1. INTRODUCTION

Infrasound waves are widely present in nature, with frequencies below the threshold of human hearing ($<20\mathrm{Hz}$) [1]. Before some natural disasters occur, infrasound waves are generated, such as seismic activity [2–5], volcanic eruptions [6], debris flows [7,8], and snow avalanches [9,10]. Hence, early warning of disasters can be achieved by detecting the infrasound waves. In the military field, infrasound waves have been used as an attack weapon since they can resonate with certain organs of the human body, and thus infrasound detection/defense is also essential. On the other hand, pipeline leakage monitoring [11,12] and underwater acoustic detection [13,14] also include the infrasound frequency band. Therefore, the development of infrasound sensors is crucial for practical applications and is moving toward miniaturization, high performance, and intelligence. Different principle-based infrasound sensors transpose the infrasound signals into different measurable physical quantities through the implementation of various energy conversion mechanisms such as capacitive, piezoelectric, and optical interference [15]. In recent years, the establishment of the International Monitoring System (IMS) infrasound network has resulted in unprecedented advancements in infrasound research [16]. The commercial electrical infrasound sensor (Martec MB2005) was applied to the IMS for the Comprehensive Nuclear-Test-Ban Treaty. However, although electrical infrasound sensors are convenient to use, they are susceptible to electromagnetic interference, difficult to achieve long-distance detection and multiplexing networking [17], which limits their application in certain specific fields, such as pipeline leakage monitoring and underwater acoustic detection [18].

Fiber-optic Fabry–Perot (FP) acoustic sensors, based on the optical FP interference principle, have the advantages of high sensitivity, small structure size, long-distance detection, immunity to electromagnetic interference, easy to multiplexing networking, and so on [19–27]. For an FP acoustic sensor, acoustic-optic transducer diaphragms of various materials were usually considered to improve the detection sensitivity of acoustic pressure response, such as metallic diaphragms [28–31], silicon-related diaphragms [32–34], polymer diaphragms [35,36], two-dimensional materials diaphragms [37–40], and special structure diaphragms [41–44]. On the other hand, the detection sensitivity is directly proportional to the size of the diaphragm. When pursuing miniaturized sensors, the detection sensitivity will be sacrificed due to the small diaphragm size. Furthermore, the detection sensitivity of the sensor is not only constrained by the materials and sizes of transducer diaphragms but also depends on the back cavity structures, which are designed to improve the response characteristics and balance the influence of the environmental factors such as temperature. For infrasound detection, to decrease the low cut-off frequency, the back cavity volume can be increased, but this will also increase the size of the sensor. Another optional approach is mechanical filtering [45,46]. In 2021, Zhang et al. reported a fiber-optic FP sensor for 0.01 Hz infrasound detection by connecting the capillary glass tube through the pressure difference mechanical filtering device. The size of the capillary tube and the back cavity jointly restrict the low cut-off frequency. Nevertheless, the limitation of capillary tube length leads to an increase in the back cavity volume and the sensor size, resulting in a very large size of $>20\mathrm{cm}\times \mathrm{\Phi }8\mathrm{cm}$. In addition, the frequency response range of the sensor was only 0.01–1 Hz [46]. Therefore, for infrasound detection, it is a challenge for fiber-optic FP sensors to simultaneously achieve miniaturization, high sensitivity, and extremely low detectable frequency.

