Annihilation Of Singularities Research Articles

A Finite Element Approximation for Nematic Liquid Crystal Flow with Stretching Effect Based on Nonincremental Pressure-Correction Method.

In this paper, a new decoupling method is proposed to solve a nematic liquid crystal flow with stretching effect. In the finite element discrete framework, the director vector is calculated by introducing a new auxiliary variable w, and the velocity vector and scalar pressure are decoupled by a nonincremental pressure-correction projection method. Then, the energy dissipation law and unconditional energy stability of the resulting system are given. Finally, some numerical examples are given to verify the effects of various parameters on the singularity annihilation, stability and accuracy in space and time.

The evolution of topological singularities between real- and complex-frequency domains and the engineering of photonic bands for Hermitian and non-Hermitian photonic crystals

Singularity annihilation, generation, and evolving (SAGE) lead to the topological phase transition (TPT) in electronic, photonic and acoustic systems. Traditionally the singularity study of Hermitian systems is only focused on the real frequency domain. In this work, we systematically investigate the complicated SAGE in complex frequency domain (CFD) for one-dimensional (1D) Hermitian and non-Hermitian systems and a more general picture is revealed. First, we study the abnormal phenomenon that one singularity evolves from the first band to the zero frequency and then into the pure imaginary frequency for Hermitian 1D photonic crystals (PhCs). New results, e.g. the general condition for the singularity at zero frequency, the stricter definition of the Zak phase of first band and the phenomenon that more singularities are pushed from first band into the imaginary frequency, are found. Second, a general evolving picture of SAGE in CFD for Hermitian systems is constructed. Complicated processes of singularities in CFD are observed, such as the SAGE not only on the real frequency axis but also on the imaginary frequency axis, the closed evolving loops for singularities which connected imaginary-frequency axis and real-frequency axis. Even more, when the PhCs is degenerated since the permittivity on one kind layer becomes same as the neighbor layer, the singularities on the integral reduced frequency will move to infinite far away and come back with half-integral shift. Third, when gain or absorption is introduced in, the SAGE on a tilted axis is also observed. The phenomenon of one singularity moving back to real frequency axis for non-Hermitian systems means that the stable states with resonance could be realized. Such complicated and general singularity evolving picture in CFD opens a new window for the studies of TPT and the rich new topological phenomena could be expected. Besides the theoretical importance, the evolution of singularity can also be used to engineer the band properties of PhCs. Some novel applications, such as the super-broadband sub-wavelength high-transmission layered structure and the broadband deep-sub-wavelength absorber, are proposed.

Phase singularity annihilation in plasmonic nano-apertures via epsilon-near-zero metamaterials

In this work, we performed an extensive theoretical and experimental study to unveil the underlying mechanisms related to the intensified transmittance in epsilon-near-zero (ENZ)-integrated plasmonic nano-apertures. The occurrence of phase singularities at the incident side of plasmonic nano-apertures results in the reduction in transmittance. We show that transmittance enhancement in ENZ-integrated nano-slits is attributed to the annihilation of the phase singularities by the ENZ layer and subsequently the modification in plasmonic fields of metallic apertures. The singularity annihilation via ENZ metamaterials eliminates the abrupt changes in the phase of the Poynting vector underneath the slit, which results in a stronger cavity resonance inside the aperture. Due to this fact, a nano-aperture on top of an ENZ metamaterial is the only effective configuration to enhance the transmittance in ENZ-integrated nano-apertures. Hence, our study reveals the physics of the transmittance enhancement through ENZ-integrated plasmonic nano-apertures.

A projection‐based time‐splitting algorithm for approximating nematic liquid crystal flows with stretching

AbstractA numerical method is developed for solving a system of partial differential equations modeling the flow of a nematic liquid crystal fluid with stretching effect, which takes into account the geometrical shape of its molecules. This system couples the velocity vector, the scalar pressure and the director vector representing the direction along which the molecules are oriented. The scheme is designed by using finite elements in space and a time‐splitting algorithm to uncouple the calculation of the variables: the velocity and pressure are computed by using a projection‐based algorithm and the director is computed jointly to an auxiliary variable. Moreover, the computation of this auxiliary variable can be avoided at the discrete level by using piecewise constant finite elements in its approximation. Finally, we use a pressure stabilization technique allowing a stable equal‐order interpolation for the velocity and the pressure. Numerical experiments concerning annihilation of singularities are presented to show the stability and efficiency of the scheme.

A linear mixed finite element scheme for a nematic Ericksen–Leslie liquid crystal model

In this work we study a fully discrete mixed scheme, based on continuous finite elements in space and a linear semi-implicit first-order integration in time, approximating an Ericksen–Leslie nematic liquid crystal model by means of a Ginzburg–Landau penalized problem. Conditional stability of this scheme is proved via a discrete version of the energy law satisfied by the continuous problem, and conditional convergence towards generalized Young measure-valued solutions to the Ericksen–Leslie problem is showed when the discrete parameters (in time and space) and the penalty parameter go to zero at the same time. Finally, we will show some numerical experiences for a phenomenon of annihilation of singularities.

