Geometric Limit Research Articles

Extending the Depth of Field beyond Geometrical Imaging Limitations Using Phase Noise as a Focus Measure in Multiwavelength Digital Holography

Digital holography is a well-established technology for optical quality control in industrial applications. Two common challenges in digital holographic measurement tasks are the ambiguity at phase steps and the limited depth of focus. With multiwavelength holography, multiple artificial wavelengths are used to extend the sensor’s measurement range up to several millimeters, allowing measurements on rough surfaces. To further extend the unambiguous range, additional highly stabilized and increasingly expensive laser sources can be used. Besides that, unwrapping algorithms can be used to overcome phase ambiguities—but these require continuous objects. With the unique feature of numerical refocusing, digital holography allows the numerical generation of an all-in-focus unambiguous image. We present a shape-from-focus algorithm that allows the extension of the depth of field beyond geometrical imaging limitations and yields unambiguous height information, even across discontinuities. Phase noise is used as a focus criterion and to generate a focus index map. The algorithm’s performance is demonstrated at a gear flank with steep slopes and a step sample with discontinuities far beyond the system’s geometrical limit. The benefit of this method on axially extended objects is discussed.

Silicon Thin-Film Solar Cells Approaching the Geometric Light-Trapping Limit: Surface Texture Inspired by Self-Assembly Processes

A new device design of microcrystalline silicon thin-film solar cell allows for approaching the geometric light-trapping limit. The solar cell is based on triangular textured surfaces in combination with optimized front and back contacts with very low optical losses. In comparison to crystalline silicon solar cells with record energy conversion efficiency the material usage of the thin-film solar cells is reduced to 1–2%, while exhibiting the potential to achieve short circuit current densities of more than 80% of their counterparts. The short circuit current density of the thin-film solar cells is approaching the geometric light-trapping limit commonly known as the Yablonovitch limit under perpendicular incidence. The design of the solar cell is described considering the electrical and optical properties of the textured solar cell.

Growth competition between columnar dendritic grains – Cellular automaton versus phase field modeling

Cellular Automaton (CA) simulations of two-dimensional growth competition among columnar dendritic grains are carried out for a succinonitrile - 0.4 wt% acetone alloy. This is achieved by computing the Grain Boundary (GB) orientation during directional solidification of a bi-crystal in a frozen temperature gradient approximation, each crystal being defined by its own orientation. Comparisons are subsequently conducted with recent Phase Field (PF) results derived under the same conditions as well as with the Geometrical Limit (GL) criterion and the Favorably Orientated Grain (FOG) criterion. The GL criterion is defined mathematically considering infinitely small branching within each grain in directions perpendicular to the main dendrite arms growth directions. The FOG criterion states survival of the grain having the growth direction best aligned with the temperature gradient. The GB orientation is investigated by CA simulations as a function of the cell size, the cell neighborhood and the position used to compute the growth velocity. Results reveal that sufficiently small cells lead to the convergence of the GB orientation towards the GL criterion, while sufficiently large cells lead to the FOG criterion. Within a range of intermediate cell size, excellent agreement is found with a revised version of the FOG criterion (rev-FOG) extracted from PF simulations over a wide range of grain orientations. The cell size needs to be of the order of the maximum step between primary stationary dendrite tips of the two competing grains. The Moore neighborhood provides better results than the von Neumann neighborhood. Noticeable improvement is also observed when computing the growth velocity at the leading dendrite tip positions compared to using the cell center approximation. With computational times several orders of magnitudes lower than PF, the CA method offers a realistic and useful alternative for direct simulations of solidification grain structures in casting processes. This work is also an example of upscaling between models, showing how PF dedicated to model phenomena at the scale of the solid-liquid interface and sidebranching competition can be used to evaluate and calibrate CA developed for large scale simulations.

Spatial super-resolution of colored images by micro mirrors

In this paper, we present two methods of dealing with the geometric resolution limit of color imaging sensors. It is possible to overcome the pixel size limit by adding a digital micro-mirror device component on the intermediate image plane of an optical system, and adapting its pattern in a computerized manner before sampling each frame. The full RGB image can be reconstructed from the Bayer camera by building a dedicated optical design, or by adjusting the demosaicing process to the special format of the enhanced image.

