# Reviews on Corrugated Diaphrags in Miniature Fiber-Optic Pressure Sensors
## 2 With initial stress
### Mechanical Properties of the single deep corrugated diaphragm (SDCD)  
Figure 6a and b show the schematic cross section of the SDCD structure with geometrical parameters and a three-dimensional cross-section of the microphone fabricated by SDCD [45], respectively. \(h_{d}\) is the thickness of the diaphragm, \(2a\) is the side length of the flat bottom square diaphragm, and \(H\) is the vertical distance between the bottom surface and the fixed point. The SDCD consists of a FD which is suspended all around with free sidewalls. The flat bottom region is believed to behave just like a normal flat square diaphragm with lower stiffness. The suspending sidewalls are expected to release the residual stress of the SDCD structure.  
The clamped square diaphragm's deflection without initial stress can be written as [41]  
\[\frac{Pa^{4}}{E{h_{d}}^{4}}=\frac{4.2}{(1-\
u^{2})}\frac{\omega_{0}}{h_{d}}+ \frac{1.58}{(1-\
u^{2})}\frac{\omega_{0}^{3}}{h_{d}^{3}}, \tag{12}\]  
2. With initial stress  
For large initial stress, the deflection equation of a flat circular diaphragm can be represented by [41]  
\[\frac{PR^{4}}{E{h_{d}}^{4}}=\frac{4\sigma R^{2}}{E{h_{d}}^{2}}\frac{\omega_{0} }{h_{d}}, \tag{13}\]  
By placing \(R^{2}\) with \(4a^{2}/\pi\) in (13), this resistance to bending due to initial stress can be added to (12) using the principle of superposition. The deflection equation of the square diaphragm with initial stress is [46]  
\[\frac{Pa^{4}}{E{h_{d}}^{4}}=\frac{4.2}{(1-\
u^{2})}\frac{\omega_{0}}{h_{d}}+ \frac{\pi\alpha a^{2}}{E{h_{d}^{2}}}\frac{\omega_{0}}{h_{d}}+\frac{1.58}{(1- \
u^{2})}\frac{\omega_{0}^{3}}{h_{0}^{3}}, \tag{14}\]  
For small deflection, mechanical sensitivity of the square diaphragm with initial stress can be obtained as [46]  
\[S_{m}=\frac{\omega_{0}}{P}=\frac{a^{2}}{\pi h_{d}\Big{[}\frac{4.2E{h_{d}}^{2} }{\pi a^{2}(1-\
u^{2})}+o\Big{]}}, \tag{15}\]  
For the SDCD structure, in order to reflect the effect of single deep corrugation, corrective factors \(C_{1}\)(H) and \(C_{2}\)(H) are introduced into (15). Here, corrective factors \(C_{1}\)(H) and \(C_{2}\)(H) reflect the improved mechanical compliance by decreasing mechanical stiffness of the diaphragm and releasing the initial stress, respectively. The mechanical sensitivity of a SDCD is then modified as [46]  
\[S_{m}=\frac{a^{2}}{\pi h_{d}\Big{[}4.2C_{1}(H)E{h_{d}^{2}}/\pi a^{2}(1-\
u^{2} )+C_{2}(H)\sigma\Big{]}}, \tag{16}\]  
The values of the two corrective factors are obtained by finite element model (FEM) simulation, and it is confirmed that both of the two factors are less than 1 and will decrease with the increase of the corrugation's depth [46]. Therefore, the single deep corrugation will release the diaphragm's initial stress and lower the diaphragm's stiffness, which will both contribute to improving the diaphragm's mechanical sensitivity.  
Figure 6: (**a**) Schematic cross section of the SDCD structure with geometrical parameters; (**b**) The three-dimensional cross-section of the microphone fabricated by SDCD (Reproduced from [45]).  
Through the above analysis, it can be concluded that the mechanical sensitivity of the FD depends on the initial stress in the diaphragm. The smaller the initial stress is, the higher the sensitivity is. However, due to the limitations of the manufacturing technology, there will be initial stress in the FD that is difficult to control. The shallow corrugated structure can reduce the influence of the initial stress to improve the mechanical performance sensitivity of the diaphragm and increase the linear range of the diaphragm; the SDCD structure can improve the mechanical sensitivity of the diaphragm by releasing the initial stress and reducing the bending stiffness. Compared with controlling the parameters of the diaphragm deposition process, the method of introducing the corrugated structure is much simpler and more accurate, which greatly reduces the complexity of manufacturing technology and production cost.