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Abstract
Transducer components are crucial in optimizing the sensitivity of microphones. Cantilever structure is commonly used as a structural optimization technique. Here, we present a novel Fabry-Perot (F-P) interferometric fiber-optic microphone (FOM) using a hollow cantilever structure. The proposed hollow cantilever aims to reduce the effective mass and spring constant of the cantilever, thereby enhancing the sensitivity of the FOM. Experimental results demonstrate that the proposed structure outperforms the original cantilever design in terms of sensitivity. The sensitivity and minimum detectable acoustic pressure level (MDP) can reach 91.40 mV/Pa and 6.20 µPa/$\sqrt {\textrm {Hz}}$ at 1.7 kHz, respectively. Notably, the hollow cantilever provides an optimization framework for highly sensitive FOMs.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Microphones are electro-acoustic transducers that convert acoustic energy into electricity, underpinning techniques such as industrial measurement [1], sonar [2], photoacoustic spectroscopy [3], photoacoustic imaging [4], and gas leakage [5]. In the past few decades, microphones can be primarily divided into three types: dynamic microphones, piezoelectric microphones, and condenser microphones [6]. However, conventional electro-acoustic microphones are plagued by low sensitivity and electrical noise interference [7,8]. Recently, fiber-optic microphones (FOMs) [9–11] have attracted wide interest due to advantages such as high sensitivity, high signal-to-noise ratio (SNR), quick response, compact size, immunity to electromagnetic interference, and the feasibility of multiplexing.
Sensitivity is key to the microphone applications. FOMs rely on photons rather than electrons for acoustic sensing, thereby enabling exceptionally high sensitivity [12]. In the literature, the sensitivity optimization of FOMs encompasses each stage of the optical signals, namely transmission, modulation, and transduction. The optimization of optical transmission aims to minimize phase loss in the medium and improve fringe visibility, which can be achieved via some techniques such as reflective coating, photonic crystal fiber [13,14], and collimators [15,16]. Nevertheless, the fabrication cost and complexity pose significant drawbacks to these optimization strategies.
The optical signal can be modulated by using external and internal techniques. The former is realized using optical devices such as electro-optic modulators and piezoelectric transducers, which produce frequency shifts and mitigate noise or other interfering signals [17–19]. However, these techniques result in a large sensing system being applied in narrow spaces [20]. In comparison, internal modulation is realized within FOMs, whereby the device itself functions as an optical modulator through phase-modulating, intensity-modulating, and polarization-modulating mechanisms [21–25]. Among these configurations, the F-P interferometric fiber-optic microphone (FPIFOM) [26,27], typically formed by a cleaved fiber end face and the inner surface of a thin rigid diaphragm, offers a more compact size, simpler fabrication, and higher sensitivity.
The effectiveness of the optical transduction is largely dependent on the performance of a transducer component (e.g., diaphragm), which is determined by its material properties and structural designs. Various materials, including metal [28–30], silicon [31], graphene [32], etc., have been investigated for acoustic-sensitive diaphragms. However, the sensitivity is hindered by the inner tension and the radial stretching of the diaphragm material in response to pressure variations [33]. To overcome these limitations, a structural design known as the cantilever has been researched [34,35], which is typically a rectangular plate with one fixed side and three sides that are free to vibrate. As a classic structure, the cantilever exhibits ideal performance in elastic deformation and linear response. First, its free-end displacement amplitude is potentially two orders of magnitude greater than that of a diaphragm subjected to the same acoustic pressure [36]. Second, the cantilever only bends but does not stretch as the pressure changes, thereby eliminating the non-linear response in the diaphragm. The impact of cantilever dimensions, such as length, width, and thickness has been discussed in Ref. [37,38]. However, few works have been reported on the use of tailored structures, such as hollow cantilevers, to improve the sensitivity of FOMs; despite the fact that hollow cantilevers act excellently as a versatile probe (or receptor) in atomic force microscopy (AFM) [39] and biosensing [40].
This paper presents, for the first time, a sensitivity-enhanced FPIFOM using a hollow cantilever. The proposed FPIFOM can achieve a flat frequency response ($\pm$3 dB) from 700 Hz to 1.9 kHz, as well as a noise-limited minimum detectable pressure (MDP) level of 6.20 $\mathrm {\mu }$Pa/$\sqrt {\textrm{Hz}}$ at 1.7 kHz. The experimental results indicate that the proposed FPIFOM, featuring a hollow stainless steel cantilever structure, yields a sensitivity of 91.40 mV/Pa, which is significantly higher than that of the original FPIFOM with an identical structure, that achieves a sensitivity of 28.47 mV/Pa. Compared to the dimension designs of the original cantilever, such as increasing the length, decreasing the width, and so on, the proposed hollow cantilever has not hindered the resonant frequency. Therefore, this finding suggests a structural optimization framework for highly sensitive FOMs.
