Phase Demodulation Method Research Articles

Angular Momentum Superposition Beams‐Based Interferometer

AbstractOrbital‐angular‐momentum (OAM) beams‐based interferometer has been found various applications in high‐precision metrology. However, most of its applications reported to date are limited to 1D ones, and the simultaneous measurements for multidimensional displacement have rarely been explored. Here an angular momentum superposition beams‐based interferometer is first proposed and demonstrated, in which the phases‐changes induced by rotational and linear displacements, are independently carried by an angular momentum superposition (SOAMS) beam containing different spin and orbital angular momenta, and thus can be simultaneously detected. In accordance with the proposed interferometry, a new phase‐demodulation method is experimentally demonstrated, which is operated in the domain of the OAM complex spectrum and thus enables accurate yet high efficient extraction of the phase‐change information from the captured OAM interferograms. As typical results, rotational and linear displacements are simultaneously and accurately measured with accuracies of 2.89° and 16.04 nm, respectively. The proposed interferometry and the corresponding phase demodulation method may open a new way to realize the OAM beams‐based multidimensional metrology.

A Review of Optical Interferometry for High-Precision Length Measurement.

Optical interferometry has emerged as a cornerstone technology for high-precision length measurement, offering unparalleled accuracy in various scientific and industrial applications. This review provides a comprehensive overview of the latest advancements in optical interferometry, with a focus on grating and laser interferometries. For grating interferometry, systems configurations ranging from single-degree- to multi-degree-of-freedom are introduced. For laser interferometry, different measurement methods are presented and compared according to their respective characteristics, including homodyne, heterodyne, white light interferometry, etc. With the rise of the optical frequency comb, its unique spectral properties have greatly expanded the length measurement capabilities of laser interferometry, achieving an unprecedented leap in both measurement range and accuracy. With regard to discussion on enhancement of measurement precision, special attention is given to periodic nonlinear errors and phase demodulation methods. This review offers insights into current challenges and potential future directions for improving interferometric measurement systems, and also emphasizes the role of innovative technologies in advancing precision metrology technology.

Accelerating the phase demodulation process based on a parallel Hilbert transform with overlapping computation for a laser heterodyne Doppler vibrometer.

A parallel Hilbert transform arctangent phase demodulation (PHT-ATAN) method based on overlapping computation is proposed for phase demodulation of laser heterodyne Doppler vibrometers. The method suppresses the end point effects by utilizing overlapping computation and data concatenation and accelerates phase demodulation through parallel processing. Simulation and experimental results demonstrate that when the algorithm's parallelism is ≥4, the computation speed of this method increases by over 100% compared to traditional methods, while maintaining the signal-to-noise ratio and accuracy of the phase demodulation results. Therefore, this method offers potential advantages in accelerating the phase demodulation process and is highly suitable for large bandwidth, embedded, and real-time sensing applications.

Phase demodulation method for stable long-haul frequency transmission based on the Michelson interferometer.

In this Letter, we present a fiber-optic radio frequency (RF) transmission scheme based on phase modulation with an interferometric detection structure. A self-developed Michelson interferometer (MI) is used to demodulate the frequency signal via an electrically controlled optical shifter. The two complementary outputs from the interferometer are detected using a balanced detector, which suppresses the common-mode noise of the fiber link. The structure is tested in the laboratory using a frequency transfer system over a 560 km fiber link. Experimental results show that a stable 2.4 GHz frequency transmission with a fractional frequency instability of 3.9 × 10-14 at 1 s and 6.2 × 10-17 at 10,000 s is achieved. Compared with the frequency transmission system based on intensity modulation and direct detection, the frequency instability is improved from 7.3 × 10-14 to 3.9 × 10-14 at 1 s. We believe that the proposed method will be useful for the construction of time-frequency synchronous fiber networks.

Quasi-distributed acoustic sensing based on optical path difference demodulation for high-frequency and large-amplitude using an optical fiber interferometer array.

