# From Far-Field to Near-Field Micro- and Nanoparticle Optical Trapping
## 3 Plasmonic Optical Tweezers
### Self-Induced Back Action Effect
This gave rise to the _self-induced back action_ (SIBA) effect and the first experimental trapping utilising this effect, where polystyrene spheres of 100 and 50 nm size were successfully trapped with incident powers as low as 0.7 and 1.9 mW, respectively [54], pushing further the boundaries of plasmonic nanotweezers. A comprehensive mathematical analysis of the SIBA effect has been done by [PERSON] et al. [62], for a small dielectric particle trapped in a plasmonic nanocavity. They demonstrated the additional restoring forces that act on the particle as it tries to escape from the trap. Below, we present the basic principle of the SIBA effect following the analysis done in [60]. The gradient force experienced by a nanosphere with radius \(r\ll\lambda\) of the incident light, is given by Equation (9), as mentioned previously. If we assume small displacements of the particle from the centre of the trap (\(|x|\ll\lambda\)), then according to the overdamped Langevin equation the particle's motion inside the trap is given by  
\[\gamma\dot{x}(t)+\kappa_{tot}x(t)=\xi(t), \tag{17}\]  
where \(\gamma\) is the viscous damping [63], assuming that the particle exists in a liquid environment, \(\kappa_{tot}\) is the stiffness of the trap, indicating how strongly the particle is confined in the trap, and \(\xi\) represents thermal fluctuations [64]. Due to their strong interaction there is a dispersive coupling between the cavity and the particle. Thus, the motion of the particle in the trap causes the plasmon resonance frequency of the cavity to shift by \(\delta\omega_{0}(x_{p})\), where \(x_{p}\) indicates the frequency dependence on the particle's position. Then, for a cavity with mode volume, \(V_{m}\), and intensity profile, \(f(x_{p})\), normalised to 1 for maximum intensity, the perturbation theory for shifts much smaller than the cavity eigenfrequency, \(\omega_{c}\), yields [65]  
\[\delta\omega_{0}(x_{p})=\omega_{c}\frac{\alpha_{d}}{2V_{m}\epsilon_{0}}f(x_{p}), \tag{18}\]  
with \(\alpha_{d}\) being the effective polarizability, from Equation (6). Note that the magnitude of the shift strongly depends on the relative size of the particle and the cavity and, as expected from Equation (18),  
Figure 9: **(a)** The presence of a dielectric material of higher refractive index than the surroundings causes an increase in the transmission of light through the subwavelength aperture. **(b)** Shift of the transmission line due to existence of the dielectric material, leading to higher transmission \(T_{2}\), compared to the transmission of light \(T_{1}\) in the case of non-filling. Figure inspired by [59]. a decrease in the particle's size (decrease in \(\alpha_{d}\), see Equations (5) and (6)), decreases the magnitude of the shift [66]. Now, for incident laser frequency, \(\omega\), and \(\Delta\equiv\omega-\omega_{c}\) being the cavity detuning, the intracavity intensity, \(I(\omega)\), is given, on Taylor expansion, as  
\[I(\omega)=I_{0}\frac{(\Gamma/2)^{2}}{(\Delta-\delta\omega_{0})^{2}+(\Gamma/2 )^{2}}\approx I_{opt}-\frac{2\delta\omega_{0}(x_{p})\Delta}{\Delta^{2}+( \Gamma/2)^{2}}I_{opt}+... \tag{19}\]  
where \(\Gamma\) is the cavity linewidth and \(I_{opt}=I_{0}\frac{(\Gamma/2)^{2}}{\Delta^{2}+(\Gamma/2)^{2}}\) is the empty cavity profile. As can been seen from Equation (19), the intensity of light inside the cavity, to a first order approximation, is given by the term related to the intensity of the empty cavity, plus the one related to the frequency shift caused by the presence of the particle. This second term is the one that causes the SIBA effect and modifies the optical potential. Following Equation (19), we can also write the total trapping stiffness, \(\kappa_{tot}\), as [62]  
\[\kappa_{tot}=\kappa_{opt}+\kappa_{SIBA}, \tag{20}\]  
where \(\kappa_{opt}\) is constant and depends on the cavity resonance profile and \(\kappa_{SIBA}\) is a function of the particle's displacement.