The Scattering Domain feature is used to model the interaction of rays with a dilute phase of particles in the medium the rays pass through. Some examples of this include the following:
Let np (dimensionless) be the absolute refractive index of a spherical particle, and
na (dimensionless) the absolute refractive index of the surrounding medium. For this analysis,
na is assumed to be real (non-absorbing) but
np may be complex-valued,
where np,i > 0 for absorbing particles.
where λ0 (SI unit: m) is the vacuum wavelength.
where R (SI unit: m) is the radius of the scattering particle.
although caution should be used because n denotes the absolute index in other sections of this manual.
Consider an incident beam of light with cross sectional area A0 (SI unit: m
2) and intensity
I0 (SI unit: W/m
2). This beam illuminates a surface, conferring an amount of energy equal to
I0A0. Now suppose that a single spherical particle obstructs part of the beam, so that the energy reaching the surface is now
I0A1. Then the difference A
0 − A
1 is the extinction cross section
σext (SI unit: m
2) of the particle.
Looking closer at the particle, the extinction or energy removal can be separated into two phenomena: scattering and absorption. The amount of energy absorbed by the particle and converted to other forms of energy (predominantly heat) is I0σabs, where
I0σabs (SI unit: m
2) is the absorption cross section. The amount of energy that radiates outward from the particle in directions other than the direction of the incident beam is
I0σsca, where
σsca (SI unit: m
2) is the scattering cross section.
It is convenient describe the scattering behavior in terms of the dimensionless quantities x,
Qext,
Qsca, and
Qabs, rather than the particle radius, wavelength, and cross sections. Another dimensionless variable is sometimes needed: the product of the size parameter with the relative refractive index is denoted
y,
The primes in Equation 3-35 indicate differentiation with respect to the argument of the function,
The general workflow of Mie scattering calculation is to first compute the Riccati–Bessel functions ψl and
ζl; then the scattering coefficients
al and
bl; and finally the scattering efficiency factors
Qext and
Qsca.
Efficient evaluation of the efficiency factors from Mie theory means that the scattering coefficients al and
bl from
Equation 3-35 must be computed as quickly and robustly as possible. Although they are infinite series, in practice the terms start to become negligibly small when the index
l is somewhat larger than
x, so the series can be taken to a finite upper bound
L; one recommendation for the value of L (
Ref. 21) is
When scattering particles are very large (x >> 1000), the extinction efficiency factor from Mie theory asymptotically approaches a value of 2. In other words, the extinction cross section of a large particle is double its geometrical cross section.
The apparent contradiction can be resolved (see for example Ref. 20) by noting that the interpretation of the extinction cross section requires observation of the scattered light to be made at a very large distance from the scattering particle, where regions of light and shadow are not so distinct. According to Babinet’s principle, when a certain amount of light is blocked by the surface of an object, an equal amount of light is diffracted around the edges of the object. For close-range interactions such as light illuminating a piece of furniture in a room, it is extremely difficult to distinguish the diffracted light from the light that continues in the forward scattering direction (
θ = 0) because the diffracted light concentrated in a very narrow cone about the forward scattering direction. The scattering amplitudes for diffracted light as a function of scattering angle are
where J1 is the Bessel function of the first kind; so if
x is very large, most of the energy is contained within a very small angle.