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. 2018 Apr 10;9:1349. doi: 10.1038/s41467-018-03839-z

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© The Author(s) 2018

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. 1 — Traditional space-coiling metamaterial unit cell and its equivalent model. a Space-coiling metamaterial structure with the overall length of t and overall width of a is depicted here. d and w are channel width and coil’s wall thickness respectively; acoustic wave trajectory is shown as the dashed line in which L represents the wave trajectory length within each coil. b Equivalent model of the coil structure featuring a single straight channel filled with medium of different refractive index (n_rn₀)