Valley-contrasting physics in all-dielectric photonic crystals: Orbital angular momentum and topological propagation

The valley has been exploited as a binary degree of freedom to realize valley-selective Hall transport and circular dichroism in two-dimensional layered materials, in which valley-contrasting physics is indispensable in making the valley index an information carrier. In this Rapid Communication, we reveal valley-contrasting physics in all-dielectric valley photonic crystals. The link between the angular momentum of light and the valley state is discussed, and unidirectional excitation of the valley chiral bulk state is realized by sources carrying orbital angular momentum with proper chirality. Characterized by the nonzero valley Chern number, valley-dependent edge states and the resultant broadband robust transport is found in such an all-dielectric system. Our work has potential in the orbital angular momentum assisted light manipulation and the discovery of valley-protected topological states in nanophotonics and on-chip integration.

Figure 1

Valley-contrasting OAM in VPC. (a) Schematic of the VPC with the unit cell of two silicon rods embedded in an air background, e.g., A and B rods in the dashed rhomboid or dashed hexagon. The rod diameters are $d_{\mathrm{A}}=0.5a$ and $d_{\mathrm{B}}=0.38a$, where $a$ is the lattice constant. (b) Two lowest bulk bands of transverse-magnetic modes, in which four valley states carrying intrinsic OAM are marked by either counterclockwise or clockwise arrows. Valley-contrasting chiral OAM and the phase distributions of $E_z$, i.e., arg($E_z$), at the frequency of 0.244 $c/a$ at the (c) $K^{\prime }$ point and (d) $K$ point. The phases of $E_z$ for the $K^{\prime }$ valley state decrease counterclockwise by $2\pi$, indicating the OAM with a topological charge of $l=-1$ and further leading to LCP spin angular momentum at the center of the unit cell. While for the $K$ valley state, arg($E_z$) decreases clockwise and has a topological charge of $l=1$ (denoted as RCP OAM). (e) Three-dimensional bulk band structure, illustrating the complete gap and two inequivalent but time-reversal valleys.

Figure 2

Unidirectional excitation of valley chiral states by $E_z$ source with chiral OAM. (a) Schematic of the hexagonal VPC with six zigzag boundaries, surrounded by a homogeneous $\mathrm{Si}{\mathrm{O}}_2$ medium. Inset: yellow marker indicates the circular $E_z$ line sources around the center of the sample. (b) Schematic of the circular $E_z$ line sources with the same magnitudes but phases carrying LCP OAM (left panel) or RCP OAM (right panel) at the frequency of 0.244 $c/a$. (c) Electric intensity and (d) $E_z$ phases of the excited $K^{\prime }$ valley state when applying the LCP source. (e) Electric intensity and (f) $E_z$ phases of the excited $K$ valley state when the source chirality is RCP. Note that it is experimentally detectable for the distinct differences of refracted beam directions at each boundary and the phase distributions inside the sample, verifying the unidirectional excitation of valley chiral states.

Figure 3

Valley-dependent topological index distribution and phase transition. (a) Schematics of topological index distributions for VPC with $\delta d<0a$ and $\delta d>0a$. The white $+$ and $-$ symbols represent the topological index for Berry curvature with $+1/2$ and $-1/2$ in each valley, respectively. Nonzero valley Chern numbers, $C_v=\left(C_K-C_{K^{\prime }}\right)$, are used to distinguish the topology. (b) Frequency spectra of valley chiral states as a function of the inversion-asymmetry amplitude described by $\delta d=\left(d_{\mathrm{A}}-d_{\mathrm{B}}\right)/2$. When $\delta d$ decreases, the frequencies are boosted for both the LCP $K^{\prime }$ valley state and the RCP $K$ valley state, while they drop for the other two valley states. The state exchange occurs at $\delta d=0a$, resulting in phase transition and the two topologically distinct VPCs. In (b), $K^{\prime }$ and $K$ valley states are, respectively, represented by blue and red circular arrows, according to the OAM chirality.

Figure 4

Broadband robust transmission and valley-dependent edge states. (a) Schematic of the edge constructed by a VPC with $\delta d=-0.06a$ (orange) and the other with $\delta d=0.06a$ (green). (b) Valley-dependent edge states, including one with negative velocity at the $K^{\prime }$ valley and the other with positive velocity at the $K$ valley. (c) Schematic of the $\mathsf{Z}$-shape bend, with an incident $E_z$ source (red) and the transmission detector (gray). (d) Transmission spectra for the $\mathsf{Z}$-shape bend (red) and the straight channel (black) and broadband robust transmission are found from 0.244 to 0.272 $c/a$. (e,f) Electric field intensity at 0.25 and 0.26 $c/a$, illustrating the smooth propagating waves through the $\mathsf{Z}$ sharp corners.