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.

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.

Wave propagation in two-dimensional generalized honeycomb lattices is studied. By employing the tight-binding (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 Schr\"odinger equations when the underlying group velocities are nearly zero. With semiclassical approximations, all the parameters are estimated analytically.

The linear spectrum and corresponding Bloch modes of shallow honeycomb lattices near Dirac points are investigated. Via perturbation theory, the dispersion relation is found to have threefold degeneracy at leading order with eigenvalue splitting at the following two orders; i.e., the threefold eigenvalue splits into single and double values. Multiscale perturbation methods are employed to describe the nonlinear dynamics of the associated wave envelopes. The dynamics of the envelope depends on different asymptotic balances whereupon a three-level nonlinear Dirac-type equation or a two-level nonlinear Dirac equation is derived. The analysis agrees well with direct numerical simulations.