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Optical tweezers
The resulting force exerted by a single ray of light on a transparent particle is the result of reflections and refractions of the incident ray (see Fig. 2.1b). A transparent particle is made of a material that does not absorb light (i.e., the index of refraction is a real -not complex- number) but the particle can reflect and refract light. The linear momentum carried by the incident light (~pi ) is split into the two outcoming rays: the reflected and the transmitted ray, each one carrying a different amount of linear momentum (~pr and p~i respectively). The balance between them is the resulting linear momentum transferred to the bead p~b : p~i = p~r + p~t + p~b p~b = p~i − (~pr + p~t ).
(2.1)
The amount of linear momentum transferred per unit of time determines the force applied to the bead (f~b ): d~pb = f~b . dt
(2.2)
Usually, the ray reflected on a transparent particle is much weaker compared to the transmitted ray. So in most cases the reflected ray can be neglected. The resulting force is usually split into two perpendicular components (see Fig. 2.1c). The first one is the axial or scattering force (f~scat ) and it is parallel to the original direction of the beam. The second component one is called radial or gradient force (f~grad ). The calculation of the force exerted by a beam of light is performed by repeating the previous computation for all the rays of the beam and summing their contributions (see Fig. 2.1a). This exerted force can be calculated numerically [78]. The summation of the radial forces for each ray gives the total gradient force exerted by the light beam. And so for the total scattering force. The calculation is strongly dependent of each particular case: position, collimation, intensity of the laser beam; and position, shape and index of refraction of the particle (see Figure 2.2). Besides, the optical forces depend on the polarization of the laser beam because the reflectivity and transmissivity on the surface of the particle are given by the Fresnel reflection and transmission coefficients respectively, which are polarization dependent. Thus, the calculation is usually performed on circularly polarized light, where the resulting trapping force is the average of the parallel and perpendicular polarizations. The forces calculated in the ray optics regime are independent of the particle size although we know that this is not true in a typical experimental setup. However, the ray optics regime provides a good qualitative description of the trapping phenomenon.