Field tracing in graded-index media

Abstract

The graded-index (GRIN) media have been widely used in modern optical systems, such as imaging lenses, optical beam delivery components, and beam shaping elements. Accurate and efficient modeling of electromagnetic fields propagating in GRIN media is essential for developing innovative and high-quality optical products. It is the goal of this thesis. We first give an overview of several existing modeling techniques, including the rigorous ones to model symmetric GRIN structures, e.g., Mie theory (spherically symmetric) and several beam propagation methods (BPMs) with approxiamations to model general GRIN structures, which are numerically more efficient than rigorous methods. However, many of BPMs are limited by considering only a small variance of the refractive index. To overcome the limitations, we develop a unified field solver, known as the Runge-Kutta (RK) k-domain field propagation method. It can be used to calculate general input fields propagating through an arbitrary GRIN medium, accurately, without physical approximations, such as the scalar field approximation or the paraxial approximation. We convert Maxwells equations in spatial (x-) domain into ordinary differential equations (ODEs) in the spatial-frequency (k-) domain, which can be solved iteratively using the RK method. In our numerical calculation, taking advantage of the convolution theorem, the total numerical effort is linear in the number of sampling points. As many GRIN components are modeled or designed using ray tracing techniques, the field passing through the GRIN components must show negligible diffraction, which satisfies the geometric field assumption. By substituting geometric field ansatz into the ODEs of RK k-domain field propagation method and Maxwells equations, we obtain two ODEs in x-domain. After solving them by using again RK method, the RK x-domain field propagation method is established.