Shadows of structured beams in lenslike media.

The self-healing phenomenon of structured light beams has been comprehensively investigated for its important role in various applications including optical tweezing, superresolution imaging, and optical communication. However, for different structured beams, there are different explanations for the self-healing effect, and a unified theory has not yet been formed. Here we report both theoretically and experimentally a study of the self-healing effect of structured beams in lenslike media, this is, inhomogeneous lenslike media with a quadratic gradient index. By observing the appearance of a number of shadows of obstructed structured wave fields it has been demonstrated that their self-healing in inhomogeneous media are the result of superposition of fundamental traveling waves. We have found that self-healing of structured beams occurs in this medium and, interestingly enough, that the shadows created in the process present sinusoidal propagating characteristics as determined by the geometrical ray theory in lenslike media. This work provides what we believe to be a new inhomogenous environment to explain the self-healing effect and is expected to deepen understanding of the physical mechanism.

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.

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.

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.

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.

Probing the reciprocal lattice associated with a triangular slit to determine the orbital angular momentum for a photon.

The orbital angular momentum conservation of light reveals different diffraction patterns univocally dependent on the topological charge of the incident light beam when passing through a triangular aperture. It is demonstrated that these patterns, which are accessed by observing the far-field measurement of the diffracted light, can also be obtained using few photon sources. In order to explain the observed patterns, we introduce an analogy of this optical phenomenon with the study of diffraction for the characterization of the crystal structure of solids. We demonstrate that the finite pattern can be associated with the reciprocal lattice obtained from the direct lattice generated by the primitive vectors composing any two of the sides of the equilateral triangular slit responsible for the diffraction. Using the relation that exists between the direct and reciprocal lattices, we provide a conclusive explanation as to why the diffraction pattern of the main maxima is finite. This can shed a new light on the investigation of crystallographic systems.

Intelligent-assistant system for scleral spur location.

A system based on the use of two artificial neural networks (ANNs) to determine the location of the scleral spur of the human eye in ocular images generated by an ultrasound biomicroscopy is presented in this paper. The two ANNs establish a relationship between the distance of four manually placed landmarks in an ocular image with the coordinates of the scleral spur. The latter coordinates are generated by the expert knowledge of a subject matter specialist. Trained ANNs that generate good results for scleral spur location are incorporated into a software system. Statistical indicators and results yield an efficiency performance above 95%.

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.

Three-dimensional ray tracing in spherical and elliptical generalized Luneburg lenses for application in the human eye lens.

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.

Composite modified Luneburg model of human eye lens.

A new lens model based on the gradient-index Luneburg lens and composed of two oblate half spheroids of different curvatures is presented. The spherically symmetric Luneburg lens is modified to create continuous isoindicial contours and to incorporate curvatures that are similar to those found in a human lens. The imaging capabilities of the model and the changes in the gradient index profile are tested for five object distances, for a fixed geometry and for a fixed image distance. The central refractive index decreases with decreasing object distance. This indicates that in order to focus at the same image distance as is required in the eye, a decrease in refractive power is needed for rays from closer objects that meet the lens surface at steeper angles compared to rays from more distant objects. This ensures a highly focused image with no spherical aberration.

Unveiling a truncated optical lattice associated with a triangular aperture using light's orbital angular momentum.

We show that the orbital angular momentum can be used to unveil lattice properties hidden in diffraction patterns of a simple triangular aperture. Depending on the orbital angular momentum of the incident beam, the far field diffraction pattern reveals a truncated optical lattice associated with the illuminated aperture. This effect can be used to measure the topological charge of light beams.

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.

Experimental observation of interfering Bessel beams.

We experimentally verified the interference resulting of the superposition of two Bessel beams propagating in free space and showed for first time the self imaging effect using nondiffracting beams. Our results are supported by numerical simulations and possible applications are discussed.

New asymmetric propagation invariant beams obtained by amplitude and phase modulation in frequency space.

In this paper, we demonstrate, numerically and experimentally that using the mask-lens setup used by Durnin to generate Bessel beams Durnin [Phys. Rev. Lett. 58, 1499 (1987)], it is possible to generate different kinds of propagation invariant beams. A modification in the amplitude or phase of the field that illuminates the annular slit is proposed that corresponds to modulation in frequency space. In particular, we characterize the new invariant beams that were obtained by modulating the amplitude of the annular mask and when the incident field was modulated with a one-dimensional quadratic or cubic phase. Experimental results using an amplitude mask are shown in order to corroborate the numerical predictions.

Optical realization of quantum-mechanical invariants.

We study the problem of paraxial propagation in two grade-index media by using invariant techniques that allow a continuous solution of the problem. By using the well-known fact that this problem is analogous to the time-dependent harmonic oscillator in quantum mechanics, known methods there may be imported producing on the one hand a solution to the propagation problem and on the other hand a realization of a quantum-mechanical invariant.

Elliptic vortices of electromagnetic wave fields.

We demonstrate the existence of elliptic vortices of electromagnetic scalar wave fields. The corresponding intensity profiles are formed by propagation-invariant confocal elliptic rings. We have found that copropagation of this kind of vortex occurs without interaction. The results presented here also apply for physical systems described by the (2+1) -dimensional Schrödinger equation.

Alternative formulation for invariant optical fields: Mathieu beams.

Based on the separability of the Helmholtz equation into elliptical cylindrical coordinates, we present another class of invariant optical fields that may have a highly localized distribution along one of the transverse directions and a sharply peaked quasi-periodic structure along the other. These fields are described by the radial and angular Mathieu functions. We identify the corresponding function in the McCutchen sphere that produces this kind of beam and propose an experimental setup for the realization of an invariant optical field.

Interference of traveling nondiffracting beams.

We study numerically the interference resulting from the superposition of two Bessel beams propagating in free space. We discuss how to obtain such beams and show the existence of the self-imaging effect during propagation. The evolution of the superimposed Bessel beams is analyzed on the basis of the evolution of the individual beams. Our exact numerical predictions contradict previous approximated analytical treatments, showing that they can lead to quantitatively wrong results and misleading conclusions.

Experimental evidence of modulation instability in a photorefractive Bi(12)TiO(20) crystal.

We present experimental results on the propagation of an interference pattern of two He-Ne laser beams of unequal amplitudes through a photorefractive Bi(12)TiO(20) crystal in the presence of drift nonlinearity. The phenomenon that we have observed is the focusing of the fringes as the nonlinearity of the crystal is increased. We show that such a phenomenon can be quantitatively interpreted in the framework of modulation instability theory.