Caustic Structure Research Articles

Ray and caustic structure of Ince-Gauss beams

The Ince-Gauss beams, separable in elliptic coordinates, are studied through a ray-optical approach. Their ray structure can be represented over a Poincaré sphere by generalized Viviani curves (intersections of a cylinder and a sphere). This representation shows two topologically different regimes, in which the curve is composed of one or two loops. The overall beam shape is described by the ray caustics that delimit the beams’ bright regions. These caustics are inferred from the generalized Viviani curve through a geometric procedure that reveals connections with other physical systems and geometrical constructions. Depending on the regime, the caustics are composed either of two confocal ellipses or of segments of an ellipse and a hyperbola that are confocal. The weighting of the rays is shown to follow the two-mode meanfield Gross–Pitaevskii equations, which can be mapped to the equation of a simple pendulum. Finally, it is shown that the wave field can be accurately estimated from the ray description.

Influence of Traveling Ionospheric Disturbances on the Caustic Structure of Radio Waves during Low Inclination Sounding of the Ionosphere

Influence of Traveling Ionospheric Disturbances on the Caustic Structure of Radio Waves during Low Inclination Sounding of the Ionosphere

The gravitational force field of proto-pancakes

It is well known that the first structures that form from small fluctuations in a self-gravitating, collisionless, and initially smooth cold dark matter (CDM) fluid are pancakes. We studied the gravitational force generated by such pancakes just after shell crossing and have found a simple analytical formula for the force along the collapse direction, which can be applied to both the single- and multi-stream regimes. We tested the formula on the early growth of CDM proto-haloes seeded by two or three crossed sine waves. Adopting the high-order Lagrangian perturbation theory (LPT) solution as a proxy for the dynamics, we confirm that our analytical prediction agrees well with the exact solution computed via a direct resolution of the Poisson equation, as long as the local caustic structure remains sufficiently one-dimensional. These results are further confirmed by comparisons of the LPT predictions performed this way to measurements in Vlasov simulations performed with the public code ColDICE. We also show that the component of the force orthogonal to the collapse direction preserves its single-stream nature – it does not change qualitatively before or after the collapse – allowing sufficiently high-order LPT acceleration to be used to approximate it accurately as long as the LPT series converges. As expected, solving the Poisson equation on the density field generated with LPT displacement provides a more accurate force than the LPT acceleration itself, as a direct consequence of the faster convergence of the LPT series for the positions than for the accelerations. This may provide a clue as to how we can improve standard LPT predictions. Our investigations represent a very needed first step in the study of gravitational dynamics in the multi-stream regime analytically: we estimate, at the leading order in time and space, the proper backreaction on the gravitational field inside the pancakes.

Extended-depth-of-focus wavefront design frompseudo-umbilical space curves.

Designing extended-depth-of-focus wavefronts is required in multiple optical applications. Caustic location and structure analysis offer a powerful tool for designing such wavefronts. An intrinsic limitation of designing extended-depth-of-focus wavefronts is that any smooth surface, with a non-constant mean curvature, unavoidably introduces a separation between caustic sheets, which is proportional to the ratio of change of the mean curvature along a curve embedded in the wavefront. We present a method to obtain extended-depth-of-focus wavefronts where the mean curvature variation ratio is reduced thanks to using a long circle-involute space curve effectively filling the wavefront surface. Additionally, we present a variant of the method in which the wavefront is modified within a small tubular neighborhood of the circle involute in order to partially meet the umbilical condition along that tubular region. Finally, we provide some numerical results showing the potential of our method in an application example.

Multiplane gravitational lenses with an abundance of images

We consider gravitational lensing of a background source by a finite system of point-masses. The problem of determining the maximum possible number of lensed images has been completely resolved in the single-plane setting (where the point masses all reside in a single lens plane), but this problem remains open in the multiplane setting. We construct examples of K-plane point-mass gravitational lens ensembles that produce ∏i=1K(5gi−5) images of a single background source, where gi is the number of point masses in the ith plane. This gives asymptotically (for large gi with K fixed) 5K times the minimal number of lensed images. Our construction uses Rhie’s single-plane examples and a structured parameter-rescaling algorithm to produce preliminary systems of equations with the desired number of solutions. Utilizing the stability principle from the differential topology, we then show that preliminary (nonphysical) examples can be perturbed to produce physically meaningful examples while preserving the number of solutions. We provide numerical simulations illustrating the result of our construction, including positions of lensed images and the structure of critical curves and caustics. We observe an interesting “caustic of multiplicity” phenomenon that occurs in the nonphysical case and has a noticeable effect on the caustic structure in the physically meaningful perturbative case.