In this work, a fiber-optic FP sensor with a very small size of $12\mathrm{mm}\times \mathrm{\Phi }12.7\mathrm{mm}$ was proposed and demonstrated for highly sensitive mHz infrasound detection. A chromium-silver-gold (Cr-Ag-Au) composite metal diaphragm was selected as the acoustic-optic transducer diaphragm, which has high sensitivity and stable performance. To decrease the low cut-off frequency and reduce the sensor size, the equivalent circuit of the acoustic sensing system was analyzed, and a miniaturized spiral micro-flow hole was designed and fabricated by micro electromechanical system (MEMS) technique to be employed as the connecting hole with a long length. The low detectable frequency can reach 0.01 Hz and the flat response range was from 0.01 to 2500 Hz with a sensitivity fluctuation of $\pm 1.5\mathrm{dB}$. The infrasound signals were demodulated accurately in real time by using the white light interferometry (WLI) Fourier phase demodulation method. The achieved performances of the designed sensor were superior with respect to the reported results, in terms of acoustic pressure sensitivity, minimum detectable pressure (MDP), and low detectable frequency as well as sensor size. Such an infrasound sensor provides a promising solution for the application in natural disaster warning, underwater acoustic detection, pipeline monitoring, and other domains.

2. PRINCIPLES AND SIMULATION

A schematic diagram of the proposed fiber-optic FP acoustic sensor is shown in Fig. 1(a). The end surface of the fiber and the transducer diaphragm formed an FP interference resonator (FP cavity), which can be simplified as a two-beam interferometer due to the low reflectivity of the fiber end surface. When the acoustic wave acts on the sensitive transducer diaphragm, the diaphragm is elastically deformed, and thus the reflection interference spectrum is modulated by the acoustic wave.

 

Fig. 1. (a) Schematic diagram of the fiber-optic FP acoustic sensor. (b) Electrical-mechanical-acoustic equivalent model of the sensor.

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When the thin diaphragm vibrates, the edge experiences minimal displacement while the center experiences maximum since the surrounding of the diaphragm is clamped. For an FP sensor, the fiber end surface must align with the diaphragm center to determine the displacement sensitivity of the diaphragm by calculating the ratio of the shape variable at the midpoint to the applied acoustic pressure. The diaphragm’s tension is $T=Ph$, where $P$ represents the tensile stress and $h$ represents the thickness, and the sensitivity can be expressed as [47]

When selecting diaphragm materials, theoretical analysis and calculations were conducted. On the other hand, for silicon-related diaphragms or polymer diaphragms, due to their low reflectivity, a layer of metal needs to be coated on them to improve reflectivity, which undoubtedly increases the production cost and process difficulty. As for two-dimensional materials, such as graphene, the large-scale preparation and the consistency of sensors are difficult to guarantee. By the calculation and simulation of diaphragm vibration theory, the Cr-Ag-Au composite metal diaphragm was selected as the acoustic-optic transducer diaphragm, which has high sensitivity and stable performance. Cr was selected as the bottom layer because Cr has excellent corrosion resistance and wear resistance, and Cr was easier to grow on the photoresist during evaporation. The temperature of Ag can be kept at a constant level at a low temperature during the coating process, which can control the residual thermal stress of the diaphragm caused by temperature change. As a result, the aggregation of thermal stress can be reduced, and thus the sensitivity can be further improved. Consequently, Ag was selected as the main deformation layer. However, Ag is prone to oxidation in the air to form silver oxide, which can affect the stability of the sensor performance. Hence, a layer of Au diaphragm was evaporated on the upper layer due to the good chemical stability of Au. In addition, Au can provide a high reflectivity for optical interferometry measurement. Furthermore, Ag is a face-centered cubic crystal structure and Cr is a body-centered cubic crystal structure. The interface composed of the two different crystal structures has a strong ability to hinder dislocation motion [48].

