Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods

For a periodic array of dielectric rods of circular cross section, the formulation is given for the band structure of a photon and the reflectivity or transmittivity of light, using the vector cylindrical waves as the basis function for expansion. As a key quantity assuring the fast convergence, the expression of the structure factor is given for two- and three-dimensional arrays and for a lattice with a complex unit cell containing a number of rods inside. In terms of the calculated band energies, we check the reliability of the widely used plane-wave methods and estimate the error involved in them. Also, we present some numerical results for the band structure and the transmittivity of light for parameters beyond the reach of the plane-wave approach.