Local Ellipticity Research Articles

Enhancement of Optical Spin-Orbit Coupling in Anisotropic ENZ Metamaterials

Vector vortex beams which possess complex intensity and polarisation patterns, and phase or polarisation singularities are important in a variety of applications ranging from high resolution imaging and data transmission to nonlinear optics and quantum technologies. Here, we theoretically analyse the effect of vector vortex beams propagation through strongly anisotropic metamaterials exhibiting epsilon-near-zero behaviour. The interaction of the metamaterial with longitudinal field of the vector beams results in a strong modification of the polarisation distribution, with the metamaterial acting as an azimuthal polariser. The dependence on the local ellipticity of the field is also investigated revealing conversion of spin angular momentum to orbital angular momentum and polarisation structure vorticity, mediated by the longitudinal field. The results show the promise of an anisotropic metamaterial platform for manipulation of polarisation properties of and spin-orbit coupling in vector vortex beams.

Focus shaping by tailoring arbitrary hybrid polarization states that have a combination of orthogonal linear polarization bases.

We report a focus shaping method by tailoring hybrid states of polarization of arbitrary polarized beams that have a combination of orthogonal linear polarization bases. Such hybridly polarized beams comprising linear, elliptical, and circular polarizations in the beam cross section, have completely different optical properties compared to the scalar and locally linear-polarized vector beams. We demonstrate that, apart from the orientation of the local polarization state, another two degrees of freedom including the local ellipticity and the handedness in the beam cross section can be used in focus shaping. Square-shaped patterns, multiple foci, three-dimensional optical cages, optical needles, and channels can be obtained due to the increased control without any additional phase or amplitude modulations.

Strongly degenerate time inhomogeneous SDEs: Densities and support properties. Application to Hodgkin–Huxley type systems

In this paper, we study the existence of densities for strongly degenerate stochastic differential equations (SDEs) whose coefficients depend on time and are not globally Lipschitz. In these models, neither local ellipticity nor the strong Hormander condition is satisfied. In this general setting, we show that continuous transition densities indeed exist in all neighborhoods of points where the weak Hormander condition is satisfied. We also exhibit regions where these densities remain positive. We then apply these results to stochastic Hodgkin–Huxley models with periodic input as a first step towards the study of ergodicity properties of such systems in the sense of Meyn and Tweedie (Adv. in Appl. Probab. 25 (1993) 487–517; Adv. in Appl. Probab. 25 (1993) 518–548).

Competition of single and double rescattering in the strong-field photoemission from dielectric nanospheres.

Nanostructures exposed to ultrashort waveform-controlled laser pulses enable the generation of enhanced and highly localized near fields with adjustable local electric field evolution. Here, we study dielectric SiO2 nanospheres (d = 100–700 nm) under strong carrier-envelope phase-controlled few-cycle laser pulses and perform a systematic theoretical analysis of the resulting near-field driven photoemission. In particular, we analyze the impacts of charge interaction and local field ellipticity on the near-field driven electron acceleration. Our semiclassical transport simulations predict strong quenching of the electron emission and enhanced electron energies due to the ionization induced space charge. Though single surface backscattering remains the main emission process for the considered parameter range, we find a substantial contribution of double rescattering that increases with sphere size and becomes dominant near the cutoff energy for the largest investigated spheres. The growing importance of the double recollision process is traced back to the increasing local field ellipticity via trajectory analysis and the corresponding initial to final state correlation. Finally, we compare the carrier-envelope phase-dependent emission of single and double recollision electrons and find that both exhibit a characteristic directional switching behavior.

A numerical study of the Rayleigh wave particle motions excited by a point source and Poisson's ratio for lateral inhomogeneous half-spaces

In this paper, the dependence of Rayleigh wave particle motions excited by a point source on the lateral inhomogeneous Poisson's ratio of propagation half-spaces is studied based on the theoretical analysis and wave field numerical simulation method. First, two types of time signals named single frequency harmonic and broadband Ricker wavelet are used respectively as a point source in the simulation of homogenous half-spaces. The results show that in the far-field region absolutely dominated by Rayleigh waves, or where body waves are well removed either by muting or polarization filtering, the ellipticity of particle motion can be approximated to be a simple inverse linear and frequency-independent relation with Poisson's ratio of rocks and soils, which has a linear correlation coefficient of over 0.9775 with the exact analytic solutions derived from theoretical formula for a homogeneous half-space. Then, two types of lateral inhomogeneous models, a vertical rock interface and a local heterogeneous body with various Poisson's ratios, are simulated. The results show that a local inhomogeneity can cause a local and wavelength-dependent ellipticity anomaly. The ellipticity is still in inverse proportion of Poisson's ratio but distorted with a high nonlinearity for the lateral inhomogeneous. The interferences of converted body waves are also studied, and the results suggest that the ellipticity estimation of Rayleigh waves suffers little from the body waves converted by the scattering of lateral inhomogeneity. The study shows that the dependency of Rayleigh wave ellipticity on Poisson's ratio is potentially beneficial for Poisson's ratio estimation, which is based on the inverse linear approximation for a homogeneous region within several half-wavelengths, and in surveys for space geometry and elastic properties of the lateral inhomogeneity, which are based on the frequency-dependent and inverse proportional properties that are all of the main goals of geophysical surveys and non-destructive tests.

