Closed-form formulae of hyperbolic metamaterial made by stacked hole-array layers working at terahertz or microwave radiation

Abstract: A metamaterial made by stacked hole-array layers known as a fishnet metamaterial behaves as a hyperbolic metamaterial at wavelength much longer than hole-array period. However, the analytical formulae of effective parameters of a fishnet metamaterial have not been reported hindering the design of deep-subwavelength imaging devices using this structure. We report the new closed-form formulae of effective parameters comprising anisotropic dispersion relation of a fishnet metamaterial working at terahertz or microwave frequency. These effective parameters of a fishnet metamaterial are consistent with those obtained by quasi-full solutions using known effective parameters of a hole-array layer working at frequency below its spoof plasma frequency with the superlattice period much smaller than the hole-array period. We also theoretically demonstrate the deep-subwavelength focusing at {\lambda}/83 using the composite structure of a slit-array layer and a fishnet metamaterial. It is found that the focused intensity inside a fishnet metamaterial is several times larger than that without the fishnet metamaterial, but the transmitted intensity is still restricted by large-wavevector difference in air and a fishnet metamaterial. Our effective parameters may aid the next-generation deep-subwavelength imaging devices working at terahertz or microwave radiation.