Symmetry-protected transport in a pseudospin-polarized waveguide

If a system possesses a spin or pseudospin, which is locked to the linear momentum, spin-polarized states can exhibit backscattering-immune transport if the scatterer does not flip the spin. Good examples of such systems include electronic and photonic topological insulators. For electromagnetic waves, such pseudospin states can be achieved in metamaterials with very special artificial symmetries; however, these bulk photonic topological insulators are usually difficult to fabricate. Here we propose a paradigm in which the pseudospin is enforced simply by imposing special boundary conditions inside a channel. The symmetry-protected pseudospin states are guided in air and no bulk material is required. We also show that the special boundary conditions can be implemented simply using an array of metallic conductors, resulting in spin-filtered waveguide with a simple structure and a broad working bandwidth. We generate several conceptual designs, and symmetry-protected pseudospin transport in the microwave regime is experimentally indicated.


Symmetry protected transport in PEC-PMC waveguide
In order to clarify the robustness of the PEC-PMC waveguide in Figure 1b, we compare the wave propagations through a sharp bend in our PEC-PMC waveguide and in a conventional PEC waveguide. Supplementary Figure 1a simulates a 150° bend of the PEC-PMC waveguide with a configuration shown in Figure 1b in the text. EM wave is incident from the lower left entrance and then passes through the sharp bend without reflection. As a control calculation, we consider the case in a conventional PEC waveguide, which also has square cross section but all the four walls are bounded by PEC. Simulated electric field pattern is shown in Supplementary Figure 1b.
The transmission through the bend is very weak and most of the EM waves are reflected.
In the cases discussed in the main text, deformations that preserve the ( ) ( )  zz  symmetry will not introduce backscattering in spin-filtered waveguide. But when the deformation or scatterer breaks this symmetry, the two pseudospin states will couple to each other and backscattering occurs. Supplementary Figure 2

Supplementary Note 2 Number of TEM modes in the waveguide with PEC and PMC boundaries
In the main text, we claimed that the spin-filtered feature is determined by the number of PEC/PMC boundaries and the ( ) ( )  zz  symmetry. This claim is elaborated here. Figure 2a depicts a general configuration of a waveguide consisting of two PECs and two PMCs. Its geometry respects the ( ) ( )  zz  symmetry, as the PEC (PMC) in upper half space and the PMC (PEC) in lower half form a pair of mirror images. The domain inside the waveguide is filled with air. We focus ourselves on the TEM modes, whose eigen fields can be written as h h e e (the plus or minus sign depends on propagation direction). Then the TEM mode problem is equivalent to an electrostatics problem for e in a 2D space bounded by PEC and PMC, where we can determine the number of TEM waveguide mode by applying the uniqueness theorem for Poisson's equation. The uniqueness theorem states that the solution e is unique when all boundaries satisfy certain types of conditions. These conditions include (i) the electric field perpendicular to the boundary is well defined and (ii) the electric potential  at the boundary is well defined. The two PMC boundaries in Figure  Maxwell equations reduce to two equations for two pseudospins. Hence the forward TEM mode (propagating along x direction) must be spin-up polarized or spin-down polarized, i.e., its z E and z H components must be in-phase or out-of-phase. If the forward TEM mode is spin-up, then the backward mode must be spin-down by applying time reversal, and vice versa. Therefore we arrive at the conclusion that the waveguide shown in Figure 2a must be spin-filtered as long as the symmetry is preserved and the numbers of PEC are 2, independent of the cross-section shape.
Next we consider a waveguide with three PECs and three PMCs as depicted in Figure 2b.
According to the uniqueness theorem, the electric field e is unique once the potential differences  Figure 2b has two TEM modes for each propagation direction. However, the spin-filter feature cannot be guaranteed in this case, because the two TEM modes for forward (backward) direction can belong to different spins.
We have proved that a waveguide with two PECs and two PMCs has only one TEM mode for each direction and that its spin-filtered feature does not rely on the particular shape of PEC or PMC. Here we give some examples in the circular waveguide with PEC and PMC boundaries.

