Key words: dielectric spheres; gradient force; Mie scattering; optical sorting; standing-wave laser field.
Accepted: August 30, 2016Published: September 7, 2016 http://dx.doi.org/10.6028/jres.121.020
SummaryRecently, as part of a study of the ability of a Gaussian standing-wave interference field to sort small dielectric spheres entrained in a moving fluid, we implemented a two-length-scale method to determine the trajectories of the particles. In the first step, effective velocities and diffusion constants are determined by solving the Fokker-Planck equation in a force field determined by the Maxwell stress tensor. In the second step, these effective constants of motion are used to predict positions of particles swept across the interference field. Here, we present the code which was used in a companion article [1].The fine scale tracks a domain of typically 16 interference fringes with a spatial resolution typically 1/16 of a fringe. The Fokker-Planck equation tracks the evolution of the probability distribution of a sphere in a period set of one-dimensional sinusoidal wells. Physically, these sinusoidal wells exist because of the interaction between the dielectric particle and the electromagnetic standing wave. The effective velocity is determined by finding the mean position of the sphere as a function of time. After allotting time for intrawell equilibration, the mean position becomes a linear function of time. The slope of this line is the velocity. Similarly, the diffusion constant is related to the time evolution of the second moment of the distribution. The effective velocity and diffusion constant needs to be found for a sphere of a given radius for a range of optical intensities.The coarse scale is given by the size of the width of the standing wave, typically 3 mm. Although the position is a continuous variable, the solution is unaware of the fringes except to the extent that there is a position-dependent effective velocity and effective diffusion constant which is determined on the fine scale above. The crossing time is a simple function of the transverse distance between the starting position and end line. A simple approximation to the distribution of crossing times (induced by diffusion) is made, however, uniform motion across the interference field is assumed. The distance travelled longitudinally is found by integrating the effective velocity across the Gaussian envelope of the interference field. The