Diffraction Catastrophe Research Articles

Manipulating self-focusing beams induced by high-dimensional parabolic umbilic beams

The excellent property that the intensity of autofocusing beams in the focal plane increases exponentially, makes them especially beneficial for biomedical treatment. We experimentally generate tunable (2+1)-dimensional circular parabolic umbilic beams (CPUBs) for the first time based on the high-dimensional diffraction catastrophe integral, which is determined by the potential function. Such CPUBs have circular and symmetrical intensity distributions in space through radial symmetry transformation. Due to the flexibility of high dimensionality, these light beams have rich light field structures and self-focusing property. Where, the maximum intensity suddenly increases by orders of magnitude in the focal plane. Unlike the classical circular Airy beams, CPUBs exhibit multiple self-focusing points along the optical axis and a needle-like structure during propagation, which can be affected by manipulating the control parameters. The rich properties provide a new perspective for exploring the novel physical mechanisms and phenomena in autofocusing beams. The experimental results verify the correctness of the numerical simulations. CPUBs greatly enrich the autofocusing beam family and will be advantageous for medical treatments, optical micromanipulation, and microscopic imaging.

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.

Sharp-edge diffraction under Bessel beam illumination: a catastrophe optics perspective.

Boundary diffraction wave theory and catastrophe optics have already proved to be a formidable combination in computational optics. In the present paper, a general paraxial theory aimed at dealing paraxial diffraction of Bessel beams by arbitrarily shaped sharp-edge apertures is developed. A key ingredient of our analysis is the δ-like nature of the angular spectrum of nondiffracting beams. This allows the diffracted wavefield to be effectively represented through two-dimensional integrals defined onto rectangular domains, whose numerical evaluation is easily achievable via standard Monte Carlo techniques. As a byproduct of the present analysis, a simple explanation of a recently observed property of some "heart-like" apertures to flatten the axial intensity of apodized Bessel beams is also provided.

Generation and control of tornado waves by means of ring swallowtail vortex beams.

Tornado waves (ToWs), which refer to a light that accelerates and twists over both the radial and the angular directions, have gained a great deal of interest since the concept was introduced by Brimis et al [Opt. Lett.45, 280 (2020)10.1364/OL.45.000280]. In this paper, we superimpose two pairs of ring swallowtail vortex beams (RSVBs) to generate ToWs and we call them tornado swallowtail waves (ToSWs). Each pair consists of RSVBs while carrying orbital angular momentum of opposite helicity and slightly different with the radius of the main ring of RSVBs. The waves spiral forward and reveal intensity maxima, exhibiting a tornado-like intensity profile during propagation. Meanwhile, the angular acceleration of the ToSWs is illustrated via tracing the angular position of the high-intensity main lobes. It is found that ToSWs present very high values of angular acceleration. Compared with typical tornado waves, ToSWs are more diverse and tunable, giving a new degree of freedom to tailor the propagation dynamics due to the flexibility of the swallowtail diffraction catastrophe. In addition, we confirm such waves experimentally and the results match well with the numerical ones. Also, we demonstrate the ability of optical manipulation of ToSWs for the first time in that they allow for particles not only to be trapped but also to be rotated. Finally, we analyze the poynting vectors and power exchange of ToSWs to demonstrate convincingly the physical mechanism.

Swallowtail-type diffraction catastrophe beams.

We demonstrate a universal approach for generating high-order diffraction catastrophe beams, specifically for Swallowtail-type beams (abbreviated as Swallowtail beams), using diffraction catastrophe theory that was defined by potential functions depending on the control and state parameters. The three-dimensional curved caustic surfaces of these Swallowtail catastrophe beams are derived by the potential functions. Such beams are generated by mapping the cross sections of the high-order control parameter space to the corresponding transverse plane. Owing to the flexibility of the high-order diffraction catastrophe, these Swallowtail beams can be tuned to a diverse range of optical light structures. Owing to the similarity in their frequency spectra, we found that the Swallowtail beams change into low-order Pearcey beams under given conditions during propagation. Our experimental results are in close agreement with our simulated results. Such fantastic catastrophe beams that can propagate along curved trajectories are likely to give rise to new applications in micromachining and optical manipulation, furthermore, these diverse caustic beams will pave the way for the tailoring of arbitrarily accelerating caustic beams.

Abruptly autofocusing circular swallowtail beams.

In this Letter, to the best of our knowledge, we report the first experimental demonstration of a new family of autofocusing beams, circular swallowtail beams (CSBs), based on the high-order swallowtail catastrophe, which were determined by potential functions depending on the state and control parameters. The dynamics of the CSBs is discussed here. These types of CSBs tend to automatically focus without external components. Numerical results showed the focal intensity increased significantly, and it was as much as 110 times in the initial plane when the radius of the main ring was 40. Additionally, in contrast to previous circular Pearcey and Airy beams, these CSBs appeared to have more diversity and tunability due to having more propagation trajectories and intensity distribution structures due to high-order diffraction catastrophe. The numerical simulations were verified by our experimental results. These diverse CSBs could have new applications in flexible optical manipulation. These various CSBs could be beneficial for potential applications in optical trapping, medical treatment, or micromachining.

