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Abstract
To solve the problem of light source jitter and asymmetric 3 × 3 coupler, a phase demodulation method with the combination of an auxiliary reference interferometer and elliptic fitting algorithm is proposed, which is verified by simulation and experiment. By introducing additional phase modulation in the auxiliary reference interferometer, the parameters of the sensing arm can be calibrated in real time, which ensures the effective operation of elliptic fitting algorithm in small signal measurement. Consequently, the experiments show that the self-calibration scheme enables a higher signal to noise and distortion ratio with an average increase of 1.65 dB and 10.47 dB compared with the traditional Arctan and cross multiplication differential, respectively. Meanwhile, the self-calibration scheme can also effectively suppress the harmonic distortion, with a total harmonic distortion of -33.64 dB in the case of small signal.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Distributed optical fiber interferometry has have been one of the most active research areas, characterized by numerous advantages, such as high sensitivity, diverse structure, immune response to electromagnetic interference and wide dynamic range. With the development of technology, the application of distributed optical fiber interferometry involves perimeter security, pipeline monitoring, and civil fields [1–4]. To achieve optimum accuracy of sensing system, a considerable amount of research effort has been expended over the past decade on the improvement of demodulation schemes. The commonly used phase demodulation methods can be broadly divided into 3 × 3 fiber coupler method [5,6] and phase generated carrier (PGC) technology [7–10].
The phase demodulation scheme based on a 3 × 3 coupler is a comparable reliable and practical method because of the passive detection, simple structure, high stability and low requirement for light source [5,11–13]. However, the 3 × 3 coupler in application is asymmetric, which means that the three-arm split ratio and phase difference of the 3 × 3 coupler is deviate from theoretical value. Consequently, the demodulation results of the 3 × 3 coupler algorithm will be distorted, and directly leads to low reproducibility of the retrieved phase. The researchers suggest several ways to effectively reduce the nonlinear distortion. Todd et al. [14] improved the arctangent algorithm by introducing a gain factor to eliminate the distortion effects caused by the asymmetric device. However, this method is extremely complex. Some studies have used elliptic fitting algorithm (EFA) in 3 × 3 coupler demodulation scheme [15–19], to compensate for nonlinear distortion and phase difference instability caused by the symmetric devices. Fan et al. [18] used EFA to realize the demodulation of 0.8-250 Hz acoustic pressure signals in Michelson interferometric sensor. Xia et al. [17] introduced the EFA in extrinsic Fabry-Perot interferometer system and realized the stable demodulation. Zhang et al. [19] proposed a morphological filtering based on elliptic pattern to overcome the signal distortion caused by polarization fading. However, these methods suffer from the requirement that amplitude of vibration signal should be greater than π/2 rad due to the invalid fit caused by the incomplete ellipses [8]. There are ways to solve this problem. Yan et al. [20] employed a precision PGC demodulation with a combined sinusoidal and triangular phase modulating scheme for homodyne interferometer.
In addition, the light intensity disturbance (LID) in optical fiber interferometer is an essential factor limiting the performance. Simultaneously, the phase noise caused by the fluctuation light source frequency also contributes to the demodulation distortion. However, the high cost of a highly stable light source suppresses the application in engineering. In the PGC scheme, the effective method to reduce the phase noise is to introduce the auxiliary reference interferometer [21]. Nonetheless, this scheme is rarely used in 3 × 3 demodulation scheme at present. Furthermore, the method requires that the two interferometers must have the same optical path difference (OPD). Otherwise, the different OPD leads to different phase modulation depths, increasing the instability of the demodulation system.
In this paper, a self-calibration and highly stable phase demodulation scheme based on auxiliary ellipse fitting is proposed. This scheme adopts an auxiliary reference interferometer based on the 3 × 3 coupler, in which the feed OPD between the sensing interferometer and the auxiliary reference interferometer is unnecessary. By introducing additional phase modulation, the relevant parameters of the sensing arm obtain real-time calibration, which ensures the effective operation of EFA in small signal condition. Besides, the demodulation signal distortion caused by LID and asymmetric 3 × 3 coupler is eliminated, and the accuracy of phase demodulation is effectively improved.
