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Abstract
For the optical fiber sensing system using phase generated carrier (PGC) technology, it is very important to eliminate the nonlinear effect of phase modulation depth (C) fluctuation on the demodulation results in the actual environment. In this paper, an ameliorated phase generated carrier demodulation technique is presented to calculate the C value and suppress its nonlinear influence on the demodulation results. The value of C is calculated out by the fundamental and third harmonic components with the equation fitted by the orthogonal distance regression algorithm. Then the Bessel recursive formula is used to convert the coefficients of each order of Bessel function contained in demodulation result into C values. Finally, the coefficients in demodulation result are removed by the calculated C values. In the experiment, when the C ranges from 1.0 rad to 3.5 rad, the minimum total harmonic distortion and maximum phase amplitude fluctuation of the ameliorated algorithm are 0.09% and 3.58%, which are far superior to the demodulation results of the traditional arctangent algorithm. The experimental results demonstrate that the proposed method can effectively eliminate the error caused by the fluctuation of the C value, which provides a reference for signal processing in practical applications of fiber-optic interferometric sensors.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Fiber-optic interferometric sensors (FOISs), due to their high sensitivity, strong anti-electromagnetic interference ability, and long-distance sensing capability, are widely used in hydroacoustic, oil pipeline, building structure, coal mine safety monitoring, and other fields [1–4]. The demodulation methods used by the FOISs directly affect the demodulation performance of the system. The commonly used demodulation methods mainly include heterodyne demodulation [5,6] and homodyne demodulation. The homodyne demodulation mainly includes 3 × 3 fiber coupler demodulation [7] and phase generated carrier (PGC) demodulation technology [8]. Owing to its advantages of good linearity, high accuracy of phase measurement, and high sensitivity [9,10], PGC demodulation is widely used in FOISs systems. The traditional arctangent demodulation algorithm (PGC-ATAN) can overcome the influence of light intensity disturbance (LID), intensity noise, and light source low frequency drift, but it requires a specific phase modulation depth (2.63 rad) to work normally [11–13]. However, for multi-channel FOISs systems, due to the influence of optical path difference and temperature, it is difficult to ensure the consistency and stability of the phase modulation depth of each channel [14]. Since the traditional PGC-ATAN algorithm cannot achieve satisfactory demodulation results in the actual working environment, the optimization and improvement of its demodulation method have become a popular research topic.
To overcome the effect of the phase modulation depth (C) fluctuation, researchers have proposed many ameliorated PGC demodulation schemes. Jun He et al. proposed a PGC demodulation algorithm based on arctangent function and differential self-multiplication (DSM) [15], which solved the effect of C value fluctuation. But the result of the differential operation will inevitably be overwhelmed by noise when the phase change rate is too low [16]. Zhiyu Qu et al. and Changbo Hou et al. introduced the ellipse fitting algorithm (EFA) and Kalman filter into PGC-ATAN respectively, successfully suppressing the influence of C value and improving the demodulation performance of the system. Nevertheless, these two improved algorithms cannot demodulate normally when the signal is small [17,18]. Anton V. Volkov et al. use four harmonic components to evaluate and correct C value [19]. L Yan et al. mixed the interference signal with the fundamental frequency, second frequency, and third frequency to enhance the system stability [20]. G Fang et al. proposed an approach to calculate and eliminate C values by J1(C)/J3(C) [21]. Both algorithms can accurately calculate the C value and eliminate it from the demodulation results, but the process of calculating the C value is complex, requires a lot of calculations and the results may be unstable.
This paper proposes an ameliorated demodulation technique (PGC-ODR-ATAN), which combines the nonlinear curve fitting orthogonal distance regression (ODR) algorithm and PGC to simultaneously calculate and compensate the C value. The method has the characteristics of high calculation accuracy. and the advantage of low resource consumption, so it is especially suitable for multi-channel FOISs systems. The accuracy of the improved algorithm in calculating C value and the effectiveness of eliminating C value are theoretically analyzed and experimentally verified as follows.
