Caustics of Light Rays and Euler's Angle of Inclination

Physical Review A, 2015

In this paper, a meaningful classification of optical caustic beams in two dimensions is presented. It is demonstrated that the phase symmetry of the beam's angular spectrum governs the optical catastrophe, which describes the wave properties of ray singularities, for cusp (symmetric phase) and fold (antisymmetric phase) caustics. In contrast to the established idea, the caustic classification arises from the phase symmetry rather than from the phase power, thus breaking the commonly accepted concept that fold and cusp caustics are related to the Airy and Pearcey functions, respectively. Nevertheless, the role played by the spectral phase power is to control the degree of caustic curvature. These findings provide straightforward engineering of caustic beams by addressing the spectral phase into a spatial light modulator or glass plate.

Engineering Transactions, 2013

The shape of the “initial curve”, i.e. the locus of material points, which if properly illuminated provide (under specific conditions) the “caustic curve”, is explored. Adopting the method of complex potentials improved formulae for the shape of the “initial curve” are obtained. Application of these formulae for two typical problems, i.e. the mode-I crack and the infinite plate with a finite circular hole under uniaxial tension, indicates that the “initial curve” is in fact not a circular locus. It is either an open curve or a closed contour, respectively, the actual shape of which depends also on the in-plane displacement field.

Journal of Mathematical Sciences and Modelling

One can often see caustic by reflection in nature but it is rather hard to understand the way of how caustic arise and which geometric properties of a mirror surface define geometry of the caustic. The caustic by reflection has complicated topology and much more complicated geometry. From engineering point of view the geometry of caustic by reflection is important for antenna's theory because it can be considered as a surface of concentration of the reflected wave front. In this paper we give purely geometric description of the caustics of wave front (flat or spherical) after reflection from mirror surface. The description clarifies the dependence of caustic on geometrical characteristics of a surface and allows rather simple and fast computer visualization of the caustics in dependence of location of the rays source or direction of the pencil of parallel rays.

2016

Journal of the Optical Society of America, 2017

The aim of the present work is to obtain an integral representation of the field associated with the refraction of a plane wave by an arbitrary surface. To this end, in the first part we consider two optical media with refraction indexes n 1 and n 2 separated by an arbitrary interface, and we show that the optical path length, Φ, associated with the evolution of the plane wave is a complete integral of the eikonal equation in the optical medium with refraction index n 2. Then by using the k function procedure introduced by Stavroudis, we define a new complete integral, S, of the eikonal equation. We remark that both complete integrals in general do not provide the same information; however, they give the geometrical wavefronts, light rays, and the caustic associated with the refraction of the plane wave. In the second part, using the Fresnel-Kirchhoff diffraction formula and the complete integral, S, we obtain an integral representation for the field associated only with the refraction phenomena, the geometric field approximation, in terms of secondary plane waves and the k-function introduced by Stavroudis in solving the problem from the geometrical optics point of view. We use the general results to compute: the wavefronts, light rays, caustic, and the intensity associated with the refraction of a plane wave by an axicon and plano-spherical lenses.

Analysis and Mathematical Physics, 2021

In this paper we study global properties of the Wigner caustic of parameterized closed planar curves. We find new results on its geometry and singular points. In particular, we consider the Wigner caustic of rosettes, i.e. regular closed parameterized curves with non-vanishing curvature. We present a decomposition of a curve into parallel arcs to describe smooth branches of the Wigner caustic. By this construction we can find the number of smooth branches, the rotation number, the number of inflexion points and the parity of the number of cusp singularities of each branch. We also study the global properties of the Wigner caustic on shell (the branch of the Wigner caustic connecting two inflexion points of a curve). We apply our results to whorls—the important object to study the dynamics of a quantum particle in the optical lattice potential.

2008

Journal of the Optical Society of America A, 2013

The aim of the present work is twofold: first we obtain analytical expressions for both the wavefronts and the caustic associated with the light rays reflected by a spherical mirror after being emitted by a point light source located at an arbitrary position in free space, and second, we describe, in detail, the structure of the ronchigrams when the grating or Ronchi ruling is placed at different relative positions to the caustic region and the point light source is located on and off the optical axis. We find that, in general, the caustic has two branches: one is a segment of a line, and the other is a two-dimensional surface. The wavefronts, at the caustic region, have self intersections and singularities. The ronchigrams exhibit closed-loop fringes when the grating is placed at the caustic region.

Optik, 2020

In this study, we investigate the special curve that is formed by the reflection of the light rays emitted from the point light source on the unit sphere from a spherical curved mirror. Also, spherical caustic curves are defined as the geometrical location of the focusing requests of the reflected light rays. To examine the trajectories of the light rays emitted from the point light source, we are using the Sabban frame apparatus of the spherical curved mirror on the sphere. Also, the contact points of these curves are examined in terms of the Sabban frame apparatus. Then, the singularity conditions of these curves are examined and the shapes in which they are diffeomorphic are characterized. Finally, we give an example, which is an application of our theorems and definitions, and we visualized the shapes of curved mirrors and point light source in the example with the help of Mathematica program.