6 III. ANALYTICAL MODEL AND INTERPRETATION OF THE RESULTS ω$_{LC}$ is the magnetic resonance frequency of the system and γ is the dumping factor, representing all the losses and the scattering mechanisms. The inductance L$_{e}$ can be easily calculated by calculat- ing the kinetic energy of the electrons, E$_{k}$ = N$_{e}$m$_{e}$v 2 $_{e}$/ 2 = V$_{w}$n$_{e}$m$_{e}$v 2 $_{e}$/ 2, and expressing the velocity through the current, I = ewt$_{m}$n$_{e}$v$_{e}$ ( n$_{e}$ is the number density of free electrons, e is the electron charge, and V$_{w}$ = wt$_{m}$l is the volume of the metallic slab - see Fig. 1(e)). This way, one can obtain L$_{e}$ = lm$_{e}$/wt$_{m}$e $^{2}$n$_{e}$ = ( l/wt$_{m}$ )(1 /ω 2 $_{p}$ε$_{0}$ ), where ω$_{p}$ = √ e $^{2}$n$_{e}$ /m$_{e}$ε$_{0}$ is the plasma frequency of the bulk metal. Note that exactly the same results as in Eq. (3) can be obtained by considering, instead of the kinetic energy for the derivation of L$_{e}$ , the dispersive behavior of the metal conductivity. Indeed, starting with the frequency dependent Drude-type conductivity To understand and explain the results presented in the previous section, we use the common approach of describ- ing the artificial magnetic structures (at the resonant magnetic response regime) as equivalent effective RLC circuits. Using circuit theory and basic electromagnetic considerations $^{2}$, one can easily obtain an expression for the frequency dependence of the effective magnetic per- meability, µ ( ω ), for a lattice of “magnetic” elements $^{2}$. For the explanation of the high frequency magnetic re- sponse we follow the approach of Ref. $^{30}$, based on the consideration of the kinetic energy of the electrons in the metal, E$_{k}$ , besides the magnetic energy of the resulting electromagnetic field. Thus, we replace the magnetic in- ductance of the system in the effective RLC circuit de- scription by the total inductance resulting as a sum of the magnetic field inductance, L , and the kinetic inductance, L$_{e}$ , where L$_{e}$ is defined by E$_{k}$ = N$_{e}$m$_{e}$v 2 $_{e}$/ 2 = L$_{e}$I $^{2}$/ 2 ( N$_{e}$ is the total number of electrons, m$_{e}$ is the electron mass and v$_{e}$ the average electron velocity). Assuming a slab-pair system like the one shown in Fig. 1(e), of length l and slabs separation t , excited by a magnetic field of the form H = H$_{0}$e − $^{iωt}$and direction as shown in Fig. 1(e), and applying the Kirchhoff voltage rule, one can obtain φ is the external magnetic flux, φ = µ$_{0}$ltH , and R and C the resistance (frequency independent, accounting for the ohmic losses) and the capacitance of the system, respec- tively. The obvious solution of Eq. (1) is I = I$_{0}$ e − $^{iωt}$, with Using R$_{tot}$ = R − iωL$_{e}$ in Eq. (1) in the place of R and only the magnetic field inductance (in order to avoid counting twice the kinetic inductance L$_{e}$ ), one can obtain the same current solution as in Eq. (2). In the following, to simplify our discussion we will con- sider the magnetic field inductance to be given by the inductance of a solenoid of area lt and length w (see Fig. 1), i.e. L = µ$_{0}$lt/w , and the capacitance by that of a parallel plate capacitor of area wl/ 2, plate separation t and dielectric core of relative dielectric constant ε$_{b}$ , i.e. C = ε$_{0}$ε$_{b}$ ( wl ) /t . (Note that these formulas are appropri- ate for the case of wide slabs while describe much less satisfactory the narrow slabs case; in any case there is a numerical correction factor of the order of one - see also Section V. Note also that in our discussion we will not consider any inter-unit-cell capacitance $^{21}$, which is important in the case of quite long slabs, i.e. slabs that approach the unit cell boundaries along E direction.) With the above considerations, the equation (3) for the magnetic permeability takes the form Having the current one can easily obtain the pair magnetic dipole moment, m = area × current = ltI , and the magnetization M = ( N$_{LC}$/V ) ltI = (1 /V$_{uc}$ ) ltI , N$_{LC}$ is the number of “RLC” circuits in the volume V , and V$_{uc}$ = a$_{E}$a$_{H}$a$_{k}$ is the volume per unit cell, where a$_{E}$, a$_{H}$, a$_{k}$ are the system lattice constants along the E , H , and k direction respectively ( a$_{H}$ ≥ w, a$_{E}$ > l, a$_{k}$ ≥ t ) - see Fig. 1). Finally, using M = χ$_{m}$ ( ω ) H, µ ( ω ) /µ$_{0}$ = 1 + χ$_{m}$ ( ω ), with χ$_{m}$ the magnetic susceptibility, one obtains that Using the above formulas and their behavior going to small length scales we will show in the following that one can reproduce and explain all the high frequency mag- netic response of artificial magnetic structures. with µ = µ$_{0}$ $^{[1}$− F $^{′}$ω 2 ω 2 − ω 2 LC + iωγ ] , (7) F ′ = F L L + L$_{e}$ , F = ltw V$_{uc}$ = inter-pair volume unit cell volume . (8) I$_{0}$ = − ω $^{2}$[ µ$_{0}$lt/ ( L + L$_{e}$ )] ω 2 − 1 / ( L + L$_{e}$ ) C + iωR/ ( L + L$_{e}$ ) H$_{0}$. (2) µ ( ω ) = µ$_{0}$ $^{[1}$− (1 /V$_{uc}$ )( µ$_{0}$ ( lt ) $^{2}$/ ( L + L$_{e}$ )) ω 2 ω 2 − ω 2 LC + iωγ ] , (3) ω$_{LC}$ = 1 √ ( L + L$_{e}$ ) C , γ = R L + L$_{e}$ . (4) with we obtain for the total resistance σ = iε$_{0}$ ω 2 p ω + iγ$_{m}$ , (5) R$_{tot}$ = 1 σ l S = ( γ$_{m}$ ε$_{0}$ω 2 p − i ω ε$_{0}$ω 2 p ) l S = R − iωL$_{e}$. (6) ( L + L$_{e}$ ) ¨ I + 1 C I + ˙ IR = − ¨ φ = ω $^{2}$µ$_{0}$ ltH$_{0}$e − $^{iωt}$. (1)