2. Theory and principles
2.1 Process of the traditional PGC demodulation algorithm
For interferometers using high frequency cosine carrier for phase modulation, the expression of the interference signal generated is
where A is the DC component proportional to the input optical power at the photoelectric detector. B is related to the interference signal visibility. C is the phase modulation depth, ω0 is the carrier frequency, φ(t) is the signal to be measured. The interference signal V(t) is multiplied by the first and second cosine harmonic components of the carrier signal. Then, a pair of in-phase and quadrature components are obtained through low pass filtering, which can be expressed as where J1(C) and J2(C) are the first order and second order Bessel functions of C, respectively. Then, the demodulated result ψ(t) of the traditional PGC-ATAN is obtained byIt is obvious that the demodulation result contains the coefficient J1(C)/J2(C) determined by the value of C. When C≠2.63 rad, this coefficient will lead to the performance of the demodulation system degradation or even failure.
2.2 Process of the ameliorated algorithm
Therefore, we propose an ameliorated PGC demodulation algorithm to calculate and compensate the C value. The schematic is shown in Fig. 1.
Fig. 1. The schematic of the ameliorated PGC demodulation technique combining the ODR algorithm and ATAN. (LPF: low-pass-filter, ODR: orthogonal distance regression, ATAN: arctangent algorithm).
The interference signal and the third cosine harmonic components of the carrier signal are mixed, and then low pass filtered
Divide Eqs. (5) and (2) then negative the result to get the following expression
In order to obtain the C value from K value, the ODR algorithm is introduced. ODR is an optimization algorithm, which is very suitable for nonlinear curve fitting [22,23]. The ODR algorithm uses the minimum sum of squared residuals of the orthogonal distance as the criterion for curve fitting, while considering the errors in the independent and dependent variables. In geometric sense, it is more accurate to use ODR to fit the curve. The basic implementation process of the ODR algorithm is shown in the following equation
The relationship between the actual data points and the fitting curve is shown in Fig. 2. The ODR algorithm is used to fit the conversion relationship Eq. (8) between the K value and the C value. Through this relationship, the C value of the system can be obtained from the K value
The following equation can be established by using the recursive relation J3(C)+J1(C) = 4J2(C)/C of the Bessel function [19]
Substituting the calculated C into Eq. (9), the demodulated phase signal of ameliorated technique can be obtained by
Equation (10) indicates that the C value is removed from the demodulation result of the ameliorated algorithm through calculation and compensation. Hence, the nonlinear effect caused by the fluctuation of C value is eliminated in PGC-ODR-ATAN.
3. Simulation
In this section, we will compare the result of C value calculation, dynamic range, linearity, and other system performance with other implemented methods through simulation. The frequency response of PGC-ODR-ATAN was also analyzed. A simulated interference fringe V(t) was performed using the following settings, and Gaussian white noise was added to it to simulate A/D quantization noise. The sampling frequency fs was set to 400 kHz. The carrier frequency ω0 was set to 20 kHz, consistent with the experiment in the following section. All calculation processes are done in LabVIEW (2015) software on the same computer.
3.1 Comparison of system performance
Setting the amplitude of the analog carrier signal gives the system a theoretical C value of 2.63 rad. The C value calculated by Eq. (8) is then compared with the results of the three-component method [20] and the four-component method [19] proposed by previous researchers. The results in Table 1 show that the algorithm proposed in this paper has the fastest speed in calculating the value of C and the smallest error between the calculated value and the theoretical value. To show the stability of the C-value calculation of the three methods, the C-value was calculated 50 times consecutively, and the results are shown in Fig. 3. The results in Table 1 and Fig. 3 demonstrate that the proposed method in this paper has significantly improved accuracy, stability and speed compared with the previous two calculation methods.
Table 1. Comparison of different C calculate methods in simulation
The other system performance of PGC-ODR-ATAN was then tested by varying the analog signal. The results in Table 2 show that the minimum detectable signal, dynamic range, and linearity of the four algorithms are close to each other. However, the C value calculation results of the three-component and four-component methods often have distortion points, leading to unstable system performance. And the PGC-ATAN deteriorates the system performance when the C value deviates from the optimal operating point.
