Ellipsoidal Lens Research Articles

Gravitational lensing by an ellipsoidal Navarro–Frenk–White dark-matter halo: An analytic solution and its properties

Context. The analysis of gravitational lensing by galaxies and galaxy clusters typically relies on ellipsoidal lens models to describe the deflection of light by the involved dark-matter halos. These models are most often based on the isothermal density profile – not an optimal description of the halo, but easy to use because it leads to an analytic deflection-angle formula. Aims. Dark-matter halos are better described by the Navarro–Frenk–White (hereafter NFW) density profile. We set out to study lensing by a general triaxial ellipsoidal NFW halo, with the aim of providing an analytic model that would be more consistent with the current understanding of dark-matter halos. Methods. We computed the conversion between the properties of a triaxial ellipsoidal lens model and its elliptical surface-density profile. In the case of the NFW lens model, its angular scale is defined by the projected scale semi-major axis of the halo, while its lensing regime depends on two parameters: the projected eccentricity e and the convergence parameter κs. We employed the Bourassa & Kantowski formalism to compute the complex scattering function of the model, which yields the deflection-angle components when separated into its real and imaginary parts. Results. We present the obtained closed-form expressions for the deflection-angle components, valid for an arbitrary eccentricity of the surface-density profile. We use them to compute and describe the lensing properties of the model, including: the shear, its components, and the phase; the critical curves, caustics, and the parameter-space mapping of their different geometries; the deformations and orientations of images. Conclusions. The analytically solved ellipsoidal NFW lens model is available for implementation in gravitational lensing software. The techniques introduced here such as the image-plane analysis can prove to be useful for understanding the properties of other lens models as well.

Functional effects of the spatial-varying lens mechanical properties in accommodation

Lens biomechanical properties are critical for our eyes to accommodate. While it is well understood that lens mechanical properties change with age, different experimental techniques have been used over the years, with varying results on how the lens modulus changes. In this study, we developed a spatial-varying elasticity model to characterize the overall elastic modulus of the lens and establish its effect on accommodation. First, to validate the model, ex vivo porcine lenses underwent compression testing using biopsy punches of different diameters to change the percentage of nucleus within samples. Importantly, we found that, indeed, changing nucleus/cortex spatial ratio produces dramatic (∼7-fold) increase in overall sample modulus. Comparing the model with human lens spatial ratios, we demonstrate how changing spatial mechanics are more influential than peak modulus changes on overall elastic modulus. Next, in vivo clinical measurements of the spatial-varying lens modulus were used to generate a simplified mechanical-optical model of accommodation. We defined an ellipsoid lens with patient-derived modulus and geometry measurements, and a statics simulation and ray tracing analysis were performed through the deformed and undeformed lens. The resulting accommodation estimates agree with general accommodation expectations.

High-Spatial-Resolution Benchtop X-ray Fluorescence Imaging through Bragg-Diffraction-Based Focusing with Bent Mosaic Graphite Crystals: A Simulation Study.

X-ray fluorescence imaging (XFI) can localize diagnostic or theranostic entities utilizing nanoparticle (NP)-based probes at high resolution in vivo, in vitro, and ex vivo. However, small-animal benchtop XFI systems demonstrating high spatial resolution (variable from sub-millimeter to millimeter range) in vivo are still limited to lighter elements (i.e., atomic number Z≤45). This study investigates the feasibility of focusing hard X-rays from solid-target tubes using ellipsoidal lens systems composed of mosaic graphite crystals with the aim of enabling high-resolution in vivo XFI applications with mid-Z (42≤Z≤64) elements. Monte Carlo simulations are performed to characterize the proposed focusing-optics concept and provide quantitative predictions of the XFI sensitivity, in silico tumor-bearing mice models loaded with palladium (Pd) and barium (Ba) NPs. Based on simulation results, the minimum detectable total mass of PdNPs per scan position is expected to be on the order of a few hundred nanograms under in vivo conform conditions. PdNP masses as low as 150 ng to 50 ng could be detectable with a resolution of 600 μm when imaging abdominal tumor lesions across a range of low-dose (0.8 μGy) to high-dose (8 μGy) exposure scenarios. The proposed focusing-optics concept presents a potential step toward realizing XFI with conventional X-ray tubes for high-resolution applications involving interesting NP formulations.

