Open Access
Add to BibSonomy
Abstract
Phase generated carrier (PGC) is widely applied in interferometric phase estimation for distance, vibration and velocity measurements. However, traditional PGC methods suffer from nonlinear effects, causing limitations to demodulation of signal. Modified PGC methods, such as ellipse fitting algorithm (EFA), resolves these issues, but usually requires additional phase shift. With our proposed method in this paper, only one period of signal and one test point is required to attain accurate depth of phase modulation and phase. We use a photodiode to calibrate light intensity in data acquisition, and develop a Levenburg-Marquadt algorithm to estimate values of PGC parameters. An improved algorithm is also proposed to avoid local optimization based on prior information to ensure measurement stability. Less than 5 × 10−3 rad phase measurement uncertainty and over 55 dB Signal to Noise and Distortion Ratio (SINAD) is obtained in experiment.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, due to its simple configuration and high sensitivity, phase generated carrier (PGC) technology has been widely applied in various fiber optic sensors [1,2]. Traditional PGC demodulation methods include PGC-arctan demodulation and PGC-DCM (differential-and-cross-multiplying) demodulation [3,4]. These two demodulation methods can achieve good demodulation results when the modulation depth C is known and the signal phase delay can be ignored. The modulation depth C is related to the detection distance and the light source, both of which are constantly changing and difficult to calibrate individually. As the modulation frequency of the DFB laser current increases, the effect of signal delay will become increasingly significant. In addition, when using a distributed-feedback (DFB) laser diode (LD) as the light source, the variation of operating current of the DFB LD will lead to additional intensity modulation effects. These three factors result in nonlinear errors in PGC-arctan and PGC-DCM demodulation, which may even cause demodulation failure. Ni et al. proposed an improved PGC demodulation algorithms introducing the ellipse fitting algorithm (EFA) to estimate and correct the nonlinear effect [5,6]. The principle of the EFA is to collect a series of measurement points with different phases, calculate the fundamental and second harmonic components of these signals, and fit the nonlinear parameters using algebraic relationships between them to obtain the closest approximation. Thus, efforts should be made to generate additional phase shifts. Yan et al. proposed a method to obtain the points for EFA algorithm by add an additional triangular wave current modulation in the working current of the DFB LD [7]. Qu et al. proposed another way to generate the fitting points by fine-tuning the distance at the beginning of the measurement process and continuously implementing calibration while measuring [8]. All these EFA-based algorithms require multi-point calculations, which often require many data points of different phases to ensure accuracy. Due to the characteristics of the ellipse fitting algorithm, when the selected fitting points are unevenly distributed on the ellipse, the algorithm will produce larger errors. In addition, due to the need for multi-point fitting calculations, the real-time performance of the EFA is insufficient. Volkov and Plotnikov proposed a PGC phase modulation depth calibration method by analyzing the first four order components of signal harmonic frequency [9]. They used the relationship among the Bessel function to estimate the C value. However, such method suffered from unstable points, which means incorrect results will be obtained when certain special phases are measured. Yan et al. also proposed a PGC signal demodulation method to compensate the C value and phase delay based on the first three order of harmonic frequency components and it also has the disadvandage of unstable points [10]. Fang et al. proposed the J1/J3 method, utilizing the first and third harmonic frequency components [11]. This method is a rough estimate that is difficult to meet the requirements in high-precision applications. This can cause inconvenience in practical measurement and cannot guarantee accuracy in some special cases, such as high-speed displacement or long-term static situations.
To handle these problems, we propose a PGC signal processing method based on nonlinear parameter estimation. By applying the Levenburg-Marquadt (LM) algorithm [12] and selecting an appropriate function model, we can determine all parameters in one signal cycle. This feature that makes it possible to more easily and accurately calculate the phase modulation depth C. As there are seven nonlinear parameters to be determined, which is out of the capability of nonlinear fitting algorithms, an auxiliary probe is added in the setup to calibrate the parameters of light intensity modulation effect. Since calibration of nonlinear parameters does not require multiple test points, it can perform better in high-speed or long-term static measurement applications compared with the EFA based method, greatly improving measurement bandwidth. In addition, this method does not require any restrictions for phase modulation depth, signal delay, etc. The proposed method can calculate the correct value of continuously drifting parameters, so it is suitable for different occasions.
