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Fourier Optics and Signal Processing|502 Article(s)
Pulse Wave Denoising Based on Improved Complementary Ensemble Empirical Mode Decomposition
Yong Chen, Zhimin Yao, Huanlin Liu, Junpeng Liao, Li Xu, and Yanqing Feng
ObjectiveThe cardiovascular health status of the human body can be reflected through pulse waves. Important physiological parameters such as heart rate, blood pressure, and the degree of vascular sclerosis can be obtained through the analysis of these waves. The sensor predominantly used for pulse measurement is the photoelectric sensor, which is capable of detecting pulses at various measurement positions, thus making it extensively used in wearable sports equipment for heart rate detection. However, during the process of measuring pulse waves with photoelectric sensors, there are often various noise interferences such as motion artifacts, power interference, and respiratory effects. Moreover, this measurement method is primarily invasive, which can make people uncomfortable. Therefore, it is necessary to select appropriate sensors to avoid discomfort to the human body during the measurement process and denoise the collected signals.MethodsWe designed a pulse wave signal acquisition platform based on fiber Bragg grating (FBG) sensors. The platform was composed of FBG sensors embedded in nylon wristbands. Initially, the FBG wristband was secured at the radial artery of the left hand to gather pulse wave signals for demodulation. The collected pulse wave signals were subject to baseline drift. Hence, integrated empirical mode decomposition (EMD) and cubic spline interpolation were used for detrending prior to denoising. Subsequently, the amplitude of Gaussian white noise added to the complementary ensemble EMD (CEMMD) was optimized using particle swarm optimization (PSO) algorithm. The CEEMD algorithm decomposed the pulse wave signal into a series of intrinsic mode function (IMF) components. An improved wavelet threshold function was then applied to process these IMF components. The correlation coefficient between each IMF component and the original pulse wave signal was calculated, and this coefficient was used to determine the effectiveness of each component. Finally, all effective signals were reconstructed to obtain a smooth pulse wave signal.Results and DiscussionsTo validate the performance of the method proposed in this study, simulation experiments are conducted using three comparative algorithms. The denoising performance is evaluated using signal noise ratio (SNR) and root-mean-square error (RMSE). Gaussian white noise with an SNR ranging from 5 to 25 dB is added to the simulation signal. The denoising performance is also verified on actual collected pulse wave signals. The simulation results (Table 1 and Table 2) show that even when 5 dB noise is added, the SNR after denoising can still reach 15.785 dB, and RMSE can be reduced to 1.251. When 25 dB noise is added, the SNR after denoising is 31.959 dB, and RMSE is 0.215. Even if the SNR is low, compared with other methods, the algorithm proposed in this study performs better on these two evaluation indicators and has better denoising performance. The results of determining the amplitude of Gaussian white noise (Fig. 4) intuitively display that when the amplitude of Gaussian white noise added in CEEMD is 0.35, the average mutual information of IMF components is the lowest. This indicates that the denoising effect is the best at this time. The actual experimental results are shown in Fig. 9. The signal obtained after denoising by the proposed algorithm is smoother; the amplitude is not distorted, and it effectively removes spikes and high-frequency noise in the signal. This is because the PSO algorithm optimizes the amplitude of white noise added to CEEMD, overcoming problems such as modal aliasing, endpoint effects, and new harmonic components introduced by inappropriate Gaussian white noise in the decomposition process of CEEMD. Using correlation coefficients to select valid and invalid signals successfully removes most invalid signals (Table 5). In general, the proposed algorithm can better remove noise in signals than other algorithms.ConclusionsWe propose a method for collecting pulse wave signals using FBG sensors. By considering the various noise interferences in the pulse wave signal, a joint denoising algorithm of PSO-CEEMD-IWT is proposed. Different amplitudes of white noise are added to both the simulation signal and the actual signal. We determine 0.35 as the optimal amplitude of white noise added to CEEMD, which further suppresses the modal aliasing phenomenon, compared with the amplitude selected based on experience. The average mutual information obtained by the method in this paper is lower than that obtained by selecting the white noise amplitude according to experience. The results show that the SNR, RMSE, and other indicators obtained by the proposed algorithm are the best; there is no waveform and amplitude distortion, and the denoised signal is smoother, which proves that the performance of the pulse wave denoising proposed in this paper is more outstanding. The signal has a higher degree of restoration to the pulse wave signal, which is of great significance for later combination with feature extraction and objectification of pulse diagnosis. We also propose a feasible way to obtain high-quality pulse waves. ObjectiveThe cardiovascular health status of the human body can be reflected through pulse waves. Important physiological parameters such as heart rate, blood pressure, and the degree of vascular sclerosis can be obtained through the analysis of these waves. The sensor predominantly used for pulse measurement is the photoelectric sensor, which is capable of detecting pulses at various measurement positions, thus making it extensively used in wearable sports equipment for heart rate detection. However, during the process of measuring pulse waves with photoelectric sensors, there are often various noise interferences such as motion artifacts, power interference, and respiratory effects. Moreover, this measurement method is primarily invasive, which can make people uncomfortable. Therefore, it is necessary to select appropriate sensors to avoid discomfort to the human body during the measurement process and denoise the collected signals.MethodsWe designed a pulse wave signal acquisition platform based on fiber Bragg grating (FBG) sensors. The platform was composed of FBG sensors embedded in nylon wristbands. Initially, the FBG wristband was secured at the radial artery of the left hand to gather pulse wave signals for demodulation. The collected pulse wave signals were subject to baseline drift. Hence, integrated empirical mode decomposition (EMD) and cubic spline interpolation were used for detrending prior to denoising. Subsequently, the amplitude of Gaussian white noise added to the complementary ensemble EMD (CEMMD) was optimized using particle swarm optimization (PSO) algorithm. The CEEMD algorithm decomposed the pulse wave signal into a series of intrinsic mode function (IMF) components. An improved wavelet threshold function was then applied to process these IMF components. The correlation coefficient between each IMF component and the original pulse wave signal was calculated, and this coefficient was used to determine the effectiveness of each component. Finally, all effective signals were reconstructed to obtain a smooth pulse wave signal.Results and DiscussionsTo validate the performance of the method proposed in this study, simulation experiments are conducted using three comparative algorithms. The denoising performance is evaluated using signal noise ratio (SNR) and root-mean-square error (RMSE). Gaussian white noise with an SNR ranging from 5 to 25 dB is added to the simulation signal. The denoising performance is also verified on actual collected pulse wave signals. The simulation results (Table 1 and Table 2) show that even when 5 dB noise is added, the SNR after denoising can still reach 15.785 dB, and RMSE can be reduced to 1.251. When 25 dB noise is added, the SNR after denoising is 31.959 dB, and RMSE is 0.215. Even if the SNR is low, compared with other methods, the algorithm proposed in this study performs better on these two evaluation indicators and has better denoising performance. The results of determining the amplitude of Gaussian white noise (Fig. 4) intuitively display that when the amplitude of Gaussian white noise added in CEEMD is 0.35, the average mutual information of IMF components is the lowest. This indicates that the denoising effect is the best at this time. The actual experimental results are shown in Fig. 9. The signal obtained after denoising by the proposed algorithm is smoother; the amplitude is not distorted, and it effectively removes spikes and high-frequency noise in the signal. This is because the PSO algorithm optimizes the amplitude of white noise added to CEEMD, overcoming problems such as modal aliasing, endpoint effects, and new harmonic components introduced by inappropriate Gaussian white noise in the decomposition process of CEEMD. Using correlation coefficients to select valid and invalid signals successfully removes most invalid signals (Table 5). In general, the proposed algorithm can better remove noise in signals than other algorithms.ConclusionsWe propose a method for collecting pulse wave signals using FBG sensors. By considering the various noise interferences in the pulse wave signal, a joint denoising algorithm of PSO-CEEMD-IWT is proposed. Different amplitudes of white noise are added to both the simulation signal and the actual signal. We determine 0.35 as the optimal amplitude of white noise added to CEEMD, which further suppresses the modal aliasing phenomenon, compared with the amplitude selected based on experience. The average mutual information obtained by the method in this paper is lower than that obtained by selecting the white noise amplitude according to experience. The results show that the SNR, RMSE, and other indicators obtained by the proposed algorithm are the best; there is no waveform and amplitude distortion, and the denoised signal is smoother, which proves that the performance of the pulse wave denoising proposed in this paper is more outstanding. The signal has a higher degree of restoration to the pulse wave signal, which is of great significance for later combination with feature extraction and objectification of pulse diagnosis. We also propose a feasible way to obtain high-quality pulse waves.