Table 2. Comparison of system performance in simulation
The total harmonic distortion (THD) results of the different PGC demodulation methods are shown in Fig. 4. As the C-value increases from 1 rad to 3.5 rad, the traditional PGC-ATAN method will reach the lowest at C = 2.63 rad, but it has a wide range of THD fluctuations. The other three improved algorithms all maintain low THD levels at different C values.
The performance comparison of the four algorithms shows that the demodulation results of the PGC-ODR-ATAN algorithm are superior to the conventional algorithm and the other two improved algorithms.
3.2 Frequency response of PGC-ODR-ATAN
The frequency response of the PGC-ODR-ATAN algorithm was tested by adjusting the frequency of the simulated signal to be measured. Figure 5 shows the dynamic range variation of the PGC-ODR-ATAN algorithm from 20 to 1500 Hz. The dynamic range of the algorithm can reach 129dB@100 Hz, and although it decreases with increasing frequency, the overall dynamic range is basically above 100 dB.
According to the simulation results, the main drawback of the traditional algorithm is the limitation of C-value, while the C-value calculation method proposed in this paper is more superior to the previous methods, and the results of C-value elimination are very effective.
4. Experimental setup and results
4.1 Experimental setup and C value calculation process
To verify the accuracy and effectiveness of the ameliorated algorithm proposed in this paper, an optical fiber sensing system based on the Michelson interferometer is constructed as shown in Fig. 6. Firstly, the semiconductor laser (RIO) with a central wavelength of 1550 nm is applied as the light source and internal modulated by a cosine signal with a frequency of 20 kHz. The arm length difference of Michelson interferometer is 5 m. Then, the signal to be measured with an amplitude of 300 mv and frequency of 300 Hz is applied to the interferometer through a piezoelectric transducer (PZT). Finally, the interference light enters the photodetector (PD, THORLABS PDB450C) from the circulator and is collected by the data acquisition card (PXIE 5170R), and then enters the computer for processing.
To demonstrate the accuracy of the ODR algorithm fitting formula in a real environment, we constantly adjust the amplitude of the internal modulation signal through the signal generator to change the C value of the system. Subsequently, the K value is obtained by Eq. (6) and substituted into Eq. (8) to calculate the C value. The results are shown in Fig. 7. When the amplitude of the internal modulated signal varies between 170 mv and 580 mv, the calculated C value is approximately linear with the amplitude of the modulated signal, which proves the accuracy of the formula in calculating the C value.
Figures 8(a) and (b) show the data point density plots of C values calculated by Eq. (8) for C = 3.0 rad and C = 3.5 rad, respectively. The sampling rate of the data acquisition card used in this paper is 1 M/s, and we substitute the data points collected within 0-0.05 s into Eq. (8) to calculate the C value. As can be seen from the figure, most of the calculated C-value data points are concentrated around the theoretical value, and the calculated results do not fluctuate with time. The above results demonstrate the stability of the C-values calculated of Eq. (8).
Fig. 8. Data point density plots of the calculated C under different C values: (a) C = 3.0.rad, (b) C = 3.5 rad.
In the next four parts, by comparing the THD, spectra, demodulation phase amplitude, and linearity of the two algorithms, the influence of C value fluctuation on the demodulation results and the effectiveness of the ameliorated technique to eliminate C value are demonstrated.
4.2 THD
When C fluctuates between 1.0 rad and 3.5 rad, the THD curves of the two algorithms are shown in Fig. 9. The THD of the traditional PGC-ATAN algorithm reaches the optimal value of 0.5% when C = 2.63 rad, but when C deviates from 2.63 rad, the demodulation result will show serious harmonic distortion. Compared with the traditional PGC-ATAN algorithm, the minimum THD of the PGC-ODR-ATAN is 0.09% and achieves 0.6% on average, which is far superior to the traditional algorithm. It can be seen that the ameliorated algorithm greatly suppresses the serious harmonic distortion caused by the deviation of C value from the optimal operating point on the demodulation results.
