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Abstract
A high-sensitivity ultralow-frequency fiber optic interferometric seismometer using phase feedback control is proposed and demonstrated. The principle of sensitivity improvement using feedback is described, and the characteristics of the seismometer, including the ultralow-frequency vibration sensing with Michelson interferometer with and without feedback control, are analyzed in terms of the amplitude response and phase response. The phase feedback control loop is designed and implemented, and higher sensitivity for very low frequency vibration is achieved. The efficacy of the new approach is demonstrated experimentally, showing that the weak vibration signal originally buried in noise can be observed unambiguously.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Fiber optic interference based sensing methodologies combine the advantages of passivity, electromagnetic interference immunity, small footprint, and ready adaptability to networked deployment, with superior performance in terms of high sensitivity, wide frequency response band, and large dynamic range. In recent decades, fiber optic interferometer has been introduced to increasingly diverse fields and applications, such as fiber optic accelerometers [1–3], liquid level sensors [4,5], and hydrophones [6]. Traditional geophones and seismometers are placed in caves or deep boreholes to avoid cultural and ambient noises and to improve the signal-to-noise ratio. However, there are several disadvantages to the electronics of these sensing platforms in such environment. High humidity in the caves and high temperature in the deep boreholes limit the lifetimes of conventional electronic components applications. The electrical cable linking the sensors to the hosts exposes the downhole electronics to the risk of damage from lightning strikes [7,8]. Naturally, fiber optic sensing technology is an effective way to tackle satisfactorily the problems mentioned above. Especially, fiber optic interferometric seismometer having high sensitivity comparable to electrical seismometers might represent the development direction of a new generation of seismometers.
In 1987, the first fiber optic seismometer based on Michelson interferometer (MI) was developed by Gardner et al. [9], which consisted of a seismic mass of 520 g supported by two rubber mandrels, each wound with a single layer of optical fiber 6.5 m long. The sensor operates as an accelerometer with a measured sensitivity of 10,500 rad/g, has a resonance frequency of about 310 Hz and a detection threshold of 1 $\textrm{ng}/\sqrt {\textrm{Hz}} $ in the 10∼100 Hz band. In 2002, Shindo et al. developed a fiber optic accelerometer based on MI to observe earthquakes on the sea floor [10]. The system has a resonance frequency of about 200 Hz and could observe micro earthquake with a noise equivalent signal of about 30 $\textrm{ng}/\sqrt {\textrm{Hz}} $ at 10 Hz. In 2004, Zeng et al. presented a 3-component fiber optic accelerometer for well logging [11]. The accelerometer is based on the MI arrangement and push-pull mass-spring structure. The measurement frequency range is 3∼800 Hz. The on-axis sensitivity, cross-sensitivity and minimum detectable level are 39 dB re rad/g, 14 dB re rad/g and 39.3 µg$/\sqrt {\textrm{Hz}} $, respectively. In 2010, Zumberge et al. designed an interferometric seismometer by adding bulk optical elements to a modified mechanical-electrical STS-1 vertical seismometer [7]. The system has a resonance frequency of about 0.02 Hz, a working frequency range of 0.001∼1 Hz, and a dynamic range of 180 dB, which can compete with the classical very broad band seismometer STS-1. But an all fiber optic design was not achieved in the system. In 2014, Han et al. proposed an all-metal 3-component optical fiber MI seismometer [12]. The axis sensitivity is about 41 dB re rad/g, while the transverse sensitivity is about -40 dB re rad/g. The minimum detectable acceleration is 90 $\textrm{ng}/\sqrt {\textrm{Hz}} $ in 5∼1000 Hz. In 2017, our group proposed a fiber optic interferometric tri-component geophone for ocean floor seismic monitoring [13–15]. The system has a resonance frequency of about 250 Hz, a sensitivity of 57 dB re rad/g, with the operation frequency range of 0.002∼50 Hz and the minimum detectable acceleration of 43.4 $\textrm{ng}/\sqrt {\textrm{Hz}} $. As far as we know, the system has the best low frequency detection performances of all fiber optic seismometers. More importantly, a tri-component version of our fiber optic interferometric seismometer was installed in the seismic observation cave at the Earthquake Administration Bureau of Jilin Province, China, and has been operating without interruption for more than two years, in a side-by-side comparison with the state-of-the-art mechanical-electrical seismometer CTS-1, obtaining comparable results in the cases of two low-grade natural earthquakes [16].
However, the all fiber optic interferometric seismometers still have an obvious technology gap to fill compared with traditional mechanical-electrical seismometers in very low frequency working performance. Generally, the minimum detectable acceleration calculated by sensitivity and phase noise of a fiber optic interferometer is higher than the standard minimum signal level for frequencies lower than 1 Hz, in which frequency band the system detection capacity is critical for monitoring naturally occurring weak earthquakes. Theoretically, the fiber length in the vibration transducer could be increased indefinitely to achieve better sensitivity without consideration of the additional noise, elevating the sensitivity to phase variation due to very low frequency weak natural earthquakes so that the corresponding signal would be higher than the system phase noise level. Unfortunately, the vibration transducer size would be too large to use in practical applications.
