  We study imaging of point sources with a quadrupole gravitational lens while
focusing on the formation and evolution of the Einstein cross formed on the
image sensor of an imaging telescope. We use a new type of a diffraction
integral that we developed to study generic, opaque, weakly aspherical
gravitational lenses. To evaluate this integral, we use the method of
stationary phase that yields a quartic equation with respect to a Cartesian
projection of the observer's position vector with respect to the vector of the
impact parameter. The resulting quartic equation can be solved analytically
using the method first published by Cardano in 1545. We find that the resulting
solution provides a good approximation of the electromagnetic (EM) field almost
everywhere in the image plane, yielding the well-known astroid caustic of the
quadrupole lens. The sole exception is the immediate vicinity of the caustic
boundary, where a numerical treatment of the diffraction integral yields better
results. We also convolve the quartic solution for the EM field on the image
plane with the point-spread function of a thin lens imaging telescope. By doing
so, we are able to explore the direct relationship between the algebraic
properties of the quartic solution for the EM field, the geometry of the
astroid caustic, and the geometry and shape of the resulting Einstein-cross
that appear on the image plane of the thin lens telescope. The new quartic
solution leads to significant improvements in numerical modeling as evaluation
of this solution is computationally far less expensive than a direct numerical
treatment of the new diffraction integral. In the case of the solar
gravitational lens (SGL), the new results drastically improve the speed of
numerical simulations related to sensitivity analysis performed in the context
of high-resolution imaging of exoplanets.