2. Principle
2.1 Optical configuration description
As shown in Fig. 1, the optical configuration of the self-calibrated scheme based on the auxiliary reference interferometer combined with EFA. The light source is divided into two beams with wavelengths ${\lambda _\textrm{s}}$ and ${\lambda _\textrm{r}}$ and enters the sensing interferometer and the reference interferometer respectively. Passing through the 3 × 3 coupler to interfere, the interference light with wavelength ${\lambda _\textrm{s}}$ is divided into two beams and detected by PD1 and PD3, which are recorded as ${I_{s1}}(t )$ and ${I_{s2}}(t )$ after analog-to-digital conversion (ADC). Similarly, the interference light ${I_{r1}}(t )$ and ${I_{r2}}(t)$ with wavelength ${\lambda _\textrm{r}}$ is detected by PD2 and PD4. The received interference signals are as follows:
Fig. 1. The optical configuration of self-calibration fiber interferometer based on a reference interferometer. SLD: superluminescent diodes. WDM: wavelength division multiplex. PZT: piezoelectric ceramics. FRM: Faraday rotating mirror.
Obviously, the dynamic changes of $A_{s(r)1}(t), A_{s(r)2}(t), B_{s(r)1}(t), B_{s(r)2}(t)$ cause nonlinear distortion of the retrieved signal. When the vibration range of the measured signal is large enough, the Lissajous graph of ${I_{s(r )1}}(t )$ and ${I_{s(r )2}}(t )$ is a fully ellipse, and the distortion can be eliminated by obtaining the parameters through EFA in real time. It has been illustrated that the EFA works properly when the data forms at least 1/4 ellipse. Moreover, when 3/4 ellipses are formed, the accuracy required for a complete ellipse can be approached. Therefore, the amplitude of vibration signal should be at least π/2 rad and better be 3π/2 rad. In order to have accurate fitting of the small signal model, a self-calibration demodulation method based on auxiliary reference interferometer is proposed in this paper.
2.2 Theory of operation
In the self-calibration scheme, in order to generate a continuous and fit-operable interference signal, we introduce a modulation signal at the auxiliary interferometer end with an amplitude greater than 3π/2 to satisfy the accuracy requirements of EFA. For two interferometers with the same parameters, the amplitude noise caused by the power instability of the light source is almost the same, then the DC bias and AC amplitude have the synchronous fluctuation. Therefore, ${I_{r1}}$ and ${I_{r2}}$ are utilized to estimate the parameters by elliptic fitting, and then the signal with the vibration information we interest is self-calibrated to obtain higher precision demodulation that can be employed normally under the special condition of small signal.
In practical application, ${A_{s1}} \ne {A_{r1}}$, ${B_{s1}} \ne {B_{r1}}$, ${A_{s2}} \ne {A_{r2}}$, ${B_{s2}} \ne {B_{r2}}$. For the purpose of the reference arm parameters to successfully match with the sensing arm, the DC bias and AC amplitude of the two interferometers can be calibrated a priori to obtain the correction factors. Firstly, a large-amplitude vibration signal is applied to the sensing interferometer to complete the correct ellipse fitting and the initial parameters are ${A_{s1ini}}$, ${A_{s2ini}}$, ${B_{s1ini}}$ and ${B_{s2ini}}$. Similarly, the EFA is performed on the reference interferometer, and then we figure out ${A_{r1ini}}$, ${A_{r2ini}}$, ${B_{r1ini}}$, ${B_{r2ini}}$. As a result, the correction factors ${\gamma _{A1}}$, ${\gamma _{A2}}$, ${\gamma _{B1}}, {\gamma _{B2}}$, ${\gamma _{delt}}$ are obtained as:
Figuring out the correction factors, it can be used to self-calibrate the measurement signal to eliminate the noise and demodulation distortion caused by the instability of the light source. The proposed scheme flowchart combining self-calibration and ellipse fitting is shown in Fig. 2. The EFA is carried out to obtain the ellipse parameters $g = {[{a,b,c,d,e,f} ]^T}$. Furthermore, the parameters of the reference interferometer are ${A_{r1}}(t )$, ${A_{r2}}(t )$, ${B_{r1}}(t )$, ${B_{r2}}(t )$. Combining the correction factors with Eqs. (1) and (2), the signals can be rewritten as:
