Vertically stacked narrow metal‐dielectric multilayered structures known as vertical hyperbolic metamaterials (vertical HMMs) are proposed for broadband negative refraction. The thickness of each layer is required to be subwavelength for effective medium theory (EMT) to be applicable. Therefore, each layer should be thin, forcing the aspect ratio to become extremely high. This requirement makes the fabrication of vertical HMMs difficult. Thus, it is important to determine the limit of the layer thickness for the realization of vertical HMM. In this study, the critical layer thicknesses of vertical HMMs are investigated numerically at the wavelengths of 400, 500, 600, and 700 nm. It is concluded that the critical layer thicknesses for both vertical and horizontal HMMs depend on the permittivity along the perpendicular direction of the layers. The maximum critical layer thickness of vertical HMMs, with a value of 0.22, is produced by a structure composed of gold and silicon dioxide at the wavelength of 500 nm, while the critical layer thickness tends to be smaller if silver is employed. It is believed that this study can provide a useful database as quantitative indicators which can be exploited in designing nanostructures using EMT.

中文翻译：

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垂直堆叠双曲线超材料的临界层厚度分析，可有效产生负折射。

提出了垂直堆叠的窄金属-电介质多层结构，称为垂直双曲线超材料（垂直HMM），用于宽带负折射。为了使有效介质理论（EMT）适用，每层的厚度必须为亚波长。因此，每一层都应该薄，迫使纵横比变得极高。该要求使得垂直HMM的制造困难。因此，重要的是确定层厚度的极限以实现垂直HMM。在这项研究中，垂直HMM的关键层厚度在400、500、600和700 nm的波长下进行了数值研究。可以得出结论，垂直和水平HMM的临界层厚度都取决于沿层垂直方向的介电常数。垂直HMM的最大临界层厚度（值为0.22）是由金和二氧化硅在500 nm波长处构成的，而如果使用银，则临界层厚度往往会变小。相信这项研究可以提供有用的数据库作为定量指标，可以在使用EMT设计纳米结构时加以利用。