7 l/a , t ′ = t/a , t ′ m = t$_{m}$/a , one can obtain the dependence of ω$_{LC}$ from the geometrical characteristics of the struc- ture, as well as an expression for the saturation value, ω sat $_{LC}$: For that, it is important to notice that by scaling-down the structures uniformly, i.e. all the lengths scale propor- tionally to a basic length a (= a$_{k}$ here), both the capac- itance and the magnetic inductance scale proportionally to a , while the kinetic inductance and the resistance scale proportionally to 1 /a , i.e. A correction to the above formula (11) can be obtained if one takes into account also the potential energy of the electrons inside the metal slabs, through an equivalent capacitance $^{31}$, C$_{e}$ = ε$_{0}$wt$_{m}$/l (the capacitance of the ca- pacitor formed inside the metal), added to the slabs ca- pacitance, C . In this case the saturation value for the magnetic resonance frequency is found as The above formulas shows the increasingly pronounced role that kinetic inductance (also the resistance) plays in the smaller scales. Specifically, one can see that the ratio L/L$_{e}$ is of the order of 40 tt$_{m}$/λ 2 $_{p}$, with the typical value of λ$_{p}$ ( λ$_{p}$ = 2 πc/ω$_{p}$ ) being around 85 nm for Al and 130 for Ag $^{41}$. Thus, for √$_{tt}$$_{m}$ smaller than 100 nm the kinetic inductance, L$_{e}$ , becomes appreciable and may dominate as the length scale becomes smaller and smaller. showing that the absolute upper limit for the saturation frequency is not arbitrary high but it is restricted by the plasma frequency of the bulk metal. Since in the following we will consider systems with both l $^{′}$/t′ and l $^{′}$/t′ m larger than unity, where the simpli- fied equation (11) is still valid, we will omit the electron potential energy in the following discussion, keeping into account though that this energy/capacitance sets a finite upper limit for the saturation value of the magnetic res- onance frequency, which is the plasma frequency of the bulk metal. Taking into account the expression for the magnetic resonance frequency in Eq. (4), in combination with Eqs. (9), it can be shown that the magnetic resonance frequency of a slab-pair system has the following scale dependence: with A$_{1}$ , A$_{2}$ constants (depending on the geometrical characteristics of the structure). Eq. (10) shows that as the slabs length-scale becomes smaller the magnetic resonance frequency does not continuously increase, but beyond a length-scale it saturates to a constant value. The saturation of the magnetic resonance frequency is exactly what is observed in Fig. 2, and is exclusively due to the existence of kinetic inductance (note that the resis- tance, R , representing the ohmic losses, does not appear in the above formula (10)). This inductance originates from the electrons inertia and represents the “difficulty” of electrons to follow high frequency motions, i.e. their difficulty to respond to high frequency fields. As had been shown in Section II, by reducing the length-scale of the artificial magnetic structures the mag- netic permeability resonance becomes more and more weak, unable to lead to negative µ values beyond a length scale. This weakening is revealed in both the real part of µ (where smaller absolute values of the Max[Re( µ )] and Min[Re( µ )] are observed) and the imaginary part (where smaller Max[Im( µ )], at ω = ω$_{LC}$ , is observed) - see Fig. 3. A detailed examination of Eq. (7) reveals that the strength of the resonance is determined from both fac- tors γ and F $^{′}$. On the other hand, the width of the neg- ative permeability regime, Δ ω , seems to be much more sensitive to the factor F $^{′}$, and almost unaffected from γ ( γ has only small influence on the lower limit of the negative permeability band, while F ′ strongly affects the upper limit of this band). Note that in the absence of losses, i.e. γ = 0, the upper limit of the negative perme- ability band is ω$_{LC}$$_{/}$√$_{1}$ − F ′ ; the lower limit is simply ω$_{LC}$ , i.e. Note that without the consideration of L$_{e}$ the second term in the square root of Eq. (10) would not exist, leading to a linear dependence ω ∼ 1 /a (as occurs in mi- crowaves and larger scales) making unable to explain the saturation behavior observed in Fig. 2. Note also that the kinetic inductance L$_{e}$ does not influence the response of the structure only at the magnetic resonance regime but at all high-frequency regimes, including permittivity resonances. Substituting Eqs. (9) in Eq. (10) and defining nor- malized (dimensionless) geometrical parameters, i.e. l ′ = Δ ω = ω$_{LC}$ ( 1 √$_{1}$ − F ′ − 1) . (13) B. Magnetic permeability resonance A. Magnetic resonance frequency ω$_{LC}$ = 1 √ ε$_{b}$l $_{′}$2 c 2 a 2 + ε$_{b}$l $_{′}$2 t ′ t ′ $_{m}$ω 2 p −→ a → 0 $_{ω$_{p}$}$√ t ′ t ′ m l ′ √ ε$_{b}$ = ω sat $_{LC}$. (11) ω sat LC = ω$_{p}$ √ ε$_{b}$l ′ 2 t ′ t ′ m + 1 , (12) C = ε$_{0}$ε$_{b}$ wl t ∼ a, L = µ$_{0}$ tl w ∼ a, R = γ$_{m}$ ω 2 $_{p}$ε$_{0}$ l wt$_{m}$ ∼ 1 a , and L$_{e}$ = l wt$_{m}$ 1 ω 2 $_{p}$ε$_{0}$ ∼ 1 a . (9) ω$_{LC}$ = 1 √ ( L + L$_{e}$ ) C ∝ 1 √$_{A}$$_{1}$$_{a}$ 2 + A$_{2}$ , (10)