# Plasmonic Tweezers towards Biomolecular and Biomedical Applications
## 2 Optical Forces
### Self-Induced Back-Action Effect  
Although the enhanced near field can be used to provide nanoscale gradient field for nanoparticle trapping and nano-aperture designs can avoid the heat accumulation, other possible damage e.g., photo damage still presents. A novel trapping mechanism is needed to provide similar level of trap stiffness at an extremely low intensity. The self-induced back-action (SIBA) effect is a trapping mechanism utilizing the trapped particle as an active role to modify its own trapping potential dynamically. The presence of the particle shifts the resonance frequency of the trapping system and this shift is dynamically decided by the position of the nanoparticle in the plasmonic cavity. Compared to the trapping mechanism based on the field gradient (static trapping), trapping using a SIBA effect is dynamic and an off-resonance frequency is used. SIBA based on planar PC cavities was theoretically proposed by [PERSON] and [PERSON] in 2006 [48]. In 2009, the SIBA phenomenon was demonstrated by using a nanohole in a metal thin film on trapping single 50 nm polystyrene particles with 1.9 mW incident power [49]. In 2013, experimentally using PCs to trap 500 nm dielectric particles with SIBA effect was achieved by [PERSON] et al. using a waveguide power lower than 0.12 mW [50]. The presence of the particle at position \(\mathbf{r}\) in a plasmonic adiabatic cavity causes a shift in the resonance frequency, \(\delta\omega(\mathbf{r})=\omega_{c}(\mathbf{r})-\omega_{c,0}\) and the intra-cavity intensity profile \(I(\mathbf{r})\) is expressed as Equation (4), where \(\omega\) is the frequency of the trapping laser, \(\Gamma\) is the empty cavity linewidth, \(\omega_{c,0}\) is the resonance frequency of the empty cavity, \(\Delta\equiv\omega-\omega_{c,0}\) is the cavity detuning, \(I_{opt}\equiv I_{0}\frac{\left(\frac{\Gamma}{2}\right)^{2}}{\Delta^{2}+\left( \frac{\Gamma}{2}\right)^{2}}\) is the empty cavity profile when there is no particle presence in the cavity, \(I_{0}\) is the incident intensity, and a Taylor expansion is used to expand the modified cavity profile [51]. When the particle cannot shift the resonance frequency large enough, Equation (4) reduced to \(I_{opt}\), since the particle influence on the system is negligible. Then, we discuss the cases when the particle can introduce significant resonance frequency shift to the plasmonic cavity:  
\[I(\mathbf{r})=I_{0}\frac{\left(\frac{\Gamma}{2}\right)^{2}}{\left(\Delta-\delta \omega(\mathbf{r})\right)^{2}+\left(\frac{\Gamma}{2}\right)^{2}}\approx I_{opt}- \frac{2\delta\omega(\mathbf{r})\Delta}{\Delta^{2}+\left(\frac{\Gamma}{2}\right)^{2 }}I_{opt}+\cdots\cdots. \tag{4}\]Since the trap stiffness is proportional to the intensity of the intra-cavity intensity, the total trap stiffness, \(k_{tot}\), can be expressed as in Equation (5) when only the first two terms in Equation (4) are considered when \(\delta\omega(\mathbf{r})\) is much smaller than \(\omega_{c,0}\):  
\[k_{tot}=k_{opt}+k_{SIBA}, \tag{5}\]  
where \(k_{opt}\) depends on the empty cavity resonance profile and \(k_{SIBA}\) originates from the coupling between the motion of the particle and the resonance of the plasmonic cavity. Since a bioparticle always have a slightly higher refractive index compared to an aqueous surrounding, \(\delta\omega\) is always positive. To make the second term positive, \(\Delta\) needs to be negative. A red cavity detuning (\(\omega<\omega_{c,0}\)) should be selected to generate a positive SIBA effect. Examine the expression of \(k_{SIBA}\) in Equation (4), when \(\Delta=\delta\omega(\mathbf{r_{max}})\) is satisfied the value of \(k_{SIBA}\) is maximized and this position is defined as resonant position of the particle. The definition of the back-action parameter is shown in Equation (6), where \(V_{m}\) is the mode volume and \(Q\) is the quality factor of the empty cavity [52]. This parameter is predefined by the properties of the particle and the plasmonic cavity.