Ray tracing in spherical Luneburg lenses has always been represented in 2D. All propagation planes in a 3D spherical Luneburg lens generate the same ray tracing, due to its radial symmetry. A geometry without radial symmetry generates a different ray tracing. For this reason, a new ray tracing method in 3D through spherical and elliptical Luneburg lenses using 2D methods is proposed. The physics of the propagation is shown here, which allows us to make a ray tracing associated with a vortex beam. A 3D ray tracing in a composite modified Luneburg lens that represents the human eye lens is also presented.

We present a new design of a plasmonic Luneburg lens made from a gradient-index metasurface that was constructed with an array of nanometer-sized holes in a dielectric thin film. The fabricated structure consists of a planar lens with a diameter of 8.7 μm composed of a rectangular array of holes with a periodicity of 300 nm. The experimental characterization includes leakage-radiation microscopy imaging in the direction and frequency space. The former allows for characterization of the point spread function and phase distribution, whereas the latter grants access to qualitative measurements of the effective mode indices inside the plasmonic lens. The experimental results presented here are in good agreement with the expected average performance predicted by the numerical calculations. Nevertheless, the robustness of the characterization techniques presented here is also exploited to determine deviations from the design parameters.

Ray tracing in gradient-index (GRIN) media has been traditionally performed either by using the analytical or numerical solutions to the Eikonal equation or by creating a layered medium where Snell’s law is calculated in each layer. In this paper, an exact general method to perform ray tracing in GRIN media is presented based on the invariants of the system as stated by Fermat’s principle when the media presents symmetries. Its advantage, compared with other methods reported in the literature, relies on its easy implementation. Besides the GRIN distribution and the initial conditions of the incident ray, once the invariants of the system are stated the resulting math is simple to solve and interpret. To benchmark the algorithm, ray tracing in typical cases of GRIN media is calculated, finding minimal discrepancies between the analytical solutions and our simulations. The used media are axial refractive index and parabolic index fiber and lenses with spherical gradient-index symmetry, such as: Luneburg’s, Gutman’s, generalized Maxwell’s Fish-eye, Eaton’s, and concentrator lenses. Our method can be further applied to distributions with symmetries associated with other common curvilinear orthogonal coordinate systems, in particular to those associated to the separability of the Helmholtz equation that would allow us to investigate wave optics in these GRIN media with the associated geometries.

In this paper, we demonstrate that by using a mathematical physics approach—focusing attention on the physics and using mathematics as a tool—it is possible to visualize the formation of the transverse modes inside a cylindrical waveguide. The opposite (physical mathematics) approach looks at the mathematical problem and then tries to impose a physical interpretation. For cylindrical waveguides, the physical mathematics route leads to the Bessel differential equation, and it is argued that in the core of the waveguide there are only Bessel functions of the first kind in the description of the transverse modes. The Neumann functions are deemed non-physical due to their singularity at the origin and are eliminated from the final description of the solution. In this paper, by combining geometric optics and wave optics concepts, we show that the inclusion of the Neumann function is physically necessary to describe fully and properly the formation of the propagating transverse modes. With this approach, we also show that the field outside a dielectric waveguide arises in a natural way.

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.