Limits of Geometries

A geometric transition is a continuous path of geometric structures that changes type, meaning that the model geometry, i.e., the homogeneous space on which the structures are modeled, abruptly changes. In order to rigorously study transitions, one must define a notion of geometric limit at the level of homogeneous spaces, describing the basic process by which one homogeneous geometry may transform into another. We develop a general framework to describe transitions in the context that both geometries involved are represented as sub-geometries of a larger ambient geometry. Specializing to the setting of real projective geometry, we classify the geometric limits of any sub-geometry whose structure group is a symmetric subgroup of the projective general linear group. As an application, we classify all limits of three-dimensional hyperbolic geometry inside of projective geometry, finding Euclidean, Nil, and Sol geometry among the limits. We prove, however, that the other Thurston geometries, in particular H 2 × R \mathbb {H}^2 \times \mathbb {R} and S L 2 R ~ \widetilde {\mathrm {SL}_2\mathbb {R}} , do not embed in any limit of hyperbolic geometry in this sense.

Vacuum polarization in Siklos spacetimes

We study the effect of one-loop vacuum polarization on photon propagation in Siklos spacetimes in the geometric optics limit. We show that for photons with a general polarization in the transverse plane, the quantum correction vanishes in spacetimes with $H_{xy}=0$. For photons polarized along a transverse axis, subluminal and superluminal solutions are admitted for certain subclasses of Siklos spacetimes. We investigate the results in the Kaigorodov and Defrise spacetimes and obtain explicit expressions for the phase velocities. In Kaigorodov spacetime with $H\sim x^3$, photons polarized along $x$-axis are subluminal in regions where $H$ is positive, and superluminal in regions where $H$ is negative, while photons polarized along $y$-axis are superluminal in $H>0$ regions and subluminal in $H<0$ regions. In Defrise spacetime, $H\sim x^{-2}$, $x$-polarized and $y$-polarized photons are superluminal for $H<0$ and subluminal for $H>0$. We comment on motion in other Siklos spacetimes.

Two families of astrophysical diverging lens models

In the standard gravitational lensing scenario, rays from a background source are bent in the direction of a foreground lensing mass distribution. Diverging lens behaviour produces deflections in the opposite sense to gravitational lensing, and is also of astrophysical interest. In fact, diverging lensing due to compact distributions of plasma has been proposed as an explanation for the extreme scattering events (ESEs) that produce frequency-dependent dimming of extra-galactic radio sources, and may also be related to the refractive radio-wave phenomena observed to affect the flux density of pulsars. In this work we study the behaviour of two families of astrophysical diverging lenses in the geometric optics limit, the power-law and the exponential plasma lenses. Generally, the members of these model families show distinct behaviour in terms of image formation and magnification, however the inclusion of a finite core for certain power-law lenses can produce a caustic and critical curve morphology that is similar to the well-studied Gaussian plasma lens. Both model families can produce dual radial critical curves, a novel distinction from the tangential distortion usually produced by gravitational (converging) lenses. The deflection angle and magnification of a plasma lens varies with the observational frequency, producing wavelength-dependent magnifications that alter the amplitudes and the shape of the light curves. Thus, multi-wavelength observations can be used to physically constrain the distribution of the electron density in such lenses.

Statistical Detection of the He ii Transverse Proximity Effect: Evidence for Sustained Quasar Activity for &gt;25 Million Years

The reionization of helium at z~3 is the final phase transition of the intergalactic medium and supposed to be driven purely by quasars. The HeII transverse proximity effect - enhanced HeII transmission in a background sightline caused by the ionizing radiation of a foreground quasar - therefore offers a unique opportunity to probe the morphology of HeII reionization and to investigate the emission properties of quasars, e.g. ionizing emissivity, lifetime and beaming geometry. We use the most-recent HST/COS far-UV dataset of 22 HeII absorption spectra and conduct our own dedicated optical spectroscopic survey to find foreground quasars around these HeII sightlines. Based on a set of 66 foreground quasars, we perform the first statistical analysis of the HeII transverse proximity effect. Despite a large object-to-object variance, our stacking analysis reveals an excess in the average HeII transmission near the foreground quasars at 3 sigma significance. This statistical evidence for the transverse proximity effect is corroborated by a clear dependence of the signal strength on the inferred HeII ionization rate at the background sightline. Our detection places, based on the transverse light crossing time, a geometrical limit on the quasar lifetime of t_Q > 25 Myr. This evidence for sustained activity of luminous quasars is relevant for the morphology of HI and HeII reionization and helps to constrain AGN triggering mechanisms, accretion physics and models of black hole mass assembly. We show how future modeling of the transverse proximity effect can additionally constrain quasar emission geometries and e.g. clarify if the large observed object-to-object variance can be explained by current models of quasar obscuration.

