The Casimir Energy for a Hyperboloid Facing a Plate in the Optical   Approximation
  We study the Casimir energy of a massless scalar field that obeys Dirichlet boundary conditions on a hyperboloid facing a plate. We use the optical approximation including the first six reflections and compare the results with the predictions of the proximity force approximation and the semi-classical method. We also consider finite size effects by contrasting the infinite with a finite plate. We find sizable and qualitative differences between the new optical method and the more traditional approaches. 
O. Schroeder, A. Scardicchio and R. L. Jaffe