Singularities Of Wave Fronts Research Articles

Multifunctional computational fluorescence self-interference holographic microscopy

Fluorescence microscopy is crucial in various fields such as biology, medicine, and life sciences. Fluorescence self-interference holographic microscopy has great potential in bio-imaging owing to its unique wavefront coding characteristics; thus, it can be employed as three-dimensional (3D) scanning-free super-resolution microscopy. However, the available approaches are limited to low optical efficiency, complex optical setups, and single imaging functions. The geometric phase lens can efficiently manipulate the optical field’s amplitude, phase, and polarization. Inspired by geometric phase and self-interference holography, a self-interference fluorescent holographic microscope-based geometric phase lens is proposed. This system allows for wide-field, 3D fluorescence holographic imaging, and edge-enhancement from the reconstruction of only one complex-valued hologram. Experiments demonstrate the effectiveness of our method in imaging biological samples, with improved resolution and signal-to-noise ratio. Furthermore, its simplicity and convenience make it easily compatible with existing optical microscope setups, making it a powerful tool for observing biological samples and detecting industrial defects.

Unified wavefront singularity characterization of three-dimensional elastodynamic time-domain half-space Green's function under impulsive boundary and internal loads

Founded on a novel analytical formulation that led to a rigorous yet compact path-integral representation of the time-domain elastodynamic half-space Green's function, a unified analysis of the possible occurrence of different singular wavefront behaviour in the response under arbitrary impulsive internal or surface point loads at arbitrary source-receiver locations is presented. With the decomposition of the general solution into distinct initiating and reflected wave group integrals that share a common factored format and simple contour definitions, the mathematical framework is shown to allow a straightforward identification of the specific conditions and the particular wave groups that are responsible for the singular wavefront phenomena without resorting to advanced analytic function theories or asymptotic methods. Analytic characterizations of the nature, strength and direction of all intrinsic singular wavefront behaviours of the three-dimensional Green's function in three canonical cases of source-receiver configurations are given in a dual integral-closed form format to facilitate their theoretical understanding as well as computational applications. Graphical illustrations of their variation with the source-receiver configuration and the medium's Poisson's ratio together with relevant comparison and clarifications of some classical treatments are included.

Extendibility and boundedness of invariants on singularities of wavefronts

Extendibility and boundedness of invariants on singularities of wavefronts

Spin-decoupled excitation and wavefront shaping of structured surface waves via on-chip terahertz metasurfaces.

Surface waves (SWs) are of great importance in terahertz (THz) photonics applications due to their subwavelength properties. Hence, it is crucial to develop surface wavefront shaping techniques, which is urgent in modern information technologies. In this paper, a new scheme is proposed to realize SW excitation and spin-decoupled wavefront shaping with an ultracompact planar meta-device working in the THz range. The meta-device is composed of two parts: meta-atoms (in the center) and plasmonic metals (on the left and right sides). By carefully setting the geometry size and rotation angle of each meta-atom, the encoded spin-decoupled phase distributions for both left circularly polarized (LCP) and right circularly polarized (RCP) incident THz waves are determined. In this way, circularly polarized (CP) incident THz waves can be converted to SWs propagating along plasmonic metals with unique wavefront profiles, i.e., Bessel and focusing profiles. Full-wave simulations and THz near-field scanning experiments were performed to verify the functionalities of the meta-device, both of which are in great agreement with theoretical predictions. Our findings may provide more solutions to design THz integrated photonic devices and systems.

Causality condition relevant functions-orientated analytical treatment on singularities in 3D TD-BEM

A direct analytical treatment on singularities in the 3D TD-BEM formulation is proposed, where the wavefront singularity and the dual singularity are analytically expressed. In the process of the solution of the spatial singularity, the integration domain is divided into the regular part and the singular part. The singularities in the singular part are analytically eliminated by the direct method of the concept of the finite part of an integral (Hadamard principal integral), while the singular integrals in the regular part are solved by the convenient Gaussian integration. Due to the increase of the dimension and the additional causality relevant function, the 3D TD-BEM formulation is further more intricate than the 2D one. In order to reduce the complexity, in the process of the solution of the boundary integral equation in the 3D TD-BEM formulation, a new coordinate transformation method is proposed to analytically transform the coefficient integrals on the spatial surface element into the 2D plane element. The 3D TD-BEM formulation based on the proposed analytical treatment on singularities is verified to be correct by three examples.