Moreover, to improve the low-frequency response of the FP acoustic sensor, the electrical-mechanical-acoustic equivalent model was applied to the sensor design, as shown in Fig. 1(b). In the studied acoustic system, the size of each element is much smaller than the wavelength of the infrasound wave, which can be analyzed as a lumped system [49]. For a diaphragm-based acoustic sensor, the diaphragm’s performance can be easily affected by external environments such as temperature if the sensor cavity is closed, so an open cavity structure can eliminate the influence of external factors by making a connecting hole in the sensor cavity. However, the low-frequency acoustic pressure response of the sensor will be reduced due to the existence of the connecting hole. In the equivalent circuit model, the actual response of the acoustic sensor is determined by the sensitivity $S_d$ of the diaphragm and the transfer function $H$. When the sensor is placed in an acoustic field with an acoustic pressure of $p_{\mathrm{in}}$, only the acoustic pressure acting on the diaphragm can cause the deformation of the diaphragm, which is the effective acoustic pressure $p_{\mathrm{out}}$. The ratio of $p_{\mathrm{out}}$ to $p_{\mathrm{in}}$ is defined as the transfer function $H$. The elements in Fig. 1(b) are the acoustic mass $M_d$ of the diaphragm, the acoustic compliance $C_d$ of the diaphragm, the radiation acoustic mass $M_{\mathrm{rad}}$ of the diaphragm, the radiation acoustic impedance $R_{\mathrm{rad}}$ of the diaphragm, the acoustic impedance $R_{\mathrm{hole}}$ of the connected hole, the acoustic mass $M_{\mathrm{hole}}$ of the balanced hole, and the acoustic compliance $C_{\mathrm{cav}}$ of the cavity. The complete electrical-mechanical-acoustic equivalent model is depicted in Appendix B. It can be seen from the transfer function expression that the low cut-off frequency $f$ of the sensor can be expressed as follows:

Figures 2(a) and 2(b) show the simulation results for the effect of connecting holes size on the transfer function. The presence of the connecting hole does not affect the sensor response in the flat area. However, in the low-frequency range, both the length and radius of the connecting hole greatly influence the frequency response of the sensor. Specifically, shorter connecting hole length and larger connecting hole radius can result in more severe low-frequency attenuation. In addition, the effects of back cavity volume on the transfer function and low cut-off frequency are shown in Figs. 2(c) and 2(d), respectively. It can be seen that a large back cavity volume can lower the cut-off frequency, but a large back cavity volume means a large sensor size. Hence, to enhance the sensor’s response sensitivity in the low-frequency band and minimize attenuation, it is recommended to increase the length of the connecting hole and decrease its radius as much as possible within the constraints of the structure and process, while ensuring that the sensor size does not increase.

 

Fig. 2. (a) Effects of connecting hole radius on transfer function when connecting hole length is fixed. (b) Effects of connecting hole length on transfer function when connecting hole radius is fixed. (c) Effects of back cavity volume on transfer function. (d) Effects of back cavity volume on low cut-off frequency.

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3. EXPERIMENTAL SETUP AND RESULTS

A. Device Design

The MEMS technique was applied to the processing of the precision structures of the composite diaphragm. As shown in Fig. 3(a), the photoresist was used as the sacrificial layer in a silicon wafer and the electron beam evaporation process was adopted to complete the preparation of the high-performance composite metal diaphragms. The thickness was regulated to 20 nm, 500 nm, and 30 nm of Cr, Ag, and Au layers, respectively, by controlling the process parameters, and the density of the composite diaphragm was $10.856\mathrm{g}/{\mathrm{cm}}^3$. A custom metal encapsulation was bonded to the composite diaphragm by epoxy glue and placed on a heating table for heating the epoxy glue to cure. The curing temperature was 80°C and the curing time was 60 min; then the diaphragm was transferred to the metal part by soaking in acetone. The effective sensing radius of the diaphragm was 4 mm.

 

Fig. 3. (a) Fabrication process of the diaphragm. (b) Fabrication process of the spiral micro-flow hole and assembly of the sensor. (c) Left: picture of the single layer of silicon chip before bonding. Right: scanning electron microscope (SEM) image of the spiral groove structure. (d) Left: picture of the dual layers of large-area silicon wafers after bonding. Right: picture of the fiber-optic FP sensor.

