Fig. 3: Photonic implementation of a spin-valley Hall phase based on silicon-on-insulator platform.

a Sketch of a spin-valley Hall phase (SVHP) design by using Si microring resonators (MRRs) on a SiO₂ substrate with a SiO₂ cladding. The outer radii of MRR A (red) and B (blue) are 6.45 \(\mathrm{\mu m}\) and 6.35 \(\mathrm{\mu m}\); both are 500 nm wide and 220 nm high; the gap distance between the MRR A and B is 250 nm. TE-polarized AM modes with \({m}_{A}=61\) at \({\lambda }_{A}=1546.7\,\mathrm{nm}\), and \({m}_{B}=60\) at \({\lambda }_{B}=1546.9\,\mathrm{nm}\) are supported, giving \(\mathrm{mod}\left({W}_{B\to A}^{\uparrow },4\right)=-1\). b Projected band diagram of an interface between two adjacent SVHP copies. The gap distance between the MRRs across the interface [see the dashed line in a] is 150 nm. The band frequency is offset by the central frequency of the involved modes, corresponding to a wavelength of \({\lambda }_{A}/2+{\lambda }_{B}/2=1546.8\,\mathrm{nm}\). The pseudo-spin index of the lower SVHP system in a is used to label the pseudo-spin configuration of the entire system. c Sketch of simulation setting for examining transport properties of ESs. A perturbation region that potentially induces pseudo-spin/valley flipping is included. d Graphical representation of transitions between edge states (ESs) with two pseudo-spin DOFs and two valley DOFs induced by various perturbations. e-g, Normalized power distributions of ESs after scatterings of valley flipping (e), pseudo-spin flipping (f) and both pseudo-spin and valley flipping (g) [see Supplementary Fig. 4 for detailed information of defect settings]. The perturbation strength is normalized by the A–B coupling strength \({t}_{0}=18.6\,\mathrm{GHz}\). The incident ES is set to \({E}_{{\boldsymbol{\Gamma }}}^{\uparrow }\), the other three ESs \({E}_{{\boldsymbol{\Gamma }}}^{\downarrow }\), \({E}_{{\bf{M}}}^{\uparrow }\), \({E}_{{\bf{M}}}^{\downarrow }\) are induced by the defects.