In Section 2, the principle of PGC distance measurement based on DFB LD, the basic principle of LM algorithm and its application in PGC signal processing are introduced. In Section 3, the feasibility of the algorithm is verified through simulation. In Section 4, the actual effect of the algorithm is verified through experiments, and the characteristics of the algorithm are discussed.
2. Principle
2.1 Principle of a PGC position sensor
When fabricating a PGC position sensor, an unbalanced arm Michelson or a Fabry-Perot interferometer with a DFB LD light source is usually adopted because of its ease to integrate and free of fiber-induced noise [13]. By coupling a small high-frequency current input to the operating current of the DFB LD, the wavelength of the DFB laser is sinusoidally modulated around a certain wavelength. Correspondingly, the interference phase of the two beams is also modulated proportionally. The amplitude of the phase modulation depends on the optical path difference (OPD) between the two light beams and the magnitude of the modulation current [14].The schematic diagram of the laser interferometric displacement sensor based on PGC is shown in Fig. 1.
Fig. 1. Unbalanced Michelson interferometer setup with an auxiliary photo diode to calibrate laser intensity modulation.
In this work, a compact Michelson interferometer with an additional photodiode is adopted to compensate for errors caused by changes in light source intensity. The laser emitted from the DFB LD is split into two paths by a 2 × 2 fiber coupler, one for interference measurement and the other for intensity calibration.
For the test configuration as shown in Fig. 1, when no wavelength modulation is performed, the interference signal can be simply expressed as
where I0 is the intensity coefficient, ν represents the visibility of interference fringes, and φm is determined by the difference in optical path length between the center wavelength of the laser and the interference light, i.e. φm = OPD/λ×2π.After adding a small modulation current to the operating current of the DFB LD, the corresponding laser wavelength changes and the interference signal becomes a phase-modulated signal considering the effect of the intensity modulation [15]
2.2 Selection of the fitting algorithm
The feasibility of nonlinear parameter estimation depends on the selection of the function model and optimization algorithm. Assuming a function model $y = f({x_1},{x_2}, \cdots ,{x_n};{\beta _1},{\beta _2}, \cdots ,{\beta _k})$, a set of parameters β is to be determined. According to the principle of least squares, the residual sum of squares (RSS) $R = {\sum\limits_i {({y_i} - {{\hat{y}}_i})} ^2}$ of fitting is minimized. It can be described as follows
Although there are multiple numerical methods to achieve the optimal solution, considering factors such as accuracy, reliability and convergence speed, we select Levenburg-Marquadt (LM) algorithm for parameter estimation [12,16]. LM algorithm is one of the most widely applied nonlinear parameter estimation algorithms, which combines the advantages of Gauss-Newton method [17] and gradient descent algorithm [18]. Although the Gauss-Newton method can find a faster direction for parameter optimization, it often fails because it is based on the first order expansion of Taylor series. The first order approximation is only effective within a small range. The performance of gradient descent algorithm is more stable, but it is easy to fall into local optimum and the optimization speed is slow. LM algorithm combines the two algorithms with a universal equation, and the optimization direction and iteration step are determined by a damping coefficients. The following briefly introduces the process of the LM algorithm.
Step 1: Select a set of initial fitting parameters b0, an initial optimization factor Λ, and an scaling factor ν > 1.
Step 2: Calculate optimization trial vector.
where A is the Jacobi matrix, δ is the trial vector, g is the target vector and I represents identity matrix.Step 3: Select the trial vector according to the optimization effect, and use the scaling factor ν to adjust Λ. Repeat the optimization process until convergence.
