Narrow Results

We revisit the extended Feynman formula for the harmonic oscillator beyond and at caustics. The extension has been made by some authors, however, it is not obtained by the discrete formulation of path integral, which we consider the most reliable regularization of it. We derive the result by, especially at caustics, more rigorous method than previous.

The aim of this paper is to develop on the asymptotics of some one-dimensional nonlinear Schrödinger equations from both the theoretical and the numerical perspectives, when a caustic is formed. We review rigorous results in the field and give some heuristics in cases where justification is still needed. The scattering operator theory is recalled. Numerical experiments are carried out on the focus point singularity for which several results have been proved rigorously. Furthermore, the scattering operator is numerically studied. Finally, experiments on the cusp caustic are displayed, and similarities with the focus point are discussed. Several shortcomings of spectral time-splitting schemes are investigated.

Spectral rendering, or the synthesis of images by taking into account the constituent wavelengths of white light, enables the rendering of iridescent colors caused by phenomena such as dispersion, diffraction, interference and scattering. Caustics, the focusing and defocusing of light through a refractive medium, can be interpreted as a special case of dispersion where all the wavelengths travel along the same paths. In this paper we extend Adaptive Caustic Mapping (ACM), a previously proposed caustics mapping algorithm, to handle physically-based dispersion. Because ACM can display caustics in real-time, it is amenable to extension to handle the more general case of dispersion. We also present a novel algorithm for filling in the gaps that occur due to discrete sampling of the spectrum. Our proposed method runs in screen-space, and is fast enough to display plausible dispersion phenomena at real-time and interactive frame rates.

A new concept of ridges, ravines and related structures (skeletons) associated with a surface in three-dimensional space is introduced. The concept is based on the investigation of relationships between the singularities of the distance function from the surface and singularities of caustic of the normals to the surface. It involves both local and global geometric surface properties and leads to the effective procedure for ridges and ravines recognition.

This paper presents a numerical method which locates caustics of classical trajectories on-the-fly. The method is conceptually simple and is applicable to a system of arbitrary dimensions. The efficiency of the method is demonstrated by determining caustics of trajectories in the 2-D Henon–Heiles potential and of trajectories used to simulate a triatomic reaction process for J (total angular momentum) = 0.

Optical caustics are formed by the focalisation of light rays. They are observed in nature as well as in experimental physics. They constitute a concrete and visual realization of the abstract notion of singularity, or better of Lagrangian singularity, as defined in the modern theory of singularity. Caustics appear naturally in applications to physics, and more particularly in visualization techniques. The main problem is then to extract physical informations from the observation of the caustic, that is to say to clarify the relation between the caustic and the refractive index field producing it. In this difficult program, the first step is to be able to calculate, from a given index field, the structure of a caustic, i.e. its decomposition into different types of singularities: folds, cusps, swallowtails, umbilics. We show, using the example of the visualization of convective structures in nematic liquid crystals, how this first step may be efficiently realized with the help of Thom-Boardman classes.

Optical phenomena visible to everyone have been central to the development of, and abundantly illustrate, important concepts in science and mathematics. The phenomena considered from this viewpoint are rainbows, sparkling reflections on water, mirages, green flashes, earthlight on the moon, glories, daylight, crystals and the squint moon. And the concepts involved include refraction, caustics (focal singularities of ray optics), wave interference, numerical experiments, mathematical asymptotics, dispersion, complex angular momentum (Regge poles), polarisation singularities, Hamilton’s conical intersections of eigenvalues (‘Dirac points’), geometric phases and visual illusions.

Caustics are a common occurrence in optics, describing the bright lines seen in a well-lit coffee cup or the dancing networks of light seen at the bottom of a swimming pool on a sunny day. Recently, caustic formation in electron trajectories was identified as the mechanism driving strong current modulations in accelerated charged-particle beams. Under certain conditions, neighboring electron trajectories coalesce to form caustics, resulting in current spikes. Caustic lines and surfaces are regions of maximum electron density, and are often witnessed in accelerator physics as folds in the transverse or longitudinal phase space distribution. Knowledge of caustics can allow us to shape the longitudinal beam shape to our advantage.

We present the first successful application of the method of Matched Expansions for the calculation of the self-force on a point particle in a curved spacetime. We investigate the case of a scalar charge in the Nariai spacetime, which serves as a toy model for a point mass moving in the Schwarzschild black hole background. We discuss the singularity structure of the Green function beyond the normal neighbourhood and the interesting effect of caustics on null wave propagation.