Ray-tracing methodology: application of spatial analytic geometry in the ray-optic model of optical tweezers

Traditional solid geometry ray-tracing method is complex in analyzing the orientation of gradient forces and calculating incident angle of optic rays upon a microsphere. We present a new ray-tracing methodology based on spatial analytic geometry in the ray-optic model. For a single ray upon a microsphere, the directions of transmission and trapping forces are depicted by spatial vectors in a Cartesian coordinate system. At the same time, the polarized direction of a single focused ray can be transformed by a matrix of rotational coordinates. According to the relations of vectors, the trapping forces can be expressed identically. We use this new method to discuss differences of trapping forces in the cases of various states of unpolarized and polarized beams, and also show the reasons for differences in transverse force between measurement and theoretical results. Our simulative results show that this method can be applied identically to calculating both transverse and axial trapping forces, and also for different polarizations of a laser beam.