# From Far-Field to Near-Field Micro- and Nanoparticle Optical Trapping
## 2 Conventional Optical Tweezers
### Dipole Approximation \((r\ll\lambda)\)
The second term is the scattering force pointing in the direction of the Poynting vector, \(\vec{S}\), and arises from absorption and scattering phenomena that cause momentum transfer from the field to the particle. The first term is the gradient force and depends on the particle's polarizability and the intensity gradient of the electric field. Since \(I=\frac{1}{2}c\epsilon_{0}|\vec{E}|^{2}\), the gradient force can be written as  
\[\vec{F}_{grad}(\vec{\tau}_{d})=\frac{1}{2}\frac{\alpha_{d}^{\prime}}{c\epsilon _{0}}\
abla I(\vec{\tau}_{d}). \tag{9}\]  
From Equation (9) we see that, for particles with positive polarizability (i.e., a higher refractive index than its environment) this force acts towards the direction of the field's higher intensity, i.e., the focal point. At the focal point of a Gaussian beam with a beam waist, \(w_{0}\), and radial coordinate, \(\rho\), we can approximate the intensity distribution as  
\[I(\rho)=I_{0}e^{-2\rho^{2}/w_{0}^{2}}, \tag{10}\]  
and, for small radial displacements, we can Taylor expand to get  
\[I(\rho)\approx I_{0}\left(1-2\frac{\rho^{2}}{w_{0}^{2}}\right), \tag{11}\]  
and substitute into Equation (9). Thence,  
\[\vec{F}_{grad}(\vec{\tau}_{d})=\frac{1}{2}\frac{\alpha_{d}^{\prime}}{c\epsilon _{0}}\frac{\partial}{\partial\rho}\left[I_{0}\left(1-2\frac{\rho^{2}}{w_{0}^{2 }}\right)\right]\hat{\rho}=-2\frac{\alpha_{d}^{\prime}}{c\epsilon_{0}}\frac{I _{0}}{w_{0}^{2}}\rho\hat{\rho}. \tag{12}\]  
By comparing with the restoring force of the classical harmonic oscillator, \(\vec{F}(x)=-\kappa x\hat{x}\), we get the trapping constant \[\alpha_{0}=4\pi r^{3}\epsilon_{0}\frac{\epsilon_{r}-1}{\epsilon_{r}+2}. \tag{5}\]0  
and, by integrating Equation (12), we get the trapping potential  
\[\alpha_{0}=4\pi r^{3}\epsilon_{0}\frac{\epsilon_{r}-1}{\epsilon_{r}+2}. \tag{5}\]1  
which is plotted in \[\alpha_{0}=4\pi r^{3}\epsilon_{0}\frac{\epsilon_{r}-1}{\epsilon_{r}+2}. \tag{5}\]2 Note that similar analysis can be used in order to obtain the potential and the trapping constant along the axial direction. We now consider the main limitation of conventional optical tweezers. [PERSON] et al. (2012) did some basic calculations and showed that, in order to create a stable optical trap, resisting the Brownian motion of particles in a liquid environment, a potential well as deep as \(10k_{B}T\) is required, where \(T\) is the temperature in Kelvin. Although in some cases this is easy to achieve, for subwavelength particles, as they become smaller in size, the gradient force scales down very quickly, making it impossible to satisfy this requirement. By replacing Equation (5) into (6) and simplifying the complex term, we see that \(\alpha_{d}^{\prime}\propto r^{3}\), which means that, when the radius of the particle decreases by a factor of 10, the polarizability of the particle and, consequently, the gradient force (Equation (9)) decrease by a factor of 1000. The trapping potential is no longer deep and tight enough to hold the particle (Figure 4) and the trap is inefficient. Equations (5), (6) and (13) can be used to calculate the change in the trapping stiffness if the particle has a radius of 0.8\(r\):  
\[\alpha_{0}=4\pi r^{3}\epsilon_{0}\frac{\epsilon_{r}-1}{\epsilon_{r}+2}. \tag{5}\]3  
Calculations show that for \(r=100\) nm, there is a 53% decrease in the trapping stiffness when the particle's radius decreases to 0.8\(r\). From Equation (13) we see that, in order to compensate for this effect and increase the trapping constant and the gradient force, we can either increase the intensity of the incident field (\(I_{0}\)) or focus tighter (\(w_{0}\)). However, even though in some cases it is experimentally possible to increase the intensity of the field by a factor of 1,000, the heat accumulation will be very large and could eventually destroy the particle, in particular if it is a biological sample.