Waves_in_Metamaterials

Fig. 3.28 Metallic stripe. (a) Sketch of the geometry. (b), (c) Dispersion with thickness of the first six modes. Also shown the ab and sb modes for w → ∞ for comparision. From Berini (2000b). Copyright c 2000 by the American Physical Society

which they cannot propagate and some of the modes also have a cuto thickness.

For a stripe width of w = 0.5 μm the normalized propagation and attenuation coe cients are plotted in Figs. 3.28(b) and (c) for the first six modes as a function of stripe thickness. The sb and ab modes of the 2D structure are also shown for comparison. As may be expected, the antisymmetric modes have higher attenuation than the symmetric modes. The most relevant conclusion is that the 1D stripe geometry is in no way worse than the amply analyzed 2D geometry. In fact, it may be much better, at least in theory. The attenuation of two of the higher modes may be seen to decrease rapidly before they reach cuto . Presumably, it happens at the expense of confinement. The ss0b mode has probably more practical significance. If we compare it with the sb mode at the lowest thicknesses shown, below 20 nm, we find that the attenuation of the stripe mode is very considerably below that of the 2D planar mode, and the attenuation can be further reduced by decreasing the stripe width.

Stripes having di erent dielectrics for substrates and superstrates have also been investigated by Berini (2001). Small di erences in the upper and lower dielectric constants have been found to lead to larger di erences in propagation properties. There has been no reply given as yet to

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the question posed by the very large propagation lengths (see Fig. 3.25) obtained theoretically by Wendler and Haupt (1986), whether they can be realized in stripe form.

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For the z components of the k vector in the TM case we obtain by using eqns (3.57), (3.58) and (3.64)

3.7.2Domains of existence of SPPs for a single interface

It follows immediately from eqn (3.60) that, as both κ1 and κ2 are to be real and positive, for TM modes to exist the condition

must be satisfied. For TE modes, as follows from eqn (3.62) the condition is

The result for the TM polarization makes good sense. We are talking here about the electric field with its Ez component changing sign across the boundary, a condition compatible with accumulation of charges at the surface, as we discussed earlier for metals. It is important to note that this condition holds for any value of μ: surface TM modes exist only if ε changes sign at the boundary.

The same argument could be applied for the TE polarization, but it is somewhat di cult to provide a simple physical interpretation of the requirement that μ1 and μ2 must be of di erent sign for a TE surface mode to exist. The reason why it is di cult to picture what is going on is that in the case of negative permeability we cannot appeal to our common-sense expectations. Negative-ε materials have received much more attention in the past and the condition εr1εr2 < 0 for the existence of TM surface modes has been known ever since Fano’s article over sixty years ago. It is quite di erent with negative-μ materials. All such materials, constructed so far, have negative permeability within a frequency

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3.7 SPP for arbitrary ε and μ 111

Table 3.1 Numerical values for the frequencies at which surface modes start or stop.

F = 0.56

modes transform into each other by the mutual substitution ε ↔ μ. As we will see in this section the symmetry is broken when εr2 and μr2 have di erent types of frequency dependence.

Here, we will look at the dispersion curves of the possible surface modes assuming that the dielectric constant of medium 2 follows the Drude model,

and that the permeability has a resonant behaviour that can be described as (see Section 2.8)

These forms of εr2 and μr2 were frequently used in describing the response of metamaterials comprising rods (Pendry et al., 1996) and SRRs (Pendry et al., 1999). The choice of the parameters, of the plasma frequency, ωp, of the magnetic resonance frequency ω0 and of the filling factor, F , has a crucial e ect on the behaviour of the metamaterial, in

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Fig. 3.31 SPP dispersion for single interface. F = 0.56, ω0 = 0.4 ωp. (a) ω–kx diagram. (b) μ–ε diagram

general, and on the character of the surface modes, in particular. Let us recall the most important asymptotes and limits for permeability and permittivity,

We will now give examples of four quite di erent scenarios for SPP modes using typical sets of parameters. In all examples we keep F = 0.56 and choose for ω0/ωp values of 0.4, 0.6, 0.8 and 1. In all four

cases, the frequency at which ε = −1 is the surface plasmon frequency,

√ r

ωp/ 2 0.71ωp and the frequency at which εr = 0 is of course the plasma frequency, ωp. Table 3.1 gives numerical values for the frequencies of interest from eqn (3.85) in the four cases.

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Fig. 3.33 SPP dispersion for single interface. F = 0.56, ω0 = 0.8 ωp. (a) ω–kx diagram. (b) μ − ε diagram

The TE region is entered at A and this mode exists until the frequency is high enough to reach B. This is the mode we have denoted by (ii). Increase ω again and we move from B to C, which is in the TM mode region corresponding to mode (iii). The curve continues beyond C as ω increases but no further surface modes are possible. It is interesting to note that this novel presentation in terms of geometrical loci in the ε,μ plane is not only capable of accounting for all the modes but also makes it clear how the various surface modes arise.

Next, let us consider what happens when ω0 increases, i.e. the magnetic resonance moves up in frequency. It may be shown that both the TE mode and the upper TM mode we have seen in Fig. 3.31(b) become flatter and then disappear altogether. For case (b), when ω0/ωp = 0.6 is taken, only the lower TM mode survives, as shown in the dispersion curve of Fig. 3.32(a). The geometrical locus of this TM mode in the εμ plane resembles the one in Fig. 3.31(b), but has shifted to the right, as illustrated in Fig. 3.32(b). What happened to the other two modes? Could we get our answer from the geometrical loci? What happened is that the PQ curve also moved to the right. At this particular value of ω0/ωp = 0.6 it crosses the ε = −1, μ = −1 point at A without ever crossing the area in which either surface mode exists.

When ω0 increases further there is an inversion of the upper TE and TM modes. First comes a backward TM mode and further up (although starting again at the same point) a forward TE mode. These changes are indicated by the next set of curves taken for ω0/ωp = 0.8 (Fig. 3.33). Note that ω0 > ωs, hence the magnetic resonance does not a ect the lower TM mode. Its range is, as in the case when μ is frequencyindependent, between ω = 0 and ω = ωs. The dispersion curves of the three modes are plotted in Fig. 3.33(a) and the corresponding curves in the ε,μ plane in Fig. 3.33(b). It may be clearly seen that in the interval