Modern mechanical-electrical seismometers are the results of more than a century of experimentation, and there are many lessons learnt that are worthwhile for designers of fiber optic interferometric seismometers to heed. One of the most effective performance-enhancement implements is the force balance feedback technique prevalent in the mechanical-electrical seismometer designs. In 1982, Wielandt and Streckeisen improved the force balance feedback technique used in an earlier experimental seismometer [17], built a small size one-component wideband seismometer that could replace the bulky conventional long period instruments, which has a flat velocity response over a very wide frequency band [18]. Actually, feedback techniques have long been used in fiber optic interferometers as well, albeit almost exclusively in the context of laser phase noise suppression. In 1986, Davis et al. presented a portable laboratory-grade fiber-optic seismometer, and utilized a phase-locked-loop (PLL), which is realized by driving the piezoelectric expander with the amplitude output of the photodetector, to track phase variation for improving the dynamic range of detection [19]. In 1989, Dagenais et al. designed a Mach-Zehnder fiber-optic magnetometer, and used quadrature control feedback to keep the interferometer at quadrature for minimizing the total sideband noise [20]. In 1991, Suzuki et al. proposed a phase-locked laser diode interferometer, where phase lock was achieved by a feedback control system for controlling the injection current of the laser diode [21,22]. In 2003, Bahoura and Clairon applied an optimal frequency control loop to a Mach-Zehnder interferometer (MZI) for diode laser frequency stabilization. The results show that the laser phase noise can be suppressed by choosing a modulation frequency equal to half of the free spectral range of the MZI [23]. In 2011, Wang et al. designed a feedback control system based on a double MZI for suppressing the polarization fading in a fiber optical sensor [24]. In 2013, Duan et al. designed a system that takes advantage of the MZI structure and close-loop feedback model, achieving precise control of the projected fringe phase with AC phase tracking to realize fringe projection [25]. In 2016, Okamoto et al. designed a feed-forward scheme for using phase error to correct the laser phase noise, and realized a phase noise reduction of about 20 dB in the 100 Hz ∼ 3 MHz band [26]. To date, feedback techniques implemented in fiber optic interferometers are mostly devoted to reducing phase noise. Due to the ubiquitous presence of flicker noises, i.e. 1/f noises, it is impractical to attempt to reduce very low frequency phase noise using feedback-loop techniques in fiber optic interferometric seismometers. Nevertheless, the frequency response of a fiber optic interferometric seismometer could be modified by implementing feedback techniques directly in the phase stabilization and detection of a vibration sensor. The original small phase variation caused by very low frequency weak vibration then emerges from the phase noise. In this work, a scheme for phase feedback control to modify the frequency response of a fiber optic interferometric seismometer is designed and demonstrated, showing drastically increased system sensitivity down to extremely low frequencies. First, the principle of modifying the system response by phase feedback control is explicated. Secondly, the particulars of our feedback technique including demodulation delay optimization, loop design and reference arm phase control by a piezoelectric (PZT) transducer are discussed. Finally, the efficacy of the proposed approach is demonstrated experimentally with a prototypical fiber optic interferometric seismometer.
2. Principle
A schematic diagram of the proposed fiber optic interferometric seismometer with phase-generated carrier (PGC) and phase feedback control is shown in Fig. 1. The narrow-linewidth external cavity laser (ECL) emits continuous coherent light with a frequency modulation in the form
Fig. 1. Schematic diagram of the proposed fiber optic interferometric seismometer with phase feedback control.
Light from the ECL is coupled into the two arms of the MI through the second fiber optic coupler. The light wave interacts with the ambient vibration imposed on the vibration sensor that is normally placed in a remote observing site where working conditions are poor. The vibration sensor in our system adopts the push-pull structure of a pair of compliant cylinders, having a resonance frequency of 258.83 Hz and sensitivity of 57 dB re rad/g [13], as shown in Fig. 2. The sensor head is mainly made up of an inertial mass, and two compliant cylinders. The ends of the optical fibers are terminated with Faraday rotation mirrors to eliminate polarization related fading. The inertial mass is made of brass, the compliant cylinders of silicon rubber (Type-601 silicon rubber, Beijing Hagibis Technology Co., Ltd., Beijing, China), and the optical fiber used is single-mode communication fiber (SM-28, Corning, USA).
