# Optical accelerometers for Detecting Low-Frequency Micro-Vibrations
## 5 Optical Phase Modulation  
Optical phase-modulated accelerometers use the phase change of an optical wave in an optical fiber caused by external vibrations to detect vibrations. Optical accelerometers based on optical phase modulation generally consist of two parts: an optical fiber interferometer and a vibration-sensitive elastic mechanism. Fiber optic interferometers can be classified into Michelson interferometers (MSIs), Mach-Zehnder interferometers (MZIs), and Fabry-Perot interferometers (FPIs) in accordance with the interference principle. An MSI is a dual-beam interferometer in which coherent light from a laser is incident on two single-mode fibers, one of which is the detection fiber while the other is the reference fiber, as shown in Figure 12. [PERSON] et al. proposed an optical accelerometer by winding optical fibers on the top and bottom of an elastic cylinder with a mass block in the middle [49]. This accelerometer achieves differential vibration measurements and improved sensitivity, as shown in Figure 13a. [PERSON] et al. fixed the detection fiber of \begin{table}
\begin{tabular}{c c c c c c} \hline
**Reference** & **Principle** & **Structure** & **Resonant Frequency (Hz)** & **Frequency Response Range (Hz)** & **Sensitivity** \\ \hline [PERSON] et al. [31] & OWG & Cantilevered fiber optics & 125 & – & 6\%/g \\ [PERSON] et al. [32] & OWG & Cantilevered fiber optics & 3.3 k & 20–2000 & 3.3 mV/g \\ [PERSON] et al. [33] & OWG & Symmetrical flexure hinge & 5.5 k & \textless{}1000 & 2.46 mV/g \\ [PERSON] et al. [34, 35] & OWG & Symmetrical flexure hinge & 51 & – & 32.14 dB/g \\ [PERSON] et al. [36] & OWG & Symmetrical flexure hinge & – & – & 1.8 V/g \\ [PERSON] et al. [37] & OWG & Symmetrical flexure hinge & – & – & 13.1 dB/g \\ [PERSON] et al. [38] & GS & Flexure hinge & – & – & 0.04 V/g \\ [PERSON] et al. [39] & GS & Flexure hinge & 60 & 10–1.5 k & – \\ [PERSON] et al. [40] & GS & Flexure hinge & 36 & – & 590 V/g \\ [PERSON] et al. [44, 45] & OSS & Cantilevered flexure hinge & 550 & \textless{}400 & 0.6 V/g \\ [PERSON] et al. [46] & OSS & Inverse spring & 2.3 & \textless{}0.02 & 41 ng/Hz\({}^{1/2}\) \\ [PERSON] et al. [47] & OSS & Asymmetric spring & 3 & \textless{}0.1 & 8.1 ng/Hz\({}^{1/2}\) \\ [PERSON] et al. [48] & OSS & V-shape elastic beam & 8.24 & \textless{}1 & 3800 μm/g \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters of the accelerometer based on optical intensity modulation. Figure 11: OSS-based accelerometer: (**a**) cantilevered and (**b**) asymmetric spring. Figures are reprinted with permission from: (**a**) Ref. [44]. Copyright 2000 Elsevier; (**b**) Ref. [47]. Copyright 2019 Springer Nature. an MSI to the seismic mass of an optical accelerometer. The mass is fixed onto the base of the accelerometer by elastic beams, as shown in Figure 13b. The frequency response range of this accelerometer is 5-500 Hz, which is a wide spectral response range [50,51]. In addition to elastic cylinders and elastic beams, elastic diaphragms with low resonance frequencies are also used as vibration-sensitive structures for optical accelerometers. [PERSON] et al. achieved vibration measurement by fixing the detection fiber of an MSI onto a different elastic structure and using the bending stress of the diaphragm vibration to load onto the detection fiber and change its phase [52, 53, 54]. MZI is similar to MSI. Although both are two-beam interferometers, the difference is that MZI does not use end-face reflection, but instead, couples two beams to interfere for a transmission interferometer, as shown in Figure 14. The phase shift \(\Delta\varphi\) from the strain caused by the acceleration \(a\) is  
\[\varphi(a)=\frac{2\pi}{\lambda_{0}}n_{f}\Delta L(a), \tag{11}\]  
where \(\lambda_{0}\) is the wavelength of incident light, \(n_{f}\) is the effective refractive index of the optical fiber, and \(\Delta L(a)\) is the amount of change in the detection fiber caused by the acceleration. The MZI optical accelerometer structure is also similar to that of MSI. [PERSON] et al.