2. Theory and sensor design
The proposed FPIFOM is an innovative device that exploits a single-mode fiber (SMF) and a 3D-printed epoxy shell to detect mechanical vibrations from sound waves. The device consists of a cleaved fiber end face and the inner surface of a reflective cantilever to form the F-P interferometer cavity. A schematic of the F-P structure in FOM is shown in Fig. 1(a). The deflections in the cantilever induce changes in the length of the F-P cavity, leading to a phase shift in the reflected light. The interference model can be simplified as a two-beam interferometer, and the total electric field $E_r$ reflected from the two surfaces can be described as [15]:
Fig. 1. (a) Schematic of the F-P interference in FPIFOM. (b) Model of the hollow cantilever. (c) Bending deflection analysis of the hollow cantilever.
The diaphragm of the FPIFOM is flexible but constrained by a frame or a shell, which limits its sensitivity and causes non-linearity in its response. To address this issue, a hollow cantilever structure with two penetration holes placed on the surface has been developed, as shown in Fig. 1(b). This design aims to decrease the effective mass and reduce the spring constant of the cantilever, thereby enhancing its sensitivity to acoustic vibrations. The schematic deflection of a hollow cantilever is shown in Fig. 1(c). When a load $P$ is applied at a point of position $x_1$ from the fixed end of the cantilever, it is transformed into a vibration signal. The displacement $z_1$, $z_2$, and $z$ from the equilibrium can be expressed as [41]:
where $E$ is Young’s modulus of the cantilever material, and $I$ is the moment of inertia, $I = wt_c^3/24$. $L$, $w$, and $t_c$ are the length, width, and thickness of the hollow cantilever, respectively. In this case, two identical penetration holes were fabricated on the surface of cantilever to enhance its performance. Assume $w_1 = w/4$, $l_1 \approx {L}$, where $w_1$ and $l_1$ are the width and length of the hole, respectively. When the same load applies equally on the entire length of the cantilever, the equivalent displacement $z_{eq}$ and the spring constant $k_c$ of the hollow cantilever can be expressed as:Considering the impact of air damping, the motion of the hollow cantilever can be described as a point mass model using a harmonic oscillator [33]. The equation of displacement $z$ is
where $F{\textrm{cos}}(\omega t)$ is the sinusoidal force cause by acoustic wave, $m_{eff}$ is the effective mass of the hollow cantilever, $\beta$ is the damping constant, $k$ is the total spring constant. The phenomenon of air compression within the FPIFOM due to the bending of the cantilever in response to acoustic waves results in the emergence of an incremental spring constant, denoted by $k_1$. The total spring constant $k$ in Eq. (5) can be expressed as a sum of $k_c$ and $k_1$:Here, $S$ is the area of the hollow cantilever, $S=Lw/2$, and $\kappa$ is the specific heat ratio. It is important to note that the hollow structure reduces both the area and mass of the cantilever, making it easier to bend. Therefore, the solution z($\omega$) of Eq. (5) gives the amplitude of the hollow cantilever as:
where the first-order resonance occurs at frequency $\omega _0=2\pi f_0$, and the $f_0$ is:The optimization of the sensitivity of FPIFOM heavily relies on the parameters and the structure of the cantilever. According to Eq. (7) and Eq. (8): First, increasing the length, as well as decreasing the width and thickness, can increase the deflection under the same external force. However, such parameter variations result in a decrease in the resonant frequency, thereby leading to a reduction in frequency response bandwidth. Second, the cantilever must not be too long as the free-end would collapse due to gravity. On the other hand, the impact of air damping cannot be overlooked when the thickness scale decreases to micrometers. In contrast, the proposed hollow structure reduced the effective mass and spring constant of the original cantilever, leading to increased displacement without significantly affecting the bandwidth. The proposed FPIFOM and its assembly components are shown in Fig. 2.
Fig. 2. (a) FPIFOM assembly components and sensor head. (b)-(c) The hollow and original FPIFOMs. (d)-(e) Simulations of the hollow and original cantilevers.
