# Plasmonic Tweezers towards Biomolecular and Biomedical Applications
## 2 Optical Forces  
The generation of optical force for OTs and PTs is the same, transferring momentum from a photon to a particle. To have a trap, a strong field gradient is needed. Compared to the wavelength (\(\lambda\)) of the trapping laser, particles are divided into Mie particles (\(d\gg\lambda\), ray optics), Lorentz-Mie particles (\(d\sim\lambda\), Maxwell stress tensor), and Rayleigh particles (\(d\ll\lambda\), dipole approximation) for OTs, where \(d\) is the diameter of the particle [38; 39; 40]. Optical force applied on Rayleigh particles are discussed for OTs and PTs. Due to the fact that a Rayleigh particle is much smaller than the diffraction limited focused laser beam for OTs, the particle can be treated as a dipole or a collection of dipoles. Take a spherical particle as an example. The polarizability of the particle can be expressed as in Equation (1), where \(k\) is the field wavevector, \(\alpha_{0}\) is the point-like particle polarizability from Clausius-Mossotti relation [41], \(\varepsilon_{0}\), \(\varepsilon\) and \(\varepsilon_{b}\) is the vacuum dielectric permittivity, and relative permittivity of the particle and the background medium, respectively. As we can observe that the polarizability of the particle is proportional to the cubic of the diameter of the particle and the difference between the relative permittivity of the particle and the one of the surrounding medium:  
\[ \begin{split}\alpha(\omega)=\frac{a_{0}(\omega)}{1-\frac{k^{3}a _{0}(\omega)}{a_{0}t_{0}}},\\ \alpha_{0}(\omega)=4\pi\varepsilon_{0}\big{(}\frac{d}{2}\big{)} \frac{3}{\varepsilon(\omega)-\varepsilon_{b}(\omega)}{\varepsilon(\omega)+2 \varepsilon_{b}(\omega)}.\end{split}  \tag{1}\]  
Based on the dipole approximation, the time averaged force applied on the particle from the light field is shown in Equation (2), where \(\sigma\) is the extinction cross-section [42]:  
\[\langle\mathbf{F}\rangle=\frac{1}{4}\text{Re}(\alpha)\
abla|\mathbf{E}|^{2}+\frac{ \sigma}{2\varepsilon}\text{Re}(\mathbf{E}\times\mathbf{H}^{\prime})+\frac{\sigma \varepsilon\varepsilon_{0}}{4\omega i}\
abla\times\mathbf{E}\times\mathbf{E}^{*}. \tag{2}\]  
The angular brackets denote time average. The first term in Equation (2) is from the intensity gradient which pulls the particle to the highest intensity location. For a particle having a largerpolarizability, the gradient force is larger when the same incident intensity is used. The second term corresponds to the radiation pressure which pushes the particle along the light propagation direction. The third term is from the polarization gradient, which corresponds to the chiral optical force that is discussed in Section 2.2. As we can observe from this equation, the intensity gradient force is proportional to the polarizability of the particle and intensity of the field. When the size of the particle is reduced 10 times, 1000 times higher intensity is needed to maintain the trap at the same potential depth. To compare the trap stiffness among different OTs and PTs, the radius and dielectric constant of the particle should also be factored out when the same incident intensity is used. Since the nanoscale enhanced near fields generated by PTs having similar size as the nanoparticles, dipole approximation is not viable. Maxwell stress tensor (MST) \(\overleftrightarrow{T}\) is used to obtain the optical force applied on the nanoscale particle from this arbitrary electromagnetic field as shown in Equation (3) [43]. The integral is taken over a surface \(S\) enclosing the particle, and \(\mathbf{n}(\mathbf{r})\) is the outward unit vector normal to the integration surface. Besides a few cases that involve simple plasmonic designs [44], it is extremely difficult to have analytical results of optical forces. To obtain as large field gradients as possible, most designs of PTs contain very complicated and irregular 3D geometries. Numerical simulation is mainly used to obtain the near field distribution.