High-Order Hilbert Curves: Fractal Structures with Isotropic, Tailorable Optical Properties

Fractals are promising candidates as nonperiodic, nonresonant structures exhibiting a homogeneous, isotropic, and frequency-independent effective optical response. We present a comprehensive optical investigation of a metallic Hilbert curve of fractal order N = 9 in the visible and near-infrared spectral range. Our experiments show that high-order fractal nanostructures exhibit a nearly frequency independent reflectance and an in-plane isotropic optical response. The response can be simulated in the framework of a simple effective medium approximation model with a limited number of parameters. It is shown that high-order Hilbert structures can be considered as a transparent in-plane metal, the dielectric function of which is modified by the filling factor f, hence creating a tunable conductive effective metal with tailorable plasma frequency and variable reflectance without going through an insulator-to-metal transition.