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

2.4.4 Relation to Surface Plasmon Polaritons
2.4. VECTORIAL MODEL
In this section the relation of the vectorial model to scattering of surface plasmon polaritons
is described. Until now the propagation of EM radiation has been described by a direct and
an indirect propagator without specifically separating out the contribution from SPPs. The
propagation of EM fields by SPPs is contained within the indirect propagator together with
the indirect propagation of s and p polarized waves. Separating the indirect propagator gives
the following expression for the total propagator
ˆG (r, r ′ ) = ˆ D (r, r ′ ) + Îs-pol (r, r ′ ) + Îp-pol (r, r ′ ) + ˆ GSPP (r, r ′ ) , (2.69)
where the last term on the right hand side describes the propagation of SPPs. In Fig. 2.11 the
different propagation paths are shown. The FF (far-field) indirect propagator is the propagation
of EM radiation via reflection of s- and p- polarized EM waves in the surface. The SPP
propagator is propagation via excitation of SPPs by near fields of the source dipole.
If both the source and the observation points are placed close to the surface and far from each
other, the angle of incidence of the FF s- and p-polarized waves at the surface will be close to 90
degrees. Under these conditions the Fresnel reflection coefficients becomes approximately -1.
This means that the two waves (the direct and the FF-indirect propagating ones) when they
meet at the observation point will be approximately out of phase. Therefore the direct and
FF-indirect part of the propagator does not contribute to the field at the observation point,
and the total propagator is approximately given by
ˆG (r, r ′ ) ≈ ˆGSPP (r, r ′ ) . (2.70)
This is an important result because it allows one to express the total propagator analytically.
It is not possible to find an analytical expression for the indirect propagator but the SPP propagator
can be described analytically. The SPP propagator is given in cylindrical coordinates
by [Evlyukhin and Bozhevolnyi, 2005]
ˆGSPP (r, r ′ ) =
(1)
−iakSPPH 0 (kSPPρ) e−akSPP(z+z′ )
2(1 − a4 )(1 − a2 ×
)
ˆzˆz + a 2 ˆρˆρ + ia(ˆz ˆρ − ˆρˆz) , (2.71)
where a = εr/ − εm and ˆρ and ˆz are cylindrical unit vectors. Using this the scattered field
can be calculated.
Direct propagator
FF-Indirect propagator
SPP propagator
Figure 2.11: Direct and indirect propagation paths taking into account the scattering to SPPs. The
SPP propagation path is shown along the surface since SPPs propagate along the surface.
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