Solving the eikonal equation by the FSM method in Julia language

Abstract

There are two main approaches to the numerical solution of the eikonal equation: reducing it to asystemofODES(methodofcharacteristics)andconstructingspecializedmethodsforthenumericalsolutionof this equation in the form of a partial differential equation. The latter approach includes the FSM (Fast sweeping method) method. It is reasonable to assume that a specialized method should have greater versatility. The purpose of this work is to evaluate the applicability of the FSM method for constructing beams and fronts. The implementation of the FSM method in the Eikonal library of the Julia programming language was used. The method was used for numerical simulation of spherical lenses by Maxwell, Luneburg and Eaton. These lenses were chosen because their optical properties have been well studied. A special case of flat lenses was chosen as the easiest to visualize and interpret the results. The results of the calculations are presented in the form of images of fronts and rays for each of the lenses. From the analysis of the obtained images, it is concluded that the FSM method is well suited for constructing electromagnetic wave fronts. An attempt to visualize ray trajectories based on the results of his work encounters a number of difficulties and in some cases gives an incorrect visual picture.

Keywords

• eikonal equation
• geometric optics
• wave optics
• julia language
• fast sweeping method