Axisymmetric Lens Research Articles

Mechanical method to reduce aberrations for a two-sided coating axisymmetric optical lens based on the specialized finite element method.

Two-sided coated optical lenses are important in optical applications. A film-stress-induced aberration can adversely affect the lens performance. In this paper, a mechanical method has been developed to reduce this aberration. The proposed method uses a specialized finite element method with an easy modeling process and high versatility to analyze the impact of film parameters (including stress, the thickness, and the coating range) on aberrations under different lens geometric parameters. Theoretically, by selecting the property film parameters within the range of an application's requirements can reduce the aberrations. The proposed method could reduce film-stress-induced aberrations to make the aberration compensation easier.

Effect of pole piece structure on magnetic lens electrospinning system: Experimental and simulation study

Magnetic lenses can effectively reduce the fiber diameter and whipping-circle width to control the random behavior of electrospinning. Although the axisymmetric magnetic lens for electrostatic spinning have several potential applications, the effect of its pole-piece structure on the electrostatic spinning process has not been adequately discussed. In this study, we explored the influence of changing the pole piece structure of the magnetic lens on the electrospinning process using experimental and numerical simulation methods. The results showed that for a 12 wt% polyacrylonitrile (PAN) solution, when the applied voltage was 15 kV, the feed rate was 1 mL/h, and current of the magnetic lens was 1 A, the gap width of the magnetic lens had a notable effect on the fiber deposition efficiency of the electrostatic spinning range of the magnetic lens. Appropriately reducing the gap width in the range of 1–10 mm can effectively reduce the deposition range of the fibers and improve fiber film thickness. Furthermore, when the angle of α is 45°, it exerts the most significant effect on the electrostatic spinning spray range and deposition efficiency, and the fiber film thickness and average fiber diameter attain their highest values. When the angle of α increases or decreases from 45°, the fiber film thickness and fiber diameter both show a decreasing trend. A change in the magnetic lens structure can effectively change the ejection range and fiber diameter during electrostatic spinning, which shows great potential in the field of nanofiber production.

Detectability of strongly lensed gravitational waves using model-independent image parameters

Strong gravitational lensing of gravitational waves (GWs) occurs when the GWs from a compact binary system travel near a massive object. The lensed waveform is given by the product of the lensing amplification factor $F$ and the unlensed waveform. For many axisymmetric lens models such as the point mass and singular isothermal sphere that we consider, $F$ can be calculated in terms of two lens parameters, the lens mass ${M}_{L}$ and source position $y$. In the geometrical-optics approximation, lensing in these models produces at most two discrete images which can be parametrized by two image parameters, the flux ratio $I$ and time delay $\mathrm{\ensuremath{\Delta}}{t}_{d}$ between images. In the macrolensing regime for which $\mathrm{\ensuremath{\Delta}}{t}_{d}$ is large compared to the time $T$ they spend within the sensitivity band of GW detectors, it is natural to parametrize lensing searches in terms of these image parameters. The functional dependence of the lensed signal on these image parameters is far simpler, facilitating data analysis for events with modest signal-to-noise ratios, and constraints on $I$ and $\mathrm{\ensuremath{\Delta}}{t}_{d}$ can be inverted to constrain ${M}_{L}$ and $y$ for any lens model. We propose that this use of image parameters can be extended to the microlensing regime ($\mathrm{\ensuremath{\Delta}}{t}_{d}<T$) in which the two interfering images are observed as a single GW event. We use image parameters to determine the detectability of gravitational lensing in GW the microlensing regime and find that for GW events with signal-to-noise ratios $\ensuremath{\rho}$ and total mass $M$, lensing should in principle be identifiable for flux ratios $I\ensuremath{\gtrsim}2{\ensuremath{\rho}}^{\ensuremath{-}2}$ and time delays $\mathrm{\ensuremath{\Delta}}{t}_{d}\ensuremath{\gtrsim}{M}^{\ensuremath{-}1}$.

