Optical Interface States Protected by Synthetic Weyl Points

Weyl fermions have not been found in nature as elementary particles, but they emerge as nodal points in the band structure of electronic and classical wave crystals. Novel phenomena such as Fermi arcs and chiral anomaly have fueled the interest in these topological points which are frequently perceived as monopoles in momentum space. Here, we report the experimental observation of generalized optical Weyl points inside the parameter space of a photonic crystal with a specially designed four-layer unit cell. The reflection at the surface of a truncated photonic crystal exhibits phase vortexes due to the synthetic Weyl points, which in turn guarantees the existence of interface states between photonic crystals and any reflecting substrates. The reflection phase vortexes have been confirmed for the first time in our experiments, which serve as an experimental signature of the generalized Weyl points. The existence of these interface states is protected by the topological properties of the Weyl points, and the trajectories of these states in the parameter space resembles those of Weyl semimetal “Fermi arc surface states” in momentum space. Tracing the origin of interface states to the topological character of the parameter space paves the way for a rational design of strongly localized states with enhanced local field.

Popular Summary

Solids can traditionally be classified as metals, semiconductors, and semimetals. A Weyl semimetal is a special kind of semimetal with unusual properties, such as always behaving as a semimetal even if perturbed. At the heart of Weyl semimetals are Weyl points, where energy bands touch at a nodal point. Important as they are, their experimental realizations are based on complex three-dimensional structures that are difficult to make, which limits the exploration of their properties. Recent theoretical work suggested that synthetic (man-made) dimensions allow for flexible control over system parameters and can hence facilitate observations of many novel phenomena. Here, we show that Weyl point physics can be explored easily using the concept of synthetic dimensions.

In particular, we consider topological nodal points in a mixed space of momentum and real space structural parameters. Such synthetic nodal points enable us to study Weyl point physics and their topological consequences in simple layer-by-layer structures, which are much easier to fabricate and characterize than ordinary Weyl crystals. As such, it enables the experimental investigation of Weyl physics in the optical regime, which is otherwise very challenging to realize. In addition, the existence of surface states in one-dimensional photonic crystals with complex unit cells can now be understood as topological consequences of nodal points in a higher-dimensional synthetic space.

Looking ahead, our approach not only increases the flexibility of realizing topological physics, but it also provides the possibility of manipulating topological matter in real time.

Figure 1

Realization of Weyl points in a parameter space. (a) Photonic crystals (PCs) with different $p$, $q$ values. The $p$, $q$ form a parameter space. The inset shows one unit cell of the PC, where the first and the third layers are made of ${\mathrm{H}}_{\text{f}}{\mathrm{O}}_2$ (blue), and the second and the fourth layers are made of ${\mathrm{S}}_{\text{i}}{\mathrm{O}}_2$ (red). The thickness of each layer depends on its position in the $p\text{−}q$ parameter space. (b) The band dispersion of PCs with different unit cells (red line for a PC with only two layers and blue line for a PC the unit cell of which consists of two unit cells used for the red line). Here, $d_a=97\mathrm{nm}$ and $d_b=72\mathrm{nm}$. (c) The dispersion of PCs in the $p\text{−}q$ space with $k=0.5k_0$, and $k_0=\pi /(d_a+d_b)$. Here, two bands form a conical intersection. Panels (b) and (c) together show that the band dispersions are linear in all directions around the degeneracy point, indicating that it is an analogue of the Weyl point. (d) Berry phases defined on the spherical surface with a fixed $\theta$ (see the inset), where blue and red represent Berry phases on the lower and the upper band, respectively. The radius of the sphere is 0.001.

Figure 2

The band dispersions in the parameter space at different k points. (a) Bands 1 and 2 at $k=0.5k_0$. (b) Bands 2 and 3 at $k=0$. (c) Bands 3 and 4 at $k=0.5k_0$. (d) Bands 4 and 5 at $k=0$. The conical intersections in (a)–(d) all correspond to Weyl points with positions and charges (“$+$” for charge $+1$, “$-$” for charge $-1$) marked in the insets, in which the dashed lines denote $p=0$ or $q=0$. Here, we use the same parameters as those in Fig. 1.