Acta Optica Sinica
- Publication Date: Apr. 10, 2024
- Vol. 44, Issue 7, 0707001 (2024)
Chirp Noise Analysis in Laser Doppler Vibration Measurement and Its Suppression Methods
Yahao Wang, Yangyi Shen, Xinxin Kong, and Wenxi Zhang
ObjectiveBased on the physical principle of the heterodyne laser Doppler vibration measurement process, we analyze the measurement process of high-frequency and low-speed movement in the presence of low-frequency and high-speed background movement. In the process, the measurement noise generated by the presence of stray light exhibits chirp characteristics, with the effects and patterns of chirp noise explained. In response to the chirp noise, we propose a derivative preprocessing method for demodulation. The theoretical analysis shows the method exerts a significant effect on suppressing chirp noise, which is verified by simulations and experiments. Meanwhile, we set a heterodyne laser Doppler vibration measurement system with stray light and measure the target vibration. The normal method and derivative preprocessing method are adopted respectively for demodulation. The experimental results verify the existence of chirp noise and the effectiveness of the derivative preprocessing method in suppressing chirp noise, which decreases the chirp noise power by about 81.8%. The method can effectively reduce the influence of stray light on vibration measurement.Heterodyne laser Doppler vibration measurement technology is a widely adopted non-contact and non-destructive movement measurement method, with the advantages of fast response speed and high resolution. It has a strong detection ability for single frequency movement and can quickly identify the characteristic frequency of target movement. However, in the presence of low-frequency high-speed background movement, the measurement of high-frequency low-speed movement is severely affected by chirp noise, which is caused by stray light and closely related to low-frequency high-speed background movement. The chirp noise can seriously affect the measurement of high-frequency low-speed movement, with errors even reaching tens of times larger than those of real movement. There is a lack of research on the principle of chirp noise caused by stray light and suppression methods of chirp noise. We deeply analyze the principle of chirp noise caused by stray light and propose a novel demodulation method called the derivative preprocessing method (DPM). This demodulation method is easy to implement and exhibits a good effect for suppressing chirp noise. This demodulation method is expected to provide a reliable noise suppression method for measuring high-frequency low-speed movement in the presence of low-frequency high-speed background movement. This plays a significant role in analyzing high-frequency vibration modes of precision devices in some special measurement scenarios, such as in the presence of background movement.MethodsOur study consists of theoretical analysis, simulation, and experimental verification. Firstly, the working principle and demodulation method of the heterodyne laser Doppler vibration measurement system, which is in the presence of stray light, are deeply analyzed. According to the analysis, the stray light would generate chirp noise in the process of normal demodulation method (ARCTAN). Based on the generation and characteristics of the chirp noise, a new demodulation method DPM is proposed. Then, the influence of chirp noise on the measurement of target movement velocity and the effect of DPM on chirp noise suppression are simulated. In the simulation, a low-frequency sinusoidal movement is utilized as the background movement, while a high-frequency sinusoidal movement is employed as the target movement. In the simulation, the background movement generates corresponding chirp noise to affect demodulation results severely. The normal demodulation method DPM is leveraged to restore the target movement by demodulating the overall movement and performing high-pass filtering. Finally, a heterodyne laser Doppler vibration measurement experiment is conducted to utilize a piezoelectric ceramic plate fixed on the pendulum device. In the experiment, the piezoelectric ceramic plate vibrates at a single and high frequency, which is regarded as the target movement, and the pendulum's movement is considered as the background movement. According to the experiment results, the existence of chirp noise is verified, and the DPM suppresses the chirp noise too.Results and DiscussionsIn the process of the normal demodulation method (ARCTAN), the cause of chirp noise is the combination of stray light and background movement. The frequency of chirp noise changes in real time with the background movement speed, and specifically, it is proportional to the absolute speed of background movement and inversely proportional to the laser wavelength (Formula 11). When the background movement is low-frequency high-speed movement and the target movement is high-frequency low-speed movement, the chirp noise will seriously affect the measurement of high-frequency part movement [Fig. 2(b)]. Compared with the normal demodulation method, DPM can effectively restore target movement [Fig. 2(c)], but will generate erroneous velocity spikes in the positions that are near zero speed locations. Since adopting the normal demodulation method's results to partly replace the demodulation results of the DPM at the corresponding positions (near zero speed locations), the velocity spikes can be suppressed, and the demodulation results approach target movement more closely. In the experiment, the background movement of the pendulum indeed generates corresponding chirp noise [Figs. 4(a) and (b)], and DPM can effectively suppress the chirp noise [Fig. 4(c)]. DPM has a significant suppression effect on chirp noise, which reduces the power of chirp noise by about 81.8% at the peak of the chirp noise (Fig. 5).ConclusionsThe principle of heterodyne laser Doppler vibration measurement is deeply analyzed. It is pointed out that chirp noise is generated due to stray light and background movement. The frequency of the chirp noise changes in real time with the background movement speed, which is proportional to the absolute background movement speed and inversely proportional to the laser wavelength. Simulations and experiments have confirmed the existence of chirp noise and its frequency variation pattern. A novel demodulation method DPM has been proposed for this type of chirp noise. Simulations and experiments prove that DPM can effectively suppress chirp noise. Above the target movement frequency, the chirp noise power can be reduced by about 81.8%. ObjectiveBased on the physical principle of the heterodyne laser Doppler vibration measurement process, we analyze the measurement process of high-frequency and low-speed movement in the presence of low-frequency and high-speed background movement. In the process, the measurement noise generated by the presence of stray light exhibits chirp characteristics, with the effects and patterns of chirp noise explained. In response to the chirp noise, we propose a derivative preprocessing method for demodulation. The theoretical analysis shows the method exerts a significant effect on suppressing chirp noise, which is verified by simulations and experiments. Meanwhile, we set a heterodyne laser Doppler vibration measurement system with stray light and measure the target vibration. The normal method and derivative preprocessing method are adopted respectively for demodulation. The experimental results verify the existence of chirp noise and the effectiveness of the derivative preprocessing method in suppressing chirp noise, which decreases the chirp noise power by about 81.8%. The method can effectively reduce the influence of stray light on vibration measurement.Heterodyne laser Doppler vibration measurement technology is a widely adopted non-contact and non-destructive movement measurement method, with the advantages of fast response speed and high resolution. It has a strong detection ability for single frequency movement and can quickly identify the characteristic frequency of target movement. However, in the presence of low-frequency high-speed background movement, the measurement of high-frequency low-speed movement is severely affected by chirp noise, which is caused by stray light and closely related to low-frequency high-speed background movement. The chirp noise can seriously affect the measurement of high-frequency low-speed movement, with errors even reaching tens of times larger than those of real movement. There is a lack of research on the principle of chirp noise caused by stray light and suppression methods of chirp noise. We deeply analyze the principle of chirp noise caused by stray light and propose a novel demodulation method called the derivative preprocessing method (DPM). This demodulation method is easy to implement and exhibits a good effect for suppressing chirp noise. This demodulation method is expected to provide a reliable noise suppression method for measuring high-frequency low-speed movement in the presence of low-frequency high-speed background movement. This plays a significant role in analyzing high-frequency vibration modes of precision devices in some special measurement scenarios, such as in the presence of background movement.MethodsOur study consists of theoretical analysis, simulation, and experimental verification. Firstly, the working principle and demodulation method of the heterodyne laser Doppler vibration measurement system, which is in the presence of stray light, are deeply analyzed. According to the analysis, the stray light would generate chirp noise in the process of normal demodulation method (ARCTAN). Based on the generation and characteristics of the chirp noise, a new demodulation method DPM is proposed. Then, the influence of chirp noise on the measurement of target movement velocity and the effect of DPM on chirp noise suppression are simulated. In the simulation, a low-frequency sinusoidal movement is utilized as the background movement, while a high-frequency sinusoidal movement is employed as the target movement. In the simulation, the background movement generates corresponding chirp noise to affect demodulation results severely. The normal demodulation method DPM is leveraged to restore the target movement by demodulating the overall movement and performing high-pass filtering. Finally, a heterodyne laser Doppler vibration measurement experiment is conducted to utilize a piezoelectric ceramic plate fixed on the pendulum device. In the experiment, the piezoelectric ceramic plate vibrates at a single and high frequency, which is regarded as the target movement, and the pendulum's movement is considered as the background movement. According to the experiment results, the existence of chirp noise is verified, and the DPM suppresses the chirp noise too.Results and DiscussionsIn the process of the normal demodulation method (ARCTAN), the cause of chirp noise is the combination of stray light and background movement. The frequency of chirp noise changes in real time with the background movement speed, and specifically, it is proportional to the absolute speed of background movement and inversely proportional to the laser wavelength (Formula 11). When the background movement is low-frequency high-speed movement and the target movement is high-frequency low-speed movement, the chirp noise will seriously affect the measurement of high-frequency part movement [Fig. 2(b)]. Compared with the normal demodulation method, DPM can effectively restore target movement [Fig. 2(c)], but will generate erroneous velocity spikes in the positions that are near zero speed locations. Since adopting the normal demodulation method's results to partly replace the demodulation results of the DPM at the corresponding positions (near zero speed locations), the velocity spikes can be suppressed, and the demodulation results approach target movement more closely. In the experiment, the background movement of the pendulum indeed generates corresponding chirp noise [Figs. 4(a) and (b)], and DPM can effectively suppress the chirp noise [Fig. 4(c)]. DPM has a significant suppression effect on chirp noise, which reduces the power of chirp noise by about 81.8% at the peak of the chirp noise (Fig. 5).ConclusionsThe principle of heterodyne laser Doppler vibration measurement is deeply analyzed. It is pointed out that chirp noise is generated due to stray light and background movement. The frequency of the chirp noise changes in real time with the background movement speed, which is proportional to the absolute background movement speed and inversely proportional to the laser wavelength. Simulations and experiments have confirmed the existence of chirp noise and its frequency variation pattern. A novel demodulation method DPM has been proposed for this type of chirp noise. Simulations and experiments prove that DPM can effectively suppress chirp noise. Above the target movement frequency, the chirp noise power can be reduced by about 81.8%.
