Statistics of Gravitational Microlensing Magnification. II.   Three-Dimensional Lens Distribution
  In Paper I we studied the theory of gravitational microlensing for a planar distribution of point masses. In this second paper, we extend the analysis to a three-dimensional lens distribution. First we study the lensing properties of three-dimensional lens distributions by considering in detail the critical curves, the caustics, the illumination patterns, and the magnification cross-sections sigma(A) of multiplane configurations with two, three, and four point masses. For N_\ast point masses that are widely separated in Lagrangian space, we find that there are ~ 2^{N_\ast} - 1 critical curves in total, but that only ~ N_\ast of these produce prominent caustic-induced features at the high magnification end of sigma(A).   In the case of a low optical depth random distribution of point masses, we show that the multiplane lens equation near a point mass can be reduced to the single plane equation of a point mass perturbed by weak shear. This allows us to calculate the caustic-induced feature in the macroimage magnification distribution P(A) as a weighted sum of the semi-analytic feature derived in Paper I for a planar lens distribution. The resulting semi-analytic caustic-induced feature is similar to the feature in the planar case, but it does not have any simple scaling properties, and it is shifted to higher magnification. The semi-analytic distribution is compared to the results of previous numerical simulations for optical depth tau ~ 0.1, and they are in better agreement than a similar comparison in the planar case. We explain this by estimating the fraction of caustics of individual lenses that merge with those of their neighbors. For tau=0.1, the fraction is approx 20%, much less than the approx 55% for the planar case. 
Man Hoi Lee, Arif Babul, Lev Kofman, and Nick Kaiser