On the kinematic morphology around haloes

ABSTRACT In this paper, we report an interesting kinematic phenomenon around the haloes’ edge related to the ‘splashback’ radius. After the shell-crossing, cosmic flow exhibits various rotational morphologies via stream-mixing. Vorticity is generated in a particular way that coincides with the large-scale structure. Notably, one specific flow morphology, which is spiralling inward and compressing in the third direction, concentrates around haloes. A detailed examination that reveals a sharp change in the logarithmic derivative of its volume fraction coincides with the location of the splashback radius Rsp defined as the outermost caustic structure. Such a feature encodes valuable phase-space information and provides a new perspective on understanding the dynamical evolution of haloes. As a volume-weighted quantity, the profile of flow morphology is purely kinematic. And unlike other related studies, the rotational flow morphologies capture the anisotropic phase structure in the multistream region.

Caustics and wavefront of Swallowtail-Gauss catastrophe beams

We analytically derive the propagating wavefront form of Swallowtail-Gauss beams (SGBs), and we demonstrate the propagation characteristics of SGBs experimentally and numerically. During propagation, one type of SGBs gradually evolves into a deformed SGBs, which is very similar to a Pearcey-Gauss beam, and other types of SGBs evolve into deformed Butterfly-Gauss beams. The experimental results are basically consistent with the simulation results. The caustic structure of a propagating SGBs is demonstrated and the propagation properties of SGBs are explained with the deformation of the control space in the language of catastrophe theory. These types of SGBs with excellent propagation characteristics could be beneficial for potential applications in optical trapping and micromachining.

Caustics in quantum many-body dynamics

We describe a new class of nonequilibrium quantum many-body phenomena in the form of networks of caustics that dominate the many-body wave function in the semiclassical regime following a sudden quench. It includes the light cone-like propagation of correlations as a particular case. Caustics are singularities formed by the birth and death of waves and form a hierarchy of universal patterns whose natural mathematical description is via catastrophe theory. Examples in classical waves range from rainbows and gravitational lensing in optics to tidal bores and rogue waves in hydrodynamics. Quantum many-body caustics are discretized by second quantization (``quantum catastrophes'') and live in Fock space, which can potentially have many dimensions. We illustrate these ideas using the Bose Hubbard dimer and trimer models, which are simple enough that the caustic structure can be elucidated from first principles and yet run the full range from integrable to nonintegrable dynamics. The dimer gives rise to discretized versions of fold and cusp catastrophes whereas the trimer allows for higher catastrophes including the codimension-3 hyperbolic and elliptic umbilics, which are organized by, and projections of, an 8-dimensional corank-2 catastrophe known as ${X}_{9}$. These results describe a hitherto unrecognized form of universality in quantum dynamics organized by singularities that manifest as strong fluctuations in mode population probabilities.

Effect of acoustic diffraction phase shift on caustic structure in deep sea convergence zone

