Investigation of the power spectral density of a scaled model simulation of an optical tweezer

Abstract

The stochastic differential equations (SDEs) representing a bead’s motion in an optical tweezer are stiff, meaning that the ratio of bead’s inertia to the viscous force from the surrounding fluid is extremely large. A scaling technique can be used to improve the computational time required to solve these SDEs numerically using adaptive SDE solvers. However, this scaling changes the SDEs as well as the power spectral density (PSD) of the bead’s position that the SDEs predict. This work shows that the scaling technique can be used effectively without too much loss of information but with significant reduction in computational time. The model uses Mie Scattering theory to compute the laser beam force, while Stokes’ law is used to calculate the drag force. Adaptive Stiff solvers, available in Julia programming language, are used to solve the SDEs. The PSD analysis is done in MATLAB. Experimental datasets for 2000nm, 1950nm, 990nm and 500nm diameter polystyrene beads are compared with the numerical results. We focus on the 2000nm bead because it is the only case where we can obtain PSD directly from the experimental data; in the other cases the PSD is indirectly obtained from experimental data. Interestingly, the 990nm and 500nm beads overshoot the focal point of the optical trap. This work presents the PSD analysis for these cases in addition to the reduction in computational time using the proposed scaling approach.