Acta Optica Sinica
- Publication Date: Mar. 10, 2024
- Vol. 44, Issue 5, 0507001 (2024)
Tip-Tilt Error Detection for Segments via Phase Transfer Function
Lu Zhang, Weirui Zhao, Yuejin Zhao, and Juan Liu
ObjectiveIn order to observe more distant and fainter objects with a better resolution and signal-to-noise ratio, larger primary mirror telescopes are required to improve the diffraction limit and increase the collected light energy. This leads to problems of manufacture, testing, transportation, and launch for monolithic primary mirrors. At present, it is hard to build a monolithic primary mirror with a diameter of 8 m or larger. The segmented primary mirror is thus adopted to address these issues. However, tip-tilt errors between segments must be eliminated to meet the requirements of the light-collecting capacity and resolution. The existing tip-tilt error detection approaches mainly include the centroid detection method, phase retrieval/phase diversity (PR/PD) method, Shack-Hartmann phase sensing method, and other methods based on interferometry. In tip-tilt error detection, the centroid detection method is usually used in the coarse stages, and the PR/PD is used to eliminate the uncertainty of the centroid detection method in the fine stages. The Shack-Hartmann phase sensing method is separately used in coarse and fine stages, which also involve special-purpose hardware, complex structure, and unstable factors.MethodsIn this paper, a novel method, for detecting tip-tilt errors in a large capture range with a better accuracy via phase transfer function (PTF), is proposed. A mask with a sparse subpupil configuration is set on the segments' conjugate plane and serves as the entrance pupil of the tip-tilt error detection system. Then, the optical transfer function (OTF) with separated sidelobes can be obtained by the Fourier transform of the point spread function (PSF) recorded in the charge-coupled device (CCD) of the detection system, which makes it possible to separate the information of tip-tilt errors overlapped in the PSF. By analyzing the OTF sidelobes, the relationship between the phase distribution gradient of the OTF sidelobes and tip-tilt can be derived and used to extract the tip-tilt error without the measurement uncertainty of the centroid detection method, which makes the tip-tilt error detection realized with better accuracy in a large dynamic range. Simulations and experiments are conducted to verify the correctness of the proposed method. We set up a two-segmented system as shown in Fig. 2, and the tip-tilt errors are introduced from different ranges. In the small range, we introduce the tip-tilt errors from 0 to 0.4λ by the step of 0.008λ. In the large range, the tip-tilt errors are introduced from 0.4λ to 2.4λ by the step of 0.04λ. In the experiment, we verify the method on the basis of the active cophasing and aligning testbed with segmented mirrors as shown in Fig. 6. The tip-tilt errors can be obtained by calculating the differences between every two centroid positions of the images formed by the segments on the focal plane. Through this experimental platform, the tip-tilt error detection method proposed in this paper is compared with the centroid detection method to achieve correctness verification. For this purpose, the mask of the tip-tilt error detection module (TEDM) is redesigned, and the original hole D is replaced with three discrete holes, as shown in Fig. 7. We have also performed preliminary simulations of the effects caused by CCD noise and figure error on the method described in this paper.Results and DiscussionsSimulation results show that the tip-tilt error can be detected with high accuracy over a large dynamic range as shown in Fig. 4 and Fig. 5, and the root-mean-square (RMS) has the order of magnitude of 10-15λ, which conforms to the detection requirements of the tip-tilt errors. Compared with the existing methods, this method does not need to divide the error detection into two stages and can effectively eliminate the measurement uncertainty of the center-of-mass detection. On the active cophasing and aligning testbed with segmented mirrors we set up before, experiments have been carried out to verify the feasibility of the method, and the RMS of detection accuracy of the method is 2.99×10-3λ, which meets the cophasing requirement of segmented telescopes. The experiment results are given in Table 1, Table 2, and Table 3. In addition, some factors affecting the detection accuracy of the proposed method, such as CCD noise and figure error of the tested segments, are analyzed by simulations, and the results in Table 4 and Table 5 show that in order to meet the cophasing requirement of λ/40 (RMS), the signal-to-noise of CCD and the figure error of segments should be better than 40 dB and 0.05λ (RMS), respectively.ConclusionsBecause of the setting of the sparse subpupil configuration and the intervention of the Fourier transform, the method in this paper effectively separates the tip-tilt errors of the segmented system in the spatial frequency domain. Then, the uncertainty of the centroid detection method during the measurement of the small errors is eliminated. The detection accuracy of the tip-tilt errors is ensured and improved. The tip-tilt error detection method simplifies the detection process and eases the demanding hardware required in existing sensing methods, and cophasing is no longer divided into coarse and fine stages that involve separate dedicated hardware solutions. This method can be adapted to any segmented primary mirror and sparse-aperture telescope system with any shape of the sub-mirror. ObjectiveIn order to observe more distant and fainter objects with a better resolution and signal-to-noise ratio, larger primary mirror telescopes are required to improve the diffraction limit and increase the collected light energy. This leads to problems of manufacture, testing, transportation, and launch for monolithic primary mirrors. At present, it is hard to build a monolithic primary mirror with a diameter of 8 m or larger. The segmented primary mirror is thus adopted to address these issues. However, tip-tilt errors between segments must be eliminated to meet the requirements of the light-collecting capacity and resolution. The existing tip-tilt error detection approaches mainly include the centroid detection method, phase retrieval/phase diversity (PR/PD) method, Shack-Hartmann phase sensing method, and other methods based on interferometry. In tip-tilt error detection, the centroid detection method is usually used in the coarse stages, and the PR/PD is used to eliminate the uncertainty of the centroid detection method in the fine stages. The Shack-Hartmann phase sensing method is separately used in coarse and fine stages, which also involve special-purpose hardware, complex structure, and unstable factors.MethodsIn this paper, a novel method, for detecting tip-tilt errors in a large capture range with a better accuracy via phase transfer function (PTF), is proposed. A mask with a sparse subpupil configuration is set on the segments' conjugate plane and serves as the entrance pupil of the tip-tilt error detection system. Then, the optical transfer function (OTF) with separated sidelobes can be obtained by the Fourier transform of the point spread function (PSF) recorded in the charge-coupled device (CCD) of the detection system, which makes it possible to separate the information of tip-tilt errors overlapped in the PSF. By analyzing the OTF sidelobes, the relationship between the phase distribution gradient of the OTF sidelobes and tip-tilt can be derived and used to extract the tip-tilt error without the measurement uncertainty of the centroid detection method, which makes the tip-tilt error detection realized with better accuracy in a large dynamic range. Simulations and experiments are conducted to verify the correctness of the proposed method. We set up a two-segmented system as shown in Fig. 2, and the tip-tilt errors are introduced from different ranges. In the small range, we introduce the tip-tilt errors from 0 to 0.4λ by the step of 0.008λ. In the large range, the tip-tilt errors are introduced from 0.4λ to 2.4λ by the step of 0.04λ. In the experiment, we verify the method on the basis of the active cophasing and aligning testbed with segmented mirrors as shown in Fig. 6. The tip-tilt errors can be obtained by calculating the differences between every two centroid positions of the images formed by the segments on the focal plane. Through this experimental platform, the tip-tilt error detection method proposed in this paper is compared with the centroid detection method to achieve correctness verification. For this purpose, the mask of the tip-tilt error detection module (TEDM) is redesigned, and the original hole D is replaced with three discrete holes, as shown in Fig. 7. We have also performed preliminary simulations of the effects caused by CCD noise and figure error on the method described in this paper.Results and DiscussionsSimulation results show that the tip-tilt error can be detected with high accuracy over a large dynamic range as shown in Fig. 4 and Fig. 5, and the root-mean-square (RMS) has the order of magnitude of 10-15λ, which conforms to the detection requirements of the tip-tilt errors. Compared with the existing methods, this method does not need to divide the error detection into two stages and can effectively eliminate the measurement uncertainty of the center-of-mass detection. On the active cophasing and aligning testbed with segmented mirrors we set up before, experiments have been carried out to verify the feasibility of the method, and the RMS of detection accuracy of the method is 2.99×10-3λ, which meets the cophasing requirement of segmented telescopes. The experiment results are given in Table 1, Table 2, and Table 3. In addition, some factors affecting the detection accuracy of the proposed method, such as CCD noise and figure error of the tested segments, are analyzed by simulations, and the results in Table 4 and Table 5 show that in order to meet the cophasing requirement of λ/40 (RMS), the signal-to-noise of CCD and the figure error of segments should be better than 40 dB and 0.05λ (RMS), respectively.ConclusionsBecause of the setting of the sparse subpupil configuration and the intervention of the Fourier transform, the method in this paper effectively separates the tip-tilt errors of the segmented system in the spatial frequency domain. Then, the uncertainty of the centroid detection method during the measurement of the small errors is eliminated. The detection accuracy of the tip-tilt errors is ensured and improved. The tip-tilt error detection method simplifies the detection process and eases the demanding hardware required in existing sensing methods, and cophasing is no longer divided into coarse and fine stages that involve separate dedicated hardware solutions. This method can be adapted to any segmented primary mirror and sparse-aperture telescope system with any shape of the sub-mirror.
Acta Optica Sinica
- Publication Date: Feb. 25, 2024
- Vol. 44, Issue 4, 0407001 (2024)
A Multi-Phase Phase-Generated Carrier Demodulation Algorithm for Stability Improvement of Noise Transfer Coefficient for Fiber Optic Hydrophone
Qingkai Hou, Qiong Yao, Hu Chen, and Shuidong Xiong
ObjectiveInterferometric fiber optic hydrophone is a relatively mature solution in the current fiber optic hydrophone system and features high sensitivity, large dynamic range, strong anti-interference ability, and easy array formation. Meanwhile, it is suitable for underwater targets and is widely employed in fields such as detection and underwater energy exploration. In recent years, the application scenarios of fiber optic hydrophones have gradually developed into complex scenarios such as far-reaching seawater acoustic detection and mobile platform deployment. These scenarios pose more challenges to the signal detection performance and noise stability of hydrophones. Phase-generated carrier (PGC) demodulation is a commonly adopted signal detection method for interferometric fiber optic hydrophones. Since the operating point and carrier modulation depth are greatly affected by external environmental changes, the PGC demodulation system has unstable output phase signals. In particular, the system's self-noise stability fluctuates greatly with environmental changes. This problem has become an important factor limiting the performance of fiber optic hydrophone systems.MethodsCentering on the noise stability of interferometric fiber optic hydrophones based on PGC demodulation, we build a noise transfer model of the interferometric fiber optic hydrophone based on PGC demodulation and focus on analyzing changes in the two parameters of the carrier modulation depth and operating point. Meanwhile, the mechanism of influence on the stability of PGC demodulation noise is studied. A new multi-phase PGC demodulation scheme is proposed, where a 3×3 coupler is introduced into the traditional PGC demodulation architecture for multi-phase detection, and the three interference signals are fused by phase shift characteristics of the coupler. The multi-phase PGC demodulation algorithm performs PGC demodulation on the outputs of three 3×3 couplers respectively, and then averages the demodulation results of the three channels. Since the measured phase signals in the three demodulated output signals are the same, the averaging operation has no effect on them, while the noise signals can be suppressed. Additionally, as the initial phases of the three interference signals differ by 2π/3, the noise influence exerted by the initial phase changes can be minimized by averaging regardless of whether the working point of the interference signals changes or not. Therefore, the demodulation noise can be relatively stable. As the working point of the hydrophone changes, this scheme can reduce fluctuations in the noise transfer coefficient of the light source intensity noise.Results and DiscussionsWe conduct simulation experiments to verify the performance of the multi-phase PGC demodulation algorithm. The simulation results show that sound noise stability can be achieved under different carrier modulation depth (C) values. Under different C values, the fluctuation of the noise transfer coefficient is less than 0.5 dB, and compared with the traditional PGC demodulation algorithm, the stability of demodulation noise of multi-phase PGC demodulation algorithm is significantly improved (Figs. 3 and 4). A multi-phase PGC demodulation system based on 3×3 coupler is built, and the demodulation phase noise performance of the system is experimentally verified. A multi-channel synchronous sampling analog-to-digital converter (ADC) is employed to acquire the three outputs of the coupler. The traditional PGC demodulation method and the multi-phase PGC demodulation algorithm are utilized to demodulate the original data collected by the system. Additionally, we calculate the noise spectrum levels of the demodulated signals of the two methods at 1 kHz frequency separately and analyze the noise fluctuation characteristics of the system. The experimental results show that the self-noise fluctuation obtained by demodulating the three outputs of the 3×3 coupler using the traditional PGC demodulation method