## 4 Radiation Pressure-Driven PhC Cavity Sensors
### Optomechanical Effects for Sensing Applications  
The starting point of subsequent radiation pressure-driven opto-mechanical interaction is briefly referred to cavity optomechanics. Radiation pressures, also known as optical forces, become obvious in the presence of strong gradients of optical mode field confined within nanomechanical holes or slots [17]. Here we take a model based on F-P cavity to discuss optomechanical phenomena, as illustrated in Figure 13a. In the optomechanical system, light with frequency \(\omega_{cav}\) is confined inside a high Q-factor \(Q_{opt}\) optical cavity. Meanwhile, one of the mirrors of optical cavity with the effective mass \(m_{eff}\) is coupled with a mechanical oscillator represented with a spring constant \(k=2m_{eff}\Omega_{m}^{2}\). The mechanical oscillator with its mechanical quality factor \(Q_{m}\) vibrates at its mechanical frequency \(\Omega_{m}\). The motion \(x\) of mechanical oscillator brings about a variation in the optical path length to alter cavity resonance \(\omega_{cav}\), which in turn causes a change of light amplitude, thereby varying the radiation pressure acting on the motion to close this optomechanical backaction. We refer to this feedback-loop backaction as dynamic backaction, as presented in Figure 13b. In this feedback-loop, it starts with a mechanical oscillator subject to an input force \(dF\) (thermal forces or applied signal forces) to drive a mechanical movement \(x\). Due to the optomechanical coupling \(g_{om}\), mechanical movement tunes optical cavity field \(d\alpha\), wherein \(d\alpha_{out}\) contains information on the mechanical position. Besides, the amplitude fluctuation \(d\alpha\) is also coupled back to the mechanical oscillator via laser detuning, resulting in a backaction. Finally, the light amplitude change-induced radiation pressure takes effect on the mechanical oscillator, closing the feedback loop. As described above, this dynamic backaction in optomechanical system can be represented by the coupled optical and mechanical resonator by an interaction Hamiltonian function as \(\hat{H}_{int}=\hbar g_{om}x\overset{\text{-}\dagger}{a}=\hbar g_{0}a\overset{ \text{-}\dagger}{a}\left(\overset{\text{-}\dagger}{b+b}\right)\), where \(g_{0}=g_{om}x_{ZPF}\), (\(x_{ZPF}=\sqrt{\hbar/2m_{eff}\Omega_{m}}\) is the zero-point fluctuation  
Figure 13: (**a**) Canonical schematic of a cavity optomechanical system with a laser-driven optical cavity and a vibrating mechanical cavity; (**b**) schematic of optomechanical backaction loop [17]. and \(g_{om}=\partial\omega_{av}/\partial x\)), denotes the single photon coupling strength; \(\dot{a}\) and \(\dot{b}\) denote photon and phonon annihilation operators with \(a\)\(a\) the circulating photon number. The cavity optomechanics induced radiation pressure can be given by the derivation \(\dot{H}_{int}\) to mechanical motion \(x\), given as \(F_{opt}=-\frac{\partial\dot{H}_{int}}{\partial x}=-\dot{\imath}g_{om}a\). In optomechanical sensing applications, this optical force can be used to develop optomechanical sensors with intrinsic advantages over traditional mechanically tunable methods on bandwidth, speed and efficiency [105]. Considering the effect of \(F_{opt}\), the optomechanical interaction equation to indicate coupled light amplitude with mechanical position is given as,  
\[ \begin{cases}\dot{\alpha}=-\frac{\kappa}{2}\alpha+i(\Delta+Gx)\alpha+\sqrt{ \kappa_{ex}}\alpha_{in}\\ m_{eff}\dot{x}=-m_{eff}\Omega_{m}^{2}x-m_{eff}\Gamma_{m}x+\hbar G|\alpha|^{2} \end{cases}  \tag{5}\]  
where \(\alpha\) and \(\alpha_{in}\) are the classical correspondence of cavity mode \(a\) and input field mode \(a_{in}\), respectively; \(\kappa\) and \(\kappa_{ex}\) refer to the total and external decay rates of cavity mode; \(\Delta=\omega_{L}-\omega_{av}\) represents laser detuning between laser and optical cavity.