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To decrease the low cut-off frequency of the FP infrasound sensor, a design of a spiral micro-flow hole was proposed based on the MEMS technique, which can effectively increase the connecting hole length and reduce the sensor size, as shown in Figs. 3(b) and 3(c). First, the photoresist as a mask was coated on two silicon wafers. Second, the designed patterns of two mirror-symmetrical miniature grooves with a same diameter of about 48 μm were transferred to the photoresist in the two silicon wafers by photoetching, respectively. Third, the two grooves were etched on the two silicon wafers, respectively. The spiral shape extended the total length of each groove to 96.6 mm. Then the photoresist was cleaned off. Fourth, the starting point of one spiral groove and the end point of another spiral groove were etched into two through-holes by using the deep silicon etching method, respectively. The through-holes for optical fiber penetration were simultaneously etched out. Finally, through the silicon–silicon bonding process for the two silicon wafers, the spiral micro-flow holes were formed by the mirror-symmetrical surface-etched spiral grooves inside two layers of silicon wafers. Due to the limitation of the silicon–silicon bonding process, two large-area silicon wafers were used in the above steps. Each small silicon chip with a size of $6\mathrm{mm}\times 6\mathrm{mm}$ containing a spiral micro-flow hole was separated from the large-area silicon wafer by the slicing method after bonding. Figure 3(c) shows the picture of the single layer of silicon chip before bonding and the scanning electron microscope (SEM) image of the spiral groove structure. Figure 3(d) left shows the picture of the dual layers of large-area silicon wafers after bonding. A limit slot was designed on the metal encapsulation to fix the chip, as shown in Figs. 3(a) and 3(b). The physical picture of the completed fiber-optic FP sensor is displayed in Fig. 3(d) right. The volume of the packaged sensor back cavity is $417{\mathrm{mm}}^3$.

B. Experimental Testing

According to the principle of acoustic metrology, we constructed an aeroacoustic testing system and conducted a performance test on the fiber-optic FP sensor, as shown in Fig. 4. Since the coupling cavity can suppress the low-frequency noise in the environment, we utilized the coupling cavity comparison method to measure the sensor response in the low-frequency band (0.01–100 Hz). Moreover, the free-field comparison method was employed to measure the sensor response in the high-frequency band (100 Hz to 5 kHz) due to the uneven distribution of acoustic field in the coupling cavity.

 

Fig. 4. Infrasound sensor calibration experimental system.

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A C-band amplified spontaneous emission (ASE) light source module was implemented as the probe light source, with a wavelength range of 1526–1563 nm. Moreover, the WLI Fourier phase demodulation method was employed for acoustic signal extraction [50,51]. The ASE light was coupled into the fiber-optic FP sensor through a variable optical attenuator (VOA) and a fiber-optic circulator, and the reflected light signals were imported to a fiber Bragg grating analyzer (FBGA). The spectral signal was collected in real-time and input into the computer via the FBGA, which has a high sampling rate of 20 kHz to detect acoustic signals in a wider frequency range. The spectral signals were then processed by a LabVIEW program that integrated a WLI demodulation algorithm to obtain the phase change information.

A low-frequency coupling cavity was used as an acoustic source for the sensor’s low-frequency response measurement from 0.01 to 100 Hz. A commercial B&K 4193-L-004 microphone was exploited as the reference. Both the FP infrasound sensor and the reference microphone were simultaneously positioned at the two input ports of the low-frequency coupling cavity. To measure the high-frequency response from 100 Hz to 5 kHz, a commercial B&K 4190-L-001 microphone and a commercial speaker were used as the reference and the acoustic source, respectively. The FP infrasound sensor and the reference microphone were symmetrically placed on either side of the acoustic source axis at the free field with a distance of approximately 2.0 m from the test acoustic source. To reduce the influence of the environment, all experimental devices were placed in a fully anechoic chamber. When testing at both high frequency and low frequency, the signal generator of a B&K 3160-A-042 acoustic analyzer produced a sinusoidal signal and drove the test sound source to generate acoustic waves through a power amplifier with a gain of 18 dB. The output electrical signal of the reference microphone was collected and analyzed by the acoustic analyzer to obtain the value of test acoustic pressure. Furthermore, the detection sensitivity of the FP infrasound sensor can be obtained by the WLI Fourier phase demodulation algorithm. The photos of the sensor calibration system are shown in Appendix C.