Scale distortion correction of a digital micromirror device using diffraction caustics

A Digital Micromirror Device or DMD is an instrument used to encode information on a light beam by modulating its amplitude. This device is composed of an array of tiny mirrors that can be controlled individually and can thus act as binary apertures to control the amplitude of light. DMD’s with a diamond pixel configuration have stretching distortion to the uploaded patterns. Although the patterns can be rescaled before uploading to the DMD, there is no straightforward method to determine the proper scaling factor. Here, we propose a method to find the scaling factor for the DMD by looking at diffraction caustics. We uploaded elliptical slits in the DMD that results in astroids caustics which collapse to a point when the correct scaling is matched. The scaling factor we found corrects both the inherent distortion caused by the DMD and the distortion caused by a slight misalignment of the DMD. This method offers a faster and more sensitive determination of the scale ratio instability of diffraction catastrophes and the very nature of diffraction. This method can be used to determine aspect ratios of other similar devices especially when these devices use differently shaped mirrors or when the device’s orientation is constrained such that when it is tilted.

Exceptional points and photonic catastrophe.

Exceptional points (EPs) with a global collapse of pairs of eigenfunctions are shown to arise in two locally coupled and spatially extended optical structures with balanced gain and loss. The global collapse at the EP deeply changes light propagation, which becomes very sensitive to small changes of initial conditions or system parameters, similar to what happens in models of classical or quantum catastrophes. The implications of global collapse for light behavior are illustrated by considering discrete beam diffraction and Bloch oscillation catastrophe in coupled waveguide lattices.

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.

Diffraction catastrophes and semiclassical quantum mechanics for Veselago lensing in graphene

We study the effect of trigonal warping on the focussing of electrons by n-p junctions in graphene. We find that perfect focussing, which was predicted for massless Dirac fermions, is only preserved for one specific sample orientation. In the general case, trigonal warping leads to the formation of cusp caustics, with a different position of the focus for graphene's two valleys. We develop a semiclassical theory to compute these positions and find very good agreement with tight-binding simulations. Considering the transmission as a function of potential strength, we find that trigonal warping splits the single Dirac peak into two distinct peaks, leading to valley polarization. We obtain the transmission curves from tight-binding simulations and find that they are in very good agreement with the results of a billiard model that incorporates trigonal warping. Furthermore, the positions of the transmission maxima and the scaling of the peak width are accurately predicted by our semiclassical theory. Our semiclassical analysis can easily be carried over to other Dirac materials, which generally have different Fermi surface distortions.

Optical catastrophes of the swallowtail and butterfly beams

We experimentally realize higher-order catastrophic structures in light fields presenting solutions of the paraxial diffraction catastrophe integral. They are determined by potential functions whose singular mapping manifests as caustic hypersurfaces in control parameter space. By addressing different cross-sections in the higher-dimensional control parameter space, we embed swallowtail and butterfly catastrophes with varying caustic structures in the lower-dimensional transverse field distribution. We systematically analyze these caustics analytically and observe their field distributions experimentally in real and Fourier space. Their spectra can be described by polynomials or expressions with rational exponents capable to form a cusp.

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.

Toward pruning theory of the Stokes geometry for the quantum Hénon map

The Stokes geometry for the propagator of the quantum Hénon map is studied in the light of recent developments of the exact WKB analysis. As the simplest possible situation the Hénon map satisfying the so-called horseshoe condition is closely analyzed, together with listing up local bifurcation patterns of the Stokes geometry. This is exactly in the same spirit as pruning theory for the classical horseshoe system, and the present paper is placed as the first step to establish pruning theory of the Stokes geometry for chaotic systems. Our analysis reveals that the birth and death of the WKB solutions caused by the Stokes phenomenon do not occur in a local but entirely global manner, reflecting topological nature encoded in the Stokes geometry. We derive an explicit general formula to enumerate the number of WKB solutions in the asymptotic region and obtain its growth rate, which is shown to be less than the topological entropy of the corresponding classical dynamics. The relations to the diffraction catastrophe integrals and one-step multiply folding maps are also discussed.

Generation of a family of Pearcey beams based on Fresnel diffraction catastrophes

Based on the theory of differential geometry and Fresnel diffraction catastrophes, we theoretically prove that the cusped caustic of Pearcey beams are the evolute of a parabola and thus identify the key factor determining the optical structure of Pearcey beams. We numerically simulate and experimentally generate a family of Pearcey beams with various optical topological structures using different parabolas. We then investigate their optical structures and propagation properties.

