Symmetric gradient-index media reconstruction.

Ray tracing in gradient-index (GRIN) media has been thoroughly studied and several ray tracing methods have been proposed. Methods are based on finding the ray path given a known GRIN. In recent decades, the inverse problem, which consists of finding the GRIN distribution for a given light ray path, has been gaining attention. Given that it is not an easy task, the methods proposed in the literature vary in degrees of difficulty. In this work, an alternative method is presented to derive symmetric GRIN distributions whose implementation can be considered the simplest to date. Since it is based on invariants, which result from the symmetries of the system as stated by Fermat's principle, it is an exact numerical method, i.e., the physical system is not approximated. The robustness of the method permits the reconstruction of the GRIN distribution from a ray propagating in three-dimensions. In order to demonstrate its operation, different known symmetric GRIN media are reconstructed using rays that propagate in two and three dimensions.

Generalization of ray tracing in symmetric gradient-index media by Fermat's ray invariants.

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.

What are the traveling waves composing the Hermite-Gauss beams that make them structured wavefields?

To the best of our knowledge, at the present time there is no answer to the fundamental question stated in the title that provides a complete and satisfactory physical description of the structured nature of Hermite-Gauss beams. The purpose of this manuscript is to provide proper answers supported by a rigorous mathematical-physics framework that is physically consistent with the observed propagation of these beams under different circumstances. In the process we identify that the paraxial approximation introduces spurious effects in the solutions that are unphysical. By removing them and using the property of self-healing, that is characteristic to structured beams, we demonstrate that Hermite-Gaussian beams are constituted by the superposition of four traveling waves.

Single function crystalline lens capable of mimicking ciliary body accommodation.

The lens is a complex optical component of the human eye because of its physiological structure: the surface is aspherical and the structural entities create a gradient refractive index (GRIN). Most existent models of the lens deal with its external shape independently of the refractive index and, subsequently, through optimization processes, adjust the imaging properties. In this paper, we propose a physiologically realistic GRIN model of the lens based on a single function for the whole lens that accurately describes different accommodative states simultaneously providing the corresponding refractive index distribution and the external shape of the lens by changing a single parameter that we associate with the function of the ciliary body. This simple, but highly accurate model, is incorporated into a schematic eye constructed with reported experimental biometric data and accommodation is simulated over a range of 0 to 6 diopters to select the optimum levels of image quality. Changes with accommodation in equatorial and total axial lens thicknesses, as well as aberrations, are found to lie within reported biometric data ranges.

Kepler's law for optical beams.

It is well known that optics and classical mechanics are intimately related. One of the most important concepts in classical mechanics is that of a particle in a central potential that leads to the Newtonian description of the planetary dynamics. Within this, a relevant result is Kepler's second law that is related to the conservation of orbital angular momentum, one of the fundamental laws in physics. In this paper, we demonstrate that it is possible to find the conditions that allow us to state Kepler's second law for optical beams with orbital angular momentum by analyzing the streamlines of their energy flow. We find that the optical Kepler's law is satisfied only for cylindrical symmetric beams in contrast to the classical mechanics situation that is satisfied for the other conic geometries, namely, parabolic, elliptical and hyperbolic. We propose a novel approach to confirm our analytic results: we observe the propagation of the Arago's spot created by a beam with orbital angular momentum as a local "light-tracer" instead of looking at the propagation of the whole beam. The observed patterns fully agree with the prediction of our formalism.

Topographic synthesis of arbitrary surfaces with vortex Jinc functions.

When a circular aperture is uniformly illuminated, it is possible to observe in the far field an image of a bright circle surrounded by faint rings known as the Airy pattern or Airy disk. This pattern is described by the first-order Bessel function of the first type divided by its argument expressed in circular coordinates. We introduce the higher-order Bessel functions with a vortex azimuthal factor to propose a family of functions to generalize the function defining the Airy pattern. These functions, which we call vortex Jinc functions, happen to form an orthogonal set. We use this property to investigate their usefulness in fitting various surfaces in a circular domain, with applications in precision optical manufacturing, wavefront optics, and visual optics, among others. We compare them with other well-known sets of orthogonal functions, and our findings show that they are suitable for these tasks and can pose an advantage when dealing with surfaces that concentrate a considerable amount of their information near the center of a circular domain, making them suitable applications in visual optics or analysis of aberrations of optical systems, for instance, to analyze the point spread function.