In this study, we designed a circular stainless steel diaphragm of 10 mm diameter and 10 $\mathrm {\mu }$m thickness, featuring a 2 mm $\times$ 0.5 mm rectangular cantilever in the middle, and two 1.5 mm $\times$ 0.125 mm rectangular penetration holes placed parallel to the cantilever. The gap between the cantilever and the circular diaphragm was 40 $\mathrm {\mu }$m, with the square free-end of the cantilever positioned at the center of the circle of the diaphragm. The diaphragm using a hollow cantilever was clamped by two circular stainless steel supports, each having the same diameter as the diaphragm and a thickness of 100 $\mathrm {\mu }$m. Both clamping supports featured a 2.2 mm $\times$ 1 mm rectangular square hole in the middle, which allowed the cantilever beam to swing freely. We encapsulated this three-layer structure into a 3D-printed epoxy shell, as shown in Fig. 2(a) and (b). As a comparison group, we fabricated another FPIFOM using the original 2 mm $\times$ 0.5 mm cantilever in an identical structure, as shown in Fig. 2(c). This three-layer design helped to suppress the thin diaphragm from collapsing, thus improving optical interference and suppressing the non-linear vibration of the diaphragm. The diaphragm and supports were attached to the shell by UV adhesive. Finite element analysis was used to assist the design of the FPIFOM, taking into consideration the vibration damping of the cantilever. According to the vibration characteristics analysis performed using COMSOL, as shown in Fig. 2(d) and (e), the hollow cantilever exhibited a first resonant frequency of 1479 Hz, whereas the original cantilever had a first resonant frequency of 1824 Hz. The proposed structure caused a slight shift of 345 Hz in the resonant frequency. Furthermore, under a uniform 1 mN external force distributed over the entire surface, the displacement of the hollow cantilever at positions 1.6 mm and 2.0 mm along the x-axis were simulated to be 0.38 nm and 0.28 nm, respectively. Simulation results demonstrated that the displacement of the hollow cantilever was greater than the original one, indicating an improvement in sensitivity.
After the assembly of the diaphragm and shell, the reflected F-P interference spectrum was detected by a spectrometer (FBGA, BaySpec) for both FPIFOMs. The ceramic ferrule in the shell was adjusted to reach maximum fringe visibility. Figure 3 presents the interference spectra of the hollow (red line) and original (black line) FPIFOMs using a SLED source under identical condition. As depicted in Fig. 3, the spectral combs exhibit the same fringe visibility (5dB) near 1540 nm, indicating the strong F-P interference within the FPIFOMs. Considering the use of a 1574 nm laser source in Section 3, we estimated the quadrature (Q) points $Q_1$ (1573.2 nm) and $Q_2$ (1575.1 nm) near 1574 nm, as shown in the inset. These Q points were further adjusted in later experiments.
3. Experimental data
The experimental setup utilized for testing the hollow FPIFOM in the context of acoustic sensing is depicted in Fig. 4. A near-infrared DFB laser (1574 $\pm$ 1 nm, Fitel, Japan) was used in the experiments. To tune the wavelength, temperature control and current control were employed, with a multi-wavelength meter (Keysight 86120D) utilized for observing wavelength movement. An acoustic wave was generated by a speaker, which received a signal from a signal generator and was amplified by a power amplifier. To measure the sound pressure, a commercial condenser microphone (Type 4966, BK, 47.8 mV/Pa) was positioned next to the sensor head. The sensor head, speaker, and commercial microphone were all enclosed within a sound isolation box. During the experiment, incident light was transmitted through a circulator, reflected by the sensor head, and detected by a photodetector (PDB450C, Thorlabs). The electrical signal was acquired by a DAQ (NI 9250, National Instrument) and presented on a PC for further analysis.
In this study, we conducted experiments on two FPIFOMs, i.e., one with a hollow cantilever and the other with the original cantilever under identical conditions. The applied sinusoidal sound field had a frequency range of 300 Hz to 3kHz, with the sound pressure divided into eight steps and controlled by the power amplifier. In the test group, the hollow sensor head was initially connected. The wavelength of the laser is tuned to be near $Q_1$ so that the output voltage exhibits the highest slope under the acoustic field. Next, a single-frequency acoustic signal was applied via the signal generator, and the sound pressure was increased step by step through the power amplifier. The output voltage of the sensor and commercial microphone was recorded at each step of the sound pressure. To assess the response of the sensor under the sound field, we varied the sound frequency and repeated the aforementioned steps. As a comparison group, the same procedure was utilized for testing the original FPIFOM.