According to the mechanical model of a damped and driven vibration system with one degree of freedom, the relationship between the fiber optic length variation and ambient vibration-induced acceleration is
Table 1. Physical characteristics of the sensor materials
In the reference arm, the fiber optic wound PZT, which is placed in a local monitoring station along with everything else to the left of the vertical dotted line in Fig. 1, is used as an electro-optical actuator in the feedback loop. To minimize the phase noise in the MI with optimal modulation depth, the interference arm length difference is set to 0.38 m [27]. The left part of the reference arm also extends to the remote observation site. A dual-core armored fiber cable is used as the transmission medium of the sensing and reference arms, which are subject to the same ambient temperature variation and vibration disturbance simultaneously. Then in a differential detection scheme such as those found in MI’s, the ambient noise in both the sensing and reference arm can be considered as common-mode noise, and will be suppressed efficiently.
2.1 Working principle without feedback
Without feedback, the phase shift in the sensing arm and reference arm is respectively
where ${L_{ra}}(t )$ is the fiber optic length variation of the sensing arm relative to the reference arm in time domain, $n = 1.4679$ the fiber optic refractive index, ${L_A}$ and ${L_B}$ the fiber optic initial length in the sensing and reference arm, c the speed of light.With ideal couplers and Faraday mirrors, the light interference intensity at the output port of the second coupler is
Invoking the classical DCM phase demodulation approach, the phase term ${\varphi _{SN}}(t )= {\varphi _S}(t )+ {\varphi _n}({t + {t_1}} )- {\varphi _n}({t + {t_2}} )$ related to the ambient vibration and laser phase noise can be obtained, where ${\varphi _S}(t )= \frac{{4\pi {\nu _C}n{L_{ra}}(t )}}{c}$. The DCM phase demodulation method has five core steps, mixing, low-pass filtering, DCM, integral and high-pass filtering, as shown in Fig. 3.
For the ambient vibration acceleration, there are two gain networks from a system point of view, as shown in Fig. 4, and the transform function is
For vibration frequencies much lower than the resonance frequency, which is the frequency region of interest here, the corresponding ${A_1}$ is 157.72 m/g.
The magnitude-frequency as well as phase-frequency curves (Bode plots) of the acceleration response $\frac{{{C_\varphi }(s )}}{{{R_a}(s )}}$ and velocity response $\frac{{{C_\varphi }(s )}}{{{R_v}(s )}}$ are shown in Fig. 5.
The U.S. Geological Survey’s New Low Noise Model (NLNM) and New High Noise Model (NHNM) [28,29], along with the acceleration noise power spectral density (PSD) of our seismometer without phase feedback control calculated from the interferometric phase noise and sensitivity are shown in Fig. 6. And from the perspective of numerical computation, the phase noise in frequency domain ${N_{p1psd}}(f )$ is obtained from the phase noise in time domain by a fast Fourier transform operation,
where ${N_{p1}}(t )$ represents the measurements of phase noise without feedback in time domain, ${N_{sample}}$ the number of samples, ${f_{res}}$ the frequency resolution in frequency domain, which is determined by the sampling rate and ${N_{sample}}$ jointly.Furthermore, the acceleration noise PSD ${N_{a1psd}}(f )$ can be obtained as
where ${S_{nofb}}(f )= 57$ dB represents the sensitivity without feedback in frequency domain.NLNM and NHNM are the standard global noise models established by the U.S. Geological Survey, which are commonly accepted as global references. They are the average low and high seismic background noise power spectra obtained from a worldwide network of seismograph stations, and have been taken as baselines for evaluating and comparing station site characteristics, for defining instrument specifications, and for predicting the response of a sensor system under quiet and noisy background conditions. For the practical seismometer employed for observing weak natural earthquakes, the acceleration noise PSD should be lower than NHNM, if not NLNM. It is clearly seen from Fig. 6 that the best existing fiber optic interferometric seismometer without feedback does not meet this requirement.
2.2 Working principle with feedback
Without feedback, the original fiber optic interferometric seismometer is essentially a second-order system with a certain resonance frequency and a damping ratio. For weak natural earthquakes, whose frequencies are much lower than the resonance frequency, the seismometer has a flat acceleration response. To improve the system response to weak natural earthquakes, the natural period should be increased to as high as 360 s by a negative feedback loop, just as what the traditional mechanical-electrical seismometers did. To introduce the negative feedback loop to the original seismometer, the fiber length of the reference arm must be adjusted by the fiber optic interferometric phase ${\varphi _S}$. Suppose the transform functions of the forward path and feedback loop are $G(s )$ and $H(s )$ respectively, the closed-loop transform function can be written as
For the ambient vibration acceleration, the closed-loop transform function is
Table 2. Parameter settings with feedback
According to the second factor in Eq. (12) , the closed-loop characteristic equation is $1 + G(s ){\; }H(s )$, i.e., ${s^2} + \frac{1}{{{T_{d2}}}}s + \frac{1}{{{T_{d2}}{T_i}}} + \frac{1}{{\beta {K_p}{A_2}{T_{d1}}{T_{d2}}}}$, meaning that there is a pair of poles, at $- 0.01235 \pm j1.233 \times {10^{ - 2}}$. When the open-loop gain changes from zero to infinity, the root locus plot with parameters from Table 2 is shown in Fig. 8. The root locus starts at poles and ends at zeros, and one root locus ends at infinity. Since the root locus does not enter the right half of the complex plane, the closed-loop system is stable for all possible gains. To further analyze the whole system stabilization issue, all the closed-loop poles in the transform function described in Eq. (12) are considered. Besides the roots of the closed-loop characteristic equation, there are two other poles at $- \zeta {\omega _n} \pm j\sqrt {1 - {\zeta ^2}} {\omega _n}$. It is clearly seen that all the closed-loop poles are located on the left half of the complex plane, ergo the system is stable.
