In one aspect, a composition of matter is disclosed, which comprises a photonic crystal comprising a plurality of 2D or 3D periodically repeating structures, where the structures are configured and arranged relative to one another such that the photonic crystal exhibits a Dirac cone at the center of the Brillouin zone of its reciprocal lattice, e.g., at one frequency in the optical regime. In some embodiments, the structures are formed of a dielectric material. In another aspect, a photonic structure is disclosed, which comprises a substrate, a plurality of periodically repeating cavities formed in said substrate, where the cavities are sized and arranged relative to one another such that said photonic structure exhibits a substantially vanishing refractive index (preferably a zero refractive index) for at least one wavelength of electromagnetic radiation propagating through said photonic structure, for example, for at least one wavelength of the electromagnetic radiation in a range of about 400 nm to about 100 microns.