Index of refraction of a photonic crystal of carbon nanotubes and homogenization of optically anisotropic periodic composites

We consider the long-wavelength limit for a periodic arrangement of carbon nanotubes. Using the Fourier expansion method, we develop an effective-medium theory for a photonic crystal of aligned optically anisotropic cylinders. Exact analytical formulas for the effective dielectric constants for the E (TM) and H (TE) eigenmodes are obtained for arbitrary 2D Bravais lattice and arbitrary cross-section of anisotropic cylinders. It is shown that, depending on the symmetry of the unit cell, a photonic crystal of anisotropic cylinders behaves in the low-frequency limit like a uniaxial or biaxial optical crystal. The developed theory of homogenization is in a good agreement with existing experimental results for the dielectric tensor of photonic crystals of carbon nanotubes.