Anomalous Quantum Hall Effect of Light in Bloch-Wave Modulated Photonic Crystals

Effective magnetic fields have enabled unprecedented manipulation of neutral particles including photons. In most studied cases, the effective gauge fields are defined through the phase of mode coupling between spatially discrete elements, such as optical resonators and waveguides in the case for photons. Here, in the paradigm of Bloch-wave modulated photonic crystals, we show the creation of effective magnetic fields for photons in conventional dielectric continua for the first time, via Floquet band engineering. By controlling the phase and wave vector of Bloch waves, we demonstrated the anomalous quantum Hall effect for light with distinct topological band features due to delocalized wave interference. Based on a cavity-free architecture, in which Bloch-wave modulations can be enhanced using guided resonances in photonic crystals, the study here opens the door to the realization of effective magnetic fields at large scales for optical beam steering and topological light-matter phases with broken time-reversal symmetry.

Figure 1

Effective magnetic fields in the Floquet photonic crystal. (a) The Floquet photonic crystal consists of a static photonic crystal and the permittivity modulations of three Bloch waves with momenta equal to the three $K$ points and phases ${\varphi }_{1,2,3}$. Black, gray, and white areas in the static photonic crystal have relative permittivities of 12, 2, and 1, respectively. (b) Band structure of the TM modes of the static photonic crystal. The permittivity modulation with frequency $\omega$ and Bloch momentum $\mathbf{q}$ (green arrow) is used to couple the two highlighted bands (red). (c) Out-of-plane electric field in the unit cell corresponding to the modes of band 2 at $\mathrm{\Gamma }$ and $K$ points. (d) Distribution of the effective magnetic field ${\mathbf{B}}_{\mathrm{eff}}$ induced by the Bloch-wave modulations for ${\varphi }_1=0$, ${\varphi }_2=2\pi /3$, and ${\varphi }_3=4\pi /3$. The net effective magnetic flux through the Wigner-Seitz unit cell is zero. (e) Floquet band structure of the modulated photonic crystal, which is formed from hybridization of static bands 1 and 2 of Fig. 1. The Chern number of each band is indicated for the modulation parameters specified in the text. (f) Berry curvature in the Brillouin zone for the top blue Floquet band in Fig. 1.

Figure 2

Anomalous topological edge mode. (a) Full photonic structure used in the FDTD simulation with three Bloch-wave modulations with phase lagging of $2\pi /3$. The star indicates the location of the point source. (b) Propagation of the electric field $E_z$ excited by the TM-polarized continuous-wave point source. A left-propagating one-way mode exists on the edge ($y\sim 6\mu \mathrm{m}$) of the Floquet photonic crystal. (c) Reflection immunity of the one-way edge mode in the presence of a defect. Here, the defect is created by removing a small dielectric rod on the edge.

Figure 3

Phase diagram of the Floquet photonic crystal. (a) Chern number of the higher frequency band of the Floquet two-band model for varying ${\varphi }_2$ and ${\varphi }_3$ (${\varphi }_1=0$). (b) Effective magnetic field in the Wigner-Seitz cell corresponding to the phases indicated by points 1 $(4\pi /3,2\pi /3)$, 2 $(3\pi /4,\pi /4)$, and 3 $(\pi ,\pi )$. Figure 1 corresponds to point $(2\pi /3,4\pi /3)$. For the calculation of the effective magnetic field, we used $\overline{u}(\mathbf{r})\propto \epsilon (\mathbf{r}{)}^2$ to remove the higher order texture due to static permittivity distribution in the photonic crystal.