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γ$_{m}$ = 12 . $^{18}$× 10 $^{10}$/ 2 π . The results concerning the wide- slab-pair system are shown in Fig. 6. 

Specifically, Fig. 6(a) shows the saturation of the mag- netic resonance frequency and Fig. 6(b) the real part of the resonant permeability response for the smaller length scales. Comparing the result of Fig. 6 with those for the non-reduced value of γ$_{m}$ , one can see that the satura- tion of the magnetic resonance frequency seems to be totally unaffected by the value of the metal collision fre- quency. This indicates that the loss-factor of the metal employed is not able to affect the highest achievable mag- netic resonance frequency of each specific design. On the other hand, the metal loss-factor can affect the minimum length scale able to give negative permeability response, as shown comparing Fig. 6(b) with Fig. 3(a). Indeed, in the small γ$_{m}$ cases the magnetic permeability reso- nance is quite stronger, maintaining negative values up to smaller length scales. Although the strength of the resonance (as measured, e.g., by the minimum value of the Re( µ )) seems to be strongly affected by the γ$_{m}$ value, calculating the width of the negative permeability regime (in the cases that such a regime exists), Δ ω - not shown here -, it is observed that it is only slightly affected by the γ$_{m}$ value; it tends to zero for both high and low val- ues of γ$_{m}$ , indicating that even in the absence of resistive losses one can not go to arbitrarily high-frequency nega- tive permeability response over a practical band-width. 

The results presented above raise many questions con- cerning the high-frequency magnetic metamaterial re- sponse and the main phenomena and factors determin- ing this response. Before attempting an interpretation of this response we will summarize here the main effects observed so far and the main questions that one needs to address as to clarify and to be able to predict the ex- istence or performance of optical negative magnetic per- meability response: 

(a) The magnetic resonance frequency of slab-pair- 

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based systems, while in mm scale structures scales in- versely proportional to the structure length-scale, in sub- µ m scale structures saturates to a constant value, de- pending mainly on the geometry of the structure and independent of the resistive losses in the component ma- terials. Among the structures that have been studied here the fishnet design leads to the higher saturation fre- quency. Moreover, while in the slabs&wires design the magnetic resonance frequency is almost unaffected by the presence of the wires, it is not the same in the fishnet de- sign, where the saturation value is quite higher than the saturation value for the the slabs only case. 

(b) The magnetic permeability resonance in sub- µ m scale structures becomes more and more weak by reduc- ing the structure length-scale, and ultimately ceases to reach negative values. The length scale at which µ < 0 stops to exist depends on the structure geometry (design) and on the resistive losses in the component materials, especially in the metallic parts. Among the structures that we have examined, negative permeability in smaller length scales is achieved in the slab-pair structures, not associated though with higher magnetic resonance fre- quency. Negative permeability at higher frequencies is achieved in the fishnet design. 

(c) Concerning the relative spectral width of the nega- tive permeability regime, this width, while it is constant in mm scale structures, as one goes towards nm scales it becomes smaller and smaller, up to almost vanishing. This width depends on the structure geometry and seems only slightly dependent on the resistive metallic losses. 

