# Photonic Topological Insulator Based on Frustrated Total Internal Reflection in Array of Coupled Prism Resonators

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Problem

- Introducing a defect into an infinite FTIR array (creating a hole), Figure 2d.
- Creating a finite FTIR array by discarding the neighborhood of an infinitely distant point from the structure.

## 3. Methodology and Limitations

## 4. Results and Discussion

## 5. Conclusions

- Beyond Rudner’s solutions, fundamentally new edge solutions were found. This new class of solutions is infinite and can be classified by rational numbers, where Rudner’s solution is a particular case corresponding to the number 1. Most solutions correspond to ratios with an odd numerator and denominator.
- The ray penetrates into the array crossing several resonator layers. This behavior differs from Rudner’s solutions which are completely localized inside the surface layer of resonators. At the same time, the ray cannot penetrate deeper, since only closed trajectories with zero group velocity can occur in an infinite array (the gapless insulator case).
- The new class of trajectories breaks the array symmetry. This behavior is also different to Rudner’s solutions that bear the same symmetry as the resonator array.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

FTIR | Frustrated Total Internal Reflection |

TI | Topological Insulator |

RI | Reversible Topological Insulator |

LC | Loop Conductor |

## References

- Haldane, F.D.; Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett.
**2008**, 100, 013904. [Google Scholar] [CrossRef] [PubMed][Green Version] - Khanikaev, A.B.; Shvets, G. Two-dimensional topological photonics. Nat. Photonics
**2017**, 11, 763–773. [Google Scholar] [CrossRef] - Kane, C.L.; Mele, E.J. Quantum Spin hall effect in graphene. Phys. Rev. Lett.
**2005**, 95, 226801. [Google Scholar] [CrossRef] [PubMed][Green Version] - Grushevskaya, H.; Krylov, G. Vortex dynamics of charge carriers in the quasi-relativistic graphene model: High-energy k→·p→ approximation. Symmetry
**2020**, 12, 261. [Google Scholar] [CrossRef][Green Version] - Grushevskaya, H.V.; Krylov, G.G.; Kruchinin, S.P.; Vlahovic, B.; Bellucci, S. Electronic properties and quasi-zero-energy states of graphene quantum dots. Phys. Rev. B
**2021**, 103, 235102. [Google Scholar] [CrossRef] - Sergeev, A.G. On mathematical problems in the theory of topological insulators. Theor. Math. Phys.
**2021**, 208, 1144–1155. [Google Scholar] [CrossRef] - Kitagawa, T.; Berg, E.; Rudner, M.; Demler, E. Topological characterization of periodically driven quantum systems. Phys. Rev. B
**2010**, 82, 235114. [Google Scholar] [CrossRef][Green Version] - Farrelly, T. A review of quantum cellular automata. Quantum
**2020**, 4, 368. [Google Scholar] [CrossRef] - Wolfram, S. Statistical mechanics of cellular automata. Rev. Mod. Phys.
**1983**, 55, 601. [Google Scholar] [CrossRef] - McMullen, C.T. Billiards, heights, and the arithmetic of non-arithmetic groups. Invent. Math.
**2022**, 228, 1309–1351. [Google Scholar] [CrossRef] - Maczewsky, L.J.; Zeuner, J.M.; Nolte, S.; Szameit, A. Observation of photonic anomalous Floquet topological insulators. Nat. Commun.
**2017**, 8, 13756. [Google Scholar] [CrossRef] [PubMed][Green Version] - Gao, F.; Gao, Z.; Shi, X.; Yang, Z.; Lin, X.; Xu, H.; Joannopoulos, J.D.; Soljaaia, M.; Chen, H.; Lu, L.; et al. Probing topological protection using a designer surface plasmon structure. Nat. Commun.
**2016**, 7, 11619. [Google Scholar] [CrossRef] [PubMed] - Leykam, D.; Chong, Y.D. Edge Solitons in Nonlinear-Photonic Topological Insulators. Phys. Rev. Lett.
**2016**, 117, 143901. [Google Scholar] [CrossRef] [PubMed][Green Version] - Leykam, D.; Smirnova, D.A. Probing bulk topological invariants using leaky photonic lattices. Nat. Phys.
**2021**, 17, 632–638. [Google Scholar] [CrossRef] - Zhu, S.; Yu, A.W.; Hawley, D.; Roy, R. Frustrated total internal reflection: A demonstration and review. Am. J. Phys.
**1986**, 54, 601–607. [Google Scholar] [CrossRef] - Shvartsburg, A.B. Tunneling of electromagnetic waves: Paradoxes and prospects. Physics-Uspekhi
**2007**, 50, 37. [Google Scholar] [CrossRef] - Galperin, G.; Zemlyakov, A. Mathematical Billiards; Library “Kvant”: Moscow, Russia, 1990; Volume 77. [Google Scholar]
- Wright, A. From rational billiards to dynamics on moduli spaces. Bull. Am. Math. Soc.
**2016**, 53, 41–56. [Google Scholar] [CrossRef] - Li, M.; Zhirihin, D.; Gorlach, M.; Ni, X.; Filonov, D.; Slobozhanyuk, A.; Alù, A.; Khanikaev, A.B. Higher-order topological states in photonic kagome crystals with long-range interactions. Nat. Photonics
**2020**, 14, 89–94. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) According to the Snell’s law, we have ${n}_{1}sin{\varphi}_{1}={n}_{0}sin{\varphi}_{0}$, where ${n}_{1}$ is the refractive index of a medium, e.g., glass (colored in gray), from which a ray falls at angle ${\varphi}_{1}$, and ${n}_{0}$ is the refractive index of a medium, for example, air (colored in orange), into which the ray colored in blue falls, refracting at angle ${\varphi}_{0}$. At ${n}_{0}<{n}_{1}$, for angles of incidence greater than the critical angle, i.e., at $\varphi >{\varphi}_{c}=arcsin{n}_{0}/{n}_{1}$, the total internal reflection (TIR) effect is observed: the ray colored in red is completely reflected from the interface between the media. (

