Multiple Lens Systems Research Articles

Light-curve calculations for triple microlensing systems

ABSTRACT We present a method to compute the magnification of a finite source star lensed by a triple lens system based on the image boundary (contour integration) method. We describe a new procedure to obtain continuous image boundaries from solutions of the tenth-order polynomial obtained from the lens equation. Contour integration is then applied to calculate the image areas within the image boundaries, which yields the magnification of a source with uniform brightness. We extend the magnification calculation to limb-darkened stars approximated with a linear profile. In principle, this method works for all multiple lens systems, not just triple lenses. We also include an adaptive sampling and interpolation method for calculating densely covered light curves. The C++ source code and a corresponding python interface are publicly available.

‘Upper-limit lensing’: constraining galaxy stellar masses with singly imaged background sources

Strong gravitational lensing can provide accurate measurements of the stellar mass-to-light ratio $\Upsilon$ in low-redshift ($z$ $\lesssim$ 0.05) early-type galaxies, and hence probe for possible variations in the stellar initial mass function (IMF). However, true multiple imaging lens systems are rare, hindering the construction of large nearby lens samples. Here, we present a method to derive upper limits on $\Upsilon$ in galaxies with single close-projected background sources, where no counter-image is detected, down to some relative flux limit. We present a proof-of-principle application to three galaxies with integral field observations from different instruments. In our first case study, only a weak constraint on $\Upsilon$ is obtained. In the second, the absence of a detectable counter-image excludes stellar masses higher than expected for a Salpeter IMF. In the third system, the current observations do not yield a useful limit, but our analysis indicates that deeper observations should reveal a counter-image if the stellar mass is any larger than expected for a Milky Way IMF. We discuss how our method can help enlarge the current samples of low-$z$ galaxies with lensing constraints, both by adding upper limits on $\Upsilon$ and by guiding follow-up of promising single-image systems in search of fainter counter-images.

Line-of-sight effects in strong lensing: putting theory into practice

We present a simple method to accurately infer line of sight (LOS) integrated lensing effects for galaxy scale strong lens systems through image reconstruction. Our approach enables us to separate weak lensing LOS effects from the main strong lens deflector. We test our method using mock data and show that strong lens systems can be accurate probes of cosmic shear with a precision on the shear terms of ± 0.003 (statistical error) for an HST-like dataset. We apply our formalism to reconstruct the lens COSMOS 0038+4133 and its LOS. In addition, we estimate the LOS properties with a halo-rendering estimate based on the COSMOS field galaxies and a galaxy-halo connection. The two approaches are independent and complementary in their information content. We find that when estimating the convergence at the strong lens system, performing a joint analysis improves the measure by a factor of two compared to a halo model only analysis. Furthermore the constraints of the strong lens reconstruction lead to tighter constraints on the halo masses of the LOS galaxies. Joint constraints of multiple strong lens systems may add valuable information to the galaxy-halo connection and may allow independent weak lensing shear measurement calibrations.

Approach for simultaneous measurement of two-dimensional angular distribution of charged particles. III. Fine focusing of wide-angle beams in multiple lens systems

This paper provides a way of focusing wide-angle charged-particle beams in multiple lens systems. In previous papers [H. Matsuda, Phys. Rev. E 71, 066503 (2005); 74, 036501 (2006)], it was shown that an ellipsoidal mesh, combined with electrostatic lenses, enables correction of spherical aberration over wide acceptance angles up to +/-60 degrees . In this paper, practical situations where ordinary electron lenses are arranged behind the wide-angle electrostatic lenses are taken into account using ray tracing calculation. For practical realization of the wide-angle lens systems, the acceptance angle is set to +/-50 degrees . We note that the output beams of the wide-angle electrostatic lenses have somewhat large divergence angles which cause unacceptable or non-negligible spherical aberration in additional lenses. A solution to this problem is presented showing that lens combinations to cancel spherical aberration are available, whereby wide-angle charged-particle beams can be finely focused with considerably reduced divergence angles less than +/-5 degrees .