Fig. 8. Root locus plot of the feed-back control network in the complex frequency plane, with parameters given in Table 2.
Additionally, the velocity response is flat around the natural period, including periods encompassing 360 s, which is a hallmark of very low frequency seismometers capable of detecting weak natural earthquakes. The seismometer with feedback is essentially a vibration velocity sensor, and the conversion from the acceleration’s flat response to the velocity’s flat response is necessary and expedient, as the corresponding Bode diagram with feedback (Fig. 9) shows.
The magnitude-frequency response with feedback in the case of the original interferometric phase noise results in the acceleration noise PSD of our seismometer with feedback, as shown in Fig. 10. It clearly shows that the acceleration noise PSD is now lower than the NHNM, and the common weak natural earthquakes whose frequencies are lower than 1 Hz can be observed in theory. From the perspective of numerical computation, the phase noise in frequency domain ${N_{p2psd}}(f )$ is calculated from the phase noise with feedback in time domain using a FFT operation,
where ${N_{p2}}(t )$ represents the measurements of phase noise with feedback in time domain.Furthermore, the acceleration noise PSD with feedback ${N_{a2psd}}(f )$ could be obtained as
where ${S_{fb}}(f )$ represents the sensitivity with feedback in frequency domain, which can be obtained directly from the closed-loop transform function.2.3 Interferometric phase noise
In the original fiber optic interferometric seismometer, the main phase noise source is the laser, more precisely the first-order differential phase noise of the laser, ${\varphi _n}({t + {t_1}} )- {\varphi _n}({t + {t_2}} )$ [30], which is the system internal noise and determined by the length difference $\Delta L$ between the two interferometric arms and the modulation depth in the form [27]:
where $\sigma _{{I_{out}}}^2$ represent the interferometric phase noise power, ${v_{F0}}$ the laser minimum linewidth, $\chi $ a constant coefficient.Since the light frequency modulation depth ${v_c}{K_\nu }$ has been determined, $\sigma _{{I_{out}}}^2$ is only related to $\Delta L$. Furthermore, in either systems with or without feedback, $\Delta L$ could be expressed as
where ${L_A} - {L_B} = 0.38$ m, ${L_{ra}}(t )$ and ${L_{rb}}(t )$ are the fiber optic length variations in the sensing and reference arms respectively. Both ${L_{ra}}(t )$ and ${L_{rb}}(t )$ are at the micrometer level and much smaller than ${L_A} - {L_B}$. Therefore, the interferometric phase noise remains basically unchanged with or without feedback. With phase feedback control, the phase measurements induced by weak natural earthquakes are improved, while the interferometric phase noise remains unchanged. And the weak vibration signal buried in noise originally can now be detected.3. Feedback loop design
The phase feedback control loop includes the DCM demodulation, PID network and PZT driver. Since the response speed is a key indicator for system performance with feedback, the DCM demodulation should adopt the analog method, instead of the more commonly used digital method. With the limitation of the operation amplifier specifications, the available bandwidth and the dynamic range in analog DCM demodulation are much poorer than those with digital DCM demodulation. Fortunately, both the narrower available bandwidth and smaller dynamic range are acceptable for weak natural earthquakes monitoring. The common frequency range of weak natural earthquakes is 360 s ∼ 10 Hz, which is not a problem for operation amplifiers even if the observed vibrations is modulated to a higher frequency. Since available strong vibration seismometers, e.g. an accelerometer, can detect strong earthquakes, the fiber optic interferometric seismometer with feedback could focus on the much more demanding task of detecting weak teleseismic signals, with relatively smaller dynamic range.
3.1 PGC demodulation without integral error
The integrator in the DCM phase demodulation method ineluctably introduces low frequency errors. To simulate such an integral error, a standard cosine signal with 0.1 Hz frequency and 1 rad amplitude is considered. Suppose that the differential signal in PGC demodulation, i.e. the output of differential amplifier, has a Gaussian random phase noise with $1 \times {10^{ - 3}}$ rad variance. When the sampling rate is 100 Hz and sampling time is 100 s, the simulation results of the original standard signal, the differential and integral signals in PGC demodulation are shown in Fig. 11. It clearly shows that the differential signal has ripples induced by noise, while the integral signal features smoother waveform. However, the low-frequency-noise PSD of the integral signal is much higher than the noise PSD of the differential signal, as shown in Fig. 12.
