Mie scattering

Mie scattering is the scattering of light by a homogeneous sphere of any size, as solved exactly by Gustav Mie in 1908 from Maxwell's equations. In practice the name refers to the regime where the particle radius $a$ is comparable to or larger than the wavelength, where the simple $\lambda^{-4}$ law of Rayleigh scattering no longer applies. Typical Mie scatterers are fog and cloud droplets (1–20 µm), smoke and dust aerosols, biological cells, and polystyrene calibration beads. Their scattering depends only weakly on wavelength, is concentrated in the forward direction, and can remove up to about four times the power falling on the particle's geometric cross-section.

Size parameter and efficiency

Mie theory depends on two numbers: the size parameter and the relative refractive index $m = n_{\text{particle}}/n_{\text{medium}}$, which may be complex for an absorbing particle:

$$x = \frac{2\pi a\, n_{\text{medium}}}{\lambda}.$$

The result is expressed as efficiencies, the ratio of each cross-section to the geometric area $\pi a^2$: $Q_{\text{ext}}$ for extinction, $Q_{\text{sca}}$ for scattering, and their difference for absorption. The solution is an infinite series of multipole coefficients $a_n$ and $b_n$ built from spherical Bessel functions; about $x + 4x^{1/3} + 2$ terms are needed for convergence.

For a water droplet in air ($m = 1.33$), a direct evaluation of the series gives:

Here $g$ is the asymmetry parameter, the mean cosine of the scattering angle. At $x = 0.1$ the result matches the Rayleigh formula $Q = \tfrac{8}{3}x^4\left|\tfrac{m^2-1}{m^2+2}\right|^2$ to better than 0.1%. The row $x = 5.7$ is a droplet 1 µm in diameter at 550 nm, close to the first maximum of $Q_{\text{ext}} \approx 4.0$ at $x \approx 6.5$ (diameter 1.14 µm). The row $x = 114$ is a 10 µm-radius cloud droplet, already close to the large-particle limit.

The extinction paradox

For large particles $Q_{\text{ext}}$ approaches 2, twice the geometric area. Half of the extinguished power is light that strikes the particle and is reflected, refracted or absorbed; the other half is light diffracted around the edge into a narrow forward lobe whose first dark ring lies at $1.22\lambda/(2a)$. A detector with a small acceptance angle, far from the particle, sees both parts removed from the beam. A detector with a wide aperture collects much of the diffracted light and measures an efficiency closer to 1.

Resonances and colour

Between the Rayleigh and geometric limits, $Q_{\text{ext}}$ oscillates with $x$. The broad oscillations come from interference between light transmitted through the sphere and light diffracted around it; the anomalous-diffraction approximation of van de Hulst places the first maximum at a phase shift $2x(m-1) \approx 4.1$. Superimposed on these are narrow ripple peaks, whispering-gallery (morphology-dependent) resonances where light circulates inside the sphere by total internal reflection. Because the efficiency near the first maximum depends on wavelength, a narrow size distribution of droplets can scatter coloured light, as in the blue or green Sun seen through some volcanic and forest-fire aerosols.

Measurement and use

Laser diffraction particle sizers illuminate a dilute sample and record the angular scattering pattern on a ring detector, then invert it with Mie theory using the known index of the particles. Flow cytometers use forward scatter (roughly particle size) and side scatter (internal structure) of cells. Optical particle counters classify aerosols by the power scattered from single particles. For a polystyrene bead 1 µm in diameter in water at 633 nm, $m = 1.59/1.33 = 1.20$, $x = 6.6$, and $Q_{\text{sca}} = 2.6$ with $g = 0.92$, a mean scattering angle cosine that places most of the scattered light in a narrow forward cone.

Atmospheric optics, lidar and free-space links depend on Mie scattering from haze and fog, which has only weak wavelength dependence; moving from 850 nm to 1550 nm helps much less in fog than the $\lambda^{-4}$ law would suggest.

Pitfalls

Mie theory is exact only for homogeneous spheres. Irregular dust, coated particles and cells need extensions (layered spheres, T-matrix or discrete-dipole methods), and nonspherical particles depolarize the scattered light. The relative index must use the surrounding medium, so a bead measured in water scatters less, and with a different angular pattern, than the same bead in air. Real samples have a size distribution, which washes out the ripple structure. Dense suspensions show multiple scattering, and the single-particle efficiencies no longer give the attenuation directly.

Common questions

What is the difference between Rayleigh and Mie scattering?

Rayleigh scattering is the small-particle limit of Mie theory ($x \ll 1$), with $\lambda^{-4}$ dependence and nearly symmetric angular distribution. Mie scattering covers all sizes and, for particles near and above the wavelength, gives weak wavelength dependence and strong forward scattering.

Why are clouds white while the sky is blue?

Cloud droplets have size parameters of order 100, where $Q_{\text{ext}} \approx 2$ for every visible wavelength, so they scatter all colours of sunlight equally. Air molecules are Rayleigh scatterers and favour blue.

Does Mie scattering depend on wavelength?

Yes, through the size parameter, but for particles much larger than the wavelength the dependence is weak. For a fixed particle the extinction oscillates with wavelength near $x \approx 1$–10.

References: G. Mie, Ann. Phys. 330, 377 (1908); C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983); H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).