Topological photonics: From crystals to particles

Photonic crystal topological insulators host protected states at their edges. In the band structure these edge states appear as continuous bands crossing the photonic band gap. They allow light to propagate unidirectionally and without scattering. In practice it is essential to make devices relying on these effects as miniature as possible. Here we study photonic topological insulator particles (finite crystals). In such particles the edge state frequencies are discrete. Nevertheless, the discrete states support pseudospin-dependent unidirectional propagation. They allow light to bend around sharp corners similarly to the continuous edge states and act as topologically protected whispering gallery modes, which can store and filter light as well as manipulate its angular momentum. Though we consider a particular all-dielectric realization that does not require a magnetic field, the results in the findings are general, explaining multiple experimental observations of discrete transmission peaks in photonic topological insulators.

Figure 1

(a) Top panel: An edge state of a photonic TI crystal. Bottom panel: The crystal band structure with blue regions representing the bulk bands. Red bands inside the bulk band gap are the topologically protected edge states. (b) Spectrum of a photonic TI particle, where the red circles represent edge states that reside inside the bulk gap. (c) Top view of the photonic TI used here. (d) Six dielectric rods arranged in a hexagon make up an artificial atom (encircled).

Figure 2

(a) Interface between topological and trivial photonic crystals supports protected edge states. (b) Frequency spectrum of a hexagonal photonic TI particle (169 atoms) embedded in trivial matrix contains six pairs of TM edge states. They are labeled by the number of nodes in azimuthal direction. (c) $E_z$ fields for the states are shown (one state from each pair); $+$ and $-$ denote the relative phase of the field. (d) The frequency of the state labeled 2 decreases linearly with the particle radius. (e) The $E_z$ field averaged over azimuthal angle for the state labeled 2. The resulting radial distribution peaks close to the edge and decays exponentially away from it.

Figure 3

The top (bottom) figure shows the $E_z$ field of the highest (lowest) edge state for (a) the “inverse” case of a trivial particle (91 atom) inside the topological matrix containing four pairs of TM edge states labeled by red circles. (b) A topological particle of the same size inside the trivial matrix. The field distributions are very similar and so are the spectra, except for a shift in frequency. (c), (d) Triangular and rhombic particles (66 and 64 atoms). The edge states exist for particles of different shapes and are able to bend around sharp corners.

Figure 4

The top (bottom) figure shows the $E_z$ field of the highest (lowest) state marked by a red circle for each particle (91 atoms). (a) A trivial particle inside the trivial matrix without edge states. (b) A TI particle inside the trivial matrix with three atoms removed. There is a 1:1 correspondence to unperturbed states with an additional pair emerging from the bulk. (c) Three topological atoms are replaced by trivial ones. (d) A TI particle with positional disorder. The rods in topological atoms are randomly moved by $10\%$. This weakly perturbs the states, suggesting they will be robust against manufacturing imperfections.

Figure 5

Time-domain simulations of the photonic TI particle (91 atom) edge states. (a) $E_z$ field of a pulse travels in opposite directions depending on its pseudospin. (b) Inset: The particle, source (S), and detector (D). The transmission spectrum shows discrete peaks, positive for transmission in $x$ direction. Blue (red) solid lines correspond to $H_x+i{H}_y$ ($H_x-i{H}_y$) pseudospin. Dashed lines show corresponding results with $10\%$ disorder in positions of rods constituting the particle (see main text). The black dotted line shows the profile of the input Gaussian pulse (magnified). (c) Horizontal lines show real parts of the state frequencies. (d) The width of the states depends on the thickness of the surrounding trivial matrix. Making the matrix two (four) trivial atoms thinner than above decreases the largest quality factor down to 500 (200), as indicated by the dashed (dotted) line.