Mechanical-acoustical analogy: From laboratory to home during the COVID-19 pandemic.

A clear comprehension of the oscillatory nature of sound for acoustics undergraduate students is of paramount importance. In this paper, two online experiments were implemented to aid teaching of the oscillatory nature of sound through the analogy between a mechanical mass-spring model and a Helmholtz resonator. The study was conducted among undergraduate students taking a science course in the Electronic and Electrical Engineering career curriculum. These in-class experiments were conducted during the COVID-19 pandemic via the Zoom platform. Students measured the Helmholtz resonant frequency of a plastic bottle with a smartphone application and compared its oscillatory behavior with that of a conventional harmonic oscillator under a professor-student collaborative environment. The results of this study suggest that, with careful experiment design, students can effectively benefit from the use of common technology tools, which, in turn, poses these methodologies as a rather satisfactory alternative to face-to-face laboratory sessions.

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.

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.