Multipole Gravitational Lensing and High-order Perturbations on the
  Quadrupole Lens
  An arbitrary surface mass density of gravitational lens can be decomposed
into multipole components. We simulate the ray-tracing for the multipolar mass
distribution of generalized SIS (Singular Isothermal Sphere) model, based on
the deflection angles which are analytically calculated. The magnification
patterns in the source plane are then derived from inverse shooting technique.
As have been found, the caustics of odd mode lenses are composed of two
overlapping layers for some lens models. When a point source traverses such
kind of overlapping caustics, the image numbers change by \pm 4, rather than
\pm 2. There are two kinds of images for the caustics. One is the critical
curve and the other is the transition locus. It is found that the image number
of the fold is exactly the average value of image numbers on two sides of the
fold, while the image number of the cusp is equal to the smaller one. We also
focus on the magnification patterns of the quadrupole (m = 2) lenses under the
perturbations of m = 3, 4 and 5 mode components, and found that one, two, and
three butterfly or swallowtail singularities can be produced respectively. With
the increasing intensity of the high-order perturbations, the singularities
grow up to bring sixfold image regions. If these perturbations are large enough
to let two or three of the butterflies or swallowtails contact, eightfold or
tenfold image regions can be produced as well. The possible astronomical
applications are discussed.
