Types Of Caustics Research Articles

Combinatorics of periodic ellipsoidal billiards

We study combinatorics of billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in d-dimensional Euclidean and pseudo-Euclidean spaces. Such partitions uniquely codify the sets of caustics, up to their types, which generate periodic trajectories. The period of a periodic trajectory is the largest part while the winding numbers are the remaining summands of the corresponding partition. In order to take into account the types of caustics as well, we introduce weighted partitions and provide closed forms for the generating functions of these partitions.

Diffraction of a cylindrical wave by a perfectly conducting cylinder enclosed in a metamaterial shell with a negative refractive index

The problem of diffraction of a plane (cylindrical) wave by a perfectly conducting cylinder enclosed in a metamaterial shell is rigorously solved. The influence of the geometric dimensions of the shell, the value of the negative refractive index of the metamaterial medium, and the position of a cylindrical wave source on the field structure in the near zone of the scatterer is investigated. It is found that, in the quasi-optical range of the problem parameters, this structure does not exhibit ideal focusing. It is shown that, there are two types of caustics inside the shell. The first type is related with rays reflected by the surface of the interior cylinder and has one cusp point, and the second type is formed by the geometric-optics rays that are refracted by the outer boundary of the shell and do not fall on the surface of the interior perfectly conducting cylinder. The spatial distribution of the total field amplitude and of the equal-amplitude lines in the near zone of the scatterer is reported. The obtained numerical results are correctly interpreted from the physical viewpoint.

Effect of Caustic Type on Bitumen Extraction from Canadian Oil Sands

The role of different types of caustics in bitumen extraction from Canadian oil sands is investigated by analyzing their effect on subprocesses involved in bitumen extraction. Both sodium hydroxide and ammonium hydroxide were shown to improve bitumen liberation, which is favorable for bitumen recovery, but increased induction time of bitumen–bubble attachment, which is harmful for bitumen recovery. Although a similar performance was observed at pH of ∼8.5, at pH 11.3, the use of ammonium hydroxide led to a shorter induction time of bitumen–bubble attachment and, hence, better bitumen recovery than the use of sodium hydroxide to achieve the same slurry pH. The better performance of ammonium hydroxide is attributed to a more hydrophobic surface and a less negative zeta potential of bitumen in the process water because of the release of less amount of natural surfactants in ammonium hydroxide solutions than in sodium hydroxide solutions of the same solution pH. Overall, ammonium hydroxide was found to be a suitable replacement for sodium hydroxide. Although this study was mainly concerned with mining–extraction operations, the major findings derived from this study could be applied readily to in situ thermal bitumen production operations, because ammonia can be delivered with steam to oil sands formations.

Properties of Planetary Caustics in Gravitational Microlensing

Although some of the properties of the caustics in planetary microlensing have been known, our understanding of them is mostly from scattered information based on numerical approaches. In this paper, we conduct a comprehensive and analytic analysis of the properties of the planetary caustics, which are one of the two sets of caustics in planetary microlensing, those located away from the central star. Under the perturbative approximation, we derive analytic expressions for the location, size, and shape of the planetary caustic as a function of the star-planet separation and the planet/star mass ratio. Based on these expressions combined with those for the central caustic, which is the other set of caustics located close to the central star, we compare the similarities and differences between the planetary and central caustics. We also present the expressions for the size ratio between the two types of caustics and for the condition of the merging of the two types of caustics. These analytic expressions will be useful in understanding the dependence of the planetary lensing behavior on the planet parameters and thus in interpreting the planetary lensing signals

Time-Domain Mapping of Electromagnetic Ray Movement Inside Anisotropic Spherical Resonator

This paper presents the analytical proof of “Time-Domain Mapping Method” for the spherical resonator made up of uniaxial crystal. In this way, the main types of caustics inside the spherical resonator made up of uniaxial crystal, which were investigated numerically before, are confirmed analytically. It is engraved that the problem of the ray flow inside the spherical resonator can be reduced to the problem of the ray flow inside metal cavity shaped as spheroid.