Symmetry breaking of optical vortices: birth and annihilation of singularities in the evanescent field

The interaction of optical vortices (or phase singularities, or screw dislocations) with ordinary matter is treated with a simple approach. Using the total internal reflection phenomenon and the superposition of four plane waves incident on a material with a refractive index lower than the original propagating medium, we are able to show the birth and annihilation of optical vortices in an evanescent field with curved topological features. Until now, this phenomenon has been explored only in free space propagation. By a suitable tuning process involving the incident angles and the amplitudes of the incident plane waves, it is possible to create unusual topological features of optical vortices in the vicinity of the material. We believe that this work can open new aspects of curved optical vortex manipulation in near-field optics.

Phase singularity annihilation of focused partially coherent beams

Expressions for focused Gaussian Schell-model (GSM) beams are derived and used to study the annihilation and subwavelength structures of phase singularities in the focal region, and to compare with the case of fully coherent Gaussian beams. It is found that the truncation parameter δ and normalized coherence length ε both affect the presence and spatial distribution of phase singularities in the focal plane. Additionally, during the creation and annihilation process the saddle point near the phase singularity does not disappear in the focal plane for GSM beams.

Spatial correlation properties and phase singularity annihilation of Gaussian Schell-model beams in the focal region

By using the generalized Debye diffraction integral, this paper studies the spatial correlation properties and phase singularity annihilation of apertured Gaussian Schell-model (GSM) beams in the focal region. It is shown that the width of the spectral degree of coherence can be larger, less than or equal to the corresponding width of spectral density, which depends not only on the scalar coherence length of the beams, but also on the truncation parameter. With a gradual increase of the truncation parameter, a pair of phase singularities of the spectral degree of coherence in the focal plane approaches each other, resulting in subwavelength structures. Finally, the annihilation of pairs of phase singularities takes place at a certain value of the truncation parameter. With increasing scalar coherence length, the annihilation occurs at the larger truncation parameter. However, the creation process of phase singularities outside the focal plane is not found for GSM beams.

Optical diabols in singular laser beams

Polarisation singularities and optical diabols in laser beams are studied experimentally by the developed method of digital Stokes-polarimetry. Three possible morphological forms of singularities with circular polarisation (C points) are found and their statistics in the speckle pattern of photoinduced scattering is determined. Singular fields with controllable parameters are realised and used for studying topological transformations, creation and annihilation of singularities and optical diabols. Both possible types of distributions of the major and minor semiaxes of polarisation ellipses in the beam cross section around C points (optical diabols) are measured. It is shown that, in accordance with the theory, the annihilation of pairs of elliptical diabols is preceded by their transformation to hyperbolic diabols.

The creation–annihilation process of phase singularities of flat-topped beams in the focal plane

The creation–annihilation process of phase singularities of flat-topped beams in the focal plane is studied within the framework of the scalar paraxial approximation, where the new expression for flat-topped beams introduced by Li is taken as the beam model. It is shown that by suitably varying the truncation parameter, the phase singularities reorganize themselves, which leads to the annihilation of singularities in the focal plane, the creation of singularities outside the focal plane and subwavelength structures. Apart from the truncation parameter, the beam order of flat-topped beams additionally affects the spatial distribution of phase singularities.

Integration over simplexes for accurate domain and boundary integral evaluation in boundary element methods

The accuracy of boundary element methods is heavily dependent on the accurate determination of singular domain and boundary integrals on element domains. Many methods have been developed for accurate integral determination but none that is universally applicable across the range of boundary element applications. This paper is concerned with a semi-analytical method that applies to simplexes of arbitrary dimension. The method involves the careful employment of multiple integration where the inner integral is performed along a radial direction. Evaluation of the radial-inner integral on a simplex of dimension n provides n+1 integrals on simplex domains of dimension n−1. It is shown in the paper that provided closed form solutions exist for the inner integrals the method can be repeated on these n+1 simplexes providing a recursive method of integration. Performing radial inner integration on a simplex is shown to be ideal for the correct treatment of the radial singularities present. If a closed form solution is not available then a radial function is added and subtracted to facilitate the use of numerical integration. In addition to singularity annihilation the form of this radial function is selected to facilitate the continued recursive application of radial integration. The method is tested on domain and boundary integrals present in thermoelastostatic, elastodynamic and on simple test problems for which known analytical solutions exist. The results obtained using the semi-analytical simplex approach are shown to be considerably more accurate than those obtained using standard quadrature methods.

The viscous damping prevents propagation of singularities in the system of viscoelasticity

SynopsisWe show that the linear viscous damping Δut, is so strong that it altogether prevents propagation of singularities of the gradient of solutions to the system of viscoelasticity. Moreover, no creation or annihilation of singularities is possible in finite time.

Eddy genesis in the double-lid-driven cavity

Using an integral-equation method which employs the authors' new and widely-applicable technique of singularity annihilation, we investigate the eddy genesis in the benchmark problem of steady viscous flow in a lid-driven cavity, two of whose opposite edges move with different velocities. The method enables us to investigate the eddy structure in the resulting flow with confidence since the singularity-annihilation method results are demonstrated to be virtually independent of the boundary discretization, so effective is the suppression of the singularity. In the event that only one of the edges moves, we provide convincing quantitative corroboration of the geometrical symmetry of the corner eddies first predicted by Moffatt (1). When two edges move with a relative speed ratio, we present an interesting study of the genesis of a variety of topologically-distinct flows, and the way in which these metamorphose as the speed ratio is altered. Some remarkable corner-eddy flow structures are presented, and a full explanation of all observed flow structures is proffered.