The phase-sensitive optical time domain reflectometry scheme has attracted great research interest in distributed acoustic sensing (DAS) but suffers from a trade-off between the dynamic range and signal frequency. In this paper, an optical path difference (OPD) demodulation method is applied to an interferometer array composed of an ultra-weak fiber Bragg grating (UWFBG) array and an imbalanced Michelson interferometer to solve this problem. As far as we know, this is the first time that OPD demodulation has been applied to DAS. The UWFBG array is interrogated by a frequency-modulated pulse, and the acoustic signal sensed by the fiber between any two adjacent UWFBGs can be retrieved by demodulating the variation of residual OPD of the interferometer formed by them. The proposed method is analyzed theoretically and validated experimentally; compared with the phase demodulation method, a 100-times boost in bandwidth is achieved for a signal with an amplitude of 0.4 µε. Results also show that the proposed method offers increasing signal-to-noise ratios as the frequency increases.

Phase Demodulation under Ultra-low sampling rate in heterodyne coherent [formula omitted]-OTDR

Heterodyne coherent phase-sensitive optical time-domain reflectometry(Φ-OTDR) requires a higher data sampling rate to demodulate phase in the “distance” direction effectively. Aiming at the problem of large amount of demodulated data, we propose a phase demodulation method that can effectively recover disturbance information from ultra-low sampling data. This method demodulates the phase of the two-dimensional reconstructed signal in the “time” direction. Thus, the influence of spectrum aliasing in the “distance” direction caused by undersampling is avoided. The undersampling rate is not limited by the spectrum aliasing effect of the detected signal when using this method to demodulate phase. Therefore, accurate phase information can be retrieved under ultra-low sampling rates, significantly reducing the data for phase demodulation in heterodyne coherent Φ-OTDR. In the experiment, three sampling rates (100 MSa/s, 10 MSa/s, 1 MSa/s) within the restricted area of the traditional undersampling demodulation method were selected to acquire data, and the proposed method can also accurately demodulate phase information.

Three-Frame Random Phase-Shifting Algorithm Based on VU Decomposition Method and Ellipse Fitting

Abstract To enhance the precision and robustness of the three-frame phase shift technique, a new three-frame random phase-shifting fringe pattern phase demodulation method that combines VU decomposition and ellipse fitting techniques without the need for pre-filtering is proposed. The proposed method first performs VU decomposition on the interferograms to establish two orthogonal components of the fringe pattern. Then, the ellipse fitting method is used to obtain the relevant ellipse coefficients, thereby achieving precise phase demodulation. This method does not require an accurate phase-shifting process, thereby relaxing the stringent requirements on the phase shifter. Compared with traditional algorithms, the proposed method performs better under conditions of non-uniform background intensity and modulation amplitude. Numerical simulation experiments demonstrate that the proposed method maintains good stability across 20 repeated tests. Within the SNR range of 30 to 50 dB, the RMSE of the suggested approach is approximately 0.006 rad. This proves the effectiveness of the algorithm and provides a new approach for the development of three-frame phase-shifting technology.

Frequency-modulated continuous-wave laser interferometry measurement method based on cross-correlation

Frequency-modulated continuous-wave (FMCW) laser interferometry technology holds significant potential for applications in the fields of ultraprecision manufacturing and high-precision sensing. This paper proposes a novel approach among current phase demodulation methods is based on cross-correlation to address the challenge of this technology. On the basis of nonlinear correction of a distributed feedback laser, the intercepted beat frequency signal was first preprocessed with Z-score signal normalization and a smoothing filter. Subsequently, the interference beat signal was subjected to processing using a correlation method to derive the correlation function. Finally, the phase difference between adjacent beat signals was determined by pinpointing the maximum value of the cross-correlation function, enabling accurate displacement demodulation. Experimental validation was performed by constructing an FMCW laser interferometric displacement measurement system. The results indicated that the standard deviation of the displacement error for the cross-correlation method was 2.41 nm during static measurements. Compared to conventional maximum-point method, the static measurement error of the cross-correlation method has been reduced by 1.43 times. In dynamic measurements in the 500 μm range, The measurement error of the cross-correlation method has been reduced by 6.04 times, avoiding the dynamic measurement positioning problem of conventional feature point demodulation methods and making the measurement results more accurate. This advancement holds substantial practical value in the realm of phase demodulation in laser interferometry.