Micro X-ray fluorescence device based on monocapillary ellipsoidal lens for thin film thickness measurement

X-ray fluorescence (XRF) is a crucial non-destructive technique for determining the thickness of thin films or coating materials to ensure their optimal performance. In order to further expand the application of XRF in thin film thickness analysis, we have developed an inner-wall coated elliptical monocapillary X-ray lens (EMXRL), which has a larger acceptance angle and higher utilization efficiency for light sources compared to traditional inner-wall uncoated EMXRL. In addition, we have built a laboratory micro-X-ray fluorescence (Micro-XRF) thickness measurement device based on coated EMXRL. Compared to common XRF devices, it has a smaller X-ray spot size and higher intensity gain, resulting in smaller spatial resolution and higher detection efficiency in XRF thickness measurement applications. We established a standard curve in the range from 0.85μm to 56.5μm for measuring the thickness of Cu coating on silicon using this Micro-XRF device. The experimental results show that the micro-area thickness measurement device, based on the standard curve method, can measure the thickness of Cu coatings from 0.85μm to about 30.00μm with a relative deviation of about 3%. Within the thickness range from about 30.0μm to 56.5μm, the relative deviation of the measured thickness increased and was larger than 5%. This larger relative deviation mainly resulted from the greater absorption from the thicker coating. Furthermore, We used the device to perform mapping scans on scratched areas of Cu-plated circuit board to verify its usefulness. The results show that the device has the advantages of simplicity and speed, and has potential applications in fields such as semiconductors.

Stereolithography 3D printing of transparent resin lens for high-power phosphor-coated WLEDs packaging

Advanced lens fabricating techniques are urgently required for the development of white light-emitting diodes (WLEDs) packaging, due to the traditional LED lenses molding possesses presents many disadvantages like high cost, long-fabrication period, less compactness and low accuracy. The current study demonstrated the fabrication of lenses with transparent resin utilizing stereolithography (SLA) 3D printing. The SLA 3D printing parameters were optimized to accurately regulate the spheric and ellipsoidal lenses with the smooth surface and the acceptable transmittance. The illuminances, lighting efficiency (LE) and lighting distribution curves of the SLA-fabricated lens packaged WLEDs highly depended on the lens structure. The WLEDs with lower lens height exhibited more intense lighting energy density, higher LE and wider light output angle. SLA 3D printing offers a flexible approach of fabricating WLED lens, which provides a novel technical route for WLEDs to meet different lighting requirements.

A lensed radio jet at milliarcsecond resolution I: Bayesian comparison of parametric lens models

ABSTRACT We investigate the mass structure of a strong gravitational lens galaxy at z = 0.350, taking advantage of the milliarcsecond (mas) angular resolution of very long baseline interferometric (VLBI) observations. In the first analysis of its kind at this resolution, we jointly infer the lens model parameters and pixellated radio source surface brightness. We consider several lens models of increasing complexity, starting from an elliptical power-law density profile. We extend this model to include angular multipole structures, a separate stellar mass component, additional nearby field galaxies, and/or a generic external potential. We compare these models using their relative Bayesian log-evidence (Bayes factor). We find strong evidence for angular structure in the lens; our best model is comprised of a power-law profile plus multipole perturbations and external potential, with a Bayes factor of +14984 relative to the elliptical power-law model. It is noteworthy that the elliptical power-law mass distribution is a remarkably good fit on its own, with additional model complexity correcting the deflection angles only at the ∼5 mas level. We also consider the effects of added complexity in the lens model on time-delay cosmography and flux-ratio analyses. We find that an overly simplistic power-law ellipsoid lens model can bias the measurement of H0 by ∼3 per cent and mimic flux ratio anomalies of ∼8 per cent. Our results demonstrate the power of high-resolution VLBI observations to provide strong constraints on the inner density profiles of lens galaxies.

Three-dimensional acoustic metamaterial Luneburg lenses for broadband and wide-angle underwater ultrasound imaging

This paper introduces spherical and ellipsoidal metamaterial lenses designed based on the Luneburg lens and transformation optics for underwater ultrasound imaging, which enhances signals over a broadband frequency range and detects objects at a wide-angle. The meta-atoms were analyzed with dispersion curves, equi-frequency contours, and effective acoustic parameters to design the lenses and predict the focusing performance. The three-dimensional lenses made of stainless steel were demonstrated numerically and experimentally in an underwater environment to determine the focal length and full width at half maximum. The spherical lens is omnidirectional. The focal length and the full width at half maximum of the spherical lens are 0λ and 0.6λ at 80 kHz, respectively. The ellipsoidal lens is thinner on the x-axis than that of the spherical lens. This lens has broadband properties by achieving the focal length and the full width at half maximum of 1.5λ to 0λ and 0.95λ to 0.59λ at frequencies between 60 kHz and 160 kHz, respectively. These lenses can potentially be applied to sound navigation ranging or medical transducers, providing the opportunities to view the underwater world with improved resolution at broadband and wide-angle.