Fig. 11. Plot of the original standard signal (a), the differential (b), and integral (c) signals in PGC demodulation.
The common method to eliminate the integral error is to use a high-pass filter. However, the lower the frequency of vibration the seismometer can respond to, the better the system performance. Obviously, the error suppression level and the detection performance in low frequency present contradictory demands. With our design, the transform function with feedback has a differential term in the numerator, which can be easily realized using inherent differential expression in the PGC demodulation. Then the low-frequency error induced by the integrator is completely removed.
3.2 PID network and PZT driver
Theoretically, the PID network could be realized by either digital method or analog method. The digital PID network has a great advantage of flexibility in parameter configuration, and is suitable for the experimental and primary design stage; while the analog PID network has better stability and is more suitable for practical engineering applications. With consideration of the analog PID realization, both the differential and integral time constant are limited in the range of ${10^{ - 6}}\sim 100\; s$, which could be easily realized by operation amplifiers with passive components, such as resistance and capacitance. Since our seismometer with feedback is at the experimental stage, the PID is realized by digital method for the time being.
The PZT driver is critical to the realization of the phase feedback control loop. The PZT used in our system is PZ1-SMF4-APC-E (Optiphase, USA). The fiber optic length in the PZT is 12.261 m and the fiber stretch is 0.14 µm/V, i.e. the electro-optical modulation index $\beta $ of the PZT is $\beta = 1.4 \times {10^{ - 7}}$ [31]. The ideal fiber stretch curve is a flat line at frequencies much lower than its resonance frequency, which is about 57 kHz [31]. However, actual PZT fiber stretch experiments show that there is obvious non-flatness in the very low frequency range. Both the theoretical and actual test PZT fiber stretch curves are shown in Fig. 13.
Since frequencies below 10 Hz constitute the working frequency band of the seismometer, the non-flatness region must be compensated, and the fiber stretch should be adjusted to a constant value. In our experimental system, the non-flatness compensation is achieved by a digital low-pass filter with a suitable cutoff frequency and roll-off velocity. The electro-optical modulation index $\beta $ decreases linearly from 10 Hz to 0.1 Hz, and the gain at 0.1 Hz should be about $\frac{{1.4 \times {{10}^{ - 7}}}}{{9.5 \times {{10}^{ - 8}}}}$=1.474, i.e. 3.368 dB. Therefore, a first-order filter having a roll-off of 20 dB/decade is sufficient. As a simple example, the analog first order low-pass filter using an operational amplifier (LPV511, Texas Instruments, USA) having 0.1 Hz cutoff frequency and 40 dB pass band gain is designed, and the schematic is shown in Fig. 14. The resulting magnitude response and phase response are shown in Fig. 15. Additionally, a voltage-follower with a $1.474\%$ passive voltage divider is used to remove the redundant gain introduced by the filter.
4. Experiments
To demonstrate the effectiveness of the proposed fiber optic interferometric seismometer with phase feedback control, an experimental system is setup, as shown in Fig. 16. The low frequency weak vibration is simulated by use of a signal generator, a power amplifier and a vibration platform. The signal generator (AFG3102C, Tektronix, USA) generates the source with arbitrary frequency and amplitude, which is amplified by the power amplifier (TST5708, Test Electronics, China). The vibration platform (TST50, Test Electronics, China) driven by the power amplifier generates weak vibration in the vertical direction, and the vibration acceleration and velocity could be measured accurately by a triaxial DC accelerometer (2470-002, Silicon Design, USA). The system host contains the ECL, a fiber optic coupler, a set of photoelectric detectors (PDs), and two data acquisition cards (DAQ)s, and the associated electronics and processors. One of the DAQs is a four-channel analog output device (PCI-6731, National Instruments, USA) for generating the modulation signal, the first and second order frequency-doubling signals for demodulation, and the PZT driving signal with unity phase-voltage conversion gain. The other DAQ is an eight-channel analog input device (PCI-6133, National Instruments, USA) for sampling the interferometric signal and the frequency-doubling signals. Currently, the DCM demodulation, PID network and PZT driver are all realized digitally in the central processing unit (CPU) of the experimental system to facilitate debugging and easy modification of parameters.
4.1 Interferometric phase noise measurement
The interferometric phase noise with and without feedback is measured in the wee hours to minimize the influence of environmental and cultural noises, and the vibration sensor is placed in a soundproof box. The measured interferometric phase noise PSD curves, obtained by spectral analysis of the measured phase noise in time domain, are shown in Fig. 17. As discussed above, the PSD curve remains unchanged either with or without phase feedback control, and it clearly shows that the feedback loop does not increase the phase noise level.
