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Abstract
In the area of fiber-optic sensors (FOSs), the past decade witnessed great efforts to challenge the thermal-noise-level sensing resolution for passive FOS. Several attempts were reported claiming the arrival of thermal-noise-level resolution, while the realization of thermal-noise-level resolution for passive FOSs is still controversial and challenging. In this paper, an ultrahigh-resolution FOS system is presented with a sensing resolution better than existing high-resolution passive FOSs. A fiber Fabry-Perot interferometer as the sensing element is interrogated with an ultra-stable probe laser by using the Pound-Drever-Hall technique. Both strain and temperature measurements are carried out to validate the performance of the sensor. The measured noise floor agrees with the theoretical thermal noise level very well.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Well-known for their high resolution, good linearity, immunity to electromagnetic interference, multiplexing and remote-sensing ability, fiber-optic sensors (FOSs) are widely adopted for strain/temperature sensing, for example, in many applications including underwater acoustic detection [1], crustal deformation measurement [2], structural health monitoring [3,4], etc. As is one of the most important performances, great efforts have been devoted to enhance the sensing resolution of FOSs to fulfill the requirements of various applications. In the early development of FOSs, many researchers improved the resolution by adopting fiber elements of higher sensitivity. For instance, by introducing periodic variations in the refractive index of the core of optical fiber, fiber Bragg grating (FBG) is fabricated for fiber-optic sensing [5]. Via monitoring the Bragg wavelength shifts of FBG, nano-strain level resolution was demonstrated. The sensing resolution is mainly limited by the wide high-reflection bandwidth of FBG. Along with the maturity of FBG fabrication technique, high-Q FBG-based resonant FOSs, such as $\pi$-phase shift FBG (PSFBG) [6], fiber Fabry-Perot interferometer (FFPI) [2,7,8], fiber laser (FL) [9], are commercially accessible. The transmission spectra of the resonant FOSs have very sharp spectral fringes known as resonant peaks. Tracking the narrow resonant peaks for sensor readout brings in much better accuracy and thus the sensing resolution is promoted to pico-strain level [10,11].
In resonant FOS systems, a narrow-linewidth laser is usually adopted as a spectral reference to measure the resonance shifts of the sensing elements. Along with the development of higher-Q fiber resonant devices and demodulation techniques, the phase noise of the laser source becomes dominating in high-resolution FOS systems. To realize sub-pico-strain resolution requires the laser linewidth to be better than 100 Hz [12]. As a result, the suppression of the laser noise is crucial in most ultrahigh resolution FOS systems. In 2010, Gagliardi et. al. stabilized a diode laser against an ultra-stable optical frequency comb (OFC) and then employed it to interrogate an FFPI sensor by using Pound-Drever-Hall (PDH) technique [10]. A strain resolution of 220 f$\varepsilon$/${\surd }$Hz at 1.5 kHz was reported and claimed reaching the ultimate resolution limited by the thermal noise in optical fiber. In spite of the excellent experimental results, the claim is quite disputed at that time. In 2018, Liu et. al. established a low-noise random fiber laser based on stimulated Brillouin backscattering and employed it for ultrahigh resolution strain sensing with the same method [13]. An even better strain resolution of 140 f$\varepsilon$/${\surd }$Hz at 1 kHz was realized, which is against with Gagliardi’s previous claim. In fact, the realization of thermal-noise-level resolution for passive resonant FOSs is still controversial and very challenging.
In this paper, an ultra-stable laser is established to probe an FFPI sensor. By using the PDH technique, a strain resolution of 14 f$\varepsilon$/${\surd }$Hz at 1 kHz is realized, which is limited by the thermal noise in optical fiber. Thanks to the long-term stability of the interrogation system, the proposed FOS is also adopted for high-resolution temperature sensing. The measured resolution is at $\mu$K level and the measurement range is up to 100 K. To the best of our knowledge, above results are the best fiber-optic strain/temperature sensing resolution in low-frequency domain.
