Anomalous Topological Phases and Unpaired Dirac Cones in Photonic Floquet Topological Insulators

We propose a class of photonic Floquet topological insulators based on staggered helical lattices and an efficient numerical method for calculating their Floquet band structure. The lattices support anomalous Floquet topological insulator phases with vanishing Chern number and tunable topological transitions. At the critical point of the topological transition, the band structure hosts a single unpaired Dirac cone, which yields a variety of unusual transport effects: a discrete analogue of conical diffraction, weak antilocalization not limited by intervalley scattering, and suppression of Anderson localization. Unlike previous designs, the effective gauge field strength can be controlled via lattice parameters such as the interhelix distance, significantly reducing radiative losses and enabling applications such as switchable topological waveguiding.

Figure 1

A square staggered lattice of helical waveguides. (a) Schematic of two neighboring waveguides, twisting clockwise along the propagation axis $z$ with relative phase shift $\pi$. (b) Cross section of the lattice potential at each helix quarter cycle, with the circular trajectories overlaid. (c) Phase diagram of the tight-binding model, in terms of the sublattice asymmetry $\mathrm{\Delta }$ and coupling strength ${\theta }_c$. Red dots indicate the parameters for the band diagrams in Fig. 2. The nontrivial phase forms an AFI along the $\mathrm{\Delta }=0$ line and a Chern insulator for $\mathrm{\Delta }\ne 0$. (d) Tight-binding bulk spectrum at the black square in (c), along the phase boundary.

Figure 2

Band structures for a semi-infinite strip ten unit cells wide. Blue points are obtained from the continuum model, and red curves from the tight-binding model. (a) Trivial insulator ($a=25\mu \mathrm{m}$, ${\theta }_c\approx 0.17\pi$, $\mathrm{\Delta }=0$). (b) Anomalous Floquet insulator ($a=20\mu \mathrm{m}$, ${\theta }_c\approx 0.4\pi$, $\mathrm{\Delta }=0$). (c) Critical phase ($a=23\mu \mathrm{m}$, ${\theta }_c\approx \pi /4$, $\mathrm{\Delta }=0$). (d) Chern insulator ($a=23\mu \mathrm{m}$, ${\theta }_c\approx 0.15\pi$, $\mathrm{\Delta }\approx \pi /4$).

Figure 3

Output intensity profile after propagation through $5Z$, with one edge site initially excited (red arrow). Reducing the lattice period causes a transition into an anomalous Floquet insulator phase with topological edge states.

Figure 4

Conical diffraction arising from the unpaired Dirac point. In the upper panels, the initial excitation is Gaussian; in the lower panels, only one unit cell is excited. [(a) and (d)] Input and [(b) and (e)] output beam intensity. [(c) and (f)] Phase profile of the output cross-polarized pseudospin component, showing clockwise phase circulation (black arrows). The lattice size is $16\times 16$ unit cells, and the propagation distance is $L=6Z$.

Figure 5

Disorder insensitivity at the critical point. (a) Fourier intensity of a broad (width $5a$) probe beam, after propagating $60Z$ through a weakly disordered [44] $100\times 100$ lattice, averaging over 20 disorder realizations. A weak antilocalization dip occurs in the backscattering direction (white arrow); the color scale saturates in the forward direction. (b) Participation number $P$ normalized by number of waveguides $2N^2=1800$, for the tight-binding eigenmodes of a strongly disordered lattice. Localization is absent for $\mathrm{\Delta }=0$, but Lifshitz localization tails appear for $\mathrm{\Delta }=\pi /8$.