Umbilic Caustics Research Articles

Generation of high-dimensional caustic beams via phase holograms using angular spectral representation.

Using angular spectral representation, we demonstrate a generalized approach for generating high-dimensional elliptic umbilic and hyperbolic umbilic caustics by phase holograms. The wavefronts of such umbilic beams are investigated via the diffraction catastrophe theory determined by the potential function, which depends on the state and control parameters. We find that the hyperbolic umbilic beams degenerate into classical Airy beams when the two control parameters are simultaneously equal to zero, and elliptic umbilic beams possess an intriguing autofocusing property. Numerical results demonstrate that such beams exhibit clear umbilics in 3D caustic, which link the two separated parts. The dynamical evolutions verify that they both possess prominent self-healing properties. Moreover, we demonstrate that hyperbolic umbilic beams follow along a curve trajectory during propagation. As the numerical calculation of diffraction integral is relatively complex, we have developed an effective approach for successfully generating such beams by using phase hologram represented by angular spectrum. Our experimental results are in good agreement with the simulations. Such beams with intriguing properties are likely to be applied in emerging fields such as particle manipulation and optical micromachining.

Catastrophe optics theory unveils the localised wave aberration features that generate ghost images

Monocular polyplopia (ghost or multiple images) is a serious visual impediment for some people who report seeing two (diplopia), three (triplopia) or even more images. Polyplopia is expected to appear if the point spread function (PSF) has multiple intensity cores (a dense concentration of a large portion of the radiant flux contained in the PSF) relatively separated from each other, each of which contributes to a distinct image. We present a theory that assigns these multiple PSF cores to specific features of aberrated wavefronts, thereby accounting optically for the perceptual phenomenon of monocular polyplopia. The theory provides two major conclusions. First, the most likely event giving rise to multiple PSF cores is the presence of hyperbolic, or less probably elliptical, umbilic caustics (using the terminology of catastrophe optics). Second, those umbilic caustics formed on the retinal surface are associated with certain points of the wave aberration function, called cusps of Gauss, where the gradient of a curvature function vanishes. However, not all cusps of Gauss generate those umbilic caustics. We also provide necessary conditions for those cusps of Gauss to be fertile. To show the potential of this theoretical framework for understanding the nature and origin of polyplopia, we provide specific examples of ocular wave aberration functions that induce diplopia and triplopia. The polyplopia effects in these examples are illustrated by depicting the multi‐core PSFs and the convolved retinal images for clinical letter charts, both through computer simulations and through experimental recording using an adaptive optics set‐up. The number and location of cores in the PSF is thus a potentially useful metric for the existence and severity of polyplopia in spatial vision. These examples also help explain why physiological pupil constriction might reduce the incidence of ghosting and multiple images of daily objects that affect vision with dilated pupils. This mechanistic explanation suggests a possible role for optical phase‐masking as a clinical treatment for polyplopia and ghosting.

Distorted mirror images organised by cuspoid and umbilic caustics

The anatomy of the distorted and topologically disrupted images of an extended object after reflection by curved mirrors, and of their refraction analogues, e.g. in gravitational lensing, is determined by the caustic surfaces enveloping the ray family issuing from each point of the object. Simulations are presented of reflections of the same object when organised by each of the five caustics (elementary catastrophes) that are stable in three-dimensional space: the fold, cusp, swallowtail, elliptic and hyperbolic umbilic catastrophes. The object is a picture of a face, whose familiar features make the image distortions and disruptions easier to understand. An important distinction is between mirrors curved in one direction and those curved in two.

Cones of silence, complex rays and catastrophes: high-frequency flow–acoustic interaction effects

In this paper we develop a novel ray solver for the time-harmonic linearized Euler equations used to predict high-frequency flow–acoustic interaction effects from point sources in subsonic mean jet flows. The solver incorporates solutions to three generic ray problems found in free-space flows: the multiplicity of rays at a receiver point, propagation of complex rays and unphysical divergences at caustics. We show that these respective problems can be overcome by an appropriate boundary value reformulation of the nonlinear ray equations, a bifurcation-theory-inspired complex continuation, and an appeal to the uniform functions of catastrophe theory. The effectiveness of the solver is demonstrated for sources embedded in isothermal parallel and spreading jets, with the fields generated containing a wide variety of caustic structures. Solutions are presented across a large range of receiver angles in the far field, both downstream, where evanescent complex rays generate the cone of silence, and upstream, where multiple real rays are organized about a newly observed cusp caustic. The stability of the caustics is verified for both jets by their persistence under parametric changes of the flow and source. We show the continuation of these caustics as surfaces into the near field is complicated due to a dense caustic network, featuring a chain of locally hyperbolic umbilic caustics, generated by the tangency of rays as they are channelled upstream within the jet.

Hyperbolic umbilic caustics from oblate water drops with tilted illumination: Observations

Various groups have reported observations of hyperbolic umbilic diffraction catastrophe patterns in the far-field scattering by oblate acoustically levitated drops with symmetric illumination. In observations of that type the drop's symmetry axis is vertical and the illuminating light beam (typically an expanded laser beam) travels horizontally. In the research summarized here, scattering patterns in the primary rainbow region and drop measurements were recorded with vertically tilted laser beam illumination having a grazing angle as large as 4 degrees. The findings from these observations may be summarized as follows: (a) It remains possible to adjust the drop aspect ratio (diameter/height) = D/H so as to produce a V-shaped hyperbolic umbilic focal section (HUFS) in the far-field scattering. (b) The shift in the required D/H was typically an increase of less than 1% and was quadratic in the tilt. (c) The apex of the V-shaped HUFS was shifted vertically by an amount proportional to the tilt with a coefficient close to unity. The levitated drops had negligible up-down asymmetry. Our method of investigation should be useful for other generalized rainbows with tilted illumination.