The constant phase shift <inline-formula><tex-math id="M1">\begin{document}$- {{\text{π}}}/{2}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220763_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20220763_M1.png"/></alternatives></inline-formula> generated by a turning point is an ideal approximation under high frequency conditions. In fact, when sound waves pass through a turning point, it will travel horizontally for a certain distance in the form of inhomogeneous plane wave and then be refracted back. At this time, a functional phase shift should also be included at the turning point. In addition, a refracted ray will bring about reflection phase shift when the turning point is close to the waveguide edge. These two kinds of phase shifts which are associated with the diffraction phenomenon are called diffraction phase shifts in this paper and are more notable when the frequency goes down. In order to accurately obtain the caustic structure at low frequency, the diffraction phase shift is used to correct the classical ray theory in calculating ray skip distance, traveling time, and group velocity in this work. On this basis, a simple and explicit analytical model is proposed which is suitable for calculating the low frequency deep water caustics. The numerical study of the first upper convergence zone in the complete deep water sound channel shows that there are three caustic lines in the refracted-refracted (RR) convergence zone and four caustic lines in the refracted surface-reflected (RSR) convergence zone under the condition of high frequency hypothesis. When the diffraction phase shifts at low frequency are included, and compares with high frequency results, it is found that the horizontal beam displacement caused by the reflection phase shift will make the RR caustics horizontally move towards the source and also lead the RSR rays to generate several new caustics, while the beam displacement caused by sound propagating in the form of inhomogeneous plan wave will make the RR caustics horizontally move away from the sound source. Accompanied with the increased frequency, the diffraction effect will decrease, and the caustic structure gradually tends to the classical ray theory results. The reliability of the correction results is verified by the normal mode theory. The model proposed in this work makes up for the deficiency of classical ray theory.

Lower turning point convegence zone in deep water with an incomplete channel

In a deep sea sound channel, rays will bend due to the sound speed profile, and convergence zone will occur when the rays are intensive. Transmission loss in the convergence zone is smaller and it is conducive to acoustic detection and communication. Therefore the study of acoustic characteristics in convergence zone is always the focus of deep-sea acoustics. A long-range sound propagation experiment is conducted in the South China Sea. An equivalent broadband explosive sound source of 1 kg is placed at a depth of 200 m, and the hydrophone receives the data at 3146 m far. The processing and analysis of the experimental data indicate that there is a convergence zone below the sound channel axis in the incomplete deep channel. Compared with the upper turning point convergence zone near the surface, this convergence zone has a high convergence gain at a long distance. The caustic lines of refracted type and refracted surface-refleted type are determined by means of ray-normal mode theory. It is found that the location of the deep convergence zone observed in the experiment is consistent with the position of the refracted caustic line. It is proved that the convergence zone is a lower turning point convergence zone formed by the superposition of a large number of normal modes in the same phase, and it has a convergence effect at a certain depth below the sound channel axis in the deep sea. The formation conditions of the convergence zone and the influence of sound source depth on the caustic structure of the convergence zone are studied. The comparisons of the transmission loss and the width between the upper and lower turning point convergence zone at a long distance aremade. The analysis shows that the convergence gain in the seventh lower turning point convergence zone is still no less than 10 dB. The influence of the vertical structure of sound velocity on the lower turning point convergence zone is studied. The theoretical analysis results are in good agreement with the experimental data.

Simulation of the Field in the Vicinity of Caustics of Ordinary and Extraordinary Waves during Ionospheric Propagation

The caustic structure of the electromagnetic field is mathematically modeled, when it appears near the ground surface during the decimeter radio wave propagation in the ionospheric plasma. The caustic structures of the fields of the ordinary and extraordinary waves are compared. The influence of the extraordinary wave on the amplitude structure of the ordinary wave field is also estimated in the vicinity of the caustic. The wave fields calculated with the help of the caustic approximation and with the help of the irregular ray approximation are compared. It is suggested to estimate the maximum value of the field amplitude in the vicinity of the caustic on the basis of the ray approximation.

Abruptly autofocusing beams from phase perturbations having forced symmetry.

A controllable manipulation of the energy distribution of caustic beams possessing rectangular symmetry is presented. The beams are designed from the spectral phase by adding a linear and/or quadratic perturbation having forced symmetry. This approach breaks the overall caustic structure into branches, allowing a fully controllable displacement of each branch. The caustic breaking leads to peculiar propagation configurations for rectangular beams. Among them, we highlight the abruptly autofocusing beam, which until now was exclusively associated to caustic beams with circular symmetry. Thereby, the abruptly autofocusing effect can be yielded for one-dimensional light sheets, contrary to what happens for circularly symmetric beams. The theoretical predictions are supported by experiments. Besides, the focus width of such rectangular beams can be reduced beyond the standard diffraction limit.