2. Sensor response to strain and temperature signals
An FFPI is adopted as the sensor head. Its configuration is shown in Fig. 1. The sensor consists of two identical FBGs fabricated on panda-type polarization maintaining fiber (PMF). The length of the two FBGs is $\sim$10 mm, and the distance between the FBGs is $\sim$1.26 m. The reflectivity of the FBGs is above 99.5$\%$ and their central wavelength lies around 1550 nm with 1-nm high-reflection band. The reflection spectrum of the FFPI is also given in Fig. 1, where many narrow transmission resonances appear at the high-reflection band with constant frequency interval [2,14]. The resonant frequency $\nu$ is expressed as:
where $c$ is the speed of light, $n$ the modal index, L the effective cavity length, and M the order of the resonant mode, respectively. The free spectral range (FSR) and the full width at half maximum (FWHM) of the resonance peak are measured to be about 81.657 MHz and 350 kHz, respectively, corresponding to a finesse of about 230.
Fig. 1. Configuration of the fiber Fabry-Perot interferometer (FFPI) and its reflection spectrum. $L$: effective cavity length; FSR: free spectral range.
When the FFPI sensor is exposed to temperature drift $\Delta T$ and strain $\varepsilon$, its modal index $n$ and cavity length L will vary, and thus the resonant frequency $\nu$ change accordingly [2]. The variation of the FFPI resonant frequency to temperature and strain signal can be obtained:
where $\alpha$ is the thermal expansion coefficient, $\xi$ the thermo-optic coefficient, and $p_{\varepsilon }$ the effective elasto-optic constant of the fiber, respectively. Equation (2) denotes that $\Delta \nu$ responses linearly to temperature and strain signal. In the experiment, known temperature and strain signals are applied to the FFPI sensor respectively. At the same time, the drift of the FFPI spectrum is obtained by using a narrow-linewdith tunable laser (TSL-710). The strain and temperature sensitivity of the FFPI sensor is tested to be about 141 kHz/n$\varepsilon$ and −1.309 GHz/K, respectively.3. Sensor interrogation with a closed-loop PDH locking system
To measure the FFPI resonance drifts for sensor readout, a frequency-stabilized probe laser is required as a stable spectral reference. In the experiment, a narrow-linewidth fiber laser (NKT E15, nominal linewidth: 100 Hz) is adopted as the light source. It is further stabilized against an optical ultra-stable cavity (OUC) to suppress the free-running laser noise by using the classical PDH locking system [14,15]. The OUC adopted here (provider: Stable Laser System) is fabricated with ultralow thermal expansion quartz glass and is protected with active temperature stabilization and passive vibration isolation. Its vacuum cavity length is 50 mm, corresponding to a FSR of 3 GHz. It has numerous ultra-narrow resonant peaks with a linewidth of 5 kHz, and the finesse is up to 600,000. Its nominal stability is better than 100 kHz/day.
The frequency noise of the OUC-stabilized and unstabilized laser is evaluated with a Mach-Zehnder interferometer (MZI) with arm length difference of 1 km [16]. The MZI is placed into a sealed box for environmental noise isolation. A balanced photodetector (Thorlabs BPD 480) receives the light power, and the laser frequency noise is obtained from the interference signal, as shown in Fig. 2. It is found that the frequency noise floor below 10 kHz is greatly suppressed after the active stabilization. The noise floor of the OUC-stabilized laser is better than 1 Hz/${\surd }$Hz at 1 kHz. However, at lower frequency domain (1 Hz $\sim$ 1 kHz), the measured noise floor of the stabilized laser is not valid because of environmental interference that coupled on the MZI. Unless another ultra-stable optical reference (such as another OUC) is employed, the actual low-frequency (below 1 kHz) frequency noise of the OUC-stabilized laser is unmeasurable.
In most laser stabilization systems, noise floor at lower frequency is usually lower due to the active feedback and suppression. Therefore, the actual low-frequency (below 1 kHz) frequency noise of the OUC-stabilized laser is estimated to be better than 1 Hz/${\surd }$Hz. The long-term stability of the OUC-stabilized laser is determined by the OUC, which has the nominal frequency drift of about 100 kHz/day. After the frequency stabilization, this probe laser is stable enough for the following high-resolution strain and temperature sensing.
