What is Mie scattering?

Rayleigh scattering is the small-particle limit: particles much smaller than the wavelength scatter as tiny dipoles, with an intensity proportional to d⁶/λ⁴, so blue light is scattered far more than red. Mie scattering is the exact solution for spheres of any size; it contains Rayleigh scattering as its small-size limit and, for larger particles, the forward-peaked, nearly colour-independent scattering of droplets and dust.

Air molecules are about a thousand times smaller than the wavelength of light, so they Rayleigh-scatter: blue light (450 nm) is scattered about 5–6 times more than red (700 nm), and the scattered skylight looks blue. Cloud droplets are 10–20 µm across, far larger than the wavelength; in this Mie / geometric regime all colours scatter about equally, so clouds look white. At sunset the long path through the air removes the blue, leaving red and orange.

The scattering cross section C_sca is the area that, multiplied by the incident intensity, gives the total power the particle scatters. Likewise C_abs for the power it absorbs, and the extinction cross section C_ext = C_sca + C_abs for the power removed from the beam (what a UV-vis spectrometer measures). Divided by the geometric cross section πr² they give the efficiencies Q_sca, Q_abs, Q_ext, which can exceed 1 at a resonance.

With the size parameter x = 2πr n_m/λ and the Mie coefficients a_n, b_n:

Q_ext = (2/x²) Σ (2n+1) Re(a_n + b_n),   Q_sca = (2/x²) Σ (2n+1) (|a_n|² + |b_n|²),   Q_abs = Q_ext − Q_sca, and C = Q πr².

The coefficients are ratios of Riccati–Bessel functions of x and mx (m: the particle's index relative to the medium). The Mie scattering calculator evaluates them for you, with stable recursions, for spheres and core–shell spheres.

In a metal nanoparticle much smaller than the wavelength, light drives the conduction electrons into a collective oscillation. Its polarizability is α = 4πr³ (ε − ε_m)/(ε + 2ε_m), which resonates when Re ε = −2ε_m (the Fröhlich condition): near 520 nm for gold and 390 nm for silver in water. This is the localized surface plasmon resonance; it sets the colour of gold and silver colloids and concentrates light into near-field hot spots.

Online, in one click, with the Mie scattering calculator on this site: choose the material (from a built-in database of measured optical constants), the medium and the diameter, and get the extinction, scattering and absorption spectra; the near-field calculator maps the field around the particle. In code, open-source packages such as miepython and PyMieScatt (Python) or Mätzler's MATLAB functions implement the same series.

When the particle is much smaller than the wavelength inside it: roughly x < 0.1–0.3 and |m|x ≪ 1. For a 20 nm gold sphere in visible light it is close; for 60 nm gold, or any silicon or titanium-dioxide particle over about 100 nm (which have magnetic Mie resonances), it fails and the full Mie series is needed.

When light hits a particle about as big as its wavelength, the particle re-radiates it in a pattern that depends on its size and material: that is Mie scattering, named after Gustav Mie. Simply put, small particles scatter blue light more and in all directions, while particles comparable to or larger than the wavelength scatter all colours more equally and mostly forwards. Its laws are Maxwell's equations solved for a sphere; there is no single power law like Rayleigh's 1/λ⁴.

It is caused by particles comparable to the wavelength of light (roughly 0.1–10 µm for visible light): water droplets, dust, smoke, pollen, fat droplets in milk, and pigment particles. Everyday examples are white clouds and fog, the whitish haze around the sun, the white of milk and paint, the glow of headlights in fog and the visible beam of a laser in smoke.

In remote sensing three regimes are named: Rayleigh scattering (particles much smaller than the wavelength, strongly selective, 1/λ⁴), Mie scattering (comparable sizes, weakly wavelength-dependent) and non-selective scattering (particles much larger than the wavelength, such as cloud droplets, scattering all wavelengths equally). Mie theory itself covers all three; the "non-selective" case is its large-particle limit.

Laser-diffraction particle sizers fit measured angular scattering with a model. The Fraunhofer approximation treats particles as opaque disks and needs no refractive index; it works for particles much larger than the wavelength (above about 25–50 µm). Mie theory needs the particle's and medium's refractive indices but is exact at all sizes, so ISO 13320 recommends it for particles below about 50 µm and for transparent particles.

Dynamic light scattering measures an intensity-weighted size distribution; converting it to volume or number distributions uses Mie theory to weight each size by how strongly it scatters. In infrared (FTIR) spectroscopy of cells and particles, Mie scattering distorts the baseline and band shapes ("Mie scattering artefacts"), which correction algorithms remove with Mie-based models.

Yes: MiePlot (Philip Laven, Windows), Python packages such as miepython, PyMieScatt and scattnlay, Mätzler's MATLAB functions and many codes on GitHub, including command-line (CLI) tools, compute Mie scattering. This site runs the same exact theory online with no code, and adds core–shell, spheroids, cylinders, dimers, substrates, focused beams, arrays, emitters and heating.

