Photonic Chern insulators made of gyromagnetic hyperbolic metamaterials

Controlling light propagation using artificial photonic crystals and electromagnetic metamaterials is an important topic in the vibrant field of photonics. Notably, chiral edge states on the surface or at the interface of photonic Chern insulators can be used to make reflection-free waveguides. Here, by both theoretical analysis and electromagnetic simulations, we demonstrate that gyromagnetic hyperbolic metamaterials (GHM) are photonic Chern insulators with superior properties. As a novel mechanism, the simultaneous occurrence of the hyperbolic and gyromagnetic effects in these metamaterials is shown to open the large topological band gaps with a gap Chern number of one. Importantly, the GHM Chern insulators possess nonradiative chiral edge modes on their surfaces, and thus allow us to fabricate unidirectional waveguides without cladding metals which generally incur considerable Ohmic loss. Furthermore, the photonic edge states in the proposed Chern insulators are robust against disorder on a wide range of length scales, in strong contrast to crystalline topological insulators, and the light flow direction on the surface of the Chern insulators can be easily flipped by switching the direction of an applied magnetic field. Fascinatingly, we find that negative refraction of the topological surface wave occurs at the boundary between the GHMs with the opposite signs of gyromagnetic parameters. Finally, we show that compared with other photonic topological materials such as chiral hyperbolic materials, the present GHM Chern insulators can be much easier to fabricate.

Figure 1

EFS evolution from (a) an isotropic medium (${\epsilon }_{xx}={\epsilon }_{zz}=2$ and $\gamma =0$) to (d) a GHM (${\epsilon }_{xx}=2, {\epsilon }_{zz}=-1$, and $\gamma =0.8$) via either (b) a hyperbolic metamaterial (${\epsilon }_{xx}=2, {\epsilon }_{zz}=-1$, and $\gamma =0$) or (c) a gyromagnetic medium (${\epsilon }_{xx}={\epsilon }_{zz}=2$ and $\gamma =0.8$). TE (TM) and RCP (LCP) denote TE (TM) polarization and right-handed (left-handed) circular polarization, respectively.

Figure 2

Polarization analysis of eigenstates. Calculated polarization of EFS dispersion for (a) and (b) the hyperbolic metamaterial, (c) and (d) the GHM, and (e) and (f) the gyromagnetic medium. RCP (LCP) and TE (TM) denote right-handed (left-handed) circular polarization and TE (TM) polarization, respectively.

Figure 3

Berry curvature distribution on the EFSs of the gyromagnetic hyperbolic metamaterials with (a) ${\epsilon }_{xx}={\epsilon }_{zz}=2$ and $\gamma =0.8$ as well as (b) ${\epsilon }_{xx}=2, {\epsilon }_{zz}=-1$, and $\gamma =1.2$. The calculated Chern number ($C$) for each EFS is also given.

Figure 4

One-way propagating edge states. (a) A GHM slab in air. (b) Calculated EFS dispersions for the GHM slab in air (a). Black, red, and blue lines denote dispersions of bulk state, top, and bottom edge states, respectively. The dashed green circle denotes light cone. (c) and (d) Simulated propagations of light emitted by a line source at $x=0$ on the top surface at $k_z=\pm 1.6k_0$ with $\gamma =0.8$ and $\gamma =-0.8$, respectively. (e) The same as in (d) but on the bottom surface. (f) The same as in (c) except that the surface is now uneven, which does not stop light from propagating along the negative $x$ direction. (g) and (h) 3D stimulated propagation of the chiral edge mode on the surface of a lossy cuboid-shaped GHM with (g) $\gamma =0.8+0.01i$ and (h) $\gamma =-0.8-0.01i$, emitted by a radiation source at $k_z=1.6k_0$. Note that here the wave propagates along the $z$ direction with the decreasing amplitude because of the small added loss (see Appendix pp2 for more discussion on the effect of the dissipation loss).

Figure 5

(a) EFS dispersions and (b) decay rate of the electromagnetic waves for the GHM slab in the air. In (a), the solid black and dashed blue curves denote the dispersions of the bulk states and the air, respectively. In (a) and (b), the yellow regions denote the overlap regions of the air dispersion and bulk gap regions, while the gray regions denote the complete gap regions. (c) and (d) Electric field distributions of the edge states with (c) $k_z=1.06k_0$ and (d) $k_z=0.88k_0$, respectively. Note that in (c), the electric field is completely confined to the edge of the GHM slab and hence there is no radiative energy loss, because the $k_z$ is outside the light EFS sphere. In contrast, in (d), because the $k_z$ is within the light EFS sphere, the electromagnetic wave of the edge state propagates strongly into the air, resulting in a large radiative energy loss.

Figure 6

Negative refraction of the topological surface states at the interface between two different GHMs. (a) Schematic of the GHM system consisting of a slab made of two GHMs with negative and positive gyromagnetic ratio. (b) Illustration of the negative refraction through an EFS analysis, where $v_{g,x}$ and $v_{\text{ph},x}$ denote the $x$ components of the group and phase velocities, respectively. (c) Stimulated negative refraction, where the incident, reflected, and transmitted waves at the interface between the two different GHMs are denoted as 1, 2, and 3, respectively.

Figure 7

Proposed gyromagnetic hyperbolic metamaterial as a bilayer superlattice composed of a gyromagnetic slab and a metal slab.

Figure 8

Calculated EFS of the bilayer superlattice shown in Fig. 7 made of a YIG slab as the gyromagnetic medium and an InSb layer as the metal slab.

Figure 9

Calculated band gap map on the $\gamma -({\epsilon }_{zz}/{\epsilon }_{xx})$ plane of the bilayer superlattice shown in Fig. 7, which is made of a YIG slab as the gyromagnetic medium and an InSb layer as the metal slab.

Figure 10

Bulk EFS (cyan) of lossy GHMs with (a) $\gamma =-1+0.1i$ and (b) $\gamma =-1+1i$. Here the yellow spheres indicate the EFS of air.

Figure 11

(a) The eigenmode distribution of a 2D slab of the GHM with $d=\lambda$, (b) the electric field distribution of a bulk mode of $k_z=1.89k_0$, and (c) the electric field distribution of the edge mode with $k_z=1.53k_0$. (d) The eigenmode distribution of a 2D slab of the GHM with $d=0.25\lambda$, (e) the electric field distribution of a bulk mode of $k_z=1.63k_0$, and (f) the electric field distribution of the edge mode with $k_z=0.99k_0$.