Higher-order finite volume methods II: Inf–sup condition and uniform local ellipticity

The main purpose of this paper is to study the construction of higher-order finite volume methods (FVMs) of triangle meshes. We investigate the relationship of the three theoretical notions crucial in the construction of FVMs: the uniform ellipticity of the family of its discrete bilinear forms, its inf–sup condition and its uniform local ellipticity. Both the uniform ellipticity of the family of the discrete bilinear forms and its inf–sup condition guarantee the unique solvability of the FVM equations and the optimal error estimate of the approximate solution. We characterize the uniform ellipticity in terms of the inf–sup condition and a geometric condition on the bijective operator mapping from the trial space onto the test space involved in the construction of FVMs. We present a geometric interpretation of the inf–sup condition. Moreover, since the uniform local ellipticity is a convenient sufficient condition for the uniform ellipticity, we further provide sufficient conditions and necessary conditions of the uniform local ellipticity.

Circular dichroism nano-imaging of two-dimensional chiral metal nanostructures

Here we report on a nanoscale circular dichroism (CD) imaging for a two-dimensional chiral pair of nanostructures to elucidate the relationship between nanoscale chirality and CD activity. The chiral pair exhibited local ellipticity as high as 42 degrees in the CD signal at the center, with signals of both handedness coexisting in one nanostructure.

The Cauchy-Dirichlet Problem for a Class of Linear Parabolic Differential Equations with Unbounded Coefficients in an Unbounded Domain

We consider the Cauchy-Dirichlet problem in [0,∞)×D for a class of linear parabolic partial differential equations. We assume that D⊂ℝd is an unbounded, open, connected set with regular boundary. Our hypotheses are unbounded and locally Lipschitz coefficients, not necessarily differentiable, with continuous data and local uniform ellipticity. We construct a classical solution to the nonhomogeneous Cauchy-Dirichlet problem using stochastic differential equations and parabolic differential equations in bounded domains.

Existence and uniqueness to the Cauchy problem for linear and semilinear parabolic equations with local conditions

We consider the Cauchy problem in R d for a class of semilinear parabolic partial dier- ential equations that arises in some stochastic control problems. We assume that the coecients are unbounded and locally Lipschitz, not necessarily dierentiable, with continuous data and local uniform ellipticity. We construct a classical solution by approximation with linear parabolic equations. The linear equations involved can not be solved with the traditional results. Therefore, we construct a classical solution to the linear Cauchy problem under the same hypotheses on the coecients for the semilinear equation. Our approach is using stochastic dierential equations and parabolic dierential equations in bounded domains. Finally, we apply the results to a stochastic optimal consumption problem. R

WEAK REGULAR REFLECTION FOR THE NONLINEAR WAVE SYSTEM

We prove local existence of a solution to a Riemann problem for the two-dimensional nonlinear wave system using the approach by Čanić, Keyfitz, Kim and Lieberman. We consider initial data resulting in weak regular shock reflection. By writing the problem in self-similar coordinates, we obtain a mixed type system and a free boundary problem for the subsonic flow and the position of the reflected shock. We reformulate this problem using a second order equation for density with mixed boundary conditions and an ordinary differential equation describing the location of the reflected shock. The main difficulty in the study of this second order problem for density is that the operator is degenerate elliptic along the sonic circle. We regularize and modify the operator, and we show existence of solutions to the regularized problems. We prove that the sequence of solutions to the regularized problems is precompact, and, using uniform local ellipticity, covering and diagonalization techniques, we obtain a local solution to the original problem.

Lower bounds for the density of locally elliptic Itô processes

We give lower bounds for the density pT(x, y) of the law of Xt, the solution of dXt=σ(Xt) dBt+b(Xt) dt, X0=x, under the following local ellipticity hypothesis: there exists a deterministic differentiable curve xt, 0≤t≤T, such that x0=x, xT=y and σσ*(xt)>0, for all t∈[0, T]. The lower bound is expressed in terms of a distance related to the skeleton of the diffusion process. This distance appears when we optimize over all the curves which verify the above ellipticity assumption. The arguments which lead to the above result work in a general context which includes a large class of Wiener functionals, for example, Itô processes. Our starting point is work of Kohatsu-Higa which presents a general framework including stochastic PDE’s.

