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. 2017 Aug 10;7:7742. doi: 10.1038/s41598-017-08171-y

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© The Author(s) 2017

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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Transverse spin ( ${\vec{S}}_{s u r f}$ ) and the spin-orbit locking in an ENG/MNG waveguide. For the spin-up $({\vec{S}}_{s u r f} > 0)$ , the wave-vector or momentum in the MNG (ENG) metamaterial is positive (negative). However, for the spin-down $({\vec{S}}_{s u r f} < 0)$ , the momentum in the MNG (ENG) metamaterial is negative (positive). The 2D magnetic-field distributions in the z direction at frequency of 2.7 GHz for rotating source with different handedness located in ENG or MNG metamaterial are shown in the four insets.