Analogs of quantum-Hall-effect edge states in photonic crystals

Photonic crystals built with time-reversal-symmetry-breaking Faraday-effect media can exhibit chiral edge modes that propagate unidirectionally along boundaries across which the Faraday axis reverses. These modes are precise analogs of the electronic edge states of quantum-Hall-effect (QHE) systems, and are also immune to backscattering and localization by disorder. The Berry curvature of the photonic bands plays a role analogous to that of the magnetic field in the QHE. Explicit calculations demonstrating the existence of such unidirectionally propagating photonic edge states are presented.

Figure 1

A representation of the phase fields of the photonic Bloch states in a two-dimensional Brillouin zone using two component rotors. The entire set of six electric and magnetic fields is associated with a single phase at each point in the Brillouin zone. The Chern invariant simply represents the winding number of this phase along the Brillouin zone boundary and is also given by the integral of the Berry curvature ${\mathcal{F}}^{xy}$ over the two-dimensional Brillouin zone. The phases in (a) correspond to bands with both inversion and time-reversal symmetries, and the phases of the band can be chosen to be real everywhere in the Brillouin zone. For bands having nonzero Chern invariants (b), the phase around the zone boundary winds by an integer multiple of $2\pi$ and there is a phase-vortex-like singularity somewhere in the Brillouin zone where the Berry connection cannot be defined, due to the occurrence of quasidegeneracies.

Figure 2

The number of forward-minus the number of backward-moving edge modes equals the difference of the Chern number of the band across the interface.

Figure 3

As we tune some parameter $\lambda$ of the Hamiltonian across a critical point where accidental degeneracies occur, and two bands touch in a linear fashion forming a Dirac point, Chern numbers of bands may be exchanged.

Figure 4

In the conventional Faraday effect used in optical isolators (a), light travels in the same direction as the applied magnetic field, resulting in the rotation of its polarization plane. However, in the photonic analog of a 2DEG heterojunction proposed here (b), light travels in a direction perpendicular to the applied field.

Figure 5

Photon bands in the $k_z=0$ plane of a 2D hexagonal lattice of cylindrical dielectric rods. Electromagnetic waves are propagating only in the $x\text{−}y$ plane (Brillouin zone shown in the lower right). As in Ref. 23, the rods have a filling fraction $f=0.431$, $ϵ=14$, and the background has $ϵ=1$. The band structure contains a pair of Dirac points at the zone corners $\left(J\right)$.

Figure 6

Phase diagram of the photonic system as a function of inversion- and time-reversal-symmetry breaking. In regions I and III, the gap openings of both Dirac points are primarily due to inversion-symmetry breaking, whereas in regions II and IV, the breaking of time-reversal symmetry lifts the degeneracy of the bands that formed the Dirac point. In all four regions, the two bands of interest have nonzero Berry curvature, but only in regions II and IV do they contain nonzero Chern numbers.

Figure 7

The band gap (in units $\omega a∕2\pi c$) opened by time-reversal-symmetry breaking as a function of the strength of the Faraday coupling $\Lambda$ (same units as ${ϵ}_b^{-1}$) showing that the gap is linearly proportional to $\Lambda$.

Figure 8

In a system with periodic boundary conditions, there are necessarily two domain walls separating regions with different Faraday axes. We study the gap of the spectrum at the (now nondegenerate) Dirac points as a function of the distance $\mathit{x}$ between the two walls.

Figure 9

The spectral gap (in units of $\omega a∕2\pi c$) between the two bands which split apart due to the breaking of time-reversal symmetry. The spectrum is computed on a torus for the extended system consisting of 30 copies of the hexagonal unit cell. Furthermore, domain walls, across which the sign of the Faraday axis flips, are introduced, and the spectrum is plotted as a function of the separation $\mathit{x}$ between the walls (see also Fig. 8).

Figure 10

The spectrum of the composite system consisting of 30 copies of a single hexagonal unit cell duplicated along the direction ${\mathit{R}}_{\perp }$. Both inversion and time-reversal symmetries are present, and the Dirac points are clearly visible. While the composite system has a spectrum containing many bands, only two bands touch at the Dirac point. The dispersion is computed in $\mathit{k}$ space along the direction parallel to the wall. The units of the horizontal axis are arbitrary: here we have discretized the points in $\mathit{k}$ space along a line in the $K_{\text{parallel}}$ direction which contains both Dirac points and present the photon spectrum at each point.