Fig. 9. THD of the traditional algorithm and ameliorated algorithm when C drifts between 1.0 rad and 3.5 rad.
4.3 SNR
The influence of the fluctuation of C value on signal to noise ratio (SNR) of the demodulation system is displayed through the spectra of the two algorithms under different C values. Set the system C values to 1.5 rad, 2.63 rad and 3.5 rad respectively and draw the spectra as Fig. 10. It can be seen from Fig. 10(b) when the system is at the optimal operating point (C = 2.63 rad), the SNR of the traditional algorithm and the ameliorated algorithm is basically the same. However, when the C value deviates from the optimal operating point, the traditional algorithm not only increases the harmonic component but also decreases the SNR of the system, as shown in Figs. 10(a) and (c). The SNR of the ameliorated algorithm maintains around 83.3 dB, which enhances the stability of the system.
Fig. 10. Spectra of the two PGC demodulation technique at different C values: (a) 1.5 rad; (b) 2.63 rad; (c) 3.5 rad.
4.4 Amplitude accuracy and deviate
The demodulation phase amplitude precision of the demodulation algorithm under different C values is also a significant indicator of algorithm performance. Figures 11(a) and (b) show the phase amplitude and deviation of the same signal to be measured demodulated by two algorithms under different C values. It can be seen from Fig. 11(a) that when C = 2.63 rad, the amplitude of the demodulation results of the two algorithms is basically the same (0.86 rad), which also demonstrates the accuracy of the demodulation results of the ameliorated algorithm. The deviation between the demodulation amplitude of other points and 0.86 rad is plotted in Fig. 11(b). The amplitude fluctuation of traditional algorithm demodulation results is very serious, with the maximum fluctuation exceeding 0.36 rad and the maximum deviation exceeding 22.16%, which is difficult to adapt to the needs of the actual system. The ameliorated algorithm realizes the precise phase amplitude demodulation, with the maximum amplitude fluctuation being less than 0.03 rad and the average deviation being less than 3.58%.
Fig. 11. Phase amplitude and deviate demodulated by two PGC demodulation techniques at different C values: (a) phase amplitude; (b) phase deviation from standard value 0.86 rad.
4.5 Linearity and demodulation error
Set the C value of the system to 3.0 rad, the linearity and demodulation error between the demodulation results of the two algorithms and the test signal is measured. Through the signal generator, the amplitude of the signal to be measured is constantly changed from 100 mv to 1000 mv, and the measured result is plotted in Fig. 12. From Figs. 12(a) and (b), it can be clearly seen that the demodulation amplitude of the ameliorated PGC-ODR-ATAN algorithm basically coincides with the linear fitting curve, the demodulation error is mostly less than 1%, and the linearity exceeds 99.99%. However, the PGC-ATAN algorithm's linearity is only 99.61%, and the demodulation error fluctuates sharply, losing part of the linearity.
Fig. 12. Response of two algorithms using different amplitude signals when C = 3.0 rad: (a) the linearity of the two PGC demodulation algorithms; (b) error between demodulation result and linear fitting.
5. Conclusion
As an ameliorated algorithm, this paper proposes a simple method to calculate the value of C by applying the formula fitted by the ODR algorithm. The calculated C value and the recursive relationship of the Bessel function are used to eliminate the C value from the demodulation result to reduce the impact of C value fluctuation and enhance the stability of the system. The experimental results show that the accuracy of the ameliorated algorithm in calculating C value exceeds 99.8%. When the C value ranges from 1.0 rad to 3.5 rad, the ameliorated algorithm effectively suppresses its nonlinear influence on the demodulation results and achieves an average THD of 0.6%, precise phase amplitude with a deviation less than 3.58%. The ameliorated algorithm provides a reference for the FOISs system in the actual environment to eliminate the influence of C value fluctuation.
Funding
The University Synergy Innovation Program of Anhui Province (GXXT-2020-050, GXXT-2020-052).
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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