Lensing Effects of Misaligned Disks in Dark Matter Halos

In this paper we study the observational signatures of the lensing signal produced by dark matter halos with embedded misaligned disks. This issue is of particular interest at present, since several of the observed multiple-lens systems have magnification ratios and image geometries that are not simultaneously well fitted by standard mass models. The presence of substructure exterior to the lens has been invoked by several authors in the context of cold dark matter (CDM) models in order to explain the anomalous magnification ratios. We emphasize that the anomalous magnification ratios may be an artifact of the simple mass models currently in use; the inclusion of a misaligned disk may be able to mimic the effect of substructure. These slight spatial offsets between the dark matter halo and the disk, which are likely to occur as a consequence of interactions or mergers, lead to complex image configurations and nonstandard magnification ratios. We investigate the effects of disk misalignment on two illustrative lenses: a spiral disk embedded within a dark matter halo, and a compact, disklike component within an early-type galaxy. The expected fraction of galaxies with a misaligned disk is estimated to be of the order of the merger fraction, since all mergers are expected to progress via a transiently misaligned phase. We find that there is a significant change in all observed lensing properties due to the added complexity of such mass profiles. In particular cases, the resulting lensing geometries are unusual, with high image multiplicities. The caustic structures—both radial and tangential—are drastically modified, and the magnification ratios differ fromexpectations from standard lens models. The additional parameters required to specify the relative alignment of multiple mass components in the primary lens introduce yet another source of uncertainty in the mass modeling of gravitational lens systems.

Astrometric microlensing: a channel to detect multiple lens systems

If a source star is gravitationally microlensed by a multiple lens system, the resulting light curve can have significant deviations from the standard form of a single lens event. The chance of producing significant deviations becomes important when the separations between the component lenses are equivalent to the combined angular Einstein ring radius of the system. For multiple lens systems composed of more than two lenses, however, this condition is difficult to meet because the orbits of such systems are unstable. Even if events are caused by a multiple lens system with stable orbits where a pair of lenses are closely located and the other component (a third body) has a wide separation from the pair, identifying the lens multiplicity photometrically will be difficult because the event will be identified by either a binary lens event caused by the close pair of lenses or a single lens event caused by the third body. In this paper, we show that if a seemingly binary lens event is followed up astrometrically using future high-precision interferometers, the existence of an additional third body can be identified via a repeating event. We show that the signatures of third bodies can be unambiguously identified from the characteristic distortions they make in the centroid shift trajectories. We also show that owing to the long-range astrometric effect of third bodies, the detection efficiency will be considerable even for third bodies with large separations from their close lens pairs.

Strong gravitational lensing by multiple galaxies

We discuss strong gravitational lensing by multiple objects along any line of sight. The probability for strong gravitational lensing by more than one lens is small, but a number of strong lens systems in which more than one separate lens contribute significantly to the lensing potential will be detected in the large sample of lens systems compiled with new instruments. Using multi-lens ray-tracing, we estimate the likelihood for gravitational lensing by two lenses at different redshifts and investigate typical image geometries and magnification cross-sections. We find that, for a cosmology with Omega (M) = 0.3 and Omega (Lambda) = 0.7, about one in 20 lens systems consists of two lenses with merging caustics. Multiple lens systems differ from single lenses as the presence of a second lens in close proximity along the line of sight leads to a strongly asymmetric potential, which increases the multiple imaging cross-section and significantly changes the image configuration. The external shear induced by a second nearby galaxy, group or cluster can significantly affect image positions even for more widely separated lens pairs. Both of these effects must be accounted for in lens modelling. We also show how the presence of aligned discs in the pair of lensing galaxies can lead to very large high-magnification cross-sections. Lensing by more than one galaxy along the line of sight can lead to interesting image configurations. Such systems will be important in future, both for constraining lens models of individual systems and for statistical lensing.

Analysis of aggregate surgically induced refractive change, prediction error, and intraocular astigmatism

