Frequency range dependent topological phases and photonic detouring in valley photonic crystals

Guo-Jing Tang, Xiao-Dong Chen, Fu-Long Shi, Jian-Wei Liu, Min Chen, and Jian-Wen Dong

Phys. Rev. B 102, 174202 – Published 4 November 2020

Abstract

The recent exploration of the valley degree of freedom in photonic systems has enriched the topological phases of light and brought the robust transport of edge states around sharp bends. The two and more simultaneous band gaps in valley-Hall systems have attracted researchers’ attention for enlarging the working bandwidth. However, band gaps with frequency range dependent topology were not reported and the demonstrated flow of electromagnetic waves is limited to the robust transport of edge states. Here, the frequency range degree of freedom is introduced into valley photonic crystals with dual band gaps. Based on the high-order plane-wave expansion model, we derive an effective Hamiltonian which characterizes dual band gaps. Metallic valley photonic crystals are demonstrated as examples in which all four topological phases are found. At the domain walls between topologically distinct valley photonic crystals, frequency range dependent edge states are demonstrated and a broadband photonic detouring is proposed. Our findings provide the guidance for designing the frequency range dependent property of topological structures and show its potential applications in wavelength division multiplexers.

Received 22 April 2020
Revised 9 September 2020
Accepted 12 October 2020

DOI:https://doi.org/10.1103/PhysRevB.102.174202

Physics Subject Headings (PhySH)

Metamaterials Photonic crystals Topological effects in photonic systems

Atomic, Molecular & Optical

Authors & Affiliations

Guo-Jing Tang ^1,*, Xiao-Dong Chen ^1,*, Fu-Long Shi¹, Jian-Wei Liu¹, Min Chen^1,2,†, and Jian-Wen Dong ^1,‡

¹School of Physics & State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou 510275, China
²Department of Physics, College of Science, Shantou University, Shantou 510063, China

^*These authors contributed equally to this work.
^†Corresponding author: stscm@mail.sysu.edu.cn
^‡Corresponding author: dongjwen@mail.sysu.edu.cn

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Vol. 102, Iss. 17 — 1 November 2020

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Images

Figure 1
Dual band gaps in valley photonic crystals (VPCs). (a) The two-dimensional triangular VPC whose unit cell contains one circle-bar shaped metallic rod in the dielectric background with $ɛ = 2.25$ . The lattice constant is given by $a$ . The right inset shows five structural parameters of the metallic rod, i.e., ( $r, l, w$ , θ, α). Labels $p$ and $q$ indicate two different triangular-lattice centers. (b) Schematic of two sets of reciprocal-lattice vectors used for the plane-wave expansion around the $K$ points. The first set makes $| K + G_{i} | = K$ (three magenta arrows at the left) while the second set makes $| K + G_{i} | = 2 K$ (three cyan arrows at the right). Within the Brillouin zone, the inequivalent $K$ and $K^{'}$ points are colored in light and dark gray. (c) The bulk bands of VPC with $a = 20 mm, r = 6.57 mm, l = 7.53 mm, w = 0.5 mm θ = - 10^{\circ}$ , and $α = 8^{\circ}$ (named VPC1 for short). Dual band gaps, i.e., gap 1 and gap 2 are found. The first (fourth) bulk states at the $K$ point is labeled by the $p^{-} (q^{+})$ state according to the position where its $| E_{z} |$ fields vanish and the winding direction of its energy fluxes. (d) $| E_{z} |$ fields and energy fluxes of the first and fourth bulk states at the $K$ point.
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Figure 2
Phase diagram and topological phase transition. (a) The phase diagram of dual band gaps of VPCs with varied θ and α. The dashed (solid) curve shows the boundary where the closing of gap 1 and gap 2 simultaneously (only gap 2) happens. These two curves divide the diagram into four domains which are characterized by a pair of valley Chern numbers ( $C_{v 1}, C_{v 2}$ ) and indexed by the Roman numerals. Three representative VPCs in different phases are labeled by colored dots. (b) The evolution of bulk states at the $K$ point as a function of θ when α is fixed at $α = 8^{\circ}$ . The phase transition happens at $θ = 0^{\circ}$ with the closing of both gap 1 and gap 2. (c) The bulk bands of VPC2 with $θ = 10^{\circ}$ and $α = 8^{\circ} .$ The $| E_{z} |$ fields and energy fluxes of the first and fourth bulk states at the $K$ point, i.e., $q^{+}$ and $p^{-}$ states, are shown. (d) The evolution of the bulk states at $K$ point as a function of α when θ is fixed at $θ = - 10^{\circ}$ . The phase transition happens at $α = 14 . 75^{\circ}$ with the closing of gap 2 but not gap 1. (e) The bulk bands of VPC3 with $θ = - 10^{\circ}$ and $α = 20^{\circ}$ . The $| E_{z} |$ fields and energy fluxes of the first and fourth bulk states (both are $p^{-}$ states) are shown.
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Figure 3
Frequency-dependent edge states. (a)–(c) Edge dispersions of the domain wall between (a) VPC1 and VPC2, (b) VPC1 and VPC3, (c) VPC3 and VPC2. The schematics of domain wall are shown as insets. Valley-dependent edge states are found at both band gaps in (a), while they exist in only one band gap in (b) and (c). (d), (e) The $E_{z}$ fields and their Fourier transforms of two representative edge states (marked by the cyan and purple stars) of the domain wall in (a).
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Figure 4
Broadband photonic detouring. (a) Schematic of structure for the photonic detouring. The source is input from the left port 1 and received from port 2 or port 3. (b), (c) $| E_{z} |$ fields of transmitted electromagnetic waves at (b) 12.95 GHz and (c) 19.10 GHz. (d), (e) The transmission spectra from port 1 to port 2 ( $S_{21}$ ) and from port 1 to port 3 ( $S_{31}$ ) when the source operates with the frequency around (d) gap 1 and (e) gap 2. The transparent boxes highlight the frequency ranges where $S_{21}$ and $S_{31}$ have large difference.
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