Real-time self-calibration PGC-Arctan demodulation algorithm in fiber-optic interferometric sensors

Phase noise suppression of optic flexural disk accelerometer by studying the thermal stability of optical fiber ring

As the core sensing elements of ultra-long fiber interferometer, the distributed thermal strain difference of the fiber rings can cause extra noise of the flexural disk, resulting in a penalty of the deterioration accuracy. In this paper, the thermal strain distribution characteristics of the fiber ring are firstly analyzed by the finite element method (FEM), and the distribution result is consistent with that demonstrated by the Rayleigh optical frequency-domain reflectometry (R-OFDR) strain measurement. The interferometer phase noise caused by the distributed strain difference is further studied by constructing a fully symmetric polarization-maintaining fiber-ring Mach-Zehnder interferometer (MZI) with an arm length of over 100 meters. The results show that the distributed thermal strain difference of two fiber rings will cause additional phase fluctuation, which leads to higher low-frequency noise. Therefore, a dual-fiber-ring MZI with matched distributed thermal strains is proposed to suppress the phase noise caused by the thermal strain, and the best suppression is as high as 45.6 dB. This is very important for the research and design of low noise fiber seismometer.

High precision 3 × 3 coupler demodulation algorithm with an additional phase shift judgment module

Ellipse fitting algorithms (EFAs) have been widely used in 3×3 coupler demodulation systems to reduce the requirement for symmetry of the 3×3 couplers. Based on the relative stability of the splitting ratio and phase difference after the establishment of the 3×3 coupler demodulation system, we solve the problem that EFA fails to work when the stimulating signal is small. Depending on the existence of a symmetry point about the origin, an additional phase shift judgment module is used to determine whether the Lissajous figure is larger than π rad. If the elliptical arc exceeds π rad, the EFA is executed. Otherwise, the previous parameters are used to correct the ellipse. Parameters are updated in real time to ensure high precision. The experimental results show that the total harmonic distortion (THD) of the ameliorated algorithm is improved by 1.28% compared to the EFA without the judgment module with a stimulus amplitude of 30 mV. The proposed scheme can effectively improve the dynamic range of the 3×3 coupler demodulation to reach 125.64 dB @ 1 kHz and 1% THD. The algorithm ensures the effective operation of the EFA under small phase shift conditions and improves the accuracy of phase demodulation.

Nonlinear error elimination using the fusion of PGC-DCM, geometric fitting, and Atan algorithms

A phase generated carrier (PGC) demodulation scheme is always accompanied by nonlinear errors. We propose a fusion of PGC differential and cross multiplying (PGC-DCM), geometric fitting, and arctangent (Atan) algorithms for fiber optic interferometric sensors to eliminate nonlinear errors. The output amplitude of the PGC-DCM algorithm is used to judge whether the Lissajous figure of quadrature signals is larger than 1/2 ellipse arc. When the Lissajous figure exceeds 1/2 ellipse arc, the contaminated quadrature signals are corrected by the ellipse correction parameters calculated from the geometric fitting; otherwise, the previous fitting parameters are employed for correction. Geometric fitting is realized by minimizing the Sampson error, and its failure problem under small signals is solved by using the temporary stability of fitting results. Finally, desired signals are extracted from the corrected quadrature signals by the Atan algorithm. Experimental results show that the fusion combines the merits of the three algorithms and expands the application of the geometric fitting in PGC demodulation schemes.

Improved PGC demodulation algorithm for fiber optic interferometric sensors

An improved phase generated carrier arctangent demodulation algorithm based on harmonic mixing and phase quadrature technology (PGC-Arctan-HP) is proposed in this paper, which can eliminate the effects of modulation depth shift, carrier phase delay, and light intensity disturbance (LID) on the demodulation results. The simulation results are consistent with theoretical analysis, and indicate that the PGC-Arctan-HP algorithm can achieve optimal demodulation compared with other demodulation algorithms. The results of interferometric experiments show that the demodulated waveforms of the improved algorithm are relatively stable with an amplitude error of 0.0294%. The total-harmonic-distortion (THD) and the signal-to-noise-and-distortion (SINAD) can reach -60.0286 dB and 59.5388 dB.

