A Cusp—Counting Formula For Caustics Due To Multiplane Gravitational Lensing

Extract

Consider a gravitational lens system with K planes. If light rays are traced back from the observer to the light source plane, then the points on the first lens plane where a light ray either terminates, or, passes through and terminates before reaching the light source plane, are “obstruction points.” More precisely, tracing rays back to the source plane induces a K-plane lensing map η : U ⊆ R² → R² of the form η(x₁) = x₁ −∑_i=1^k α_i(x_i(x_i)). We then define an obstruction point of η to be a point a of U where lim_x1→a |α_i(x_i(x₁))| = ∞ for some “deflection angle” α_i.