Figure 11

The same system as above, but with broken time-reversal symmetry without a domain wall. There is a single Faraday axis in the rods of the entire system.

Figure 12

Same system in Fig. 11, but with a domain wall introduced corresponding to maximum separation of the walls on the torus. The two additional modes present in the gap correspond to edge modes with a “free photon” linear dispersion along the wall. There are two modes, since across the domain wall the Chern number of the band just below the band gap changes by 2.

Figure 13

The normalized real-space electromagnetic energy density profile associated with the edge modes in Fig. 12 plotted as a function of the direction perpendicular to the domain wall (and “integrated” over the direction parallel to the interface) and fitted to a Gaussian profile. The integrated energy density depicted here plays the role of the photon probability density, which confirms that light is confined to the interface.

Figure 14

In the weak-coupling approach, the free photon TE mode plane waves are perturbed by a periodic modulation in the permittivity. The plane wave frequency at the three equivalent zone corners (${\mathit{K}}_i$, $i=1,2,3$) is lifted by the permittivity in $\mathit{k}\cdot \mathit{p}$ perturbation theory into a nondegenerate singlet and a degenerate doublet.

Figure 15

Spectrum of photon dispersion in the vicinity of the zone corners. Here, the range of the horizontal axis is much smaller than a reciprocal lattice vector; we are showing the spectrum very close to the zone corner $J$ where the nearly free approximation is valid. We have arbitrarily set $\lambda <0$ so that the singlet band has a lower frequency than the doublet. Free photon spectra are given by dashed lines. Away from the zone corners, the free spectrum is not affected to leading order in $\lambda$.

Figure 16

Spectrum of the integrable Pöschl-Teller model, Eq. (75). The horizontal line corresponds to $\omega ={\omega }_D$. $\delta k_{\parallel }=0$ when the zero-energy mode crosses this line. At $\delta k_{\parallel }=0$, the first excited state has a frequency $\omega =2v_D\mid {\kappa }^{\infty }\mid ∕\xi$ and corresponds to bidirectional propagation. With the exception of the zero mode, all bound states correspond to bidirectionally propagating modes localized at the interface where the function $\kappa \left(\mathit{r}\right)=0$. The zero mode, on the other hand, is unbalanced, and furthermore it corresponds to unidirectional propagation.

Figure 17

Although the photon bands (b) cannot be filled as in the electronic case (a), they can have no analog of the bulk quantum Hall effect. However, the Chern number is a topological invariant of Bloch states independent of the constituents. With the Faraday term, we are able to tune the system such that the total Chern number below a photonic band gap changes across the interface, which gives rise to unidirectionally propagating edge modes of photons localized at the interface. These modes are direct analogues of the chiral edge modes of electronic systems which occur at interfaces between two regions having different total Chern invariants below the Fermi level (i.e., with different Hall conductances).

Figure 18

The generic discretization scheme for the photon-band-structure problem. Space is broken up into polyhedra. The local electric energy density is defined at the vertices of each polyhedron, and the electric fluxes, defined on the edges of the polyhedron, connect two electric energy sites. The volume of the polyhedron is associated with local magnetic energy density, and magnetic fluxes exist on the faces of the polyhedron. The scheme here has electric-magnetic duality in that a dual polyhedron can be defined, on the vertices of which the magnetic energy density is defined, etc. The scheme here is inspired by lattice QED, which ensures the correct long-wavelength photon dispersion; the only difference here is the absence of sources.

Figure 19

The discretized form of $\mathit{A}$, which contains the Poisson bracket relations of the fluxes. Shown here are example configurations of $\{{\Phi }_i^D,{\Phi }_j^B\}=+i$ (top), $-i$ (middle), and 0 (bottom).

Figure 20

The discretized form of $\mathit{B}$, which contains the geometric as well as the dynamics information. It couples fluxes of the same type only, and allows for anisotropy in the constitutive relations.