Fig. 1. (a) Schematic illustration of helical waveguide array with detuning between two sublattices, shown with red and blue colors. Green hexagons highlight honeycomb structure of the array. Band structure of bulk waveguide array with (b) r=0 and δ=0.1, (c) r=0.4 and δ=0, and (d) r=0.4 and δ=0.1.

Fig. 2. Berry curvature B of the bottom band of the lattice with unpaired Dirac cones. Parameters in (a) are: a=1.6, Z=6, r=0.3, ℘=0.1, t=0.18, and A(z)=rΩ[−cos(Ωz),sin(Ωz)]. Parameters in (b) are: a=1.6, Z=6, r=0.3, ℘=−0.1, t=0.18, and A(z)=rΩ[−cos(Ωz),−sin(Ωz)]. The dashed hexagon represents the first Brillouin zone.

Fig. 3. (a) Schematic representation of composite helical waveguide array with two domain walls that are indicated by the vertical dashed green lines. Only waveguides of one type (blue or red) fall onto domain wall. The array is periodic in the y direction. Both detuning and direction of waveguide rotation are opposite in A and B arrays forming the domain wall. (b) A-B-A array configuration with two domain walls. (c) Quasi-propagation constant versus Bloch momentum ky for B-A-B configuration in (a). Two longitudinal Brillouin zones are shown to stress vertical periodicity of the spectrum. Red and blue dots correspond to the edge states appearing on red and blue domain walls, respectively. Gray dots correspond to bulk modes. (d) Field modulus distribution |ψ| of the edge state (three y-periods are shown) with ky/K=0.3 residing on the red domain wall in (c) at different propagation distances within one period Z. (e) Edge states with ky/K=0.35 from blue domain wall at different distances. In all cases Z=6, r0=0.3, and δ=0.07.

Fig. 4. Zigzag-shaped domain walls with sharp corners that are adopted for demonstration of the topological protection of the edge states from the (a) red branch and (b)  blue branch of Fig. 3(c). The arrows indicate the direction of motion of the edge states with initial broad envelope at different stages of propagation. Corresponding field modulus distributions at different distances are shown too. The two edge states correspond to ky/K=0.3 (a) and ky/K=0.35 (b).

Fig. 5. First-order derivative b′ of the quasi-propagation constant of the valley Hall edge states at Z=6 and r0=0.3.

Fig. 6. (a) A zigzag domain wall with a defect in the form of missing waveguide indicated by the green circle. (b)–(e) Spatial field modulus distributions in the edge state at different propagation distances. Cyan dashed circle highlights theoretical radius of the conical diffraction pattern. (f) Total spatial spectrum of the edge state at z=300. (g) Part of the spectrum at z=300 where only contribution from regions around K points is left (see white dashed circles), while spectral intensity around K′ points is set to zero. (h) Field modulus distribution in the form of conical wave produced by spectrum in (g). (i) Part of the spectrum at z=300 where only contribution from regions around K′ points is left (see dashed circles), while spectral intensity around K points is set to zero. (j) Field modulus distribution of the edge state produced by spectrum in (i). White dashed line represents the zigzag domain wall. The parameters are the same as in Fig. 4.

Fig. 7. Dynamics of excitation of the valley Hall edge state with a tilted elliptical Gaussian beam. (a) Input field modulus distribution at z=0 and field modulus distributions at progressively increasing propagation distances, indicating that the excited state bypasses zigzag bend of the domain wall, indicated with white dashed line. (b) Spectra in momentum space corresponding to spatial distributions in (a). Circles in (b) highlight the valleys in six corners of the first Brillouin zone. Parameters of the array are the same as in Fig. 4. The input Gaussian beam has width wx=10, wy=205 and initial momentum kx=0, ky=−4π/33a.

Fig. 8. Setup similar to that of Fig. 4(a), but for the edge state with ky=0.2K.