% and 2.39 dB when the Lissajous figure is about a 1/2 ellipse arc, respectively. The experimental results are consistent with the simulation.
Keep decreasing the stimulating amplitude to 100 mV (the induced radian amplitude is 0.14 rad), the original and corrected Lissajous figures of the interference signals are plotted in Fig. 9. As the desired phase shift signal is small, the original Lissajous figure is smaller than 1/8 ellipse arc. Signals of V1 and V2 are corrected by the fitting parameters output from IRESF and the fixed parameters. The dashed circle is a standard circle with the radius of 1 V. The accuracy of IRESF is not high when the Lissajous figure is too small, and the Lissajous figure corrected by the fitting parameters deviates from the dashed circle. In contrast, the Lissajous figure corrected by the fixed parameters overlaps with the dashed circle well.
Demodulated signal 1 (DS1) is extracted from the two quadrature signals corrected by the fitting parameters and DS2 is extracted from the two quadrature signals corrected by the fixed parameters. Time domain waveforms and frequency spectra of DS1 and DS2 are plotted in Fig. 10. It can be seen clearly that DS2 is superior to DS1, no harmonic distortions are excited in the frequency spectrum of DS2. The THDs of DS1 and DS2 are 2.3161% and 0.0430%, and the SINADs of DS1 and DS2 are 34.05 dB and 62.93 dB, respectively. Compared to the algorithm without radian judging module, the THD and SINAD are improved by 2.2731% and 28.88 dB, respectively.
In order to test the response of the proposed ameliorated demodulation algorithm to both small and large signals, the stimulating amplitude is increased from 1 mV to 10 V. The relationship between the amplitude of the output signal and the amplitude of the stimulus is shown in Fig. 11. The experimental results show that the response slope is 1.4345 rad/V and the response linearity is as high as 99.999%. The results shows that the proposed ameliorated demodulation algorithm has a good response to both small and large signals. The reason of the high linearity is that the nonlinear errors have been effectively eliminated by the proposed algorithm and the stimulus has been demodulated with high fidelity.
The frequency spectra of the demodulated signals under no stimulus and a large stimulus are plotted in Fig. 12. Figure 12(a) is the real time frequency spectrum of the demodulated signal without stimulus, which is the result of one single measurement. It can be seen that the noise floor is about −112 dB re rad/√Hz, thus, the minimum detectible phase shift of the system is 2.51 µrad. As mentioned in Ref. [25], the calculated minimum detectible displacement is 2.71 × 10−4 nm. Figure 12(b) shows the frequency spectrum of the demodulated signal with the stimulating amplitude of 10 V, the amplitude at 1 kHz is 23.15 dB (14.37 rad), the THD and SINAD of the demodulated signal are 0.0488% and 60.87 dB, respectively.
In order to test the system response to an abrupt signal, a bead drop test is conducted. A steel ball falls at a height of 10 centimeters and lands on the table. The vibration signals are detected by the sensor system, and the demodulated waveform and frequency spectrum are plotted in Fig. 13. It can be seen clearly that all the frequency components in the vibration signals are well demodulated.
4.3 Discussion
We compared our proposed demodulation algorithm with different methods mentioned in literatures and the comparison of different 3 × 3 coupler-based demodulation algorithms is listed in Table 1. The improved NPS or NRL algorithms described primarily focus on the removal of DC components and normalization of AC components, without requiring extra modulations. However, these algorithms are unable to completely eliminate all nonlinear errors, resulting in unsatisfactory demodulation performance. EFAs have been widely used to improve the demodulation performance in other literatures, however, most of the EFAs are realized based on the conventional DLS method that is not accuracy when the input the input point sequence covers only a small part of the ellipse circumference. Extra modulation is required to obtain a large Lissajous figure to guarantee the ellipse fitting accuracy, which increases the complexity of the system and introduces an auxiliary amplitude modulation. The post-processing technique demonstrated is not a real-time algorithm and the ellipse curve fitting is realized by the Origin Pro manually. Compared with other algorithms, our proposed algorithm can work under small signals without extra modulation and has a comparable noise floor and a better response linearity because of the Lissajous figure judging module and IRESF.
Table 1. Comparison of different 3 × 3 coupler-based demodulation algorithms.
5. Conclusion
In this paper, an ameliorated 3 × 3 coupler-based demodulation algorithm using IRESF is proposed for the first time, to our best knowledge. Computer vision technologies of iterative reweight and ellipse specific fitting are combined in the IRESF. IRESF decreases the weights of high noise points until the sum of squared algebraic distances is minimum and always outputs ellipse solutions. With the help of the Lissajous figure radian judging module, stability of the parameters (h, k, a, b and δ) is used to correct the two signals under a small stimulus. Experimental results show that the proposed algorithm is superior to the algorithm based on DLS, the THD and SINAD of the demodulated signal using IRESF are improved by 0.1027% and 15.54 dB when the Lissajous figure is about a 1/4 ellipse arc. Compared to the algorithm without radian judging module, the THD and SINAD are improved by 2.2731% and 28.88 dB, respectively. The noise floor of the demodulator is −112 dB re rad/√Hz and the large signal can reach 23.15 dB at 1 kHz when the THD is 0.0488%. And the response linearity is as high as 99.999%. The proposed ameliorated 3 × 3 coupler-based demodulation algorithm using IRESF has promising application prospects in fiber optic interferometric sensors and φ-ODTRs due to its merits of high accuracy, good linearity, passive detection, low cost and large dynamic range.
Funding
Natural Science Foundation of Anhui Provincial Higher Education (2022AH050964); Open Fund of Information Materials and Intelligent Sensing Laboratory of Anhui Province (IMIS202104, IMIS202212); Research Start-up Fund of Anhui Polytechnic University (2021YQQ057, 2022YQQ098); Student Research Project of Anhui Polytechnic University (2023DZ12).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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