2.2.1 Basic definitions
Electromagnetic waves, such as light undergo scattering if they come in contact with small particles. The Mie theory is a useful tool to describe characteristic properties of materials. It explains the visual appearance of objects, such as the blue color of sky and white color of clouds, and why objects reflect or absorb light.
When light impinges on a particle, electric charges in the particle are excited into oscillatory motion. These excited charges radiate energy in form of light in all directions. This process is called scattering. A part of the incident light may be absorbed by the particle, which is called absorption.
The amount of energy per unit area per second scattered is the scattering in-tensity, and has unit W/m2. The incident intensity of radiation is represented as I0
The total energy scattered by a particle depends on the incident intensity:
Escat= CscatI0 (2.27)
In equation 2.27, Escatis the energy of the scattered light, Cscatrepresents the scatter cross section. The scatter cross section is defined as the area transverse to the relative motion within which they must meet to scatter from each other.
It is essentially the effective area that quantifies the likelihood when a beam of light strikes a particle.
Similarly, the absorption intensity can be defined in the same manner.
Eabs= CabsI0 (2.28)
In equation 2.28 Eabs and Cabs are the absorption intensity and cross section respectively
According to the conservation of energy, the loss of energy after the incident beam of light reached a particle must be caused by absorption. The total effect of scattering and absorption is called extinction. An extinction coefficient (Cext) can be defined by
Cext= Cscat+ Cabs (2.29)
Extinction can also be represented in terms of scatter and absorption efficiency.
Efficiency is the ratio between the physical cross section and geometrical
crossec-tion of the particle itself:
Q =C
A (2.30)
For spherical particles, A equals πa2
By this definition, equation 2.29 can be expressed in following terms:
Qext= Qscat+ Qabs (2.31)
The ratio of Qscatto Qext, noted as ω is called single scatter albedo. This ratio is the fraction of light extinction that is scattered, and therefore 1 − ω is the fraction of the light extinction that is absorbed.
Scattering can be categorized in two main forms, elastic and inelastic scatter-ing. When light scatters elastically, the wavelength of incident light is the same as light scattered. For inelastic scattering, the wavelength changes when light strikes a particle.
2.2.2 Scatter parameters
Elastic electromagnetic scattering can be further categorized for three different domains based on the size of the particle. The size is typically described with a dimensionless quantity called size parameter. The size parameter is defined as the circumference of the particle and wavelength of the incident light, as described in formula 2.32
ρ =2πa
λ (2.32)
a is the particle radius and λ is the wavelength of the incident light. Based on the values of the size parameter, the following three domains can be defined.
If ρ << 1, Rayleigh scattering theory applies. The particle is much smaller than the wavelength.
For ρ ≈ 1, the particle size has about the same order of magnitude as the wavelength and the Mie scattering theory applies. The Rayleigh and Mie theory will be further elaborated.
If the size parameter ρ >> 1, geometric optics apply. The size far exceeds the wavelength of the light. Scattering theory is no longer relevant in this domain.
2.2.3 Index of refraction
The size parameter, wavelength of the incident light and the index of refraction of the particle are the three fundamental parameters that determine the
scatter-ing and adsorption of light. The latter factor, index of refraction is a complex number, denoted by a capital N.
N = n + ik (2.33)
In above equation (2.33) n is the real part of the refractive index and k the imaginary part. Both the real and imaginary part depend on the wavelength λ The real and imaginary parts represent the non-absorbing and absorbing com-ponents respectively. [10]
2.2.4 Derivation of scatter and absorption cross section
In order to understand the scattering behavior, the scatter and absorption cross-sections are determined. Two scenarios are considered: neutral and charged particles.
The solution for scattering particles is obtained by solving the Laplace equation for the scalar electric potential:
∇2Φ = 0 and E = −∇Φ
Certain boundary conditions hold depending whether the particle is charged or not. Furthermore, the electronic affinity, which tells if electrons are trapped inside a particle or in a surface around it affect the boundary conditions[10].
The following 5 relations hold at the interface between a spherical particle and it’s surrounding medium, denoted by index 1 and 2 respectively:
(0E~2− 0E~1) · ~n = η0+ η the stationary charge density, and η the fluctuating charge. ~K is the surface current, and ~n is a unit vector perpendicular to the particles surface. The first four equations in 2.34 are derived from Maxwell equations, and the last one is the continuity equation. Particles are assumed ideal and satisfy the well-known linear equations for ~D and ~B: k= kE~k and ~Bk: k = µkH~k
If χ is negative, the conduction band inside the dielectric is above the potential outside the barrier. This means electrons cannot penetrate into the dielectric, and are trapped in the so called image potential in front of the surface. [8]. This image potential is a potential of a so called ’image charge’. When an electron approaches a particle, the conduction electrons of the particle screen the charge
of the particle. The screening effect is typically described as a positive image charge inside the particle.[5].
For positive electron affinity, electrons penetrate into the wall and a negative space-charge layer develops in the interior of the dielectric.
The electronic affinity therefore alters the boundary conditions for the electric and magnetic field given by equation 2.34. The surface charges may sustain a surface current, denoted by the boundary condition for the magnetic field. For χ < 0, the following boundary condition holds for the magnetic field.
ˆ
er(Hi+ Hr− Ht) =4π
c K (2.35)
in which K = σsE|| is the surface current. The equation is represented in terms of Hi, Hr, Ht: these are incident, refracting and transmitting components of the H field respectively. σs is the surface conductivity, and E|| is the parallel component of the E field. If χ > 0, there is no surface current, and therefore K in equation 2.35 equals zero.
Applying the boundary conditions leads to the following general formulas for Qscat and Qext
ak and bk are coefficients which consist of spherical Bessel and Neumann func-tions, which are functions of the dimensionless surface conductivity and size parameter . For sake of clarity, ak and bk are not further elaborated here, a full derivation can be found in Klacka et al [12].
These equations serve as the basis for computational procedure of Mie theory The Mie theory can be simplified and approximated depending on the value of ρ.
2.2.5 Rayleigh Scattering
The physical understanding of scattering by small particles was established by Lord Rayleigh in the 19th century. He calculated the scattering intensity of molecules which are much smaller than the wavelength of incident light.
In order to do this, the scattering Phase function is utilized. It is the scat-ter intensity at a particular angle θ (relative to the incident beam), and then
normalized by the integral of the scatter intensity of all angles, see equation index of the medium surrounding the particle, which is effectively unity. ρ is the size parameter.
Several useful parameters can be derived from the phase function. It is not shown explicitly, see for a complete derivation.
The asymmetry parameter g is defined as the intensity-weighted average of the cosine of the scattered angle.
The factor g in equation 2.39 describes how much scattered light deviates from uniform scattering in all directions. If all light is scattered forward (θ = 0) then g equals 1. Likewise, at θ = π (backward scattering), g equals -1. For a uniform scatter distribution, θ equals 0.
Another useful parameter called hemispheric backscatter ratio can be defined.
It is the fraction of scatter intensity of the light scattered backwards and light scattered in all directions, and defined by equation 2.40
b = Rπ
π/2P (θ) sin θdθ Rπ
0 P (θ) sin θdθ (2.40)