where ${\phi _{s(r )}} = {\varphi _{s(r )}}(t )+ {\varphi _0} + {\varphi _e}$. In non-ideal 3 × 3 coupler, ${A_1}(t )\ne {A_2}(t )$, ${B_1}(t )\ne {B_2}(t )$, $\Delta \varphi = \frac{{2\pi }}{3} - \delta $, $\delta $ is phase deviation from 3π/2 caused by asymmetric coupler.When the disturbance signal of the sensing interferometer is small (the amplitude of vibration signal is less than π/2), the coefficients ${A_{1(2 )}}(t )$, ${B_{1(2 )}}(t )$ cannot be accurately obtained by elliptic fitting. In this case, we employ the correction factors and fitting parameters of the reference interferometer for calibration. The corrected two-way signal successfully eliminates the DC bias and AC amplitude, written as:
The demodulation result can be obtained by dividing Eq. (12) by Eq. (11) and then conducting inverse trigonometric and phase unwrapping operation:
2.3 Ellipse fitting description
The signals ${I_{r1}}(t )$ and ${I_{r2}}(t )$ of the reference interferometer are modulated with the same frequency of change and constant phase difference. Therefore, the two signals meet the conditions for the formation of Lissajous pattern. In general, the DC bias and AC amplitude parameters tends to change slowly. When the ellipse fitting rate is fast enough, it can be regarded that the parameters remain stable during each fitting period. The process can be described as when the ith ellipse fitting period is performed, i.e. $t \in ({{t_i},{t_i} + \Delta t} )$, ${A_{r1(2 )}}(t )= A_{r1(2 )}^i$, ${B_{r1(2 )}}(t )= B_{r1(2 )}^i$, where $\Delta t$ is the sampling interval. The elliptic recessive equation can be expressed as:
where, $g = {[{a,b,c,d,e,f} ]^T}$ is the coefficient vector of the ellipse. The least square method [22] is selected to obtain the coefficient vector, which can be solved correspondingly to obtain the signal parameters:In the proposed scheme, the fitting period is corresponding to the sampling interval, and the data newly collected in each sampling interval will be utilized by the EFA. This pipeline calculation enables real-time tracking of phase and light source jitter and phase demodulation. By substituting the corrected parameters into the sensing signal for calibration, two mutually orthogonal signal $\cos ({{\phi_s}} )$ and $\sin ({{\phi_s}} )$ are obtained. Using the 3 × 3 coupler Arctan algorithm, we can retrieve the variation signal.
3. Simulation analysis
Aiming to verify the effectiveness of the 3 × 3 coupler demodulation scheme based on the auxiliary interferon calibration, the ellipse fitting is firstly simulated and analyzed in this paper. The performance for nonlinear distortion, high stability and noise resistance of the proposed scheme will be demonstrated by comparing with traditional demodulation methods.
In the numerical simulation, we use a single-frequency trigonometric signal ${\varphi _s} = D\cos ({2\pi {f_s}t} )$ as the signal to be measured. Figure 3(a) shows the Lissajous figure and the corresponding ellipse fitting results when the amplitude D is 5 rad, π/2 rad and 1 rad, respectively. Obviously, when the amplitude of the signal to be measured is less than π/2, the fitting result deviates from the real situation, so the traditional method is not applicable for the small signal case. Figure 3(b) shows the self-calibration results with the measured signal amplitudes of 5 rad, π/2 rad, and 1 rad. After correction, the two signals of the sensing interferometer form a standard circle.
Fig. 3. (a) Ellipse fitting results with different signal amplitudes in the sensing interferometer. (b) Lissajous figure of different signal amplitudes after self-calibration while the carrier signal of the reference interferometer is kept at 5 rad.
