Geometrical-light-propagation in non-normalized symmetric gradient-index media

Abstract

Typically, as a means to obtain a less complicated ray tracing method on a gradient-index (GRIN) medium, a normalization is done. This normalization is based upon the fact that the values of the refractive index on the surface of the GRIN medium and the value of the refractive index medium where it is immersed are the same. In this paper, a Fermat’s-ray-invariants-based ray tracing method in a non-normalized GRIN medium is presented. This method is so simple that it is no longer necessary to perform normalization to reduce ray tracing complications in GRIN media. In order to observe its operation, the method is implemented in different GRIN media with rectangular, cylindrical, and spherical symmetry. Also, the method is implemented in two different axicon GRIN lenses. MATLAB codes for the implementations are provided as supplementary files. Finally, it is possible to observe that the Fermat’s ray invariant is still preserved even outside the lens. This implies that it is not necessary to apply Snell’s law when the ray leaves the GRIN medium, because the conservation of the Fermat’s ray invariant performs the refraction as if Snell’s law had been applied.