Technology for developing a high-aperture four-mirror lens with aspherical mirrors

Subject of study: Methods for optimizing the design, assembly, and alignment of a high-aperture wide-angle four-mirror axisymmetric lens are presented. Aim of study: The aim is the development of a high-tech lens design considering the relationship between the calculated tolerances for the decentering of mirrors and the technological capabilities of their fabrication, assembly, and alignment. Method: The methods include selection of an image quality criterion and calculation of its permissible decrease, distribution of the fabrication errors between the decentering errors and surface shape errors, and calculation of the tolerances based on the obtained ratios. Moreover, the motivation for choosing a lens design with a minimum number of adjustment stages based on the optimization of the calculated permissible decentering is provided considering the technological tolerances for positioning the optical elements. In addition, the possibilities of the scheme to compensate for the residual aberrations resulting from the fabrication and positioning of the mirrors are investigated. At the final stage, the method of reducing the evaluation function for the final adjustment of the optical elements in the lens is used. Main results: The design and technological solutions enabling the optimization of the requirements for the tolerances for deviation of the surface shape of the mirrors from their calculated profile, as well as decentering of the optical elements considering the technological capabilities of production, are examined. The criteria for the distribution of tolerances for permissible deviations of optical system parameters affecting the image quality are formulated. The possibility of designing a high-tech drop-in structure of a high-aperture four-mirror lens comprising a case of two blocks and a single linear adjustment stage of the second mirror to compensate for residual aberrations is demonstrated using the proposed method. Practical significance: The proposed technical solutions were tested by designing a high-resolution lens comprising four aspherical mirrors. The positive results of fabricating a lens with a high aperture and resolution allow the proposed solutions to be used for the design of multimirror axisymmetric lenses.

Evolution of an accelerated charged vortex particle in an inhomogeneous magnetic lens

We present a detailed analysis of the capture and acceleration of a nonrelativistic charged vortex particle (electron, positron, proton, etc.) with an orbital angular momentum in the field of an axisymmetric electromagnetic lens, typical for a linear accelerator. We account for the acceleration and the inhomogeneity of both electric and magnetic fields that may arise from real-life imperfections. We establish the conditions for this wave packet to be captured and successfully transported through the lens. We describe the transition process and explain how a free Laguerre-Gaussian packet can be captured into the Landau state of the lens and how its structure is preserved for all moments in time. We illustrate the developed formalism with several representative examples.

Measuring lens dimensionality in extreme scattering events through wave optics

ABSTRACT Compact radio sources have been observed to undergo large, frequency-dependent changes in intensity due to lensing by structures in the interstellar medium, in so-called ‘extreme scattering events’ (ESEs). While the study of astrophysical plasma lensing has primarily focused on the geometric limit of optics, coherent radio sources such as pulsars exhibit wave effects when lensed. The additional phase information provided by interference effects in the wave regime may yield more information about the lens than could be obtained in the geometric regime. In this paper, we show that, using wave effects, one can potentially distinguish a one-dimensional lens (where ‘one-dimensional’ includes both highly elongated lenses, as well as perfectly axisymmetric lenses) from a fully two-dimensional lens, with minimal assumptions on the form of the lensing potential.

Imaging point sources with the gravitational lens of an extended sun

We study the optical properties of the solar gravitational lens (SGL) while treating the Sun as an extended, axisymmetric and rotating body. The gravitational field of the Sun is represented using a set of zonal harmonics. We develop an analytical description of the intensity of light that is observed in the image plane in the strong interference region of a realistic SGL. This formalism makes it possible to model not only the point-spread function of point sources, but also actual observables, images that form in the focal plane of an imaging telescope positioned in the image plane. Perturbations of the monopole gravitational field of the Sun are dominated by the solar quadrupole moment, which results in forming an astroid caustic on the image plane. Consequently, an imaging telescope placed inside the astroid caustic observes four bright spots, forming the well-known pattern of an Einstein cross. The relative intensities and positions of these spots change as the telescope is moved in the image plane, with spots merging into bright arcs when the telescope approaches the caustic boundary. Outside the astroid caustic, only two spots remain and the observed pattern eventually becomes indistinguishable from the imaging pattern of a monopole lens at greater distances from the optical axis. We present results from extensive numerical simulations, forming the basis of our ongoing study of prospective exoplanet imaging with the SGL. These results are also applicable to describe a large class of gravitational lensing scenarios involving axisymmetric lenses that can be represented using zonal harmonics.