Axis-symmetric ellipsoidal lens antenna design with independent E and H radiation pattern beamwidth

In this paper, a 3D ellipsoidal lens is designed based on the transformation optics (TO) method. The conventional Luneburg and Gutman lenses are mapped to an ellipsoidal lens to make a fan-beam radiation pattern. The Beamwidth of vertical and horizontal planes can be designed separately. The lens structure and its refractive index are axis-symmetric. So the fan-beam scanning can be resulted by rotating a source around the symmetric axis of the Lens in the horizontal plane. This structure can be used in search radar to estimate the azimuth angle of the targets. Full-wave simulations are performed using the software of CST.

A wide‐scanning ellipsoid lens antenna fed by phased array antenna

Luneburg lens antennas have been widely applied to many communication systems for their multibeam scanning and high gain characteristics. However, the large profiles of conventional Luneburg lens antennas restrict their applications. To solve these issues, an ellipsoid lens antenna fed by phased array antenna (PAA) operating at Ka-band is proposed in this article. The profile of the ellipsoid lens is reduced by half compared to the conventional Luneburg lens, based on the transformation optics theory. The wide-angle scanning property is obtained by optimizing the amplitude and phase distributions of the PAA, combined with the element pattern superposition principle. The proposed five-layer ellipsoid lens antenna is able to scan up to ±45° with less than 3 dB scanning loss at 28 GHz. Finally, the lens was easily fabricated through three dimensional printing technology and computer numerical control. Experimental results are in good agreement with the numerical results, thus validating the proposed design.

Gaia GraL: Gaia DR2 Gravitational Lens Systems

Aims. In this work, we aim to provide a reliable list of gravitational lens candidates based on a search performed over the entire Gaia Data Release 2 (Gaia DR2). We also aim to show that the astrometric and photometric information coming from the Gaia satellite yield sufficient insights for supervised learning methods to automatically identify strong gravitational lens candidates with an efficiency that is comparable to methods based on image processing. Methods. We simulated 106 623 188 lens systems composed of more than two images, based on a regular grid of parameters characterizing a non-singular isothermal ellipsoid lens model in the presence of an external shear. These simulations are used as an input for training and testing our supervised learning models consisting of extremely randomized trees (ERTs). These trees are finally used to assign to each of the 2 129 659 clusters of celestial objects extracted from the Gaia DR2 a discriminant value that reflects the ability of our simulations to match the observed relative positions and fluxes from each cluster. Once complemented with additional constraints, these discriminant values allow us to identify strong gravitational lens candidates out of the list of clusters. Results. We report the discovery of 15 new quadruply-imaged lens candidates with angular separations of less than 6″ and assess the performance of our approach by recovering 12 of the 13 known quadruply-imaged systems with all their components detected in Gaia DR2 with a misclassification rate of fortuitous clusters of stars as lens systems that is below 1%. Similarly, the identification capability of our method regarding quadruply-imaged systems where three images are detected in Gaia DR2 is assessed by recovering 10 of the 13 known quadruply-imaged systems having one of their constituting images discarded. The associated misclassification rate varies between 5.83% and 20%, depending on the image we decided to remove.

Gaia GraL: Gaia DR2 gravitational lens systems

Context. Multiply imaged gravitationally lensed quasars are among the most interesting and useful observable extragalactic phenomena. Because their study constitutes a unique tool in various fields of astronomy, they are highly sought, but difficult to find. Even in this era of all-sky surveys, discovering them remains a great challenge, with barely a few hundred systems currently known. Aims. We aim to discover new multiply imaged quasar candidates in the recently published Gaia Data Release 2 (DR2), which is the astrometric and photometric all-sky survey with the highest spatial resolution that achieves effective resolutions from 0.4″ to 2.2″. Methods. We cross-matched a merged list of quasars and candidates with Gaia DR2 and found 1 839 143 counterparts within 0.5″. We then searched matches with more than two Gaia DR2 counterparts within 6″. We further narrowed the resulting list using astrometry and photometry compatibility criteria between the Gaia DR2 counterparts. A supervised machine-learning method, called extremely randomized trees, was finally adopted to assign a probability of being lensed to each remaining system. Results. We report the discovery of two quadruply imaged quasar candidates that are fully detected in Gaia DR2. These are the most promising new quasar lens candidates from Gaia DR2 and a simple singular isothermal ellipsoid lens model is able to reproduce their image positions to within ~1 mas. This Letter demonstrates the discovery potential of Gaia for gravitational lenses.

