# Forces in the Interaction of Light with Matter
## 4 Forces in Optical Tweezers  
An experiment conducted in 1970 by [PERSON], which eventually led to their shared award of the 2018 Nobel Prize in Physics, represents the birth of the powerful, and now widely used, technique known as _optical tweezers_. In the original experiment, [PERSON] succeeded in manipulating the motion of tiny rubber beads using a laser beam [57]. In the form we now know it, the instrumentation of optical tweezers was fully devised by [PERSON] and [PERSON], who succeeded in both trapping and controllably moving miniature dielectric beads [142]. Nowadays, a very diverse and broad range of applications in biology, micromechanics, modelling of thermodynamic systems, colloidal systems, and lab-on-a-chip microchemistry are based on the use of optical tweezers [84]. Optical tweezers have the capacity to control the motion of micro- and nanoparticles. When the dimensions of such a particle are smaller than the wavelength of laser light (the so called _Rayleigh regime_), its motion can be attributed to forces associated with the following relation [143]:  
\[\mathbf{F}=\mathbf{F}_{grad}+\mathbf{F}_{diss}. \tag{24}\]The first force, \(\mathbf{F}_{grad}\), is the the gradient force given by Equation (19). It is a conservative force responsible for confinement in optical tweezers, and it arises from the potential energy of a dipole in the electric field. The second force, \(\mathbf{F}_{diss}\), is given by  
\[\mathbf{F}_{diss}=\frac{1}{c}\sigma_{\text{ext}}\,\mathbf{S}(\mathbf{r})-\frac {1}{2}\sigma_{\text{ext}}c\
abla\times\mathbf{s}, \tag{25}\]  
where \(\mathbf{S}=\frac{1}{2}\Re\{\mathbf{E}\times\mathbf{B}^{*}\}\) is the real part of the time-averaged Poynting vector, \(\mathbf{s}=i\frac{\xi_{0}}{2\omega}\Im\{\mathbf{E}\times\mathbf{E}^{*}\}\) is the time-averaged spin density of the incoming wave, and \(c\) is the speed of light. The quantity \(\sigma_{ext}=\frac{\kappa a^{\prime\prime}}{\xi_{0}}\) is the extinction cross section, and \(a^{\prime\prime}\) is the imaginary part of the complex polarizability. It will be evident, on comparing the expression given in Equation (19) with the one in Equation (25), that there is a significant difference. In fact, the former formula is a special case of the latter, being applicable to point-like particles--as we considered an atom when calculating radiation forces. In that regime, as has been extensively discussed, the scattering force is directly proportional to the Poynting vector [144,145]. In the present, more general formulation, the form of the dissipative force shows that it is not simply proportional to the Poynting vector whenever the spin density of the field has a non-vanishing curl. This term has been frequently associated with a third force component [146, 147, 148], but it has been shown that it does not play a role in the radiation force. Indeed, the contribution of the Poynting vector can be broken down into an orbital part and a spin part; the spin part is equal and opposite to the spin curl term and cancels it [143,149,150]. In this case, \(\mathbf{F}_{diss}=\mathbf{F}_{orbital}\), with \(F_{orbital}=\frac{1}{c}\sigma_{\text{ext}}\,\mathbf{S}(\mathbf{r})-\frac{1}{2 }\sigma_{\text{ext}}\
abla\times\mathbf{s}\). The canonical orbital momentum density alone is responsible for the radiation pressure force. The two non-conservative components of the radiation pressure force owe their origin to the time-averaged momentum, which in free space can again be separated into two parts, one independent of the polarization and the other dependent upon a degree of circular polarization. The theoretical justification of this distinction rests upon [PERSON]'s symmetrization of the canonical stress energy tensor, which makes the results gauge independent. The part of the dissipative force which depends on the spin-curl density is zero in homeogeneous fields and emerges when spatial inhomogeneities of polarization and intensity are present [145].