is greater than 4.5 dB (Fig. 6). The noise spectrum levels obtained by the multi-phase PGC demodulation method are significantly reduced, and the noise fluctuation during the entire test cycle is less than 1.8 dB (Fig. 6). The experimental results verify the effectiveness of the multi-phase PGC demodulation algorithm.ConclusionsWe build a noise transfer model for interferometric fiber optic hydrophones, analyze and derive the noise transfer model of system noise sources on demodulation results, and propose a multi-phase PGC demodulation algorithm. Compared with traditional PGC demodulation algorithm, the proposed algorithm can suppress the fluctuation of light intensity noise transfer coefficient under the changing operating point, and improve the noise stability of demodulation results. Simulation and experimental results are consistent with the theoretical analysis results of the model. In applications such as deep-sea exploration and long-distance target detection which have increasingly stringent noise performance requirements for fiber optic hydrophones, the noise transfer model and the multi-phase PGC demodulation algorithm based on 3×3 coupler proposed in our study have research and practical significance. ObjectiveInterferometric fiber optic hydrophone is a relatively mature solution in the current fiber optic hydrophone system and features high sensitivity, large dynamic range, strong anti-interference ability, and easy array formation. Meanwhile, it is suitable for underwater targets and is widely employed in fields such as detection and underwater energy exploration. In recent years, the application scenarios of fiber optic hydrophones have gradually developed into complex scenarios such as far-reaching seawater acoustic detection and mobile platform deployment. These scenarios pose more challenges to the signal detection performance and noise stability of hydrophones. Phase-generated carrier (PGC) demodulation is a commonly adopted signal detection method for interferometric fiber optic hydrophones. Since the operating point and carrier modulation depth are greatly affected by external environmental changes, the PGC demodulation system has unstable output phase signals. In particular, the system's self-noise stability fluctuates greatly with environmental changes. This problem has become an important factor limiting the performance of fiber optic hydrophone systems.MethodsCentering on the noise stability of interferometric fiber optic hydrophones based on PGC demodulation, we build a noise transfer model of the interferometric fiber optic hydrophone based on PGC demodulation and focus on analyzing changes in the two parameters of the carrier modulation depth and operating point. Meanwhile, the mechanism of influence on the stability of PGC demodulation noise is studied. A new multi-phase PGC demodulation scheme is proposed, where a 3×3 coupler is introduced into the traditional PGC demodulation architecture for multi-phase detection, and the three interference signals are fused by phase shift characteristics of the coupler. The multi-phase PGC demodulation algorithm performs PGC demodulation on the outputs of three 3×3 couplers respectively, and then averages the demodulation results of the three channels. Since the measured phase signals in the three demodulated output signals are the same, the averaging operation has no effect on them, while the noise signals can be suppressed. Additionally, as the initial phases of the three interference signals differ by 2π/3, the noise influence exerted by the initial phase changes can be minimized by averaging regardless of whether the working point of the interference signals changes or not. Therefore, the demodulation noise can be relatively stable. As the working point of the hydrophone changes, this scheme can reduce fluctuations in the noise transfer coefficient of the light source intensity noise.Results and DiscussionsWe conduct simulation experiments to verify the performance of the multi-phase PGC demodulation algorithm. The simulation results show that sound noise stability can be achieved under different carrier modulation depth (C) values. Under different C values, the fluctuation of the noise transfer coefficient is less than 0.5 dB, and compared with the traditional PGC demodulation algorithm, the stability of demodulation noise of multi-phase PGC demodulation algorithm is significantly improved (Figs. 3 and 4). A multi-phase PGC demodulation system based on 3×3 coupler is built, and the demodulation phase noise performance of the system is experimentally verified. A multi-channel synchronous sampling analog-to-digital converter (ADC) is employed to acquire the three outputs of the coupler. The traditional PGC demodulation method and the multi-phase PGC demodulation algorithm are utilized to demodulate the original data collected by the system. Additionally, we calculate the noise spectrum levels of the demodulated signals of the two methods at 1 kHz frequency separately and analyze the noise fluctuation characteristics of the system. The experimental results show that the self-noise fluctuation obtained by demodulating the three outputs of the 3×3 coupler using the traditional PGC demodulation method is greater than 4.5 dB (Fig. 6). The noise spectrum levels obtained by the multi-phase PGC demodulation method are significantly reduced, and the noise fluctuation during the entire test cycle is less than 1.8 dB (Fig. 6). The experimental results verify the effectiveness of the multi-phase PGC demodulation algorithm.ConclusionsWe build a noise transfer model for interferometric fiber optic hydrophones, analyze and derive the noise transfer model of system noise sources on demodulation results, and propose a multi-phase PGC demodulation algorithm. Compared with traditional PGC demodulation algorithm, the proposed algorithm can suppress the fluctuation of light intensity noise transfer coefficient under the changing operating point, and improve the noise stability of demodulation results. Simulation and experimental results are consistent with the theoretical analysis results of the model. In applications such as deep-sea exploration and long-distance target detection which have increasingly stringent noise performance requirements for fiber optic hydrophones, the noise transfer model and the multi-phase PGC demodulation algorithm based on 3×3 coupler proposed in our study have research and practical significance.
Acta Optica Sinica
- Publication Date: Jan. 25, 2024
- Vol. 44, Issue 2, 0207001 (2024)
Adaptive Non-Iterative Linearization Technique for Broadband Multi-Carrier Microwave Photonic Link
Bing Lu, Kang Chen, Weigang Hou, Yifan Bai, Jiaxin Zhang, and Lei Guo
ObjectiveMicrowave photonic technology has an important potential in future high-speed microwave/millimeter-wave communication systems due to its large bandwidth, low loss, and immunity to electromagnetic interference. However, due to the inherent cosine response of the electro-optic modulators, the output signals of the broadband multi-carrier microwave photonic link (MPL) will suffer from nonlinear distortions, mainly including harmonic distortions (HD), cross-modulation distortion (XMD), and third-order intermodulation distortion (IMD3). Since HD can be filtered out by a suitable filter, the XMD and IMD3 are the main factors limiting the system performance. We build a nonlinear distortion model for in-band third-order IMD3 and out-of-band XMD compensation of a broadband MPL. Despite various optical and electrical methods are proposed to compensate for the IMD3, few methods can quickly compensate for both XMD and IMD3 of a broadband MPL spontaneously. Thus, a nonlinear distortion model is presented for compensating the in-band IMD3 and out-of-band XMD in the wideband MPL. This method does not require priori parameters of the system and signals, and a complicated training and iterative optimization process, which is more practical.MethodsWe provide a nonlinear distortion model for a broadband multi-carrier MPL. Firstly, due to large frequency differences between the HD signal and the fundamental frequency signal, the HD signal can be easily filtered by a digital filter. Then, the XMD and IMD3 signals are extracted, which are the opposite sign to the fundamental frequency signal. Thus, it is easy to obtain that the cubic power of the XMD and IMD3 signals is also the opposite sign of the fundamental frequency signal. Based on the characteristic, a cost function with a closed-form solution can be constructed, where an optimal linearization coefficient is obtained quickly and adaptively. Finally, this optimal linearization coefficient is introduced to compensate the XMD and IMD3 simultaneously in the digital domain.Results and DiscussionsSimulation experiments are built to verify the performance of XMD and IMD3 suppression. Figure 2 shows the signal spectra before and after linearization as two-tone signals are received. The XMD and IMD3 are suppressed by more than 35 dB and 29 dB respectively. The power of the fundamental frequency signal is found to remain unchanged, but the power of the XMD term increases linearly with the slope change of 2 (Fig. 3). Additionally, after compensation by the proposed algorithm, all the XMDs are suppressed below the noise and the compensation effect does not decrease with the increasing input fundamental signal power. As the power of the input fundamental signal increases, the powers of the fundamental signal and the IMD3 signal of the pre-compensation in-band signal rise linearly with slopes of 1 and 3 respectively. Meanwhile, the power of the XMD term after linearization increases linearly at a slope of 5. The spurious-free dynamic range of the compensated system is improved by more than 21.5 dB (Fig. 4). According to the simulation experiment, after algorithmic compensation, the error vector magnitudes (EVMs) of single-carrier orthogonal frequency division multiplexed signal (OFDM) and multi-carrier OFDM signals are optimized by 6.1% and 5.9% respectively (Figs. 6 and 7). As multi-carrier OFDM signals with different Vpp are input (Fig. 8), the best compensation effect is at 1 V, and the EVM is optimized by 7.2%.ConclusionsA nonlinear distortion model is presented for the XMD and IMD3 generated in a broadband multi-carrier MPL. Then based on the characteristic that the XMD and IMD3 signals have the opposite sign to that of the fundamental frequency signals, the out-of-band XMD and the in-band IMD3 can be suppressed. Compared with the traditional XMD and IMD3 compensation methods, this method does not require priori parameters of the system and signals, and a complicated training and iterative optimization process. Simulation results show that the XMD and IMD3 are suppressed by more than 35 dB and 29 dB respectively, and the spurious-free dynamic range is improved by about 22 dB as the multi-tone signal is transmitted. When a multi-carrier OFDM signal is transmitted, the EVM of the signal is optimized from 8.1% to 2.2%. ObjectiveMicrowave photonic technology has an important potential in future high-speed microwave/millimeter-wave communication systems due to its large bandwidth, low loss, and immunity to electromagnetic interference. However, due to the inherent cosine response of the electro-optic modulators, the output signals of the broadband multi-carrier microwave photonic link (MPL) will suffer from nonlinear distortions, mainly including harmonic distortions (HD), cross-modulation distortion (XMD), and third-order intermodulation distortion (IMD3). Since HD can be filtered out by a suitable filter, the XMD and IMD3 are the main factors limiting the system performance. We build a nonlinear distortion model for in-band third-order IMD3 and out-of-band XMD compensation of a broadband MPL. Despite various optical and electrical methods are proposed to compensate for the IMD3, few methods can quickly compensate for both XMD and IMD3 of a broadband MPL spontaneously. Thus, a nonlinear distortion model is presented for compensating the in-band IMD3 and out-of-band XMD in the wideband MPL. This method does not require priori parameters of the system and signals, and a complicated training and iterative optimization process, which is more practical.MethodsWe provide a nonlinear distortion model for a broadband multi-carrier MPL. Firstly, due to large frequency differences between the HD signal and the fundamental frequency signal, the HD signal can be easily filtered by a digital filter. Then, the XMD and IMD3 signals are extracted, which are the opposite sign to the fundamental frequency signal. Thus, it is easy to obtain that the cubic power of the XMD and IMD3 signals is also the opposite sign of the fundamental frequency signal. Based on the characteristic, a cost function with a closed-form solution can be constructed, where an optimal linearization coefficient is obtained quickly and adaptively. Finally, this optimal linearization coefficient is introduced to compensate the XMD and IMD3 simultaneously in the digital domain.Results and DiscussionsSimulation experiments are built to verify the performance of XMD and IMD3 suppression. Figure 2 shows the signal spectra before and after linearization as two-tone signals are received. The XMD and IMD3 are suppressed by more than 35 dB and 29 dB respectively. The power of the fundamental frequency signal is found to remain unchanged, but the power of the XMD term increases linearly with the slope change of 2 (Fig. 3). Additionally, after compensation by the proposed algorithm, all the XMDs are suppressed below the noise and the compensation effect does not decrease with the increasing input fundamental signal power. As the power of the input fundamental signal increases, the powers of the fundamental signal and the IMD3 signal of the pre-compensation in-band signal rise linearly with slopes of 1 and 3 respectively. Meanwhile, the power of the XMD term after linearization increases linearly at a slope of 5. The spurious-free dynamic range of the compensated system is improved by more than 21.5 dB (Fig. 4). According to the simulation experiment, after algorithmic compensation, the error vector magnitudes (EVMs) of single-carrier orthogonal frequency division multiplexed signal (OFDM) and multi-carrier OFDM signals are optimized by 6.1% and 5.9% respectively (Figs. 6 and 7). As multi-carrier OFDM signals with different Vpp are input (Fig. 8), the best compensation effect is at 1 V, and the EVM is optimized by 7.2%.ConclusionsA nonlinear distortion model is presented for the XMD and IMD3 generated in a broadband multi-carrier MPL. Then based on the characteristic that the XMD and IMD3 signals have the opposite sign to that of the fundamental frequency signals, the out-of-band XMD and the in-band IMD3 can be suppressed. Compared with the traditional XMD and IMD3 compensation methods, this method does not require priori parameters of the system and signals, and a complicated training and iterative optimization process. Simulation results show that the XMD and IMD3 are suppressed by more than 35 dB and 29 dB respectively, and the spurious-free dynamic range is improved by about 22 dB as the multi-tone signal is transmitted. When a multi-carrier OFDM signal is transmitted, the EVM of the signal is optimized from 8.1% to 2.2%.