Ameliorated 3 × 3 coupler-based demodulation algorithm using iteratively reweighted ellipse specific fitting

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.

Phase noise suppression technique based on an improved reference interferometer scheme

The reference interferometer scheme is an effective noise reduction method, but the optical path length difference (OPD) of the two interferometers must be strictly equal, which limits its application in practical environments. In this paper, an improved reference interferometer demodulation technique without strictly equal OPDs is proposed to suppress phase noise. By introducing a reference interferometer, the phase noise can be removed from the demodulation results. The combination of the differential self-multiplication algorithm and the fitted phase modulation depth calculation formula can evaluate the phase modulation depth of both interferometers in real time and simultaneously eliminate the nonlinear distortion caused by phase modulation depth drift and the effect of different OPDs on the reference interferometer scheme. The experimental results show that the technique can obtain highly stable and accurate demodulation results even if the OPDs of the two reference interferometers are different. The phase modulation depth calculation error is less than 0.57%, the maximum phase noise reduction is 15 dB, the average reduction is 9 dB, the minimum total harmonic distortion is 0.17%, and the SINAD reaches 35.90 dB.

Improved topography measurement with a high dynamic range using phase difference sensing technology

Phase difference sensing technology (PDST) is employed for topography measurement, and two interference structures are proposed to achieve upper-limit adjustment and high resolution in the measurement range: a dual-wavelength system with a single Fabry-Perot (FP) cavity and a single-wavelength system with dual FP cavities. The phase difference between the two interference signals is determined by an elliptic fitting algorithm (EFA), and this change in phase difference is utilized to characterize the step height. Experimental results indicate that the measurement upper-limit can be adjusted to either 410 µm, 187 µm, or 108 µm by varying the wavelength difference in the dual-wavelength system, which gives a measurement error of 2.96%. In contrast, while offering a measurement resolution of 3.47 nm, the single-wavelength system exhibits a measurement error of 5.38%. The proposed method is capable of satisfying the measurement requirements during micro-electromechanical system (MEMS) processing with proficiency.

Direct nonlinear parameter estimation method for a phase generated carrier position sensor

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^-3rad phase measurement uncertainty and over 55 dB Signal to Noise and Distortion Ratio (SINAD) is obtained in experiment.

Improved ellipse-fitting phase demodulation technique to suppress the effect of light source intensity noise in interferometric system

An improved ellipse-fitting algorithm phase demodulation (EFAPD) technique is proposed to reduce the influence of light source intensity noise on a system. In the original EFAPD, the sum of the intensities of coherent light (ICLS) is an important part of the interference signal noise, which makes the demodulation results suffer. The improved EFAPD corrects the ICLS and fringe contrast quantity of the interference signal by an ellipse-fitting algorithm, and then calculates the ICLS based on the structure of pull-cone 3 × 3 coupler, so as to remove it in the algorithm. Experimental results show that the noise of the improved EFAPD system is significantly reduced compared with that of the original EFAPD, with a maximum reduction of 35.57 dB. The improved EFAPD makes up for the deficiency of the original EFAPD in suppressing light source intensity noise, and promotes the application and popularization of EFAPD.

Ameliorated PGC demodulation technique based on the ODR algorithm with insensitivity to phase modulation depth

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.

Self-calibration phase demodulation scheme for stabilization based on an auxiliary reference fiber-optic interferometer and ellipse fitting algorithm

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.

