Electromagnetic energy and energy flows in photonic crystals made of arrays of parallel dielectric cylinders

We consider electromagnetic propagation in two-dimensional photonic crystals, formed by parallel dielectric cylinders embedded a uniform medium. The frequency band structure is computed using the standard plane-wave expansion method, and the corresponding eigenmodes are obtained subsequently. The optical flows of the eigenmodes are calculated by a direct computation approach, and several averaging schemes of the energy current are discussed. The results are compared to those obtained by the usual approach that employs a group velocity calculation. We consider both the case in which the frequency lies within passing band and the situation in which the frequency is in the range of a partial band gap. The agreements and discrepancies between various averaging schemes and the group velocity approach are discussed in detail. The results indicate that the group velocity can be obtained by an appropriate averaging method. Existing experimental methods are also discussed.