Emergence of Type-II Dirac Points in Graphynelike Photonic Lattices

We theoretically demonstrate that a type-II class of tilted Dirac cones can emerge in generalized two-dimensional anisotropic lattice arrangements. This is achieved by introducing a special set of graphynelike exchange bonds by means of which the complete spectrum of the underlying Weyl Hamiltonian can be realized. Our ab initio calculations demonstrate a unique class of eigensolutions corresponding to a type-II class of Dirac fermionic excitations. Based on our approach, one can systematically synthesize a wide range of strongly anisotropic band diagrams having tilted Dirac cones with variable location and orientation. Moreover, we show that asymmetric conical diffraction, as well as edge states, can arise in these configurations. Our results can provide a versatile platform to observe, for the first time, optical transport around type-II Dirac points in two-dimensional optical settings under linear, nonlinear, and non-Hermitian conditions.

Figure 1

Dispersion band diagrams near the DPs for: (a) type-I DCs with a pointlike Fermi surface and (b) type-II DCs with a conical-like Fermi surface. (c) The centered-square lattice is shown in 3D (left), while the constituent $A$ (blue) and $B$ (red) square sublattices, along with their inter- and intracouplings $(t_1,t_2,t_3,t_4,t^{\prime })$, are depicted in a magnified 2D view (right).

Figure 2

(a) 1D chain of waveguides (left) and equivalent model (right), consisting now of only the two edge waveguide cores coupled via an effective hopping parameter $\tilde{t}$. (b) The centered-square lattice, after the inclusion of the main waveguide chains, is shown on the left. The corresponding dispersion diagram, together with a magnified view of the linear dispersion in the vicinity of a degenerate type-I DP, is shown on the right. (c) The lattice after incorporating the secondary waveguide chains along the $y$ direction (left panel). For the mitigation of undesired effects, stemming from the asymmetric couplings between adjacent waveguide chains, we slightly alter the geometry (middle panel) to locally attain a more symmetric topology in the vicinity of the main lattice points, as depicted on the right.

Figure 3

Dispersion curves (type-I DP) along $k_x=2\pi /3$ (left) and $k_y=2\pi /3$ (right), corresponding to the noncentrosymmetric graphynelike arrangement, depicted in the right panel in Fig. 2. Each graph is related to a different value of the refractive index difference of the secondary chains $(\mathrm{\Delta }{n}_{\mathrm{SC}}=1,1.03,1.07\times \mathrm{\Delta }{n}_0)$. In all cases, the refractive index of the main chains is kept identical to that of the main lattice nodes $(\mathrm{\Delta }{n}_{\mathrm{MC}}=\mathrm{\Delta }{n}_0=0.002)$. By increasing the value of $\mathrm{\Delta }{n}_{\mathrm{SC}}$, this tilt becomes pronounced enough that tipping of the cones takes place (type-II DCs). This becomes evident for $\mathrm{\Delta }{n}_{\mathrm{SC}}=1.3\mathrm{\Delta }{n}_0$ in (b), which depicts the respective bulk band structure throughout the Brillouin zone (right), along with a magnified view of the type-II DPs (left, inset: hyperboliclike intersection with the $\beta =0$ plane). The linear dispersion near the Dirac point, located at $(k_{x_0},k_{y_0})=(2\pi /3,2\pi /3)$, is shown in (c). In (d), asymmetric and symmetric conical diffraction (inset) is demonstrated, after exciting (near the generate point) the lattices of Figs. 2 ($\mathrm{\Delta }{n}_{\mathrm{SC}}=1.3\mathrm{\Delta }{n}_0$) and 2(b) ($\mathrm{\Delta }{n}_{\mathrm{SC}}=0$), respectively. In (e) and (f), the band diagram and respective eigenmode distribution are depicted for top (left) and bottom (right) ribbonlike edge states, supported by the lattice in the right panel in Fig. 2 for $\mathrm{\Delta }{n}_{\mathrm{SC}}=1.3\mathrm{\Delta }{n}_0$.