Why the high-frequency inverse scattering by topological sensitivity may work.

This study deciphers the topological sensitivity (TS) as a tool for the reconstruction and characterization of impenetrable anomalies in the high-frequency regime. It is assumed that the anomaly is simply connected and convex, and that the measurements of the scattered field are of the far-field type. In this setting, the formula for TS-which quantifies the perturbation of a cost functional due to a point-like impenetrable scatterer-is expressed as a pair of nested surface integrals: one taken over the boundary of a hidden obstacle, and the other over the measurement surface. Using multipole expansion, the latter integral is reduced to a set of antilinear forms featuring Green's function and its gradient. The remaining expression is distilled by evaluating the scattered field on the surface of an obstacle via Kirchhoff approximation, and pursuing an asymptotic expansion of the resulting Fourier integral. In this way, the TS is found to survive upon three asymptotic lynchpins, namely (i) the near-boundary approximation for sampling points close to the 'exposed' surface of an obstacle; (ii) uniform expansions synthesizing the diffraction catastrophes for sampling points near caustic surfaces, lines and points; and (iii) stationary phase approximation. Within the framework of catastrophe theory, it is shown that, in the case of the full source aperture, the TS is asymptotically dominated by the (explicit) near-boundary term-which explains the previously reported reconstruction capabilities of this class of indicator functionals. The analysis further shows that, when the (Dirichlet or Neumann) character of an anomaly is unknown beforehand, the latter can be effectively exposed by assuming point-like Dirichlet perturbation and considering the sign of the leading term inside the reconstruction.

Experimental validation of the vectorial complex ray model on the inter-caustics scattering of oblate droplets.

We report the first experimental validation of the Vectorial Complex Ray Model (VCRM) using the scattering patterns of large oblate droplets trapped in an acoustic field. The two principal radii and refractive index of the droplets are retrieved with a minimization method that involves VCRM predictions and experimental light scattering patterns. The latter are recorded in the droplet equatorial plane between the primary rainbow region and the associated hyperbolic-umbilic diffraction catastrophe. The results demonstrate that the VCRM can predict the fine and coarse stuctures of scattering patterns with good precision, opening up perspectives for the characterization of large non-spherical particles.

Production of accelerating quad Airy beams and their optical characteristics.

Based on a geometric caustic argument and diffraction catastrophe theory, we generate a novel form of accelerating beams using a symmetric 3/2 phase-only pattern. Such beams can be called accelerating quad Airy beams (AQABs) because they look very much like four face-to-face combined Airy beams. Optical characteristics of AQABs are subsequently investigated. The research results show that the beams have axial-symmetrical and centrosymmetrical transverse intensity patterns and quasi-diffraction-free propagation features for their four main lobes while undergoing transverse shift along parabolic trajectories. Moreover, we also demonstrate that AQABs possess self-construction ability when local areas are blocked. The unique optical properties of these beams will make them useful tools for future scientific applications.

Diffraction catastrophes threaded by quantized vortex skeletons caused by atom-optical aberrations induced in trapped Bose-Einstein condensates

We propose a nonlinear atom-optics experiment to create diffraction catastrophes threaded by quantized vortex skeletons in Bose-Einstein condensed matter waves. We show how atom-optical aberrations induced in trapped Bose-Einstein condensates evolve into specific caustic structures due to imperfect focusing. Vortex skeletons, whose cross-sections are staggered vortex lattices, are observed to nucleate inside the universal diffraction catastrophes. Our observations shed further light on the structure and dynamics of Bose-novae and suggest applications of matter wave diffraction catastrophes, including detection of the order parameter pairing symmetry in cold gas experiments.

Electron Vortex Production and Control Using Aberration Induced Diffraction Catastrophes

An aberration corrected electron microscope is used to create electron diffraction catastrophes, containing arrays of intensity zeros threading vortex cores. Vortices are ascribed to these arrays using catastrophe theory, scalar diffraction integrals, and experimentally retrieved phase maps. From measured wave function phases, obtained using focal-series phase retrieval, the orbital angular momentum density is mapped for highly astigmatic electron probes. We observe vortex rings and topological reconnections of nodal lines by tracking the vortex cores using the retrieved phases.

Numerical computation of diffraction catastrophes with codimension eight

A computational strategy, aimed at evaluating diffraction catastrophes belonging to the X(9) family is presented. The approach proposed is based on the use of power series expansions, suitably derived for giving meaningful representation of the whole (0)X(9) subfamily, jointly with a powerful sequence transformation algorithm, the so-called Weniger transformation. The convergence features of the above series expansions are investigated, and several numerical experiments are carried out to assess the effectiveness of the retrieving action of the Weniger transformation, as well as the ease of implementation of the whole approach.