Additionally, another interferometric phase noise experiment is performed to demonstrate the effectiveness of suppressing the noise induced by temperature variation. The vibration sensor is put into an electro-thermostatic water cabinet, and the temperature rises from 20 ${\circ}^{C}$ to 25 ${\circ}^{C}$. In the process of temperature rise, phase noise measurements of PSD with feedback are carried out at 20 ${\circ}^{C}$, 22.5 ${\circ}^{C}$, and 25 ${\circ}^{C}$, as shown in Fig. 18. Since the phase noise measurements are almost exactly the same, the noise induced by ambient variation, as a kind of common-mode noise for our seismometer, should be suppressed completely, and indeed it is, as evidenced by Fig. 18.
4.2 Demonstration of feasibility of phase feedback control
To clearly demonstrate the feasibility of phase feedback, two PZTs are used as the sensing arm and reference arm, the single proportional network is used as a feedback loop whose structure diagram is shown in Fig. 19, where V represents the driving voltage for the PZT in the sensing arm.
The transfer function without and with feedback is respectively
where the measurement value of $\beta $ at 1 Hz is $1.38 \times {10^{ - 7}}$ m/V.Values inside the parentheses represent the normalized values base on the measured phase without feedback.
A cosine signal with 1 Hz and 1 ${\textrm{V}_{\textrm{pp}}}$ is used to drive the PZT in the sensing arm, and the phase measurement curves with ${A_3} = 1$, $0.75,$ and $0.5$ are shown in Fig. 20. The specific measurements are listed in Table 3. And the theoretical phase values are achieved by the triaxial DC accelerometer, while the measurements are achieved by our seismometer with feedback.
Table 3. Phase measurements with proportional feedback
To focus on the errors induced by the feedback control only, both the theoretical and measurement values are normalized. And the errors in Table 3 are determined by
where ${\varphi _{mn}}$ and ${\varphi _{tn}}$ represent the normalized results of measurement values and theoretical values respectively.4.3 Sensitivity
To test the sensitivity of the proposed system with phase feedback control, the velocity measurements obtained from the acceleration vibration sensor, and the phase measurements, are recorded in the frequency range of 0.1∼10 Hz, and the corresponding sensitivity measurements are listed in Table 4, with a near-constant value of about 90 dB re rad/(m/s) across the entire band. The theoretical and measurement curves of sensitivity are also graphed in Fig. 21.
Fig. 21. Theoretical and measured sensitivities of the interferometric fiber seismometer with phase feedback control.
Table 4. Sensitivity measurements with phase feedback control
According to the sensitivity measurements, the experimental phase data corresponding to the NLNM and the NHNM could be obtained. Since the frequency points in the NLNM and NHNM data are not necessarily coincident with the test frequency points of our sensitivity measurements, the required sensitivity values at frequency points where NLNM and NHNM have no values are obtained by the use of linear interpolation [32]. The equivalent phase PSD curves of NLNM, NHNM, and our measured noise are shown in Fig. 22. It is clearly shown that the NHNM buried in noise originally could now be observed with the phase feedback control technique.
Fig. 22. Equivalent phase PSD curves of NLNM, NHNM, and measured noise of the interferometric fiber seismometer with phase feedback control.
5. Conclusion
A high sensitivity fiber optic interferometric seismometer with phase feedback control is conceived and developed. The principle of the seismometer with phase feedback control for improving sensitivity is described, and the characteristics of the seismometer with and without feedback are analyzed in terms of the amplitude response and phase response. The design of the phase feedback control loop, including the DCM demodulation, PID network and PZT driver, is discussed in detail. The efficacy of the new approach is demonstrated experimentally, showing that the low-frequency weak vibration signals buried in noise originally could be observed with the new system.
With an eye for practical weak natural earthquake monitoring applications, the realization of the phase feedback control below 0.1 Hz is also needed to achieve improved sensitivity at very low frequencies. However, with the limitation of very low frequency modulation performance of the PZT used in our system, the feedback loop below 0.1 Hz has poor performances at the present time. A homemade PZT with tunable length of fiber is currently under development and will be used for adjusting the fiber stretch factor $\beta $, and the very low frequency phase feedback control is expected to be achieved. Additionally, the DCM demodulation and PID network are all realized by digital means currently, with adequate computation quality, resulting in several seconds of system response time. In the future, we plan to switch to analog methods to further improve system response time.
The approach presented in this paper has potential applications beyond weak earthquake monitoring: The principle of phase feedback control is readily extendable to other fiber optic interferometric systems, such as hydrophones and water level sensors and should bring about enhanced performances to those systems as well.
Appendix
The narrow linewidth external cavity laser (ECL) emits continuous light with frequency modulation.
