# From Far-Field to Near-Field Micro- and Nanoparticle Optical Trapping
## 2 Conventional Optical Tweezers  
The Nobel Prize in Physics 2018 was awarded (50%) to [PERSON] _\"for the optical tweezers and their application to biological systems\"_. The whole research field started when [PERSON] calculated that _\"a power \(P=1\) W of cw (continuous wave) argon laser light at \(\lambda=0.5145\)\(\mu m\) focussed on a lossless dielectric sphere of radius \(r=\lambda\) and density \(=1\) gm/cc gives a radiation pressure force \(F_{rad}=2 qP/c=6.6\times 10^{-5}\) dyn, where \(q\), the fraction of light effectively reflected back, is assumed to be of order 0.1. The acceleration \(=1.2\times 10^{8}\) cm/sec\({}^{2}\cong 10^{5}\) times the acceleration of gravity\"_[7]. In SI units, this is equivalent to a radiation pressure force \(F_{rad}=0.66\) nN, leading to an acceleration of \(1.2\times 10^{6}\) m/s\({}^{2}\). In the same work, he demonstrated the first experimental approach to test his calculations on transparent, micron-sized latex spheres in liquids and gas and found that the radiation pressure exerted on the particles from a focussed laser beam was able to accelerate them along the direction of the beam. The measured velocities of the accelerated particles were in very good agreement with the theoretical predictions.  
[PERSON] next demonstrated trapping of particles using one laser beam and the wall of a glass cell, as well as using two counter-propagating beams with the same characteristics [7]. A few years later, [PERSON] et al. [12], reported trapping of dielectric particles (10 \(\upmu\)m-25 nm) using a single beam by focussing argon-laser light at 514.5 nm through a high numerical aperture objective lens (\(NA=1.25\)). This achievement is attributed to the existence of a force additional to that caused by the radiation pressure (from now on called the scattering force) which originates from the axial beam intensity gradient. It then becomes apparent that, whereas the scattering force depends on the optical intensity and has the direction of the incident beam, the gradient force depends on the intensity gradient and is directed along it from low to high intensities (for the case of particles with higher refractive index than the surrounding medium). Stable optical trapping can occur when these two forces are balanced.  
The theoretical mechanism that explains this observation, depends on the relative size of the particle (radius, \(r\)) with respect to the wavelength of the laser light (\(\lambda\)). For \(r\gg\lambda\), ray optics can be used and the reflection and transmission of the beam from the particle can give rise to the two forces. For \(r\ll\lambda\), Rayleigh scattering is assumed and the particle is treated like a dipole in an external electromagnetic field. The two regimes are analysed below, where we assume spherical dielectric particles with refractive index higher than that of the surrounding environment. For the calculation of optical forces acting on particles with arbitrary shapes, the reader is encouraged to study other works [13; 14]. Finally, there is an intermediate regime where the particle size is of the same order of magnitude as the wavelength. In this case, the approximations mentioned above cannot be used and, in order to evaluate the forces arising, [PERSON]'s stress tensor, which relates the interactions between electromagnetic forces and mechanical momentum [10; 15], should be used. To handle this complicated mathematical analysis different algorithms have been established, such as the transition matrix (T-matrix) method [16] and the discrete dipole approximation (DDA) [17]. Many works have followed various approaches for the calculation of the forces in this regime [18; 19; 20; 21; 22; 23]. Due to its complexity this regime is not analysed here.