8 it is rather the result of the kinetic inductance (affecting through the factor F $^{′}$). Finally, it is important to point out that both γ and F ′ depend not only on the “quality” (i.e. plasma fre- quency and damping factor) of the metal used for the fabrication of metamaterials but also on the geometrical parameters of the structures (see Eqs. (16) and (17)). This reveals the possibility to modify these factors, and thus to enhance the performance of the high frequency metamaterials, by modifying the geometry. Using Eqs. (8) and (4) in combination with with Eqs. (9), one can obtain the scaling dependence of both F ′ and γ , as ( B constant) and F ′ = F L L + L$_{e}$ ∝ BF a 2 (14) γ = R L + L$_{e}$ ∝ 1 D$_{1}$a 2 + D$_{2}$ (15) IV. OPTIMIZED DESIGNS γ = γ$_{m}$ 1 + ( ω 2 $_{p}$/c $^{2}$) t$_{m}$t = γ$_{m}$ 1 + (2 π ) $^{2}$t$_{m}$ t/λ 2 p . (17) F ′ = F 1 1 + c $^{2}$/ ( ω 2 $_{p}$t$_{m}$ t ) = F 1 + λ 2 $_{p}$/ [(2 π ) $^{2}$tt$_{m}$ ] . (16) From the analytical formulas and the discussion of the previous section it becomes clear that for achievement of optimized high frequency (e.g. optical) magnetic meta- materials one should require: (a) Highest possible sat- uration value for the magnetic resonance frequency; (b) larger possible parameter F $^{′}$, determining the strength of the magnetic resonance and the width of the negative permeability regime; (c) smallest possible loss factor γ . The general requirements for meeting the above con- ditions can be easily concluded based on Eqs. (10), (14) and (15). They demand: Structures of small capacitance, C ; structures of small kinetic inductance L$_{e}$ , compared to magnetic inductance, L ; structures of low resistance, R . On the other hand, the role of the magnetic field induc- tance is more puzzling: while low inductance facilitates the achievement of high magnetic resonance frequency, it results to “lower-quality” resonance, i.e. weaker reso- nance and higher losses (compared to a higher inductance structure of the same length scale). Thus, the inductance optimization should be based on the specific requirements for the designed metamaterial. To translate the above optimization conditions to spe- cific geometrical and material conditions for the slab- pair-based systems we can use Eqs. (11), (16) and (17). From these equations can be concluded that optimized slab-pair-based systems are favored from structures of thick metal (high t$_{m}$ ) and thick separation layer between the slabs of the pair (i.e. high t - not as high though as to cancel the interaction between the slabs); also structures of wide slabs (i.e. large width w - to maximize the struc- ture volume fraction F appearing in Eq. (16)). More- over, high quality optical metamaterials demand metals of the highest possible plasma frequency and the lowest possible collision frequency. The role of the slab-length is not one-way: while shorter slabs (compared to the corresponding unit cell side) facilitate high attainable magnetic resonance fre- quencies they lead to narrower negative permeability regime (due to reduced F in Eq. (16)), and vice versa. Finally, we should emphasize here that for enhanced metamaterial performance one should target operation below the saturation regime, as to ensure large negative permeability width and lower losses. Eq. (17) shows that the saturation value of γ is the colli- sion frequency of the bulk metal as considered in the free- electron (Drude) description of the metal $^{42}$. The satura- tion regime for γ is approached simultaneously with the magnetic resonance frequency saturation, showing that deep in the saturation regime the weakening of the res- onance is not mainly the result of the ohmic losses but Concerning the loss factor γ , from Eq. (15) one can see that γ increases as the length-scale decreases, justi- fying the higher losses in the smaller length scales. This increase though does not continue up to the smallest scales, but beyond a length-scale it approaches a satu- ration value. The geometrical dependence of γ for the slab-pair design is obtained analogously with that of F $^{′}$, as: The geometrical dependence of the factor F ′ for the slab-pair design can be easily obtained by substituting Eqs. (9) to Eq. (14): The scaling behavior of F ′ also implies that the relative bandwidth of the negative permeability regime, while it is maintained almost constant before the staring of the saturation (see Fig. 4), it becomes smaller and smaller deeper in the saturation regime, indicating that working before the saturation regime favors the widest bandwidth of each specific structure. ( D$_{1}$ , D$_{2}$ constants). From Eq. (14) one can derive two important conclusions: (a) The factor F $^{′}$, which mainly determines the frequency width of the negative perme- ability regime, is independent of the resistance, R , thus independent of any loss mechanisms. (b) With the con- sideration of the kinetic inductance the factor F ′ from scale independent (if L$_{e}$ is negligible) becomes scale de- pendent and tends to zero as the size of the structure becomes smaller and smaller ( a → 0). This means that even in the absence of ohmic losses, it would be impos- sible to get negative permeability values of band-width substantially larger than ( a $^{2}$/λ2 $_{p}$) ω$_{LC}$ in arbitrarily small length scales.