## 4 Radiation Pressure-Driven PhC Cavity Sensors
### Optomechanical Effects for Sensing Applications
In the presence of optomechanical interaction, the dynamical backaction can be deduced from the change of mechanical susceptibility as  
\[\sum(\omega)=\chi_{m,m_{eff}}^{-1}-\chi_{m}^{-1}=2m_{eff}\Omega_{m}g^{2}\bigg{[} \frac{1}{(\Delta+\omega)+i\kappa/2}+\frac{1}{(\Delta-\omega)-i\kappa/2}\bigg{]} \tag{6}\]  
To observe dynamical behaviors arising from optomechanical coupling, the real and imaginary parts of Equation (6) can yield the mechanical eigenfrequency shift \(\delta\Omega_{m}(\omega)\) and optical damping rate \(\Gamma_{opt}(\omega)\) in the strong optomechanical coupling regime (\(g>\kappa/2\) and \(\kappa\ll\Omega_{m}\), i.e., resolved sideband) as  
\[\delta\Omega_{m}(\omega)=g^{2}\frac{\Omega_{m}}{\omega}[\frac{\Delta+\omega}{ \left(\Delta+\omega\right)^{2}+\kappa^{2}/4}+\frac{\Delta-\omega}{\left( \Delta-\omega\right)^{2}+\kappa^{2}/4}] \tag{7}\]  
\[\Gamma_{opt}(\omega)=g^{2}\frac{\Omega_{m}}{\omega}[\frac{\kappa}{\left( \Delta+\omega\right)^{2}+\kappa^{2}/4}+\frac{\kappa}{\left(\Delta-\omega \right)^{2}+\kappa^{2}/4}] \tag{8}\]  
Equation (7) indicates optical spring effect, where shift of mechanical frequency can be deemed to modify mechanical stiffness as \(\Omega_{m}=\sqrt{k/m_{eff}}\). Since \(\Gamma_{opt}\) in Equation (8) causes the change in total effective mechanical damping as \(\Gamma_{eff}=\Gamma_{m}+\Gamma_{opt}\), extra optical damping leads to cooling while anti-damping can lead to amplification. Both optical spring and damping behaviors can be observed in experiments under different ratios of \(\Omega_{m}/\kappa\)[106]. In Doppler regime (\(\kappa\gg\Omega_{m}\)), mechanical frequency shift caused by the pump light yields \(\left.\delta\Omega_{m}(\Delta=\omega_{L}-\omega_{av})\right|_{\kappa\gg\Omega _{m}}=g^{2}\frac{2\Delta}{\Delta^{2}+\kappa/4}\). It implies that the spring of mechanical oscillator will be softened under the red laser detuning (\(\Delta<0\)) and hardened under the blue laser detuning (\(\Delta>0\)). However, mechanical frequency shift becomes remarkably different when it works in the resolved sideband regime (\(\Omega_{m}\gg\kappa\)), where optical spring effect vanishes at certain laser detuning and radiation pressure contributes only to cooling or amplification function [107]. When it comes to sensing applications, excellent sensing performance can be achieved through the optical readout of mechanical characteristics from optomechanical system. Aim to obtain the maximum mechanical signal readout, the probing laser wavelength should be placed at the point of the highest slope of optical resonance, as depicted in Figure 14a [108,109]. As light cycles inside the cavity, a radiation pressure induced by optomechanical coupling can interplay with mechanical resonator. Under the blue-detuned \((\Delta_{L}=\lambda_{L}-\lambda_{av}<0)\) laser wavelength, the motion of mechanical resonator is strengthened due to optomechanical interaction. This dynamic backaction can result in the optical spring effect, namely, setting of laser detuning \(\Delta_{L}\) determines the behavior of mechanical frequency \(\Omega_{m}\). The concept of optomechanical spring sensing is represented in Figure 14b,c where a high optical Q-factor cavity is employed. Any tiny perturbation induced by nanoparticle adhesion or physical motion causing optical resonance shift \(\delta\lambda\) could be converted to the mechanical frequency shift as \(\delta\Omega_{m}=-\frac{d\Omega_{m}}{d\Delta_{L}}\delta\lambda\). Hence, it provides a valid optomechanical sensing mechanism through amplifying the optical resonant wavelength.