The stabilized laser is then adopted to probe the FFPI sensor by using the same PDH locking system [2,11,15], as shown in Fig. 3. After the frequency shifting by an acousto-optical modulator (AOM), the light passes through a phase modulator (PM) and a circulator (CIR) and injects into the FFPI sensor. A photodetector (PD) receives the reflected light from the FFPI. If the probe laser is frequency aligned with one FFPI resonance, the reflected light will exhibit a pattern of intensity modulation. An error signal (the so-called PDH curve), which indicates the resonant frequency variation of the FFPI in real-time, is extracted from the output of PD with a lock-in amplifier (LIA) and the PDH technique. The driven signal of the AOM is generated by a direct digital frequency synthesizer (DDS). A field programmable gate array (FPGA, NI USB 7855R) module is employed to sample the error signal, calculate the laser frequency deviation, feedback to control the AOM frequency and communicate with a personal computer (PC).
Fig. 3. Interrogation system setup (a) and principle (b). AOM: acousto-optic modulator; PM: phase modulator; PD: photodetector; FPGA: field programmable gate array; DDS: direct digital frequency synthesizer; CIR: circulator; PC: personal computer; LIA: lock-in amplifier; $f_\textrm{AOM}$: the working frequency of the AOM.
In the PDH system, the phase modulator (MPZ-LN-10) is driven with a sinusoidal signal (6 MHz, 3 Vpp), which is generated by the LIA. When the AOM frequency sweeps, the FPGA module acquires the error signal at the same time and a PDH curve is obtained, as shown in Fig. 4. The signal-to-noise-ratio (SNR) of the PDH curve is about 30 dB. Within the central region of the PDH curve, error signal is proportional to frequency deviation. The slope is $\sim$170 kHz/V. Therefore, a proportion-integration-differentiation (PID) servo loop can be established on the FPGA module to tune the AOM frequency, so that the laser can be locked to the zero-crossing point of the PDH curve to perform closed-loop measurement [12,14]. In the PID loop, the integrated anolog-digital converter (ADC) on the FPGA acquires the error signal with a sampling rate of 1 MSa/s. The frequency deviation is then calculated according to the slope of the PDH curve in Fig. 4 and the AOM frequency is tuned correspondingly. The bandwidth of this PID is 500 kHz. The amplitude of the AOM driven signal is also adjusted via controlling the amplitude of DDS signal so that the transmission power after the AOM is constant.
In the laser locking loop, the AOM operates between 145-235 MHz. The tuning range is larger than the FSR of the FFPI sensor. Therefore, with the help of the AOM, the probe laser can always find a nearest FFPI resonance to lock. Once the FFPI resonance under locking moves far away from the laser, the AOM frequency ($f_{\textrm{AOM}}$) can jump a FSR to lock another resonance. For instance, when $f_{\textrm{AOM}}$ reaches 235 MHz, the FPGA controls the DDS frequency to jump from 235 to (235-FSR) MHz, and the laser is exactly aligned to another resonance again. The AOM frequency is tuned with 500 kHz updating rate and better than 1 Hz resolution. So the stabilized laser and FFPI resonances are locked accurately, and their frequency deviation is equal to the AOM frequency. The resonance trace ($f_{\textrm{R}}$) of the FFPI can be taken from AOM frequency and be expressed as:
where N stands for the number of resonances that have passed through the laser. The resonance trace ($f_{\textrm{R}}$) is the final sensor output.4. Strain test
At first, the proposed FOS is employed for strain measurement. In the experiment, the FFPI is wound around a piezoelectric transducer (PZT) and they are inserted into a small metal tank. By using a commercial temperature-control module, its internal temperature is monitored and then stabilized with a resolution of 0.01 K. Afterwards, the metal tank is placed into a thick glass sealed box, which is fabricated with vacuum-level sealing condition. Further, they are all placed on an active vibration isolation platform (Meiritsu, ME40-0506N). After these isolation actions, the environmental temperature and vibration interference is minimized.