In order to show the superiority of the LM algorithm [16], the parameters in Eq. (2) were estimated using these two algorithms with the same initial values.
Figure 2 shows that in an optimization process, due to the flexibility of the LM algorithm, a faster convergence can be obtained, and it avoids falling into local optimization, while the gradient descent algorithm has a high probability of falling into a nearby local optimal solution. Because of the randomness of the initial parameter selection, the distance from the optimal point is far, the Gauss-Newton method cannot obtain reasonable results. In summary, the LM algorithm is a more suitable algorithm for this application.
2.3 Applying LM algorithm to PGC signal processing
An unreasonable selection of initial parameters may cause iteration to fall into a local minimum. There are five parameters that need to be calculated, namely intensity coefficient I0, fringe visibility v, phase modulation depth C, signal phase delay φd, and measured phase φm. It is reasonable to choose the initial values to be close to the real ones, so initial value of I0 can be set as the average value of the signal, and the initial value of v can be set as half of the peak-to-valley (PV) value of the signal divided by the average value of the signal. The initial value of C is estimated by the modulation current, the OPD of the interferometer arm. Because φd and φm have strong randomness, it is difficult to determine their initial values directly. Another choice is to use the previous frame's data as the initial values.
The most important parameter in the LM algorithm is the scaling factor ν. It controls the change of the optimization step and direction. Selecting a reasonable ν value can improve the algorithm performance, make the iteration enter the convergence state faster, and achieve higher accuracy. The value of ν is set between 5 and 10.
When the scaling factor ν and the initial parameters are set to specific values, the optimization may get stuck in a local optima in certain φm. For measurement systems, occurrence of local optima may bring about measurement errors.
The PGC signal can be written in polynomial form by Jacobi-Angel expansion as follows:
It can be seen that the parameters with higher nonlinearity are C and φm. I0 and v are scaling factors, and phase delay φd is a horizontal shift of the signal waveform along the coordinate axis. In such case, the situation of falling into a local optimal solution is likely to occur. An optimization process stuck in a local optimum is shown in Fig. 3.
Fig. 3. (a) Change of fitting RSS when local optimum state occurs. (b) Change of phase error when local optimum state occurs.
As shown in Fig. 4, when falling into local optimization, the fitted signal differs greatly from the normal signal. Therefore, restrictions can be added during the iterative optimization process to avoid this situation. Regarding the nonlinear parameters of the signal, especially I0 and v, these parameters can be approximately obtained by analyzing the characteristics of the signal directly. According to the initial values, convergence restrictions are set for I0 and v.
Taking these factors into consideration, several functionalities can be added to the original LM algorithm to avoid optimization falling into a local minimum. A function is added to judge whether the iteration reach a stable state. After convergence, the fitting parameters are checked if they meet the requirements. If not, change the initial parameter setting and re-iterate.
2.4 Consideration of phase modulation depth
Although the LM algorithm can obtain signal parameters of any value by fitting, inappropriate selection of C may lead to poor performance. From Eq. (5). it can be seen that when C decreases, the higher-order harmonic components will decay. With certain values of φm, the function is act like a sinoidal function. In this case, C and ν are difficult to separate, and the process of reaching the global optimal solution is very tortuous, requiring a long iteration to find the correct solution (exceed 100 loop times). As shown in Fig. 5, when C = 1.5 and the number of iterations is insufficient 50 times, and the fitted function is similar to the target but with a large error in the high-frequency component.
Fig. 5. Fitting signal of an insufficient 50 times iteration when C = 1.5 and comparison with target signal.
In practical use, a phase modulation depth C above 2.5 is suitable, and algorithm performance is better when C is greater than π.
