The purpose of this tutorial is to give a general guide to calculating projected band structures of PBG structures and defect modes. In particular, we show how to scan over the k-vector in the plane of the photonic crystal. The examples in this tutorial are based on the analysis in Ref  by Mekis et al.
Creating the Structure
This example consists of a 2D cubic array of dielectric GaAs circles with refractive index of 3.4 in the XZ plane in a background material of air. The radii of the circles will be 18% of the period. A line defect is created in this lattice to form the waveguide along which modes can propagate.
The simulation results for this structure are:
This result shows the projected band structure for the defect mode in this waveguide, and is the result of a scanning over the k-vector along the propagation direction.
 Attila Mekis, Shanhui Fan, and J. D. Joannopoulos, Bound states in photonic crystal waveguides and waveguide bends, Phys. Rev. B 58, 4809 (1998).