Pretreatment of Ultra-Weak Fiber Bragg Grating Hydrophone Array Based on Cubic Spline Interpolation Using Intensity Compensation

The demodulation algorithm based on 3 × 3 coupler in a fiber-optic hydrophone array has gained extensive attention in recent years. The traditional method uses a circulator to construct the normal path-match interferometry; however, the problem of increasing the asymmetry of the three-way signal to be demodulated is easily overlooked. To provide a solution, we report a pretreatment method for hydrophone array based on 3 × 3 coupler demodulation. We use cubic spline interpolation to perform nonlinear fitting to the reflected pulse train and calculate the peak-to-peak values of the single pulse to determine the light intensity compensation coefficient of the interference signal, so as to demodulate the corrected three-way interference signal. For experimental verification, ultra-weak fiber Bragg gratings (uwFBGs) with reflectivity of -50 dB are applied to construct a hydrophone array with 800 sensors, and a vibratory liquid column method is set up to generate a low-frequency hydroacoustic signal. Compared to the traditional demodulation algorithm based on a 3 × 3 coupler, the pretreatment method can improve the consistency of interference signals. The Lissajous figures show that cubic spline interpolation can improve the accuracy of monopulse peak seeking results by about 1 dB, and intensity compensation can further lead to a much lower noise density level for the interference pulse amplitude-specifically, more than 7 dB at 5~50 Hz-and the signal-to-noise ratio is improved by approximately 10 dB at 10 Hz. The distinct advantages of the proposed pretreatment method make it an excellent candidate for a hydrophone array system based on path-match interferometry.

High-stability PGC demodulation technique with an additional sinusoidal modulation based on an auxiliary reference interferometer and EFA

In the reference interferometer demodulation scheme, it's difficult to guarantee in practice that both interferometers have the same optical path length difference (OPD), which makes the phase modulation depth different in different interferometers with the same laser modulation. The random shift of phase modulation depth also affects the demodulation results. An improved phase-generated carrier (PGC) technique is proposed based on an auxiliary reference interferometer and the ellipse fitting algorithm (EFA). The technique ensures the correct fitting of the EFA for small amplitude signals by introducing a sinusoidal signal as an additional phase modulation. The combination of the reference interferometer and EFA can eliminate the effect of different phase modulation depths of the two interferometers caused by different OPDs, the non-linear distortion caused by phase modulation depth shifts, and improve the accuracy of the demodulation results. The experiment results are consistent with the theoretical analysis, and the method extends the application of the EFA in the reference interferometer phase demodulation technique.

Active linearized PGC demodulation with fusion of PGC-Arctan and PGC-DCM schemes for nonlinear error elimination in SPM interferometer

To eliminate the nonlinear error of phase generated carrier (PGC) demodulation in sinusoidal phase modulating interferometer (SPMI), an active linearized PGC demodulation with fusion of differential-and-cross-multiplying (PGC-DCM) and the arctangent (PGC-Arctan) schemes is proposed. In this method, the periodic integer multiple of π (π-integer phases) of PGC-Arctan without nonlinear error and the corresponding PGC-DCM results recorded at the same time are fused to obtain a calibration coefficient for PGC-DCM demodulation. Combining the accurate π-integer phases of PGC-Arctan and the calibrated fractional phase in the range of π of PGC-DCM, a linearized PGC demodulation result can be achieved, effectively eliminating the nonlinear error caused by drifts of phase demodulation depth (m) and carrier phase delay (θ). The distinct advantage of the proposed method is that it actively and linearly calibrates the fractional result of PGC-DCM without needing to measure or compensate m and θ. Simulation and displacement measurement experiments with different m and inherent arbitrary θ are performed to validate the proposed method. The experimental results show that nonlinear error of the proposed method can be reduced to about 0.1 nm with real-time linearization.

Ultra-low noise phase measurement of fiber optic sensors via weak value amplification

The noise floor is a vital specification that determines the minimum detectable signal in the phase measurement. However, the noise floor in optical phase measurement conducted via conventional optical interferometry tends to approach the intrinsic limit. In this study, a low noise phase measurement of a fiber optic sensor conducted via weak value amplification is experimentally demonstrated. The system has a flat, wideband frequency response from 0.1 Hz to 10 kHz, as well as adequate linearity. The operating band is wider than the present sensor using the same mechanism. In particular, the system noise floor is measured to be -98 dB at 1 Hz and -155 dB at 1 kHz. The results indicate that the minimum detectable signal can reach as low as 5.6 × 10^-6 rad at 1 Hz and 8 × 10^-9 rad at 1 kHz. In addition, it is demonstrated that the noise result of the proposed system is two-order of magnitude lower than that of the typical interferometric fiber optic sensors through the comparison experiment. With the characteristic of low-noise, the system is promising in the field of weak signal detection such as underwater acoustic signal detection, seismic wave detection, and mineral resource exploration.

