Robust valley polarized states beyond topology

Valley-contrast physics 1–5 has gained considerable attention, particularly for realizing photonic topological insulators (PTIs) 5–11 that support reflection-free valley-polarized edge modes (VPEMs) in the absence of inter-valley scattering. It is an open question whether similar robust states can exist in the absence of topological order. We propose a new approach to VPEMs based on a line defect 12 in a topologically-trivial, C 6 υ -symmetric photonic crystal (PhC). The VPEMs result from opposing orbital angular momenta (OAM) due to a local valley Hall effect (LVHE), where the valley polarization is locally defined as opposed to being fixed throughout the bulk of the PhC. We fabricate our device on a silicon-on-insulator (SOI) slab and characterize it at near-infrared frequencies showing robust transmission through sharp bends. Our results present a new perspective to the existence of gapless chiral edge (kink) states and outlines a new waveguiding mechanism applicable to the electromagnetic spectrum as well as other wave systems including plasmonics, mechanics and acoustics.

Generally, it is assumed that a valley-contrast response requires a transition to a topological valley phase, and so a reduction of the lattice symmetry to C3υ symmetry. In addition, VPEMs have only been observed at the interface between valley PTIs with opposite valley Chern numbers. Here we assess the degree to which these caveats are practically important, demonstrating valley-contrast response and robust VPEMs in the absence of these criteria.

LVHE design
We consider a two-dimensional (2D) dielectric PhC slab with a triangular lattice of circular holes as shown in Fig. 1a. This PhC is widely used for waveguide applications due to its inherent bandgap of TE modes, spanning a frequency range from the K point at the first band to the M point at the higher band. The dashed line rhombus illustrates the Wigner-Seitz unit cell of the lattice with vertices located at the holes' centers. The surface phase distribution map of the outof-plane magnetic field (Hz) at the extrema of the first band -coinciding with the K point-is plotted in Fig. 1b. Opposite OAM states, as indicated by the curled arrows in the inset, are evenly distributed throughout the bulk. OAM states correspond to phase singular points, around 3 which the phase incrementally increases either in clockwise (cw) or counter-clockwise (ccw) direction. Fig. 1c shows the Wilson-loop 34 diagram of the PhC, which plots the Berry phase, θ, for the first band along the loop kx ∈ [−π, π] for a given ky. The lack of winding phase as ky goes from −π to π indicates that the PhC is topologically trivial. In contrast, in a valley PhC, θ varies by ±π/2 near the K and K′ valleys, corresponding to positive and negative ky values, respectively.
Generally, one needs to break spatial inversion symmetry (SIS) to generate opposite, nonvanishing Berry curvature profiles near the K and K′ valleys leading to valley Hall effect (VHE) and topological valley phase 29,30 . Here we establish, essentially, a location-dependent version of the VHE (i.e. LVHE) that does not rely on bulk Berry curvature. This is corroborated by the fact the C6υ lattice shown in Fig. 1 has two OAM states that are evenly distributed throughout the bulk, whereas a transition to topological valley phase, achieved via reduction to C3υ symmetry, promotes only one OAM state. Intuitively, the C6υ symmetry doubly preserves the C3υ rotational symmetry, 35 hence the PhC in Fig. 1   This proves that our PhC supports valley-contrast response that is analogous to conventional VHE and valley PTIs. Note that K and K' valleys are related by time-reversal symmetry; hence the fields at the K' valley could be deduced by applying a time-reversal symmetry operation to the fields at the K valley, which reverses the direction of the energy flux and the phase rotation, and hence the polarity of the OAM and CP states.

Valley-polarized edge modes
While the bulk-boundary correspondence principle prohibits topological edge states between a valley-projected topological phase 6,31 and a topologically trivial phase, the domain wall between crystals with half-integer valley-Chern numbers of opposite signs allow for edge states (also referred to as kink states for distinction). 31 Accordingly, we infer that the conservation of the binary valley DOF 30,36 is responsible for VPEMs and that VPEMs must appear at the interface between opposite OAMs. Where only a local region is concerned, as is the case for a waveguide scenario, the LVHE upholds valley DOF. Therefore, despite the bandgap in our PhC not being the result of a broken SIS like in a valley PTI, we should expect VPEMs to appear if we could enforce opposite OAM polarities across some defect line.  in our PhC. We further prove the nature of these edge modes and the reason for their robustness by testing the wave routing through a magic-T junction, as shown in Fig. 2f. The surface field map shows that when a wave is injected from port 2, it is routed into ports 1 and 3 but, counterintuitively, not port 4. This can be explained by the edge mode being valley polarized. As marked in Fig. 3d, the guided mode in the input port 1 belongs to the K valley, which is of the same valley polarization as that of the output ports 1 and 3. On the other hand, the valley polarization of the output port 4 belongs to the K′ valley, so light cannot be coupled into this port. As such, the edge states in our PhC indeed share the same origin as the VHE and inherit similar features to topological edge states in valley PTIs.