C. Experimental Result

Figures 5(a)–5(c) show the measured frequency spectra of the sensor output signals by the WLI Fourier phase demodulation algorithm under the acoustic waves at 0.01 Hz, 0.05 Hz, and 0.1 Hz, respectively, which demonstrate the superior frequency resolution of the sensor. The demodulated 0.1 Hz acoustic signal with an excellent sine fitting, as shown in Fig. 5(d), means that the acoustic signal was accurately restored. In addition, the linear response of the sensor was also measured, as shown in Fig. 5(e). The linearity of the sensor reached 0.9999 at 5 Hz acoustic wave.

 

Fig. 5. Frequency spectra of the sensor output signals under the acoustic waves with different frequencies: (a) 0.01 Hz, (b) 0.05 Hz, (c) 0.1 Hz. (d) Sinusoidal fitting of time domain signal of 0.1 Hz. (e) Linearity of the sensor at 5 Hz acoustic wave.

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The measured acoustic pressure sensitivity of the sensor from 0.01 Hz to 5 kHz is shown in Fig. 6. The results indicate that the designed fiber-optic FP acoustic sensor can achieve 0.01 Hz acoustic detection and has a flat response from 0.01 to 2500 Hz with a sensitivity fluctuation of $\pm 1.5\mathrm{dB}$, while the average acoustic pressure sensitivity reached $-123.19\mathrm{dB}\mathrm{re}1\mathrm{rad}/\mathrm{\mu Pa}$ at 5 Hz.

 

Fig. 6. Acoustic pressure sensitivity of the fiber-optic FP acoustic sensor from 0.01 Hz to 5 kHz.

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The MDP is another critical index for an acoustic sensor, which is determined by both sensitivity and noise. To accurately assess the sensor noise characteristic, the sensor was placed in a fully anechoic chamber to effectively suppress the potential influence of environmental noise. The noise output signal, denoted as $n(t)$, was collected over a specific period. Subsequently, a frequency domain analysis of $n(t)$ was conducted, and the noise spectrum $N(f)$ was derived through Fourier transformation, as depicted in Fig. 7(a). Given that the total noise sampling time is $T_{\mathit{s}}$, the frequency spectrum abscissa point interval (or frequency resolution) $\mathrm{\Delta }f=1/T_{\mathit{s}}$ post-Fourier transformation. The MDP of the acoustic sensor with phase demodulation at each frequency can be calculated using Eq. (3). For instance, the MDPs were $1.2\mathrm{mPa}/{\mathrm{Hz}}^{1/2}$ at 5 Hz and $29.7\mathrm{\mu Pa}/{\mathrm{Hz}}^{1/2}$ at 250 Hz. The MDPs under different frequencies are shown in Fig. 7(b):

Compared with the currently reported fiber-optic infrasound sensor, the developed sensor in this work has a higher acoustic pressure sensitivity, a better MDP, and a lower detectable frequency. The comparison between different fiber-optic infrasound sensors is listed in Table 1.

 

Fig. 7. Noise test results. (a) Noise spectrum, (b) minimum detectable pressure (MDP) under different frequencies.

