# From Far-Field to Near-Field Micro- and Nanoparticle Optical Trapping
## 2 Conventional Optical Tweezers
### Dipole Approximation \((r\ll\lambda)\)  
In this case, the electric field that the particle experiences is approximately spatially constant over its dimensions, and assuming a dielectric particle, we can treat the entire particle as a collection of induced point dipoles in a homogeneous electric field. Early theoretical work on radiation forces and scattering effects for subwavelength dielectric media can be found in [24; 29]. Based on this work and the electromagnetic theory for electromagnetically induced dipoles, we can describe the optical forces and the trapping potential that arises. The situation for a homogeneous particle can be briefly described as follows (the analysis presented is taken from [14]): The oscillating electromagnetic field from the laser beam causes each of the particle's point dipoles to have a dipole moment  
\[\vec{p}=\alpha_{0}\vec{E}, \tag{4}\]  
where \(\vec{E}\) is the electric field and \(\alpha_{0}\) is the polarizability of the particle, given by the _Clausius-Mossotti relation_  
\[\alpha_{0}=4\pi r^{3}\epsilon_{0}\frac{\epsilon_{r}-1}{\epsilon_{r}+2}. \tag{5}\]  
Here, \(r\) is the particle's radius, \(\epsilon_{0}\) is the vacuum dielectric permittivity, and \(\epsilon_{r}\) is the particle's dielectric permittivity. The external field causes the dipoles to oscillate and, thus, radiate. Now, we have to take into account the dipole's interaction not only with the external electromagnetic field, but with its own induced scattered field as well. For that reason, the _effective polarizability_, \(\alpha_{d}\), is introduced as a radiative reaction correction to the intrinsic polarizability of the particle [14] and it is given by  
Figure 3: The calculated \(Q\) factors for the gradient, scattering, and total trapping efficiencies for a single circularly polarised ray acting on a spherical dielectric particle of effective refractive index \(n_{m,eff}=1.2\), as a function of the incident angle. Figure reproduced with permission from [28]. \[\alpha_{d}=\frac{\alpha_{0}}{1-\frac{\epsilon_{r}-1}{\epsilon_{r}+2}[(k_{0}r)^{2}+ \frac{2i}{3}(k_{0}r)^{3}]}, \tag{6}\]  
with \(k_{0}\) being the vacuum wavenumber and \(i=\sqrt{-1}\). Similarly, in order to calculate the forces acting on all the dipoles, we have to take into account the Lorentz force from the external field and the radiation forces arising from the dipoles themselves. It is also convenient to calculate the time-averaged total force since it is the one that is observable (electromagnetic fields oscillate on the order of \(\sim 10^{15}\) Hz which is very fast). Rigorous calculations have been done in [14; 26; 30]. According to these works, the resulting, time-averaged force acting on a dielectric particle is given by  
\[\vec{F}=\frac{1}{4}\alpha_{d}^{\prime}\
abla|\vec{E}|^{2}+\frac{\sigma_{ext,d} }{c}\vec{S}+\frac{c\epsilon_{0}\sigma_{ext,d}}{4\omega i}\
abla\times(\vec{E} \times\vec{E}^{*}), \tag{7}\]  
where \(\alpha_{d}^{\prime}\) is the real part of the effective polarizability, \(\omega\) is the angular frequency, \(\sigma_{ext,d}\) is the _extinction cross-section_, i.e., the active area of the particle that causes part of the energy of the incident electromagnetic wave to be extinguished due to scattering and absorption from the particle, \(\sigma_{ext,d}=\frac{k_{0}}{\epsilon_{0}}Im(\alpha_{d})\). It, therefore, indicates the rate of energy loss from the incident wave. \(\vec{S}\) is the time-averaged real part of the Poynting vector of the incident wave  
\[\vec{S}=\frac{1}{2}Re\{\vec{E}\times\vec{H}^{*}\}. \tag{8}\]  
In Equation (7), we see that the force acting on a particle consists of three terms; the third term is called the _spin-curl force_ and is related to polarisation gradients in the electromagnetic field that arise when the polarisation is inhomogeneous (e.g., under conditions of tight focussing) [31; 32].