Figure 5 displays the time domain waveforms and corresponding frequency spectra detected by the hollow FPIFOM (red lines) under 40 mPa acoustic pressure, with the frequency range spanning from 600 Hz to 1.7 kHz. As the acoustic sensing system interacts with the environment and the components are not ideal, several noise components, including acceleration, acoustic, and electrical noise, can impact the measurements. The experimental results reveal that the waveforms at 1.2 kHz and 1.7 kHz are smooth, indicating the hollow FPIFOM’s better signal recovery ability. However, the waveform at 600 Hz appears slightly distorted, due to the poor performance of commercial speakers in the low-frequency sound band and electrical noise generated by the experimental instruments. Therefore, low-noise devices and shielded cable should be utilized when conducting low-frequency measurements or high-power devices are in the vicinity. Moreover, the acceleration and acoustic noises occurred, which cause small peaks as shown in Fig. 5(e). When conducting measurements near the resonant frequency, the acoustic pressure should be controlled as high-order harmonics tend to appear, as shown in Fig. 5(f). The highest SNR of 76.2 dB is observed at a frequency of 1.7 kHz, which corresponds to the minimum detectable pressure (MDP) of 6.20 $\mathrm {\mu }$Pa/$\sqrt {\textrm{Hz}}$ at 1 Hz resolution bandwidth. Notably, the SNR at 1.7 kHz is substantially higher than that at 600 Hz. For comparison, the acoustic responses of the original FPIFOM were also tested as shown in Fig.5 (black lines). The response of the hollow FPIFOM presents a quasi-sinusoidal curve (red curve) with a peak-to-peak voltage of 8 mV and the reference FPIFOM (black curve) has a smaller peak-to-peak voltage of 2 mV.
Fig. 5. The time domain waveforms (a)-(c) and the corresponding frequency spectra (d)-(f) at the frequency of 600 Hz, 1.2 kHz, and 1.7 kHz, respectively. The time domain waveforms were normalized.
Figure 6(a) illustrates the linear fits for the output voltage signals of the two FPIFOMs as a function of acoustic pressure at a frequency of 600 Hz. The estimated acoustic pressure sensitivity of the FPIFOM with a hollow cantilever is 11.4 mV/Pa (red line), which is higher than that of the reference sensor with the original cantilever (black line) at 4.6 mV/Pa. The acoustic sensitivity of both sensors in the frequency range of 300 Hz to 3 kHz is presented in Fig. 6(b). The proposed FPIFOM with a hollow cantilever shows a flat response region from 700 Hz to 1.9 kHz and a resonance peak at 2.1 kHz. The sensitivity is 91.40 mV/Pa at 1.7 kHz, which is higher than that of the original sensor (28.47 mV/Pa at 1.7 kHz). On the other hand, the frequency response range of the FPIFOM with the original cantilever is 700 Hz to 1.7 kHz, which is narrower than the proposed FPIFOM. Therefore, the proposed FPIFOM with a hollow cantilever exhibits better acoustic sensitivity and a wider frequency response range than the reference sensor with the original cantilever.
Fig. 6. (a) Fitting curves of acoustic pressure responses of the FPIFOMs with hollow (red line) and the original (black line) cantilevers at 600 Hz. (b) Frequency response sensitivity of the FPIFOMs at different frequencies from 300 Hz-3 kHz.
4. Conclusion
This paper demonstrates, for the first time, a sensitivity optimization technique for FOMs using a hollow cantilever. The proposed hollow structure is theoretically shown to reduce the effective mass and spring constant of the original cantilever, thereby enhancing its deformation displacement under acoustic pressure. This improvement comes at an acceptable cost of a slight shift in resonant frequency, which does not plague the frequency response bandwidth. Experimental results demonstrate that the proposed FPIFOM yields a sensitivity of 91.40 mV/Pa at 1.7 kHz, surpassing the sensitivity of its original counterpart of 28.47 mV/Pa. It demonstrates a flat frequency response over a broader range from 700 to 1.9 kHz. Moreover, it also exhibits a higher SNR of 76.2 dB at 1.7 kHz under the same acoustic pressure, indicating a superior MDP level. Notably, the proposed hollow cantilever represents an optimization strategy, encompassing geometries such as circular, triangular, U-shaped, ladder-shaped, and hollow cross-section designs. These findings offer a novel framework for optimizing sensitivity in future FOM designs, with significant potential for advancing the field of acoustic sensing.
Funding
National Key Scientific Instrument and Equipment Development Projects of China (62027816); National Natural Science Foundation of China (62005247, 62271451); Zhengzhou Collaborative Innovation Major Project (18XTZX12008); Henan Provincial Science and Technology Research Project (222102210163); Science and Technology Major Project of Henan Province (221100230300); Henan Provincial Postdoctoral Research Grant (NO.202102011).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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