Mie Scattering

Mie scattering is the straightforward (if somewhat painful) application of Maxwell's equations to an isotropic, homogeneous, dielectric sphere. It will be the first type of scattering examined as it is equally applicable to spheres of all sizes, refractive indices and for radiation at all wavelengths.

Briefly, Maxwell's equations are solved in spherical co-ordinates through
separation of variables. The incident plane wave is expanded in
Legendre polynomials so the solutions inside and outside the sphere can
be matched at the boundary. The solution sought is at a distance
much larger than the wavelength,
,
in the so-called
far-field zone.
The far-field solution is expressed in terms
of two scattering functions,

= | (3.25) | ||

= | (3.26) |

and where is the scattering angle. These infinite series can be physically interpreted as a multiple expansion of scattered light (Hansen and Travis, 1974). Thus, the

(3.27) | |||

(3.28) |

where

a_{n} |
= | (3.29) | |

b_{n} |
= | (3.30) |

where is called the size parameter,

Using these scattering functions, the scattered components of the
electric field can be expressed in terms of the incident components,

Using equation (2.2), equation (2.31) can
expressed in terms of an incident and scattered Stokes vector.
They are related by the Mie scattering matrix,

where the four independent Mie scattering matrix elements are,

= | (3.33) | ||

= | (3.34) | ||

= | (3.35) | ||

= | (3.36) |

The Mie scattering, absorption, and extinction cross-sections can all
be expressed in terms of the *a*_{n} and *b*_{n} coefficients,

Another useful quantity is the asymmetry factor, *g*, which is also the
first moment of the phase function,

(3.40) |

The asymmetry factor describes the shape of the phase function;

(3.41) |

The number of terms required in these summations is slightly larger than the size parameter, (Hansen and Travis, 1974). For example, for a particle of radius 1 m at 500 nm, approximately 13 terms are necessary.

Assuming initially unpolarized light,
,
after a scattered event the Stokes vector will have the form
.
Hence, the degree of linear polarization as a function of scattering
angle is,

All the Mie quantities discussed thus far have been for a single size
parameter, and hence a particular radius. As will be discussed in section
2.5,
a typical aerosol mass will contain a wide range of sizes. Hence,
Mie cross-sections and scattering matrix elements must be
averaged over a particle size distribution, *n*(*r*).

The scattering matrix elements transform as,

where

Similarly, the cross-sections and asymmetry factor transform as,

Usually, the size distribution is normalized such that the denominator in equations (2.43)-(2.45) are unity.