$C^{1,\beta}$ regularity of viscosity solutions via a continuous-dependence result

In this article, we are interested in the existence, uniqueness, and regularity of solutions of fully nonlinear parabolic equations, with initial data $u_0$, in the whole space $\mathbb R^n$. Our main result is the existence of a strictly subquadratic solution with a local $C^{1,\beta}$ regularity with respect to the space variable, assuming $C^{1,\alpha}$ regularity on $u_0$ and local uniform ellipticity of the equation. Our proof relies on a result of N. Zhu which shows the local $C^{1,\beta}$ regularity of the solution provided it is Lipschitz continuous and H\"older continuous in t, with an exponent $ {\gamma>\frac{1}{2}}$; we obtain this last property through a continuous-dependence result. Then we investigate further regularity for the solution using results of L. Wang.

Statistical Thermodynamics of Membrane Bending-Mediated Protein–Protein Attractions

Highly wedge-shaped integral membrane proteins, or membrane-adsorbed proteins can induce long-ranged deformations. The strain in the surrounding bilayer creates relatively long-ranged forces that contribute to interactions with nearby proteins. In contrast, to direct short-ranged interactions such as van der Waal’s, hydrophobic, or electrostatic interactions, both local membrane Gaussian curvature and protein ellipticity can induce forces acting at distances of up to a few times their typical radii. These forces can be attractive or repulsive, depending on the proteins’ shape, height, contact angle with the bilayer, and a pre-existing local membrane curvature. Although interaction energies are not pairwise additive, for sufficiently low protein density, thermodynamic properties depend only upon pair interactions. Here, we compute pair interaction potentials and entropic contributions to the two-dimensional osmotic pressure of a collection of noncircular proteins. For flat membranes, bending rigidities of ∼100k B T, moderate ellipticities, and large contact angle proteins, we find thermally averaged attractive interactions of order k B T. These interactions may play an important role in the intermediate stages of protein aggregation. Numerous biological processes where membrane bending-mediated interactions may be relevant are cited, and possible experiments are discussed.

Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method

We report on a new method appropriate to the study of the polarization states of the electromagnetic field at the output of nonlinear medium. The novelty here is the measurement of the elliptical parameters of the polarization state field function of the input laser intensity. New relations were found to estimate the local ellipticity e( r) and the local polarization ellipse rotation θ( r) at the output of the medium. In order to check the reliability of the proposed method, we have confronted this one to determine the third-order nonlinear susceptibility tensor components χ xyyx 〈3〉. The comparison between the values obtained by this approach and some previous values from the literature was entirely satisfying. For instance, we have obtained the χ xyyx 〈3〉 (CS 2)=6.5×10 −20 m 2V −2 with a 10% relative error.

Multicolour CCD surface photometry for E and S0 galaxies in 10 clusters

CCD surface photometry for 232 E and S0 galaxies is presented. The galaxies are observed in Gunn r and Johnson B, or Gunn r and g. For 48 of the galaxies surface photometry in Johnson U is also presented. Aperture magnitudes in Gunn v are derived for half of the galaxies. Galaxies in the following clusters have been observed: Abell 194, Abell 539, Abell 3381, Abell 3574, Abell S639, Abell S753, HydraI (Abell 1060), DC2345 - 28, Doradus and Grm15. The data are part of our ongoing study of the large-scale motions in the Universe and the physical background for the fundamental plane. We use a full model fitting technique for analysing the CCD images. This gives radial profiles of local surface brightness, colour, ellipticity and position angle. The residuals relative to the elliptical isophotes are described quantitatively by Fourier expansions. Effective radius, mean surface brightness and total magnitude are derived by fitting a de Vaucouleurs r(1/4) growth curve. We have derived a characteristic radius r(n) similar to the diameter D-n introduced by Dressier et al. The derivation of the effective parameters and of r(n) takes the seeing into account. We confirm the results by Saglia et al. that the effects of the seeing can be substantial. Seeing-corrected Values of the effective parameters and r(n) are also presented for 147 E and S0 galaxies in the Coma cluster. Colours, colour gradients and geometrical parameters are derived. The photometry is internally consistent within 0.016 mag. Comparison with the photoelectric aperture photometry from Burstein et al. shows a mean offset of 0.010 mag with an rms scatter of 0.034 mag. The global photometric parameters are compared with data from Faber et al., Lucey et al. and Lucey and Carter. These comparisons imply that the typical rms errors are as follows - log r(n): +/- 0.015; log r(e): +/-0.045; m(T): +/-0.09 mag; [mu](e): +/-0.16 mag. The rms error on the combination log r(e)- 0.35 [mu](e) which enters the fundamental plane is +/-0.020. Also, comparisons with data from Saglia et al. are presented. The accuracy of the absolute photometry, as well as the derived parameters, makes the data suitable for our investigations of the fundamental plane and of the large-scale motions in the Universe.