3. Simulation
3.1 Effectiveness verification for the algorithm
A set of signals with known parameters were generated through software, and the parameters were solved using the LM algorithm to evaluate its effectiveness. The PGC carrier frequency was set to 1 kHz, and the sampling frequency was set to 100 kHz. A one-period signal was generated for testing. The value of I0 was set to 1, m was set to 0.05, v was set to 0.5, and C was randomly set between 1.5 and 4, with the phase parameter also being randomly set. The initial value of the λ for the LM algorithm was set to 10, and the scaling factor ν was set to 7. In simulation verification, the LM algorithm converged within 50 iterations. As shown in Fig. 6, the RSS and phase errors decreased with the number of iterations during a typical data fitting process.
Fig. 6. Optimization progress (C = 3). (a) RSS decreasing when iteration goes on. (b) Change of phase error when iteration goes on.
If the difference between the measured phase initial value and the nominal value exceeds and approaches π, the optimization direction may reverse, resulting in a negative value for parameter φm when the phase measurement value differs from the nominal value by π. If the result shows a negative value for v, it should be converted to a positive value by taking its absolute value and subtracting π from φm. As shown in Fig. 5, after 50 iterations of optimization, the RSS is less than 10−4 and the phase error is less than 5 × 10−4 rad.
To verify the improved ability to avoid local optimum, the φm and C are changed, and parameter estimation is performed using both the original and improved algorithms. As shown in Fig. 7, the original algorithm may get stuck in a local optimum state at certain points, making the calculated parameters completely wrong. The improved algorithm can effectively avoid this situation and perform correct parameter calculation over the entire 2π phase range, with the phase error being less than 5 × 10−4 rad in no more than 50 iterations.
In addition to directly calculating the measured phase, another meaningful function of this algorithm is to calculate the value of C. As shown in Fig. 8, as C slowly increases from 2.5 to 4.0, the algorithm can give correct results with an absolute error not exceeding 0.01 rad. Moreover, when C exceeds $\pi $, the absolute error is less than 10−5 rad.
According to the simulation results, the LM algorithm can accurately calculate various parameters of PGC signal and achieve stable convergence. Excellent performance is observed when C is large.
3.2 Effect of intensity calibration
As shown in Fig. 9, when the companion light modulation coefficient m = 0.01, directly using Eq. (2). to estimate all of the seven parameters results in an error of up to 0.02 rad. This is expected because there is a significant coupling among the seven parameters, making it difficult for the algorithm to find the globally optimal path.
It can be seen that the additional light intensity calibration photodiode has improved the measurement accuracy by nearly 100 times, making the algorithm applicable in practical use.
3.3 Accuracy analysis
In actual cases, algorithms are often accompanied by various types of noise. In this simulation, software-generated additive white Gaussian noise is used as the simulated noise source. Figure 10 shows the changes of the maximum RSS and maximum phase error by simulating the signal with signal-to-noise ratio (SNR) ranging from 50 dB to 200 dB. When the SNR is greater than 70 dB, the algorithm can obtain accurate results, with a phase error not exceeding 5 × 10−4 rad. When the SNR of the simulated signal is less than 70 dB, the accuracy of the algorithm begins to be affected but is still within an acceptable range, which shows that the method has good anti-noise ability.
Fig. 10. (a) Fitting RSS with different amount of noise. (b) phase error with different amount of noise.
When selecting Gaussian white noise with different SNR, the RSS and phase errors also vary accordingly. In addition, due to the different choices of algorithms and signal parameters, both the RSS and phase errors will vary, lacking a unified standard for analyzing the accuracy of phase estimation. It is found that there is a significant positive correlation between RSS and phase errors, so the RSS can be used as a reference to estimate the accuracy of phase measurement. We simulated the corresponding RSS and phase errors with different background noises and C values, and put all these points into one figure. Figure 11 shows the corresponding relationship between a large number of RSS and phase errors (approximately 3 million pairs), and based on the trend shown in the figure, the corresponding relationship between RSS and the maximum phase error can be estimated. According to statistical methods, we can select the upper contour of the points as the characteristic curve for measuring errors. For simplicity, this contour is chosen to be linear.
