Topological photonic crystal with equifrequency Weyl points

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Topological photonic crystal with equifrequency Weyl points

Luyang Wang, Shao-Kai Jian, and Hong Yao

Phys. Rev. A 93, 061801(R) – Published 27 June 2016

Abstract

Weyl points in three-dimensional photonic crystals behave as monopoles of Berry flux in momentum space. Here, based on general symmetry analysis, we show that a minimal number of four symmetry-related (consequently equifrequency) Weyl points can be realized in time-reversal invariant photonic crystals. We further propose an experimentally feasible way to modify double-gyroid photonic crystals to realize four equifrequency Weyl points, which is explicitly confirmed by our first-principle photonic band-structure calculations. Remarkably, photonic crystals with equifrequency Weyl points are qualitatively advantageous in applications including angular selectivity, frequency selectivity, invisibility cloaking, and three-dimensional imaging.

Received 9 January 2016

DOI:https://doi.org/10.1103/PhysRevA.93.061801

Physics Subject Headings (PhySH)

Metamaterials Photonic crystals Topological effects in photonic systems

Atomic, Molecular & Optical

Authors & Affiliations

Luyang Wang¹, Shao-Kai Jian¹, and Hong Yao^1,2,*

¹Institute for Advanced Study, Tsinghua University, Beijing 100084, China
²Collaborative Innovation Center of Quantum Matter, Beijing 100084, China

^*yaohong@tsinghua.edu.cn

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Vol. 93, Iss. 6 — June 2016

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Images

Figure 1
Gyroid photonic crystal has a body-center cubic structure. The lattice vectors are given by $a_{1} = (- 1, 1, 1) \frac{a}{2}, a_{2} = (1, - 1, 1) \frac{a}{2}$ , and $a_{3} = (1, 1, - 1) \frac{a}{2}$ . (a) Primitive cell of single-gyroid photonic crystals. The red region for $g (\vec{r}) > 1.1$ is occupied by dielectric materials. (b) Primitive cell of double-gyroid photonic crystals. The red $(g (\vec{r}) > 1.1)$ and blue $(g (\vec{r}) < - 1.1)$ regions represent two single gyroids related by the inversion symmetry. (c) Air spheres modified double-gyroid structure. The positions of air spheres are given by $(\frac{1}{4}, - \frac{1}{8}, \frac{1}{2}) a, (\frac{1}{4}, \frac{1}{8}, 0) a, (\frac{5}{8}, 0, \frac{1}{4}) a$ , and $(\frac{3}{8}, \frac{1}{2}, \frac{1}{4}) a$ . The radius of the air spheres is $r = 0.07 a$ . (d) The first Brillouin zone of the bcc lattice.
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Figure 2
Calculated first eight bands of DG PhC with or without air spheres. (a) The QBT is shown in the $k_{z} = 0$ plane by a green spot. (b) The band dispersions of DG PhC without air spheres. The dispersions are highly symmetric owing to the $O_{h}$ symmetry and the QBT at $Γ$ point is manifest. (c) Schematic representation of equifrequency Weyl points located at the $k_{x}$ and $k_{y}$ axes. They can be understood as the splitting of the QBT by breaking the inversion symmetry. (d) The calculated band dispersion of modified DG PhC. Since the $O_{h}$ is broken to $D_{2 d}$ , the QBT at $Γ$ point splits into the four equifrequency Weyl points, located at $X − Γ$ and $Γ − Y$ .
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Figure 3
(a) The band structure near one Weyl point. The upper and lower cones have positive and negative refractive indices, respectively. (b) The $k_{y} = 0$ cross section of equifrequency surfaces in a vacuum (upper circle) and in the photonic crystal (lower circle) [46], if $ω \neq ω^{*}$ . When the light shines from the $\hat{x}$ direction and is confined in the $x z$ plane, the $k_{z}$ component is conserved at the boundary. Here, $θ_{1}$ and $θ_{2}$ are the incident angle and the refractive angle, respectively. (c) At $ω^{*}$ , only the rays at particular directions can transmit, while those at other directions are totally reflected. (d) If the Weyl cone is nearly isotropic, the refractive index is well defined and can be positive or negative depending on $ω$ , with the condition that the light shines from $\hat{x}$ or $\hat{y}$ direction. (e) A negative refractive index gives rise to a 3D real image while a positive one gives rise to a virtual image.
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Figure 4
(a) A collimated beam with a broad range of frequencies incident at an angle $θ$ , close to $θ^{*}$ . The “green” ray in the frequency range $(ω^{*} - δ ω, ω^{*} + δ ω)$ can be totally reflected while lights with other frequencies, represented by “blue” and “red” rays, can transmit. (b) Based on this novel frequency selectivity, a waveguide of light with a particular range of frequencies can be made from the photonic crystal with equifrequency Weyl points.
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