  We present a new numerical scheme to solve the Helmholtz equation in a
wave-guide. We consider a medium that is bounded in the $x_2$-direction,
unbounded in the $x_1$-direction and $\varepsilon$-periodic for large $|x_1|$,
allowing different media on the left and on the right. We suggest a new
numerical method that is based on a truncation of the domain and the use of
Bloch wave ansatz functions in radiation boxes. We prove the existence and a
stability estimate for the infinite dimensional version of the proposed
problem. The scheme is tested on several interfaces of homogeneous and periodic
media and it is used to investigate the effect of negative refraction at the
interface of a photonic crystal with a positive effective refractive index.
