Nonlinear ray tracing in focused fields, part 2. Tracing the flux: tutorial.

In this three-part paper series, we develop a method to trace the lines of flux through a three-dimensional wavefield by following a direction that is governed by the derivative of the phase at each point, a process that is best described as flux tracing but which we interchangeably name "nonlinear ray tracing." In the first part we focused on the high-speed calculation of focused three-dimensional complex wavefields in the paraxial approximation for T E M 00 and T E M 01 laser modes. The algorithms developed in the first paper are first used to generate the three-dimensional grid of samples of the complex wavefield in the focal region. In this second part, we focus on tracing a flux through this three-dimensional point cloud. For a given "ray" at an arbitrary position in the 3D volume, interpolation of the three-dimensional samples is applied to determine the derivative of the phase (normal to the direction of propagation) at the ray position, which is then used to direct the ray as it "propagates" forward in a straight line over a short distance to a subsequent plane; the process is repeated between consecutive planes. The initial origin of the ray can be chosen arbitrarily at any point, and the ray can then be traced through the volume with appropriate interpolation. Results are demonstrated for focused wavefields in the absence of aberrations, corresponding to the cases highlighted in the first paper. Some of the most interesting results relate to focused Laguerre-Gaussian beams, for which the rays are found to spiral at different rates of curvature. In the third paper we extend the application of this algorithm to the investigation of lens aberration.