Spin-flip transitions in self-assembled quantum dots

Detailed realistic calculations of the spin-flip time (T1) for an electron in a self-assembled quantum dot (SAQD) due to emission of an acoustic phonon, using only bulk properties with no fitting parameters, are presented. Ellipsoidal lens shaped As quantum dots, with electronic states calculated using 8-band strain dependent theory, are considered. The phonons are treated as bulk acoustic phonons coupled to the electron by both deformation potential and piezoelectric interactions. The dependence of T1 on the geometry of SAQD, on the applied external magnetic field and on the lattice temperature is highlighted. The theoretical results are close to the experimental measurements on the spin-flip times for a single electron in QD.

Designing an optical system of a high precision solar simulator for meteorological application

To satisfy meteorological observation and measurement needs, the general composition and optical system of a high precision solar simulator is designed to the calibration requirements for high precision solar radiation meters. Given the light source characteristics of the solar simulator, an ellipsoid convergent lens is employed so as to enhance the energy utilization ratio of the light source; the optical integrator and the collimator are optimized in design with the aid of Zemax software, resulting in improved irradiation uniformity and output beam alignment. The test results suggest that, at a working distance of 1000 mm and with an effective irradiation aperture of O60 mm, the solar simulator generates an irradiance of up to 1222 W/m2 and the irradiation nonuniformity is 0.83%, satisfying the requirements for meteorological observation and measurement.

Fabrication and shape analysis of refractive ellipsoidal lens controlled by surface tension

Ellipsoidal lens that can deliberately introduce astigmatism into the system was fabricated utilizing surface tension. A droplet of UV-curable photoresist was put on an elliptical substrate that was fabricated by 3D printing and cured by UV light when it was steady. The lens shape was calculated both analytically and numerically, indicating the relationship among lens shape, droplet volume and substrate size, which was verified by experimental data. Effects of gravity on the lens shape were finally studied by numerical simulation.

Adaptive semi-linear inversion of strong gravitational lens imaging

We present a new pixelized method for the inversion of gravitationally lensed extended source images which we term adaptive semi-linear inversion (SLI). At the heart of the method is an h-means clustering algorithm which is used to derive a source plane pixelization that adapts to the lens model magnification. The distinguishing feature of adaptive SLI is that every pixelization is derived from a random initialization, ensuring that data discretization is performed in a completely different and unique way for every lens model parameter set. We compare standard SLI on a fixed source pixel grid with the new method and demonstrate the shortcomings of the former when modeling singular power law ellipsoid (SPLE) lens profiles. In particular, we demonstrate the superior reliability and efficiency of adaptive SLI which, by design, fixes the number of degrees of freedom (NDOF) of the optimization and thereby removes biases present with other methods that allow the NDOF to vary. In addition, we highlight the importance of data discretization in pixel-based inversion methods, showing that adaptive SLI averages over significant systematics that are present when a fixed source pixel grid is used. In the case of the SPLE lens profile, we show how the method successfully samples its highly degenerate posterior probability distribution function with a single non-linear search. The robustness of adaptive SLI provides a firm foundation for the development of a strong lens modeling pipeline, which will become necessary in the short-term future to cope with the increasing rate of discovery of new strong lens systems.

Asymptotic solutions for the case of SIE lens models and application to the quadruply imaged quasar Q2237+0305

Considering a small misalignment between a point-like source, a singular isothermal ellipsoid deflector and an observer, we derive to first order simple relations between the model parameters and the lensed image positions, and an expression for the time delay between pairs of opposed images which is analogue to the one previously derived for the case of e − γ models. Combined with the first-order astrometric relations, we retrieve a simple expression for the time delays, in agreement with Witt, Mao & Keeton, which solely depends on the lensed image positions. The real advantage of using the first-order equations when dealing with symmetric gravitational lens systems is to directly test the validity of the adopted lens model without having to perform any accurate numerical fit. In this paper, we present in detail the calculations which lead to those relations between the singular isothermal ellipsoid lens model parameters and the lensed image positions. In addition, we model the well-known Einstein cross Q2237+0305 with three families of models: e − γ, singular isothermal ellipsoid and non-singular isothermal ellipsoid plus shear, using a genetic algorithm from the Qubist Optimization Toolbox. We conclude that although the non-singular isothermal ellipsoid plus shear model shows the best agreement between the calculated and the observed image positions (��x �= 0.0026 arcsec), the more simple singular isothermal ellipsoid also leads to quite satisfactory and acceptable results (��x �= 0.0059 arcsec).

