Comparison of ray-tracing methods for semitransparent slab with variable spatial refractive index

Abstract

The ray-tracing technique has the main difficulty in solving radiative transfer in the medium with variable spatial refractive index. Recently, three methods have been developed for the application of the ray-tracing technique in those medium. To compare and discuss the numerical characteristics of those methods, a semitransparent slab with variable spatial refractive index is taken as an example, and the reflectivity and the transmissivity of the slab are computed by the curved ray-tracing method, the multi-layer approach, and the discrete curved ray-tracing method, respectively. As the result, it is shown that, the discrete curved ray-tracing method gives the result with good accuracy and convergence characteristics than the multi-layer approach. Due to accounting physically inexistent reflection on the interface between sublayers, the multi-layer approach converges slowly.

Introduction

The radiative heat transfer in semitransparent medium with variable spatial refractive index is of great interest in thermo-optical systems. Because the ray goes along a curved path determined by the Fermat principle, the ray-tracing is very difficult and complex in the medium with variable spatial refractive index. Then, the ray-tracing technique has difficulty in solving radiative transfer problems in the medium with graded refractive index.

Recently, three methods have been proposed using the ray-tracing technique in the medium with varying refractive index. The first is the curved ray-tracing method presented by Ben Abdallah and Le Dez [1], [2], [3], [4] and Ben Abdallah et al. [5], in which the curved path of ray transfer is obtained directly by solving the ray equation deduced from the Fermat principle. For some simple distributions of refractive index, this method can give a simple and exact analytical expression of the ray path. For complicated distributions of refractive index, however, the analytical expression of the ray path becomes complex and the computation of the ray trajectories is very time-consuming. The second is the multi-layer approach, in which the slab is divided into many sublayers with uniform indices, and the ray transfer in the sublayers and the reflection and refraction at their interfaces are traced. This method is based on the idea of Siegel and Spuckler [6], [7], who pointed out that, by increasing the number of sublayers, the radiative behavior of multi-layer approaches to the corresponding single layer with a continuous variation of refractive index. The third is the discrete curved ray-tracing method proposed by Liu [8], Liu and Tan [9]. In this method, the semitransparent slab is divided into many sublayers. In each sublayer, the optical constants are assumed to be uniform, and the curved ray trajectory is locally treated as straight line as in the multi-layer approach, and the complicated and time-consuming computation of the curved ray trajectory can be avoided. It is noted that, inside the real absorbing-emitting medium with continuous variable optical constants, there is no the real interface that reflects rays. Then, differing from the multiplayer approach, the discrete curved ray-tracing method considers the interface reflections only for total reflection condition.

The objective of this paper is to compare these three methods. A semitransparent slab with variable spatial refractive index is taken as an example, and the reflectivity and the transmissivity of the slab are computed by curved ray-tracing method, multi-layer approach, and discrete curved ray-tracing method, respectively. The computed results are compared to each other, and the numerical characteristics of those methods are discussed.