Meanwhile, we compare the conventional 3 × 3 coupler Arctan, cross multiplication differential (DCM) [23] and the self-calibration Arctan algorithm recommended in this paper. The amplitude D is set as 5, π/2, 1 rad respectively, and the frequency ${f_s}$ is set as 1 kHz to simulate the small signal situation that cannot be handled properly by the traditional 3 × 3 coupler demodulation method. The carrier signal in the reference interferometer is ${\varphi _r} = D\mathrm{^{\prime}}\cos ({2\pi {f_r}t} )$, $D\mathrm{^{\prime}}$ is set to 5 rad, and the frequency is 190 kHz. To inspect the performance of the scheme in noisy case, we apply a Gaussian noise with a signal-to-noise ratio (SNR) of 30 dB to the signals of both the reference interferometer and the sensing interferometer. The left column of Fig. 4 shows the comparison of the demodulation phase of the above three schemes when the vibration signal amplitude is 5, π/2 and 1 rad, respectively. And the right column shows the corresponding zoom of the results. It can be seen that the traditional Arctan algorithm produces distortion for small signals, while the self-calibration algorithm maintains relatively stable state. In addition, under the small signal case, the signal amplitude obtained through the traditional Arctan does not change linearly with the real situation, which will damage the amplitude-frequency consistency of the sensing system. This is because in the traditional algorithm, we obtain the maximum and minimum value of the interference signal to obtain the parameters, so as to conduct normalization processing. In the case of small signal, the correct parameters cannot be obtained. The inverse trigonometric function contains constants that increase the value, so the amplitude of demodulation signal cannot change linearly with the actual situation.
Fig. 4. The comparison of the retrieved phase of the three demodulation methods. (a), (c) and (e) The signal amplitude is 5 rad, π/2 rad and 1 rad, respectively. (b), (d) and (f) t partial magnification corresponding to the demodulation results.
Figures 5(a) and (b) show the power spectral density (PSD) of the retrieved signals of the three algorithms for small signal vibration signal with frequency of 30 kHz and amplitude of π/2, 1 rad. It can be seen that the demodulation results of conventional algorithms have large harmonic distortion. Using total harmonic distortion (THD) as a parameter for quantitative evaluation, the THD of the self-calibration algorithm in this paper is reduced by 48.98 dB, 47.86 dB and 46.24 dB, 47.94 dB compared with the conventional Arctan and DCM algorithm under the two small signal conditions, respectively. It is obvious that the traditional Arctan algorithm produces serious harmonics, while the recommended scheme demodulates well, and the demodulation effect of proposed algorithm is more significant in the case of small signal.
Fig. 5. The PSD of retrieved signals through the traditional 3 × 3 coupler and the self-calibration demodulation algorithm for small signals. (a) The amplitude is pi/2 rad. (b) The amplitude is 1 rad.
4. Experiment and discussion
The experimental structure is shown in Fig. 6. A broadband laser source with a center wavelength of 1550 nm and a bandwidth of more than 30 nm is used. The laser is divided into ${\lambda _\textrm{s}}$ and ${\lambda _\textrm{r}}$ wavelengths of 1553.3 nm and 1548.9 nm with a bandwidth of 3 nm, respectively. PD1 and PD3 receive interference light with wavelength ${\lambda _\textrm{s}}$ and the data sampling rate is 5 M Samples/s. Similarly, PD2 and PD4 also receive interference light with wavelength ${\lambda _\textrm{r}}$. The length of the delay fiber we used is 2 km. A series of vibration experiments are performed on the 20 km sensing fiber and 2 km reference fiber. The vibration signal source adopts a PZT placed at the tip of the 20 km sensing fiber, which can be driven by an optional signal, thus causing the phase changes. To verify the feasibility, a signal generator is used to apply vibrations of different frequencies and amplitudes. Particularly, the experimental configuration can also be transformed into a single-end measurement application rather than a distributed detection according to our actual application requirements.
The amplitude and frequency of the carrier signal on the reference interferometer is adjusted to 10 V and 190 kHz, and the modulation depth is observed to ensure that EFA work properly. The reference signals (${I_{r1}},{I_{r2}}$) is used to draw the Lissajous figure and the fitting and calibration results are shown in Fig. 7(a). When the amplitude of the measured signal decreases, the complete ellipse cannot be formed, which leads to the failure of ellipse fitting. The excitation voltage is adjusted to 2 V and 1 V for small signal state corresponding to 1.2 rad and 0.6 rad, and the frequency is set to 30 kHz. The reference signal forms a fully ellipse and obtain a better ellipse fitting effect. Figures 7(b) and (c) show the ellipse fitting and calibration results under small signal condition. It can be seen that using the self-calibration scheme is a feasible solution for small signal detection.
