We study the optical properties of an oblate gravitational lens, such as the solar gravitational lens, which, in addition to a monopole, is characterized by the presence of a small quadrupole zonal harmonic. We obtain a new type of diffraction integral using our recently developed angular eikonal method. We evaluate this integral using the method of stationary phase. The resulting quartic equation can be solved algebraically using the method first published by Cardano in 1545. We find that the resulting solution provides a good approximation to the electromagnetic 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 with the point-spread function of a thin-lens optical telescope. We explore the direct relationship between the algebraic properties of the quartic, the geometry of the astroid caustic, and the geometry and shape of the resulting Einstein-cross that appear on the image sensor of the model telescope. This leads to improvements in numerical simulations as the quartic solution is computationally far less expensive than numerical integration.
تحميل البحث