After the second coupler shown in Fig. 1, the electric field intensity of the transmitted and coupled light entering the sensing arm and reference arm are respectively
where ${K_{ct}}$ and ${K_{cc}}$ represent the transmission coefficient and coupling coefficient of the coupler respectively.The electric field intensity of the reflective light from the sensing arm and reference arm at the output port of the second coupler are respectively
where ${K_{M1}}$ and ${K_{M2}}$ represent the reflectivity of the Faraday mirrors inside the vibration sensor and in the reference arm respectively, ${t_1} = \frac{{2({{L_A} + {L_{ra}}(t )} )n}}{c}$ and ${t_2} = \frac{{2{L_B}n}}{c}$ the time delay in the sensing and reference arm, c the speed of light.The amplitude terms in (26) and (27) are respectively
Provided ${K_{M1}} = {K_{M2}}$, the interference light intensity is
For ideal couplers and Faraday mirrors, ${K_{ct}} = \frac{1}{{\sqrt 2 }}$, ${K_{cc}} = j\frac{1}{{\sqrt 2 }}$, ${K_{M1}} = {K_{M2}} = 1$. In the third term of the cosine function, the phase variation induced by ${L_{ra}}(t )$ is much less than the phase variation induced by $({{L_A} - {L_B}} )$, and the ${L_{ra}}(t )$ term could be omitted. Our system has the optimal phase modulation depth 2.37 rad, corresponding to $\frac{{4\pi {\nu _C}{K_\nu }n({{L_A} - {L_B}} )}}{c} = 2.37$ rad. (32) could be rewritten without the phase constant term as
Funding
Leading Talents of Guangdong Province Program (00201507); State Oceanic Administration of China (201405026-01).
Disclosures
The authors declare no conflicts of interest.
References
1. D. A. Brown and S. L. Garrett, “Interferometric fiber optic accelerometer,” in Proc. of SPIE Microelectronic Interconnect and Integrated Processing Symposium, Feb. 1991, pp. 282–288.
2. C. H. Chen, D. L. Zhang, G. L. Ding, and Y. M. Cui, “Broadband Michelson fiber-optic accelerometer,” Appl. Opt. 38(4), 628–630 (1999). [CrossRef]
3. Z. H. Zhao, Z. H. Yu, K. Chen, and Q. X. Yu, “A fiber-optic Fabry-Perot accelerometer based on high-speed white light interferometry demodulation,” J. Lightwave Technol. 36(9), 1562–1567 (2018). [CrossRef]
4. Z. D. Wang, Y. T. Wang, P. G. Hou, and Y. J. Wang, “Study and design of a liquid level meter based on fiber optic sensing technology,” in Proc. of SPIE The International Society for Optical Engineering, Feb. 2005, pp. 489–493.
5. C. A. Rodríguez, M. R. N. Ribeiro, A. Frizera-Neto, C. E. S. Castellani, and M. J. Pontes, “Envelope-based technique for liquid level sensors using an in-line fiber Mach–Zehnder interferometer,” Appl. Opt. 55(34), 9803–9809 (2016). [CrossRef]
6. M. J. Matthew, A. Davis, and B. Redding, “Fiber-wrapped mandrel microphone for low-noise acoustic measurements,” J. Lightwave Technol. 36(16), 3205–3210 (2018). [CrossRef]
7. M. Zumberge, J. Berger, J. Otero, and E. Wielandt, “An optical seismometer without force feedback,” Bull. Seismol. Soc. Am. 100(2), 598–605 (2010). [CrossRef]
8. M. Zumberge, H. Alnes, O. Eiken, G. Sasagawa, and T. Stenvold, “Precision of seafloor gravity and pressure measurements for reservoir monitoring,” Geophysics 73(6), WA133–WA141 (2008). [CrossRef]
9. D. L. Gardner, T. Hofler, S. R. Baker, R. K. Yarber, and S. L. Garrett, “A fiber-optic interferometric seismometer,” J. Lightwave Technol. 5(7), 953–960 (1987). [CrossRef]
10. Y. Shindo, T. Yoshikawa, and H. Mikada, “A large scale seismic sensing array on the seafloor with fiber optic accelerometers,” in Proc. of the IEEE Sensors 2002, Orlando, FL, USA, 12–14 June 2002, vol. 2, pp. 1767–1770.