Figure 3

The reflection phase in the $p\text{−}q$ space. (a),(c) Reflection phase in the $p\text{−}q$ space at the frequency of the Weyl point in Figs. 2 and 2. The reflection phase shows a vortex structure with charge $-1$ around Weyl points. The white circle, triangle, square, diamond, and pentagram mark the values of $(p,q)$ of the five samples, which are, respectively, $(p,q)=(0.24,0.44)$, $(-0.11,0.49)$, $(-0.38,0.31)$, $(-0.45,0.22)$,and $(0.33,-0.37)$. These five samples have configurations located on a circle (dashed gray line) in the parameter space which encloses a Weyl point in (a) but does not enclose any Weyl point in (c). (b),(d) Reflection phases in the band gaps of the five samples in (a) and (c). The markers show the experimental results whose shape marks the position in the $p\text{−}q$ space in (a) and (c), and colored lines represent the corresponding reflection phases from numerical simulations. The black dashed lines mark the frequencies of the Weyl points in (a) and (c), and the bulk band regions of the PC with the narrowest band gap are shaded in gray.

Figure 4

The “Fermi arc” in the parameter space. (a) A sketch shows the structure used in measuring the interface states. Here, the silver slab represents the silver film, blue represents ${\mathrm{H}}_f{\mathrm{O}}_2$, and red represents ${\mathrm{S}}_i{\mathrm{O}}_2$. Wave incidents from the silver film side and the interface states are localized in the interface region between the silver slab and the semi-infinite PC. The amplitude of the wave is shown schematically in yellow. (b) The reflection phase of the PCs at the frequency of Weyl points with charge $+1$ with its dispersion shown in Fig. 2. The bulk band regions at the working frequency are shaded in gray. The white dashed lines show the trajectories of the interface states of a system consisting of a semi-infinite PC coated with a silver film, where the semi-infinite PC is truncated at the center of the first layer. These interface state trajectories are analogues of Fermi arc states. The triangles mark the $p$ and $q$ values of the four samples in the experiment. (c) The cyan line indicates the working frequency used in (b), and the red crosses label the experimental results. The projection of the bulk band as a function of $q$ is shaded in gray. The number of unit cells is 10, and $d_a=0.323\mu \mathrm{m}$ and $d_b=0.240\mu \mathrm{m}$ for the four PCs in (c). The thickness of layers in these four PCs are given by $(p,q)=(0.50,0.26)$, (0.56,0.30), (0.70,0.38), and (0.78,0.36), respectively.

Figure 5

Topological transitions occur as we change the parameter $R$ defined in the text. (a)–(i) Band structures of band 4 and band 5 with different values of $R$ at $k=0$. The values of $R$ are $0.05$, 0.15, 0.25, 0.35, 0.5, 0.65, 0.75, 0.85, and 0.95 for (a)–(i), respectively. The insets show the positions of the Weyl points (“$+$” for positive charge and “$-$” for negative charge) and “nodal lines” (black dashed lines).

Figure 6

The dispersion near the Weyl points. Panels (a)–(c) show the comparison between the numerical result (blue circles) and the effective Hamiltonian in Eq. (a16) (red dashed lines) in three directions. The parameters of the PC are given by $d_a=97\mathrm{nm}$, $d_b=72\mathrm{nm}$, and the Weyl point considered is between the first and the second bands.

Figure 7

The method to calculate the charge of the Weyl point numerically. (a) Schematic illustration of the integration paths used to calculate topological charges of Weyl points. (b) Berry phases defined on the spherical surface with fixed $\theta$, where blue and red represent Berry phases on the lower band and the upper band, respectively. The parameters of the PC are given by $d_a=97\mathrm{nm}$, $d_b=72\mathrm{nm}$, and the Weyl point considered is between the first and the second band. The radius of the sphere is 0.001.

Figure 8

The reflection phase as a function of the polar angle $\phi$ on a circle with radius 0.01 in the $p\text{−}q$ space around the Weyl point. Blue solid line and red circles represent the reflection phases calculated through the transfer matrix method and Eq. (c14), respectively. Here, we consider the same Weyl point as in Fig. 6. The parameters of the PC are given by $d_a=97\mathrm{nm}$, $d_b=72\mathrm{nm}$, and the Weyl point considered is between the first and the second bands and located at $(p,q)=(0,0)$. The PC is truncated at the starting plane of the first layer, and the working frequency is also fixed at the frequency of the Weyl point, which is $f_w=247.6\mathrm{THz}$.

Figure 9

(a) The reflection phase in the $p\text{−}q$ space for the PCs with $d_a=97\mathrm{nm}$ and $d_b=72\mathrm{nm}$. Now the PCs are truncated at the starting plane of the first layer. The working frequency is also fixed at the frequency of the Weyl point, which is $f_w=247.6\mathrm{THz}$. Compared with Fig. 3, the reflection phase in the $p\text{−}q$ space changes locally, but the charge of the vortex located at the Weyl point stays the same. (b) The dispersion of the interface states near a Weyl point, where the cyan cone represents the bulk state and the colored sheet represents the interface states. Color code of the interface states represents different frequencies. In this calculation, the photonic crystals are truncated at the starting plane of the first layer.