Laser beam shaping with an ellipsoidal lens

This paper presents a semiconductor laser beam shaping system that can collimate the irradiance profile effectively by using an ellipsoidal lens. Geometrical optics analysis based on the ray tracing method is done and the formulas to calculate the shape of ellipsoidal lens are given. Both the theoretical and experimental result show that the laser beam system works effectively; the divergence angle is reduced to less than 1° in the fast-axial direction. By using epoxy resin, this shaper collimates a semiconductor laser beam and packages the laser diode (LD) at the same time, which simplifies the manufacturing process and greatly reduces the LD volume. Because of the small volume, low-cost, high rigidity and easy fabrication, the shaper is of great value in the field of semiconductor laser diode applications.

360° scanning multi-beam antenna based on homogeneous ellipsoidal lens fed by circular array

Presented is a multi-beam antenna which can realise 360° switching scanning in azimuth. The antenna system is composed of a homogeneous ellipsoidal lens and a planar circular tapered slot antenna (TSA) array. Each TSA element is fed with a rectangular waveguide by a special waveguide-to-TSA transition. Beam scanning can be achieved by switching between the waveguide feeders. A Ku-band Teflon ellipsoidal lens fed with 20 Fermi-TSAs has been fabricated. Measurements show that the sidelobe levels of the 20 beams are all below-17 dB and the gain differences are all within +0.4 dB.

A universal magnification theorem for higher-order caustic singularities

We prove that, independent of the choice of a lens model, the total signed magnification always sums to zero for a source anywhere in the four-image region close to swallowtail, elliptic umbilic, and hyperbolic umbilic caustics. This is a more global and higher-order analog of the well-known fold and cusp magnification relations, in which the total signed magnifications in the two-image region of the fold and the three-image region of the cusp are both always zero. As an application, we construct a lensing observable for the hyperbolic umbilic magnification relation and compare it with the corresponding observables for the cusp and fold relations using a singular isothermal ellipsoid lens. We demonstrate the greater generality of the hyperbolic umbilic magnification relation by showing how it applies to the fold image doublets and cusp image triplets and extends to image configurations that are neither. We show that the results are applicable to the study of substructure on galactic scales using observed quadruple images of lensed quasars. The magnification relations are also proven for generic one-parameter families of mappings between planes, extending their potential range of applicability beyond lensing.

Constraints on the velocity profiles of galaxies from strong lensing statistics and semi-analytical modelling of galaxy formation

Semi-analytical models of galaxy formation can be used to predict the evolution of the number density of early-type galaxies as a function of the circular velocity at the virial radius, v_{c,vir}. Gravitational lensing probability and separation distribution on the other hand are sensitive to the velocity dispersion (or circular velocity) at about the effective radius. We adopt the singular isothermal ellipsoid (SIE) lens model to estimate the velocity dispersion at the effective radius. We use radio lenses from the Cosmic Lens All-Sky Survey and the PMN-NVSS Extragalactic Lens Survey to study how the velocity dispersions, \sigma_SIE, are related to v_{c,vir}. When we include both the lensing probability and separation distribution as our lensing constraints, we find \sigma_SIE /(200\kms) = [(1.17_{-0.26}^{+0.40}) v_{c,vir}/ (200\kms)]^{0.22^{+0.05}_{-0.04}} for 200\kms \la \sigma_SIE \la 260\kms; at \sigma_SIE = 200\kms, the ratio \sqrt{2} \sigma_SIE / v_{c,vir} is about 1.65^{+0.57}_{-0.37} (68% CL) but decreases to 0.65_{-0.12}^{+0.15} (68% CL) for \sigma_SIE = 260\kms. These results are consistent with those of Seljak (2002) obtained from galaxy-galaxy weak lensing for galaxies of around L_*. However, our results clearly suggest that the ratio must vary significantly as \sigma_SIE is varied and are marginally discrepant with the Seljak results at \sigma_SIE = 260\kms. The scaling \sigma_SIE \propto v_{c,vir}^{0.22\pm 0.05} is broadly consistent with those from galaxy occupation statistics studies and the most recent galaxy-galaxy weak lensing study. (Abridged)