Nonlinear Ray Tracing to Calculate the Forces of an Optical Laser Trap on a Dielectric Sphere

Nonlinear rays are described by the eikonal function, which bridges the gap between wave and geometrical optics. These nonlinear rays follow a path that is always normal to the phase front as it propagates in space. In this paper, we explore how to augment the classical geometrical ray-optics approach to calculate the forces acting on the sphere in an optical trap developed by Ashkin, such that non-linear rays can replace linear rays in the calculation. The greatest advantage of such an approach would be be the capacity to model the orbital angular momentum imparted on the sphere using a Laguerre-Gaussian spatial mode laser. The non-linear rays associated with any convering wavefield can be traced towards their intersection point on the surface of the sphere and for each one of these 'rays', the scattering force, gradient force, and torque can be derived using the Equations defined by Ashkin. Integration of these forces reveals the total three-dimensional force acting on the sphere as well as total rotational forces which can be decomposed into a 'vertical torque' and 'horizontal torque.' As well as investigating the single beam dielectric trap in the model of Ashkin, we additionally investigate the dual beam trap for all cases, which has the benefit of enhanced trapping forces.