The caustic structure near a grazing point in three-dimensional space

By definition, a grazing point is a point of the edge of a concave mirror where the incident ray is tangent to the mirror. In its neighborhood, the rays are reflected a great number of times and an ordered series of caustics is formed, a caustic of order n corresponding to n reflections by the mirror. We study, in the framework of geometrical optics in three dimensions, the structure of this remarkable infinite set of caustic surfaces passing through the grazing point. We calculate the geometrical characteristics of the caustics at the grazing point, namely, their Gaussian curvatures and their mean curvatures. We show that these characteristics tend to those of the mirror when the caustic’s order tends to infinity. We also show that the boundary of every caustic surface forms a corner at the grazing point. We calculate the angle of the corner and we show that this angle tends to zero when the caustic’s order tends to infinity. We show that, unlike the case of a grazing point in the plane, a grazing point in three dimensional space possesses the fundamental property of being stable. It thus constitutes a new generic type of optical singularity, in addition to the well known five Lagrangian singularities of space, i.e. the fold, the cusp, the swallowtail, the elliptic umbilic, and the hyperbolic umbilic.

Spitzer Microlensing Parallax for OGLE-2016-BLG-1067: A Sub-Jupiter Orbiting an M Dwarf in the Disk

Abstract We report the discovery of a sub-Jupiter-mass planet orbiting beyond the snow line of an M dwarf most likely in the Galactic disk as part of the joint Spitzer and ground-based monitoring of planetary microlensing anomalies toward the Galactic bulge. Most of the microlensing parameters are strongly constrained by the light-curve modeling, and in particular there is a Spitzer-based measurement of the microlens parallax, π E. However, there are no caustic crossings, so the angular Einstein radius has only an upper limit based on the light-curve modeling alone. Additionally, the analysis leads us to identify eight degenerate configurations: the fourfold microlensing parallax degeneracy being doubled by a degeneracy in the caustic structure present at the level of the ground-based solutions. To calculate the physical parameters, and at the same time to break the parallax degeneracy, we make use of a series of arguments: the χ 2 hierarchy, the Rich argument (stating that the small-parallax solution is more likely), and a prior Galactic model. The preferred configuration, favored by a likelihood ratio of at least 4000, is for a host at with mass , orbited by a Saturn-like planet with at projected separation , about 2.1 times beyond the system snow line. Therefore, it adds to the growing population of sub-Jupiter planets orbiting beyond the snow line of M dwarfs discovered by microlensing. Based on the rules of the real-time protocol for the selection of events to be followed up with Spitzer, this planet will not enter the sample for measuring the Galactic distribution of planets.

Caustic Skeleton & Cosmic Web

We present a general formalism for identifying the caustic structure of a dynamically evolving mass distribution, in an arbitrary dimensional space. The identification of caustics in fluids with Hamiltonian dynamics, viewed in Lagrangian space, corresponds to the classification of singularities in Lagrangian catastrophe theory. On the basis of this formalism we develop a theoretical framework for the dynamics of the formation of the cosmic web, and specifically those aspects that characterize its unique nature: its complex topological connectivity and multiscale spinal structure of sheetlike membranes, elongated filaments and compact cluster nodes. Given the collisionless nature of the gravitationally dominant dark matter component in the universe, the presented formalism entails an accurate description of the spatial organization of matter resulting from the gravitationally driven formation of cosmic structure. The present work represents a significant extension of the work by Arnol'd et al. [1], who classified the caustics that develop in one- and two-dimensional systems that evolve according to the Zel'dovich approximation. His seminal work established the defining role of emerging singularities in the formation of nonlinear structures in the universe. At the transition from the linear to nonlinear structure evolution, the first complex features emerge at locations where different fluid elements cross to establish multistream regions. Involving a complex folding of the 6-D sheetlike phase-space distribution, it manifests itself in the appearance of infinite density caustic features. The classification and characterization of these mass element foldings can be encapsulated in caustic conditions on the eigenvalue and eigenvector fields of the deformation tensor field. In this study we introduce an alternative and transparent proof for Lagrangian catastrophe theory. This facilitates the derivation of the caustic conditions for general Lagrangian fluids, with arbitrary dynamics. Most important in the present context is that it allows us to follow and describe the full three-dimensional geometric and topological complexity of the purely gravitationally evolving nonlinear cosmic matter field. While generic and statistical results can be based on the eigenvalue characteristics, one of our key findings is that of the significance of the eigenvector field of the deformation field for outlining the entire spatial structure of the caustic skeleton emerging from a primordial density field. In this paper we explicitly consider the caustic conditions for the three-dimensional Zel'dovich approximation, extending earlier work on those for one- and two-dimensional fluids towards the full spatial richness of the cosmic web. In an accompanying publication, we apply this towards a full three-dimensional study of caustics in the formation of the cosmic web and evaluate in how far it manages to outline and identify the intricate skeletal features in the corresponding N-body simulations.