Improved multi-channel interferometric fiber-optic sensor demodulation based on the Goertzel algorithm

In a multi-channel interferometric fiber-optic sensor system using space-division multiplexing (SDM) and phase-generated-carrier (PGC) demodulation, the phase delay and phase modulation depth fluctuation of each channel will affect the amplitude consistency and harmonic distortion of the demodulation results. In this paper, an improved demodulation scheme based on the Goertzel algorithm is proposed to calculate the multi-channel phase delay and phase modulation depth and to compensate for their fluctuations simultaneously. First, the carrier's 1^st to 6^th harmonic amplitudes in the interference fringe are extracted using the Goertzel algorithm. Then, the phase delay is calculated using the real and imaginary components of the 1^st harmonic amplitude. The phase modulation depth is calculated with a combinatorial operation of the 1^st to 6^th harmonic amplitudes. In addition, a reference channel is introduced to implement phase delay and modulation depth feedback control. The experimental results demonstrate that the improved scheme can effectively suppress the harmonic distortion and improve the amplitude consistency of multi-channel interferometric fiber-optic sensors with low resource consumption.

Large-Dynamic-Range and High-Stability Phase Demodulation Technology for Fiber-Optic Michelson Interferometric Sensors

A large-dynamic-range and high-stability phase demodulation technology for fiber-optic Michelson interferometric sensors is proposed. This technology utilizes two output signals from a 2 × 2 fiber-optic coupler, the interferometric phase difference of which is π. A linear-fitting trigonometric-identity-transformation differential cross-multiplication (LF-TIT-DCM) algorithm is used to interrogate the phase signal from the two output signals from the coupler. The interferometric phase differences from the two output signals from the 2 × 2 fiber-optic couplers with different coupling ratios are all equal to π, which ensures that the LF-TIT-DCM algorithm can be applied perfectly. A fiber-optic Michelson interferometric acoustic sensor is fabricated, and an acoustic signal testing system is built to prove the proposed phase demodulation technology. Experimental results show that excellent linearity is observed from 0.033 rad to 3.2 rad. Moreover, the influence of laser wavelength and optical power is researched, and variation below 0.47 dB is observed at different sound pressure levels (SPLs). Long-term stability over thirty minutes is tested, and fluctuation is less than 0.36 dB. The proposed phase demodulation technology obtains large dynamic range and high stability at rather low cost.

Strain distribution characteristics of sensing fiber and influence on sensitivity of a fiber-optic disk accelerometer

In a fiber-optic disk accelerometer, the strain distribution of sensing fiber is crucial for the improvement of sensitivity. The distribution characteristics of axial and radial strain in the sensing fiber are analyzed by the finite-element method, and the influence of strain distribution on the sensitivity of accelerometer is studied. Sensors with different outer radii of sensing fiber coils are designed and manufactured, and their performance is tested. The resonant frequencies are greater than 200 Hz, and the sensitivity increases as the outer radius of the sensing fiber coil increases. The dynamic range of the sensor with maximum sensitivity is 145.8 dB@100 Hz, and the transverse cross talk is 32.5 dB. Among the sensing fiber strain calculation methods we tested to predict the value of sensitivity, using the strain of sensing fiber directly obtained by the finite-element analysis method shows the smallest error with experimental results (within 7%). It is concluded that in the optical fiber strain disk, the absolute values of axial strain and radial strain of sensing fiber decrease with the increase of disk radius, while the signs of axial strain and radial strain of the ipsilateral sensing fiber are opposite and remain unchanged. The sensitivity can be further improved by optimizing the inner and outer radius of the sensing fiber coils, which is very important for the research and design of high-sensitivity fiber-optic accelerometers.