A sine strain signal (1 n$\varepsilon _{\textrm{pp}}$, 100 Hz) is applied on the FFPI sensor via supplying voltage on the PZT. The trace of the FFPI resonance ($f_{\textrm{R}}$) is exported as the sensor output and the blue curve in Fig. 5 shows the sensor output in frequency domain. Using the frequency-to-strain ratio of 141 kHz/n$\varepsilon$, the equivalent strain spectral density is also displayed at the right Y axis. The highest peak at 100 Hz corresponds to the applied strain signal, which is measured to be about 1 n$\varepsilon _{\textrm{pp}}$. In spite of the noise isolation, there are still noise peaks attributed to environmental vibrations. At the infrasonic domain, the noise floor rises and has the 1/$\sqrt {f}$ feature, which results from environmental temperature drifts.
Fig. 5. Sensor output in frequency domain while using unstabilized laser (red curve) and stabilized laser (blue curve). The black line is the calculated resolution limited by the thermal noise of the optical fiber.
The red curve shows the degraded sensor output if the laser stabilization is turned off. The red noise floor agrees with that of previous high-resolution FOS systems where same type of probe laser (NKT E15) was employed [12,13]. After the active laser stabilization, the sensing resolution is improved by at least 10 dB in the full displayed bandwidth. The blue noise floor reaches 2 Hz/${\surd }$Hz at 1 kHz, corresponding to a strain measuring resolution of 14 f$\varepsilon$/${\surd }$Hz. The black curve presents the theoretical thermal-noise-limit resolution of the 1.26-m FFPI sensor [17]. It is found that the resolution of the proposed FOT is generally limited by the thermal noise of optical fiber in the acoustic frequency domain.
5. Temperature sensing
A temperature sensing experiment is then carried out to validate the performance of the proposed FOS. The experiment setup is shown in Fig. 6. The FFPI sensor and an electronic thermometer (10k NTC sensor with 0.01 K resolution) are immersed into a tank which is filled with hot water (55 $^{\circ }$C). Then the water tank and the FFPI sensor are left in a stable environment ($\sim$26.55 $^{\circ }$C) to cool down. Because the FFPI sensor can only measure the relative temperature change, the initial value of the FOS is calibrated by the electronic thermometer. The temperature of water in the natural cooling process is recorded in 3000 s by these two thermometers. The temperature of water has obvious exponential characteristics when it drops from 55 $^{\circ }$C to 30 $^{\circ }$C, as shown in Fig. 6(a). First-order exponential fitting of the temperature curve predicts that it will finally fall down to 26.65 $^{\circ }$C, which is very close to the actual environment temperature. First-order exponential fitting is performed to the FOS readout in a time period of 100 s and 10 s, respectively. The fitting residuals are shown in Fig. 6(b) and Fig. 6(c). The standard deviations of the residual fluctuations are about 602 $\mu ^{\circ }$C in 100 s and 3.5 $\mu ^{\circ }$C in 10 s. The disturbance caused by water convection is believed to be responsible for this residual fluctuations.
Fig. 6. (a) Temperature readout of water cooling process using electronic and the proposed fiber-optic thermometer; (b, c) Exponential fitting residual of sensor readout in 100 s and 10 s, respectively.
Compared with the water, air condition is more stable. Therefore, the FFPI temperature sensor is also placed in air to test its self-heating effect. The isolation environment is as same as that in the strain sensing experiment. When the probe laser is frequency aligned to the cavity resonance, strong power will be accumulated in the cavity and lead to a temperature increase. The self-heating of the cavity impairs the temperature measurement, especially for small cavity sensors. The test result of the self-heating effect is shown in Fig. 7. When the light power injecting into the sensor jumps from 25 $\mu$W to 50 $\mu$W, the temperature readout increases by 36 $\mu$K. The temperature-to-power ratio is calculated to be $\sim$1.44 $\mu$K/$\mu$W, and it is relatively low compared with the other silica-based micro-ring temperature sensors, whose self-heating effect is at the level of $\sim$1 mK/$\mu$W [18]. While the injecting power is stable, the standard deviation of the short-term fluctuation of the temperature readout is about 0.16 $\mu$K, which is relatively small because of the effective temperature isolation. In a long measurement time, however, the temperature drift is drastic. Many factors would impair the long-term temperature resolution evaluation, such as the cross-sensitivity between temperature and strain, temperature stability and uniformity in the testing environment, long-term stability of the OUC. Therefore, it is very difficult to give a specific temperature resolution value for the proposed FFPI sensor.
