  We derive contributions to the trace formula for the spectral density
accounting for the role of diffractive orbits in two-dimensional polygonal
billiards. In polygons, diffraction typically occurs at the boundary of a
family of trajectories. In this case the first diffractive correction to the
contribution of the family to the periodic orbit expansion is of order of the
one of an isolated orbit, and gives the first $\sqrt{\hbar}$ correction to the
leading semi-classical term. For treating these corrections Keller's
geometrical theory of diffraction is inadequate and we develop an alternative
approximation based on Kirchhoff's theory. Numerical checks show that our
procedure allows to reduce the typical semi-classical error by about two orders
of magnitude. The method permits to treat the related problem of flux-line
diffraction with the same degree of accuracy.
