Controlling the electromagnetic properties of materials, going beyond the limit that is attainable with naturally existing substances, has become a reality with the advent of metamaterials^{1, 2, 3}. The range of various structured artificial ‘atoms’ has promised a vast variety of otherwise unexpected physical phenomena^{3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}, among which the experimental realization of a negative refractive index has been one of the main foci thus far. Expanding the refractive index into a high positive regime will complete the spectrum of achievable refractive index and provide more design flexibility for transformation optics^{9, 10, 11, 12, 13, 14}. Naturally existing transparent materials possess small positive indices of refraction, except for a few semiconductors and insulators, such as lead sulphide or strontium titanate, that exhibit a rather high peak refractive index at mid- and far-infrared frequencies¹⁸. Previous approaches using metamaterials were not successful in realizing broadband high refractive indices^{19, 20, 21}. A broadband high-refractive-index metamaterial structure was theoretically investigated only recently²², but the proposed structure does not lend itself to easy implementation. Here we demonstrate that a broadband, extremely high index of refraction can be realized from large-area, free-standing, flexible terahertz metamaterials composed of strongly coupled unit cells. By drastically increasing the effective permittivity through strong capacitive coupling and decreasing the diamagnetic response with a thin metallic structure in the unit cell, a peak refractive index of 38.6 along with a low-frequency quasi-static value of over 20 were experimentally realized for a single-layer terahertz metamaterial, while maintaining low losses. As a natural extension of these single-layer metamaterials, we fabricated quasi-three-dimensional high-refractive-index metamaterials, and obtained a maximum bulk refractive index of 33.2 along with a value of around 8 at the quasi-static limit.

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