Photonic Crystal Optics and Homogenization of 2D Periodic Composites

We study the long-wavelength limit for an arbitrary photonic crystal (PC) of 2D periodicity. Light propagation is not restricted to the plane of periodicity. We proved that 2D PC's are uniaxial or biaxial and derived compact, explicit formulas for the effective (``principal'') dielectric constants; these are plotted for silicon-air composites. This could facilitate the custom design of optical components for diverse spectral regions and applications. Our method of ``homogenization'' is not limited to optical properties, but is also valid for electrostatics, magnetostatics, dc conductivity, thermal conductivity, etc. Thus our results are applicable to inhomogeneous media where exact, explicit formulas are scarce. Our numerical method yields results with unprecedented accuracy, even for very large dielectric contrasts and filling fractions.