Volumetric Scattering Theory

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:

When a beam of light passes through a cloud of particles, some energy is subtracted from the forward-propagating beam. Two other things can happen to the subtracted energy: it is either sent out in a different direction (scattering) or converted into another form of energy within the particles, most often heat (absorption). The overall subtraction of energy from the foward-propagating direction is called extinction. By the principle of conservation of energy, the fundamental relationship between the different scattering phenomena is

The scattering particles are assumed to be homogeneous isotropic spheres. The space between them is assumed to be large compared to the electromagnetic wavelength and particle size. In addition, the particles are also assumed to be oriented randomly, not in any specific pattern (such as a rectangular array). Thus there is no coherent phase relationship between different particles, so scattering by many particles can be understood by first considering a single isolated sphere.

Simplifying assumptions can be made if the particles are very large or very small when compared to the wavelength of the electromagnetic radiation. When the particles are optically very small, the simplified Rayleigh scattering model can be used. For particles much larger than the wavelength, the ratio of extinction cross section to geometrical cross section approaches an asymptotic limit of 2.

When the particles are neither optically very large nor very small, the more general Mie scattering theory must be used. Mie theory provides an exact solution for the interaction of a plane wave with a sphere, but the calculations involved are much more computationally intensive than either of the asymptotic limits for large or small particles.

When considering whether the particles are small enough to apply Rayleigh theory, remember to consider the wavelength both inside and outside the particle; this could make a big difference if the particles are made of a highly conductive material.

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.

The wavelength in the medium surrounding the particle λ (SI unit: m) is

where λ0 (SI unit: m) is the vacuum wavelength.

For optical scattering calculations, it is convenient to work with dimensionless variables, such as the dimensionless size parameter x,

where R (SI unit: m) is the radius of the scattering particle.

The size parameter may also be expressed in terms of the wave number in the surrounding medium k (SI unit: rad/m),

For brevity, in the remainder of this section the relative refractive index of the scattering particle, that is, the ratio of the particle refractive index to that of the surroundings, will be denoted n,

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: m2) and intensity I0 (SI unit: W/m2). 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 A0 - A1 is the extinction cross section σext (SI unit: m2) of the particle.

In other words, the extinction cross section of a particle is the surface area on a detector that, if subtracted, causes the same reduction in total energy at that detector as the presence of the particle in the beam.

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: m2) 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: m2) is the scattering cross section.

By the principle of conservation of energy exemplified in Equation 3-27, the three cross sections follow the relationship

For each cross section, it is convenient to define a dimensionless efficiency factor by dividing by the geometrical cross-sectional area of the particle,

Like the scattering cross sections, the efficiency factors follow a relationship based on the energy conservation principle,

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,

Rayleigh scattering theory (not to be confused with Rayleigh-Gans theory) is a simplified scattering model for optically small particles. For Rayleigh scattering theory to be applicable, the particle must be small relative to the wavelength outside of it,

and also small relative to the wavelength inside it,

It is possible for the former to be true, but not the latter, if the scattering centers are small metallic particles with refractive indices that have very large imaginary parts.

The scattering and absorption efficiency factors in the Rayleigh limit are

As usual, the sum of these two efficiency factors yields the extinction efficiency factor.

The absorption efficiency factor is thus zero if the refractive index of the scattering particles is real-valued. However, if the refractive index has any imaginary part, the absorption efficiency factor tends to dominate because it is only proportional to the first power of the size parameter x, whereas the scattering efficiency scales with the fourth power of x; when applying Rayleigh scattering theory, x tends to be very small.

The Mie scattering theory represents an exact solution to the scattering of a plane wave by a homogeneous, isotropic spherical particle. In principle, this exact solution can be used for any value of the size parameter x. In practice, some expressions for the efficiency factors can become ill-conditioned or numerically unstable at extremely large or small x, or if the refractive index has a very large imaginary part.

A complete derivation for the scattering amplitudes will not be given here; instead, the reader is directed to Ref. 19. The extinction and scattering efficiency factors are defined as infinite series,

The absorption efficiency factor if the difference between these two expressions. The coefficients al and bl are expressions involving the Riccatti–Bessel functions

These functions can be expressed in terms of the spherical Bessel functions of the first and third kind, or in terms of the corresponding Bessel functions,

	In Ref. 20, spherical Hankel functions of the first kind are used. This is tied to the author’s choice of sign convention when setting up the electromagnetic wave equation. The convention used in COMSOL is generally to represent the transient part of an electromagnetic wave as exp(iωt), whereas Ref. 20 uses exp(-iωt). Ref. 19 uses exp(iωt) and also arrives at .

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 Riccatti-Bessel functions ψl and ζl; then the scattering coefficients al and bl; and finally the scattering efficiency factors Qext and Qsca.

	The scattering efficiency factors are computed using the built-in COMSOL functions mieextinction(n,x) and miescattering(n,x). The first input argument is the relative refractive index of the scattering particle and the second input argument is the size parameter. Although primarily used in ray optics, these functions can be accessed in any model.

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. 20) is

For example, if the size parameter is 1000, the first 1042 terms in the infinite series will be computed.

One approach to computing the scattering coefficients is via upward recurrence. The first few Riccatti-Bessel functions are

and the remainder can be obtained via the recurrence relations

When scattering particles are very large (

), 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.

This observation is sometimes called the extinction paradox because it is contrary to most day-to-day observations, in which a large object only obstructs the light that intersects its geometrical cross section.

The apparent contradiction can be resolved (see for example Ref. 19) 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