This study is concerned with the numerical solution of 2D electromagnetic wave propagation problems in the frequency domain. We present a high order discontinuous Galerkin method formulated on unstructured triangular meshes for the solution of the system of the time-harmonic Maxwell equations in mixed form. Within each triangle, the approximation of the electromagnetic field relies on a nodal polynomial interpolation and the polynomial degree is allowed to vary across mesh elements. The resulting numerical methodology is applied to the simulation of 2D propagation problems in homogeneous and heterogeneous media as well.