Statistical strong lensing

Context. The number of known strong gravitational lenses is expected to grow substantially in the next few years. The combination of large samples of lenses has the potential to provide strong constraints on the inner structure of galaxies. Aims. We investigate the extent to which we can calibrate stellar mass measurements and constrain the average dark matter density profile of galaxies by combining strong lensing data from thousands of lenses. Methods. We generated mock samples of axisymmetric lenses. We assume that, for each lens, we have measurements of two image positions of a strongly lensed background source, as well as magnification information from full surface brightness modelling, and a stellar-population-synthesis-based estimate of the lens stellar mass. We then fitted models describing the distribution of the stellar population synthesis mismatch parameter αsps (the ratio between the true stellar mass and the stellar-population-synthesis-based estimate) and the dark matter density profile of the population of lenses to an ensemble of 1000 mock lenses. Results. We obtain the average αsps, projected dark matter mass, and dark matter density slope with greater precision and accuracy compared with current constraints. A flexible model and knowledge of the lens detection efficiency as a function of image configuration are required in order to avoid a biased inference. Conclusions. Statistical strong lensing inferences from upcoming surveys provide a way to calibrate stellar mass measurements and to constrain the inner dark matter density profile of massive galaxies.

On the compound sessile drops: configuration boundaries and transitions

Abstract

Анализ фокусировки волн в усеченной линзе Гутмана

The problem of scattering of a plane electromagnetic wave by a truncated axisymmetric Gutman lens including calculation of the field inside the lens is considered. The problem is solved numerically using the hybrid projection method developed earlier for the general case of wave scattering by inhomogeneous bodies of revolution. New results are presented that demonstrate the accuracy of the method, as well as the field distribution in the Gutman lens, which characterize the features of wave focusing depending on the lens size, focal length, and position of the truncation plane.

Analysis of Wave Focusing by Axisymmetric Mikaelian Lenses

Numerical analysis of wave focusing by axisymmetric Mikaelian lens and by its generalized version with the focus located at some distance from the surface is carried out. The analysis is based on the hybrid projection method involving projection matching of the field expansions on a spherical surface introduced in the algorithm, projection of the Maxwell's equations for the fields inside the sphere on the transverse vector spherical functions, and use of the 1-D finite element method in projection form for solving the ordinary differential equations obtained for variable coefficients of expansions as a result of the projection. The algorithm is realized for the case of a circularly polarized plane wave incident on the lens in the axial direction. A number of new numerical results obtained for the field distributions characterizing wave focusing by Mikaelian lenses of different dimensions and different relations of their parameters are presented and discussed.

A ray-trace analysis of x-ray multilayer Laue lenses for nanometer focusing

Thick diffractive optical elements offer a promising way to achieve focusing or imaging at a resolution approaching 1 nm for x-ray wavelengths shorter than about 0.1 nm. Efficient focusing requires that these are fabricated with structures that vary in period and orientation so that rays obey Bragg’s law over the entire lens aperture and give rise to constructive interference at the focus. Here the analysis method of ray-tracing of thick diffractive optical elements is applied to such lenses to optimise their designs and to investigate their operating and manufacturing tolerances. Expressions are provided of the fourth-order series expansions of the wavefront aberrations and transmissions of both axi-symmetric lenses and pairs of crossed lenses that each focuses in only one dimension like a cylindrical lens. We find that aplanatic zone-plate designs, whereby aberrations are corrected over a large field of view, can be achieved by axi-symmetric lenses but not the crossed lenses. We investigate the performance of 1 nm-resolution lenses with focal lengths of about 1 mm and show their fields of view are mainly limited by the acceptance angle of Bragg diffraction, and that aberrations can limit the performance of lenses with longer focal lengths. We apply the ray-tracing formalism for a tolerancing analysis of imperfect lenses and examine some strategies for the correction of their aberrations.

Rotating planar gravity currents at moderate Rossby numbers: fully resolved simulations and shallow-water modelling