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Table 1. Comparison between Different Fiber-Optic Infrasound Sensors

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4. CONCLUSION AND DISCUSSION

In conclusion, a fiber-optic FP infrasound sensor was designed and fabricated from principle to verification, which used a highly sensitive Cr-Ag-Au composite diaphragm as the acoustic-optic transducer diaphragm and a miniaturized MEMS spiral micro-flow hole with a total length of 96.6 mm as a connecting hole. The infrasound sensor achieved 0.01 Hz infrasound detection and a flat response from 0.01 to 2500 Hz with a sensitivity fluctuation of $\pm 1.5\mathrm{dB}$. The MDP reached $1.2\mathrm{mPa}/{\mathrm{Hz}}^{1/2}$ at 5 Hz and $29.7\mathrm{\mu Pa}/{\mathrm{Hz}}^{1/2}$ at 250 Hz. Most importantly, the size of the developed infrasound sensor was only $12\mathrm{mm}\times \mathrm{\Phi }12.7\mathrm{mm}$. Hence, the sensor simultaneously achieved miniaturization, high sensitivity, mHz detectable frequency, and wide frequency response range. Due to the frequency limitation of the reference microphone, the frequency signal below 0.01 Hz was not detected, but it can also be responded in theory. Moreover, the all-fiber detection approach makes the sensor possess the advantages of all-optical passivity, intrinsic safety, immunity to electromagnetic interference, resistance to harsh environments, long-distance detection ability, and multiplexing ability. In practical applications, it is necessary to further optimize the strength of the diaphragm, and in the future, materials with high mechanical strength can be found as transducer diaphragms while ensuring a high sensitivity. Furthermore, the developed small-sized sensor is very convenient to integrate with other sensor devices to compose a multifunctional intelligent detection system. In addition, a sensor array can be formed due to the excellent consistency of the proposed MEMS-based fiber-optic infrasound sensor, and some corresponding algorithms can be combined to achieve the localization or orientation for the sound sources. This work provides a new idea and scheme for the military, disaster warning, pipeline leakage monitoring, and other fields.

APPENDIX A: PRINCIPLES OF THE DIAPHRAGM VIBRATION

Theoretical analysis of the forced vibration of the diaphragm is needed to select the appropriate material for the energy conversion diaphragm. According to the characteristics of the forced vibration of the diaphragm, the vibration equation can be obtained:

(A1)$${\nabla }_r^2\xi -\frac{1}{c^2}·\frac{{\partial }^2\eta }{\partial t^2}=-\frac{p}{c^2\sigma },$$

where $\eta$ is the amount of displacement of the diaphragm, $c=\surd T/\sigma$, $\sigma$ represents the areal density of the diaphragm, and $T$ represents the tension of the diaphragm. When the diaphragm vibrates, the surrounding of the diaphragm is clamped, and the boundary condition of its vibration is ${\xi }_{(r=a)}=0$, where $r$ is the distance from the center of the circle and $a$ is the radius of the diaphragm. The amount of displacement of the diaphragm can be obtained by solving the above equation, and the formula is expressed as follows:

(A2)$$\xi (t,r)=\frac{p_a}{k^{2}T}[\frac{J_0(kr)}{J_0(ka)}-1]·e^{j\omega t}.$$

In the formula, $k=\omega /c,J_0(kr)$ is the zeroth-order Bezier function. The maximum value of $\xi$ can be obtained when $J_0(ka)=0$. The resonant frequency equation for the forced vibration of the diaphragm and the first-order resonance frequency equation when the first-order root ${\delta }_1=2.405$ can be denoted as

(A3)$$f_n=\frac{{\delta }_n}{2\pi a}\sqrt{\frac{T}{\sigma }},$$

(A4)$$f_1=\frac{2.405}{2\pi a}\sqrt{\frac{T}{\sigma }}.$$

The formula above explains that the amount of displacement on a diaphragm when acoustic pressure is applied depends on its distance from the center. The edge experiences minimal displacement while the center experiences maximum. For FP sensors, the fiber end face must align with the center to determine the displacement sensitivity of the diaphragm by calculating the ratio of the shape variable at the midpoint to the acoustic pressure applied. The diaphragm’s tension is $T=Ph$, where $P$ represents the tensile stress and $h$ represents the thickness, and the sensitivity can be expressed as Eq. (1) in the main text.

