Conical diffraction in honeycomb lattices is analyzed. This phenomenon arises in nonlinear Schrödinger equations with honeycomb lattice potentials. In the tight-binding approximation the wave envelope is governed by a nonlinear classical Dirac equation. Numerical simulations show that the Dirac equation and the lattice equation have the same conical diffraction properties. Similar conical diffraction occurs in both the linear and nonlinear regimes. The Dirac system reveals the underlying mechanism for the existence of conical diffraction in honeycomb lattices.

Pseudospin, an additional degree of freedom inherent in graphene, plays a key role in understanding many fundamental phenomena such as the anomalous quantum Hall effect, electron chirality and Klein paradox. Unlike the electron spin, the pseudospin was traditionally considered as an unmeasurable quantity, immune to Stern-Gerlach-type experiments. Recently, however, it has been suggested that graphene pseudospin is a real angular momentum that might manifest itself as an observable quantity, but so far direct tests of such a momentum remained unfruitful. Here, by selective excitation of two sublattices of an artificial photonic graphene, we demonstrate pseudospin-mediated vortex generation and topological charge flipping in otherwise uniform optical beams with Bloch momentum traversing through the Dirac points. Corroborated by numerical solutions of the linear massless Dirac-Weyl equation, we show that pseudospin can turn into orbital angular momentum completely, thus upholding the belief that pseudospin is not merely for theoretical elegance but rather physically measurable.

Consider electromagnetic waves in two-dimensional honeycomb structured media, whose constitutive laws have the symmetries of a hexagonal tiling of the plane. The properties of transverse electric (TE) polarized waves are determined by the spectral properties of the elliptic operatordenotes the equilateral triangular lattice), and such that with respect to some origin of coordinates,where R is a 120 • rotation in the plane). A summary of our results is as follows: a) For generic honeycomb structured media, the band structure of L A has Dirac points, i.e. conical intersections between two adjacent Floquet-Bloch dispersion surfaces. b) Initial data of wave-packet type, which are spectrally concentrated about a Dirac point, give rise to solutions of the time-dependent Maxwell equations whose waveenvelope, on long time scales, is governed by an effective two-dimensional time-dependent system of massless Dirac equations. c) Dirac points are unstable to arbitrary small perturbations which break either C (complexconjugation) symmetry or P (inversion) symmetry. d) The introduction through small and slow variations of a domain wall across a line-defect gives rise to the bifurcation from Dirac points of highly robust (topologically protected) edge states. These are time-harmonic solutions of Maxwell's equations which are propagating parallel to the line-defect and spatially localized transverse to it. The transverse localization and strong robustness to perturbation of these edge states is rooted in the protected zero mode of a one-dimensional effective Dirac operator with spatially varying mass term. e) These results imply the existence of uni-directional propagating edge states for two classes of time-reversal invariant media in which C symmetry is broken: magneto-optic media and bi-anisotropic media. arXiv:1710.03389v2 [math-ph] 13 Sep 2018 PreliminariesIn this section, we outline the relevant spectral theory [22,50, 51,76] and introduce terminology and notation for discussing the symmetry properties of the (unperturbed) bulk operator L A . Fourier analysisLet {v 1 , v 2 } be a linearly independent set in R 2 . The lattice generated by {v 1 , v 2 } is the subset of R 2 :A choice of fundamental cell is the parallelogram: Ω = {θ 1 v 1 + θ 2 v 2 : 0 ≤ θ j ≤ 1, j = 1, 2}.

Wave propagation in two-dimensional generalized honeycomb lattices is studied. By employing the tightbinding (TB) approximation, the linear dispersion relation and associated discrete envelope equations are derived for the lowest band. In the TB limit, the Bloch modes are localized at the minima of the potential wells and can analytically be constructed in terms of local orbitals. Bloch-mode relations are converted into integrals over orbitals. With this methodology, the linear dispersion relation is derived analytically in the TB limit. The nonlinear envelope dynamics are found to be governed by a unified nonlinear discrete wave system. The lowest Bloch band has two branches that touch at the Dirac points. In the neighborhood of these points, the unified system leads to a coupled nonlinear discrete Dirac system. In the continuous limit, the leading-order evolution is governed by a continuous nonlinear Dirac system. This system exhibits conical diffraction, a phenomenon observed in experiments. Coupled nonlinear Dirac systems are also obtained. Away from the Dirac points, the continuous limit of the discrete equation leads to coupled nonlinear Schrödinger equations when the underlying group velocities are nearly zero. With semiclassical approximations, all the parameters are estimated analytically.