# Optical accelerometers for Detecting Low-Frequency Micro-Vibrations
## 6 Optical Wavelength Modulation  
Accelerometers based on optical wavelength modulation detect acceleration by measuring the wavelength drift of the output light under the effect of acceleration. The two main types of optical wavelength modulated accelerometers include fiber optic Bragg grating (FBG) accelerometers and photonic crystal (PHC) accelerometers. Research on FBG started as early as 1978 [71; 72]. FBG is a periodic modulation of the refractive index in the fiber core. When a broadband light is transmitted along with the fiber core, the FBG reflects the narrowband part of a specific wavelength while the rest of the broadband light passes through. The central wavelength of the reflected spectrum is defined as \(\lambda_{b}\) and can be expressed in terms of two parameters [73; 74]:  
\[\lambda_{b}=2*n_{f}*\Lambda, \tag{13}\]  
where \(n_{f}\) is the effective refractive index of the optical fiber, and \(\Lambda\) is the periodicity of the grating. The wavelength shift of \(\lambda_{b}\) is decided by the changes of \(n_{f}\) and \(\Lambda\), which are mainly influenced by the axial strain \(\Delta\epsilon\) and environment temperature \(\Delta T\). In this case, the Bragg wavelength dependency on the strain and temperature could be described by [75]  
\[\frac{\Delta\lambda_{b}}{\lambda_{b}}=(1-P_{e})*\Delta\epsilon+(\alpha_{f}+ \xi)*\Delta T, \tag{14}\]  
where \(\alpha_{f}\) is the thermal expansion coefficient, \(\xi\) is the thermal optic coefficient, and \(P_{e}\) is the elastic optical coefficient. [PERSON] et al. designed an original FBG-based optical accelerometer, shown in Figure 17a, wherein the FBG is arranged in an elastic material [76]. The measurement of vibration is achieved by measuring the central wavelength drift of the FBG. This accelerometer has a sensitivity of \(1\) ng/Hz\({}^{1/2}\) and a bandwidth of 2 kHz. In addition to placing FBG inside an elastic material, fixing FBG onto an elastic beam has also been used in the design of optical accelerometers. The optical accelerometer designed by [PERSON] et al. placed a seismic mass in the middle of two simply supported beams and fixed the FBG onto the lower surface of the lower simply supported beam, as shown in Figure 17b [77]. When external vibration is loaded onto the sensor, the deformation of the beam causes the center wavelength of the FBG to drift, and the amplitude and frequency information of the external vibration \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Reference** & **Principle** & **Structure** & **Resonant** & **Frequency Response** & **Sensitivity** \\  & & & **Frequency (Hz)** & **Range (Hz)** & **Sensitivity** \\ \hline [PERSON] et al. [49] & MSI & Elastic cylinder & 300 & \(<\)100 & – \\ [PERSON] et al. [50; 51] & MSI & Elastic beams & 400 & 10–150 & 0.66 V/g \\ [PERSON] et al. [52] & MSI & Elastic diaphragm & 1.6 k & 100–800 & 2.2 rad/g \\ [PERSON] et al. [59] & MSI & Elastic diaphragm & 3.8 k & \(<\)1 k & 35.9 dB re rad/g \\ [PERSON] et al. [54] & MSI & Cantilever beam & 490 & 10–300 & 1.13 V/g \\ [PERSON] et al. [56] & MZI & Elastic diaphragm & – & \(<\)25 k & – \\ [PERSON] et al. [57] & MZI & Elastic cylinder & 1 k & 10–800 & 38 dB re rad/g \\ [PERSON] et al. [58] & FPI & Elastic beams & 450 & \(<\)47.7 & 220 ng/Hz\({}^{1/2}\) \\ [PERSON] et al. [59] & FPI & Combined elastic beams & 11.75 k & \(<\)5 k & 0.3 mV/g \\ [PERSON] et al. [60] & FPI & Parallelogram flexible & 11 k & 1.5–12 k & 100 ng/Hz\({}^{1/2}\) \\ [PERSON] et al. [61; 62] & FPI & Cantilever beam & 520 & 20–160 & 20 V/g \\ [PERSON] et al. [63] & FPI & Symmetrical leaf spring & 144 & 20–120 & 36 dB re rad/g \\ [PERSON] et al. [64; 65] & GI & Four cantilever beams & 1 k & \(<\)200 & 619 V/g \\ [PERSON] et al. [66] & GI & Four serpentine-shaped & 34.5 & \(<\)5 & 2485 V/g \\ [PERSON] et al. [69] & GI & Eight cantilever beams & 7553 & 100–2000 & – \\ [PERSON] et al. [70] & GI & Six serpentine-shaped & 24.8 & \(<\)3 & 60 V/g \\  & & flexure beams & & & \\ \hline \hline \end{tabular}
\end{table}