Purpose To demonstrate analytical methods for evaluating the results of keratorefractive surgical procedures and emphasize the importance of intraocular astigmatism. Setting University of Texas Medical School, Houston, Texas, USA. Methods A standard data set, provided by an editor of this journal, comprising the preoperative and postoperative keratometric and refractive measurements of 100 eyes that had keratorefractive surgery was evaluated by 2 methods, vector and spheroequivalent (SEQ) analysis. The individual and aggregate surgically induced refractive changes (SIRCs) and prediction errors were determined from the refractive and keratometric measurements using both methods and then compared. The refraction vertex distance, keratometric index of refraction, and corneal asphericity were used to make the results calculated from refractive data directly comparable to those derived from keratometric data. Doubled-angle and equivalency plots as well as frequency and cumulative histograms were used to display the data. Standard descriptive statistics were used to determine the mean and standard deviation of the aggregate induced astigmatism after converting the polar values (cylinder and axis) to Cartesian ( x and y) values. Results The preoperative SEQ refractive errors were undercorrected by at least 0.25 diopter (D) in most cases (78%). Six percent were corrected within ± 0.24 D, and 16% were overcorrected by at least 0.25 D SEQ. The mean SEQ was −6.68 D ± 2.49 (SD) before and −0.61 ± 0.82 D after surgery, reflecting a SIRC SEQ of −6.07 ± 2.40 D. The defocus equivalent (DEQ) was 7.41 ± 2.53 D before and 0.96 ± 0.74 D after surgery; for a nominal 3.0 mm pupil, this corresponded to an estimated improvement in uncorrected visual acuity (UCVA) from worse than 20/200 to better than 20/25, respectively. The predictability of the treatment decreased as the attempted refractive correction increased. The average magnitude of the refractive astigmatism was 1.46 ± 0.61 D before and 0.40 ± 0.38 D after surgery. The centroid of the refractive astigmatism was +0.96 × 87.9 ± 0.85 D, ρ = 0.43 before and +0.11 × 83.1 ± 0.37, ρ = 0.49 after surgery. The decrease in the square root of the centroid standard deviation shape factor (ρ 1/2) indicated an 8% increase in the amount of oblique astigmatism in the population. The prevalence of preoperative keratometric irregular astigmatism in excess of 0.5 D in this group of patients was 13%. The correlation between keratometric and refractive astigmatism was extremely poor before ( r 2 = 0.26) and especially after surgery ( r 2 = 0.02), demonstrating the presence of intraocular astigmatism and the limitations of manual keratometry. The centroid of intraocular astigmatism at the corneal plane was +0.48 × 178 ± 0.49 D, ρ = 0.59, and was compensatory. Conclusions The 2 analytical methods are complimentary and permit thorough and quantitative evaluation of SIRCs and allow valid statistical comparisons within and between data sets. The DEQ allows comparison of refractive and visual results. The decrease in refractive predictability with higher corrections is well demonstrated by the SEQ and doubled-angle plots of the SIRC. Doubled-angle plots were particularly useful in interpreting errors of cylinder treatment amount and errors in alignment. The correlation between refractive and keratometric astigmatism was poor for preoperative, postoperative, and SIRC data, indicating the presence of astigmatic elements beyond the corneal surface (ie, intraocular astigmatism). Sources of error in refractive outcome statistics include the use of multiple lens systems in the phoropter, errors in vertex calculations, difficulty in accurately defining the axis of astigmatism, and failure to consider measurement errors when working with keratometric data. The analysis of this particular data set demonstrates the significant clinical benefits of refractive surgery: an 8-fold increase in UCVA, an 11-fold decrease in SEQ refractive error, as well as a 9-fold and nearly a 2 1/2-fold decrease in the magnitude and distribution of astigmatism, respectively.

First-order optimization of multiple magnetic lens systems for transport and focusing of annular ion beams

A standard matrix method for tracing rays through an ion optical system is extended to include a recently discovered [W. Z. Liu and F. D. Bechetti, Rev. Sci. Instrum. 60, 1228 (1989)] defocusing solenoidal lens. The results of the matrix approach are shown to correspond with results obtained by numerical integration of particle trajectories through real coil configurations. First-order radial miss distances due to radial divergence of the beam, chromatic aberration of the lens system, and physical extent of the source are compared for one and two lens systems using the matrix formulation. It is demonstrated that a two lens system in a focusing-defocusing arrangement can simultaneously correct to first order for chromatic aberration and beam divergence, producing a better focus for thin annular beams with significant energy spread and divergence. A new calculation of the focal length for the negative solenoidal lens is also included.

The design and test results for an acoustic lens with elliptic surfaces

A design of a nonspherical acoustic lens and its test results are presented. The lens has elliptic surfaces to eliminate on-axis spherical aberration. The lens was designed with the aid of third-order aberration theory. Acoustic lenses with nonspherical surfaces are easily manufactured and should prove to be a valuable addition to single or multiple element lens systems.