Nanoparticles for in-vitro and in-vivo biosensing and imaging

18 Metal nanoparticles
by the dielectric function ɛ and the magnetic permeability µ (generally, the relative
magnetic permeability of the materials under study is close to 1), that enter in the wave
number as k 2 = ω 2 ɛµ. At the boundary between the particle and the medium, ɛ and µ
are discontinuous. It follows that the normal components of the field are discontinuous
whereas tangential ones are continuous. For points x on the particle surface, we can
write (ˆn is the normal vector)
[E 2 (x) − E 1 (x)] × ˆn = 0 (1.3)
[H 2 (x) − H 1 (x)] × ˆn = 0 (1.4)
Figure 1.5: Sketch of the problem as it is treated in section . A particle with optical constants ɛ p and
µ p is embedded in a medium with optical constants ɛ m and µ m, and illuminated by a plane wave, which
generates an electric field E 1 and a magnetic field H 1 inside the particle. The particle radiates a scattered
field in all directions, which leads, together with the applied fields, to an electric field E 2 and a magnetic
field H 2 outside of the particle.
Only restricted to spherical particles, this problem is exactly solvable as shown in
1908 by Gustav Mie [36] (a complete derivation of Mie theory is given by Bohren and
Huffman [30]), and the scattering matrices can be derived. From these information about,
e.g., the direction and polarization dependence of the scattered light can be extracted.
An important parameter that can be also calculated is the cross section, a geometrical
quantity that relates the incident light to the scattered, absorbed or extincted power.
The absorption, scattering and extinction cross sections (σ abs , σ sca and σ ext respectively)
[37] [38],[39],[30] for an arbitrary spherical particle with dielectric function ɛ p are defined
as:
σ sca = P sca
I inc
σ abs = P abs
I inc
σ ext = P ext
I inc
(1.5)
Since the extincted power is the sum of the scattered and absorbed power, the absorption
cross section is simply