The light scattering of an incident plain wave can be described in principle by a combination of a part that is passing the scatterer without being disturbed and an outgoing spherical wave. The ratio of these two different parts is dependent on the scattering cross section which is defined as the fraction of the scattering events in a certain time divided by the incident photon-flux.
The scattered photon fluxjsca through a surface elementdAleads to the differential
2.1. SINGLE SCATTERING 11 scattering crossection
dσ= jsca
|jinc|dA.
The surface element dA is given by
dA=r2dΩer,
such that the integration over the whole solid angle gives a value for the total scattering cross section:
σ= Z dσ
dΩdΩ (2.1)
By integrating (2.1) one is able to calculate the scattering behavior of a particle.
This is however not a trivial problem since one needs the value of the differential crossection which has to be calculated directly from the properties of the scatterer and the scattered wave.
2.1.1 Rayleigh Scattering
For scatterers which are small compared to the wavelength of light, the incident plain wave can be considered homogenous over the size of the particle. The incident wave thus excites the scatterer to perform dipole oscillations, which have the same phase and direction with respect to the incident light. This dipole radiates intensity which is proportional to the volume V squared and to the fourth power of the incident wavevectork [7].
I(r) = 16π2V2k4
r α2E02sin2ϑ. (2.2)
The intensity is isotropic for theϕ-axis as presented in Fig.2.1 and shows a sin2θ behavior.
J j
Figure 2.1: The angular intensity distribution of a rayleigh-scatterer. It shows a sin2ϑand an isotropic behavior for itsϕ-axis. This behavior correspond to the radiation of an electromagnetic dipole.
Rayleigh Debye Gans Scattering
Using the approach of treating scatterers as dipoles, one is able to expand the model to be able to describe bigger particles. This approach does not require a certain shape but will be presented here for spherical scatterers. The idea is to split the particle into infinitesimal volume elements that can all be considered as independent Rayleigh-scatterers. After passing the scatterer the different dipole-terms have to be added coherently in order to calculate the scattered intensity.
However, the electric field inside the scatterer is assumed to be the same as that of the incident wave. This is called Born’s first approximation and is quantified by the condition that
with nsc and n0 the refractive indices of the scatterer and the incident medium respectively. As illustrated in Fig.2.2 the phase difference ∆ between two scatterers
r
D
Figure 2.2: The approach to explain the scattering coming from an arbitrary shaped scatterer.
The particle consists of many rayleigh scatterers which scatter the light isotropic. Due to the distance to each other the angular dependence of the scattered light is illustrated considering two scatterers with a distance ofr.
having a distancer turns out to be
∆ = 2krsinϑ
2 =~q·~r (2.4)
using the scattering vector |q| = 2ksinϑ2. Integrating the phase differences over the whole volume, one ends up with the so called form factor
F(q) = 1
When dealing with spherical scatterers, this integral can be solved exactly to give a form factor of the form [8]
F(a) = 9
a6[sina−acos(a)]2 (2.6) with a=qr representing the size parameter of the scatterer.
However the use of Born’s first approximation limits the method to particles with a low refractive index or small diameters. The single scattering on strongly scat-tering, large particles will briefly be explained in the following section.
2.1. SINGLE SCATTERING 13
2.1.2 Mie Scattering
In order to calculate the scattering properties in general, one has to solve Maxwell’s equations. It is not surprising that this set of partial differential equations is not analytically solvable for arbitrarily shaped scatterers. Numerical approximations for example have to be used as presented by Schuerman [9] and Barber [10].
In a paper published in 1908, Mie [11] presented an analytical solution for the scattering of a plain monochromatic wave by a homogenous sphere having an arbitrary diameter. An equivalent solution to this problem was found by Debye [12] shortly thereafter. In his work Debye used light forces on a conducting sphere in order to end up with a solution to Maxwell’s equation. The propagation of an electromagnetic wave which is represented by its wave-functionφ(~r, t), can be calculated using the Helmholtz equation:
52φ(~r, t) +k2φ(~r, t) = 0 where the boundary conditions are given by
Etangmed =Etangsc (2.7)
Htangmed =Htangsc , (2.8)
which means that the tangential components of the electric and magnetic fields have to be continuous for an electromagnetic wave entering the scatterer (sc) com-ing from a homogenous medium (med). The solution to this differential equation is presented in various textbooks e.g. [8, 13] using slightly different techniques.
The calculations end up in a scattering crossection having a nontrivial dependence on the diameter and the wavelength as presented in Fig.2.3. The following data was calculated using ”Light Lab: Far Field Mie Scattering” [14]. For fixed values
A B
Figure 2.3: Scattering cross sections for different properties of the scatterers. The scattering cross section for Mie-scatterers shows a strong dependence on the wavelength (A) and the particle diameter (B).
of those two parameters one is able to calculate the angular distribution1 of the
1Instead of using the scattering angle θone can also use the scattering vectorq=kout−kin
to be the momentum change of the photon which calculates as|q|= 2ksinθ/2.
scattered intensity known as the form factor F(θ). The form factor shows charac-teristic minima as shown in Fig.2.4 for different particle diameters such that this measure can be used to characterize spherical particles as will be shown in chapter 4.3.
d=400nm
d=300nm
d=500nm
(deg)
Figure 2.4: Form factor for different Mie-scatterers at fixed wavelength, λ = 590nm, and refractive index,n=2.7. There are several minima for different angles dependent on the diameter of the scatterers.