The flow of a gravity current of finite volume and density $\unicode[STIX]{x1D70C}_{1}$ released from rest from a rectangular lock (of height $h_{0}$) into an ambient fluid of density $\unicode[STIX]{x1D70C}_{0}$ (${<}\unicode[STIX]{x1D70C}_{1}$) in a system rotating with $\unicode[STIX]{x1D6FA}$ about the vertical $z$ is investigated by means of fully resolved direct numerical simulations (DNS) and a theoretical model (based on shallow-water and Ekman layer spin-up theories, including mixing). The motion of the dense fluid includes several stages: propagation in the $x$-direction accompanied by Coriolis acceleration/deflection in the $-y$-direction, which produces a quasi-steady wedge-shaped structure with significant anticyclonic velocity $v$, followed by a spin-up reduction of $v$ accompanied by a slow $x$ drift, and oscillation. The theoretical model aims to provide useful insights and approximations concerning the formation time and shape of wedge, and the subsequent spin-up effect. The main parameter is the Coriolis number, ${\mathcal{C}}=\unicode[STIX]{x1D6FA}h_{0}/(g^{\prime }h_{0})^{1/2}$, where $g^{\prime }=(\unicode[STIX]{x1D70C}_{1}/\unicode[STIX]{x1D70C}_{0}-1)g$ is the reduced gravity. The DNS results are focused on a range of relatively small Coriolis numbers, $0.1\leqslant {\mathcal{C}}\leqslant 0.25$ (i.e. Rossby number $Ro=1/(2{\mathcal{C}})$ in the range $2\leqslant Ro\leqslant 5$), and a large range of Schmidt numbers $1\leqslant Sc<\infty$; the Reynolds number is large in all cases. The current spreads out in the $x$ direction until it is arrested by the Coriolis effect (in ${\sim}1/4$ revolution of the system). A complex motion develops about this state. First, we record oscillations on the inertial time scale $1/\unicode[STIX]{x1D6FA}$ (which are a part of the geostrophic adjustment), accompanied by vortices at the interface. Second, we note the spread of the wedge on a significantly longer time scale; this is an indirect spin-up effect – mixing and entrainment reduce the lateral (angular) velocity, which in turn decreases the Coriolis support to the $\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}x$ slope of the wedge shape. Contrary to non-rotating gravity currents, the front does not remain sharp as it is subject to (i) local stretching along the streamwise direction and (ii) convective mixing due to Kelvin–Helmholtz vortices generated by shear along the spanwise direction and stemming from Coriolis effects. The theoretical model predicts that the length of the wedge scales as ${\mathcal{C}}^{-2/3}$ (in contrast to the Rossby radius $\propto 1/{\mathcal{C}}$ which is relevant for large ${\mathcal{C}}$; and in contrast to ${\mathcal{C}}^{-1/2}$ for the axisymmetric lens).

Dual-component plasma lens models

In contrast to the converging, achromatic behaviour of axisymmetric gravitational lenses, diverging frequency-dependent lensing occurs from refraction due to a distribution of over-dense axisymmetric plasma along an observer's line of sight. Such plasma lenses are particularly interesting from the point of view of astronomical observations because they can both magnify and dim the appearance of background sources as a function of frequency. Plasma lensing is believed to be involved in a number of separate phenomena involving the scintillation of radio pulsars, extreme scattering events of background radio sources and may also play a role in the generation of fast radio bursts. These lensing phenomena are believed to occur in dense environments, in which there may be many density perturbations between an observer and background source. In this work we generalize individual plasma lens models to produce dual component lenses using families of plasma lens models previously studied in the literature, namely the exponential and softened power-law lenses. Similar to binary gravitational lens models, these dual component plasma lenses feature a rich and complex critical and caustic morphology, as well as generate more complicated light curves. We map the number of criticals formed for a given component separation and angular size, and highlight a relevant degeneracy between two particular models. This work provides an argument in favor of close monitoring of extreme scattering events in progress in order to break such model degeneracies.

Application of coordinate measuring machines for the assembly of axisymmetric dual-mirror lenses with aspherical mirrors

Application of coordinate measuring machines for the assembly of axisymmetric dual-mirror lenses with aspherical mirrors

Limits of applicability of assembly and adjustment methods for axisymmetric two-mirror lenses with aspherical mirrors

Limits of applicability of assembly and adjustment methods for axisymmetric two-mirror lenses with aspherical mirrors

Extreme scattering events from axisymmetric plasma lenses

Frequency-dependent brightness fluctuations of radio sources, the so-called extreme scattering events (ESEs), have been observed over the last three decades. They are caused by Galactic plasma structures whose geometry and origin are still poorly understood. In this paper, we construct axisymmentric two-dimensional (2D) column density profiles for the plasma lens and explore the resulting ESEs for both point-like and extended sources. A quantity that becomes relevant is the impact parameter $b$, namely the distance of the observer's path from the lens symmetry axis. We demonstrate its effects on the shape of ESE light curves and use it for a phenomenological classification of ESEs into four main types. Three of them are unique outcomes of the 2D model and do not show a characteristic U-shaped dip in the light curve, which has been traditionally used as an identification means of ESEs. We apply our model to five well-studied ESEs and show that elongated plasma tubes or quasi-spherical clouds are favoured over plasma sheets for four of them, while the remaining one is compatible with both lens geometries.

