Observation of Hourglass Nodal Lines in Photonics

Nodal line semimetals exhibiting line degeneracies in three-dimensional momentum space have been demonstrated recently. In general, the presence of nodal line semimetals is protected by special symmetries, such as mirror symmetry. However, these symmetries are usually necessary but not sufficient conditions, as nodal lines can be annihilated even without breaking them. Very recently, nodal line semimetal possessing an hourglass-shaped band structure emerges as a more robust candidate, where line degeneracies cannot be annihilated while preserving all underlying spatial symmetries. Here, for the first time, we experimentally demonstrate the presence of an hourglass nodal line (HNL) in photonic metacrystal at microwave frequency. We observe the HNL through near-field scanning of the spatial fields, followed by subsequent Fourier transformations. The observed photonic HNL resides in a clean and large frequency interval and is immune to symmetry preserving perturbation, which provides an ideal robust platform for photonic applications, such as anomalous quantum oscillation, spontaneous emission and resonant scattering.

Figure 1

(a) Nodal lines formed by accidental degeneracy possess opposite eigenvalues of mirror symmetry. (b) HNL formed by an hourglass-shaped band dispersion that goes around a circle in the momentum space. (c) Schematically illustrating that nodal line degeneracy is removed by perturbations. (d) Schematically illustrating the robustness of HNL. Note, the blue and red bands correspond to negative ($-1$) and positive ($+1$) eigenvalue of the mirror symmetries. $\mathrm{\Gamma }$ and $Y$ denote the BZ center and edge points, respectively.

Figure 2

(a) Unit cell of metallic structure layer with in-plane period $p_x=p_y=4\mathrm{mm}$ and thickness of 2 mm are shown in the left panel. Between two structure layers, there is a spacer layer with thickness of 3 mm. The bottom or top view of the metallic layer is given in the right panel. (b) The first (bulk) BZ of tetragonal lattice with reduced (bulk) BZ boxed by red triangle prism. The $k_z=\pi /p_z$ plane is highlighted in yellow, where HNL exists. The upper blue plane shows the surface BZ. (c) The hourglass-shaped dispersion along line $Z\text{−}P$ as indicated in (b). The red and blue lines are bands corresponding to glide mirror (${\tilde{M}}_z$) eigenvalue of $+1$ and $-1$, respectively. $b1-b4$ are four bands forming an hourglass-shaped dispersion. (d) The quarter view of photonic HNL dispersion on $k_z=\pi /p_z$ plane. The HNL degeneracy is plotted in red. (e) The bulk bands dispersion along high symmetry lines.

Figure 3

(a) Photograph of top surface of the sample fabricated by printed circuit board technology. (b) Experiment setup configuration with source denoted by red cylinder and probe denoted by blue cylinder, respectively. The top surface illustrates the real part of electric field $E_z$ at frequency of 14.8 GHz. (Inset) Structure unit cell. (c) The theoretical equifrequency contour at 14.8 GHz near the HNL, which is shaped in torus. The four dashed lines $k_x=k_y\tan \theta$, $k_y=0.2\pi /p_y$, $k_y=0.4\pi /p_y$, and $k_y=0.8\pi /p_y$ labeled by D–G in the surface BZ correspond to four cases (d)–(g), respectively. (d) The experiment (left) and theoretical (right) band projection when $\theta =\pi /6$. (e)–(g) The experiment (left) and theoretical (right) band projection along $k_y=0.2\pi /p_y$, $0.4\pi /p_y$, and $0.8\pi /p_y$ line, respectively. The two lines $k_y=0.2\pi /p_y$ and $0.4\pi /p_y$ intersect with the HNL. The $k_y=0.8\pi /p_y$ line locates outside the HNL. Note that in Figs. 3, the white line (left) and red line (right) denote the light cone. The white dashed lines in the left panel illustrate hourglass-shaped band dispersion. In the right panel, the red dashed lines show hourglass-shaped band dispersion simulated on $k_z=\pi /p_z$.

Figure 4

(a)–(c) The experimental results of projected $E_z$ field at frequency of 14.8, 15.16, and 15.65 GHz, respectively. The positions of the second band ($b2$) and third band ($b3$) are pointed out by white arrows, where $b2$ and $b3$ are hourglass-shaped bands as denoted in Fig. 2. The white circles denote the light cone. (d)–(f) The theoretical results of EFCs at frequency of 14.2, 14.48, and 14.75 GHz, respectively. The red circles denote the light cone.