When analyzing the computing accuracy of a parameter estimate, the fitting RSS can be directly incorporated for estimation. Actually, most phase errors are concentrated within a range less than 5 × 10−4 rad. The error estimation method adopted here is a strict one so the obtained error is likely to be greater than the actual error.
3.4 Signal to Noise and Distortion Ratio (SINAD) analysis
The SINAD characterizes the performance of distance detectors for detecting dynamic targets [19]. The previous analysis was based on the assumption of a fixed measurement phase, however, when the detection distance changes at a certain frequency, in the process of data collection, the signal changes caused by phase changes affect the PGC signal. Therefore, the actual measured signal will be distorted compared to the ideal signal. The SINAD represents any other frequency components compared with the expected ones. The value of SINAD is related to multiple factors, such as the frequency, amplitude, and other various noises of phase vibrations, and the characteristics of algorithms. The SINAD is an indicator that comprehensively considers SNR and dynamic signal distortions. We simulate the sinusoidal phase vibration and calculate its SINAD through simulation. Among them, C = 3.5, the SNR of added white noise is 100 dB, the phase vibration frequency ranges from 1 Hz to 250 Hz, and the phase amplitude ranges from 0.001 rad to 0.5 rad. The calculation results are shown in Fig. 12.
The overall SINAD can be seen in the figure to be above 40 dB, and in most cases, SINAD is above 55 dB. Even with vibration frequency as high as 250 Hz and amplitude up to 0.5 rad, the SINAD can reach above 60 dB. This indicates the strong dynamic measurement capability of the algorithm, which can meet the accuracy requirements of most dynamic measurement situations.
4. Experiment and discussion
4.1 Experiment
To verify the effectiveness of the algorithm, we constructed a signal processing system based on a field programmable gate array (FPGA) and a digital signal processor (DSP) in this paper. One Xilinx Artix-7 FPGA chip is used for the FPGA part, and TI TMS320C6655 chip is used for the DSP part. The TMS320C6655 uses a unique hardware design to accelerate floating-point operations, making it particularly suitable for implementing complex algorithms that use a large amount of floating-point data. The FPGA part is used to drive ADC and collect data, communicate with DSP, and send data to the host computer. Data communication between the FPGA end and the DSP end is achieved through the SRIO interface. The DFB laser diode used in the experiment is the EP1550-NLW-B-100 LIDAR model made by Eblana Photonics, and the laser driver is the CLD1015 made by Thorlab. The DC bias point of the laser's working current is set to 180 mA. Consistent with the simulation, the carrier frequency of PGC in this experiment is chosen as 1kHz, and the ADC sampling frequency is 100kHz. The optical path difference of the two interference paths of the unbalanced interferometer used in this experiment is approximately 200 mm.The experimental set up is briefly shown in Fig. 13.
The ability of this algorithm to fit actual signals and estimate parameters was validated. As shown in Fig. 14, the matching degree between the measured signal and the fitting signal is very high, with the two signals almost completely overlapping each other except for some small spikes in the actual signal. It can be seen that the difference between the fitting signal and the actual signal is attributed to the mismatch caused by noise, and the SNR of the signal is estimated to be about 88.7 dB.
Fig. 14. Practical sampled data and the fitting signal with different C value. (a) C = 1.74. (b) C = 2.36. (c) C = 2.88. (d) C = 3.44.
In order to apply the error analysis in Section 3.3 to practical situations, it is necessary to normalize the fitting RSS based on the actual signal strength and fringe contrast. The normalization coefficient k = (I0v / I0'v’)2.After normalization, the fitting RSS is 1.87 × 10−4, and according to the analysis in Section 3.3, the phase measurement error is within 5 × 10−3 rad.
