2 FIG. 1: The unit cell of the four designs studied. (a): Narrow- slab-pair system; (b) wide-slab-pair system; (c) slabs&wires system; (d) fishnet design. Panel (e) is a magnification of panel (a) where the structure parameters appearing in the structure simulations are shown. The parameters of the pair are given as a function of the scale parameter a = a$_{k}$ (lat- tice constant along propagation direction; equal to the system thickness): lattice constants a$_{E}$ = 2 . 97 a$_{k}$ , a$_{H}$ = 2 . 19 a$_{k}$ , slab- length l = 2 . 19 a$_{k}$ , slab-width w = 0 . 47 a$_{k}$ , thickness of the metal t m = 0 . 25 a$_{k}$ and thickness of the substrate t = 0 . 5 a$_{k}$ . For the fishnet design (d) the width of the slabs is equal to the corresponding unit cell side ( a$_{H}$ ), while the width of the “necks” (continuous metallic parts joining the slabs along E direction) is w n = 0 . 469 a$_{k}$ . The dielectric spacer separating the metallic pairs has been considered as glass (with relative permittivity ε b = 2 . 14), while for the metal plasma frequency and damping factor the aluminum parameters have been em- ployed. the negative permittivity response can be easily obtained using metals; that is why most of the existing efforts to go to the optical LHMs start from attempts to achieve structures of only negative permeability.) Since many of the existing attempts to create high fre- quency magnetic metamaterials and LHMs are based on the scaling down of known microwave designs, there are already efforts trying to determine the possibilities and the limitations of this scaling approach $^{30–33}$. (Note that the properties of the metals, which are involved in most of todays metamaterials, are drastically different in the optical regime compared to microwaves - there, metals behave almost as perfect conductors.) Most of those at- tempts concern SRR systems and they have led to two important conclusions $^{30,34,35}$: (a) By scaling down a SRR, the frequency of its reso- nant magnetic response does not continuously increase, but after some length scale it saturates to a constant value. This value was found to be dependent on the SRR geometry employed, and with proper modifications of this geometry (e.g., adding gaps in the SRR) it could go up to a small fraction (e.g. 20%) of the plasma fre- quency, i.e. to the middle visible range. The satura- tion response of the magnetic resonance frequency was explained taking into account the contribution of the ki- netic energy of the electrons associated with the current inside the SRR ring to the magnetic energy created by the loop current (or, equivalently, taking into account the dispersive response of the metal in the conductivity). (b) The magnetic permeability resonance becomes weaker and weaker by going to smaller length-scale SRR systems, and below some length scale it ceases to reach negative values. This weakening of the permeability was attributed to the kinetic energy of the electrons (giving rise to saturation) in combination with the increased re- sistive losses in the metal as one goes to higher frequen- cies; these losses are strengthened by the resonant re- sponse, implying long-time interaction of the wave with the metallic structures. Although the existing studies are very revealing con- cerning the high frequency response of magnetic meta- materials, they examine the influence of the kinetic en- ergy (or the dispersive response of the metals) only to the magnetic resonance frequency and not to other fea- tures of the resonant magnetic response (like resonance shape and damping factor), neither clarified the role of the losses in the saturation of the magnetic resonance frequency. Moreover, the role of the various geometrical parameters in the high frequency response of metama- terials still remains to be determined, as to identify the dominant parameters determining this response and to define optimization rules for those materials. In this paper we attempt to study all the above men- tioned issues. We will be restricted to systems based on pairs of slabs, alone or in combination with continuous wires; this is mainly due to the fact that slab-pair-based systems are offered for an easy experimental demonstra- tion of negative permeability or negative index response and, moreover, have been proven up to now the most promising systems for the achievement of high frequency negative permeability and negative index metamaterials. The structures discussed here are shown in Fig. 1. Fol- lowing the approach of Refs. $^{30,35}$, we will attempt to analyze the high frequency magnetic response of those structures, to compare their performance and to propose optimization rules for them. For that we examine in de- tail the scaling behavior of the magnetic resonance fre- quency and the magnetic permeability as the structures are scaled down from mm to nm scale. The basic idea that we will use to reproduce and under- stand the small length scale (high frequency) behavior of our structures is the consideration of the kinetic energy of the current carried electrons. This kinetic energy, as