Twist compensated, high accuracy and dynamic fiber optic shape sensing based on phase demodulation in optical frequency domain reflectometry

We present a twist compensated, high accuracy and dynamic fiber optic shape sensing based on phase demodulation in Optical Frequency Domain Reflectometry (OFDR) by using multiple single core fiber based sensor (MFS). A dynamic strain sensing is realized by tracking the optical phase in OFDR and combining with the phase de-hopping filtering algorithm, and the sensing spatial resolution reaches 45 μm. In addition, in order to eliminate the influence of external twist and fluctuation of inherent spin on the shape reconstruction results, we propose an external twist compensation method and inherent spin rate calibration method, respectively. Finally, we use a circle segment method to reconstruct a 3D shape of MFS. The experimental results show that the reconstruction accuracies by the proposed external twist compensation and inherent spin rate calibration methods increase over 18 times and 20 times than those without these two methods, respectively. At the same time, comparing with the traditional cross-correlation-based method, we find that the proposed phase demodulation method has a similar reconstruction accuracy, the maximum reconstruction error is 0.61 %, whereas the shape reconstruction speed is improved by nearly 10 times. This is of great significance for the application of FOSS, which can be used for dynamic shape sensing such as intelligent soft robots, surgical robot and etc.

Phase demodulation method of high line density grating interferometric signal based on wavelet transform

The increasing line density of the reference grating and the accelerating miniaturization of ultra-precision displacement measurement technology necessitate more stable interferometric signal processing methods for high line density gratings, particularly in low signal-to-noise ratio scenarios. This paper presents a phase demodulation method for dynamic interferometric signals for high line density gratings. The Morlet wavelet transform is utilized to obtain the instantaneous frequency of the interferometric signal, integration of which yields the relative displacement, while adding adjacent relative displacements without gaps provides the absolute displacement during dynamic motion of the grating. In simulations with a signal-to-noise ratio ranging from 40 to 70 dB, the proposed method demonstrates greater robustness compared to the traditional method. By establishing a platform for repeated experiments and comparing it with traditional methods, it was found that the maximum deviation between calculation results obtained using this method and traditional methods is 0.8 nm, further confirming its potential application.

SE-FSCNet: full-scale connection network for single-shot phase demodulation

The accuracy of phase demodulation has significant impact on the accuracy of fringe projection 3D measurement. Currently, researches based on deep learning methods for extracting wrapped phase mostly use U-Net as the subject of network. The connection method between its hierarchies has certain shortcomings in global information transmission, which hinders the improvement of wrapped phase prediction accuracy. We propose a single-shot phase demodulation method for fringe projection based on a novel full-scale connection network SE-FSCNet. The encoder and decoder of the SE-FSCNet have the same number of hierarchies but are not completely symmetrical. At the decoder a full-scale connection method and feature fusion module are designed so that SE-FSCNet has better abilities of feature transmission and utilization compared with U-Net. A channel attention module based on squeeze and excitation is also introduced to assign appropriate weights to features with different scales, which has been proved by the ablation study. The experiments conducted on the test set have demonstrated that the SE-FSCNet can achieve higher precision than the traditional Fourier transform method and the U-Net in phase demodulation.

High quality phase demodulation method for direct detection Φ-OTDR.

This article discovers that the excessive correlation between the selected temporal sensing sequences will lead to phase demodulation failure in the demodulation process of direct detection Φ-OTDR in certain duration, which reduces the quality of demodulated phase. Besides, we also first discover a phase polarity flipping phenomenon in the demodulation process, which will introduce additional errors and further degrade the quality of demodulated phase. In order to obtain the real phase change caused by external intrusion, a high-quality phase demodulation strategy with multi-position compensation based on leveraging the information redundancy between each Rayleigh back-scattered temporal sequence is proposed. The optimal demodulation position is selected by calculating the cross-correlation between temporal sensing sequences. The phase demodulation failure is then compensated by phase demodulation results from multiple positions. At the same time, the phase polarity change is also determined and corrected. The experimental results show that this strategy can effectively suppress the waveform distortion and improve the signal-to-noise ratio of the demodulated phase. This scheme can effectively improve the demodulation effect and detection performance of direct detection Φ-OTDR and may promote its application.

