## 3 Mechanically Tunable PhC Cavity Sensors
### Sensing Mechanisms of Mechanically Tunable PhC Cavities Sensors
Assuming that the sensing mechanism based on RI variation by monitoring resonance shift upon an arrival event of an analyte, according to the perturbation theory, the resonant frequency shift \(\Delta\omega\) induced by a near-field probing analyte is represented by [87; 92],  
\[\frac{\Delta\omega}{\omega_{cav}}=\frac{\alpha_{\alpha}\big{|}E(r_{p})\big{|} ^{2}}{2\int\varepsilon\big{|}E(r)\big{|}^{2}V} \tag{2}\]  
Figure 2: **(a)** Configurations of waveguide-resonator coupling: **(a-1)** waveguide (WG) side-coupled [88], **(a-2)** double side-coupled [89], and **(a-3)** single side-coupled [90]; **(b)** a characteristic dip at resonance wavelength for WG side-coupled cavity. The resonance shifts from blue line to red line due to perturbation, and a linewidth broadening \(\Delta\lambda^{{}^{\prime}}-\Delta\lambda\) leads to a decreasing Q-factor [29]. where \(\alpha_{ex}\) and \(\varepsilon\) describe the polarizability of surrounding analyte and dielectric constant of cavity; and \(E(\mathbf{r})\) and \(E(\mathbf{r_{p}})\) are electric field inside cavity and at the probing analyte position \(\mathbf{r_{p}}\); it is obvious that the presence of a dielectric probe analyte can cause a red shift of resonance frequency based on the distance between probe perturbation and cavity. Besides, both a small mode volume \(V\) and intensive light confinement in the perturbation area can result in a wide \(\Delta\omega\) to achieve a preferable sensitivity. In addition, it should be noted that monitoring Q-factor can also detect an external analyte based on RI perturbation theory. Analytes-perturbated cavity sensor translates change of RI of external analyte into the change of linewidth. In the optical cavity with an approaching probe analyte, optical absorption induced cavity mode linewidth broadening has been widely exploited to probe nanoparticles [93; 94]. Moreover, displacing coupled photonic resonators are also widely used in mechanically tunable PhC sensors, namely coupled PhC cavities. The resonance frequencies of coupled supermodes can be controlled by tuning the coupling strength between two cavities. According to the temporal coupled mode theory (CMT) [95], when two identical PhC resonators are placed close to each other, their mode fields are coupled as below,  
\[\left\{ \begin{array}{l}\frac{d}{dt}a_{1}=j\omega_{ox}a_{1}-j\kappa a_{2}\\ \frac{d}{dt}a_{2}=j\omega_{ox}a_{2}-j\kappa a_{1}\end{array} \right. \tag{3}\]  
where \(\kappa\) is coupling strength between two cavities with natural frequency \(\omega_{ox}\); and \(a_{1}\) and \(a_{2}\) denote the optical modes in their individual resonators. Resonance of coupled cavities can be given as,  
\[\omega_{i}=\omega_{ox}\pm\kappa \tag{4}\]  
For coupled PCN cavities, the interaction of two cavities causes a splitting of their original uncoupled modes into corresponding pairs of even mode (symmetric mode shape) and odd mode (anti-symmetric mode shape). As can be seen from Equation (4), the coupling strength \(\kappa\) determines resonance frequency and it is sensitive to the mechanical displacement between two PCN cavities. Therefore, tuning the mechanical displacement of movable photonic resonator plays a significant impact on the mode-splitting status and it can be easily compatible with an on-chip integrated N/MEMS system for sensing applications.