Acta Optica Sinica
- Publication Date: Jun. 25, 2024
- Vol. 44, Issue 12, 1207001 (2024)
Fourier Transform Profilometry Based on Improved Goldstein Branch-Cut Algorithm
Qian You, Hui Weng, Jiang Zhao, Yuebin Li, Wenfeng Wang, Shi Lu, and Kuang Peng
ObjectiveFringe projection profilometry is a representative method for optical three-dimensional measurement and is widely applied in intelligent manufacturing, virtual reality, cultural heritage protection, biomedicine, and industrial inspection. Fringe projection profilometry mainly includes Moiré profilometry, Fourier transform profilometry, and phase measurement profilometry. Fourier transform profilometry can recover the three-dimensional surface information of the measured object through phase calculation, phase unwrapping, and phase-height mapping. It has the advantages of less data processing and a fast measurement speed, thus being widely used in three-dimensional reconstruction. The phase value obtained by phase calculation will be wrapped at (-π, π]. It is necessary to convert the wrapped phase into a continuous phase through phase unwrapping, and then the height distribution of the measured object can be determined by phase-height mapping. Therefore, the quality of phase unwrapping directly influences the reconstructed accuracy of the measured object. Among many phase unwrapping algorithms, Goldstein branch-cut algorithm is widely used because of its noise-immune ability and high efficiency. After identifying all residues in the wrapped phase map, the Goldstein branch-cut algorithm generates branch cuts by connecting the residues to optimize the phase unwrapping path. The shorter the total length of the branch cuts is, the better the result of phase unwrapping will be. However, the branch cuts constructed by Goldstein branch-cut algorithm cannot ensure the shortest total length and are easy to close, which causes incorrect phase unwrapping in some regions and finally affects the reconstructed accuracy. Therefore, Fourier transform profilometry based on an improved Goldstein branch-cut algorithm is proposed to ensure the accuracy of three-dimensional measurement.MethodsThe computer-generated grating fringes are projected onto the surface of the measured object by digital light processing, and the grating fringes are modulated by the height of the measured object. The deformed fringes containing the height information of the measured object are collected by a charge-coupled device, and the wrapped phase map is obtained through the operations of Fourier transform, fundamental frequency filtering, and inverse Fourier transform. First, all positive and negative residues are identified in the wrapped phase map. Then, the problem of constructing branch cuts with the shortest total length is transformed to a maximum weighted matching problem by constructing a weighted bipartite graph. The Kuhn-Munkres algorithm is applied to solve the maximum weighted matching problem, and the branch cuts with the shortest total length are obtained. Finally, the path that avoids branch cuts is selected for phase unwrapping. Pixels on the branch cuts can be unwrapped according to the unwrapped pixels around the branch cuts. The surface information of the measured object is recovered by phase-height mapping. This paper compares the total length of the branch cuts, the root mean square error, and the execution time of generating branch cuts between the proposed method and the Goldstein branch-cut algorithm. The root mean square error of the proposed method under different noises is studied to evaluate its noise-immune ability. In addition, three-dimensional reconstruction experiments are carried out on complex objects, and the reconstruction results show that the proposed method is suitable for the three-dimensional measurement of complex objects.Results and DiscussionsThe Goldstein branch-cut algorithm is a powerful anti-noise method, and the quality of phase unwrapping depends on the generated branch cuts. Shorter branch cuts result in a better phase unwrapping result. The simulation results show that the proposed method constructs branch cuts with a shorter total length and takes less time for generating branch cuts than the Goldstein branch-cut algorithm, bringing a lower root mean square error (Table 1). In addition, the research on the root mean square errors of the proposed method and the Goldstein branch-cut algorithm under different noises shows that the former has a stronger anti-noise ability (Table 2). In the reconstruction experiment of complex objects, the results reconstructed by the Goldstein branch-cut algorithm are poor in some areas, while the proposed method can ensure the reconstructed accuracy of complex objects (Fig. 13).ConclusionsThis paper expounds the basic principles of Fourier transform profilometry and the Goldstein branch-cut algorithm. The Goldstein branch-cut algorithm is a local nearest neighbor algorithm that may not generate the shortest branch cuts. Moreover, branch cuts are easy to close, which makes phase unwrapping incorrect in some regions and increases the reconstructed error. To ensure the reconstructed accuracy of the measured object, this paper proposes Fourier transform profilometry based on an improved Goldstein branch-cut algorithm. The simulation results show that compared with the Goldstein branch-cut algorithm, the proposed method reduces the total length of branch cuts, has a stronger noise-immune ability, and can effectively improve reconstructed accuracy. Experimental results indicate that the proposed method is suitable for the three-dimensional measurement of complex objects. ObjectiveFringe projection profilometry is a representative method for optical three-dimensional measurement and is widely applied in intelligent manufacturing, virtual reality, cultural heritage protection, biomedicine, and industrial inspection. Fringe projection profilometry mainly includes Moiré profilometry, Fourier transform profilometry, and phase measurement profilometry. Fourier transform profilometry can recover the three-dimensional surface information of the measured object through phase calculation, phase unwrapping, and phase-height mapping. It has the advantages of less data processing and a fast measurement speed, thus being widely used in three-dimensional reconstruction. The phase value obtained by phase calculation will be wrapped at (-π, π]. It is necessary to convert the wrapped phase into a continuous phase through phase unwrapping, and then the height distribution of the measured object can be determined by phase-height mapping. Therefore, the quality of phase unwrapping directly influences the reconstructed accuracy of the measured object. Among many phase unwrapping algorithms, Goldstein branch-cut algorithm is widely used because of its noise-immune ability and high efficiency. After identifying all residues in the wrapped phase map, the Goldstein branch-cut algorithm generates branch cuts by connecting the residues to optimize the phase unwrapping path. The shorter the total length of the branch cuts is, the better the result of phase unwrapping will be. However, the branch cuts constructed by Goldstein branch-cut algorithm cannot ensure the shortest total length and are easy to close, which causes incorrect phase unwrapping in some regions and finally affects the reconstructed accuracy. Therefore, Fourier transform profilometry based on an improved Goldstein branch-cut algorithm is proposed to ensure the accuracy of three-dimensional measurement.MethodsThe computer-generated grating fringes are projected onto the surface of the measured object by digital light processing, and the grating fringes are modulated by the height of the measured object. The deformed fringes containing the height information of the measured object are collected by a charge-coupled device, and the wrapped phase map is obtained through the operations of Fourier transform, fundamental frequency filtering, and inverse Fourier transform. First, all positive and negative residues are identified in the wrapped phase map. Then, the problem of constructing branch cuts with the shortest total length is transformed to a maximum weighted matching problem by constructing a weighted bipartite graph. The Kuhn-Munkres algorithm is applied to solve the maximum weighted matching problem, and the branch cuts with the shortest total length are obtained. Finally, the path that avoids branch cuts is selected for phase unwrapping. Pixels on the branch cuts can be unwrapped according to the unwrapped pixels around the branch cuts. The surface information of the measured object is recovered by phase-height mapping. This paper compares the total length of the branch cuts, the root mean square error, and the execution time of generating branch cuts between the proposed method and the Goldstein branch-cut algorithm. The root mean square error of the proposed method under different noises is studied to evaluate its noise-immune ability. In addition, three-dimensional reconstruction experiments are carried out on complex objects, and the reconstruction results show that the proposed method is suitable for the three-dimensional measurement of complex objects.Results and DiscussionsThe Goldstein branch-cut algorithm is a powerful anti-noise method, and the quality of phase unwrapping depends on the generated branch cuts. Shorter branch cuts result in a better phase unwrapping result. The simulation results show that the proposed method constructs branch cuts with a shorter total length and takes less time for generating branch cuts than the Goldstein branch-cut algorithm, bringing a lower root mean square error (Table 1). In addition, the research on the root mean square errors of the proposed method and the Goldstein branch-cut algorithm under different noises shows that the former has a stronger anti-noise ability (Table 2). In the reconstruction experiment of complex objects, the results reconstructed by the Goldstein branch-cut algorithm are poor in some areas, while the proposed method can ensure the reconstructed accuracy of complex objects (Fig. 13).ConclusionsThis paper expounds the basic principles of Fourier transform profilometry and the Goldstein branch-cut algorithm. The Goldstein branch-cut algorithm is a local nearest neighbor algorithm that may not generate the shortest branch cuts. Moreover, branch cuts are easy to close, which makes phase unwrapping incorrect in some regions and increases the reconstructed error. To ensure the reconstructed accuracy of the measured object, this paper proposes Fourier transform profilometry based on an improved Goldstein branch-cut algorithm. The simulation results show that compared with the Goldstein branch-cut algorithm, the proposed method reduces the total length of branch cuts, has a stronger noise-immune ability, and can effectively improve reconstructed accuracy. Experimental results indicate that the proposed method is suitable for the three-dimensional measurement of complex objects.