In this experiment, an AC voltage signal was generated by a signal generator. The signal was coupled onto a DC working current using the laser driver. In order to verify the algorithm's ability to calculate C, the signal amplitude was varied and the laser wavelength modulation amplitude was changed, resulting in a change in C. Figure 15 shows this correlation.
The resistance of the AC-coupled port of the driver CLD1015 is 150 mA/V. When the input voltage amplitude is 12 mV, the corresponding current modulation amplitude is 2.4 mA and wavelength modulation amplitude is 4.8pm. It can be seen that within this input signal amplitude range increasing gradually from 5 mV to 12 mV, the depth of phase modulation C and the amplitude of modulation current maintain a good linear relationship. The ability of this algorithm to accurately calculate the value of C is once again verified.
Finally, Fig. 16 shows that the frequency component in one SINAD test as the phase vibration frequency is 20 Hz. The reference phase vibration data used to calculate SINAD is generated by a closed-loop controlled displacement device. During the measurement process, the value of C varied around 3.26, and the SINAD reached 58.41 dB.
In summary, our adopted algorithm and function model can accurately fit the actual measured signal, and has the ability to accurately calculate the phase and phase modulation depth C value of the measurement.
4.2 Discussion
There is a significant difference in principle between the PGC signal processing based on fitting algorithm and the PGC demodulation algorithm based on frequency domain signal processing. Firstly, according to the PGC signal expansion, it can be seen that the PGC signal has rich high-frequency components. Traditional PGC demodulation algorithms usually only use several low-frequency components, and then obtain all parameters by additional phase shift information. On the contrary, the method based on nonlinear fitting algorithm can utilize all information in a signal, and each sampling point can help improve the result. Therefore, correct results can be obtained with a shorter signal period, which can be validated by both simulation and experimental results. The fitting-based algorithm also has the advantage of good universality, with no necessity of generating additional phase shifts, and almost no restrictions on fringe visibility and phase modulation depth, which can be applied for measurements under various conditions such as long-time stationary objects and high-speed moving objects. In addition, this method can be used as a method for calibrating nonlinear parameters, combined with traditional PGC demodulation algorithms to achieve larger data throughput and make PGC measurement technology more flexible and versatile.
When higher carrier frequencies are needed for real-time measurement, the computation can be reduced by appropriately reducing the number of sampled points per cycle. When C is less than 4, the high-frequency components of PGC signal are usually negligible above 8th harmonic. To ensure accuracy, the sampling frequency is set as 20 times the carrier frequency. According to the Nyquist theorem, the frequency components of carrier frequency up to the 10th harmonic are retained, and the algorithm performance will not result in a large decay. In this case, one iteration process needs about 1000 multiplication operations. Through simulation and experimental verification, only 25 iterations are needed for optimization to basically converge, and the phase error is within 0.01 rad, which is sufficient for general applications. Based on the above analysis, each calculation requires approximately 25,000 multiplication operations. TMS320C6655 performs a single-precision multiplication operation in one clock cycle and a double-precision multiplication operation in four clock cycles. The clock frequency of TMS320C6655 is 1 GHz. If double-precision calculation is adopted, processing one frame of data takes about 100µs. Therefore, if the carrier frequency approaches or exceeds 40kHz, real-time data processing may not be possible with existing systems alone, and more hardware resources may need to be applied for parallel processing. Since one cycle of the signal can output one phase information, the highest output frequency of this algorithm can reach 40kHz, which is considerable.
5. Conclusion
In this paper, a PGC signal processing method based on the LM algorithm is presented, and the effectiveness of the algorithm is verified through simulation and experimentation. A compact Michelson interferometer system with an optical power calibration photodetector was set up to generate the measurement signal, and the signal processing algorithm was implemented using an FPGA and a high-performance DSP hardware. By processing a period of the signal, this algorithm can obtain the precise values of the phase modulation depth C and the measured phase. The algorithm is modified to be capable of stable convergence and avoiding local optimization. Our proposed method has the advantages of simple configuration, good universality, and high accuracy.