Optical Measurements
We fabricated LVHE-based LDWG devices on a standard SOI wafer with straight, zigzag, and double zigzag pathways (see Methods) as shown by the scanning-electron-microscope (SEM) images in Fig. 3a. We chose a lattice constant of 380 nm, air hole diameter of 160 nm and slab thickness of 220 nm. The separation distance between adjacent holes at the line defect was 60 nm and the devices were covered with 3μm buried oxide layer. This gives a TE bandgap spanning the wavelength range of 1514nm to 1595nm. The PhC bandgap is in the guide part of the band diagram (i.e. below the light line), hence the VPEMs will be confined in the plane of the PhC slab. Fig. 3b shows the measured transmission spectra of the proposed waveguides in the wavelength range of 1530-1565nm, which is limited by the grating coupler performance used for testing (see Methods). The measurement results show high transmittance that is comparable for the three interfaces, as expected of VPEMs. We attribute the reported insertion losses to scattering into 7 plane waves in the high-dielectric SiO2 substrate and buried cladding layer (see methods). For reference, similar waveguide devices using a conventional valley PTI 8 were fabricated and tested, as shown in Fig. 3c (see Methods). The measured transmission spectra are comparable to the results from the LVHE-based LDWGs. In addition, we fabricated and tested typical LDWGs 12 with similar sharp bends, as shown in Fig. 2d (see Methods). As expected, light transmission is greatly deteriorated due to the sharp bends in contrast to the previous two devices.

Conclusion
We have presented a new paradigm for realizing VPEMs and experimentally confirmed their robust light transmission through sharp bends using an SOI slab at telecommunication wavelengths. These modes share similar characteristics to topological modes in VHE-based PTIs albeit happening in a bandgap PhC with no topological order. Instead, the VPEMs here can be understood as the product of a line defect in a lattice with LVHE (a general feature of C6υ point symmetry), where the defect causes a phase discontinuity in spatially-varying OAM states. Our results reveal the role of interface effects in forming gapless chiral edge (kink) states and expand how we can exploit valley DOF and engineer valleytronic devices. This includes new opportunities to develop low-loss compact delay lines 11 , on-chip isolation, slow-light optical buffers, and lasers 10,33 . Lastly, the waveguiding phenomenon demonstrated here is applicable not only to the electromagnetic spectrum and various optical systems, but also to other wave systems such as plasmonics, mechanics and acoustics.

Fabrication:
The PhCs were fabricated on an SOI wafer, with a 220-nm-thick silicon (Si) device layer over silica (SiO2). The device patterns were defined by high-resolution e-beam lithography and then transferred to the silicon device layer by plasma dry etching. Subsequently, a 3-µm-thick buried SiO2 cladding layer was deposited on top of the Si layer for protection using PECVD process.
We fabricated the reference valley PTI on the same SOI wafer using the same process. The associated PhC has a honeycomb lattice of circular air holes with diameters of 160 nm and 80 nm for A and B sites, respectively, and a periodicity of 380 nm. The layout of the waveguide interface was chosen to be the same as reported in Ref. 8. In addition, we fabricated a conventional LDWG for comparison on the same SOI wafer using the same parameters as our LVHE PhC. This LDWG's width was 658 nm, which is defined as the distance between the centers of the air holes nearest the waveguide.

Measurement:
To measure the transmission spectra of the fabricated devices, light from a tunable semiconductor laser was coupled via a single-mode fiber to the SOI waveguides through an integrated grating coupler, which had a bandwidth of ≈35 nm. The TE-polarized continuous waves at the telecommunication wavelength were coupled to a 1.55μm-width input rectangular waveguide and then launched into the PhC sample. We used a linear taper to connect the waveguide of 500 nm width at the grating coupler to the waveguide of 1550 nm width at the PhCs' facets. After passing through the PhC, the propagating wave was coupled to the output 9 rectangular waveguide and then collected by another grating coupler. The transmission spectra of the devices were obtained by sweeping the laser wavelength and simultaneously measuring the transmitted signal using a high-sensitivity optical power meter. Note that the dimensions of the strip waveguides that couple into and out of the PhC devices was chosen the same as given in