Waves_in_Metamaterials

3.4 Surface plasmon–polaritons on a slab: TM polarization 95

Two other approximations for small kx were first given by Economou (1969). The derivation is given below. For small enough kx we may expect the solution to be close to the light line, i.e. we may assume

where δ2 1. It follows then that

We may then determine δ by substituting eqn (3.39) into eqn (3.35) yielding

which leads to the approximate dispersion equation

If δ → ∞, i.e. we are back to the single interface, then the approximate curve is

Note that eqn (3.41) would further simplify when kpd/2 1 to

It can be immediately seen that the smaller d is the farther is the dispersion equation from the light line for a given ω. Next, we shall look for an approximation for the upper branch. We shall assume again that

3.4 Surface plasmon–polaritons on a slab: TM polarization 97

We shall not replot the frequency against normalized propagation constant curve—which remains practically the same. The frequency against normalized attenuation coe cient is plotted in Figs. 3.21(a) and (b) for the ω(+) and the ω(−) branches, respectively, for a range of normalized slab thicknesses. It may be seen that the waves in the upper branch are much less attenuated than the waves in the lower branch. In Fig. 3.22 we shall return to the concept of propagation length. It is plotted against the free-space wavelength for the ω(+) branch. It may be clearly seen that as the thickness declines, the propagation length increases. The reason why attenuation is lower for the upper branch is far from being obvious. We shall show first the field distributions and then discuss in quite some detail the physical reasons why the upper branch deserves the epithet of ‘long range’.

3.4.2Field distributions

Let us now see the field distributions at one of these thicknesses (kpd = 0.25) for ω/ωp = 0.3, 0.5 and 0.7. They are plotted in Figs. 3.23(a)–

(c). Let us first look at the symmetry. For the upper mode Ez and Hy (and, consequently, Sx) are symmetric as a function of z, whereas Ex is antisymmetric. For the lower mode Ez and Hy are antisymmetric (consequently, Sx is symmetric) and Ex is symmetric. It is also true, as we have already known for the single interface, that the decay away from the interfaces becomes stronger as ω/ωp increases. It also follows from the figures that Sx is positive in the dielectric and flows in the opposite direction in the metal. As ω increases towards ωs the net power flow tends to zero. We need to emphasize that for a given ω up to ωs we had two di erent values of kx belonging to two di erent branches of

3.4Surface plasmon–polaritons on a slab: TM polarization 101

is only a small di erence in the dielectric constant then we can still talk about quasi-symmetric and quasi-antisymmetric modes but we find that the field confinement is very asymmetric. The quasi-symmetric mode is localized at the boundary between the metal and the dielectric with the lower dielectric constant, whereas the quasi-antisymmetric mode is localized at the higher dielectric constant boundary. The quasi-symmetric mode inherits the mantle of the sb mode: it is still the long-range mode, it still has a lower attenuation and, as may be expected, the attenuation decreases as the slab thickness decreases. The new phenomenon is that the quasi-symmetric mode has a cuto thickness below which no propagation is possible. This is bound to occur (Berini, 2001) because the sb mode no longer has the chance quietly to convert into the TEM electromagnetic wave as d → 0 because the di erent dielectrics on the two sides do not permit it. The quasi-antisymmetric mode remains the short-range mode, its attenuation increases with decreasing slab thickness, it has not got a cuto thickness.

For detailed calculations of the propagation length in asymmetric structures see Wendler and Haupt (1986). They investigate theoretically thin silver slabs at the He-Ne wavelength of 632.8 nm with a dielectric constant of εr2 = −18−j 0.47 obtained from Johnson and Christy (1972). They first assume that both dielectrics have identical dielectric constants of εr1 = εr3 = 2.1211, then change the dielectric constant of one of them within a range of about 20%. They find that the propagation length first remains unchanged and then suddenly increases. Their

102 Plasmon–polaritons

results, quite spectacular, are shown in Fig. 3.25.

In a further study Zervas (1991) showed that long-range SPPs are supported even by highly asymmetric configurations.

3.5Metal–dielectric–metal and periodic structures

It is now an easy exercise to derive dispersion for the surface modes for a thin dielectric layer with two semi-infinite metallic media adjacent to it. The dispersion equations are very similar to those of eqns (3.35) and (3.36) and can be derived in an analogous manner. In the final result the only di erence is that κ2 is being replaced by κ1 in the arguments of the tanh and coth functions, i.e. the equations are given as

As before, we shall try to find some approximations. For the upper branch ω(+) we can take the argument of the coth function small and

The frequency at which kx = 0 may be obtained from the above equation as

i.e. the dispersion curve cuts the vertical axis just below the bulk plasma frequency.

We may again resort to the electrostatic approximation that turns out to be identical with that (eqn (3.36)) already derived for the dielectric– metal–dielectric structure.

The dispersion curves calculated from eqns (3.48) and (3.49) are plotted in Figs. 3.26(a) and (b) together with the electrostatic approximation and that given by eqn (3.50) for kpd = 0.25 and 0.5. The first thing to notice is that the ω(+) branch crosses the light line. The wave can propagate at phase velocities higher than the velocity of light. The ω(−) branch behaves as it did for the dielectric–metal–dielectric structure. It tends to ωs for kx → ∞ and to the light line as kx → 0. The electrostatic approximation is quite good for the ω(−) mode and even better for the ω(+) mode. The low kx approximation, not surprisingly, deteriorates as kx increases.

3.6 One-dimensional confinement: shells and stripes 103

Economou (1969) goes on investigating two and more metal slabs between dielectrics, including the generalization to a periodic structure (see Fig. 3.27(a)). More and more layers in the structure lead to multiple splitting, i.e. to the appearance of more and more dispersion curves with the final result that both the ω(+) and the ω(−) branches widen into bands, as shown schematically in Fig. 3.27(b).

3.6One-dimensional confinement: shells and stripes

All the problems we have investigated so far have been concerned with two-dimensional confinement of the surface waves, to a single surface or to multiple surfaces. Clearly, for any practical application we need a one-dimensional structure to take information from point A to point B. Such a structure was analyzed by Al-Bader and Imtar (1992). They considered a cylindrical metallic shell of inner radius a, and thickness t embedded in a dielectric or in two di erent dielectrics. They set up the di erential equations for the fields of a TM mode and solved them numerically for silver at the He-Ne wavelength of 633 nm. They presented their results in terms of a mode index ne = ne − j ne where the complex propagation coe cient is given as k = k0ne. Their results were similar to those found for planar structures to which they reduced in the limit of infinite radius. They were also interested in the evolution of the TM01 fibre mode into surface plasmons, the model they considered was a dielectric cylinder embedded in metal. They found that the surface