Optimization of the phase generated carrier demodulation algorithm based on support vector regression

The phase generated carrier (PGC) algorithm is often used in the demodulation of interference signals in optical fiber sensors for its high sensitivity, good linearity, and large dynamic range. However, the PGC demodulation method is often distorted by the amplitude of the interference signal and the depth of modulation. In this study, the support vector regression (SVR) method is used to compensate the distortion of the PGC demodulation schemes. Simulation results showed that the SVR algorithm can effectively reduce the nonlinear error of the PGC demodulation system. The fitting accuracy of the SVR algorithm is 97.5% and greater than 90% in noiseless and noise systems, which is better than the back propagation (BP) neural network algorithm. Also, the SVR-based algorithm can better restore the amplitude with smaller mean square error and good correlation. A vibration monitoring system has been built, and experiment results confirm that the performance of the SVR-based algorithm is better than direct PGC demodulation and BP algorithm with the mean square error of 0.0005 and relevance of 0.94.

Sinusoidal Phase-Modulated Angle Interferometer for Angular Vibration Measurement

Primary angular vibration calibration devices based on laser interferometers play a crucial role in evaluating the dynamic performance of inertial sensing devices. Here, we propose a sinusoidal phase-modulated angle interferometer (SPMAI) to realize angular vibration measurements over a frequency range of 1–1000 Hz, in which the sinusoidal measurement retro-reflector (SMR) and the phase generation carrier (PGC) demodulation algorithm are adopted to track the dynamic angle variation. A comprehensive theoretical analysis is presented to reveal the relationship between demodulation performance of the SPMAI and several factors, such as phase modulation depth, carrier phase delay and sampling frequency. Both the simulated and experimental results demonstrate that the proposed SPMAI can achieve an angular vibration measurement with amplitude of sub-arcsecond under given parameters. Using the proposed SPMAI, the frequency bandwidth of an interferometric fiber-optic gyroscope (IFOG) is successfully determined to be 848 Hz.

Phase-shifted demodulation technique with additional modulation based on a 3 × 3 coupler and EFA for the interrogation of fiber-optic interferometric sensors

A phase-shifted demodulation technique with a 3×3 coupler and ellipse fitting algorithm (EFA) for the interrogation of interferometric sensors is proposed. To reduce the error of the EFA as to measure small phase signals, additional phase modulation is introduced. The additional modulation provides a walk of the operating point along the Lissajous ellipse large enough to permit calculation of the ellipse parameters at every moment. Experimental result shows that this technique demonstrates high accuracy and stability for measuring small phase signals. The setting of this technology expands the application of the EFA in fiber-optic phase demodulation technology.

Distortion-corrected phase demodulation using phase-generated carrier with multitone mixing

We present a novel phase generated carrier (PGC) demodulation technique for homodyne interferometers which is robust to modulation depth variations and source intensity fluctuations. By digitally mixing the waveform with a multitone synthetic function (a linear combination of harmonics of the modulating signal), distortion can become negligible even in presence of large variations of the modulation depth. The technique only requires two mixers and can also provide the DC component of the phase in real time, without needing any previously recorded data or ellipse-fitting algorithms. We validate the technique with simulated waveforms and with experimental data from a wavelength metering experiment using an integrated unbalanced interferometer on-chip, showing that the technique corrects distortion without increasing the noise with respect to the standard PGC technique.

Demonstration of multi-channel fiber optic interrogator based on time-division locking technique in subway intrusion detection

Low noise, good consistency, and long-term stability are critical for a multi-channel fiber optic intrusion detection system. This paper proposes a low-noise 32-channel fiber optic interrogator based on phase generation carrier technique, which emphasizes on the analysis of key parameters related to system consistency and stability. A novel time-division locking technique of the carrier modulation depth and carrier phase delay is proposed. By locking the carrier modulation depth and carrier phase delay simultaneously, the consistency and stability of the multi-channel interferometric interrogator is significantly improved. The results show that in the multi-channel system, the noise level of each channel is below 12.6 ng/√Hz @ 5 Hz, and the noise level differences between each two channels are less than 2.6 dB, the noise level fluctuations of all channels are less than 1.4 dB over 24 hours. A subway intrusion detection is demonstrated in Shenzhen, China. Geological drilling events can be clearly recorded and identified.