6. Discussion
6.1 Resolution limit
The ultimate resolution limitation of fiber-optic sensors is set by the thermal noise of the fiber-optic sensor head. According to Duan’s theory, fiber thermal noise has two distinctive spectral characteristics [17,19]. At high-frequency domain ($>$1 kHz), the noise spectrum is well described by a thermodynamic model based on spontaneous temperature fluctuations. At low-frequency domain ($<$1 kHz), the noise floor exhibits a strong 1/f dependence, which results from the thermomechanical fluctuations in optical fibers. Combination of the thermodynamic ${S_{\varphi _T}}(f)$ and thermomechanical ${S_{\varphi _L}}(f)$ noise results in an overall thermal phase noise spectral density ${S_\varphi }(f)$, which is expressed as [17]:
For the resonant fiber-optic sensing elements like FFPI, the thermal noise is often characterized with frequency noise (Hz/${\surd }$Hz) rather than phase noise (rad/${\surd }$Hz). Therefore, thermal-noise-induced frequency noise can be expressed as:
In the numerical simulation, typical parameters of the SMF-28 fiber are used. The following typical values are assumed [17,19,21]: $T=298 K$, $n=1.457$, $\lambda =1550 \rm {nm}$, $E_0=19 \rm {GPa}$, $\varphi _0 = 0.01$, $\kappa = 1.37 W\cdot m^{-1}\cdot K^{-1}$, $\alpha _L = 5\times 10^{-7}K^{-1}$. The amplitude of the frequency noise is
According to Eq. (6), the theoretical thermal noise is shown by the black curve in Fig. 5, which agrees well with the sensor noise floor (the blue curve in Fig. 5) from 20 Hz to 1000 Hz.
In the infrasonic frequency domain, the sensing resolution is strongly affected by the environmental temperature drifts around the FFPI sensor. Unless the temperature stability of the testing environment can be controlled or compensated at the $\mu$K level, the thermal noise in passive optical fiber is still hard to be observed experimentally. Generally, with the ultra-stable laser and the advanced demodulation technique, the thermal-noise-limited resolution level of the 1.26-m FFPI sensor has been reached in the acoustic frequency domain. The achieved strain/temperature sensing resolution is the state-of-the-art for all fiber-optic sensors in this frequency domain.
6.2 Measurement range
In the experiment, the frequency of the stabilized laser is locked to one FFPI resonance with the help of an AOM. When the temperature and strain applied on the FFPI sensor varies, many sensor resonances pass through the laser center. The sensor system locks the FFPI resonances one by one via frequency jumping and counts the number. So the measurement range of the proposed FOT is determined by how many resonances the FFPI sensor can provide. Since the FFPI can provide resonances covering the 1-nm high-reflection band, the measurement range of the proposed FOS is calculated to be about 800 $\mu \varepsilon$/100 K. If the applied signal is over this range, the laser and the FFPI sensor will not be on the same wavelength, and the sensor system stops working. The measurement range can be further extended by using other fiber-optic resonators with broader bandwidth, such as the fiber-optic ring resonator.
7. Conclusion
In conclusion, this paper presents an ultrahigh-resolution FOS based on an ultra-stable probe laser and the PDH technique. It is validated with strain and temperature sensing experiment. A strain resolution of 14 f$\varepsilon$/${\surd }$Hz at 1 kHz and a temperature resolution around $\mu$K level are demonstrated. It has reached the ultimate resolution limit in the acoustic frequency domain set by the thermal noise in optical fiber. The measurement range is about 800 $\mu \varepsilon$/100 K and can be further improved with a wide-band sensing element.
Funding
National Natural Science Foundation of China (61620106015, 61875121, 61975116); National Key Research and Development Program of China (2018YFC1503703).
Disclosures
The authors declare no conflicts of interest.
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