11. N. Zeng, C. Z. Shi, M. Zhang, L. W. Wang, Y. B. Liao, and S. R. Lai, “A 3-component fiber-optic accelerometer for well logging,” Opt. Commun. 234(1-6), 153–162 (2004). [CrossRef]
12. J. Han, W. T. Zhang, D. S. Jiang, Z. G. Wang, and F. Li, “Fiber optic 3-component seismometer,” Photonic Sens. 4(2), 102–107 (2014). [CrossRef]
13. J. D. Chen, T. Y. Chang, Q. J. Fu, J. P. Lang, W. Z. Gao, Z. M. Wang, M. Yu, and H.-L. Cui, “A fiber-optic interferometric tri-component geophone for ocean floor seismic monitoring,” Sensors 17(12), 47 (2016). [CrossRef]
14. J. D. Chen, W. Z. Gao, T. Y. Chang, W. Sun, Z. M. Wang, M. Yu, Y. Yang, Q. J. Fu, and H.-L. Cui, “Method for determining the resonance frequency of a fiber optic interferometric accelerometer,” IEEE Trans. Instrum. Meas. 67(3), 698–705 (2018). [CrossRef]
15. J. D. Chen, T. Y. Chang, Y. Yang, W. Z. Gao, Z. M. Wang, and H.-L. Cui, “Ultra-low-frequency tri-component fiber optic interferometric accelerometer,” IEEE Sensors J. 18(20), 8367–8374 (2018). [CrossRef]
16. Y. Yang, Z. Wang, T. Chang, M. Yu, J. Chen, G. Zheng, and H.-L. Cui, “Seismic observation and analysis based on three-component fiber optic seismometer,” IEEE Access 8, 1374–1382 (2020). [CrossRef]
17. E. Wielandt and W. Mitronovas, “An electronic long-period seismograph for surface-wave dispersion studies,” B. Seismol. Soc. Am. 66(3), 987–996 (1976).
18. E. Wielandt and G. Streckeisen, “The leaf-spring seismometer: design and performance,” B. Seismol. Soc. Am. 72(6), 2349–2367 (1982).
19. C. M. David, J. G. Eustace, and C. J. Zarobila, “Fiber-optic seismometer,” in Proc. of SPIE Cambridge Symposium-Fiber/LASE, 1986, pp. 203–211.
20. D. M. Dagenais, F. Bucholtz, K. P. Koo, and A. Dandridge, “Detection of low-frequency magnetic signals in a magnetostrictive fiber-optic sensor with suppressed residual signal,” J. Lightwave Technol. 7(6), 881–887 (1989). [CrossRef]
21. T. Suzuki, O. Sasaki, K. Higuchi, and T. Maruyama, “Phase-locked laser diode interferometer: high-speed feedback control system,” Appl. Opt. 30(25), 3622–3626 (1991). [CrossRef]
22. T. Suzuki, O. Sasaki, K. Higuchi, and T. Maruyama, “Phase-locked laser diode interferometry for surface profile measurement,” Appl. Opt. 28(20), 4407–4410 (1989). [CrossRef]
23. M. Bahoura and A. Clairon, “Diode laser phase noise influence on the ultimate performance of its frequency stabilization to a Mach–Zehnder interferometer fringe,” IEEE Trans. Instrum. Meas. 52(6), 1846–1853 (2003). [CrossRef]
24. Y. T. Wang, R. R. Yuan, S. T. Wang, X. Gao, and X. Li, “The research of polarization feedback control system based on double Mach-Zehnder interferometer,” Adv. Mater. Res. 308-310, 1751–1754 (2011). [CrossRef]
25. X. J. Duan, F. J. Duan, and C. R. Lv, “Phase stabilizing method based on PTAC for fiber-optic interference fringe projection profilometry,” Opt. Laser Technol. 47, 137–143 (2013). [CrossRef]
26. T. Okamoto, K. Toge, and T. Manabe, “Robust laser phase noise reduction by tracking frequency discrimination change of interferometer,” J. Lightwave Technol. 34(16), 3908–3913 (2016). [CrossRef]
27. Y. Yang, Z. M. Wang, T. Y. Chang, L. Y. Cheng, M. Yu, and H.-L. Cui, “Performance optimization of fiber optic interferometric accelerometer based on phase noise analysis,” IEEE Sensors J. 19(22), 10498–10505 (2019). [CrossRef]
28. J. Peterson, “Observation and modeling of seismic background noise,” U.S. Geol. Surv. Tech. Rept. 93–322 (1993).
29. D. E. McNamara and R. P. Buland, “Ambient noise levels in the continental united states,” Bull. Seismol. Soc. Am. 94(4), 1517–1527 (2004). [CrossRef]
30. A. Dandridge, A. B. Tveten, R. O. Miles, and T. G. Giallorenzi, “Laser noise in fiber-optic interferometer systems,” Appl. Phys. Lett. 37(6), 526–528 (1980). [CrossRef]
31. Optiphase (2015), PZ1 High-speed fiber stretcher, Available at: http://www.optiphase.com/data_sheets/PZ1_Data_Sheet_Rev_G.pdf.
32. T. Blu, P. Thevenaz, and M. Unser, “Linear interpolation revitalized,” IEEE Trans. on Image Process. 13(5), 710–719 (2004). [CrossRef]