Radar Sensing of Displacement Motions With High Robustness Against Additive Noise

Doppler radar uses phase demodulation to extract the target displacement motion contained in the phase of the echo signal. The demodulation performance is vulnerable to the additive Gauss noise. Especially when the signal-to-noise ratio (SNR) is extremely low in the scenarios of long-range or subtle displacement motion sensing, it may be impossible to extract the correct phase information. In this article, a phase demodulation method with high robustness against additive noise is proposed for accurate displacement motion reconstruction. Mathematical derivations are performed to analyze the noise effect on phase demodulation, and it shows that the influence of phase noise is much smaller than the additive noise. Based on this finding, the key idea of the proposed method is to transform the additive noise of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$I$</tex-math> </inline-formula> / <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$</tex-math> </inline-formula> signals into phase noise. The simulation results validate the noise analysis and verify the high robustness of the proposed method against additive noise. Furthermore, experiments were carried out to evaluate the proposed method under different radial distances and displacement amplitudes. The experimental results show that the proposed method enables accurate motion sensing even in low-SNR scenarios. In our experiments, we observe that the relative root-mean-square error (RRMSE) of the reconstructed displacement motion can be greatly reduced with the proposed method.

Distributed dynamic strain measurement with a direct detection scheme by using a three-step-phase-shifted double pulse in a UWFBG array.

We demonstrate a stable homodyne phase demodulation method with a double pulse based on an ultra-weak fiber Bragg grating (UWFBG) array. The technique divides one of the probe pulses into three sections and introduces successive 2π/3 phase differences into each section. By using a simple direct detection scheme, it can achieve distributed and quantitative vibration measurement along the UWFBG array. Compared to the traditional homodyne demodulation technique, the proposed technique is more stable and easier to accomplish. Moreover, the reflected light from the UWFBGs can provide a signal that is modulated uniformly by the dynamic strain and multiple results for averaging, resulting in a higher signal-to-noise ratio (SNR). We experimentally demonstrate the technique's effectiveness by monitoring different vibrations. The SNR for measuring a generic 100 Hz, 0.08 rad vibration in a 3 km UWFBG array with a reflectivity of -40 to -45 dB is estimated to be ∼44.92 dB.

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.

Photothermal Gas Detection With a Dithered Low-Finesse Fiber-Optic Fabry-Pérot Interferometer

Pump-probe photothermal spectroscopy has been recently proposed for sensitive gas detection. The probe laser wavelength was usually locked at the quadrature point of an interferometer to provide a linear output of the phase signal. However, this detection scheme brings unwanted fluctuations to the interferometer. Here we report a novel phase demodulation method of the probe laser for photothermal spectroscopy to enhance the stability. We demonstrate a proof-of-concept experiment by performing photothermal detection of methane (CH <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> ) inside a hollow-core fiber, which is sandwiched by two pieces of solid-core fibers to form a low-finesse fiber-optic Fabry-Pérot interferometer (FPI). Besides the wavelength modulation of the 1.65- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">μ</i> m pump laser, we employed a dither to modulate the injection current of the probe diode laser, which was frequency-locked to the FP cavity resonance. The photothermal signal was sensitively measured from the error signal of the feedback loop by demodulating the FPI output at the dither frequency. A normalized noise equivalent absorption (NNEA) coefficient of 7.5 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−9</sup> cm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−1</sup> WHz <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−1/2</sup> was achieved in this work. A comparison to the conventional quadrature point locking technique was performed to reveal the remarkable sensor stability with immunity to optical disturbance.

