# Coherent scattering: Wikis

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# Encyclopedia

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Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of reflected radiation from the angle predicted by the law of reflection. Reflections that undergo scattering are often called diffuse reflections and unscattered reflections are called specular (mirror-like) reflections.

The types of non-uniformities which can cause scattering, sometimes known as scatterers or scattering centers, are too numerous to list, but a small sample includes particles, bubbles, droplets, density fluctuations in fluids, defects in crystalline solids, surface roughness, cells in organisms, and textile fibers in clothing. The effects of such features on the path of almost any type of propagating wave or moving particle can be described in the framework of scattering theory.

Some areas where scattering and scattering theory are significant include radar sensing, medical ultrasound, semiconductor wafer inspection, polymerization process monitoring, acoustic tiling, free-space communications, and computer-generated imagery.

## Single and multiple scattering

When radiation is only scattered by one localized scattering center, this is called single scattering. It is very common that scattering centers are grouped together, and in those cases the radiation may scatter many times, which is known as multiple scattering. The main difference between the effects of single and multiple scattering is that single scattering can usually be treated as a random phenomenon and multiple scattering is usually more deterministic. Because the location of a single scattering center is not usually well known relative to the path of the radiation, the outcome, which tends to depend strongly on the exact incoming trajectory, appears random to an observer. This type of scattering would be exemplified by an electron being fired at an atomic nucleus. In that case, the atom's exact position relative to the path of the electron is unknown and would be immeasurable, so the exact direction of the electron after the collision is unknown, plus the quantum-mechanical nature of this particular interaction also makes the interaction random. Single scattering is therefore often described by probability distributions.

With multiple scattering, the randomness of the interaction tends to be averaged out by the large number of scattering events, so that the final path of the radiation appears to be a deterministic distribution of intensity. This is exemplified by a light beam passing through thick fog. Multiple scattering is highly analogous to diffusion, and the terms multiple scattering and diffusion are interchangeable in many contexts. Optical elements designed to produce multiple scattering are thus known as diffusers. Coherent backscattering, an enhancement of backscattering that when coherent radiation is multiply scattered by a random medium, is usually attributed to weak localization.

Not all single scattering is random, however, as a well-controlled laser beam can be exactly positioned to scatter off a microscopic particle with a deterministic outcome. Such situations are encountered in radar scattering as well, where the targets tend to be macroscopic objects such as people or aircraft.

Similarly, multiple scattering can sometimes have somewhat random outcomes, particularly with coherent radiation. The random fluctuations in the multiply-scattered intensity of coherent radiation are called speckles. Speckle also occurs if multiple parts of a coherent wave scatter from different centers. In certain rare circumstances, multiple scattering may only involve small number of interactions such that the randomness is not completely averaged out. These systems are considered to be some of the most difficult to model accurately.

The description of scattering and the distinction between single and multiple scattering are often highly involved with wave-particle duality.

## Scattering theory

Major research problems in scattering often involve predicting how various systems will scatter radiation, which can almost always be solved given sufficient computing power and knowledge of the system. A widely studied but more difficult challenge is the inverse scattering problem, in which the goal is to observe scattered radiation and use that observation to determine properties of either the scatterer or the radiation before scattering. In general, the inverse is not unique; several different types of scattering centers can usually give rise to the same pattern of scattered radiation, so the problem is not solvable in the general case. Fortunately, there are ways to extract some useful, albeit incomplete, information about the scatterer, and these techniques are widely used for sensing and metrology applications (Colton & Kress 1998).

## Electromagnetic scattering

A Feynman diagram of scattering between two electrons by emission of a virtual photon.

Electromagnetic waves are one of the best known and most commonly encountered forms of radiation that undergo scattering. Scattering of light and radio waves (especially in radar) is particularly important. Several different aspects of electromagnetic scattering are distinct enough to have conventional names. Major forms of elastic light scattering (involving negligible energy transfer) are Rayleigh scattering and Mie scattering. Inelastic scattering includes Brillouin scattering, Raman scattering, inelastic X-ray scattering and Compton scattering.