Observational signature of high spin at the Event Horizon Telescope

We analytically compute the observational appearance of an isotropically emitting point source on a circular, equatorial orbit near the horizon of a rapidly spinning black hole. The primary image moves on a vertical line segment, in contrast to the primarily horizontal motion of the spinless case. Secondary images, also on the vertical line, display a rich caustic structure. If detected, this unique signature could serve as a "smoking gun" for a high-spin black hole in nature.

The caustic structure near a grazing point in the plane

A grazing point of a concave mirror illuminated by a large beam of rays is a point of the mirror's edge where the incident ray is parallel to the mirror. In the neighborhood of such point the rays are reflected a great number of times. We show that an ordered series of caustics passes through the grazing point, each caustic corresponding to a fixed number of reflections by the mirror. We study, in the framework of planar geometrical optics, the structure of this remarkable set of caustics. Our main result is a formula giving the curvature of the caustic curves at the grazing point as a function of the number of reflections. This sequence is universal in the sense that it is independent of the shape of the incident wavefront. A grazing point in the plane is an unstable point and we show how the caustic structure is modified under the effect of a small perturbation of the optical system.

Stable and unstable Airy-related caustics and beams

Optical beams with an underlying caustic structure are stable under perturbation if the caustics belong to the catastrophe-theory classification; otherwise they are unstable. The original Airy beam in two spatial dimensions, with its curved caustic, is stable in this sense. But the separable Airy-product beam in three-dimensions is unstable: under separability-breaking perturbation, it unfolds into the hyperbolic umbilic diffraction catastrophe, which is stable. By including initial phase factors, a variety of new exact solutions of the paraxial wave equation can be generated, corresponding to Pearcey and higher-catastrophe beams with stable caustics, and with the associated diffraction catastrophes appearing in their canonical forms or as deformations of these.

Internal tides and deep diel fades in acoustic intensity.

A mechanism is presented by which the observed acoustic intensity is made to vary due to changes in the acoustic path that are caused by internal-tide vertical fluid displacements. The position in range and depth of large-scale caustic structure is determined by the background sound-speed profile. Internal tides cause a deformation of the background profile, changing the positions of the caustic structures-which can introduce intensity changes at a distant receiver. Gradual fades in the acoustic intensity occurring over timescales similar to those of the tides were measured during a low-frequency (284-Hz) acoustic scattering experiment in the Philippine Sea in 2009 [White et al., J. Acoust. Soc. Am. 134(4), 3347-3358 (2013)]. Parabolic equation and Hamiltonian ray-tracing calculations of acoustic propagation through a plane-wave internal tide environmental model employing sound-speed profiles taken during the experiment indicate that internal tides could cause significant gradual changes in the received intensity. Furthermore, the calculations demonstrate how large-scale perturbations to the index of refraction can result in variation in the received intensity.

Adiabatic theory of solitons fed by dispersive waves

We consider scattering of low-amplitude dispersive waves at an intense optical soliton which constitutes a nonlinear perturbation of the refractive index. Specifically, we consider a single-mode optical fiber and a group velocity matched pair: an optical soliton and a nearly perfectly reflected dispersive wave, a fiber-optical analog of the event horizon. By combining (i) an adiabatic approach that is used in soliton perturbation theory and (ii) scattering theory from quantum mechanics, we give a quantitative account of the evolution of all soliton parameters. In particular, we quantify the increase in the soliton peak power that may result in the spontaneous appearance of an extremely large, so-called champion soliton. The presented adiabatic theory agrees well with the numerical solutions of the pulse propagation equation. Moreover, we predict the full frequency band of the scattered dispersive waves and explain an emerging caustic structure in the space-time domain.