Nonlinear ray tracing in focused fields, part 1. Calculation of 3D focused wavefields: 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 this first part, we provide a tutorial on the high-speed calculation of three-dimensional complex wavefields, which is a necessary precursor to flux tracing. The basis of this calculation is the angular spectrum method, a well-known numerical algorithm that can be used to efficiently and accurately calculate diffracted fields for numerical apertures <0.7. It is known that this approach yields identical predictions to the first Rayleigh-Sommerfeld solution. We employ the angular spectrum method to develop two algorithms that generate the 3D complex wavefield in the region of focus of a lens. The first algorithm is based on the thin lens approximation, and the second is based on the concept of an ideal lens, which can be modeled using an optical Fourier transform. We review how these algorithms can be used to calculate focused laser beams with T E M ₀₀ and T E M ₀₁ laser profiles. The three-dimensional sampling requirements of the focused field are explained in detail, and expressions for the computational and memory efficiency of the two algorithms are developed. These two algorithms generate the 3D scaffold for the flux tracing method developed in the second paper, and in the third paper we highlight the application of the method to understanding monochromatic lens aberration. Disregarding the second and third papers, this first paper will serve as a practical tutorial for anyone seeking to compute focused fields in three dimensions.