The Mie theory is a theory of absorption and scattering of plane electromagnetic waves by uniform isotropic particles of the simplest form (sphere, infinite cylinder) which are in a uniform, isotropic dielectric infinite medium. Though the initial assumptions of the Mie theory are idealized its results are widely used when solving problems of radiation heat transfer in light scattering media.
The basic aim of the theory is the calculation of efficiency coefficients (factors) for absorption (Qa), scattering (Qs) and extinction (Qe), The ratio of σi, the cross-section for the appropriate process, to the particle protected area,
defines the efficiency coefficients (factors) Qi, where r is the particle radius. The cross section σi is the ratio of the energy flux absorbed, scattered or (in sum) extinguished by a particle to the incident energy flux density (i.e., to the energy of undisturbed electromagnetic waves per unit area oriented normally to the wave front). The cross section is of area dimension while the efficiency coefficients factors are dimensionless.
According to the definition of extinction
The mathematics of the Mie theory for application to interaction with a spherical object can be divided into the following steps:
Introduction of the spherical coordinate system with the particle center as an origin.
Plane electromagnetic wave expansion in vector spherical functions.
Expansions of a spherical electromagnetic field and field inside the particle in spherical vector functions.
Determination of the coefficients of function expansions into series in vector functions: the coefficients are obtained for the field inside a particle and for the scattering field being solved.
The coefficients of scattering Qs and attenuation efficiencies Qe, are calculated by integrating the Pointing vector (expressed in terms of the electric and magnetic field expansions) with respect to angle and space variables.
In Bohren and Huffman (1983), Deirmendjian (1969) and Van de Hulst (1957) the detailed procedure is given to obtain the resulting relations of the Mie theory. The relations for scattering and extinction are of the form
where x = 2πr/λ is the diffraction parameter, Re is the real part of the sum of the complex numbers:
aj, bj are the expansion coefficients (the Mie coefficients) expressed in terrns of the Riccatty-Bessel functions Ψj(t) and ξi(t) which are expressed in terms of the Bessel functions of noninteger order
the stroke in Eqs. (4) and (5) means the differentiation with respect to an argument, ml = n1,λ + in2,λ is the complex refractive index of a particle material in the surrounding medium, n1,λ is the refractive index, n2,λ is the absorption index associated with the spectral absorption coefficient by the relative κλ = 4πn1,λ/λ.
The coefficient of absorption efficiency is determined after finding Qe and Qs with (1) taken into account. The dependence of Qe on x for water drops is given in Figure 1. For particle radii commensurable with a wave length (r = 1 μm) the typical spectral peculiarities are manifested, namely, the interference structure (large scale oscillations), ripple (irregular fine structure), and the weak spectral dependence region.
The Mie series (2), (3) are poorly converging series, especially for diffraction parameter x > 20. Numerous investigations are devoted to this problem which resulted in effective computational algorithms using formulas of direct and inverse recursion (the expression of the series subsequent terms in terms of the preceding ones) as well as computational programs [Bohren and Huffman (1983)].
Since the problem is azimuthally symmetrical the phase function pgl, according to Mie theory depends on the latitude angle θ between the scattering direction and that of undisturbed wave front propagation. The averaged cosine of a scattering angle
(where μ = cos θ) characterized the elongation degree of the phase function and is determined using the Mie coefficient
Here * means the complex conjugate quantity.
For small values of diffraction parameter, x << 1, one can retain just the first summands of the Mie series which corresponds to the Rayleigh law for scattering by particles the size of which are essentially less than the radiation wavelength. For large x ≥ 20 the efficiency coefficients are found from geometrical optics. The intermediate region for the diffraction parameter variation is called the "Mie scattering region."
The polarization characteristics of scattered radiation are rarely taken into account in the radiation heat transfer theory. However, if necessary they can be computed according to the Mie theory [Bohren and Huffman (1983)].
Volume spectral absorption coefficients involved in the radiation transfer equation are determined by formulas
where N0 is the total number of particles in unit volume, N(r) is the particle distribution (along the radius) function. For a monodisperse system of particles (of radius r0):
Bohren, C, F. and Huffman, D. R. (1983) Absorption and Scattering of Light by Small Particles. A Wiley Interscience Publication, John Wiley & Sons, Inc.
Deirmendjian, D. (1969) Electromagnetic Scattering on Spherical Polydispersions, Elsevier, New York.
Van de Hulst, H. C. (1957) Light Scattering by Small Particles, Wiley, New York.