P7 – Scattering of Surface Plasmon Polaritons by Gold ... - repetit.dk

2.2. SCALAR MODEL
For λ = 650 nm, the permittivity of gold and silver are -12.9 and -19 respectively
[Raether, 1986]. This yields for gold zdm = 25.5 nm and zda = 329.4 nm and for silver zdm =
21.3 nm and zda = 405.1 nm.
Once light has been converted into a propagating SPP on a flat metal surface it will gradually
attenuate because of losses due to absorption in the metal. This attenuation depends on the
dielectric function of the metal at the SPP wavelength. As the SPP propagates along the
surface of the metal the intensity decreases as exp(−2kxix) with kxi given by Eqn. 2.19. The
length travelled after which the intensity has decreased to 1/e is given by
δSP P = 1
2kxi
= λ
2π
εmr + εp
εmrεp
3/2 ε 2 mr
, (2.37)
εmi
where λ is the free space wavelength and εmr and εmi is the real and imaginary part of the
permittivity of the metal respectively. Silver is the metal with the lowest losses in the visible
spectrum [Barnes et al., 2003], yielding propagation lengths of around 10µm. By using light of
a longer wavelength the propagation length increases. The SPP propagation length determines
the upper limit of the size of any photonic circuit based on SPPs.
2.2 Scalar Model
When an SPP propagating on the surface of a metal hits an irregularity the SPP is scattered
in all directions. This model is a simple description of SPP scattering by nanoparticles on a
surface [Bozhevolnyi and Coello, 1998]. The model is based on three main assumptions:
1. Elastic (in the plane) scattering of SPPs dominates over inelastic (out of the plane)
scattering.
2. The scattered SPP field is an isotropic cylindrical wave propagating outwards from the
scatterer.
3. The z-component of the field is much larger than the x and y component.
The incident SPP field is described by a plane wave. In Fig. 2.7(a) scattering by a single
nanoparticle is shown. It can be seen that the wavefronts of the scattered waves are isotropic
and cylindrical. When more than one scatterer is present all the scattered waves and the
incident field interfere. The field at any arbitrary point due to this interference can be calculated
by the expression
E(r) = E0(r) +
N
αiE(ri)G(r, ri). (2.38)
i=1
In this expression αi is the polarizability of the i’th scatterer and G is the field propagater.
The field propagater describes how the field propagates on the surface and how it is scattered.
In this model the propagater is simply given by
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