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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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Effective parameters and the dispersion relationship based on the TLs. (a) Effective parameters of two kinds of materials in a waveguide (see inset). The parameters can be divided into three regions labeled by three different colors. The permittivity and permeability are shown by the dashed and solid lines, respectively. (b) Dispersion relations of guided modes in ENG/MNG waveguides for TE polarization (the red line) and TM polarization (the blue line), respectively.