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As was mentioned in the previous sections and can be easily concluded from Eq. (10), structures with reduced inductance lead to higher magnetic resonance frequen- cies. A structure based on slab-pairs which offers a con- siderable reduction in the inductance is the fishnet design - see Fig. 1(d). Studies of this design in microwaves 24 have revealed that at the magnetic resonance loop cur- rents exist not only at the slab pair but also in the necks’ part of the metallic element. This neck contribution can be taken into account in an effective LC circuit descrip- tion of the structure by considering an additional induc- tance, due to the necks, which is in parallel with the inductance of the slabs, resulting to a reduced total in- ductance, L$_{fishnet}$ , with 

( L$_{slabs}$ and L$_{necks}$ represent the inductance of the slabs and the necks part respectively; note that the L$_{slabs}$ here it is not exactly the same as that for the slab-pair-only system, due to the difference in the charge and current distribution between only-slab-pair systems and fishnet.) This reduced inductance leads to a magnetic resonance frequency 

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i.e. higher than that of only the slabs. 

By reducing the size of the structure down to submi- cron and nanometer scale, the inductance of both the slabs and the necks part gets a contribution from the ki- netic (electrons) inductance. This contribution though does not modify the general relation (19), indicating the higher magnetic resonance frequency of the fishnet de- sign than that of the component wide-slab-pair system even in the saturation regime. 

This higher attainable magnetic resonance frequency makes the fishnet design the preferential one for the achievement of optical metamaterials, something that has been already revealed from many existing ex- perimental and theoretical works concerning optical metamaterials $^{14,15}$. 

The relation (19), combined with Eqs. (9), can easily lead to a relation for the geometrical dependence of the magnetic resonance frequency for the fishnet, i.e. 

and thus for the saturation value of this magnetic reso- nance frequency. In Eq. (20) w$_{n}$ and w is the width of the neck and slab parts respectively (along H -direction) and a$_{E}$ is the lattice constant along the E direction (see Fig. 1). Eq. (20) suggests that for the achievement of high magnetic resonance frequency saturation values for fishnet, apart from the conditions for the optimization 

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## A. The fishnet design 

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of the slab-pair components one should pursue also wide neck parts; moreover, the wider the neck parts the higher the saturation value of the fishnet magnetic resonance frequency is. 

Concerning the magnetic permeability expression for the fishnet, here the situation is more complicated com- pared to only-slabs systems, due to the more complicated current picture at the magnetic resonance $^{24}$. Following the observations and conclusions of Ref. $^{24}$, according to which the fishnet unit cell can be approximated with an RLC circuit with inductance of slabs and inductance of necks connected in parallel, the magnetic permeability for the fishnet can be calculated following the same steps as the ones presented in the previous section for the slab case, with modifications in the incident flux, the mag- netic moment and the total inductance per unit cell: 

Here the incident flux, φ , can be written as φ = µ$_{0}$a$_{E}$tH$_{0}$ e − $^{iωt}$, and the modified magnetic moment as 

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where I = I$_{slabs}$ + I$_{necks}$ and IL$_{fishnet}$ = I$_{slabs}$L$_{slabs}$ = I$_{necks}$L$_{necks}$ . 

The inductances L$_{slabs}$ and L$_{necks}$ include both the magnetic field inductance and the electrons inductance for slabs and necks. 

From Eq. (21) one can see already that the presence of the necks weakens the magnetic response of the struc- ture, since the neck contribution in the magnetic moment opposes that of the slabs. 

With the above considerations, the magnetic perme- ability for the fishnet design can be expressed as 

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Comparing the above equation with Eq. (3), one can observe that the factor multiplying the ω 2 in the nu- merator, which is the main factor determining the spec- tral width of the negative permeability regime, becomes smaller than that of the slab only case ( F $^{′}$). This can explain the reduced (compared to only slabs) spectral width of the fishnet negative µ regime observed in Fig. 4. 

## V. OBTAINING QUANTITATIVE RESULTS 

In the previous sections we presented a simple RLC circuit model for obtaining qualitative results and opti- mizing the slab-pair-based metamaterial structures. It is important to note though that the relations (3)-(4) do not only have a qualitative power, but they can be used also to obtain quantitative results, if plugged with more ac- curate expressions for the capacitances and inductances involved. 

