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Compound strong gravitational lensing is a rare phenomenon, but a handful ofsuch lensed systems are likely to be discovered in forthcoming surveys. In thiswork, we use a double SIS lens model to analytically understand how theproperties of the system impact image multiplicity for the final source. Wefind that up to six images of a background source can form, but only if thesecond lens is multiply imaged by the first and the Einstein radius of thesecond lens is comparable to, but does not exceed that of the first. We thenbuild a model of compound lensing masses in the Universe, using SIE lenses, andassess how the optical depth for multiple imaging by a galaxy-galaxy compoundlens varies with source redshift. For a source redshift of 4, we find opticaldepths of $6 \times 10^{-6}$ for multiple imaging and $5 \times 10^{-8}$ formultiplicity of 6 or greater. We find that extreme magnifications are possible,with magnifications of 100 or more for $6 \times 10^{-9}$ of $z=10$ sourceswith 0.1 kpc radii. We show some of the image configurations that can begenerated by compound lenses, and demonstrate that they are qualitativelydifferent to those generated by single-plane lenses; dedicated compound lensfinders will be necessary if these systems are to be discovered in forthcomingsurveys.

Compound lensing: Einstein Zig-Zags and high multiplicity lensed images