The conductance through open quantum dots, or quantum billiards, shows fluctuations that can be explained as interference between waves following different paths between the leads of the billiard. We examine such systems by the use of a semiclassical Greens functions. In this paper we examine how the choice of boundary conditions at the lead mouths affects the diffraction. We derive a superior formula for the S-matrix element. Finally, we compare semiclassical simulations to quantum mechanical ones, and show that this formula yields superior results.