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In short, the solution of the Maxwell vector wave equations in Mie theory is obtained as follows (for example, Liou 1977). The incident plane wector wave and the unknown scattered spherical wave are expressed in spherical coordinates which entails expansion of the waves into infinite series of vector spherical harmonics. The expansion coefficients of the scattered wave series are determined from the boundary conditions linking the electric and magnetic fields of the incident and scattered waves at the sphere boundary. This leads to the following scattering matrix, M(θ), a function of the scattering angle, θ, for the sphere:
|
(1) |
where the unique non-zero elements are expressed as follows (for example, Bohren and Huffman 1983):
| M11 = ½ ( | S2 |2 + | S1 |2 ) | (2a) |
| M12 = ½ ( | S2 |2 - | S1 |2 ) | (2b) |
| M33 = ½ ( S2*S1 + S2S1* ) | (2c) |
| M34 = i/2 ( S2*S1 - S2S1* ) | (2d) |
and
| S1 = ∑n=1 to ∞ {(2n + 1) / [n(n + 1)]} (an πn + bn τn) | (3a) |
| S2 = ∑n=1 to ∞ {(2n + 1) / [n(n + 1)]} (an τn + bn πn) | (3b) |
Coefficients an and bn, derived from the spherical Bessel functions, depend on the refractive index, m, of the sphere material relative to that of the surrounding medium and on the relative particle size, x
| x = 2πa / λ | (4) |
where a is the sphere radius and λ is the wavelength of light in the homogenous, non-absorbing medium surrounding the sphere. See also Algorithms for functions of the sphere size and refractive index and Tests for function A.
Coefficients πn and τn are functions of the scattering angle, θ, only. These latter coefficients are respectively equal to P1n / cosθ and dP1n / dθ, with P1n being the associated Legendre polynomial of the 1st degree, i.e. the 1st derivative of the Legendre polynomial. See also Algorithms for functions of the scattering angle.
Finally, the integral scattering properties, as represented here by the attenuation efficiency, Qc scattering efficiency, Qb, and the backscattering efficiency, Qbb can be expressed with these simple series:
| Qc = (2 / x2) ∑n=1 to ∞ (2n + 1) Re(an + bn) | (5) |
| Qb = (2 / x2) ∑n=1 to ∞ (2n + 1) (|an|2 + |bn|2) | (6) |
| Qbb = (1 / x2) |∑n=1 to ∞ (2n + 1) (-1)n(an - bn)|2 | (7) |
The terms of all the above sums begin to decrease rapidly following certain value of the summation index, on the order of x (see Eq. 1 in Number of terms in Mie series for a more accurate estimate). Hence, the summation can be terminated at a finite value of that index.
| CITATION: Jonasz M. 2006. Mie theory: Overview (www.tpdsci.com/Tpc/MieThe.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 03-Mar-2006 Modified: 19-Apr-2006 Reviewed: PENDING |
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