Fig. 7. (a) The Lissajous figure and the self-calibration result of the reference interferometer with a stable carrier signal; (b) The Lissajous figure before and after self-calibration with the amplitude 1.2 rad. (c) The Lissajous figure before and after self-calibration with the amplitude 0.6 rad.
Firstly, the feasibility of the self-calibration method using auxiliary reference interferometer is verified. The driving voltage of the signal generator is set to 10 V. A vibration single-frequency signal with large amplitude is applied to the test interferometer, so as to initially calibrate the sensing system and obtain initial calibration factors ${\gamma _{A1}}$, ${\gamma _{A2}}$, ${\gamma _{B1}}$, ${\gamma _{B2}}$, ${\gamma _{delt}}$ as 2.5589, 2.5248, 1.8734, 0.9191 and 1.0612, respectively. In the case of large signal, the feasibility of self-calibration scheme factor according to Eq. (11) and (12) is investigated. In order to track the variation of the interference parameters in real time, the flow-line parameter fitting is carried out with the data of length 8E4. Data collected recently in each sampling interval is added to the ellipse fitting database, while the endmost bit of data is deleted to keep the database length constant and we could achieve a real-time phase demodulation. Figure 8 shows the results of the auxiliary interferometer ellipse fit parameters (red), the sensing interferometer ellipse fit parameters (black), and the parameters after self-calibration (blue). It can be seen that the estimated parameters largely coincide with the actual parameter curves. In addition, the standard deviations of the three estimated parameters and the actual parameters are calculated as 0.0089, 0.0070 and 0.0029, respectively.
Fig. 8. The comparison of fitting parameters of the auxiliary interferometer, the sensing interferometer ellipse and the parameters after self-calibration. (a) The comparison for ${A_1}(t )$. (b) The comparison for ${B_1}(t )$. (c) The comparison for $\theta (t )$.
The auxiliary reference interferometer is placed in a noise insulation and isolation container with isolation rubber. Besides, a Wiener filter is designed for the signal of auxiliary reference interferometer. In this way, the parameters obtained by EFA directly reflect the noise caused by the jitter of light source and the interference system. In order to further verify the stability of the calibration factors, we select 5 hours per day to perform the flow-line parameter fitting with the data of 8E4 length. We pay attention to the standard deviations and average value between the parameters calculated with calibration factors and the parameters with ellipse fitting, and carry out a three-day experiment. The long-time stability is shown in Fig. 9, which indicates that the correction factors are relatively stable and the standard deviations of the three parameters are 0.0124, 0.0067, and 0.0025,respectively.
Fig. 9. The average value of parameters ${A_1}(t )$, ${B_1}(t )$ and $\theta (t )$ after self-calibration over three-day measurements.
Next, we demonstrate the demodulation effect of the self-calibration scheme using reference interferometer for small signal. The scheme is performed using the parameters of the auxiliary reference instrument at a vibration signal frequency of 30 kHz and an amplitude of 0.6 rad. Figure 10 shows the retrieved signal and PSD of the measured small-signal demodulation utilizing traditional Arctan, DCM and self-calibration scheme. It is obvious that the retrieved signal of the traditional DCM produces significant drift, while the phase amplitude of the conventional Arctan method fails to linearly correspond to the actual vibration signal. In addition, the demodulation results of the traditional methods have larger harmonic distortion, and the self-calibration scheme has the smallest THD, which is 14.65 dB lower than that of the conventional Arctan.
Fig. 10. Demodulation results and PSD in the small signal case using traditional Arctan, DCM and self-calibration demodulation methods.
In addition, we apply the sinusoidal waveform of the peak voltage variation to the PZT. The peak-to-peak value of the retrieved phase after self-calibration is recorded, and we perform a linear fit to the applied voltage. As shown in Fig. 11, three experiments with vibration signal frequencies of 5 kHz, 10 kHz and 15 kHz are carried out. The applied voltage for each experiment varies from 1 V to 10 V with a step of 1 V. The slope of the linear fit of the measured phase response and the applied voltage increases with frequency, which is due to the amplitude-frequency response characteristics of the differential interferometer [24]. The correlation coefficients R are 0.9806, 0.9942 and 0.9986, respectively. According to the results given in the Fig. 11, the proposed scheme has good linearity and stability.