Ambiguities in gravitational lens models: impact on time delays of the source position transformation

The central ambition of the modern time delay cosmography consists in determining the Hubble constant H0 with a competitive precision. However, the tension with H0 obtained from the Planck satellite for a spatially flat ΛCDM cosmology suggests that systematic errors may have been underestimated. The most critical of these errors probably comes from the degeneracy existing between lens models that was first formalized by the well-known mass-sheet transformation (MST). In this paper, we assess to what extent the source position transformation (SPT), a more general invariance transformation which contains the MST as a special case, may affect the time delays predicted by a model. To this aim, we have used pySPT, a new open-source python package fully dedicated to the SPT that we present in a companion paper. For axisymmetric lenses, we find that the time delay ratios between a model and its SPT-modified counterpart simply scale like the corresponding source position ratios, Δtˆ/Δt ≈ βˆ/β, regardless of the mass profile and the isotropic SPT. Similar behavior (almost) holds for nonaxisymmetric lenses in the double image regime and for opposite image pairs in the quadruple image regime. In the latter regime, we also confirm that the time delay ratios are not conserved. In addition to the MST effects, the SPT-modified time delays deviate in general no more than a few percent for particular image pairs, suggesting that its impact on time delay cosmography seems not be as crucial as initially suspected. We also reflected upon the relevance of the SPT validity criterion and present arguments suggesting that it should be reconsidered. Even though a new validity criterion would affect the time delays in a different way, we expect from numerical simulations that our conclusions will remain unchanged.

Generalised model-independent characterisation of strong gravitational lenses

In galaxy-galaxy strong gravitational lensing, Einstein rings are generated when the lensing galaxy has an axisymmetric lensing potential and the source galaxy is aligned with its symmetry centre along the line of sight. Using a Taylor expansion around the Einstein radius and eliminating the unknown source, I derive a set of analytic equations that determine differences of the deflection angle of the perturber weighted by the convergence of the axisymmetric lens and ratios of the convergences at the positions of the arcs from the measurable thickness of the arcs. In the same manner, asymmetries in the brightness distributions along an arc determine differences in the deflection angle of the perturber if the source has a symmetric brightness profile and is oriented parallel to or orthogonal to the caustic. These equations are the only model-independent information retrievable from observations to leading order in the Taylor expansion. General constraints on the derivatives of the perturbing lens are derived such that the perturbation does not change the number of critical curves. To infer physical properties such as the mass of the perturber or its position, models need to be inserted. The same conclusions about the scale of detectable masses (of the order of 108M⊙) and model-dependent degeneracies as in other approaches are then found and supported by analysing B1938+666 as an example. Yet, the model-independent equations show that there is a fundamental degeneracy between the main lens and the perturber that can only be broken if their relative position is known. This explains the degeneracies between lens models already found in simulations from a more general viewpoint. Hence, apart from the radii and brightness distributions of the arcs, independent information on the axisymmetric lens or the perturber has to be employed to disentangle the axisymmetric lens and the perturber. Depending on the properties of the pertuber, this degeneracy can be broken by characterising the surrounding of the lens or by measuring the time delay between quasar images embedded in the perturbed Einstein ring of the host galaxy.

Eikonal Distribution on the Surface of an Axisymmetric Dielectric Lens and Minimization of Aberrations

On the basis of the one-parametric theory of aberrations developed in the paper, a formula is obtained that describes the eikonal distribution on the surface of an axisymmetric dielectric lens with two focuses arbitrarily located on the axis and a source displaced from the focus in an arbitrary direction. The error of the formula is of the fourth order of the source displacement. A numerical analysis of the accuracy of the obtained formula and its expansions is performed by comparing the results of the eikonal calculation and the magnitude of aberrations on the surface of an aplanatic lens with the results of the exact geometrical optics calculation. The equation of the focal curve and the condition for the minimum of astigmatism are obtained. As an example of application of the formulas found, a wide-angle aplanate-anastigmat with optimal parameters is synthesized. The scanning characteristics of a lens antenna based on the synthesized aplanat-anastigmat are analyzed in the physical optics approximation.