The simulation of the influence of various parameters on the diaphragm sensitivity is shown in Fig. 8. It can be seen that in order to improve the sensitivity of the diaphragm, it is necessary to reduce the tensile stress, increase the diameter, and reduce the thickness of the diaphragm. Considering that a diaphragm with a too large or too thin size is more fragile and prone to damage during practical use, a Cr-Ag-Au composite diaphragm with a radius of 4 mm and a total thickness of 550 nm was selected.

 

Fig. 8. Relationship between diaphragm sensitivity and various parameters. (a) Stress, (b) radius, (c) thickness, (d) density.

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APPENDIX B: PRINCIPLES OF THE ELECTRICAL-MECHANICAL-ACOUSTIC EQUIVALENT MODEL

When studying acoustic sensors, the parameters in acoustics can be equivalent to those in circuits, and it is possible to analyze the acoustic phenomenon by solving the equivalent circuit equation. But it should be noted that the electric acoustic equivalence model is mainly applicable to lumped systems, that is, the sizes of the acoustic objects (cavities, stubs, small holes, etc.) are much smaller than the wavelength of the acoustic wave. In the studied acoustic system, the size of each element is much smaller than the wavelength of the acoustic wave, which can be analyzed as a lumped system. For a diaphragm acoustic sensor, an open cavity structure with connecting holes can be used to eliminate sensor performance changes caused by external environments. However, the low-frequency response of the sensor will be reduced due to the connecting hole. In the equivalent circuit model, the actual response of the acoustic sensor is determined by the sensitivity $S_d$ of the diaphragm and the transfer function $H$:

(B1)$$S_{\mathrm{fact}}=H\cdot S_d.$$

The internal structure and the equivalent circuit system of the proposed FP acoustic sensor are shown in Figs. 1(a) and 1(b), respectively, among them $Z_1=R_{\mathrm{rad}}+j\omega (M_d+M_{\mathrm{rad}})$, $Z_2=1/j\omega C_d$, $Z_3=1/j\omega C_{\mathrm{cav}}$, $Z_4=R_{\mathrm{hole}}+j\omega M_{\mathrm{hole}}$. Therefore, the transfer function $H$ can be expressed as

(B2)$$H=\frac{p_{\mathrm{out}}}{p_{\mathrm{in}}}=\frac{Z_2{Z}_4}{Z_3{Z}_4+(Z_1+Z_3)(Z_3+Z_4)}.$$

The lumped element definitions are shown in Table 2.

Table 2. Lumped Element Definitions

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It can be seen from the transfer function expression that the sensing system has two resonance frequencies:

(B3)$$f_{\mathrm{high}}=\frac{1}{2\pi }\sqrt{\frac{C_{\mathrm{cav}}+C_d}{C_{\mathrm{cav}}{C}_d(M_{\mathrm{rad}}+M_d)}},$$

(B4)$$f_{\mathrm{low}}=\frac{1}{2\pi }\frac{C_{\mathrm{cav}}+C_d}{C_d{R}_{\mathrm{hole}}(1+C_{\mathrm{cav}}/C_d)}.$$

$f_{\mathrm{high}}$ is the resonant frequency, and $f_{\mathrm{low}}$ is the cut-off frequency of low frequency.

APPENDIX C: ACOUSTIC TEST PHOTOS OF THE FIBER-OPTIC FP SENSOR

The photos of the sensor calibration system are shown in Fig. 9, including high-frequency testing and low-frequency testing, respectively.

 

Fig. 9. Acoustic test photos of the fiber-optic FP sensor.

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Funding

Science Fund for Creative Research Groups of the Nature Science Foundation of Hubei (2021CFA033); National Natural Science Foundation of China (62275096); Interdisciplinary Research Program (HUST: 2023JCYJ046).

Acknowledgment

We thank engineer Jun Su in the Center of Optoelectronic Micro & Nano Fabrication and Characterizing Facility, Wuhan National Laboratory for Optoelectronics of Huazhong University of Science and Technology for the support in test and analysis.

Disclosures

The authors declare no competing interests.

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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