Numerical simulation of classical and rotating detonation waves in methane mixtures

A numerical simulations of a two-dimensional multi-front irregular structure of the detonation wave (DW) in methane-based mixtures at normal initial condition have been conducted. The computations have been performed in a wide range of channel height. From the analysis of the flow structure and the number of primary transverse waves in channel, the dominant size of the detonation cell for studied mixtures has been determined. We have simulated the cellular front structure in stoichiometric methane–air and methane–oxygen mixtures, and in a rich (equivalence ratio φ = 1.5) methane-air mixture. Based on the fundamental studies of front structure of the classical propagating DW in methane mixtures, numerical simulation of continuous spin detonation of rich (ϕ = 1.2) methane-oxygen mixture has been carried out in the cylindrical detonation chamber (DC) of the rocket-type engine. We studied the global flow structure in DC, and the detailed structure of the front of the continuous rotating DW. Integral characteristics of the detonation process - the distribution of average values of static and total pressure along the length of the DC, and the value of specific impulse have been obtained. The geometric limit of stable existence of rotating DW has been determined.

Statistical Detection of the He ii Transverse Proximity Effect: Evidence for Sustained Quasar Activity for &gt;25 Million Years

Abstract The He ii transverse proximity effect—enhanced He ii transmission in a background sightline caused by the ionizing radiation of a foreground quasar—offers a unique opportunity to probe the morphology of quasar-driven He ii reionization. We conduct a comprehensive spectroscopic survey to find quasars in the foreground of 22 background quasar sightlines with Hubble Space Telescope/COS He ii transmission spectra. With our two-tiered survey strategy, consisting of a deep pencil-beam survey and a shallow wide-field survey, we discover 131 new quasars, which we complement with known SDSS/BOSS quasars in our fields. Using a restricted sample of 66 foreground quasars with inferred He ii photoionization rates greater than the expected UV background at these redshifts ( ) we perform the first statistical analysis of the He ii transverse proximity effect. Our results show qualitative evidence for a large object-to-object variance: among the four foreground quasars with the highest only one (previously known) quasar is associated with a significant He ii transmission spike. We perform a stacking analysis to average down these fluctuations, and detect an excess in the average He ii transmission near the foreground quasars at significance. This statistical evidence for the transverse proximity effect is corroborated by a clear dependence of the signal strength on . Our detection places a purely geometrical lower limit on the quasar lifetime of . Improved modeling would additionally constrain quasar obscuration and the mean free path of He ii-ionizing photons.

Influence of spectral particle properties on radiative heat transfer in optically thin and thick media of fluidized bed combustors

In this study, influence of spectral particle properties on radiative heat transfer in the freeboard/dilute zone of fluidized bed combustors (FBCs) is investigated. The aim is to identify how important it is to involve spectral particle properties and to determine the predictive accuracy of gray Mie and gray Geometric Optics (GOA) approximations in optically thin and thick media by benchmarking their predictions against spectral solutions. For that purpose, input data required for modelling radiative heat transfer and validating its predictions are provided from three combustion tests carried out in lignite-fired 300 kWt Atmospheric Bubbling Fluidized Bed Combustor (ABFBC) and 150 kWt Circulating Fluidized Bed Combustor (CFBC) test rigs. The results show that gray particle assumption leads to accurate radiative heat flux predictions with one order of magnitude less CPU time for both optically thin and thick media while it gives acceptable accuracy in source term predictions for only optically thin medium and the error becomes significant as the optical thickness increases. Assessment of GOA in FBCs, on the other hand, reveals that applicability limit of GOA should be based on cumulative cross sectional area distribution rather than surface mean diameter or cumulative weight distribution of particles. If the majority of cumulative cross sectional area is constituted by large particles which fall into geometric optics limit, gray GOA yields satisfactory results compared to spectral solution even if the medium is optically thick without going through cumbersome spectral calculations.

