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, $d \gg \lambda$, in the so-called far-field zone. The far-field solution is expressed in terms of two scattering functions,

$$S_1(\Theta) \sum_{n=1}^{\infty}\frac{2n+1}{n(n+1)} [a_n \pi_n(\cos\Theta) + b_n \tau_n(\cos\Theta)]$$ = (3.25)

$$S_2(\Theta) \sum_{n=1}^{\infty}\frac{2n+1}{n(n+1)} [b_n \pi_n(\cos\Theta) + a_n \tau_n(\cos\Theta)]$$ = (3.26)

and where $\Theta$ is the scattering angle. These infinite series can be physically interpreted as a multiple expansion of scattered light (Hansen and Travis, 1974). Thus, the a₁ coefficient specifies the amount of electric dipole radiation. The functions $\pi_n$ and $\tau_n$ are given by,

$$\pi_n(\cos\Theta)=\frac{1}{\sin\Theta} P_n^1(\cos\Theta)$$ (3.27)

$$\tau_n(\cos\Theta)=\frac{d}{d\Theta} P_n^1(\cos\Theta)$$ (3.28)

where P_n¹ are associated Legendre polynomials of the first kind. Also,

$$\frac{\psi_n'(m\alpha)\psi_n(\alpha)-m\psi_n(m\alpha)\psi_n'(\alpha)} {\psi_n'(m\alpha)\xi_n(\alpha)-m\psi_n(m\alpha)\xi'(\alpha)}$$ a_n = (3.29)

$$\frac{m\psi_n'(m\alpha)\psi_n(\alpha)-\psi_n(m\alpha)\psi_n'(\alpha)} {m\psi_n'(m\alpha)\xi_n(\alpha)-\psi_n(m\alpha)\xi'(\alpha)}$$ b_n = (3.30)

where $\alpha=ka=2\pi a/\lambda$ is called the size parameter, m is the index of refraction, a is the particle radii, and $\psi_n$ and $\eta_n$ are related to spherical Bessel functions. They are characteristic of spherical geometry and arise when solving the wave equation. The prime indicates the first derivative with respect to r. The majority of the effort in carrying out Mie calculations is in determining the coefficients a_n and b_n.

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

$$\left[ \begin{array}{c} E_l^s \\ E_r^s \\ \end{array}\right... ...left[ \begin{array}{c} E_l^i \\ E_r^i \\ \end{array}\right].$$ (3.31)

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,

$${\b P}(\Theta) = \left[ \begin{array}{c c c c} P_{11}(\Theta... ...0 & 0 & -P_{34}(\Theta)& P_{33}(\Theta) \\ \end{array}\right]$$ (3.32)

where the four independent Mie scattering matrix elements are,

$$P_{11}(\Theta) \frac{2\pi}{k^2\sigma_s} [\vert S_1(\Theta)\vert^2 + \vert S_2(\Theta)\vert^2]$$ = (3.33)

$$P_{12}(\Theta) \frac{2\pi}{k^2\sigma_s} [\vert S_2(\Theta)\vert^2 - \vert S_1(\Theta)\vert^2]$$ = (3.34)

$$P_{33}(\Theta) \frac{2\pi}{k^2\sigma_s} [S_2(\Theta)S_1^*(\Theta) + S_1(\Theta)S_2^*(\Theta)]$$ = (3.35)

$$P_{34}(\Theta) \frac{2\pi}{k^2\sigma_s} [S_2(\Theta)S_1^*(\Theta) - S_1(\Theta)S_2^*(\Theta)]$$ = (3.36)

The Mie scattering, absorption, and extinction cross-sections can all be expressed in terms of the a_n and b_n coefficients,

  

$$\sigma_e \frac{2\pi}{k^2} \sum_{n=1}^{\infty} (2n+1) Re(a_n + b_n)$$ = (3.37)

$$\sigma_s \frac{2\pi}{k^2} \sum_{n=1}^{\infty} (2n+1) (\vert a_n\vert^2 + \vert b_n\vert^2)$$ = (3.38)

$$\sigma_a \sigma_e - \sigma_s.$$ = (3.39)

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

$$g = \int_{4\pi} P_{11}(\Theta) \cos\Theta \, d\Omega.$$ (3.40)

The asymmetry factor describes the shape of the phase function; g>1 indicates forward scattering is favoured while g<1 indicates backscattering is favoured. It is a useful parameter for characterizing the phase function independent of scattering angle. The Mie asymmetry factor can be expressed as,

$$g = 2 \sum_{n=1}^{\infty} \left[ \frac{n(n+2)}{n+1} Re(a_n a_... ... + b_n b_{n+1}^*) + \frac{2n+1}{n(n+1)} Re(a_n+b_n^*) \right].$$ (3.41)

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

Assuming initially unpolarized light, ${\b I_o}=[I_o,0,0,0]^{T}$, after a scattered event the Stokes vector will have the form ${\b I}=[P_{11}I_o,P_{12}I_o,0,0]^{T}$. Hence, the degree of linear polarization as a function of scattering angle is,

$$LP(\Theta) = - \frac{Q}{I} = - \frac{P_{21}(\Theta)}{P_{11}(\Theta)}.$$ (3.42)

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,

$$P_{ij}(\Theta) = \frac{ \int_{r_1}^{r_2} P_{ij}(\Theta,r) n(r) dr } {\int_{r_1}^{r_2} n(r) dr }$$ (3.43)

where n(r) is the particle size distribution of equation (2.60) or (2.61) and $P_{ij}(\Theta,r)$ is the ij element of the scattering matrix, explicitly showing the particle radius dependence. It is characterized by peaks in the forward scattering and, to a lesser extent, backward scattering directions. In addition, there is fine structure throughout, indicative of constructive and destructive interference effects (although when averaged over a wide size distribution much of this detailed structure is lost). The large forward scattering peak can be understood qualitatively as follows. If the smallest of scatterers can be represented by a single dipole, then larger scatterers can be represented by a collection of dipoles, with the required number increasing with larger particles size. It is a well known result that for more than one dipole, randomly distributed, the only direction in which the interference is totally constructive is the forward direction (Jackson, 1962).

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

 

$$\sigma_k \frac{ \int_{r_1}^{r_2} \sigma_k(r) n(r) dr } {\int_{r_1}^{r_2} n(r) dr }$$ = (3.44)

$$\frac{ \int_{r_1}^{r_2} g(r) n(r) dr } {\int_{r_1}^{r_2} n(r) dr }.$$ g = (3.45)

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