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. 2022 Oct 17;13:6120. doi: 10.1038/s41467-022-33896-4

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© The Author(s) 2022

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Fig. 2 — a Band structure of periodic metallic plates without perforations. b Distribution of the ideal type-II nodal ring for (a). No other nodal rings exist. c Band structure on the dashed loop denoted in (b). The radius of the circular loop is 0.1π/a₀. Solid lines and hollow dots respectively denote the results from full-wave simulations and the effective Hamiltonian. d Evolution of the nodal rings, from uniform plates to plates with blind holes and finally through holes.