Supplementary Note 4 The correspondence of the field solutions between the PEC-PMC structure and the periodic PEC structure
In this section, we elaborate the field solution correspondence between PEC-PMC structure and periodic PEC structure in detail. First, we take the PEC-PMC edge waveguide in Figure 3a  Similar to the periodic structure in Supplementary Figure 8, the robustness of the edge mode in PEC fan-shaped waveguide is protected by C 12v symmetry and the ( ) ( ) zz   symmetry of its corresponding PEC-PMC structure. We can also comprehend this in another way.
planes of this C 12v system. When we focus on the fully symmetric solution of the C 12v system, its electric field at the yellow lines should satisfy both the mirror symmetry (even under reflection) and the continuous boundary condition. These two conditions require the direction of the electric field to be parallel to these mirror planes (marked by yellow dashed lines), i.e., the electric field component perpendicular to these planes should be zero. In this way, the continuous boundary conditions lying on the mirror symmetry planes act as an effective PMC boundary. Therefore solving the fully symmetric solution (A 1 for E field) of the C 12v system in Supplementary Figure   9b is equivalent to solving the PEC-PMC problem in Supplementary Figure 9a. It is the C 12v symmetry that ensures this equivalence and the effective PMC boundary. Hence the edge transport is robust against arbitrary defect that preserves both C 12v symmetry and the hidden ( ) ( ) zz   symmetry between the PEC and the effective PMC, the latter condition ensuring the pseudo spins' decoupling.

Supplementary Note 5 Realization in periodic PEC slabs with finite periods
According to the field solution correspondence, the edge mode of PEC-PMC waveguide in Figure 3a can be realized in a periodic PEC waveguide with infinite periods. For the PEC structure with a finite number of periods, the fields in the outermost layers should deviate from the infinitely periodic structure. It is a natural question to ask as to how many periods should be enough to realize the edge mode in the PEC-PMC configuration. We numerically calculate the decay length of the edge mode at the central plane ( 0 z  ) as a function of number of periods, plotted as the black squares in Supplementary Figure 11. The edge mode decay length is defined as the distance at which the magnitude of the electric field decays to 1/e of its value at the edge.
For the extreme case of one single period (two PEC slabs), the central plane lies in the outermost layer. The eigen field of the edge mode is expected to deviate quite a bit from the periodic one.
Its decay length is 0.65a compared with the ideal value of 0.426a (marked by the red line). The "one period" structure is a system with two PECs, and as such, it support one TEM edge mode according to our discussion in Supplementary Note 2. The eigen field of the TEM edge mode can be obtained by solving the Poisson's equation when we apply different electric potentials on the two PECs. And it is straightforward to see that most of the electric field localized near the edge.

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As the number of periods increases, the central plane lies further and further away from the outermost layer and its eigen field returns to the ideal periodic one. As we can see from Supplementary Figure 11, when the central plane is 1.25a away from the outermost PEC slab (for the case of 3 periods), the decay length of the eigenfield is almost the same as the ideal one.
Here we give an example of realizing robust transport in periodic PEC structure like Figure   3h but with a finite number of periods. A periodic PEC slabs with 10 periods is simulated in Supplementary Figure 12a, where the PEC slabs have the same geometry as that in Figure 3h. Its cross sectional view is illustrated in Supplementary Figure 12b. The structure is illuminated by an y E -polarized wave with frequency of 0.3 ( / ) ca from left. The field pattern at the middle plane in Supplementary Figure 12a shows that the EM wave is guided through the zigzag edge without backscattering. All calculations are done using COMSOL.

Supplementary Note 6 Feasibility of the periodic PEC edge configuration in the high frequency regime
The dissipative loss of the metals should be taken into account in high frequency regimes and we expect that the loss will limit the propagation length of the edge mode. Supplementary Figure 13 simulates the zigzag edge of periodic gold slabs that is similar to Supplementary Figure 8     corresponds to the telecommunication wavelength of 1.55μm . As can be seen from Supplementary Figure 13, the EM wave is still robustly guided along the zigzag edge although some decay exists along the propagation direction due to the metallic loss. The robust transport can still be sustained because the field of edge mode is mainly guided in air and therefore the mode propagation is not seriously compromised by the absorption of the metal.