We explore the optical properties of periodic layered media containing left-handed metamaterials. This study is based on several analogies between the propagation of light in metamaterials and charge transport in graphene. We derive the conditions for these two problems become equivalent, i.e., the equations and the boundary conditions when the corresponding wave functions coincide. We show that the photonic band-gap structure of a periodic system built of alternating left- and right-handed dielectric slabs contains conical singularities similar to the Dirac points in the energy spectrum of charged quasiparticles in graphene. Such singularities in the zone structure of the infinite systems give rise to rather unusual properties of light transport in finite samples. In an insightful numerical experiment (the propagation of a Gaussian beam through a mixed stack of normal and meta-dielectrics) we simultaneously demonstrate four Dirac point-induced anomalies: (i) diffusion-like decay of the intensity at forbidden frequencies, (ii) focusing and defocussing of the beam, (iii) absence of the transverse shift of the beam, and (iv) a spatial analogue of the Zitterbewegung effect. All of these phenomena take place in media with non-zero average refractive index, and can be tuned by changing either the geometrical and electromagnetic parameters of the sample,or the frequency and the polarization of light.
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