Equilibrium Unfolding of Yeast Phosphoglycerate Kinase and Its Mutants Lacking One or Both Native Tryptophans: A Circular Dichroism and Steady-State and Time-Resolved Fluorescence Study

Yeast 3-phosphoglycerate kinase contains two tryptophans, both situated in the carboxy-terminal domain, and seven tyrosines, five in the amino-terminal domain, one in the domain-domain interface, and one in the carboxy-terminal domain. Site-specific mutagenesis has been used to construct two single-tryptophan mutants and one no-tryptophan mutant by replacing one or both native tryptophans, W308 and W333, with phenylalanines. The mutations have been shown to have a relatively small effect on the overall structure and enzymatic properties of the mutants. Both tryptophans are quenched in the folded state. The steady-state emission spectra and tryptophan quantum yields are the same in the single-tryptophan mutants and in the wild-type protein. Large changes in the tryptophan emission maxima and steady-state emission intensities are observed upon unfolding. Far-UV circular dichroism and steady-state as well as time-resolved fluorescence spectroscopy have been used to monitor the equilibrium unfolding transitions of these mutants and wild-type PGK. For each protein, the transitions followed by CD and steady-state fluorescence are nearly coincident, suggesting that the structural changes monitored by local fluorescence probes and ellipticity changes, which are sensitive to the changes in the overall structure, report a single cooperative transition, consistent with a two-state unfolding mechanism. Both tryptophans have three lifetimes, which follow a similar pattern as a function of denaturant concentration. The amplitude terms associated with the two longer lifetimes increase with unfolding while the short lifetime amplitude decreases. It thus appears that these population amplitudes represent markers for the unfolded and folded states, respectively.(ABSTRACT TRUNCATED AT 250 WORDS)

Probing the structure of the galactic halo with gamma rays produced by annihilations of weakly interacting massive particles.

It has recently been pointed out that the annihilations of a class of possible dark-matter halo constituents (including photinos, Higgs fermions, and heavy neutrinos or scalar neutrinos) within the halo of our galaxy produce a series of narrow gamma-ray lines. If these lines are detected, then the directional dependence of the intensity of the lines provides a very powerful probe of the halo mass distribution, and the local density, core radius, and ellipticity of the halo could be readily determined. We also briefly comment on the possibility of detecting annihilations of weakly interacting massive particles which may be in the galactic spheroid.

On the failure of ellipticity of the equations for finite elastostatic plane strain

In this paper we establish necessary and sufficient conditions, in terms of the local principal stretches, for ordinary and strong ellipticity of the equations governing finite plane equilibrium deformations of a compressible hyperelastic solid. The material under consideration is assumed to be homogeneous and isotropic, but its strain-energy density is otherwise unrestricted. We also determine the directions of the characteristic curves appropriate to plane elastostatic deformations that are accompanied by a failure of ellipticity.

Fermi surfaces in graphite-bromine, studied by Shubnikov-de Haas oscillations

Intercalation of bromine into single crystalline and well oriented pyrolytic graphite leads to the formation of several distinct, coexisting phases having compositions in the range C4600Br to C140Br. These phases are best regarded as graphitic regions of characteristic Fermi energy and dimensions, separated by condensed monolayers of Br- ions which have accepted their ionic charge from the pi electron system of the graphite matrix. Such acceptors do not markedly increase the collision broadening of the graphite Landau levels. Thus, using the transverse Shubnikov-de Haas effect, one can observe the dominant hole period which characterizes each phase of the graphite. Typically, the new hole periods are 5.27, 1.63, 1.24 and 0.21*10-6 G-1, which, using the Sloncewski-Weiss energy band model, give thicknesses for the individual tabular phases ranging from 700 to 21 AA, with corresponding Fermi energy depressions of 0.035 to 0.49 eV. In order of decreasing thickness, the Fermi surfaces of these phases evolve from the triple ellipsoids of pristine graphite through an elongated dumb-bell to an ovoid with a single extremum. The periods vary as the cosine of the angle of rotation of H away from c, confirming that in the present composition range the local ellipticity of the Fermi-surfaces remains of order 10. Landau state broadening and the relative stability of the new phases are briefly discussed.