Orbifolds, the A, D, E family of caustic singularities, and gravitational lensing

We provide a geometric explanation for the existence of magnification relations for the \documentclass[12pt]{minimal}\begin{document}$A_n (n \ge 2), D_n (n \ge 4), E_6, E_7, E_8$\end{document}An(n≥2),Dn(n≥4),E6,E7,E8 family of caustic singularities, which were established in recent work. In particular, it was shown that for families of general mappings between planes exhibiting any of these caustic singularities, and for any noncaustic target point, the total signed magnification of the corresponding preimages vanishes. As an application to gravitational lensing, it was also shown that, independent of the choice of a lens model, the total signed magnification vanishes for a light source anywhere in the four-image region close to elliptic and hyperbolic umbilic caustics. This is a more global and higher order analog of the well-known fold and cusp magnification relations. We now extend each of these mappings to weighted projective space, which is a compact orbifold, and show that magnification relations translate into a statement about the behavior of these extended mappings at infinity. This generalizes multidimensional residue techniques developed in previous work, and introduces weighted projective space as a new tool in the theory of caustic singularities and gravitational lensing.

Optical caustics observed in light scattered by an oblate spheroid

The electromagnetic fields scattered when a plane wave is incident on an oblate spheroid in the side-on orientation may be calculated using a generalization of Mie theory, and the results may be decomposed in a Debye series expansion. A number of optical caustics are observed in the computed scattered intensity for the one internal reflection portion of the Debye series for scattering angles in the vicinity of the first-order rainbow, and are analyzed in terms of the rainbow, transverse cusp, and hyperbolic umbilic caustics of catastrophe optics. The specific features of these three caustics are described, as is their assembly into the global structure of the observed caustics for spheroid scattering. It is found that, for a spheroid whose radius is an order of magnitude larger than the wavelength of the incident light, the interference structure accompanying the transverse cusp and hyperbolic umbilic caustics is only partially formed.

A universal magnification theorem for higher-order caustic singularities

We prove that, independent of the choice of a lens model, the total signed magnification always sums to zero for a source anywhere in the four-image region close to swallowtail, elliptic umbilic, and hyperbolic umbilic caustics. This is a more global and higher-order analog of the well-known fold and cusp magnification relations, in which the total signed magnifications in the two-image region of the fold and the three-image region of the cusp are both always zero. As an application, we construct a lensing observable for the hyperbolic umbilic magnification relation and compare it with the corresponding observables for the cusp and fold relations using a singular isothermal ellipsoid lens. We demonstrate the greater generality of the hyperbolic umbilic magnification relation by showing how it applies to the fold image doublets and cusp image triplets and extends to image configurations that are neither. We show that the results are applicable to the study of substructure on galactic scales using observed quadruple images of lensed quasars. The magnification relations are also proven for generic one-parameter families of mappings between planes, extending their potential range of applicability beyond lensing.

Higher-dimensionality caustics owing to competing reorientation of a liquid crystal by laser beams

Formation of hyperbolic umbilic caustics is observed experimentally in the far-field zone as a result of self- and mutual-phase modulation of laser beams by means of competing reorientation of a nematic liquid crystal. The interference pattern reveals maximum response for an unusually small (≈5°) angle of light refraction into the cell. It is thus demonstrated that higher-order diffraction catastrophes can be induced and controlled by nonlinear optical processes.

The transient wave fields in the vicinity of the elliptic, hyperbolic, and parabolic umbilic caustics

The caustics of high-frequency wave propagation may be classified using catastrophe theory. In this paper, we examine the transient wave fields in the vicinity of the elliptic, hyperbolic, and parabolic umbilic catastrophes. Analytical results are presented which describe the time-dependent wave field due to an impulsive source on and at large distances in each control direction away from the most singular point of each of these three catastrophes. Numerical results are presented which show the smooth variation of these transient wave fields as each control parameter is independently varied. These results are compared to previously derived results on the transient wave fields in the vicinity of the cuspoid catastrophes (the cuspoid and umbilic catastrophes have coranks 1 and 2, respectively). It is found that the transient wave fields in the vicinity of the cuspoid and umbilic catastrophes differ with regard to the temporal structure of the wave field on the most singular point of each catastrophe, the manner in which these temporal structures unfold, and the phase shifts which individual rays undergo as a result of touching one of the catastrophes.

Impulsive wave propagation and the elliptic, hyperbolic, and parabolic umbilic caustics

The time‐dependent wave field due to an impulsive wave in the vicinity of the elliptic, hyperbolic, and parabolic umbilic caustics is examined. Numerical results are presented which show the smooth variation of the time‐dependent acoustic field in the vicinity of all three caustics. Analytical results are given which describe the most important properties of these wave fields. The time‐dependent structures of these wave fields are qualitatively different than those associated with the cuspoid caustics. The reason for the difference stems from the difference in phase shifts which individual ray arrivals undergo as a result of touching one of the two types of caustic (cuspoid versus umbilic).