Optical Ray Movement Inside Dielectric Sphere

The paper presents results of numerical investigations of electromagnetic (optical) ray propagation inside a sphere, made of optically uniaxial crystal. The types of caustics formed by the ray trajectory inside the sphere are cleared up. It has been demonstrated the difference between types of caustics formed by the extraordinary ray trajectory inside the sphere made of optically positive and negative crystals. Results are discussed on the basis of one-to-one mapping of 2D billiard problem inside isotropic region bounded by ellipse onto anisotropic one.

Spectrophotometric Resolution of Stellar Surfaces with Microlensing

Microlensing is a powerful tool for studying stellar atmospheres, because as the source crosses regions of formally infinite magnification (caustics), the surface of the star is resolved, thereby allowing one to measure the radial intensity profile both photometrically and spectroscopically. However, caustic crossing events are relatively rare and monitoring them requires intensive application of telescope resources. It is therefore essential that the observational parameters needed to accurately measure the intensity profile are quantified. We calculate the expected errors in the recovered radial intensity profile as a function of the unlensed flux, source radius, spatial resolution of the recovered intensity profile, and caustic crossing time for the two principle types of caustics: point-mass and binary lenses. We demonstrate that for both cases there exist simple scaling relations between these parameters and the resultant errors. We find that the error as a function of the spatial resolution of the recovered profile, parameterized by the number of radial bins, increases as N, considerably faster than the naive N½R expectation. Finally, we discuss the relative advantages of binary caustic-crossing events and point-lens events. Binary events are more common, easier to plan for, and provide more homogeneous information about the stellar atmosphere; however, a subclass of point-mass events with low impact parameters can provide dramatically more information provided that they can be recognized in time to initiate observations.

Catastrophes and stable caustics in bound states of Hamiltonian systems

Caustics—envelopes of families of classical trajectories, or boundaries between classically allowed and forbidden regions—correspond to singular points of a phase-space surface called a Lagrangian manifold. According to catastrophe theory, only a limited number of types of caustics are stable under general perturbations of the manifold. Most of the caustics that are found in calculations correspond to members of the canonical list of elementary catastrophes. However, there are some exceptions—examination of trajectories of typical Hamiltonian systems shows that stable structures exist which are not in accord with the stability theorem of catastrophe theory. These exceptional cases are discussed in this paper. They arise because of the special form of the typical Hamiltonian of physical systems.

Caustics, catastrophes, and wave fields

The modern view of caustics as the singularities of mappings performed by rays is set forth. The substantial progress which has recently been achieved in research on caustic fields can be credited to progress in the theory of the singularities of differentiable mappings (catastrophe theory). This theory has generated an exhaustive classification of the structurally stable caustics and of the corresponding standard diffraction integrals which describe the fields near caustics. The standard integrals can be used to construct both local and uniform asymptotic field representations in the presence of caustics. Asymptotic methods for describing the field have also been developed for penumbral caustics associated with edge catastrophes and for several other types of caustics which arise in wave problems in optics, acoustics, radio propagation, plasma physics, atomic physics, etc.

The geometrical optics field at a caustic

The asymptotic expansion of a wave field in powers of 1/k , where k is the wave number, for large k has as its lowest order term what is commonly known as the geometrical optics field. The caustics of geometrical optics are those point sets on which the zero order teen becomes infinite. It is well known that caustics may exist even where the exact wave field is perfectly regular. An investigation of reflection from cylindrical walls of arbitrary cross section shows that the occurrence of caustic points means a change in character of the asymptotic expansion of the true field such that the lowest order term is no longer independent of k , but actually contains a factor k raised to a positive power. There also occurs a Jump in phase along a ray passing through a caustic which, as is well known, equals \pi/2 in the case of a focal point, but which may differ from \pi/2 in the case of more general types of caustics. In addition, the geometric optics field is worked out in detail for the case of a plane wave incident on a parabolic cylinder, and the field is obtained in its lowest order at the focus and in the neighborhood of the focus.