From Mie theory applied to the measured or assumed size distribution and refractive index of the aerosol: the extinction, scattering and absorption coefficients, single-scattering albedo and asymmetry parameter that climate and remote-sensing models use. Each size is computed exactly, then summed over the distribution.

The incident plane wave, the field inside the sphere and the scattered field are each expanded in vector spherical harmonics (multipoles). Requiring the tangential electric and magnetic fields to be continuous at the surface gives, for each multipole order n, two equations whose solution are the Mie coefficients a_n (electric) and b_n (magnetic). All measurable quantities, cross-sections, angular patterns and near fields, follow from them; Bohren & Huffman, chapter 4, gives the full derivation.

The phase function is the angular distribution of the scattered light, normalized over all directions. Its mean cosine is the asymmetry parameter g = ⟨cos θ⟩: 0 for symmetric (Rayleigh) scattering, close to 1 for the strongly forward scattering of large particles (about 0.85 for cloud droplets). Radiative-transfer models often replace the exact Mie phase function by the Henyey–Greenstein function with the same g.

A generalization of Mie theory: the particle is described by a transition matrix that maps the multipoles of any incident field onto those of the scattered field. For a sphere the T-matrix is diagonal with the Mie coefficients; for spheroids, clusters, particles on substrates and periodic arrays it is computed once and reused, which is how the spheroid, dimer and array calculators here work. Variants such as the invariant imbedding T-matrix method extend it to large and irregular particles.

Richard Gans extended the small-particle (quasi-static) Mie result to ellipsoids in 1912, giving each axis its own depolarization factor. It explains why gold nanorods have a transverse and a longitudinal plasmon and how the longitudinal one red-shifts with aspect ratio; the spheroid calculator goes beyond it with an exact T-matrix.

Plotted against the size parameter, the extinction efficiency of a non-absorbing sphere rises steeply, overshoots to about 4, then oscillates (the interference structure) with a fine ripple of narrow resonances, settling at 2 for large spheres. The angular (polar) plot changes from the symmetric Rayleigh pattern to a strong forward lobe with many side lobes. The calculators plot both spectra and field maps.

A typical Mie scattering experiment shines a laser on particles and records the scattered intensity versus angle (goniometry, nephelometers, laser-diffraction sizers) or versus wavelength (extinction and dark-field spectroscopy). The angle dependence and the spectrum are compared with Mie theory to find size and refractive index. Particles from tens of nanometres to the millimetre scale are covered; field maps such as those here give a visualization of what happens in and around the particle.

Yes. The scattered light has the same wavelength as the incident light, unlike Raman or fluorescence, which shift it. Mie theory describes elastic scattering and absorption by spheres.

On aerosols, dust, smoke, pollen, haze and cloud droplets, particles comparable to or larger than the wavelength. It makes haze whitish, clouds white and the glow around the sun bright, and it is what lidar and satellite remote sensing measure to retrieve aerosol size and amount.

In electromagnetism, the (electro-)quasistatic approximation applies to a particle much smaller than the wavelength: the incident field is nearly uniform across it at any instant, so the particle can be treated as if it sat in a static field that slowly oscillates. This gives the simple dipole (Rayleigh) formulas and the plasmon condition Re ε = −2ε_m; it misses the red shift, broadening and higher multipoles of larger particles, which full Mie theory includes.

Rainbows form in raindrops of about a millimetre, thousands of wavelengths across, where geometric optics (refraction and internal reflection) explains them; Mie theory, which is exact for spheres of any size, reproduces the rainbow and adds the supernumerary bows and the glory that ray optics misses. For small cloud droplets the bows wash out into a whitish fogbow.

The main scattering loss in glass fiber is Rayleigh scattering from tiny density fluctuations in the glass, which falls as 1/λ⁴ and sets the low-loss window near 1550 nm. Mie scattering comes from larger defects, bubbles or inclusions, comparable to the wavelength, and is mostly forward-directed; good fibers keep it small.

Bohren & Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), is the standard text, with derivations and code (Wiley paperback 1998, Wiley-VCH reprint 2008). Van de Hulst, Light Scattering by Small Particles (1957), is the classic on the physics, including the anomalous diffraction approximation; Mishchenko, Travis & Lacis, Scattering, Absorption, and Emission of Light by Small Particles (2002), covers the T-matrix method.

Gustav Mie published the solution in 1908 to explain the colours of colloidal gold; Ludvig Lorenz had derived an equivalent result in 1890, and Peter Debye extended it in 1909, hence "Lorenz–Mie theory" or "Lorenz–Mie–Debye theory". The standard references are Bohren & Huffman, Absorption and Scattering of Light by Small Particles (1983), and van de Hulst, Light Scattering by Small Particles (1957).