Beyond the Thin Lens Approximation
  We obtain analytic formulae for the null geodesics of Friedmann-Lema\^{\i}tre-Robertson-Walker spacetimes with scalar perturbations in the longitudinal gauge. We use these to provide a rigorous derivation of the cosmological lens equation. We obtain an expression for the magnification of a bundle of light rays in these spacetimes without restriction to static or thin lens scenarios. We show how the usual magnification matrix naturally emerges in the appropriate limits. 
Ted Pyne and Mark Birkinshaw