General Bound States in the Continuum in Momentum Space

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General Bound States in the Continuum in Momentum Space

Qiao Jiang, Peng Hu, Jun Wang, Dezhuan Han, and Jian Zi

Phys. Rev. Lett. 131, 013801 – Published 6 July 2023

Abstract

Polarization singularities including bound states in the continuum (BICs) and circularly polarized states have provided promising opportunities in the manipulation of light waves. Previous studies show that BICs in photonic crystal slabs are protected by $C_{2} T$ symmetry and hence normally exist on the high-symmetry lines of momentum space. Here, we propose an approach based on graph theory to study these polarization singularities in momentum space, especially in the region off the high-symmetry lines. With a polarization graph, it is demonstrated for the first time that BICs can stably exist off the high-symmetry lines of momentum space for both one-dimensional and two-dimensional photonic crystal slabs. Furthermore, two kinds of interesting processes, including the merging involved with this newly found BICs both on and off the high-symmetry lines, are observed by changing the geometrical parameters of photonic crystal slabs while keeping their symmetry. Our findings provide a new perspective to explore polarization singularities in momentum space and render their further applications in light-matter interaction and light manipulation.

Received 17 January 2023
Accepted 8 June 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.013801

Physics Subject Headings (PhySH)

Nanophotonics Photonic crystals Topological effects in photonic systems

Atomic, Molecular & Optical

Authors & Affiliations

Qiao Jiang ^1,2, Peng Hu ¹, Jun Wang ¹, Dezhuan Han ^1,*, and Jian Zi^3,†

¹College of Physics, Chongqing University, Chongqing 401331, China
²Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing University, Chongqing 401331, China
³Department of Physics, Key Laboratory of Micro- and Nano-Photonic Structures (MOE), and State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China

^*dzhan@cqu.edu.cn
^†jzi@fudan.edu.cn

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Issue

Vol. 131, Iss. 1 — 7 July 2023

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Images

Figure 1
Polarization graph in momentum space. (a) Schematic diagram of a polarization graph and its dual graph. In the polarization graph, the edges correspond to polarization nodal lines of the major axis of polarization ellipse (green lines: $ψ_{∥}$ , black lines: $ψ_{⊥} = ψ_{∥} + π / 2$ ). BICs and C points correspond to the intersection and joint points of edges with different colors, respectively. The dual graph is shown by the purple vertices and edges. (b) The charge of a bounded face in the polarization graph is always zero. Numbers near the red dashed contour indicate the shared charges of the corresponding vertices to this face. (c) The charge of a bounded face in the dual graph is exactly the charge of the corresponding vertex in the original graph.
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Figure 2
Polarization graph and general BICs in a PhC slab. (a) Schematic of a two-dimensional PhC slab consisting of a square lattice of circular air holes in a dielectric slab. (b) Simulated TE-like band structure along the $X − Γ − M$ direction for a PhC slab studied in Ref. [26]. (c) Polarization graph of the ${TE}_{1}$ band [red line in (b)], exhibiting one at- $Γ$ BIC and eight off- $Γ$ BICs. (d) Simulated TM-like band structure along the $Γ − M$ direction. (e) Polarization graph of the ${TM}_{3}$ band. The inset shows the far-field polarization states and $Q$ factors near the general BIC at $k_{∥} = (0.0775, 0.1115) 2 π / a$ . The white dotted line in (c) and (e) indicates the $Γ − M$ direction. The blue and red colors in (c) and (e) indicate the regions with Stokes parameter $S_{2} < 0$ and $S_{2} > 0$ , respectively.
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Figure 3
Merging of two general BICs and an off- $Γ$ BIC. (a) Schematic of a one-dimensional PhC slab and simulated TE-like band structure for $h = 1.41 a$ . (b) Simulated polarization graph for the ${TE}_{2}$ band. The inset shows far-field polarization states and $Q$ factors near the general BIC (indicated by the white dashed box). (c) Simulated polarization graphs for $h = 1.3 92 a$ , $1.3 87 a$ , and $1.3 82 a$ . The insets show the polarization states and $Q$ factors near the off- $Γ$ BIC on the $k_{y}$ axis (indicated by the white dashed boxes).
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Figure 4
Merging two general BICs off the high-symmetry lines. (a) Simulated TM band structure of a one-dimensional ${Si}_{3} N_{4}$ PhC slab with $w = 0.47 a$ and $h = 2.05 a$ . (b)–(d) Polarization graphs for (b) $h = 2.05 a$ , (c) $2.08 a$ , and (d) $2.10 a$ . The insets are far-field polarization states and the corresponding $Q$ factors near the two general BICs (marked by the white dashed box). The blue (red) dot indicates the right-handed (left-handed) C point.
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