Topological edge states of Kekul\'e-type photonic crystals induced by a synchronized rotation of unit cells

Letter

Topological edge states of Kekulé-type photonic crystals induced by a synchronized rotation of unit cells

Rui Zhou, Hai Lin, Y. Liu (刘泱杰), Xintong Shi, Rongxin Tang, Yanjie Wu, and Zihao Yu

Phys. Rev. A 104, L031502 – Published 7 September 2021

Abstract

Generating and manipulating Dirac points in artificial atomic crystals has received attention, especially in photonic systems due to their ease of implementation. In this Letter, we propose a two-dimensional photonic crystal made of a Kekulé lattice of pure dielectrics, where the internal rotation of cylindrical pillars induces optical Dirac-degeneracy breaking. Our calculated dispersion reveals that the synchronized rotation reverses bands and switches parity as well so as to induce a topological phase transition. Our simulation demonstrates that such topologically protected edge states can achieve robust transmission in defect waveguides under deformation, and therefore provides a pragmatically tunable scheme to achieve reconfigurable topological phases.

Received 12 April 2021
Accepted 10 August 2021

DOI:https://doi.org/10.1103/PhysRevA.104.L031502

Physics Subject Headings (PhySH)

Photonic crystals

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Rui Zhou¹, Hai Lin ^1,*, Y. Liu (刘泱杰)^2,†, Xintong Shi¹, Rongxin Tang³, Yanjie Wu¹, and Zihao Yu¹

¹College of Physics Science and Technology, Central China Normal University, Wuhan 430079, Hubei Province, China
²School of Physics and Electronic Sciences, Hubei University, Wuhan 430062, Hubei Province, China
³Institute of Space Science and Technology, Nanchang University, Nanchang 330021, Jiangxi Province, China

^*Corresponding author: linhai@mail.ccnu.edu.cn
^†Corresponding author: yangjie@hubu.edu.cn

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Vol. 104, Iss. 3 — September 2021

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Figure 1
Schematic diagram of a two-dimensional (2D) Kekulé lattice with a lattice constant of $a_{0}$ . (a) The arrangement of the lattice unit marked by the black solid line and the lattice red vectors. (b) Unit cell before rotation. (c) Unit cell under a rotation angle $α あるふぁ = 9 . 4^{\circ}$ .
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Figure 2
TM mode dispersion diagrams (a)–(c) and eigenstate distribution (d) when the PC unit cells rotate for different angles. (a) Rotation angle $α あるふぁ = 0^{\circ}$ . (b) $α あるふぁ = 9 . 4^{\circ}$ where band inversion (red dotted line and blue solid line represent the $d$ band and $p$ band) occurs to generate an accidental Dirac point. (c) $α あるふぁ = 12^{\circ}$ . (d) $E_{z}$ field of the $p$ band and $d$ band in (a). (e) $E_{z}$ field of the $d$ band and $p$ band in (c). The black dashed lines in (b) and (c) indicate the unrotated unit cell. (f) Working frequency for rotation angle $α あるふぁ$ with $a_{0} / R = 2.92$ . The two regions $I$ and $II$ represent two different topological states of PC: $I$ is the topological nontrivial phase, and $II$ the trivial one. The shaded region represent the band-gap width. (g) Phase diagram with rotation angle $α あるふぁ$ and lattice constant $a_{0} / R$ (inset: the first Brillouin zone of the Kekulé lattice).
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Figure 3
(a) Projected band diagram of a one-dimensional supercell, composed of both a topologically nontrivial phase $I$ ( $α あるふぁ = 0^{\circ}$ ) and topologically trivial phase region $II$ ( $α あるふぁ = 12^{\circ}$ ). A (red dot) and B (blue dot) correspond to the clockwise spiral and counterclockwise helical edge states, respectively. (b) Left panel: Electric field distribution $E_{z} (x, y)$ around the edge. Right panel: Zoom-ins for electric field $E_{z}$ . The arrows represent the average Poynting vector directions and the magnitude corresponding to points A and B for $k_{x}$ is $\pm 0.02$ , in units of $2 π ぱい / a_{0}$ , and circular thick arrows in red and blue are guides for the eyes.
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Figure 4
A defect waveguide along the interface between two types of PCs (I: $α あるふぁ = 0^{\circ}$ and II: $α あるふぁ = 12^{\circ}, a_{0} / R = 3$ ). The positive (negative) circular polarization excitation source $S_{+}$ ( $S_{-}$ ) represents the pseudo-spin-up (pseudo-spin-down) mode with frequency of 0.52 in units of $c / a_{0}$ . (a) Electric field of the edge state is excited by the pseudo-spin-up mode $S_{+}$ , showing that the EM wave propagates unidirectionally to the right. (b) In the presence of defects, the crystal lattice appears disordered on the interface between $I$ and $II$ and the spin-down mode is still able to excite reflectionless transmission. (c) On an interface with a sharp angle, EM waves can also steer around geometric bends without backscattering. (d) Poynting vector distribution of an in-line waveguide excited by the $S_{+}$ source. (e) Poynting vector distribution of the $S_{-}$ source excited in a bent waveguide. (f) Transmission spectra measured around band gaps in gray $I$ and $II$ [lines with red squares, black triangles, and blue circles respectively represent the corresponding waveguides in (a)–(c)].
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Physical Review A

covering atomic, molecular, and optical physics and quantum science

Topological edge states of Kekulé-type photonic crystals induced by a synchronized rotation of unit cells

Rui Zhou, Hai Lin, Y. Liu (刘泱杰), Xintong Shi, Rongxin Tang, Yanjie Wu, and Zihao Yu

Phys. Rev. A 104, L031502 – Published 7 September 2021

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Physical Review A

covering atomic, molecular, and optical physics and quantum science

Topological edge states of Kekulé-type photonic crystals induced by a synchronized rotation of unit cells

Rui Zhou, Hai Lin, Y. Liu (刘泱杰), Xintong Shi, Rongxin Tang, Yanjie Wu, and Zihao Yu

Phys. Rev. A 104, L031502 – Published 7 September 2021

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