Funding
National Natural Science Foundation of China (12274156); Science, Technology and Innovation Commission of Shenzhen Municipality (JCYJ20210324115812035).
Disclosures
The authors declare no conflicts of interest.
Data availability
The 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. T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Sigel, J. H. Cole, S. C. Rashleigh, and R. G. Priest, “Optical fiber sensor technology,” IEEE Trans. Microwave Theory Techn. 30(4), 472–511 (1982). [CrossRef]
2. K. Thurner, F. P. Quacquarelli, P. F. Braun, C. Dal Savio, and K. Karrai, “Fiber-based distance sensing interferometry,” Appl. Opt. 54(10), 3051–3063 (2015). [CrossRef]
3. A. Dandridge, B. T. Alan Tveten, and G. G. Thomas, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE Trans. Microwave Theory Techn. 30(10), 1635–1641 (1982). [CrossRef]
4. T. R. Christian, A. F. Philip, and H. H. Brian, “Real-time analog and digital demodulator for interferometric fiber optic sensors,” Smart Structures and Materials 1994: Smart Sensing, Processing, and Instrumentation. Vol. 2191 (1994).
5. C. Ni, M. Zhang, Y. Zhu, C. Hu, S. Ding, and Z. Jia, “Sinusoidal phase-modulating interferometer with ellipse fitting and a correction method,” Appl. Opt. 56(13), 3895–3899 (2017). [CrossRef]
6. Q. Ge, J. Zhu, G. Zhang, Y. Cui, J. Xiao, H. Wang, X. Wu, and B. Yu, “High Stability PGC-EFA-DCM Demodulation Algorithm Integrated With a PID Module,” J. Lightwave Technol. 40(24), 7961–7968 (2022). [CrossRef]
7. 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), 4818–4831 (2018). [CrossRef]
8. 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), 23593–23609 (2019). [CrossRef]
9. A. V. Volkov and M. Y. Plotnikov, “Phase modulation depth evaluation and correction technique for the PGC demodulation scheme in fiber-optic interferometric sensors,” IEEE Sensors J. 17(13), 4143–4150 (2017). [CrossRef]
10. L. Yan, Y. Zhang, and J. Xie, “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. G. Fang, T. Xu, Y. Li, and F. Li, “Research of PZT modulation factor with high precision and stability based on J1/J3 method,” Asia Communications Photonics Conf.12–15 (2013).
12. D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11(2), 431–441 (1963). [CrossRef]
13. K. Meiners-Hagen, R. Schödel, F. Pollinger, and A. Abou-Zeid, “Multi-wavelength interferometry for length measurements using diode lasers,” Measurement science review 9(1), 16 (2009). [CrossRef]
14. A. Dandridge and L. Goldberg, “Current-induced frequency modulation in diode lasers,” Electron. Lett. 18(7), 302–304 (1982). [CrossRef]
15. Y. Dong, P. Hu, H. Fu, H. Yang, R. Yang, and J. Tan, “Long range dynamic displacement: precision PGC with sub-nanometer resolution in an LWSM interferometer,” Photonics Res. 10(1), 59–67 (2022). [CrossRef]
16. H. P. Gavin, “The Levenberg-Marquardt algorithm for nonlinear least squares curve-fitting problems,” Department of Civil and Environmental Engineering, Duke University, 19 (2019).
17. H. O. Hartley, “The modified Gauss-Newton method for the fitting of non-linear regression functions by least squares,” Technometrics 3(2), 269–280 (1961). [CrossRef]
18. S. Ruder, “An overview of gradient descent optimization algorithms,” arXiv, arXiv:1609.04747 (2016). [CrossRef]
19. W. Kester, “Understand SINAD, ENOB, SNR, THD, THD+ N, and SFDR so you don’t get lost in the noise floor.” MT-003 Tutorial, (2009), www.analog.com/static/importedfiles/tutorials/MT-003.pdf.