A Direct Method for Solving Singular Integrals in Three-Dimensional Time-Domain Boundary Element Method for Elastodynamics

The analytically time integrable time-space domain (ATI-TSD) is discovered based on which the minimum time-space domain is identified for treatment on singularities in the three-dimensional time-domain boundary element method (3D TD-BEM) formulation. A direct method to solve singular integrals in the 3D TD-BEM formulation for elastodynamic problems is proposed. The wavefront singularity can be analytically eliminated in ATI-TSD, while the dual singularity can be treated by the direct method using Kutt’s quadrature in the identified minimum time-space domain. Three benchmark examples are presented to verify the correctness and the applicability of the direct method for solving the singular integrals in 3D TD-BEM.

The Hawking energy on the past lightcone in cosmology

This work studies the Hawking energy in a cosmological context. The past lightcone of a point in spacetime is the natural geometric structure closely linked to cosmological observations. By slicing the past lightcone into a one-parameter family of spacelike two-surfaces, the evolution of the Hawking energy down the lightcone is studied. Strong gravitational fields may generate lightcone self-intersections and wave front singularities. We show that in the presence of swallow-tail type singularities, the Hawking energy and its variation along the null generators of the lightcone remains well-defined and subsequently discuss its positivity and monotonicity.

Spiral spectrum of the phase singularity beam in the source plane and atmospheric turbulence

The phase singularity beam has a unique wavefront phase structure—a screw dislocation (optical vortex) or edge dislocation—which is closely related to the orbital angular momentum (OAM). Screw dislocation refers to the phase change of integer multiples of 2π around a point, while edge dislocation refers to the existing π phase shift along a curve in the wavefront phase distribution. In this study, we derived the OAM detection probability expression and mode probability density expression of the phase singularity beam propagating through atmospheric turbulence. These expressions were used to examine the spiral spectrum and the OAM mode probability density of the phase singularity beam. It was determined that the phase singularity beam carrying a noncanonical optical vortex or an edge dislocation has more than one OAM state, whereas the phase singularity beam carrying a canonical optical vortex has only one OAM state. In atmospheric turbulence transmission, atmospheric disturbances will cause crosstalk between OAM states. The larger is the detection probability of the OAM state at the source plane, the closer is the maximum value of the OAM mode probability density throughout the transmission process to the beam center. The conclusions obtained have important applications in optical communications.

Existence and uniqueness of forced waves in a delayed reaction–diffusion equation in a shifting environment

The existence and uniqueness of forced waves in a general reaction–diffusion equation with time delay under climate change is concerned in this paper. By using upper and lower solutions method, monotone iteration scheme combined with the strong maximum principle, we show that there exists a nondecreasing and unique wave front with the speed consistent with the habitat shifting speed. Our results indicate the propagation of both the leading and trailing edges of the comoving population wavefront lag behind the climate envelope, which drives the species to extinction. Three examples and their corresponding numerical simulations are also given to illustrate the universality of analytical conclusions.

Wave Fronts in a Causality-Violating Gödel-Type Metric

The light rays and wave fronts in a linear class of the Gödel-type metric are examined to reveal the causality-violating features of the space-time. Noncausal features demonstrated by the development of unusual wave front singularities are shown to be related to the nonmonotonic advance of time along the light rays, as measured by a system of observers at rest with respect to one another with synchronized clocks.

Topological optics with 3D-tangle laser solitons

We analyse, theoretically and numerically, stable localized 3D- structures with radiation wavefront singularities in lasers or laser media with fast saturable absorption; radiation has a fixed polarization state. These laser solitons’ internal structure is characterized by their skeletons, which consist of vortex lines where the field vanish and radiation phase changes by 2πm with integer m – topological charge – as a result of a round-trip around the line. The vortex lines are oriented according the direction of radiation phase increase; high-order vortex lines (m>1) are unstable and split into m lines with m=1. Complexes of fundamental (without vortices when isolated) laser solitons have unclosed vortex lines. Presented are topological laser solitons with vortex lines, which are unclosed and closed, unknotted and knotted, unlinked and linked. Various types of the solitons are stable in different, but overlapping domains of the scheme parameters; a slow variation of such parameters leads to hysteretic variations of solitons’ characteristics. Such hysteresis is reversible if soliton’s topology does not change during the parameters’ variation, and irreversible in the opposite case. The increased stability of dissipative solitons and the preservation of the topological characteristics of structures under perturbations make topological laser solitons promising for information applications.