Acta Optica Sinica
- Publication Date: Mar. 10, 2023
- Vol. 43, Issue 5, 0507001 (2023)
Phase Calibration Method of Electro-Optic Modulator Based on Cross-Correlation
Enxing He, Youhua Chen, Shunyu Xie, and Cuifang Kuang
ObjectiveElectro-optic modulators can change the phase and polarization state of incident light, therefore having a wide range of applications in many fields, such as optical communication, integrated optics, and super-resolution microscopy. They feature fast response and reliability. However, due to the differences among electro-optic modulator devices, the relationship between the applied voltage and the phase change in actual operation is not consistent with the corresponding relationship in the technical manuals. Meanwhile, when modulators are working at different wavelengths, applying the same voltage to the modulator results in different phase changes. Therefore, before utilization, it is necessary to calibrate the relationship between the phase change and voltage change of the electro-optic modulator. Since its modulation is linear, the slope of the function only needs to be obtained for subsequent experiments.Common calibration methods include the contour method and the Michelson interferometry method. The contour method employs two optical paths for interference, one of which passes through an electro-optic modulator to change the phase. Interference fringes are taken at a certain voltage interval. The displacement between two fringes is divided by the fringe period and then multiplied by 2π to obtain the phase difference which is combined with the voltage interval to get the half-wave voltage. This method is simple but requires repeated adjustment and correction, which is time-consuming and laborious. Additionally, the limitation of camera pixels reduces the accuracy. The Michelson interferometry method passes one of the interferometer arms through an electro-optic modulator, applies a voltage to produce a phase shift, and then moves this arm to change the optical path difference and make the interference fringes disappear to determine the phase difference. This method has high accuracy but requires optical path rebuilding with too much consumed time.To quickly and accurately calibrate the half-wave voltage of an electro-optic modulator without rebuilding an optical path, we study a method using cross-correlation in the frequency domain. The complex conjugate of the high-order spectrum of the previous interference fringe is multiplied by the high-order spectrum of the subsequent fringe to calculate the phase difference for calibrating half-wave voltage.MethodsWe adopt the phase angle of the cross-correlation function between multiple fringe images to determine the phase difference. After converting the fringe illumination light to the frequency domain, the high-order spectrum is extracted. In the spectra of multiple fringe patterns, the conjugate of the high-order spectrum of the previous one is multiplied by the next one to obtain the cross-correlation function of adjacent images. The angle of this function is the phase difference between two fringe images. The feasibility of this method is verified by generating fringe images with the same phase difference in Matlab. The optical path for experimental calibration is part of a structured illumination super-resolution microscopy system. This system achieves high-speed imaging based on electro-optic modulators and galvanometers. One optical path passes through a phase electro-optic modulator, while the other does not. Finally, the two beams interfere at the camera through a mirror to form fringes.Results and DiscussionsThe experiment applies voltage to the EOM through a computer-controlled acquisition card, with a voltage interval of 1 V and a voltage range of -10 V to 9 V. When the voltage is changed each time, the camera is controlled to acquire an image and save it. A total of 20 interference fringe images with equal interval displacement are collected. The 640 nm and 561 nm lasers are utilized for calibration, and the calibration results of the Michelson interferometry method serve as the correct results for accuracy consideration. To further eliminate the errors caused by interference, we take nine sets of fringe images for each laser wavelength, calculate the average value of the nine sets of results, and then compare this value with the accurate calibration value. The half-wave voltage obtained by calibrating the 640 nm using the Michelson interferometry method is 6.6 V, and the result obtained using this method is 6.57 V, with a difference of 0.03 V and an error of 0.45%. The half-wave voltage obtained by calibrating the 561 nm using the Michelson interferometry method is 5.87 V, and the result obtained by this method is 5.84 V, with a difference of 0.03 V and an error of 0.51%. After converting to phase difference, the phase difference calculated for 640 nm is 0.478 rad, the standard phase difference is 0.476 rad, and the difference is 0.002 rad. The phase difference calculated for 561 nm is 0.538 rad, the standard phase difference is 0.535 rad, and the difference is 0.003 rad. By employing the contour method to process the 640 nm image, the obtained half-wave voltage is 6 V, which has a larger error than the result obtained by this method. The half-wave voltage obtained by our method is close to that obtained by the Michelson interferometry method, with the same accuracy and faster speed.ConclusionsBefore adopting an electro-optic modulator, it is often necessary to calibrate the half-wave voltage. Previous methods such as Michelson interferometry require additional optical path construction, and the calibration process is slow and easily interfered by noise and jitter. Thus, the contour method is not accurate enough. Therefore, a method based on the high-order cross-correlation of the interference fringe frequency domain is proposed to calculate the phase difference between fringe images and calibrate the half-wave voltage of the electro-optic modulator. This method employs a specific mask to remove the 0th-order spectrum and extract the high-order spectrum. The complex conjugate of the high-order spectrum of the previous image is multiplied by the high-order spectrum of the next image to obtain the angle and then solve for the phase difference. The phase error in actual detection reaches 0.002 rad and the half-wave voltage error is 0.03 V, which meet the calibration requirements of electro-optic modulators. Since the proposed method has a large calibration speed and does not require optical path rebuilding, it can check whether the electro-optic modulator drifts at any time and whether corrections are needed or not. ObjectiveElectro-optic modulators can change the phase and polarization state of incident light, therefore having a wide range of applications in many fields, such as optical communication, integrated optics, and super-resolution microscopy. They feature fast response and reliability. However, due to the differences among electro-optic modulator devices, the relationship between the applied voltage and the phase change in actual operation is not consistent with the corresponding relationship in the technical manuals. Meanwhile, when modulators are working at different wavelengths, applying the same voltage to the modulator results in different phase changes. Therefore, before utilization, it is necessary to calibrate the relationship between the phase change and voltage change of the electro-optic modulator. Since its modulation is linear, the slope of the function only needs to be obtained for subsequent experiments.Common calibration methods include the contour method and the Michelson interferometry method. The contour method employs two optical paths for interference, one of which passes through an electro-optic modulator to change the phase. Interference fringes are taken at a certain voltage interval. The displacement between two fringes is divided by the fringe period and then multiplied by 2π to obtain the phase difference which is combined with the voltage interval to get the half-wave voltage. This method is simple but requires repeated adjustment and correction, which is time-consuming and laborious. Additionally, the limitation of camera pixels reduces the accuracy. The Michelson interferometry method passes one of the interferometer arms through an electro-optic modulator, applies a voltage to produce a phase shift, and then moves this arm to change the optical path difference and make the interference fringes disappear to determine the phase difference. This method has high accuracy but requires optical path rebuilding with too much consumed time.To quickly and accurately calibrate the half-wave voltage of an electro-optic modulator without rebuilding an optical path, we study a method using cross-correlation in the frequency domain. The complex conjugate of the high-order spectrum of the previous interference fringe is multiplied by the high-order spectrum of the subsequent fringe to calculate the phase difference for calibrating half-wave voltage.MethodsWe adopt the phase angle of the cross-correlation function between multiple fringe images to determine the phase difference. After converting the fringe illumination light to the frequency domain, the high-order spectrum is extracted. In the spectra of multiple fringe patterns, the conjugate of the high-order spectrum of the previous one is multiplied by the next one to obtain the cross-correlation function of adjacent images. The angle of this function is the phase difference between two fringe images. The feasibility of this method is verified by generating fringe images with the same phase difference in Matlab. The optical path for experimental calibration is part of a structured illumination super-resolution microscopy system. This system achieves high-speed imaging based on electro-optic modulators and galvanometers. One optical path passes through a phase electro-optic modulator, while the other does not. Finally, the two beams interfere at the camera through a mirror to form fringes.Results and DiscussionsThe experiment applies voltage to the EOM through a computer-controlled acquisition card, with a voltage interval of 1 V and a voltage range of -10 V to 9 V. When the voltage is changed each time, the camera is controlled to acquire an image and save it. A total of 20 interference fringe images with equal interval displacement are collected. The 640 nm and 561 nm lasers are utilized for calibration, and the calibration results of the Michelson interferometry method serve as the correct results for accuracy consideration. To further eliminate the errors caused by interference, we take nine sets of fringe images for each laser wavelength, calculate the average value of the nine sets of results, and then compare this value with the accurate calibration value. The half-wave voltage obtained by calibrating the 640 nm using the Michelson interferometry method is 6.6 V, and the result obtained using this method is 6.57 V, with a difference of 0.03 V and an error of 0.45%. The half-wave voltage obtained by calibrating the 561 nm using the Michelson interferometry method is 5.87 V, and the result obtained by this method is 5.84 V, with a difference of 0.03 V and an error of 0.51%. After converting to phase difference, the phase difference calculated for 640 nm is 0.478 rad, the standard phase difference is 0.476 rad, and the difference is 0.002 rad. The phase difference calculated for 561 nm is 0.538 rad, the standard phase difference is 0.535 rad, and the difference is 0.003 rad. By employing the contour method to process the 640 nm image, the obtained half-wave voltage is 6 V, which has a larger error than the result obtained by this method. The half-wave voltage obtained by our method is close to that obtained by the Michelson interferometry method, with the same accuracy and faster speed.ConclusionsBefore adopting an electro-optic modulator, it is often necessary to calibrate the half-wave voltage. Previous methods such as Michelson interferometry require additional optical path construction, and the calibration process is slow and easily interfered by noise and jitter. Thus, the contour method is not accurate enough. Therefore, a method based on the high-order cross-correlation of the interference fringe frequency domain is proposed to calculate the phase difference between fringe images and calibrate the half-wave voltage of the electro-optic modulator. This method employs a specific mask to remove the 0th-order spectrum and extract the high-order spectrum. The complex conjugate of the high-order spectrum of the previous image is multiplied by the high-order spectrum of the next image to obtain the angle and then solve for the phase difference. The phase error in actual detection reaches 0.002 rad and the half-wave voltage error is 0.03 V, which meet the calibration requirements of electro-optic modulators. Since the proposed method has a large calibration speed and does not require optical path rebuilding, it can check whether the electro-optic modulator drifts at any time and whether corrections are needed or not.
