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Monostatic radar cross section [RCS] of a perfectly conducting metal sphere as a function of frequency (calculated by Mie theory). In the low frequency Rayleigh scattering limit where the circumference is less than the wavelength, the normalized RCS is σ/(πR²) ~ 9(kR)⁴. In the high frequency optical limit σ/(πR²) ~ 1

Mie theory, also called Lorenz-Mie theory or Lorenz-Mie-Debye theory, is an analytical solution of Maxwell's equations for the scattering of electromagnetic radiation by spherical particles (also called Mie scattering). The Mie solution is named after its developer, German physicist Gustav Mie. However, Danish physicist Ludvig Lorenz and others independently developed the theory of electromagnetic plane wave scattering by a dielectric sphere.

The term "Mie theory" is misleading, since it does not refer to an independent physical theory or law. The phrase "the Mie solution (to Maxwell's equations)" is therefore preferable. Currently, the term "Mie solution" is also used in broader contexts, for example when discussing solutions of Maxwell's equations for scattering by stratified spheres or by infinite cylinders, or generally when dealing with scattering problems solved using the exact Maxwell equations in cases where one can write separate equations for the radial and angular dependence of solutions.

In contrast to Rayleigh scattering, the Mie solution to the scattering problem is valid for all possible ratios of diameter to wavelength, although the technique results in numerical summation of infinite sums. In its original formulation it assumed an homogeneous, isotropic and optically linear material irradiated by an infinitely extending plane wave. However, solutions for layered spheres are also possible.

A modern formulation of the Mie solution to the scattering problem on a sphere can be found in J. A. Stratton's Electromagnetic Theory, published in 1941. In this formulation, the incident plane wave as well as the scattering field is expanded into radiating spherical vector wave functions. The internal field is expanded into regular spherical vector wave functions. By enforcing the boundary condition on the spherical surface, the expansion coefficients of the scattered field can be computed.

[edit] Codes for electromagnetic scattering by spheres

There are many codes for electromagnetic scattering by spheres available.

[edit] Applications

Mie theory is very important in meteorological optics, where diameter-to-wavelength ratios of the order of unity and larger are characteristic of many problems regarding haze and cloud scattering. A further application is in the characterization of particles via optical scattering measurements. The Mie solution is also important for understanding the appearance of common materials like milk, biological tissue and latex paint.

[edit] Atmospheric science

Mie scattering occurs when the particles in the atmosphere are the same size as the wavelengths being scattered. Dust, pollen, smoke and water vapour are common causes of Mie scattering which tends to affect longer wavelengths. Mie scattering occurs mostly in the lower portions of the atmosphere where larger particles are more abundant, and dominates when cloud conditions are overcast.

[edit] Particle sizing

The Mie theory has been used in the detection of oil concentration in polluted waters.

[edit] Relevant lists of light scattering codes

• Discrete dipole approximation codes
• Codes for electromagnetic scattering by cylinders

[edit] References

• A. Stratton: Electromagnetic Theory, New York: McGraw-Hill, 1941.
• H. C. van de Hulst: Light scattering by small particles, New York, Dover, 1981.
• M. Kerker: The scattering of light and other electromagnetic radiation. New York, Academic, 1969.
• C. F. Bohren, D. R. Huffmann: Absorption and scattering of light by small particles. New York, Wiley-Interscience, 1983.
• P. W. Barber, S. S. Hill: Light scattering by particles: Computational methods. Singapore, World Scientific, 1990.
• G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Leipzig, Ann. Phys. 330, 377–445 (1908)
• M. Mishchenko, L. Travis, A. Lacis: Scattering, Absorption, and Emission of Light by Small Particles, Cambridge University Press, 2002.
• J. Frisvad, N. Christensen, H. Jensen: Computing the Scattering Properties of Participating Media using Lorenz-Mie Theory, SIGGRAPH 2007.
• Thomas Wriedt: Mie theory 1908, on the mobile phone 2008, Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008), 1543–1548.

[edit] External links

• Example of the Mie Theory at www.coultercounter.com, Beckman Coulter, Inc.
• English translation of “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,”
• American translation of “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,”
• JMIE (2D C++ code to calculate the analytical fields around an infinite cylinder, developed by Jeffrey M. McMahon)
• Collection of light scattering codes
• www.T-Matrix.de. Implementations of Mie solutions in FORTRAN, C++, IDL, PASCAL, Mathematica and Mathcad
• ScatLab. Mie scattering software for Windows.
• on line Mie solution calculator is available, with documentation in German and English.
• on line Mie scattering calculator produces beautiful graphs over a range of parameters.
• Mie resonance mediated light diffusion and random lasing.
• A short description of Mie scattering