Exact analytic formulas for calculating the effective permittivity, permeability, and crossed magnetoelectric tensors for photonic crystals in the long-wavelength limit are presented. The formulas are valid for arbitrary Bravais lattice and form of inclusions, which can be dielectric, magnetic, or chiral. We have applied them to study the optical anisotropy of homogenized magnetodielectric three-dimensional photonic crystals, which can be induced by the type of Bravais lattice or the form of the inclusion even in the case when the photonic crystal is composed of isotropic materials. It is established that the electromagnetic modes, propagating in such anisotropic magnetodielectric metamaterials, are extraordinary, unlike the modes in homogenized nonmagnetic dielectric composites, where at least one mode is ordinary.

Abstract-We have derived simple analytical expressions for the frequency-dependent effective permittivity tensor of a one-dimensional metal-dielectric photonic crystal in the long wavelength limit. Our results describe the transition between the regime, described by Rytov's formulas for sufficiently long waves, and that predicted by Xu et al. [6], where the effective plasma frequency is independent of the metallic-layer parameters. The derived expressions can be useful for determining the frequency intervals where such an anisotropic system can exhibit metamaterial behavior.

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The effective bianisotropic response of 3D periodic metal-dielectric structures, composed of crosses with asymmetrically-cut wires, is investigated within a general homogenization theory using the Fourier formalism and the form-factor division approach. It is found that the frequency dependence of the effective permittivity for a system of periodically-repeated layers of metal crosses exhibits two strong resonances, whose separation is due to the cross asymmetry. Besides, bianisotropic metamaterials, having a base of four twisted asymmetric crosses, are proposed. The designed metamaterials possess negative refractive index at frequencies determined by the cross asymmetry, the gap between the arms of adjacent crosses lying on the same plane, and the type of Bravais lattice.

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