Acta Optica Sinica
- Publication Date: Dec. 10, 2023
- Vol. 43, Issue 23, 2307001 (2023)
Denoising Method Based on OVMD-ICA for Fiber Current Sensor
Jianhua Wu, Xiaofeng Zhang, and Liang Chen
Results and Discussions Various optimization algorithms are compared and analyzed. When the energy spectrum entropy function is taken as the fitness function, the particle swarm optimization (PSO) algorithm has the best performance, but its time cost is too high. The grey wolf optimization (GWO) and HPO algorithms are the second best, and the HPO algorithm is better when the time cost and the iterations are taken into account. In this case, the HPO algorithm is better than the other optimization algorithms, as shown in Table 2. In addition, the main data processing methods are compared and discussed. When the signal-to-noise ratio (SNR), mean square error (MSE), and correlation coefficient are taken as the evaluation criterions, the OVMD-ICA has the highest SNR, the minimum MSE, and the largest correlation coefficient. The SNR should be greater than 30 dB according to the applicable standard of the electronic current transformer. The Wavelet (sym10), VMD-wavelet, and OVMD-ICA can suffice for the requirement, as shown in Table 3. The OVMD-ICA can achieve the optimal noise reduction effect, and the current resolution is 3 mA.ObjectiveThe fiber current sensor based on the Faraday effect and Ampère's circuital law can measure the current accurately. It has many advantages, such as excellent insulation characteristics, simultaneous measurement of the alternating current (AC) and direct current (DC), flexible sensor diameter, and digital output. However, it can hardly measure the microcurrent because the magnetic field generated by the weak current is small, and the Verdet constant of the sensing fiber is tiny (about 1 μrad/A when the wavelength is 1300 nm). Therefore, the current resolution of the fiber current sensor is limited. The methods to improve the current resolution mainly include the following: improving the optical path structure, increasing the number of optical fiber loop turns, and improving the Verdet constant of the sensing fiber. However, these methods have the disadvantages of complex operations and high costs. The data processing method is a promising scheme to improve the current resolution. To meet the requirements of information sources for independent component analysis (ICA) and improve the performance of variational mode decomposition (VMD) to deal with impact noise, this paper proposes the co-clustering algorithms of ICA and VMD with the parameters optimized by the hunter-prey optimization (HPO) algorithm.MethodsThis paper proposes the co-clustering algorithms of ICA and VMD with the parameters optimized by the HPO algorithm. Firstly, the random Gaussian noise, shot noise, impact noise, and output signal are measured. The output signal and noise characteristics of the fiber current sensor are analyzed. Secondly, the key parameters of VMD are optimized by the HPO algorithm. With the energy spectrum entropy function as the fitness function, the modal parameter K and the quadratic penalty factor α are obtained by the HPO algorithm, and VMD is realized with the two parameters. Third, the virtual channels of ICA are constructed. The mode functions are classified by the setting of the threshold of the correlation coefficient to construct the virtual channels for ICA. In this way, the application conditions of ICA are satisfied. Finally, the FastICA algorithm is applied for denoising.ConclusionsMore outstanding performance can be achieved in terms of the operation time, required iterations, and search for the globally optimal solution when the parameters of VMD are optimized by the HPO algorithm. The mode functions are classified by the setting of the threshold of the correlation coefficient to construct the virtual channels for ICA, and the FastICA algorithm is applied for denoising. The SNR of the output signal is enhanced, and the MSE is reduced by OVMD-ICA. By this algorithm, the SNR can be improved by at least 5 dB, and the resolution and measurement of 3 mA weak current can be realized. Results and Discussions Various optimization algorithms are compared and analyzed. When the energy spectrum entropy function is taken as the fitness function, the particle swarm optimization (PSO) algorithm has the best performance, but its time cost is too high. The grey wolf optimization (GWO) and HPO algorithms are the second best, and the HPO algorithm is better when the time cost and the iterations are taken into account. In this case, the HPO algorithm is better than the other optimization algorithms, as shown in Table 2. In addition, the main data processing methods are compared and discussed. When the signal-to-noise ratio (SNR), mean square error (MSE), and correlation coefficient are taken as the evaluation criterions, the OVMD-ICA has the highest SNR, the minimum MSE, and the largest correlation coefficient. The SNR should be greater than 30 dB according to the applicable standard of the electronic current transformer. The Wavelet (sym10), VMD-wavelet, and OVMD-ICA can suffice for the requirement, as shown in Table 3. The OVMD-ICA can achieve the optimal noise reduction effect, and the current resolution is 3 mA.ObjectiveThe fiber current sensor based on the Faraday effect and Ampère's circuital law can measure the current accurately. It has many advantages, such as excellent insulation characteristics, simultaneous measurement of the alternating current (AC) and direct current (DC), flexible sensor diameter, and digital output. However, it can hardly measure the microcurrent because the magnetic field generated by the weak current is small, and the Verdet constant of the sensing fiber is tiny (about 1 μrad/A when the wavelength is 1300 nm). Therefore, the current resolution of the fiber current sensor is limited. The methods to improve the current resolution mainly include the following: improving the optical path structure, increasing the number of optical fiber loop turns, and improving the Verdet constant of the sensing fiber. However, these methods have the disadvantages of complex operations and high costs. The data processing method is a promising scheme to improve the current resolution. To meet the requirements of information sources for independent component analysis (ICA) and improve the performance of variational mode decomposition (VMD) to deal with impact noise, this paper proposes the co-clustering algorithms of ICA and VMD with the parameters optimized by the hunter-prey optimization (HPO) algorithm.MethodsThis paper proposes the co-clustering algorithms of ICA and VMD with the parameters optimized by the HPO algorithm. Firstly, the random Gaussian noise, shot noise, impact noise, and output signal are measured. The output signal and noise characteristics of the fiber current sensor are analyzed. Secondly, the key parameters of VMD are optimized by the HPO algorithm. With the energy spectrum entropy function as the fitness function, the modal parameter K and the quadratic penalty factor α are obtained by the HPO algorithm, and VMD is realized with the two parameters. Third, the virtual channels of ICA are constructed. The mode functions are classified by the setting of the threshold of the correlation coefficient to construct the virtual channels for ICA. In this way, the application conditions of ICA are satisfied. Finally, the FastICA algorithm is applied for denoising.ConclusionsMore outstanding performance can be achieved in terms of the operation time, required iterations, and search for the globally optimal solution when the parameters of VMD are optimized by the HPO algorithm. The mode functions are classified by the setting of the threshold of the correlation coefficient to construct the virtual channels for ICA, and the FastICA algorithm is applied for denoising. The SNR of the output signal is enhanced, and the MSE is reduced by OVMD-ICA. By this algorithm, the SNR can be improved by at least 5 dB, and the resolution and measurement of 3 mA weak current can be realized.
Acta Optica Sinica
- Publication Date: Jan. 25, 2023
- Vol. 43, Issue 2, 0207001 (2023)
Adaptive PDH Frequency Stabilization Method with Large Linear Dynamic Range Based on Two Modulation Depths
Liping Yan, Zhewei Zhang, Jiandong Xie, Yingtian Lou, and Benyong Chen
ObjectiveTo solve the problems of narrow linear dynamic range and weak anti-interference ability of the Pound-Drever-Hall (PDH) technique, a PDH frequency stabilization method based on two modulation depths and two error signals is proposed herein. The PDH technique is widely used in the fields of laser frequency or optical resonant cavity locking. The traditional PDH technique usually utilizes a modulation depth of 1.08 rad to obtain the most sensitive error signal. However, the traditional PDH technique, used for frequency stabilization, is susceptible to environmental disturbances and loss of lock owing to the narrow linear dynamic range of error signals. In addition, only when the phase of a local demodulation signal matches the phase of an interference signal reflected by the cavity, the error signal with the highest sensitivity can be obtained. Currently, most methods manually adjust the initial phase of the local demodulation signal to achieve phase matching; these methods exhibit low accuracy and cannot realize automatic locking easily. Therefore, an adaptive locking mechanism having large modulation depth with large linear dynamic range error signal and small modulation depth with high-sensitivity error signal is developed to achieve frequency stabilization with strong anti-interference ability and high precision.MethodsFirst, a digital quadrature demodulation technique was used to accurately extract the phase of the interference signal to achieve automatic matching between the phases of the local demodulation and interference signals. Second, a new error signal (Spre) was realized using the transmitted power signal Ptran and traditional error signal SPDH to enlarge the linear dynamic range of the PDH frequency stabilization system. Then, Spre corresponding to the large modulation depth was used to realize fast capture and prelocking. Finally, SPDH corresponding to the small modulation depth was used to realize precise locking. After locking, the modulation depths and error signals could be automatically switched according to the amplitude change in Ptran, realizing frequency stabilization with a large linear dynamic range and high sensitivity in the PDH technique. A frequency stabilization control system based on a field-programmable logic gate array (FPGA) was developed, and a locking test was conducted on a Fabry-Perot cavity. The experimental results show that the adaptive locking mechanism with double modulation depths and double error signals can greatly improve the anti-interference ability of the locking system with precision locking.Results and DiscussionsConsidering the influence of phase mismatch and narrow linear dynamic range on the frequency stabilization accuracy of the PDH technique, an adaptive frequency stabilization method with a large linear dynamic range based on two modulation depths and two error signals is proposed herein. The phase of the interference signal is obtained using the digital quadrature demodulation technique to realize phase matching between the interference and local demodulation signals to improve the sensitivity of the error signal SPDH obtained using the PDH technique (Fig. 3). To improve the anti-interference ability of the locking system, Spre with a large linear dynamic range is constructed and combined with SPDH and Ptran (Fig. 4). The adaptive locking mechanism using large modulation depth to obtain Spre and small modulation depth to obtain SPDH is designed herein (Figs. 5 and 6). Thus, the proposed locking mechanism has the highest sensitivity and linear dynamic range, affording high precision and strong anti-interference locking. A locking control system based on FPGA was designed herein (Fig. 7), and a locking test was conducted on the Fabry-Perot cavity. The test results show that the linear dynamic range of Spre corresponding to β= 1.80 rad can reach 6.04 nm (Fig. 8), which is ~3.4 times that of SPDH corresponding to β= 1.08 rad. The automatic switching and locking mechanism based on two modulation depths and two error signals can realize relocking of the Fabry-Perot cavity after instantaneous detuning (Figs. 10 and 11). The long-term relative stability of the Fabry-Perot cavity is 5.72×10-9 (Fig. 12). Therefore, the proposed adaptive PDH frequency stabilization method can achieve long-term precise locking of the optical cavity/laser frequency.ConclusionsThis study proposes an adaptive frequency stabilization mechanism using two modulation depths and two error signals to modify the traditional PDH technique to achieve large linear dynamic range, high locking accuracy, and strong anti-interference ability. The test results show that the linear dynamic range of Spre corresponding to a large modulation depth of 1.80 rad can reach 6.04 nm, which is ~3.4 times that of SPDH (1.78 nm) corresponding to a small modulation depth of 1.08 rad. The adaptive switching and locking mechanism using two modulation depths and two error signals can substantially improve the anti-interference ability of the locking system, with precision locking. The relative stability of the locked cavity reaches 5.72×10-9 within 3 h. Thus, the proposed method can be widely used in fields such as laser frequency locking and resonant cavity