As shown in Fig. 12, the THD and signal to noise and distortion ratio (SINAD) with different demodulation methods are different vibration signals with different amplitudes. When the drive voltage is in the range of 1-10 V, the harmonic distortion rate of the conventional 3 × 3 coupler scheme will increase as the signal amplitude decreases. Compared with the traditional methods, the self-calibration scheme can effectively suppress the harmonic distortion with a THD of -33.64 dB in the case of small signal with a 0.6 rad amplitude. In addition, as the vibration signal changes, the self-calibration method is able to maintain the SINAD at the highest level, with an average improvement of 1.65 dB and 10.47 dB compared with the traditional Arctan and DCM, respectively. The SINAD of the self-calibrated scheme is 0.64 dB higher than that of the conventional Arctan method for an applied voltage signal amplitude of 1 V. Although the overall SINAD is not significantly improved, the phase peak-to-peak value of the traditional Arctan method do not correspond to the intensity of the real vibration signal.
5. Conclusion
Based on the demodulation method of 3 × 3 coupler, a phase demodulation method with the combination of an auxiliary reference interferometer and EFA is proposed to solve the problem of light source jitter and asymmetric 3 × 3 coupler, which is verified by simulation and experiment. By introducing additional phase modulation in the auxiliary reference interferometer, the parameters of the sensing arm can be calibrated in real time, which ensures the effective operation of EFA in small signal measurement, and also eliminates the distortion caused by unstable light source and non-ideal 3 × 3 coupler. Consequently, the accuracy of phase demodulation is effectively improved. The experimental results show that the self-calibration scheme enables a higher SINAD with an average increase of 1.65 dB and 10.47 dB compared with the traditional Arctan and DCM, respectively. Meanwhile, the self-calibration scheme can also effectively suppress the harmonic distortion, with a THD of -33.64 dB in the case of small signal with the excitation voltage of 1 V.
Funding
Science and Technology Commission of Shanghai Municipality (17DZ2280600).
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.
References
1. Q. Sun, D. Liu, J. Wang, and H. Liu, “Distributed fiber-optic vibration sensor using a ring Mach-Zehnder interferometer,” Opt. Commun. 281(6), 1538–1544 (2008). [CrossRef]
2. W. Huang, W. Zhang, J. Huang, and F. Li, “Demonstration of multi-channel fiber optic interrogator based on time-division locking technique in subway intrusion detection,” Opt. Express 28(8), 11472–11481 (2020). [CrossRef]
3. X. Huang, B. Wang, K. Liu, and T. Liu, “An Event Recognition Scheme Aiming to Improve Both Accuracy and Efficiency in Optical Fiber Perimeter Security System,” J. Lightwave Technol. 38(20), 5783–5790 (2020). [CrossRef]
4. A. Kurzych, L. R. Jaroszewicz, Z. Krajewski, B. Sakowicz, J. K. Kowalski, and P. Marc, “Fibre-Optic Sagnac Interferometer in a FOG Minimum Configuration as Instrumental Challenge for Rotational Seismology,” J. Lightwave Technol. 36(4), 879–884 (2018). [CrossRef]
5. D. Xu, F. Yang, D. Chen, F. Wei, H. Cai, Z. Fang, and R. Qu, “Laser phase and frequency noise measurement by Michelson interferometer composed of a 3 × 3 optical fiber coupler,” Opt. Express 23(17), 22386 (2015). [CrossRef]
6. W. Li and J. Zhang, “Distributed Weak Fiber Bragg Grating Vibration Sensing System Based on 3 × 3 Fiber Coupler,” Photonic Sens 8(2), 146–156 (2018). [CrossRef]
