Topological Nature of Optical Bound States in the Continuum

Optical bound states in the continuum (BICs) have recently been realized in photonic crystal slabs, where the disappearance of out-of-plane radiation turns leaky resonances into guided modes with infinite lifetimes. We show that such BICs are vortex centers in the polarization directions of far-field radiation. They carry conserved and quantized topological charges, defined by the winding number of the polarization vectors, which ensure their robust existence and govern their generation, evolution, and annihilation. Our findings connect robust BICs in photonics to a wide range of topological physical phenomena.

Figure 1

Stable bound states in the continuum as vortex centers of polarization vectors. (a) Schematics of radiation field decomposition for resonances of a slab structure. The spatially averaged Bloch part of the electric field $⟨{\mathbf{u}}_{\mathbf{k}}⟩$ is projected onto the $x\text{−}y$ plane as the polarization vector $\mathbf{c}=(c_x,c_y)$. A resonance turns into a BIC if and only if $c_x=c_y=0$. (b) Schematic illustration for the nodal lines of $c_x$ (green) and of $c_y$ (red) in a region of $\mathbf{k}$ space near a BIC. The direction of vector $\mathbf{c}$ (shown in arrows) becomes undefined at the nodal line crossing, where a BIC is found. (c) Two possible configurations of the polarization field near a BIC. Along a closed loop in $\mathbf{k}$ space containing a BIC (loop goes in counterclockwise direction, $1\to 2\to 3\to 4$), the polarization vector either rotates by angle $2\pi$ (denoted by topological charge $q=+1$) or rotates by angle $-2\pi$ (denoted by topological charge $q=-1$). Different regions of the $\mathbf{k}$ space are colored in four gray-scale colors according to the signs of $c_x$ and $c_y$. In this way, a BIC happens where all four gray-scale colors meet, and charge $q=+1$ corresponds to the color changing from white to black along the counterclockwise loop $C$, and charge $q=-1$ corresponds to the color changing from black to white.

Figure 2

Characterization of BICs using topological charges. (a) Calculated radiative quality factor $Q$ of the ${\mathrm{TM}}_1$ band on a square-lattice photonic crystal slab (as in Ref. [20]), plotted in the first Brillouin zone. Five BICs can be seen. (b) Directions of the polarization vector field reveal vortices with topological charges of $\pm 1$ at each of the five $k$ points. The area shaded in blue indicates modes below the light line and thus bounded by total internal reflection. (c) Nodal lines and gray-scale colors of the polarization vector fields [same coloring scheme as in Fig. 1].

Figure 3

Evolution of BICs and conservation of topological charges. (a) Schematic drawing of a photonic crystal slab with one-dimensional periodicity in $x$ and is infinitely long in $y$. (b) Calculated TM-like band structure along $k_x$ axis and along $k_y$ axis. The area shaded in yellow indicates the light cone, where there is a continuum of radiation modes in the surrounding medium. (c),(d) An example showing topological charges with the same sign bouncing off each other. As the slab thickness $h$ decreases, the two BICs with charge $+1$ move along the $k_x$ axis, meet at the origin, and then deflect onto the $k_y$ axis. This can be understood from the conservation of topological charges or from the evolution of nodal lines. (c),(d) An example of topological charge annihilation happening on the lowest-frequency TE-like band of the same structure with different slab thicknesses. As the slab thickness $h$ decreases, two BICs with charge $-1$ meet with a BIC with charge $+1$ at the origin. These three BICs annihilate to yield one BIC with charge $-1$, as governed by charge conservation.