Analytical calculations of optical trapping forces for drag calibration: Effects of mismatch between beam focus and particle center

Abstract

We present analytical calculations of optical forces for drag calibrations via the geometrical optics approximation (GOA) for the case in which a single laser beam is strongly focused on a polystyrene microsphere dispersed in an aqueous phase. When the beam focus is mismatched with the particle center owing to the presence of vertical forces, such as gravity, buoyancy, and the radiation force caused by the laser beam, the trapped particle would be displaced in a radial direction when the lateral drag force is applied. Based on the analytical calculations of optical trapping forces and the force balance upon dragging, we found that the critical laser power at which the beam focus matches the particle center exhibits a power-law relationship with the particle radius a according to the empirical expression log(P _crit (mW)) = 2.899 log(2a (µm)) - 2.316. We also found an empirical expression for the mechanical equilibrium position of the vertical displacement Δz ⁰ _eq (Δy=0) as functions of the laser power P and the particle radius, $\Delta z_{eq}^0 = \frac{{\left[ {{F_{gb}}\left( {pN} \right) - 0.068P\left( {mW} \right)} \right]\left[ {2.7a\left( {\mu m} \right) - 0.252} \right]}}{{P\left( {mW} \right)}}$, where F _gb is the sum of the gravitational and buoyant forces.