Multipole Expansion Model in Gravitational Lensing

Non-transparent models of multipole expansion model and two point-mass model are analyzed from the catastrophe theory. Singularity behaviours of $2^n$-pole moments are discussed. We apply these models to triple quasar PG1115+080 and compare with the typical transparent model, softened power law spheroids. Multipole expansion model gives the best fit among them.