An Improved Two-Wavelength Phase Demodulation Method for Nanosecond Pulsed Electro-Optical Sensors

Bias drift can always be a difficult problem in the application of electro-optical (EO) sensors based on intensity demodulation method, which may lead to the output jitter and distortion of the waveform. The bias drift problem can be avoided using a two-wavelength phase demodulation scheme, however the system noise restricts its use to low amplitude measurements. In this paper, the existing two-wavelength phase demodulation method is improved to overcome the limitation of low amplitude measurement for pulsed signals. The improved method can effectively avoid the bias drift problem compared with the intensity demodulation method and reduce the measurement limit by more than one order of magnitude compared with the phase demodulation method simultaneously. Based on the asymmetric two-wavelength EO modulation structure, we obtain the transfer function and its explicit demodulation formula without the bias term. Particularly, by adjusting the bias difference of the two lasers in the EO sensor to π/2, the modulated signal is simply proportional to the root mean square of the 2-channel outputs. To support and illustrate our method, a simulation, and a verification experiment are carried out and show that the demodulated signals have good consistency with each other.

Orbital-angular-momentum beams-based Fizeau interferometer using the advanced azimuthal-phase-demodulation method

A stably acquiring and accurately demodulating interferogram is crucial for the interferometer to achieve ultra-high precision and sensitivity measurements. In this study, a robust orbital angular momentum (OAM) beams-based Fizeau interferometer is proposed and experimentally demonstrated, which is more compact and stable than the OAM interferometers with other structures due to the common optical path characteristic. In accordance with such an interferometric scheme, a phase-demodulation method operated in the domain of the OAM complex spectrum is also proposed and demonstrated in this study. In contrast to other phase-demodulation techniques, the proposed phase-demodulation technique neither requires phase shifters or phase modulators nor brings spectral leakage, which provides a robust alternative enabling to accurately and quickly extract the phase from the OAM interferogram. As a proof-of-concept of the proposed scheme, tiny displacements ranging from 0 to 800 nm were measured. The proposed OAM beams-based Fizeau interferometer and the corresponding azimuthal phase demodulation method may provide a feasible way for exploring further applications of the OAM-based interferometer in metrology.

Analysis of Telescope Wavefront Aberration and Optical Path Stability in Space Gravitational Wave Detection

Space-based gravitational wave detection programs, such as the Laser Interferometer Space Antenna (LISA) or Taiji program, obtain gravitational wave signals by measuring the change in the distance between three satellites by laser. The telescope is an important part of the measurement system, and its function is to transmit and receive laser signals. Due to changes in the space environment, the telescope will inevitably introduce additional dynamic aberrations, which will bring optical path errors to the inversion of gravitational wave signals. Taking LISA as an example, to achieve pm-level measurement accuracy at the detection frequency of 0.1 mHz–1 Hz, the stability requirements of the telescope are less than 1 pm/Hz1/2. This paper theoretically deduces the aberration types that affect the telescope’s stability and conducts simulation analysis according to the actual phase demodulation method, which verifies the theory’s correctness. In addition, using this theory, it can be concluded that under the condition that the total size of the telescope aberration is determined to be stable, reducing the ratio of rotationally symmetric aberrations such as “spherical aberration” and “defocusing” among common aberrations can significantly improve the stability of the telescope. The conclusion guides the optical system design of LISA or Taiji.

Femtosecond laser written ultra-weak Fabry-Perot array for distributed absolute temperature profile sensing with high spatial resolution.

In this paper, high spatial-resolution distributed temperature sensing has been realized based on a femtosecond laser written ultra-weak Fabry-Perot Array (FPA). 50 identical Fabry-Perot cavities are fabricated in a 10 mm long optical fiber by femtosecond laser point-by-point written technology, and the corresponding spatial resolution is as high as 200 µm. Besides, by employing the total phase demodulation method, the optical path lengths (OPLs) in the ultra-weak FPA are successively demodulated based on the Rayleigh backscattering signal recorded by an optical frequency domain reflectometry (OFDR), and therefore the absolute temperature values instead of the relative ones can be obtained. When compared with the conventional single mode fiber-based OFDR, the proposed ultra-weak FPA presents both higher spatial resolution and lower temperature sensing uncertainty (0.25 °C) benefiting from the periodically enhanced Rayleigh backscattering. Furthermore, the experiments also confirm that the ultra-weak FPA can be applied for absolute temperature field profile sensing with large temperature gradient, which is particularly suitable for high-resolution temperature measurement of miniature devices.