Geometrical optics and geodesics in thin layers

Abstract

The propagation of a light ray in thin layer (film) within geometrical optics is considered. It is assumed that the ray is captured inside the layer due to reflecting walls or total internal reflection (in the case of a dielectric layer). It has been found that for a very thin film (the length scale is imposed by the curvature of the surface at a given point) the equations describing the trajectory of the light beam are reduced to the equation of a geodesic on the limiting curved surface. There have also been found corrections to the trajectory equation resulting from the finite thickness of the film. Numerical calculations performed for a couple of exemplary curved layers (cone, sphere, torus and catenoid) confirm that for thin layers the light ray which is repeatedly reflected, propagates along the curve close to the geodesic but as the layer thickness increases, these trajectories move away from each other. Because the trajectory equations are complicated non-linear differential equations, their solutions show some chaotic features. Small changes in the initial conditions result in remarkably different trajectories. These chaotic properties become less significant the thinner the layer under consideration.

The hollow Gaussian beam propagation on curved surface based on matrix optics method

“…Currently, two-dimensional surface is a common simplification model in the study of curved space, such as the two-dimensional Wolf effect of curved space [12], and other studies on geometric optics [13][14][15] and physical optics [16,17] of two-dimensional manifolds. In our research, based on geometrical optics, ABCD matrix and Collins' formula [18] are introduced to explore the light propagation characteristics on the curved surface under the paraxial approximation.…”

Guiding neutral polar molecules by electromagnetic vortex field

“…As we already know, in the case of extremely strong fields, formally capable of trapping particles, there appears another escape mechanism, mentioned in the previous sub-section (see last plots in Figures 2 and 3). It should also be added that in quantum picture, the escape can also be realized through tunneling effect [18].…”

“…For instance electric fields of order of 100 kV/cm (which is, however, still much lower than the internal field in a molecule) yield the values of the parameter γ falling within the range from 2.5 × 10 −28 for Λ 0 to 10 −12 for molecules (assuming the electromagnetic wave frequency of order of ω 0 ≈ 10 12 s −1 ). This is the typical situation when trapping and guiding neutral particles, that the effects are very tiny: this equally refers to the particles endowed with magnetic moment [17,18] in interaction with magnetic field. Figure 2 presents the numerical solutions of the equations of motion in the form of the particle trajectories for eight increasing values of γ.…”

“…As in the case of [9], for the latter ones the trapping mechanism was based on the interaction of magnetic moments of particles with the magnetic field of the vortex. Due to very tiny values of magnetic moments of elementary particles, extremely strong fields are needed to trap or guide them but in principle it should be possible [18].…”

Self-Adjoint Extension Approach to Motion of Spin-1/2 Particle in the Presence of External Magnetic Fields in the Spinning Cosmic String Spacetime

“…The presence of curvature or torsion, for instance, can affect such properties. Even the light propagation can be influenced under such conditions [5], which can be understood by the notion of geodesics [6]. A particularly relevant case involving curved spacetimes and singularities occur when a topological defect shows up in producing a singularity associated with a conical space [7].…”