The band structure of 2D photonic crystals -- a periodic material with discontinuous dielectrical properties -- and their eigenmodes can be efficiently computed with the finite element method (FEM). For second order elliptic boundary value problems with piecewise analytic coefficients it is known that the solution converges extremly fast, i.e. exponentially, when using {\em p}-FEM for smooth and {\em hp}-FEM for polygonal interfaces and boundaries. In this article we discretise the variational eigenvalue problems for photonic crystals with smooth and polygonal interfaces in scalar variables with quasi-periodic boundary conditions by means of {\em p}- and {\em hp}-FEM -- this for the transverse electric (TE) and transverse magnetic (TM) modes. Our computations show exponential convergence of the numerical eigenvalues for smooth and polygonal lines of discontinuity of dielectric material properties.