Estimating the effects of an ellipsoidal earth and topography on infrasonic propagation

Infrasonic signals often propagate significant horizontal distances so that predictions obtained using a flat ground approximation can introduce inaccuracies. Simulations of propagation in an atmospheric layer around a spherical globe have shown non-negligible deviations from flat ground predictions for both arrival locations and propagation times. Simulation predictions for flat-ground and spherical earth models will be discussed and the additional challenges of implementing a non-spherical globe model and the inclusion of topography will be discussed using the approximation of geometric acoustics. A non-spherical globe model, such as the WGS84 ellipsoid, is found to produce range dependence in the propagation medium, even in the case that a stratified local atmosphere is assumed. Further, although scattering and diffraction effects are not included in the geometric limit, variations in the ground surface level can be included in ray path computation to more accurately model propagation of infrasound. Propagation effects will be detailed in the case of a tropospheric waveguides for which interaction with the ground surface is significant as well as the case that source and receiver locations have differences in elevation.

Geometrical optics limit of phonon transport in a channel of disclinations

We show that a simple model of 1D array of topological defects in a crystalline environment is able to guide acoustic waves, depending on the angle of the incident wave. We comment on a recently proposed geophysical mechanism explaining the mantle dynamics by the presence, in some crystalline materials there, of wedge disclinations.

Guiding of 4 MeV C+ and C4+ ion beams using cylindrical glass channel

To investigate how the initial charge state affects the transmission of 4 MeV C+ and C4+ ion beams through a channel, we measured the transmission probability and kinetic energy of atoms and ions that pass through a cylindrical channel in glass, such as a narrow gap between a cylindrical convex glass surface and a cylindrical concave glass surface. Kinetic energy distributions were measured at three typical observation angles φ with the cylindrical glass channel tilted at angles θ = −3, −2, −1, 0, 1, 2, and 3° with respect to the incident ion beam. The ion beam is guided in both initial charge states; that is, the transmission maintains the initial kinetic energy at tilt angles greater than the geometric limit. However, no marked difference in transmission appears between the two initial charge states.

Interactions of calcium with the external surfaces of fullerenes and endofullerenes doped with radioactive sodium iodide.

We report first-principles calculations carried out to analyze the adsorption of calcium on the outer surface of the fullerene C60, yielding [C60 + mCa]. Geometric optimization (GO) and molecular dynamics (MD) simulation were performed using the plane-wave pseudopotential method within the framework of density functional theory (DFT) and time-dependent DFT (TD-DFT) to investigate the configurations, the associated energies in the ground state, and the stabilities of fullerenes and endofullerenes doped with radioactive sodium iodide when they interact with calcium atoms on the outer fullerene surface (i.e., [nNa131I@C60 + mCa]). The reason for investigating these calcium-functionalized (endo)fullerene systems was to gauge their potential stability when used as vectors to deliver radioiodine to cancerous tissue in the human body. In the simulations, we found that the geometric limit on the number of calcium atoms that can be physisorbed on the outer surface of an empty fullerene while maintaining its structural stability is 28 calcium atoms, which also takes into account the proportional expansion of the fullerene as the number of absorbed calcium atoms increases. However, the stability of a fullerene system during calcium adsorption also strongly depends on whether any atoms or molecules are being encapsulated by the fullerene, as these encapsulated atoms/molecules can also interact with the fullerene and influence its stability. A Mulliken electronegativity analysis revealed that, when atoms inside and/or outside the fullerene donate charge (electrons) to the fullerene, the fullerene expands. The excess charge on the carbon atoms of the fullerene weakens some of the carbon-carbon bonds, potentially causing them to break, in which case the fullerene loses its ability to encapsulate molecules and releases them. Graphical Abstract DFT simulation of a endo fullerene doped with radioactive sodium iodide interacting with 28 calcium atoms in a geometric arrangement.