locking. ObjectiveTo solve the problems of narrow linear dynamic range and weak anti-interference ability of the Pound-Drever-Hall (PDH) technique, a PDH frequency stabilization method based on two modulation depths and two error signals is proposed herein. The PDH technique is widely used in the fields of laser frequency or optical resonant cavity locking. The traditional PDH technique usually utilizes a modulation depth of 1.08 rad to obtain the most sensitive error signal. However, the traditional PDH technique, used for frequency stabilization, is susceptible to environmental disturbances and loss of lock owing to the narrow linear dynamic range of error signals. In addition, only when the phase of a local demodulation signal matches the phase of an interference signal reflected by the cavity, the error signal with the highest sensitivity can be obtained. Currently, most methods manually adjust the initial phase of the local demodulation signal to achieve phase matching; these methods exhibit low accuracy and cannot realize automatic locking easily. Therefore, an adaptive locking mechanism having large modulation depth with large linear dynamic range error signal and small modulation depth with high-sensitivity error signal is developed to achieve frequency stabilization with strong anti-interference ability and high precision.MethodsFirst, a digital quadrature demodulation technique was used to accurately extract the phase of the interference signal to achieve automatic matching between the phases of the local demodulation and interference signals. Second, a new error signal (Spre) was realized using the transmitted power signal Ptran and traditional error signal SPDH to enlarge the linear dynamic range of the PDH frequency stabilization system. Then, Spre corresponding to the large modulation depth was used to realize fast capture and prelocking. Finally, SPDH corresponding to the small modulation depth was used to realize precise locking. After locking, the modulation depths and error signals could be automatically switched according to the amplitude change in Ptran, realizing frequency stabilization with a large linear dynamic range and high sensitivity in the PDH technique. A frequency stabilization control system based on a field-programmable logic gate array (FPGA) was developed, and a locking test was conducted on a Fabry-Perot cavity. The experimental results show that the adaptive locking mechanism with double modulation depths and double error signals can greatly improve the anti-interference ability of the locking system with precision locking.Results and DiscussionsConsidering the influence of phase mismatch and narrow linear dynamic range on the frequency stabilization accuracy of the PDH technique, an adaptive frequency stabilization method with a large linear dynamic range based on two modulation depths and two error signals is proposed herein. The phase of the interference signal is obtained using the digital quadrature demodulation technique to realize phase matching between the interference and local demodulation signals to improve the sensitivity of the error signal SPDH obtained using the PDH technique (Fig. 3). To improve the anti-interference ability of the locking system, Spre with a large linear dynamic range is constructed and combined with SPDH and Ptran (Fig. 4). The adaptive locking mechanism using large modulation depth to obtain Spre and small modulation depth to obtain SPDH is designed herein (Figs. 5 and 6). Thus, the proposed locking mechanism has the highest sensitivity and linear dynamic range, affording high precision and strong anti-interference locking. A locking control system based on FPGA was designed herein (Fig. 7), and a locking test was conducted on the Fabry-Perot cavity. The test results show that the linear dynamic range of Spre corresponding to β= 1.80 rad can reach 6.04 nm (Fig. 8), which is ~3.4 times that of SPDH corresponding to β= 1.08 rad. The automatic switching and locking mechanism based on two modulation depths and two error signals can realize relocking of the Fabry-Perot cavity after instantaneous detuning (Figs. 10 and 11). The long-term relative stability of the Fabry-Perot cavity is 5.72×10-9 (Fig. 12). Therefore, the proposed adaptive PDH frequency stabilization method can achieve long-term precise locking of the optical cavity/laser frequency.ConclusionsThis study proposes an adaptive frequency stabilization mechanism using two modulation depths and two error signals to modify the traditional PDH technique to achieve large linear dynamic range, high locking accuracy, and strong anti-interference ability. The test results show that the linear dynamic range of Spre corresponding to a large modulation depth of 1.80 rad can reach 6.04 nm, which is ~3.4 times that of SPDH (1.78 nm) corresponding to a small modulation depth of 1.08 rad. The adaptive switching and locking mechanism using two modulation depths and two error signals can substantially improve the anti-interference ability of the locking system, with precision locking. The relative stability of the locked cavity reaches 5.72×10-9 within 3 h. Thus, the proposed method can be widely used in fields such as laser frequency locking and resonant cavity locking.
Acta Optica Sinica
- Publication Date: Oct. 10, 2023
- Vol. 43, Issue 19, 1907001 (2023)
Digital Compensation Method for Nonlinear Distortion of Microwave Photonic Channelized Link
Jincheng Wang, Xiaoen Chen, Min Ding, Jianping Chen, and Guiling Wu
ObjectiveMicrowave photonic channelization technology converts broadband microwave signals to the optical domain for processing, breaking through the bandwidth limitations of conventional electronics. In general, external intensity modulation based on the Mach-Zehnder modulator (MZM) is employed in microwave photonic channelized links. However, due to the intrinsic cosine response of MZM, several nonlinear distortions occur in the process of electro-optical conversion, including harmonic distortion, intermodulation distortion (IMD), and cross-modulation distortion (XMD). Since the harmonic components can be effectively removed by filters, the IMD and XMD will become the main factors limiting the system's performance. Numerous electronic and optical methods have been proposed to compensate for the IMD in conventional narrow-band links, but they are incapable of suppressing the XMD. In this study, we propose a nonlinear distortion compensation scheme based on digital domain iteration processing, which can suppress the IMD and XMD simultaneously. Moreover, compared with the previous linearization methods, the proposed scheme does not require the construction of complex compensation function models or the introduction of additional hardware devices.MethodsTheoretical analysis shows that an approximate equation can be established between the distorted intermediate frequency signal output and the linear one from each channel. The iteration process can be utilized to approach the linearized output. Specifically, the distorted output in each channel can be selected as the initial value for the first iteration. The initial value is first squared and processed by low-pass filtering. Then, it is split into two paths that are processed by different operators. Finally, the results of the different channels processed by the operator are multiplied, and the distorted output is divided by them to obtain the result of the first iteration. Similarly, the output result of the first iteration can be used for the second iteration. Therefore, the digital compensation algorithm based on iteration can gradually convert the distorted output into a linear result.Results and DiscussionsA simulation experiment is built to verify whether the simulation results are consistent with the theoretical derivation results. The signal spectra before and after the digital compensation algorithm processing are presented (Fig. 3). It can be found that rare times of iterations are sufficient to suppress the third-order intermodulation distortion (IMD3) and cross-modulation distortion (Fig. 4). With the increase in the fundamental signal power, the power of XMD and IMD3 increases with the slope of one and three, respectively [Fig. 5 (a)]. As the power of the out-of-channel signal increases, the power of the fundamental signal and IMD3 is almost unchanged, while that of XMD increases with the slope of two [Fig. 5 (b)]. According to the simulation experiment, the IMD3 and XMD can be completely suppressed [Fig. 6 (b)] when the parameter is accurate. When the parameter deviation is 5%, IMD3 and XMD have been suppressed by 15 and 16 dB, respectively [Fig. 6 (c)]. The ability of the digital compensation algorithm to suppress nonlinear distortion disappears as the parameter deviation approaches 18% [Fig. 6 (d)].ConclusionsThe linearity in microwave photonic channelized links is mainly limited by the IMD and XMD. In this study, a nonlinear distortion compensation method based on digital domain iteration processing is proposed, which jointly processes the intermediate frequency signal output from each channel in the digital domain and approaches the ideal result of linearization through iteration. It can effectively suppress the IMD and XMD in channelized links. The simulation results show that the method can completely suppress the IMD and XMD when the parameter is accurate. When the parameter deviation is 5%, the IMD3 and XMD can still be suppressed by 15 and 16 dB, respectively. ObjectiveMicrowave photonic channelization technology converts broadband microwave signals to the optical domain for processing, breaking through the bandwidth limitations of conventional electronics. In general, external intensity modulation based on the Mach-Zehnder modulator (MZM) is employed in microwave photonic channelized links. However, due to the intrinsic cosine response of MZM, several nonlinear distortions occur in the process of electro-optical conversion, including harmonic distortion, intermodulation distortion (IMD), and cross-modulation distortion (XMD). Since the harmonic components can be effectively removed by filters, the IMD and XMD will become the main factors limiting the system's performance. Numerous electronic and optical methods have been proposed to compensate for the IMD in conventional narrow-band links, but they are incapable of suppressing the XMD. In this study, we propose a nonlinear distortion compensation scheme based on digital domain iteration processing, which can suppress the IMD and XMD simultaneously. Moreover, compared with the previous linearization methods, the proposed scheme does not require the construction of complex compensation function models or the introduction of additional hardware devices.MethodsTheoretical analysis shows that an approximate equation can be established between the distorted intermediate frequency signal output and the linear one from each channel. The iteration process can be utilized to approach the linearized output. Specifically, the distorted output in each channel can be selected as the initial value for the first iteration. The initial value is first squared and processed by low-pass filtering. Then, it is split into two paths that are processed by different operators. Finally, the results of the different channels processed by the operator are multiplied, and the distorted output is divided by them to obtain the result of the first iteration. Similarly, the output result of the first iteration can be used for the second iteration. Therefore, the digital compensation algorithm based on iteration can gradually convert the distorted output into a linear result.Results and DiscussionsA simulation experiment is built to verify whether the simulation results are consistent with the theoretical derivation results. The signal spectra before and after the digital compensation algorithm processing are presented (Fig. 3). It can be found that rare times of iterations are sufficient to suppress the third-order intermodulation distortion (IMD3) and cross-modulation distortion (Fig. 4). With the increase in the fundamental signal power, the power of XMD and IMD3 increases with the slope of one and three, respectively [Fig. 5 (a)]. As the power of the out-of-channel signal increases, the power of the fundamental signal and IMD3 is almost unchanged, while that of XMD increases with the slope of two [Fig. 5 (b)]. According to the simulation experiment, the IMD3 and XMD can be completely suppressed [Fig. 6 (b)] when the parameter is accurate. When the parameter deviation is 5%, IMD3 and XMD have been suppressed by 15 and 16 dB, respectively [Fig. 6 (c)]. The ability of the digital compensation algorithm to suppress nonlinear distortion disappears as the parameter deviation approaches 18% [Fig. 6 (d)].ConclusionsThe linearity in microwave photonic channelized links is mainly limited by the IMD and XMD. In this study, a nonlinear distortion compensation method based on digital domain iteration processing is proposed, which jointly processes the intermediate frequency signal output from each channel in the digital domain and approaches the ideal result of linearization through iteration. It can effectively suppress the IMD and XMD in channelized links. The simulation results show that the method can completely suppress the IMD and XMD when the parameter is accurate. When the parameter deviation is 5%, the IMD3 and XMD can still be suppressed by 15 and 16 dB, respectively.
Acta Optica Sinica
- Publication Date: Jul. 10, 2023
- Vol. 43, Issue 13, 1307001 (2023)
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