Singularity cancellation method for time-domain boundary element formulation of elastodynamics: A direct approach

In the implementation of time-domain boundary element method for elasto-dynamic problems, there are two types of singularities: the wave front singularity arising when the product of wave velocity and time is equal to the distance between the source point and the field point, and the spatial singularity arising when the source point coincides with the field point. In this paper, the singularity of the first type in the integrand is eliminated by an analytical integration over time, Cauchy principal value and Hadamard finite part integral. Four types of singularities with different orders appear in the integrand after analytical time integration. In order to accurately calculate the integral, in which the integrand is piecewise continuous, the integral domain is subdivided into several patches based on the relation between the product of wave velocity and time and the distance. In singular patches, the integrands are separated into a regular part and a singular part. The regular part can be computed by traditional numerical integration method such as Gaussian integration, while the singular part can be analytically integrated. Using the proposed method, the spatial singular integrals can be calculated directly. Numerical examples using various kinds of elements are presented to verify the proposed method.

Examining the Kerr metric through wave fronts of null geodesics

We examine the singularities of the wave fronts of null geodesics from point sources in the Kerr Metric. We find that the wave fronts develop a tube like structure that collapses non-symmetrically, leading to cusp features in the wave front singularities. As the wave front advances, the cusps trace out an astroidal shaped caustic tube, which had been discovered previously using lens mapping and geodesic deviation methods. Thus, the wave front approach in this study helps to complete a picture of caustics and gravitational lensing in the Kerr geometry.

Simulation and Analytical Study of Optical Complex Field in Nano-corral Slits Plasmonic Lens

Although spiral plasmonic lens has been proposed as circular polarization analyzer, there is no such plasmonic nanostructure available for linear polarization. In the current work, we have designed nano-corral slits (NCS) plasmonic lens, which focuses the x- and y-polarized light into spatially distinguished plasmonic fields. We have calculated analytically and numerically the electric field intensity and phase of the emission from nano-corral slits plasmonic lens with different pitch lengths under various polarizations of the illumination. It has been shown that one can control the wave front of the output beam of these plasmonic lenses by manipulating the illumination of both circular and linear polarization. Our theoretical study in correlation with FDTD simulation has shown that NCS plasmonic lens with pitch length equal to λspp produces scalar vortex beam having optical complex fields with helical wave front and optical singularity at the center under circular polarization of light. When NCS lens (pitch = λspp) is illuminated with linearly polarized light, it exhibits binary distribution of phase with same electric field intensity around the center. However, with pitch length of 0.5λspp, NCS shows linear dichroism under linearly polarized illumination unlike spiral plasmonic lens (SPL) eliminating the use of circularly polarized light. Optical complex fields produced by these NCS plasmonic lenses may find applications for faster quantum computing, data storage, and telecommunications.

Observation of linear plasmonic breathers and adiabatic elimination in a plasmonic multi-level coupled system

We provide experimental and numerical demonstrations of plasmonic propagation dynamics in a multi-level coupled system, and present the first observation of plasmonic breathers propagating in such systems. The effect is observed both for the simplest symmetric case of a thin metal layer surrounded by two identical dielectrics, and also for a more complex system that includes five and more layers. By a careful choice of the permittivities and thicknesses of the intermediate layers, we can adiabatically eliminate the plasmonic waves in all the intermediate interfaces, thus enabling efficient vertical delivery and extraction of plasmonic signals between the top layer and deeply buried layers. The observation relies on controlling the excited mode by breaking the symmetry of excitation, which is crucial for obtaining the results experimentally. We also observe this breathing effect for transversely shaped plasmonic beams, with Hermite-Gauss, Airy and Weber wavefronts, that despite the oscillatory nature of propagation in such systems, still preserve all their unique wavefront properties. Finally, we show that such approaches can be extended to plasmonic propagation in a general multi-layered system, opening a path for efficient three-dimensional integrated plasmonic circuitry.

