Plane of reference

x Eol x Eol plane of polariza

plane of polariza

FlG. 6.4. Scattering of electric vectors of a plane wave by a homogeneous spherical scatterer. Plane of reference taken through incident and scattered wave. Plane of polarization contains the orthogonal electric field, E and magnetic field H vectors. The Poynting vector P, is in the direction of wave propagation.

scattering albedo, u>, defined by eqn (3.36), and knowing the asymmetry factor we can solve the multiple-scattering problem within an atmosphere, as we shall see in §6.4.

6.3.1 Mie scattering

Details on the electromagnetic theory and the solution of Maxwell's equations to derive the Mie solution are given in the literature (e.g. van de Hulst 1981). Here, we will give the relevant equations and definitions to allow one to compute the above particle radiative properties with a physical understanding of how these quantities are determined.

In Fig. 6.4, we depict the scattering of the electric field E of an electromagnetic wave by a homogeneous spherical particle. The propagation of the incident wave is orthogonal to both the electric field and the magnetic field, H. The electric field has two components, Eol parallel to the plane of reference (containing the directions of the incident and scattered wave) and Eor perpendicular to it. The electric-field components lie on the plane of polarization and the two generally unequal components are said to describe elliptically polarized radiation since the variation of the electric field on this plane describes an ellipse. The radiation flux (W m~2), flowing through an element of area in the direction of the incident wave is given by the Poynting vector where

The scattered wave has electric-field components Ei and Er and the Poynting vector is P. The angle between the incident and scattered radiation is 9, and < describes the azimuthal angle on the plane of polarization such that the components of the incident wave are Eol = Eo cos < and Eor = Eo sin <, where Eo is the amplitude or scalar magnitude of the incident electric field. For an EM wave the magnitude of the magnetic field, H is related to the magnitude of the electric field, E through

where e0 is the permittivity of free space and c is the speed of light in vacuum. The magnitude of the Poynting vector at any instant can be written as

The average rate of energy flow of a sinusoidal incident wave is then

In mks units, e0c is equal to 2.66x 10-3 W V-2, and the magnitude of the electric field then has units of V m-1. Thus, the magnitude of the Poynting vector has units of W m~2. We also recall that c = (eoMo), where ¡jlq is the permeability of free space.

In a medium other than vacuum, the speed of light is reduced according to v = c/n, (6.15)

where n is unity for vacuum and greater than unity for a scattering medium, such as the atmosphere or droplets of clouds. The factor n is know as the real part of the refractive index, m, which is a complex number m = n — in', where n', as we shall see, gives the absorption of the wave, and i = %/—I, so that i2 = — 1. The frequency of a wave remains the same in vacuum and the medium, so that only the wavelength changes according to X' = X/m, where A is the wavelength in the surrounding space, usually taken to be vacuum. The wave number in the medium is k' = 2n/X' where k' = km, and k is the wave number in vacuum corresponding to the angular frequency w = 2nv = kc.

In Mie theory, formal solutions for the scattered electric-field components, and hence Poynting vector, are derived based on the scattering of a plane sinusoidal wave by a spherical scatterer. We can represent the electric-field components of an incident plane sinusoidal wave travelling in the direction z in vacuum by

with amplitude Eo. This complex number notation is for mathematical expediency since eiM-kz) =cos w(t — z/u) + isinw(t — z/u), (6.18)

where u = w/k is the phase velocity, w(t — z/u) is the phase, and t — z/u is the phase angle. We recall that any complex number x + iy can also be written as r exp(i\$), where r2 = x2 + y2, and represents a vector of distance r from the origin to the point (x,y), making an angle \$ = wt with the X-axis. Thus, as t varies, r exp(iwt), represents a rotating vector at an angular frequency w. The distance r projected on the X-axis represents its amplitude, which thus varies sinusoidally.

The above complex form of the electric-field components satisfies the wave equation for a wave travelling in the Z-direction with no attenuation. If the medium surrounding the spherical scatterer has m = n — in', then we can write

so that the magnitude of the Poynting vector is reduced according to

where the optical depth for absorption in the surrounding medium is rabs = 2kn'z, after travelling a distance z. The factor 2kn' has units of inverse length and can be thought of as the absorption coefficient of the surrounding medium.

For simplicity, let us assume that the refractive index of the surrounding medium is that of vacuum, and ignore attenuation. After the plane wave undergoes scattering with the spherical particle, considered as a point source, the outgoing wave is a spherical wave with origin the particle. For a spherical wave, the wave equation can be written as where

with

Thus, the amplitude of a spherical wave decreases with the radius r from the source. However, because of refraction (scattering plus absorption) within the scattering sphere, the Mie solution involves a 2x2 scattering matrix, with diagonal components S\ and S2 with offdiagonal components zero for spherical or particles, which gives the distribution of energy that is scattered at the different angles 0, between incident and scattered wave, as shown in Fig. 6.4, so that the scattered components of the electric field are 