# Optical accelerometers for Detecting Low-Frequency Micro-Vibrations
## 5 Optical Phase Modulation
wrapped an optical fiber of MZI around the transformer core, and an acceleration value can be obtained by demodulating the change in the optical phase inside the fiber [55]. To improve the sensitivity of the MZI accelerometer, [PERSON] et al. fixed the detection fiber of the MZI interferometer in the form of a spiral on a diaphragm with a mass block to maximize the mechanical sensitivity of the MZI accelerometer, but the measurement range is small [56]. [PERSON] et al. proposed a 3D MZI optical accelerometer by mounting three-component optical accelerometers in orthogonal orientation [57]. Each single-component optical accelerometer is constructed by wrapping MZI detection fiber around an elastic cylinder, similar to the  
Figure 12: Schematic of the MSI. Figure 13: MSI-based accelerometer: (**a**) elastic cylinder and (**b**) elastic beam. design of [PERSON] et al. [49]. The frequency response of this 3D optical accelerometer is within the range of 10-800 Hz. MZI and MSI exhibit similarities in measurement, sensor structure design, and signal demodulation. However, MSI exhibits two times the sensitivity of MZI when light wave makes one round trip in fiber with the same fiber length. In addition, MSI does not have a closed fiber loop and the fiber winding process is simpler. Therefore, MSI is more widely used than MZI. In addition to MSI and MZI, which are two-optical interferometric sensing systems, the single-optical interferometric-based FPI is also widely used in optical accelerometers. The structure of FPI is shown in Figure 15. Its resonant cavity consists of two optical fiber end faces, wherein the incident light is reflected several times between the two end faces. The phase difference between two adjacent emitted lights is the same, and all reflected lights enter the detection system to produce multi-beam interference. The relationship between the end-face displacement \(\Delta L\) and the phase shift \(\Delta\varphi\) is  
\[\Delta\varphi=\frac{4\pi n_{f}\Delta L}{\lambda_{0}}, \tag{12}\]  
where \(\lambda_{0}\) is the wavelength of the incident light, and \(n_{f}\) is the effective refractive index of the optical fiber. Given that it is subject to acceleration, the inertial force of the seismic mass drives the end face of FPI, changing the reflection or transmission spectrum. The magnitude of input acceleration can be obtained by analyzing the spectrum. [PERSON] et al. proposed an optical accelerometer that consists of two identical hemispherical FPIs [58]. Two identical spherical metal mirrors are attached onto the central diaphragm (one on each side), forming the outer mirror of FPI, while the distal end of the fiber is used as the inner mirror. This optical accelerometer has a sensitivity of 2.2 \(\times\) 10\({}^{-7}\) g/Hz\({}^{1/2}\) and a resonant frequency of 450 Hz. Early FPI optical accelerometers exhibited high performance. In recent years, many researchers have started improving the mechanical design of optical accelerometers to increase their sensitivity. For example, [PERSON] et al. introduced a V-beam displacement amplification structure into the accelerometer structure, as shown in Figure 16[59]. The inertial displacement of the mass causes the V-shaped structure to compress in the \(x\)-direction and expand in the \(y\)-direction, modulating the Fabry-Perot cavity length and changing the wavelength of the output light. The bandwidth of this accelerometer is about 10 kHz, and the maximum mechanical amplification is 18.6. Figure 14: Schematic of the MZI. Figure 15: Schematic of the FPI. Improving the resonant cavity of FPI is another strategy for increasing the sensitivity of an optical accelerometer in addition to improving its mechanical design. [PERSON] et al. proposed an optical accelerometer based on an improved FPI that combines a parallelogram flexible oscillator and an optical fiber micro cavity [60]. This type of accelerometer can reach a sensitivity of \(100\)\(\mathrm{ng/Hz^{1/2}}\).