The small size of plasmonic nanostructures compared to the wavelength of light is one of their most distinct and defining characteristics. It results in the strong compression of an incident wave to intense hot spots which have been used most remarkably for molecular sensing and nanoscale lasers. But another important direction for research is to use this ability to design miniaturized interconnects and modulators between fast, loss-less photonic components. In this situation one is looking for the smallest optical nanostructure possible while trying to mitigate losses. Here we show that despite their high absorption, conductors are still the best materials to reach the sub-wavelength regime for optical antennae when compared to polar crystals and high-index dielectrics, two classes of material which have shown a lot of potential recently for nanophotonic applications. It is demonstrated through both Mie theory and numerical calculations that the smallest possible, efficient, radiating antenna has a length L > λ res /20 in all cases (this length is typically L = λ res /2 in microwave engineering), including the redshifting mechanism induced by a background or substrate refractive index, the effect of material loss and that of shape. In addition, we show that although the assembly of individual particles can further increase the miniaturization factor, it strongly increases the size-mismatch in detriment of the overall efficiency, thus making this method unfit for radiating antennae. By identifying the relevant dimensionless properties for conductors, polar materials and high index dielectrics, we present an unified understanding of the behaviour of sub-wavelength nanostructures which are at the heart of current nanophotonic research and cast the upper achievable limits for optical antennae crucial to the development of real-life implementation.
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