Dynamics-based analytical solutions to singular integrals for elastodynamics by time domain boundary element method

The singularities in 2-D time domain boundary element (TD-BEM) formulation for elastodynamics are divided into three categories: the wave front singularity, the space singularity and the dual singularity. A fully analytical procedure for dealing with the three singularities is proposed by adopting the concept of the finite part of an integral (Hadamard principle integral). In order to reduce the computation time, the conventional numerical procedure is adopted for the non-singular integrals in 2-D TD-BEM formulation. Therefore, the algorithm including the fully analytical procedure for dealing with singular integrals and the numerical procedure for dealing with non-singular integrals is implemented in this study. Two examples, 1-D rod and 2-D cavity representing the problems for the finite domain and the infinite domain respectively, are chosen to verify the effectiveness of the proposed algorithm. It shows that the results obtained from the proposed algorithm agree with the analytical solutions with good accuracy, indicating that the proposed algorithm is applicable for elastodynamics in both finite and infinite domains.

Null geodesics and wave front singularities in the Gödel space–time

We explore wave fronts of null geodesics in the Godel metric emitted from point sources both at, and away from, the origin. For constant time wave fronts emitted by sources away from the origin, we find cusp ridges as well as blue sky metamorphoses where spatially disconnected portions of the wave front appear, connect to the main wave front, and then later break free and vanish. These blue sky metamorphoses in the constant time wave fronts highlight the non-causal features of the Godel metric. We introduce a concept of physical distance along the null geodesics, and show that for wave fronts of constant physical distance, the reorganization of the points making up the wave front leads to the removal of cusp ridges.

GENERALIZED GELFAND–GRAEV REPRESENTATIONS IN SMALL CHARACTERISTICS

Let$\mathbf{G}$be a connected reductive algebraic group over an algebraic closure$\overline{\mathbb{F}_{p}}$of the finite field of prime order$p$and let$F:\mathbf{G}\rightarrow \mathbf{G}$be a Frobenius endomorphism with$G=\mathbf{G}^{F}$the corresponding$\mathbb{F}_{q}$-rational structure. One of the strongest links we have between the representation theory of$G$and the geometry of the unipotent conjugacy classes of$\mathbf{G}$is a formula, due to Lusztig (Adv. Math.94(2) (1992), 139–179), which decomposes Kawanaka’s Generalized Gelfand–Graev Representations (GGGRs) in terms of characteristic functions of intersection cohomology complexes defined on the closure of a unipotent class. Unfortunately, the formula given in Lusztig (Adv. Math.94(2) (1992), 139–179) is only valid under the assumption that$p$is large enough. In this article, we show that Lusztig’s formula for GGGRs holds under the much milder assumption that$p$is an acceptable prime for$\mathbf{G}$($p$very good is sufficient but not necessary). As an application we show that every irreducible character of$G$, respectively, character sheaf of$\mathbf{G}$, has a unique wave front set, respectively, unipotent support, whenever$p$is good for $\mathbf{G}$.

Light transport in refractive turbid media.

Light scattering in refractive media can be used for visualization of caustics and singularities of wavefronts of the incident radiation. Its modeling requires solving the radiative transfer equation. Numerical solution of the radiative transfer equation in turbid media with a spatially varying refractive index is presented and discussed in this paper. The approach is based on the self-consistent approximation of the radiative transfer equation and results in a relatively inexpensive and robust numerical algorithm. Simulations are presented for several cases including a weakly scattering medium with varying refractive index and a refractive weakly scattering medium with embedded highly scattering objects.

Solar Ground-Layer Adaptive Optics

Solar conventional adaptive optics (CAO) with one deformable-mirror uses a small field-of-view (FOV) for wave-front sensing, which yields a small corrected FOV for high-resolution imaging. Solar activities occur in a two-dimensional extended FOV and studies of solar magnetic fields need high-resolution imaging over a FOV at least 60″. Recently, solar Tomography Adaptive Optics (TAO) and Multi-Conjugate Adaptive Optics (MCAO) were being developed to overcome this problem of small AO corrected FOV. However, for both TAO and MCAO, wavefront distortions need to be tomographically reconstructed from measurements on multiple guide stars, which is a complicated and time-consuming process. Solar Ground-Layer Adaptive Optics (S-GLAO) uses one or several guide stars, and does not rely on a tomographic reconstruction of the atmospheric turbulence. In this publication, we present two unique wavefront sensing approaches for the S-GLAO. We show that our S-GLAO can deliver good to excellent performance at variable seeing conditions in the Near Infrared (NIR) J and H bands, and is much simpler to implement. We discuss details of our S-GLAO associated wavefront approaches, which make our S-GLAO a unique solution for sunspot high-resolution imaging that other current adaptive optics systems, including the solar MCAO, cannot offer.