**b**) At an angle of incidence greater than the critical angle, the ray, despite total reflection, can almost completely tunnel from a glass prism to another prism through a thin intermediate layer (air gap). To do this with anti-reflection coating, the gap thickness should be a multiple of half a light wavelength corrected for the angle of incidence [16]. (

**c**) Noncontinuous array of glass prisms (coupled prism resonators). The ray propagates through the structure via total internal reflection (FTIR) between the prisms and the TIR at the interfaces with air wells (colored in orange). Black lines represent air gaps between the prisms. It should be noted that the ray trajectory remains qualitatively the same when the air gap has zero thickness.

**Figure 2.**(

**a**–

**c**) Internal and edge modes on the photonic topological insulators assembled from triangular, quadrangular, and hexagonal glass prisms. (

**d**–

**f**) Edge modes.

**Figure 3.**(

**a**) Choosing an angle expressed through the rational fraction $p/q$. Denominator q is plotted on the x-axis and numerator p is plotted on the y-axis from the initial $(4.5,3)$ position. Red rays correspond to fractions $3/2,1,1/2$ and $1/4$. (

**b**) Loops (closed light paths) in an infinite resonator array at tangents of angles of incidence of $1,1/3,3/5$ and $5/6$. Blue circles correspond to glass prisms and empty white squares correspond to gaps. The symmetry of the loops is described by the second-order dihedral group (${D}_{2}$), although the array symmetry bears the higher (fourth) order (${D}_{4}$).

**Figure 4.**Finite array of $18\times 18$ coupled prism resonators. Blue circles correspond to glass prisms and empty white squares correspond to gaps. For tangents 1 and $1/3$ of the angle of incidence, the edge trajectories correspond to a photonic topological insulator. (

**a**) Rudner’s case $tan\varphi =1$. (

**b**) $tan\varphi =1/3$, on the left boundary, the trajectory penetrates deep into the insulator extending through five resonator layers and the approximation of the rigid coupling with the nearest neighbors is violated.

**Table 1.**The FTIR insulators were classified using the following designations: topological insulator TI (the light propagates along the edge), conductor C (the trajectory extends to infinity), loop conductor LC (there are trajectories both closed and penetrating deep into the array), and reversible topological insulator RI, in which the edge-wave propagation direction can be reversed. Symbol # denotes the reducible rational fractions accounted for earlier (for lower p and q). The classification table is symmetric with respect to the main diagonal.

$\mathit{p},\mathit{q}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | TI | C | TI | C | TI | C | TI | C | TI | C | TI | C | TI |

2 | # | C | # | C | # | C | # | C | # | C | # | C | |

3 | # | C | TI | # | TI | C | # | C | TI | # | TI | ||

4 | # | C | # | C | # | C | # | C | # | C | |||

5 | # | LC | TI | C | TI | # | TI | C | TI | ||||

6 | # | LC | # | # | # | C | # | C | |||||

7 | # | C | TI | C | TI | C | TI | ||||||

8 | # | C | # | C | # | C | |||||||

9 | # | C | RI | # | TI | ||||||||

10 | # | C | # | C | |||||||||

11 | # | LC | RI | ||||||||||

12 | # | LC | |||||||||||

13 | # |

**Table 2.**Topological and reversible FTIR insulators are observed only at odd p and q values taken from Table 1.

$\mathit{p},\mathit{q}$ | 1 | 3 | 5 | 7 | 9 | 11 | 13 |
---|---|---|---|---|---|---|---|

1 | TI | TI | TI | TI | TI | TI | TI |

3 | # | TI | TI | # | TI | TI | |

5 | # | TI | TI | TI | TI | ||

7 | # | TI | TI | TI | |||

9 | # | RI | TI | ||||

11 | # | RI | |||||

13 | # |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fedchenko, D.P.; Kim, P.N.; Timofeev, I.V. Photonic Topological Insulator Based on Frustrated Total Internal Reflection in Array of Coupled Prism Resonators. *Symmetry* **2022**, *14*, 2673.
https://doi.org/10.3390/sym14122673

**AMA Style**

Fedchenko DP, Kim PN, Timofeev IV. Photonic Topological Insulator Based on Frustrated Total Internal Reflection in Array of Coupled Prism Resonators. *Symmetry*. 2022; 14(12):2673.
https://doi.org/10.3390/sym14122673

**Chicago/Turabian Style**

Fedchenko, Dmitry P., Petr N. Kim, and Ivan V. Timofeev. 2022. "Photonic Topological Insulator Based on Frustrated Total Internal Reflection in Array of Coupled Prism Resonators" *Symmetry* 14, no. 12: 2673.
https://doi.org/10.3390/sym14122673