Light scattering is one of the two major physical processes that contribute to the visible appearance of most objects, the other being absorption. Surfaces described as white owe their appearance almost completely to the scattering of light by the surface of the object. The absence of surface scattering leads to a shiny or glossy appearance. Light scattering can also give color to some objects, usually shades of blue (as with the sky, the human iris, and the feathers of some birds (Prum et al. 1998)), but resonant light scattering in nanoparticles can produce different highly saturated and vibrant hues, especially when surface plasmon resonance is involved (Roqué et al. 2006). Light scattering can be divided into three domains based on a dimensionless size parameter, α which is defined as

$\mathit{\alpha}\mathrm{{=}}\mathrm{\frac{\mathit{\pi}\;{D_p}}{\mathit{\lambda}}}$

where πDp is the circumference of a particle and λ is the wavelength of incident radiation. Based on the value of α, these domains are:

α<<1: Rayleigh scattering (small particle compared to wavelength of light)

α≈1: Mie scattering (particle about the same size as wavelength of light)

α>>1: Geometric scattering (particle much larger than wavelength of light)

Rayleigh scattering is a process in which electromagnetic radiation (including light) is scattered by a small spherical volume of variant refractive index, such as a particle, bubble, droplet, or even a density fluctuation. This effect was first modeled successfully by Lord Rayleigh, from whom it gets its name. In order for Rayleigh's model to apply, the sphere must be much smaller in diameter than the wavelength (λ) of the scattered wave; typically the upper limit is taken to be about 1/10 the wavelength. In this size regime, the exact shape of the scattering center is usually not very significant and can often be treated as a sphere of equivalent volume. The inherent scattering that radiation undergoes passing through a pure gas is due to microscopic density fluctuations as the gas molecules move around, which are normally small enough in scale for Rayleigh's model to apply. This scattering mechanism is the primary cause of the blue color of the Earth's sky on a clear day, as the shorter blue wavelengths of sunlight passing overhead are more strongly scattered than the longer red wavelengths according to Rayleigh's famous 1/λ4 relation. Along with absorption, such scattering is a major cause of the attenuation of radiation by the atmosphere. The degree of scattering varies as a function of the ratio of the particle diameter to the wavelength of the radiation, along with many other factors including polarization, angle, and coherence.

For larger diameters, the problem of electromagnetic scattering by spheres was first solved by Gustav Mie, and scattering by spheres larger than the Rayleigh range is therefore usually known as Mie scattering. In the Mie regime, the shape of the scattering center becomes much more significant and the theory only applies well to spheres and, with some modification, spheroids and ellipsoids. Closed-form solutions for scattering by certain other simple shapes exist, but no general closed-form solution is known for arbitrary shapes.

Both Mie and Rayleigh scattering are considered elastic scattering processes, in which the energy (and thus wavelength and frequency) of the light is not substantially changed. However, electromagnetic radiation scattered by moving scattering centers does undergo a Doppler shift, which can be detected and used to measure the velocity of the scattering center/s in forms of techniques such as LIDAR and radar. This shift involves a slight change in energy.

At values of the ratio of particle diameter to wavelength more than about 10, the laws of geometric optics are mostly sufficient to describe the interaction of light with the particle, and at this point the interaction is not usually described as scattering.

For modeling of scattering in cases where the Rayleigh and Mie models do not apply such as irregularly shaped particles, there are many numerical methods that can be used. The most common are finite-element methods which solve Maxwell's equations to find the distribution of the scattered electromagnetic field. Sophisticated software packages exist which allow the user to specify the refractive index or indices of the scattering feature in space, creating a 2- or sometimes 3-dimensional model of the structure. For relatively large and complex structures, these models usually require substantial execution times on a computer.

## References

• Bohren, Craig F.; Donald R. Huffman (1983). Absorption and Scattering of Light by Small Particles. Wiley. ISBN 0-471-29340-7.
• Colton, David; Rainer Kress (1998). Inverse Acoustic and Electromagnetic Scattering Theory. Springer. ISBN 3-540-62838-X.
• Gonis, Antonios; William H. Butler (1999). Multiple Scattering in Solids. Springer. ISBN 0-387-98853-X.
• Prum, Richard O.; Rodolfo H. Torres, Scott Williamson, Jan Dyck (1998). "Coherent light scattering by blue feather barbs". Nature 396 (6706): 28–29. doi:10.1038/23838.
• Roqué, Josep; J. Molera, P. Sciau, E. Pantos, M. Vendrell-Saz (2006). "Copper and silver nanocrystals in lustre lead glazes: development and optical properties". Journal of the European Ceramic Society 26 (16): 3813–3824. doi:10.1016/j.jeurceramsoc.2005.12.024.
• Seinfeld, John H.; Pandis, Spyros N. (2006). Atmospheric Chemistry and Physics - From Air Pollution to Climate Change (2nd Ed.). John Wiley and Sons, Inc. ISBN 0471828572
• Stover, John C. (1995). Optical Scattering: Measurement and Analysis. SPIE Optical Engineering Press. ISBN 0-8194-1934-6.