Volume bounds for weaving knots

Weaving knots are alternating knots with the same projection as torus knots, and were conjectured by X.-S. Lin to be among the maximum volume knots for fixed crossing number. We provide the first asymptotically correct volume bounds for weaving knots, and we prove that the infinite weave is their geometric limit.

Essential twisted surfaces in alternating link complements

Checkerboard surfaces in alternating link complements are used frequently to determine information about the link. However, when many crossings are added to a single twist region of a link diagram, the geometry of the link complement stabilizes (approaches a geometric limit), but a corresponding checkerboard surface increases in complexity with crossing number. In this paper, we generalize checkerboard surfaces to certain immersed surfaces, called twisted checkerboard surfaces, whose geometry better reflects that of the alternating link in many cases. We describe the surfaces, show that they are essential in the complement of an alternating link, and discuss their properties, including an analysis of homotopy classes of arcs on the surfaces in the link complement.

Ligament Mediated Fragmentation of Viscoelastic Liquids.

The breakup and atomization of complex fluids can be markedly different than the analogous processes in a simple Newtonian fluid. Atomization of paint, combustion of fuels containing antimisting agents, as well as physiological processes such as sneezing are common examples in which the atomized liquid contains synthetic or biological macromolecules that result in viscoelastic fluid characteristics. Here, we investigate the ligament-mediated fragmentation dynamics of viscoelastic fluids in three different canonical flows. The size distributions measured in each viscoelastic fragmentation process show a systematic broadening from the Newtonian solvent. In each case, the droplet sizes are well described by Gamma distributions which correspond to a fragmentation-coalescence scenario. We use a prototypical axial step strain experiment together with high-speed video imaging to show that this broadening results from the pronounced change in the corrugated shape of viscoelastic ligaments as they separate from the liquid core. These corrugations saturate in amplitude and the measured distributions for viscoelastic liquids in each process are given by a universal probability density function, corresponding to a Gamma distribution with n_{min}=4. The breadth of this size distribution for viscoelastic filaments is shown to be constrained by a geometrical limit which can not be exceeded in ligament-mediated fragmentation phenomena.

Envelope equation for the linear and nonlinear propagation of an electron plasma wave, including the effects of Landau damping, trapping, plasma inhomogeneity, and the change in the state of wave

This paper addresses the linear and nonlinear three-dimensional propagation of an electron wave in a collisionless plasma that may be inhomogeneous, nonstationary, anisotropic, and even weakly magnetized. The wave amplitude, together with any hydrodynamic quantity characterizing the plasma (density, temperature, etc.) is supposed to vary very little within one wavelength or one wave period. Hence, the geometrical optics limit is assumed, and the wave propagation is described by a first order differential equation. This equation explicitly accounts for three-dimensional effects, plasma inhomogeneity, Landau damping, and the collisionless dissipation and electron acceleration due to trapping. It is derived by mixing results obtained from a direct resolution of the Vlasov-Poisson system and from a variational formalism involving a nonlocal Lagrangian density. In a one-dimensional situation, abrupt transitions are predicted in the coefficients of the wave equation. They occur when the state of the electron plasma wave changes, from a linear wave to a wave with trapped electrons. In a three dimensional geometry, the transitions are smoother, especially as regards the nonlinear Landau damping rate, for which a very simple effective and accurate analytic expression is provided.

Numerical modelling of continuous spin detonation in rich methane-oxygen mixture

A numerical simulation of a two-dimensional structure of the detonation wave (DW) in a rich (equivalence ratio φ=1.5) methane-air mixture at normal initial condition has been conducted. The computations have been performed in a wide range of channel heights. From the analysis of the flow structure and the number of primary transverse waves in the channel, the dominant size of the detonation cell for studied mixture has been determined to be 45÷50 cm. Based on the fundamental studies of multi-front (cellular) structure of the classical propagating DW in methane mixtures, numerical simulation of continuous spin detonation (CSD) of rich (φ=1.2) methane-oxygen mixture has been carried out in the cylindrical detonation chamber (DC) of the rocket-type engine. We studied the global flow structure in DC, and the detailed structure of the front of the rotating DW. Integral characteristics of the detonation process - the distribution of average values of static and total pressure along the length of the DC, and the value of specific impulse have been obtained. The geometric limit of stable existence of CSD has been determined.