7. Z. Qu, S. Guo, C. Hou, J. Yang, and L. Yuan, “Real-time self-calibration PGC-Arctan demodulation algorithm in fiber-optic interferometric sensors,” Opt. Express 27(16), 23593 (2019). [CrossRef]
8. L. Gui, X. Wu, B. Yu, D. Guang, J. Shi, C. Zuo, and W. Zhang, “High-Stability PGC Demodulation Algorithm Based On a Reference Fiber-Optic Interferometer With Insensitivity to Phase Modulation Depth,” J. Lightwave Technol. 39(21), 6968–6975 (2021). [CrossRef]
9. J. Xie, L. Yan, B. Chen, and Y. Lou, “Extraction of Carrier Phase Delay for Nonlinear Errors Compensation of PGC Demodulation in an SPM Interferometer,” J. Lightwave Technol. 37(13), 3422–3430 (2019). [CrossRef]
10. L. Yan, Y. Zhang, J. Xie, Y. Lou, B. Chen, S. Zhang, and Y. Zhou, “Nonlinear Error Compensation of PGC Demodulation With the Calculation of Carrier Phase Delay and Phase Modulation Depth,” J. Lightwave Technol. 39(8), 2327–2335 (2021). [CrossRef]
11. L. Ruan, C. He, and B. Wu, “Security monitoring system based on a line structure Sagnac interferometer with 3 × 3 coupler,” Opt. Eng. 55(6), 064108 (2016). [CrossRef]
12. S. Q. Jing, L. Huang, W. H. Dong, and J. Rong, “Positive and Negative Feedback Judgment and High-Frequency Vibration Detection by Interferometer Based on 3 X 3 Coupler,” Acta Opt. Sin. 39(7), 1 (2019). [CrossRef]
13. S. Park, J. Lee, Y. Kim, and B. H. Lee, “Nanometer-Scale Vibration Measurement Using an Optical Quadrature Interferometer Based on 3 × 3 Fiber-Optic Coupler,” Sensors 20(9), 2665 (2020). [CrossRef]
14. M. D. Todd, M. Seaver, and F. Bucholtz, “Improved, operationally-passive interferometric demodulation method using 3 × 3 coupler,” Electron. Lett. 38(15), 784–786 (2002). [CrossRef]
15. Y. Zhang, J. Wang, M. Chen, M. Wang, Y. Liang, and Z. Meng, “A Method to Eliminate the Influence of Frequency-Modulating-Induced Auxiliary Amplitude Modulation on the Calibration of 3 × 3 Coupler Asymmetric Parameters,” Sensors 20(21), 6180 (2020). [CrossRef]
16. T. Liu, J. Cui, D. Chen, L. Xiao, and D. Sun, “A new demodulation technique for optical fiber interferometric sensors with [3×3] directional couplers,” Chin. Opt. Lett. 6(1), 12–15 (2008). [CrossRef]
17. J. Xia, F. Wang, H. Luo, S. Xiong, Z. Zhao, L. Wei, Y. Liao, W. Zhang, D. Jiang, W. Wang, and K. T. V. Grattan, “Wavelength-switched phase interrogator for EFPI sensors with polarization self-calibrated,” (SPIE, 2017), p. 104642E.
18. P. Fan, W. Yan, P. Lu, W. Zhang, W. Zhang, X. Fu, and J. Zhang, “High sensitivity fiber-optic Michelson interferometric low-frequency acoustic sensor based on a gold diaphragm,” Opt. Express 28(17), 25238 (2020). [CrossRef]
19. Y. Zhang, J. Liu, L. Li, and F. Gao, “Denoising using 3 × 3 coupler demodulation,” Opt. Eng. 60(12), 127113 (2021). [CrossRef]
20. L. Yan, Z. Chen, B. Chen, J. Xie, S. Zhang, Y. Lou, and E. Zhang, “Precision PGC demodulation for homodyne interferometer modulated with a combined sinusoidal and triangular signal,” Opt. Express 26(4), 4818 (2018). [CrossRef]
21. A. D. Kersey and T. A. Berkoff, “Passive laser phase noise suppression technique for fiber interferometers,” (SPIE, 1991), pp. 310–318.
22. A. W. Fitzgibbon, M. Pilu, and R. B. Fisher, “Direct Least Squares Fitting of Ellipses,” IEEE Trans. Pattern Anal. Machine Intell. 21(5), 476–480 (1999). [CrossRef]
23. Y. Wu, H. Tang, and C. Wang, “Study on the performance of passive digital phase demodulation algorithm based on 3×3 coupler and the hardware implementation,” (SPIE, 2021), pp. 116010A–116070A.
24. Y. Wu, C. Li, H. Tang, B. Jia, and C. Wang, “Improved Differential Phase Detecting Optical Fiber Interferometer With a Low-Frequency Compensation Scheme,” IEEE Access 10, 15487–15494 (2022). [CrossRef]
