A silicon-on-insulator slab for topological valley transport

Backscattering suppression in silicon-on-insulator (SOI) is one of the central issues to reduce energy loss and signal distortion, enabling for capability improvement of modern information processing systems. Valley physics provides an intriguing way for robust information transfer and unidirectional coupling in topological nanophotonics. Here we realize topological transport in a SOI valley photonic crystal slab. Localized Berry curvature near zone corners guarantees the existence of valley-dependent edge states below light cone, maintaining in-plane robustness and light confinement simultaneously. Topologically robust transport at telecommunication is observed along two sharp-bend interfaces in subwavelength scale, showing flat-top high transmission of ~10% bandwidth. Topological photonic routing is achieved in a bearded-stack interface, due to unidirectional excitation of valley-chirality-locked edge state from the phase vortex of a nanoscale microdisk. These findings show the prototype of robustly integrated devices, and open a new door towards the observation of non-trivial states even in non-Hermitian systems.

Here, d = d1 -d2 is the diameter difference between the upper and lower holes. The purple/green line of valley phase is locked with anticlockwise/clockwise vortex, which can be viewed as a pseudospin analogous to the A-B sublattice in graphene system [1].
The red (blue) dots represent the designed VPC1 (VPC2) in Fig. 1 of the main text. Figure 1d gives Hz phase profile at K corresponding to these two specific dots. Both states display typical vortex profiles centered at the honeycomb lattice. The counterparts at K' possess opposite chirality, as required by time-reversal symmetry. We will focus on the physics at K in the detail discussion, while that at K' can be derived from timereversal symmetry.
The TE-like gap exists between the K point and the M point, which is highlighted as yellow region in Supplementary Figure 1b. Such indirect band gap will not affect on the K1 valley bulk state and the valley-induced interface transport. Due to bulk-edge correspondence, the physics is completely same as previously well-known valley states as the nontrivial topology of the gap is guaranteed by the bands below, i.e. TE1 band.
However, the indirect gap will cause some minor issue (e.g. multi-mode effect) when the K2 bulk state is excited, which is not considered in this work. Figure 1 Transition of valley topological phase with spatial-inversionsymmetry broken. a-b, Bulk band structures for d = 0 nm (d1 = d2 = 140 nm) and d = -100 nm (d1 = 81 nm and d2 = 181 nm), respectively. Here, d = d1 -d2 is the diameter difference between the upper and lower holes. The colormap indicates the linear polarization of photonic band. Yellow region: TE-like bandgap in calculation. Gray region: light cone of silica. c, Phase diagram revealed by the band-edge frequencies at K1 and K2 points. Purple/green curve implies the phase vortex locking with anticlockwise/clockwise. Red (blue) dots: the designed VPC1 (VPC2) in the main text. d, Hz phase profile at K1 and K2 points corresponding to the specific dots in c. Both states display typical vortex profiles centered at the honeycomb lattice.

Supplementary Note 2: Photonic valley Hall effect in silicon-on-insulator valley photonic crystal
In this section, we will show how to distinguish TE1 bulk states between K valley The normalized operation frequency is slightly below that of K1-VPC1 to guarantee the group velocity of propagating wave in bulk VPC as a non-zero value. Note that the region outside triangle structure is filled with silicon-rich nitride (n = 2.46) in order to match the parallel wavevectors at the interface between VPC and outside SiN waveguide [4].  Figure 4b).
A more practical case is illustrated in Supplementary Figure 4c. We introduce random bias of up to 10% in each air-hole diameter, and perform 60 different sets of simulations using 3D FDTD calculator. One can still find that broadband high transmission plateau is again, robust to those random bias settings (purple curves in Supplementary Figure 4c

Supplementary Note 4: Optical characterization setup
The experimental setup of optical characterization is shown in Supplementary microscope objective and then imaged by using an InGaAs CCD (Xenics Bobcat-640-GigE). Such vertical path was used to monitor the couple between lensed fiber and input/output waveguide. All the transmission spectra are normalized to the 1.7-m-width silicon strip waveguide located in the same writing field near the VPC devices.
As for the photonic routing experiment, the experimental setup was quite the same, only different in the output part. The incident light is also generated by the tunable lasers and then coupled to the sample with the aid of a polarization maintaining lensed fiber, after passing through the fiber polarization controller to select TE mode. There were two silicon waveguides at the left of the sample. When the incident light couple to the left/right input waveguide, it will partly convert to LCP/RCP at the designed-mircodisk, then the remaining LCP/RCP light will pass through the left/right channel. Another silicon waveguides were fabricated at the end of each channel, and the propagating wave coupled out in the z-direction thanks to the gratings at the end of the waveguide. The outof-plane radiation was collected by a 20X microscope objective and then imaged by using an InGaAs CCD. After the appropriate adjustment of the exposure time, the intensity of each end of the channels was gained from the CCD and the ratio rate can be calculated. The full-band ratio rate can be measured by tuning the operation wavelength of the excited waves.
Supplementary Figure 5 Schematic diagram of experimental setup for transmission measurements and optical microscope images. A NIR continuous wave was firstly launched into a fiber polarizer to select TE mode, and then coupled to the device with the aid of lensed fiber. After passing through device, the output signals can be divided into two parts: one is the in-plane propagating wave collected by another lensed fiber and detected by an optical power meter, the other is the out-of-plane radiation collected by a 20X microscope objective and then imaged by using an InGaAs CCD. Another purpose of out-of-plane detect path is to monitor the couple between the lensed fiber and the input/output waveguide. For the routing images measurement (Fig. 4), the in-plane detect path was removed, as the in-plane propagating waves can be turn into out-of-plane radiation by using grating couplers.

Supplementary Note 5: Details for photonic routing with a subwavelength microdisk
In this section, we will show more details about the experiments of photonic routing devices. The far-field microscope images in the second row of Supplementary Figure 6 have demonstrated in the main text, implying that the high-directional topological routing effect exists in silicon-on-insulator valley photonic crystal. Here, we offer more routing images at out-of-bangap wavelengths both for topological and normal routing devices, respectively. For topological routing, the images did not preserve unidirectional coupling outside the bandgap (see the first and third rows of Supplementary Figure 6), due to the simultaneous excitation of additional bulk states. For normal routing, the guided interface using strip waveguide is insensitive to the optical vortex of microdisk, in correspondence with low-directionality spectra. Note that the scattering spot near sample is mainly caused by out-of-plane scattering among input waveguide, subwavelength microdisk and SOI VPC. It is clear that the spots output from grating couplers have a lower intensity relative to that of the central excitation spot. In spite of this side effect, the valley-dependent directional emission is already visible to be distinguished. Supplementary Figure 7a and 7b show measured (also in Fig. 4h) and simulated directionality spectra for topological photonic routing devices. Simulated directionality spectra are in good agreement with measured results, except for the blue shift of the peak owing to fabrication error.
Supplementary Figure 6 Photonic routing profiles measured by an optical far-field microscopy (20X objective). a, topological routing device and b, normal routing device with tuning the operation wavelength at  = 1342 nm, 1400 nm and 1530 nm, respectively . Yellow region implies the TE-like bandgap in calculation. Red dashes indicate WVG1 incident cases while blue dashes are for WVG2 incidence.