Multiplane gravitational lenses with an abundance of images

Abstract

We consider gravitational lensing of a background source by a finite system of point-masses. The problem of determining the maximum possible number of lensed images has been completely resolved in the single-plane setting (where the point masses all reside in a single lens plane), but this problem remains open in the multiplane setting. We construct examples of K-plane point-mass gravitational lens ensembles that produce ∏i=1K(5gi−5) images of a single background source, where gi is the number of point masses in the ith plane. This gives asymptotically (for large gi with K fixed) 5K times the minimal number of lensed images. Our construction uses Rhie’s single-plane examples and a structured parameter-rescaling algorithm to produce preliminary systems of equations with the desired number of solutions. Utilizing the stability principle from the differential topology, we then show that preliminary (nonphysical) examples can be perturbed to produce physically meaningful examples while preserving the number of solutions. We provide numerical simulations illustrating the result of our construction, including positions of lensed images and the structure of critical curves and caustics. We observe an interesting “caustic of multiplicity” phenomenon that occurs in the nonphysical case and has a noticeable effect on the caustic structure in the physically meaningful perturbative case.