# From Far-Field to Near-Field Micro- and Nanoparticle Optical Trapping
## 2 Conventional Optical Tweezers
### Ray Optics Approximation (\(r\gg\lambda\))  
We assume spherical particles of higher refractive index than their surrounding environment, being in a liquid solution and undergoing Brownian motion. As soon as a particle, randomly moving, enters the light beam, a small fraction of light is reflected off the surface of the particle and most of it is refracted on passing through the particle (assuming no absorption). Light carries momentum and, since refraction is a light-matter interaction phenomenon, there is a momentum transfer from the photons to the particle. As known from geometrical optics, the path of the light changes due to refraction, resulting in a change in the momentum (\(\vec{p}\)) of the photons, over time (\(t\)). Obviously, from conservation of momentum for the light-particle system, there should also be a change in the momentum of the particle and this creates a force (\(\vec{F}\)) acting on the particle, \(\vec{F}=d\vec{p}/dt\). To get a first insight, we can initially assume that there is no reflection and the entire beam is refracted inside the sphere as shown in Figure 1a. According to the work done in [24; 25], the magnitude of the force on the particle due to the momentum change of a single ray is given by  
\[F=\frac{n_{m}}{c}P, \tag{1}\]  
where \(n_{m}\) is the refractive index of the particle, \(c\) is the speed of light and \(P\) is the power of the incident ray. For a Gaussian beam profile, there is more power towards the centre of the beam than in the tail regions, thus the resultant force from ray 2 (\(F_{2}\)) is stronger than that from ray 1 (\(F_{1}\)), as shown in Figure 1a. The forces acting on the particle can be reduced into a longitudinal component parallel to the incident ray and a transversal one perpendicular to it. As shown in the figure, the longitudinal components of the two forces add up to create a scattering force, whereas the transversal components subtract leading to a gradient force towards the beam's regions of higher intensity. Thence, the particle moves towards the centre of the Gaussian beam and along its axis. Note that, for particles with a lower refractive index than the surroundings, the forces reverse and the particle moves away from the centre of the beam. Figure 1: The scattering and gradient forces acting on a dielectric particle in the ray optics regime, arising from a free space (**a**) and a tightly focussed (**b**) Gaussian beam. In the subplot (**a**) the forces push the particle towards the centre of the beam and along the direction of propagation, whereas in (**b**) the forces drag the particle towards the focus of the beam. The upper panel in subplot (**a**) shows the proposed geometry for calculating the forces using Fresnel coefficients. The bottom panel shows trapping using counter-propagating beams with the same characteristics so that the scattering forces cancel each other to achieve a stable trap. Figure reproduced with permission from [26]. If we want to describe the process in a more accurate and mathematically rigorous way, we have to take into account multiple internal reflections and refractions of the rays, as shown in the top inset of Figures 1a and 2. The forces exerted on a particle were first calculated by [PERSON] [27] by considering Fresnel's reflection (\(R\)) and transmission (\(T\)) coefficients. For a detailed derivation see [14; 28].