Wave-optical study of the Einstein cross formed by a quadrupole gravitational lens
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.