# Topological Refraction in Kagome Split-Ring Photonic Insulators

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## Abstract

**:**

## 1. Introduction

## 2. Model and Calculation Method

#### 2.1. Model

#### 2.2. Valley Chern Number

## 3. Results

#### 3.1. Band Inversion of Topological Valley-Hall-like States

#### 3.2. Valley Topological Refraction

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**Dependence of topological phase transition at K point on global $\alpha $ and individual $\theta $ rotations. Specifically, the cross sections formed by the cyan frame and the red frame are shown in Figure 2a,c.

## Appendix B

**Figure A2.**(

**a**) Supercell and (

**b**) bands of topological insulator with $\alpha ={30}^{\circ},\theta =-{180}^{\circ}$ (gray) and $\alpha ={30}^{\circ},\theta =-{110}^{\circ}$ (light red). (

**c**) The k-space analysis on the out-coupling of ${K}^{\prime}$ valley and projected edge states along negative-type (Zigzag) interface. (

**d**) Supercell and (

**e**) bands of topological insulator with $\alpha ={30}^{\circ},\theta =-{70}^{\circ}$ (light red) and $\alpha ={30}^{\circ},\theta ={0}^{\circ}$ (light blue). (

**f**) The k-space analysis on the out-coupling of ${K}^{\prime}$ valley and projected edge states along negative-type (Zigzag) interface. In (

**b**,

**e**), dashed black curves represent the bulk modes. The dashed red/blue curves represent the negative-type/positive-type interfaces edge states. In (

**c**,

**f**), the white solid hexagon represents the first Brillouin zone. The red solid circle shows the dispersion in background material Si. Simulated distribution of fields at frequency $f=0.1$ THz and $f=0.101$ THz are separately illustrated in the bottom panels.

## Appendix C

**Figure A3.**Simulated propagation of a light beam through the Zigzag interface with $f=0.102$ THz. In (

**a**), the tentagram, square, and triangle represent source, impurity, and defect, respectively. In (

**b**), the tentagram and rectangle represent source and defect, respectively. In (

**c**), the tentagram represents source.

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**Figure 1.**(

**a**) The 2D unit cell and first Brillouin zone of ${C}_{3v}$ Kagome SRPI. The geometric parameters are taken as $a=400$ $\mathsf{\mu}$m, ${r}_{1}=0.1a$, ${r}_{2}=0.16a$. (

**b**) ${C}_{3v}$ symmetry-broken unit cell, $\alpha $ and $\theta $ are the global and individual rotation of the split rings. (

**c**) Photonic bands of the SRPI for $\alpha ={0}^{\circ}$, $\theta ={0}^{\circ}$ (blue solid curves) and $\alpha ={30}^{\circ}$, $\theta ={0}^{\circ}$ (red solid curves).

**Figure 2.**(

**a**) Dependence of topological phase transition on global $\alpha $ as $\theta ={0}^{\circ}$ at the K point; the region I and $II$ represent bandgaps with $m>0$ and $m<0$, respectively. (

**b**) The field distribution of the two eigenstates at the K point (denoted as p and q) for $\alpha =-{30}^{\circ}$ and ${30}^{\circ}$. (

**c**) Dependence of topological phase transition on individual $\theta $ as $\alpha ={30}^{\circ}$ at the K point; the region $III$, $IV$ ($VI$) and V represent bandgaps with $m<0$, $m>0$, and $m<0$, respectively. (

**d**) The field distribution of three eigenstates at the K point (denoted as s, p, and q) for $\theta =-{180}^{\circ}$, $-{110}^{\circ}$, $-{70}^{\circ}$, and ${0}^{\circ}$. The Poynting vectors are represented by red arrows.

**Figure 3.**(

**a**) Supercell and (

**b**) bands of topological insulator with $\alpha =-{30}^{\circ},\theta ={0}^{\circ}$ (light red) and $\alpha ={30}^{\circ},\theta ={0}^{\circ}$ (light blue). In (

**b**), dashed black curves represent the bulk modes and the dashed red/blue curves represent the negative-type/positive-type interfaces edge states. (

**c**) The distribution of field for ${A}_{+}$, ${A}_{-}$, ${B}_{+}$, ${B}_{-}$ in (

**b**), and the Poynting vectors are represented by red arrows.

**Figure 4.**(

**a**) The k-space analysis on out-coupling of K valley projected edge states along the positive-type (Zigzag) interface. (

**b**) The k-space analysis on the out-coupling of ${K}^{\prime}$ valley projected edge states along the negative-type (Zigzag) interface. The white solid hexagon represents the first Brillouin zone, and the red solid circles show the dispersion in background material Si. The simulated distribution of fields at the frequency $f=0.102$ THz (in bandgap) are separately illustrated in the bottom panels. The light-red and light-blue regions represent $m>0$ and $m<0$, respectively.

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## Share and Cite

**MDPI and ACS Style**

Li, H.; Luo, C.; Zhang, T.; Xu, J.; Zhou, X.; Shen, Y.; Deng, X.
Topological Refraction in Kagome Split-Ring Photonic Insulators. *Nanomaterials* **2022**, *12*, 1493.
https://doi.org/10.3390/nano12091493

**AMA Style**

Li H, Luo C, Zhang T, Xu J, Zhou X, Shen Y, Deng X.
Topological Refraction in Kagome Split-Ring Photonic Insulators. *Nanomaterials*. 2022; 12(9):1493.
https://doi.org/10.3390/nano12091493

**Chicago/Turabian Style**

Li, Huichang, Chen Luo, Tailin Zhang, Jianwei Xu, Xiang Zhou, Yun Shen, and Xiaohua Deng.
2022. "Topological Refraction in Kagome Split-Ring Photonic Insulators" *Nanomaterials* 12, no. 9: 1493.
https://doi.org/10.3390/nano12091493