Compound lensing: Einstein Zig-Zags and high multiplicity lensed images
  Compound strong gravitational lensing is a rare phenomenon, but a handful of
such lensed systems are likely to be discovered in forthcoming surveys. In this
work, we use a double SIS lens model to analytically understand how the
properties of the system impact image multiplicity for the final source. We
find that up to six images of a background source can form, but only if the
second lens is multiply imaged by the first and the Einstein radius of the
second lens is comparable to, but does not exceed that of the first. We then
build a model of compound lensing masses in the Universe, using SIE lenses, and
assess how the optical depth for multiple imaging by a galaxy-galaxy compound
lens varies with source redshift. For a source redshift of 4, we find optical
depths of $6 \times 10^{-6}$ for multiple imaging and $5 \times 10^{-8}$ for
multiplicity of 6 or greater. We find that extreme magnifications are possible,
with magnifications of 100 or more for $6 \times 10^{-9}$ of $z=10$ sources
with 0.1 kpc radii. We show some of the image configurations that can be
generated by compound lenses, and demonstrate that they are qualitatively
different to those generated by single-plane lenses; dedicated compound lens
finders will be necessary if these systems are to be discovered in forthcoming
surveys.
