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

2.3. SINGLE PARTICLE SCATTERING
E0(r) = e −y2 /w 2
e i(kSPPx−ωt) , (2.41)
where w is the beam waist, y is the distance from the waist of the beam and kSPP is the SPP
propagation constant. When the beam waist approaches infinity the wave becomes a plane
wave of infinite extent in the direction perpendicular to the direction of propagation.
The polarizability α of the scatterers is determined experimentally, in the sense that the polarizability
is simply adjusted until the theoretically predicted results looks similar to the
experimental ones. From [Bozhevolnyi and Coello, 1998] the polarizability is approximately
α = 3. Later in Sec. 2.4.5 an analytical expression for the polarizability will be derived.
2.3 Single Particle Scattering
Single particle scattering is considered analytically in this section in order to illustrate some
properties of scattering of surface plasmons.
The starting point is Eqn. 2.38. In the case of only one scatterer this equation reduces to
E(r) = E0(r) + αE(r1)G(r, r1). Using Eqn. 2.40 is is seen that the field at the scatterer is
given simply by E(r1) = E0(r1). Therefore, using the expression Eqn. 2.41 with an infinite
beam waist for the incident field (the time dependence of the field is ignored, it would cancel out
anyway when calculating the intensity) and the expression Eqn. 2.39 for the field propagator,
the field at an arbitrary position is given by
E(r) = E0(r) + αE0(r1)G(r, r1)
= e ikSPPx i ikSPPx1
+ αe
4 H(1) 0 (kSPP |r − r1|). (2.42)
In the far field limit the Hankel function can be approximated by
Using this approximation the field is then given by
H (1)
2 eix
0 (x) ≈ e−iπ/4 √ . (2.43)
π x
E(r) = e ikSPPx
α2 +
8π eikSPPx1e iπ/2 e −iπ/4 eikSPP|r−r1|
kSPP |r − r1|
= e ikSPPx
α2 e
+
8π
i(kSPPx1+kSPP|r−r1|+π/4)
. (2.44)
kSPP |r − r1|
The intensity is now calculated assuming that kSPP is a purely real number corresponding to
no damping of the propagating field. Since the damping does not matter for the purposes of
this analytical treatment, this is not a limitation
I(r) = |E(r)| 2 = 1 + C 2 + 2C cos(kSPPx1 + kSPP |r − r1| − kSPPx + π/4), (2.45)
25